issn 2148-838x j. algebra comb. discrete appl. 7(1) received: 29 december 2019 journal of algebra combinatorics discrete structures and applications editor’s note. special issue algebraic coding theory: new trends and its connections nuh aydin, bahattin yildiz dear colleagues the purpose of this special issue of journal of algebra,combinatorics, discrete structures and applications was to collect a sample of papers in active areas of research in algebraic coding theory and its connections to other areas. a number of researchers submitted manuscripts to the special issue. after a thorough review process, six articles have been selected to appear in the special issue. we thank all researchers who submitted an article. their contributions are sincerely appreciated, regardless of whether they have been accepted for publication or not. we are particularly grateful to our small number of dedicated reviewers who did a meticulous job of reviewing in a short period of time. the articles selected for this special issue are a representative sample of the current research trends in algebraic coding theory. in their article "construction of quasi-twisted codes and enumeration of defining polynomials", gulliver and venkaiah enumerate all twistulant matrices of a given size and use that information to construct quasi-twisted (qt) codes with better parameters and they start new databases over gf(17) and gf(19). qt codes have been studied extensively in coding theory and they continue to yield useful results. in the article "locally recoverable codes from planar graphs" haymaker and o’pella construct codes that are locally recoverable from 3-connected regular and almost regular graphs. furthermore, they present methods of constructing regular and almost regular planar graphs. in the paper "constructions of mds convolutional codes using superregular matrices", lieb and pinto show how to obtain mds convolutional codes from superregular matrices with certain properties. they provide explicit ways of constructing generator matrices of mds convolutional codes from superregular matrices. in the paper titled "g-codes over formal power series rings", korban et al. introduce g-codes over an infinite ring, using tools from group rings. they study the duality properties of these codes and show that their projections are g-codes over finite chain rings. they prove similar results for the lifts of codes over finite chain rings as well. nuh aydin; department of mathematics kenyon college, usa (email: aydinn@kenyon.edu). bahattin yildiz; department of mathematics northern arizona university, usa (email: bahattin.yildiz@nau.edu) 1 n. aydin and b. yildiz / j. algebra comb. discrete appl. 7(1) (2020) 1–2 in "zq(zq + uzq)-linear skew constacyclic codes", melakhessou et al. consider zq(zq+uzq) skew constacyclic codes where q is a prime power and u2 = 0. they describe the generator polynomials, the minimal spanning sets, and sizes of these codes. they also obtain some new z4-codes from the gray images of these codes. in "weight distributions of some constacyclic codes over a finite field and isodual constacyclic codes", singh describes the weight distribution of a family of constacyclic codes over fq. singh also constructs a family of non-binary isodual-constacyclic codes of a special length and gives specific examples of the constructions. algebraic coding theory continues to be an active area of research with many theoretical and applied aspects. we believe that this special issue will help disseminate recent results to a broad audience in an open access journal and promote further research. 2 issn 2148-838xhttps://doi.org/10.13069/jacodesmath.867644 j. algebra comb. discrete appl. 8(1) • 53–57 received: 24 may 2020 accepted: 19 october 2020 journal of algebra combinatorics discrete structures and applications two families of graphs that are cayley on nonisomorphic groups∗ research article joy morris, josip smolčić abstract: a number of authors have studied the question of when a graph can be represented as a cayley graph on more than one nonisomorphic group. the work to date has focussed on a few special situations: when the groups are p-groups; when the groups have order pq; when the cayley graphs are normal; or when the groups are both abelian. in this paper, we construct two infinite families of graphs, each of which is cayley on an abelian group and a nonabelian group. these families include the smallest examples of such graphs that had not appeared in other results. 2010 msc: 05c25, 05c60 keywords: cayley graphs, nonisomorphic groups 1. introduction a cayley graph cay(g,s) on a group g with connection set s, is the graph whose vertices are the elements of g, with two vertices g1 and g2 adjacent if and only if g2 = sg1 for some s ∈ s. in order to ensure that this is a graph rather than a directed graph, we must require that s = s−1; that is, s is closed under inversion; if we omit this condition, we obtain digraphs (and an arc from g1 to g2 rather than an edge between them). conventionally we also generally assume that the identity e of g is not in s; this avoids having loops at every vertex. cayley graphs and digraphs are a major area of study, as their symmetries lead to many useful properties in the networks they represent. we use standard notation for graphs. in particular, in a graph γ, v (γ) represents its vertex set, while we use v ∼ u to indicate that there is an edge between the vertices v and u. it is well known (first observed by sabidussi) that a (di)graph can be represented as a cayley (di)graph on the group g if and only if its automorphism group contains a subgroup isomorphic to g ∗ this work was supported by the natural science and engineering research council of canada (grant rgpin2017-04905). josip smolčić worked on this project as a summer research experience supported out of this grant. joy morris (corresponding author), josip smolčić; department of mathematics and computer science, university of lethbridge, lethbridge, canada (email: joy.morris@uleth.ca, josip.smolcic@uleth.ca). 53 https://orcid.org/0000-0003-2416-669x https://orcid.org/0000-0002-7456-3100 j. morris, j. smolčić / j. algebra comb. discrete appl. 8(1) (2021) 53–57 in its regular action. however, a particular representation of a cayley (di)graph may not be its only representation, either on a fixed group, or on different groups. sometimes a particular representation may be more useful for practical purposes than a different representation, so it is of interest to understand all possible representations. in this paper, we construct two infinite families of graphs, each of which is cayley on an abelian group and a nonabelian group. these families include the smallest examples of such graphs that had not appeared in other results. the so-called "cayley isomorphism" (ci) problem studies whether or not all representations for a given cayley graph on some fixed group g can be determined purely algebraically. it is therefore a large part of the question of when a cayley graph on a group g is isomorphic to another cayley graph on the same group g (or, equivalently, when there are two distinct regular subgroups isomorphic to g in the automorphism group of the graph). the ci problem has been extensively studied by many researchers. for example, the papers [1–3, 10, 15, 16] amongst many others, and the survey article [9] all deal with this question. the question of when a cayley graph on g can be represented as a cayley graph on some nonisomorphic group h has also received some attention. joseph in 1995 [7] determined necessary and sufficient conditions for a cayley digraph of order p2 (where p is prime), to be isomorphic to a cayley digraph of both groups of order p2 ([6, lemma 4] provides a group theoretic version of this result). the first author [13, 14] subsequently extended this result and determined necessary and sufficient conditions for a cayley digraph of the cyclic group of order pk, k ≥ 1 and p an odd prime, to be isomorphic to a cayley digraph of some other group of order pk. the equivalent problem for p = 2 (when both groups are abelian) was solved by kovács and servatius [8]. in these cases, graphs that could be represented on both groups are all "wreath" (or "lexicographic") products, and their automorphism groups are significantly larger than the number of vertices. in contrast, when neither group is cyclic, [12] shows that it is often possible to find cayley digraphs that can be represented on two nonisomorphic p-groups (one abelian and the other not) whose automorphism group is only slightly larger than the original groups. recent work by dobson and the first author [5] considers graphs that are cayley on more than one abelian group when the number of vertices is not a prime power. digraphs of order pq that are cayley graphs of both groups of order pq, where q | (p− 1) and p,q are distinct primes were determined by dobson in [4, theorem 3.4]. marušič and the first author studied the question of which normal circulant graphs of square-free order are also cayley graphs of a nonabelian group [11]. some of the graphs in our families fall into each of these categories, but neither of our families is limited to square-free orders. 2. the families the first of these families may be known to researchers, but to the best of our knowledge no proof has previously appeared in the literature. a circulant graph is a cayley graph on a cyclic group, and we use dk to denote the dihedral group of order 2k. proposition 2.1. let γ be a circulant graph on n = 2k vertices. then γ is a cayley graph on dk and cn. proof. let γ = cay(cn,s), where s ⊂ cn is closed under inverses, and cn = 〈c〉. by assumption, γ is a cayley graph on cn. we must show that γ is also a cayley graph on dk. we do this by finding a regular subgroup of aut(γ) that is isomorphic to dk. define α and β by α(z) = zc2 and β(z) = z−1c−1 for z ∈ v (γ) = cn. we first show that α and β are automorphisms. for every u,v ∈ v (γ) with u ∼ v, there exists s ∈ s 54 j. morris, j. smolčić / j. algebra comb. discrete appl. 8(1) (2021) 53–57 such that su = v. it is not hard to see that sα(u) = suc2 = vc2 = α(v). also, since s is closed under inverses and u and s are both elements of the abelian group cn, we have s−1β(u) = s−1u−1c−1 = (us)−1c−1 = v−1c−1 = β(v) as desired. since n = 2k is the order of c, it is clear that α has order k. also β2(z) = β(z−1c−1) = (z−1c−1)−1c−1 = czc−1 = z, thus β has order 2. finally, β−1αβ(z) = βα(z−1c−1) = β(z−1c) = zc−2 = α−1(z), so β inverts α. we conclude that 〈α,β〉∼= dk is a subgroup of aut(γ). it remains to observe that 〈α,β〉 acts regularly on the vertices of γ. since |〈α,β〉| = n, it is sufficient to show that 〈α,β〉 is acting transitively on the vertices of γ. let u = ci and v = cj be arbitrary vertices of γ. if i and j have the same parity, then cj = ci+2m for some m ∈ z, in which case it is easy to see that v = cj = αm(ci) = αm(u). on the other hand, if i and j have opposite parities, then β(u) = u−1c−1 = c−i−1 and −i− 1 has the same parity as j. again this implies that there exists some integer m such that v = cj = αm(c−i−1) = αmβ(u). thus the action of 〈α,β〉 is regular, and therefore γ is a cayley graph on dk. the second family has slightly more restrictions, but is at the same time potentially more interesting. to understand it, we must define the family of generalised dihedral groups. definition 2.2. let a be an abelian group. define the group dih(a,x) = 〈a,x〉, where x2 = 1 and x−1ax = a−1 for every a ∈ a. notice that dih(a,x) ∼= a o c2 where c2 acts by inversion. in the special case where a is cyclic, this is the usual dihedral group. notice that the group dih(a,x) is abelian if and only if a is an elementary abelian 2-group, in which case dih(a,x) is the elementary abelian 2-group whose rank is one higher than the rank of a. the following theorem shows that for a particular slightly restricted family of connection sets, cayley graphs on the generalised dihedral group dih(a,x) = a o c2 are also cayley graphs on a×c2. theorem 2.3. let a be an abelian group, and let d = dih(a,x) be the corresponding generalised dihedral group. let s ⊆ d be closed under inversion, and let γ = cay(d,s). suppose that there is some y ∈ xa such that for every a ∈ a we have ya ∈ s ∩ xa if and only if ya−1 ∈ s ∩xa. then aut(γ) has a regular subgroup isomorphic to a×c2, so γ is also a cayley graph on the abelian group a×c2. proof. first note that if a is an elementary abelian 2-group, then dih(a,x) ∼= a × c2 so there is nothing to prove. for every a ∈ a, define the map αa on the vertices of γ by αa(z) = za, and define the map β by β(z) = yz for all z ∈ v (γ) = d. let h = 〈αa,β : a ∈ a〉. we claim that h ∼= a × c2 is a regular subgroup of aut(γ). first we show that h ∼= a × c2. it should be clear that 〈αa : a ∈ a〉 ∼= a. furthermore, since y ∈ xa we have y = xa for some a ∈ a, so y2 = xaxa = a−1a = 1, meaning that β has order 2. it remains only to show that h is abelian. again, since a is abelian, we really only need to show that β commutes with every αa. this is easy, since βαa(z) = β(za) = yza = αa(yz) = αaβ(z). for the remainder of the proof, we must show that h consists of automorphisms of γ. let u,v ∈ v (γ), where u ∼ v, so there is some s ∈ s such that v = su. it is easy to see that for any αa ∈ h, αa is an element of the right regular action of dih(a,x) on γ, and is therefore an automorphism of γ. 55 j. morris, j. smolčić / j. algebra comb. discrete appl. 8(1) (2021) 53–57 to show that β is also an automorphism of γ, we will require the extra conditions we assumed for s: that s is inverse-closed (which is necessary for γ to be a graph rather than a digraph) and also that for every a ∈ a, we have ya ∈ s if and only if ya−1 ∈ s. we will also need the observation that for every a ∈ a, y−1ay = yay = a−1; this follows immediately from the definitions of y and x and the fact that a is abelian. again, we take u,v ∈ v (γ) where u ∼ v, so there is some s ∈ s such that v = su. we deal separately with the possibilities that s ∈ a or s ∈ xa = ya. suppose first that s ∈ a, so that s−1y = ys. since s is closed under inverses s−1β(u) = s−1yu = ysu = yv = β(v). thus β(u) ∼ β(v) if and only if u ∼ v, meaning that β is an automorphism of γ. now suppose that s ∈ xa = ya, say s = yb where b ∈ a. then yb−1 is also in s, and yb−1β(u) = yb−1yu = y(yb)u = ysu = yv = β(v). thus β(u) ∼ β(v) if and only if u ∼ v, meaning that β is an automorphism of γ. we have shown that h ≤ aut(γ). since |h| = 2|a| = |d|, in order to show that h is a regular subgroup of aut(γ) we need only show that it is transitive. it should be apparent that 〈αa : a ∈ a〉≤ d,h is transitive on each coset of a in v (γ), so we need only show that β interchanges the cosets. let u ∈ v (γ). if u = a ∈ a, then β(u) = ya ∈ xa since y ∈ xa. likewise, if u = xa ∈ xa, then β(u) = yxa ∈ a. thus, h is indeed a regular subgroup of aut(γ). in the case where a is not an elementary abelian 2-group, we have shown that such graphs are cayley graphs on both the abelian group a×c2 and the nonabelian group dih(a,x), which are nonisomorphic. it is easy to construct examples of graphs that satisfy our restriction on the connection set; for example, any connection set that contains exactly one element of xa will have this property, or any connection set that contains all but one element of xa. a connection set that contains exactly two elements y1 and y2 of xa will have this property if and only if y−11 y2 is a square in a. it would be nice to completely characterise the cayley graphs on dih(a,x) that are also cayley on a×c2. this would, however, require a fairly deep understanding of the full automorphism group of any such graph (for example, whether or not the cosets of a are blocks of imprimitivity for the automorphism group will be important) that is beyond the scope of this project. acknowledgment: the authors thank the 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[16] m. muzychuk, a solution of the isomorphism problem for circulant graphs, proc. london math. soc. 88 (2004) 1–41. 57 https://doi.org/10.1002/jgt.20625 https://doi.org/10.1002/jgt.20625 https://doi.org/10.1016/s0012-365x(01)00438-1 https://doi.org/10.1007/s10801-006-0052-1 https://doi.org/10.1007/s10801-006-0052-1 https://doi.org/10.1002/jgt.20088 https://doi.org/10.1002/jgt.20088 https://doi.org/10.26493/2590-9770.1254.266 https://doi.org/10.26493/2590-9770.1254.266 https://doi.org/10.1002/(sici)1097-0118(199908)31:4%3c345::aid-jgt9%3e3.0.co;2-v https://doi.org/10.1002/(sici)1097-0118(199908)31:4%3c345::aid-jgt9%3e3.0.co;2-v https://doi.org/10.1016/s0012-365x(99)90119-x https://doi.org/10.1016/s0012-365x(99)90119-x https://doi.org/10.1112/s0024611503014412 https://doi.org/10.1112/s0024611503014412 introduction the families references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.863113 j. algebra comb. discrete appl. 8(1) • 23–29 received: 23 january 2020 accepted: 13 september 2020 0 journal of algebra combinatorics discrete structures and applications clique polynomials of 2-connected k5-free chordal graphs research article hossein teimoori faal abstract: in this paper, we give a generalization of the author’s previous result about real rootedness of clique polynomials of connected k4-free chordal graphs to the class of 2-connected k5-free chordal graphs. the main idea is based on the graph-theoretical interpretation of the second derivative of clique polynomials. finally, we conclude the paper with several interesting open questions and conjectures. 2010 msc: 05c31, 05c69, 30c15 keywords: chordal graphs, clique polynomials, clique roots, clique value 1. introduction polynomials with only real roots are appeared in many branches of theoretical and applied mathematical sciences. in combinatorics, having all roots real property is equivalent to being log-concave and unimodal using newton’s well-known theorem [3]. indeed if a = {ak}k≥0 is a combinatorial sequence and it’s generating function pa(x) = ∑n k=0 akx k has only real roots, then the sequence a′ = { ak (nk) } k≥0 is log-concave. this immediately implies that a is also log-concave. moreover, if a positive and log-concave then a is also unimodal. a complete subgraph of a graph g with i vertices is called an i-clique. the (ordinary) generating function of i-cliques of g denoted by c(g,x) is called the clique polynomial of g [4]. a real root of c(g,x) is called a clique root of g. in [4] hajiabolhassan and mehrabadi proved that any simple graph g has a clique root in [−1, 0). they also showed that the class of triangle-free graphs has only clique roots. this immediately implies another algebraic proof of mantel’s theorem [1]. hossein teimoori faal; department of mathematics and computer science, allameh tabataba’i university, tehran, iran (email: hossein.teimoori@atu.ac.ir). 23 https://orcid.org/0000-0001-5861-6287 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 in [8], the author obtained a generalization of clique-rootedness of the class of triangle-free graphs. indeed, he showed that an intersting subclass of chordal graphs has only clique roots. indeed, using the idea of breadth-first search tree the author of this paper proved the following result. theorem 1.1. the class of k4 free connected chordal graphs has only clique roots. in particular, this class has always a clique root −1. as a consequence of the above theorem, he also gave an algebraic proof of turán’s graph theorem for k4 free graphs. it is not hard to show that the class of connected k5 free chordal graphs has not always only clique roots. therefore, characterizing those subclasses of connected k5 free chordal graphs which have only clique roots is an interesting and challenging problem. in this paper, we continue the above line of research by proving that the class of 2-connected k5-free chordal graphs has only clique roots. our main motivation for introducing a new class of graphs with only clique roots is to move forward in the direction of finding a new algebraic proof of turán’s graph theorem. it seems that this idea can be more generalized to prove turán-type extremal results. the paper is organized, as follows. in section two, we quickely review the basics of clique polynomials. in particular, we introduce the graph-theoretical interpretations of the first and the second derivatives of clique polynomials. we also review the basic theory of chordal graphs. in section three, we state and prove the main result of our paper which asserts that the class of 2-connected k5-free chordal graphs has only clique roots. finally, we conclude the paper with several interesting open questions and conjectures. 2. basic definitions and notations we assume that our graphs are simple, finite and undirected. we also assume that they are connected, unless otherwise stated. for a given graph g = (v,e) and an arbitrary finite set s ⊆ v (g), the subgraph induced by s denotes by g[s]. for a vertex v ∈ v (g), it’s open neighborhood n(v) is defined as the set of vertices adjacent to v. a new graph obtained by deleting the vertex v from g is called the vertexdeleted subgraph of g and is denoted by g−v. one can similarly define the edge-deleted subgraph g−e of g. we also recall that the collection of all vertex-deleted subgraphs {g − v}v∈v (g) of g is called a vertex-deck of g. the collection of edge-deleted subgraphs {g−e}e∈e(g) of g is called the edge-deck of g. 2.1. clique polynomials for a given graph g, the clique polynomial of g denoted by c(g,x) is defined as c(g,x) = 1 +∑ω(g) i=1 ci(g)x i, where ci(g) denotes the number of i-cliques of g and ω(g) is the size of the largest clique in g. using a simple counting argument, one can obtain the following vertex-recurrence relation for clique polynomials. lemma 2.1 (see [4]). let g be a graph and v ∈ v (g) be a vertex of g. then, we have c(g,x) = c(g−v,x) + xc(g[n(v)],x). (1) we also have the following vertex-deck identity for the number of cliques. the proof is a straight forward double-counting argument. lemma 2.2. let g = (v,e) be a graph on n vertices. then, we have (n− i)ci(g) = ∑ v∈v (g) ci(g−v), (i ≥ 1). (2) 24 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 in a similar way, one can also get the following edge-recurrence relation for clique polynomials. we will use the notation n(e) = n(u) ∩n(v), for an edge e = {u,v}∈ e(g). lemma 2.3 (see [4]). let g = (v,e) be a graph and e ∈ e(g) be an edge of g. then, we have c(g,x) = c(g−e,x) + x2c(g[n(e)],x). (3) one can also similarly obtain the following edge-deck identity for the number of cliques. lemma 2.4. let g = (v,e) be a graph on n vertices and m edges. then, we have( m− ( i 2 )) ci(g) = ∑ e∈e(g) ci(g−e), (i ≥ 2). (4) the first derivative of subgraph-counting polynomials has been already studied in the literature [6]. here, we have a graph-theoretical interpretation for the case of clique polynomials. proposition 2.5. let g = (v,e) be simple graph, then we have d dx c(g,x) = ∑ v∈v (g) c(g[n(v)],x). we can generalize the above formula as the following interesting formula. to the best of our knowledge, the formula is new. proposition 2.6. let g = (v,e) be simple graph, then we have d2 dx2 c(g,x) = 2 ∑ e∈e(g) c(g[n(e)],x). proof. we first multiplying both sides of equation (4) by xi and then summing over all i (i ≥ 0), we get ∑ i≥0 ( m− ( i 2 )) ci(g)x i = ∑ i≥0 ∑ e∈e ci(g−e)xi, or equivalently, by interchanging the summation order, we obtain m ∑ i≥0 ci(g)x i −x2 ∑ i≥2 ( i 2 ) ci(g)x i−2 = ∑ e∈e  ∑ i≥0 ci(g−e)xi   . (5) next, by the definition of a clique polynomial of a graph, it’s second derivative is equal to d2 dx2 c(g,x) = 2 ∑ i≥2 ( i 2 ) ci(g)x i−2. (6) therefore, considering relation (6) and the definition of a clique polynomial, we can rewrite equation (5) as follows mc(g,x) −x2 ( 1 2 d2 dx2 c(g,x) ) = ∑ e∈e c(g−e,x). or equivalently, d2 dx2 c(g,x) = 2 ∑ e∈e c(g,x) −c(g−e,x) x2 . (7) finally, equations (7) and (3) imply the desired result. 25 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 2.2. chordal graphs in this subsection, we briefly review the basics of the class of chordal graphs. for more information about chordal graphs, one can refer to [7]. we first recall that a graph g is called a chordal graph if any induced cycle of g of length greater than three has a chord. by a chord, we simply mean an edge connecting two non-adjacent vertices of a cycle. we also need the following important definition from the paper [8]. for given graphs g1 and g2, we say that a graph g arises from g1 and g2 by pasting along s if we have g1 ∪g2 = g and g1 ∩g2 = s. we also use the notation g1 ∪s g2 whenever g1 and g2 are pasted along s. it is worth to note that the above definition immediately implies that any chordal graph can be recursively constructed by successively pasting complete graphs along cliques. we also recal the following key result [8]. theorem 2.7. let g be a chordal graph defined as a pasting of the complete graphs {gi}ri=1 of sizes ni’s, respectively. that is, g = g1 ∪q1 g2 ∪q2 · · · ∪qr−1 gr, where {qj} r−1 j=1 are cliques of sizes lj’s, respectively. then, we have c(g,x) = r∑ i=1 (x + 1)ni − r−1∑ j=1 (x + 1)lj. (8) the above theorem immediately implies the following interesting conclusion. corollary 2.8. every r-connected chordal graph g has always a clique root −1 . the multiplicity of this root is equal to r. 3. main results in this section, we give a generalization of the result of the paper [8], as follows. we also recall that a real root α of a polynomial pn(x) of degree n is called of multiplicity 2, if pn(x) can be written as (x−α)2qn(x), where qn−2(x) is another polynomial of degree n− 2. theorem 3.1. the class of 2-connected k5-free chordal graphs has only clique roots. moreover, it has r = −1 as a clique root of multiplicity 2. here is the idea of the proof. we first prove that the quartic clique polynomial c(g,x) has a real root r = −1 of multiplicity 2. therefore, we get c(g,x) = (1 + x)2q(g,x) in which q(g,x) is a quadratic polynomial. thus, we only need to prove that q(g,x) has at least one real root. it is not hard to show that this is equivalent to prove that d 2 dx2 c(g,x) has at least a real root. proof. first of all we note that the last part of the result follows form corollary 2.8, for r = 2. next we note that since g is a k5-free chordal graph the induced graph g[n(e)], for each e ∈ e(g), is a collection of some trees or even isolated vertices. hence, we get 1 2 d2 dx2 c(g,x) = ∑ e∈e c(g[ng(e)],x), = k1∑ i=1 c(g[ng(ei)],x) + k2∑ j=1 c(g[ng(ej)],x) − (k1 + k2 −m), = k1∑ i=1 (1 + x)(1 + lix) + k2∑ j=1 (1 + kjx) − (k1 + k2 −m), 26 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 where k1 and k2 denotes the number of trees (on li > 1 vertices) and isolated vertices, respectively. here, m denotes the number of edges. thus, considering the fact that kj ≥ 1 and k1 + k2 ≥ m, we finally obtain d2 dx2 c(g,−1) ≤ 0. this completes the proof, considering the fact that d 2 dx2 c(g, 0) = 1 > 0 and intermediate value theorem. recall that a sequence {ai}i≥0 of real numbers is called log-concave [2] if for every i ≥ 1, we have ai−1ai+1 ≤ a2i . we recall the well-known netwon’s theorem about the real-rootedness of polynomials. lemma 3.2. if pn(x) = a0 + a1x + · · · + anxn ∈ r[x] is a real-rooted polynomial, then the sequence {ai}i≥0 is log-concave. moreover, the following sequence a0( n 0 ), a1(n 1 ), · · · , an(n n ), is also log-concave. a generalization of turán’s theorem is the following which is due to zaykov [9]. theorem 3.3 (see [9]). for all integers k ≥ l ≥ 0, the maximum number of l-cliques in a graph on n vertices and no k + 1-cliques is ( k l ) (n k )l. the following result is an immediate consequence of theorem 3.1 which is a zykov theorem for the class of 2-connected k5-free graphs. corollary 3.4. for 0 ≤ l ≤ 4, the number of l-cliques of the class of 2-connected k5-free chordal graphs is ( 4 l ) (n 4 )l . proof. let c(g,x) = 1 + c1x + c2x2 + c3x3 + c4x4 be the clique polynomial of a 2-connected k5-free graphs. then by theorem 3.1, the clique polynomial c(g,x) is real-rooted. now, by lemma 3.2, we immediately conclude the following inequalities 1 1 ≤ c1 4 ≤ c2 6 ≤ c3 4 ≤ c4 1 . (9) now, the proof for the cases l = 0, 1 are obvious. for l = 2, from inequalities 1 1 ≤ c1 4 ≤ c2 6 , using log-concavity we immediately conclude that c2 ≤ 3n 2 8 . this is indeed the turan’s graph theorem for k5-free graphs. for l = 3, again inequalities c14 ≤ c2 6 ≤ c3 4 (using log-concavity) imply c3 ≤ n 3 16 . the case l = 4 is a little tricky. first of all, from inequalities c2 6 ≤ c3 4 ≤ c4 1 by log-concavity, we obtain that c4 c2 6 ≤ c 2 3 16 . now, by multiplying both sides to c1 4 and consdering the inequality c1 4 c3 4 ≤ c 2 2 36 , we get c1c4 ≤ c2c36 . finally, considering the two previous cases l = 2, 3 we obtain c2c3 6 ≤ n 5 162 . thus, the two last inequalities immediately imply c4 ≤ n 4 162 . 4. open questions and conjectures in this section, we present some interesting open questions and problems. the following question naturally arises. question 1. is it true that the class of 2-connected k6-free chordal graphs has only clique roots? 27 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 4 3 1 2 5 6 g1 : figure 1. the clique polynomial of g1 is c(g1, x) = 1 + 6x + 12x2 + 11x3 + 5x4 + x5 here is a counter-example. let g1 be the graph depicted in figure 1. it is not hard to see that d3 dx3 c(g1,x) = (5)(4)(3)(1 + x) 2 + (3)(2) > 0, which immediately implies that not all roots of c(g1,x) are real. in this direction, we made the following conjecture. conjecture 1. the class of 3-connected k6-free chordal graphs has only clique roots. the following question is also of our special interest. question 2. which classes of 2-connected k5-free non-chordal graphs have only clique roots. a graph g is called an r-triangulated graph if each edge of g belongs to at most r + 1 triangles of g. we come up with the following seemingly interesting conjecture. conjecture 2. the class of 2-connected 2-triangulated graphs has only clique roots. it seems that one obvious way to extend the result of our paper is to think about some generalizations of the edge-deck counting formula (4). thus the question that we encounter is as follows. question 3. is there any anlogue of edge-deck identity for the case of triangles? more specifically, does the following formula ( t− ( i 3 )) ci(g) = ∑ δ∈∆(g) ci(g− δ) (i ≥ 3), (10) hold in general? note that here t shows the number of triangles of g and ∆(g) denotes the set of all triangles of g. we will call the above identity the triangle-deck formula. here, we have a counter-example for triangle-deck identity in general. let g2 be a graph which is shown in figure 2. hence, for i = 2, we obtain t = 3, c2(g2) = 8, c2(g2 −δ1) = c2(g2 −δ2) = c2(g2 − δ3) = 5. thus, we get (3 − 0)8 6= 15, as required. we define a triangle graph of a given graph g which is denoted by t(g) as a graph with the vertex set ∆(g). two triangles δ1 and δ2 are connected in t(g) if their share an edge in g. we believe that the following conjecture is true. conjecture 3. let g be a graph such that it’s triangle graph t(g) is an empty graph (a collection of isolated vertices), then the triangle-deck identity is true. 28 h. t. faal / j. algebra comb. discrete appl. 8(1) (2020) 23–29 3 4 1 2 5 6 g2 : figure 2. the triangle-deck identity is not true for g2 for more on triangle graphs see the reference [5]. finallly, we are interested to answer to the following question. the reason is clear, becuase it will give us a proof of the well-known zykov theorem [9] for k5-free graphs. question 4. can we relax the condition of being biconnected with connected in corollary 3.4? references [1] j. a. bondy, u. s. r. murty, graph theory, springer gtm 244 (2008). [2] p. branden, unimodality, log-concavity, real-rootedness and beyond, handbook of enumerative combinatorics, crc perss (2018). [3] l. comet, advanced combinatorics, 200. reidel, dordrecht-boston (1974). [4] h. hajiabolhassan, m. l. mehrabadi, on clique polynomials, australasian journal of combinatorics 18 (1998) 313–316. [5] p. haxell, a. kostochka, s. thomasse, packing and covering triangles in k4-free planar graphs, discrete applied mathematics 28 (2012) 653–662. [6] x. li, i. gutman, a unified approach to the first derivatives of graph polynomials, discrete applied mathematics 587 (1995) 293–297. [7] t. a. mckee, f. r. mcmorris, topics in intersection graph theory (monographs on discrete mathematics and applications), society for industrial and applied mathematics (1987). [8] h. teimoori, clique roots of k4-free chordal graphs, electronic journal of graph theory and applications 7(1) (2010) 105–111. [9] a. a. zykov, on some properties of linear complexes, mat. sbornik n.s. 24(66) (1949) 163–188. 29 https://www.springer.com/gp/book/9781846289699 https://www.taylorfrancis.com/books/handbook-enumerative-combinatorics-miklos-bona/e/10.1201/b18255 https://www.taylorfrancis.com/books/handbook-enumerative-combinatorics-miklos-bona/e/10.1201/b18255 https://www.springer.com/gp/book/9789401021982 https://ajc.maths.uq.edu.au/pdf/18/ajc-v18-p313.pdf https://ajc.maths.uq.edu.au/pdf/18/ajc-v18-p313.pdf https://doi.org/10.1007/s00373-011-1071-9 https://doi.org/10.1007/s00373-011-1071-9 https://doi.org/10.1016/0166-218x(95)00121-7 https://doi.org/10.1016/0166-218x(95)00121-7 https://dx.doi.org/10.5614/ejgta.2019.7.1.8 https://dx.doi.org/10.5614/ejgta.2019.7.1.8 http://mi.mathnet.ru/eng/msb5974 introduction basic definitions and notations main results open questions and conjectures references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056555 j. algebra comb. discrete appl. 9(2) • 71–78 received: 17 march 2021 accepted: 25 november 2021 journal of algebra combinatorics discrete structures and applications growth of harmonic functions on biregular trees research article francisco javier gonzález vieli abstract: on a biregular tree of degrees q + 1 and r + 1, we study the growth of two classes of harmonic functions. first, we prove that if f is a bounded harmonic function on the tree and x, y are two adjacent vertices, then |f(x) − f(y)| ≤ 2(qr − 1)‖f‖∞/((q + 1)(r + 1)), thus generalizing a result of cohen and colonna for regular trees. next, we prove that if f is a positive harmonic function on the tree and x, y are two vertices with d(x, y) = 2, then f(x)/(qr) ≤ f(y) ≤ qr ·f(x). 2010 msc: 31c20, 05c05 keywords: biregular tree, growth, harmonic function 1. introduction a tree is homogeneous (or regular) if all its vertices have the same degree; it is biregular if any vertices x and y whose distance is even have the same degree, which we will assume greater than two. regular and biregular trees are infinite. a complex valued function f defined on the vertices of a graph is harmonic at a vertex x if its value at x is the arithmetical mean of its values at the neighbours of x; the function is harmonic on the graph if it is harmonic at every vertex of the graph. the study of harmonic functions on graphs pertains to such diverse domains as probability [7], potential theory [1], graph theory or harmonic analysis. in particular, since the seminal work of cartier [4] the properties of harmonic functions on regular trees have been thoroughly investigated (see for example [5], [1] and the references therein). although they are quite straightforward generalizations of regular trees, biregular trees have not attracted a similar attention. here we study the growth of harmonic functions on biregular trees using only elementary reasonings. in contrast to rn [2, p.31], there are non-constant bounded harmonic functions on a regular tree of degree q + 1; but any such function f has its growth limited by the inequality |f(x) − f(y)| ≤ 2(q − 1)‖f‖∞/(q + 1) when x and y are adjacent [5, theorem 1, p.65]. here we generalize this result to francisco javier gonzález vieli; montoie 45, 1007 lausanne, switzerland (email: francisco-javier.gonzalez @gmx.ch). 71 https://orcid.org/0000-0002-2245-5757 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 biregular trees: if f is a bounded harmonic function on a biregular tree t of degrees q + 1 and r + 1, and x, y are adjacent vertices, then |f(x)−f(y)| ≤ 2(qr −1)‖f‖∞/((q + 1)(r + 1)). an example shows that this inequality is the best possible. also in contrast to rn [2, p.45], there are non-constant positive harmonic functions on a biregular tree but they cannot grow too rapidly. we prove that if f is a positive harmonic function on t and x, y are vertices with d(x,y) = 2, then f(x)/(qr) ≤ f(y) ≤ qr · f(x). we give an example of a positive harmonic function which has maximal growth on a given infinite path (in the terminology of [3]) in t. 2. bounded harmonic functions let t be a biregular tree and fix a vertex x0 in t. we suppose that q + 1 is the degree of x0 (and of all vertices z with d(x0,z) even), and that r + 1 is the degree of all q + 1 neighbours of x0 (and of all vertices y with d(x0,y) odd). an easy induction shows that the number of points on the sphere with centre x0 and radius ρ (i.e. the set of vertices in t at distance ρ from x0) is given, for ρ ∈ n,ρ ≥ 1, by |s(x0,ρ)| = (q + 1)rbρ/2cqb(ρ−1)/2c, where we define bac = max{k ∈ z : k ≤ a} for any a ∈ r. it follows that the number of points on the ball with centre x0 and radius ρ (i.e. the set of vertices in t at distance ≤ ρ from x0) is given, for ρ ∈ n,ρ ≥ 1, by |b(x0,ρ)| = 1 + ρ∑ j=1 (q + 1)rbj/2cqb(j−1)/2c. when the radius is even, this can be written |b(x0,2k)| = 1 + (q + 1)(r + 1) k−1∑ l=0 rl ql = 1 + (q + 1)(r + 1) (qr)k −1 qr −1 , and this result does not depend on the particular degree of x0. take now a function f harmonic on t; this means that, for all x ∈ t, f(x) = 1 |s(x,ρ)| ∑ y∈s(x,ρ) f(y) when ρ = 1. an induction shows that this also holds for any ρ ∈ n, and then f(x) = 1 |b(x,ρ)| ∑ y∈b(x,ρ) f(y) for any ρ ∈ n. proposition 2.1. let t be a biregular tree of degrees q+1 and r+1. if f is a bounded harmonic function on t and x, y are two adjacent vertices in t, then |f(x)−f(y)| ≤ 2(qr −1) (q + 1)(r + 1) ‖f‖∞. (1) proof. take k ∈ n, k ≥ 1. we calculate |f(x)−f(y)| = ∣∣∣∣∣∣ 1|b(x,2k)| ∑ z∈b(x,2k) f(z)− 1 |b(y,2k)| ∑ z∈b(y,2k) f(z) ∣∣∣∣∣∣ 72 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 ≤ 1 |b(x,2k)| ∑ z∈b(x,2k)4b(y,2k) |f(z)| ≤ ‖f‖∞ |b(x,2k)4b(y,2k)| |b(x,2k)| = ‖f‖∞ 2(qr)k 1 + (q + 1)(r + 1)((qr)k −1)/(qr −1) , where b(x,2k)4b(y,2k) is the symmetric difference of the two balls. when k tends to +∞, we arrive at (1). remark 2.2. for regular trees (q = r), this was first established in [5, theorem 1, p.65]. the proof here is an adaptation of [6]. is the inequality (1) the best possible? to answer this, we must construct a bounded harmonic function g such that, for some vertices x0 and x1, |g(x0) −g(x1)| is as large as possible with respect to ‖g‖∞. we take x0 ∈ t of degree q + 1 and x1 a neighbour of x0 (of degree r + 1). we put g(x0) = 0 and g(x1) = 1. the vertex x1 has r neighbours x (1) 2 , . . .x (r) 2 other than x0; in order to have ‖g‖∞ as low as possible, we must choose g to take the same value α2 at all these vertices, the harmonicity of g at x1 implying that α2 is given by the equation 1 r + 1 (0 + r ·α2) = 1. hence α2 = (r + 1)/r. every vertex x (j) 2 has q neighbours other than x1. again we must choose g to take the same value α3 at all these vertices, the harmonicity of g at x (j) 2 implying that α3 is given by the equation 1 q + 1 (1 + q ·α3) = r + 1 r , and we find α3 = (1 + q + qr)/qr. proceeding in this way, we construct a harmonic function g on t by defining g to take the same value αn at all vertices which are at distance n (n ∈ n, n ≥ 1) from x0 and at distance n−1 from x1, this value being αn = ∑n−1 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c if n is even, αn = ∑n−1 j=0 q b(j+1)/2crbj/2c qbn/2crb(n−1)/2c if n is odd; and by defining g to take the same value α−n at all vertices which are at distance n (n ∈ n, n ≥ 1) from x0 and at distance n + 1 from x1, this value being α−n = − ∑n−1 j=0 q bj/2crb(j+1)/2c qbn/2crbn/2c if n is even, α−n = − ∑n−1 j=0 q b(j+1)/2crbj/2c qb(n+1)/2crbn/2c if n is odd. the function g is bounded because supx∈t g(x) is equal to lim k→+∞ α2k = lim k→+∞ (1 + r) ∑k−1 j=0 (qr) j r(qr)k−1 73 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 = lim k→+∞ (1 + r)((qr)k −1)/(qr −1) r(qr)k−1 = (1 + r)q qr −1 and infx∈t g(x) is equal to lim k→+∞ α−2k = lim k→+∞ − (1 + r) ∑k−1 j=0 (qr) j (qr)k = lim k→+∞ − (1 + r)((qr)k −1)/(qr −1) (qr)k = − 1 + r qr −1 . let now f = g − 1 2 (supx∈t g(x) + infx∈t g(x)), that is, f = g − 1 2 (q −1)(r + 1) qr −1 . then f is harmonic on t, |f(x0)−f(x1)| = |g(x0)−g(x1)| = 1 and ‖f‖∞ = sup x∈t f(x) = − inf x∈t f(x) = (1 + r)(1 + q) 2(qr −1) , conclusion 2.3. the inequality (1) is the best possible, being in fact, for the function f just defined and the vertices x0 and x1, an equality. 3. positive harmonic functions lemma 3.1. let x0 and x1 be two adjacent vertices in a biregular tree t, with x0 of degree q + 1. if f is a real valued harmonic function on t, there exists an infinite path (x0,x1, . . . ,xm, . . .) in t such that, for all m ∈ n, m ≥ 2, we have f(xm) ≤ (r + 1)f(xm−1)−f(xm−2) r if m is even, (2) f(xm) ≤ (q + 1)f(xm−1)−f(xm−2) q if m is odd. (3) proof. among the r neighbours of x1 which are not x0, we write x2 the one where f takes its smaller value, and x(1)2 , . . . ,x (r−1) 2 the other neighbours: f(x2) ≤ f(x (j) 2 ) for all j = 1, . . . ,r − 1. since f is harmonic, f(x0) + f(x2) + f(x (1) 2 ) + · · ·+ f(x (r−1) 2 ) r + 1 = f(x1); (4) hence f(x2) + f(x (1) 2 ) + · · ·+ f(x (r−1) 2 ) = (r + 1)f(x1)−f(x0). 74 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 from the choice of x2, we get f(x2) ≤ (r + 1)f(x1)−f(x0) r . this establishes the assertion for the case m = 2. among the q neighbours of x2 which are not x1, we write x3 the one where f takes its smaller value. a similar argument as above gives f(x3) ≤ (q + 1)f(x2)−f(x1) q . this establishes the assertion for the case m = 3. proceeding in this way we construct the needed path step by step. lemma 3.2. let x0 and x1 be two adjacent vertices in a biregular tree t, with x0 of degree q + 1. let f be a real valued harmonic function on t, and (x0,x1, . . . ,xm, . . .) an infinite path in t such that, for all m ≥ 2, (2) and (3) hold. then, for all n ∈ n, n ≥ 2, f(xn) ≤ ∑n−1 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x1)− ∑n−2 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x0) (5) if n is even, and f(xn) ≤ ∑n−1 j=0 q b(j+1)/2crbj/2c qbn/2crb(n−1)/2c f(x1)− ∑n−2 j=0 q b(j+1)/2crbj/2c qbn/2crb(n−1)/2c f(x0) (6) if n is odd. proof. by induction on n. the case n = 2 is the hypothesis (2). we next turn to the case n = 3. by assumption f(x3) ≤ 1 q [ (q + 1)f(x2)−f(x1) ] and f(x2) ≤ 1 r [ (r + 1)f(x1)−f(x0) ] . hence f(x3) ≤ 1 q [ (q + 1) 1 r { (r + 1)f(x1)−f(x0) } −f(x1) ] ≤ 1 q [ 1 r (q + 1)(r + 1)f(x1)− 1 r (q + 1)f(x0)−f(x1) ] ≤ 1 q [ (q + 1)(r + 1)−r r f(x1)− q + 1 r f(x0) ] ≤ 1 + q + qr qr f(x1)− 1 + q qr f(x0). the case n = 3 is proved. we suppose now that n > 3 and that the lemma is true for n− 1. we first consider the case where n is even. then (x1,x2, . . . ,xn) is a path of odd length n−1 in t which satisfies (2) and (3) if we invert the roles of q and r. the induction hypothesis then gives (we invert the roles of q and r in (6)) f(xn) ≤ ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c f(x2)− ∑n−3 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c f(x1). 75 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 but f(x2) ≤ 1 r [ (r + 1)f(x1)−f(x0) ] = ( 1 + 1 r ) f(x1)− 1 r f(x0). hence f(xn) ≤ ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c (( 1 + 1 r ) f(x1)− 1 r f(x0) ) − ∑n−3 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c f(x1) = ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c f(x1) + ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2cr f(x1) − ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2cr f(x0)− ∑n−3 j=0 r b(j+1)/2cqbj/2c rb(n−1)/2cqb(n−2)/2c f(x1) = rb(n−2+1)/2cqb(n−2)/2c rb(n−1)/2cqb(n−2)/2c f(x1) + ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n+1)/2cqb(n−2)/2c f(x1) − ∑n−2 j=0 r b(j+1)/2cqbj/2c rb(n+1)/2cqb(n−2)/2c f(x0) = rbn/2cqb(n−1)/2c rbn/2cqb(n−1)/2c f(x1) + ∑n−2 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x1) − ∑n−2 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x0) = ∑n−1 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x1)− ∑n−2 j=0 q bj/2crb(j+1)/2c qb(n−1)/2crbn/2c f(x0), where, for the last equality but one, we have used the fact that, since n is even, b(n + 1)/2c = bn/2c and b(n−1)/2c = b(n−2)/2c. we have thus proved (5). the case n odd can be handled in a similar manner. proposition 3.3. let t be a biregular tree of degrees q+1 and r+1, and x0 and x1 two adjacent vertices in t with x0 of degree q + 1. if f is a positive harmonic function on t, then q + 1 q(r + 1) f(x0) ≤ f(x1) ≤ r(q + 1) (r + 1) f(x0). proof. by lemma 3.1, there exists an infinite path (x0,x1, . . . ,xm, . . .) in t such that, for all m ≥ 2, (2) and (3) hold. we can then use (6) with n = 2k + 1 and find f(x2k+1) ≤ (qr)k + (1 + q) ∑k−1 j=0 (qr) j (qr)k f(x1)− (1 + q) ∑k−1 j=0 (qr) j (qr)k f(x0). since f(x2k+1) is positive, this implies (qr)k + (1 + q) ∑k−1 j=0 (qr) j (qr)k f(x1) ≥ (1 + q) ∑k−1 j=0 (qr) j (qr)k f(x0). 76 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 and so f(x1) ≥ (1 + q) ∑k−1 j=0 (qr) j (qr)k + (1 + q) ∑k−1 j=0 (qr) j f(x0) = (1 + q)((qr)k −1)/(qr −1) (qr)k + (1 + q)((qr)k −1)/(qr −1) f(x0) = (1 + q)((qr)k −1) (qr)k(qr −1) + (1 + q)((qr)k −1) f(x0) = (1 + q)((qr)k −1) qk+1rk+1 −qkrk + qkrk + qk+1rk −1−q f(x0) = (1 + q)((qr)k −1) qk+1rk+1 + qk+1rk −1−q f(x0) = (qr)k(1 + q)− (1 + q) q(qr)k(r + 1)− (1 + q) f(x0). when k tends to +∞, we find f(x1) ≥ 1 + q q(r + 1) f(x0). inverting the roles of x0 and x1, as well as the roles of q and r, we get f(x0) ≥ 1 + r r(q + 1) f(x1). the conclusion follows. corollary 3.4. let t be a biregular tree of degrees q + 1 and r + 1. if f is a positive harmonic function on t and x, y are two vertices in t with d(x,y) = 2, then 1 qr f(x) ≤ f(y) ≤ qr ·f(x). proof. first, we suppose that x is of degree q + 1. let z be the vertex in t with d(x,z) = 1 = d(z,y). by proposition 3.3 we get f(y) ≥ r + 1 r(q + 1) f(z) ≥ r + 1 r(q + 1) · q + 1 q(r + 1) f(x) = 1 rq f(x). the other inequality is obtained by inverting the roles of x and y. for x of degree r + 1, invert the roles of q and r in the preceding reasoning. let ζ = (. . . ,x−2,x−1,x0,x1,x2, . . .) be an infinite path in t, with x0 of degree q + 1. is it possible to find a positive harmonic function f on t such that its growth along this path is maximal? if we fix f(x0) = 1, then we deduce from proposition 3.3 and corollary 3.4 that we must set, for all k ∈ z, f(x2k) = (qr) k and f(x2k+1) = (qr)k r(q + 1) r + 1 . (7) consider now the r−1 neighbours of x1 other than x0 and x2. if we suppose that f takes the same value α on all these vertices, then the harmonicity of f at x1 implies that (f(x0)+f(x2)+(r−1)α)/(r+1) = f(x1) or 1 r + 1 (1 + qr + (r −1)α) = r(q + 1) r + 1 . 77 f. j. gonzález vieli / j. algebra comb. discrete appl. 9(2) (2022) 71–78 hence 1 + qr + (r−1)α = rq + r and finally α = 1 = f(x0). this also shows that f must take this same value on these r−1 neighbours, because if it was not the case, the value on at least one neighbour, y say, would be less than 1, contradicting proposition 3.3 applied to y and x1. consider next the q − 1 neighbours of x0 other than x−1 and x1. if we suppose that f takes the same value β on all these vertices, then the harmonicity of f at x0 implies that (f(x−1) + f(x1) + (q − 1)β)/(q + 1) = f(x0) or 1 q + 1 ( q + 1 q(r + 1) + r(q + 1) r + 1 + (q −1)β ) = 1, whose solution is β = (q + 1)/(q(r + 1)) = f(x−1). this also shows that f must take this same value on these q − 1 neighbours, because if it was not the case, the value on at least one neighbour, y say, would be less than (q + 1)/(q(r + 1)), contradicting proposition 3.3 applied to y and x0. conclusion 3.5. given an infinite path ζ = (. . . ,x−2,x−1,x0,x1,x2, . . .) in t with x0 of degree q + 1, there exists one and only one positive harmonic function f on t with f(x0) = 1 such that f has maximal growth along ζ. on ζ, f is given by (7) and on a vertex y not in ζ it is defined as follows: let x be the vertex in ζ closest to y, and n the distance between x and y; then f(y) =   f(x) · (qr)−k if n = 2k, f(x) · (qr)−k q + 1 q(r + 1) if n = 2k + 1. remark 3.6. proposition 3.3 and corollary 3.4 are in fact true for positive superharmonic functions on t. (a function is superharmonic at a vertex x if its value at x is greater or equal to the arithmetical mean of its values at the neighbours of x.) this follows from the fact that lemma 3.1 and lemma 3.2 are true for real valued superharmonic functions. indeed, we can modify the proof of lemma 3.1 by changing (4) to f(x0) + f(x2) + f(x (1) 2 ) + · · ·+ f(x (r−1) 2 ) r + 1 ≤ f(x1); hence f(x2) + f(x (1) 2 ) + · · · + f(x (r−1) 2 ) ≤ (r + 1)f(x1) −f(x0). the end of the proof needs no change, and neither do the other proofs need one. references [1] v. anandam, harmonic functions and potentials on finite and infinite networks, springer, heidelberg, bologna (2011). [2] s. axler, p. bourdon, w. ramey, harmonic function theory, springer-verlag, new york (2001). [3] n. l. biggs, discrete mathematics, clarendon press, oxford university press, new york (1985). [4] p. cartier, fonctions harmoniques sur un arbre, sympos. math. 9 (1972) 203–270. [5] j. m. cohen, f. colonna, the bloch space of a homogeneous tree, bol. soc. mat. mex. 37 (1992) 63–82. [6] e. nelson, a proof of liouville’s theorem, proc. amer. math. soc. 12(6) (1961) 995. [7] w. woess, random walks on infinite graphs and groups, cambridge university press (2000). 78 https://doi.org/10.1007/978-3-642-21399-1 https://doi.org/10.1007/978-3-642-21399-1 https://doi.org/10.1007/978-1-4757-8137-3 https://dl.acm.org/doi/10.5555/533687#secabs https://mathscinet.ams.org/mathscinet-getitem?mr=0353467 https://mathscinet.ams.org/mathscinet-getitem?mr=1317563 https://mathscinet.ams.org/mathscinet-getitem?mr=1317563 https://doi.org/10.2307/2034412 https://doi.org/10.1017/cbo9780511470967 introduction bounded harmonic functions positive harmonic functions references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1000852 j. algebra comb. discrete appl. 8(3) • 213–218 received: 27 november 2020 accepted: 16 april 2021 journal of algebra combinatorics discrete structures and applications subsocles and direct sum of uniserial modules research article ayazul hasan abstract: suppose m is a qtag-module with a subsocle s such that m/s is a direct sum of uniserial modules. our global aim here is to investigate an interesting connection between the structure of m/s and the qtag-module m. specifically, the condition s = soc(n) for some h-pure submodules n of m allows m to inherit the structure of m/s. 2010 msc: 16k20, 13c12, 13c13 keywords: qtag-modules, h-pure submodules, socles 1. introduction and background material modules are the natural generalizations of abelian groups. the results which hold good for abelian groups need not be true for modules. by putting some restrictions on rings/modules these results hold good for modules too. in 1976 singh [16] started the study of tag-modules satisfying the following two conditions while the rings were associative with unity. (i) every finitely generated submodule of any homomorphic image of m is a direct sum of uniserial modules. (ii) given any two uniserial submodules u and v of a homomorphic image of m, for any submodule w of u, any non-zero homomorphism f : w → v can be extended to a homomorphism g : u → v , provided the composition length d(u/w) ≤ d(v/f(w)). in 1987 singh made an improvement and studied the modules satisfying only the condition (i) and called them qtag-modules. other authors also have considered the problem of detecting finite direct sums of uniserial modules [2, theorem 4]. the study of qtag-modules and their structure began with work of singh in [17]. the structure theory of such modules has been developed by various authors. different notions and structures of qtag-modules have been studied, and a theory was developed, ayazul hasan; college of applied industrial technology, jazan university, jazan, kingdom of saudi arabia (email: ayazulh@jazanu.edu.sa, ayaz.maths@gmail.com). 213 https://orcid.org/0000-0002-2895-8267 a. hasan / j. algebra comb. discrete appl. 8(3) (2021) 213–218 introducing several notions, interesting properties, and different characterizations of submodules. many interesting results have been surfaced, but there is still a lot to explore. all rings below are assumed to be associative and with nonzero identity element; all modules are assumed to be unital qtag-modules. a uniserial module m is a module over a ring r, whose submodules are totally ordered by inclusion. this means simply that for any two submodules n1 and n2 of m, either n1 ⊆ n2 or n2 ⊆ n1. a module m is called a serial module if it is a direct sum of uniserial modules. an element x ∈ m is uniform, if xr is a non-zero uniform (hence uniserial) module and for any r-module m with a unique decomposition series, d(m) denotes its decomposition length. for a uniform element x ∈ m, e(x) = d(xr) and hm (x) = sup { d ( yr xr ) | y ∈ m, x ∈ yr and y uniform } are the exponent and height of x in m, respectively. hk(m) denotes the submodule of m generated by the elements of height at least k and hk(m) is the submodule of m generated by the elements of exponents at most k. let us denote by m1, the submodule of m, containing elements of infinite height. the module m is h-divisible if m = m1 = ∞⋂ k=0 hk(m) and it is h-reduced if it does not contain any h-divisible submodule. in other words, it is free from the elements of infinite height. the module m is said to be bounded [16], if there exists an integer k such that hm (x) ≤ k for every uniform element x ∈ m. moreover, a module m is called σ-uniserial [5], if it is isomorphic to a direct sum of uniserial modules. the sum of all simple submodules of m is called the socle of m, denoted by soc(m) and a submodule s of soc(m) is called a subsocle of m. a submodule n of m is h-pure in m if n ∩hk(m) = hk(n), for every integer k ≥ 0. for an ordinal σ, a submodule n of m is said to be σ-pure [14], if hβ(m)∩n = hβ(n) for all β ≤ σ. a submodule b ⊆ m is a basic submodule [13] of m, if b is h-pure in m, b = ⊕bi, where each bi is the direct sum of uniserial modules of length i and m/b is h-divisible. a submodule n ⊂ m is nice [11] in m, if hσ(m/n) = (hσ(m) +n)/n for all ordinals σ, i.e. every coset of m modulo n may be represented by an element of the same height. imitating [12], the submodules hk(m),k ≥ 0 form a neighborhood system of zero, thus a topology known as h-topology arises. closed modules are also closed with respect to this topology. thus, the closure of n ⊆ m is defined as n = ∞⋂ k=0 (n + hk(m)). therefore the submodule n ⊆ m is closed with respect to h-topology if n = n. for qtag-modules m and m′, a homomorphism f : m → m′ is said to be small if kerf contains a large submodule of m. the set of all small homomorphisms from m to m′, denoted by homs(m,m′) is a submodule of hom(m,m′). mehran et al. [14] proved that the results which hold for tag-modules are also valid for qtagmodules. many results of this paper are the generalization of [7]. in what follows, all notations and notions are standard and will be in agreement with those used in [3, 4]. 2. main concepts and results the problem with which the present article is concerned is to find those classes of qtag-modules, which possess the following property: m/s is a direct sum of uniserial modules such that s = soc(n) for some h-pure submodules n of m, then m is a direct sum of uniserial modules and n is a direct summand of m. singh [16] proved that a qtag-module m is a direct sum of uniserial modules if and only if m is the union of an ascending chain of bounded submodules. this indicates that m is a direct sum of uniserial modules if and only if soc(m) = ⊕ k∈ω sk and hm (x) = k for every x ∈ sk. in [9] it was seen that any h-pure submodule n of a qtag-module m with a submodule k of n containing soc(n), and for some elementary summand t of soc(k) in soc(m), (t ⊕k)/k = soc(l/k) where l/k is an h-pure submodule of m/k which is a direct sum of uniserial modules. then n is 214 a. hasan / j. algebra comb. discrete appl. 8(3) (2021) 213–218 a summand of of m and m/n is a direct sum of uniserial modules. we also indicate the well-known generalizations to the last fact that if n is an h-pure submodule of a qtag-module m such that m/k, where k is a submodule of n generated by uniform elements of exponent at most k for some positive integer k, is a direct sum of uniserial modules then m is a direct sum of uniserial modules. as a culmination of a series of such claims, the following excellent statement-improvement of the preceding one, concerning the subsocles and a direct sum of uniserial modules, namely: corollary 2.1. [9, corollary 3]. let s be a subsocle of a qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is a direct sum of uniserial modules then so is m. from the above discussion, it is naturally to pose the following actual in this topic problem. given s is a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s belongs to any fixed sort of qtag-modules, then does this imply that m belongs to the same module sort, whenever n is a direct summand of m? to develop the study, we need to find certain kinds of qtag-modules, and we start with the following subsection. 2.1. elementary completeness in qtag-modules the purpose of this subsection is to explore some structural consequences of completeness in qtagmodules. first, we recall the following notions from [1, 6, 8], respectively. a qtag-module m is said to be quasi-complete, if the closure n of every h-pure submodule n of m, is h-pure in m and a qtag-module m is called semi-complete, if it is the direct sum of a closed module and a direct sum of uniserial modules. moreover, a qtag-module m is called h-pure-complete, if for every subsocle s of m there exists an h-pure submodule n of m such that s = soc(n). now we are ready to deal with the following theorem. theorem 2.2. let s be a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is h-pure-complete, then m is h-pure-complete and n is a summand of m. proof. by hypothesis, soc(m)/s supports an h-pure submodule k/s for some h-pure submodules k of m. by [1, proposition 1, proposition 2], m/s = n/s ⊕ k/s. note that if m = l ⊕ t and m is h-pure-complete, then m/soc(t) ' l⊕h1(t) is h-pure-complete. consequently, (m/s)/soc(k/s) is h-pure-complete. but (m/s)/soc(k/s) = (m/s)/(soc(m)/s) ' m/soc(m) ' h1(m). now m is h-pure-complete if and only if hk(m) is h-pure-complete for some integer k. consequently, m is h-pure-complete. we continue with other statement, namely corollary 2.3. let s be a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is h-pure-complete such that m/s has an unbounded direct sum of uniserial modules summand, then m has an unbounded direct sum of uniserial modules summand and n is a summand of m. proof. note that if m = n ⊕ k for some h-pure submodules k of m, and m has an unbounded direct sum of uniserial modules summand, then either n or k has such a summand. if h1(n) has an unbounded direct sum of uniserial modules summand, then n has such a summand. now, we proceed by proving corollary 2.4. let s be a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is quasi-complete, then m is quasi-complete and n is a summand of m. 215 a. hasan / j. algebra comb. discrete appl. 8(3) (2021) 213–218 proof. clearly, quasi-complete modules are h-pure-complete and summands of quasi-complete modules are quasi-complete. also, h1(m) quasi-complete implies that m is quasi-complete, and we are done. nevertheless, in certain specific cases, the following direct summand property holds. corollary 2.5. let s be a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is a direct sum of closed modules, then m is a direct sum of closed modules and n is a summand of m which is a direct sum of closed modules. proof. if m is a direct sum of closed modules and soc(m) = s ⊕ k, where hm (x + y) = min{hs(x),hk(y)} for all x ∈ s and y ∈ k, then s and k support summands of m which are direct sums of closed modules. then soc(m)/s supports a summand l/s in m/s which is a direct sum of closed modules. observe that a summand of a direct sum of closed modules is a direct sum of closed modules. note that if h1(n) is a direct sum of closed modules, then n is such a direct sum. consequently, we get that m is a direct sum of closed modules and n is a summand of m. the proof is over. as immediate consequence, we yield the following. corollary 2.6. let s be a subsocle of the qtag-module m such that s = soc(n) for some h-pure submodules n of m. if m/s is semi-complete, then m is semi-complete and n is a summand of m. analysis. the condition s = soc(n) for some h-pure submodules n of m is essential. it has been seen that m/s is a direct sum of uniserial modules, but m is not such a direct sum of uniserial modules. it is also easy to see that soc(m)/s is not a subsocle of m/s with s = soc(n). for some submodules l of m, consider the h-pure resolution l � m � n, where n is a module which is not a direct sum of uniserial modules and m is a direct sum of uniserial modules. let s = soc(l). if soc(m)/s is a subsocle of m/s with s = soc(n), then l is a summand of m. but this contradicts the fact that n is not a direct of uniserial modules. 2.2. large submodules and ht -modules the study of large submodules and its numerous characterizations makes the theory of qtagmodules more enlightening. a fully invariant submodule l ⊆ m is large [13], if l + b = m, for every basic submodule b in m. it is well-known that any large submodule l of m, l1 = m1. likewise, it is evident that l/m1 is large in m/m1 if and only if l is large in m. we also discuss a significant class of qtag-modules which properly contains the closed module. the members of such a class are termed as ht-modules. a qtag-module m is said to be a ht-module (see [5]) if for any σ-uniserial module m′, a homomorphism f : m → m′ is small. in fact, a qtag-module m is a ht-module if there exists k ∈ z+ with n ⊃ soc(hk(m)) whenever m/n is a direct sum of uniserial modules. nontrivial examples of ht-modules are the quasi-complete modules and, in particular, the closed modules. we come now to a significant characterization of large submodule. lemma 2.7. let s be a subsocle of the qtag-module m. if l is a large submodule of m, then (l+s)/s contains a large submodule of m/s. proof. as we have noted earlier, a submodule n of m contains a large submodule l of m if and only if for each integer k there is an integer tk such that sock(htk (m)) ⊆ n. let k and tk be the appropriate integers for l in m. for (l + s)/s in m/s, let nk = tk+1 for each integer k. it is easy to see that sock(hnk (m/s)) ⊆ (l + s)/s. consequently, (l + s)/s contains a large submodule of m/s. next, we concentrate on the following theorem. 216 a. hasan / j. algebra comb. discrete appl. 8(3) (2021) 213–218 theorem 2.8. let s be a subsocle of the qtag-module m. then m is a ht-module if and only if m/s is a ht-module. proof. suppose k is a submodule of m, and let f : m/s → m′ be a map such that ker(f) = k/s. thus, the composition map m g −→m/s f −→m′. since m is a ht-module, it follows that k ⊇ l, where l is large in m. the submodule k/s contains (l+s)/s which contains a large submodule of m/s. consequently, m/s is a ht-module. the converse follows from lemma 2.7 and the following relation soc(m)/s � m/s � m/soc(m) ' h1(m). 2.3. totally projective modules this brief subsection is devoted to the exploration of so-called totally projective modules [10]. an h-reduced qtag-module m is said to be totally projective if it possesses a collection n consisting of nice submodules of m which satisfies the following three conditions: (i) 0 ∈n ; (ii) if {ni}i∈i is any subset of n , then ∑ i∈i ni ∈n ; (iii) given any n ∈ n and any countable subset x of m, there exists k ∈ n containing n ∪x, such that k/n is countably generated. totally projective modules have a significance in the theory of qtag-modules, which may be characterize in several ways. we require only for information a few more comprehensive characterizations (see [15]): let σ be a limit ordinal such that σ = ω + β. a qtag-module m is called σ-projective, if there exists a submodule n ⊂ hβ(m) such that m/n is a direct sum of uniserial modules. a qtag-module m is totally projective, if and only if m/hσ(m) is σ-projective for every ordinal σ. and so, we prepare to prove the following. lemma 2.9. let s be a subsocle of the qtag-module m such that s = soc(n) for some (σ + 1)-pure submodules n of m, where σ is the length of m/s. if m/s is totally projective, then m is totally projective and n is a summand of m. proof. since n/s is (σ + 1)-pure in m/s which is σ-projective and so n/s is a summand of m/s. consequently, m/n is totally projective and since n/s ' h1(n), n is totally projective. consider the exact sequence n � m � m/n. now n is σ-pure in m and m/n is σ-projective. thus, the preceding exact sequence splits and n is a summand of m and m is totally projective. as a direct consequence of the preceding lemma we have the following corollary. corollary 2.10. let s be a subsocle of the qtag-module m such that s = soc(n) for some (σ + 1)pure submodules n of m, where σ is the length of m/s. if m/s is a direct sum of countably generated h-reduced modules, then m is a direct sum of countably generated h-reduced modules and n is a summand of m. 217 a. hasan / j. algebra comb. discrete appl. 8(3) (2021) 213–218 3. open problems in closing, we pose the following left-open questions: problem 3.1. let f be a concrete class of qtag-modules. does it follow that m ∈ f ⇔ s ∈ f, whenever s is a subsocle of the qtag-module m and m/s is a direct sum of uniserial modules. problem 3.2. does the class f defined above indeed contains all h-reduced qtag-modules? problem 3.3. is it true that if one large submodule of a qtag-module is a direct sum of uniserial module, then all large submodules are also direct sum of uniserial modules? acknowledgment: the author owes his sincere thanks to the specialist referees for the constructive criticism and helpful suggestions made, and to the editor, for the expert editorial advice. references [1] m. ahmad, a. h. ansari, m. z. khan, on subsocles of s2-modules, tamkang j. math. 11(2) (1980) 221-229. 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[11] a. mehdi, m.y. abbasi, f. mehdi, nice decomposition series and rich modules, south east asian j. math. & math. sci. 4(1) (2005) 1-6. [12] a. mehdi, s. a. r. k. naji, a. hasan, small homomorphisms and large submodules of qtagmodules, sci. ser. a. math sci. 23 (2012) 19–24. [13] a. mehdi, f. sikander, s. a. r. k. naji, generalizations of basic and large submodules of qtagmodules, afr. mat. 25(4) (2014) 975-986. [14] h. a. mehran, s. singh, on σ-pure submodules of qtag-modules, arch. math. 46(6) (1986) 501– 510. [15] f. sikander, a. hasan, a. mehdi, on n-layered qtag-modules, bull. math. sci. 4(2) (2014) 199-208. [16] s. singh, some decomposition theorems in abelian groups and their generalizations, ring theory: proceedings of ohio university conference, marcel dekker, new york 25 (1976) 183-189. [17] s. singh, abelian groups like modules, act. math. hung, 50 (1987) 85-95. 218 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=696923 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=696923 https://doi.org/10.1080/00927879008823928 https://doi.org/10.1080/00927879008823928 https://doi.org/10.1007/s13370-015-0318-7 https://doi.org/10.4153/cjm-1971-005-7 https://doi.org/10.4153/cjm-1971-005-7 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=539804 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=539804 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=506442 https://mathscinet-ams-org.ep.bib.mdh.se/mathscinet-getitem?mr=2961700 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=2208767 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=2208767 https://doi.org/10.1007/s13370-013-0167-1 https://doi.org/10.1007/s13370-013-0167-1 https://doi.org/10.1007/bf01195018 https://doi.org/10.1007/bf01195018 https://doi.org/10.1007/s13373-014-0050-x https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=435146 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=435146 https://doi.org/10.1007/bf01903367 introduction and background material main concepts and results open problems references issn 2148-838x j. algebra comb. discrete appl. 9(3) • 149–159 received: 7 november 2020 accepted: 21 march 2022 journal of algebra combinatorics discrete structures and applications the covering numbers of the mclaughlin group and some primitive groups of low degree research article michael epstein abstract: a finite cover of a group g is a finite collection c of proper subgroups of g with the property that⋃ c = g. a finite group admits a finite cover if and only if it is noncyclic. more generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. if c is a finite cover of a group g, and no cover of g with fewer subgroups exists, then c is said to be a minimal cover of g, and the cardinality of c is called the covering number of g, denoted by σ(g). here we investigate the covering numbers of the mclaughlin sporadic simple group and some low degree primitive groups. 2010 msc: 20d60 keywords: covering numbers, mclaughlin group, primitive groups 1. introduction a finite collection c of proper subgroups of a group g with the property that ⋃ c = g is called a finite cover of g. if no cover of g with fewer subgroups exists, then c is called a minimal cover and the number of subgroups in c is called the covering number of g, denoted σ(g). a number of interesting results about covering numbers were proven in [6], where it was conjectured that the covering number of a finite, noncyclic, solvable group is of the form 1 + q, where q is the order of a chief factor of the group. this conjecture was proven by m. j. tomkinson in [18]. in light of this result, much of the more recent work on covering numbers of finite groups has focused on nonsolvable groups, and in particular on simple and almost simple groups. a number of results can be found in [2, 3, 5, 10, 13–17]. in this article, we investigate group covering numbers in some outstanding open cases from [11] and [13]. in section 31, we determine the covering number of the mclaughlin sporadic simple group. this was first attempted by p. e. holmes in [13], where it was shown that 24541 ≤ σ(mcl) ≤ 24553. we will show that the upper bound of 24553 is in fact the correct covering number. michael epstein; department of mathematics, colorado state university, fort collins, colorado 80523, usa (email: michael.epstein@colostate.edu). 1 section 3 of this article is based on a portion of the author’s dissertation [9], submitted in partial fulfillment of the requirements for the degree of doctor of philosophy at florida atlantic university. 149 https://orcid.org/0000-0002-5541-2323 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 in [11], the authors investigated integers which are not covering numbers of groups, and in the process determined the covering numbers of many primitive groups of degree less than or equal to 129. however, a few difficult cases remain unsolved (see table 2 of [11] for upper and lower bounds). in section 4 we consider some of these open cases. in particular, we find the exact covering numbers of l5(3), pσl2(121), a5 wr 3, (a5 ×a5 ×a5).6, pγl2(125), and l7(2) and give improved bounds for the covering numbers of a7 wr 2, l3(4).222, hs : 2, l2(11) wr 2, and pgu3(5). 2. preliminaries in this article we follow [8] for basic group theoretic terminology and notation. we use the notation of [7] for simple groups and group structures. the basic method we use for determining the covering numbers of finite groups is based on two simple observations: first, that one need only consider covers consisting of maximal subgroups, and second, that in order to prove that a collection c of proper subgroups is a cover of a group g it is sufficient to show that each of the maximal cyclic subgroups is contained in a subgroup from c. we call the maximal cyclic subgroups of g the principal subgroups of g and a generator of a principal subgroup is called a principal element of g. in light of these observations, the first step in determining the covering number of a group g is to determine the maximal subgroups and principal subgroups of g up to conjugacy. the second step is to determine which principal subgroups are contained in which maximal subgroups. consider a bipartite graph whose vertex set is the union of a conjugacy class p of principal subgroups of g and a conjugacy class m of maximal subgroups of g, with an edge between c ∈p and h ∈m if and only if c ≤ h. the group g acts transitively on both p and m by conjugation, so any two vertices in p have the same degree ap,m and any two vertices in m have the same degree bp,m. moreover, counting the number of edges in the graph in two ways yields the equation ap,m|p| = bp,m|m|. (1) consequently, one can construct two matrices a = (ap,m) and b = (bp,m), with rows indexed by the conjugacy classes of principal subgroups of g and columns indexed by the classes of maximal subgroups, describing the incidence between the principal subgroups of g and the maximal subgroups up to conjugacy. in this article we give only the matrix a for each group as the entries tend to be smaller, and as noted above the entries of b can easily be computed from those of a. in order to compute the entries of the matrices a and b, we make use of the characters of the permutation representations of g acting on the sets of right cosets of representatives of each conjugacy class of maximal subgroups. this is justified by the following well-known fact from character theory: proposition 2.1. let h be a subgroup of g, x ∈ g, k be the conjugacy class of x, and θ be the permutation character of the action of g on the right cosets of h. then, |k ∩h| = θ(x)|k| |g : h| . with this proposition we can compute the entries of a and b as follows: proposition 2.2. let x ∈ g, p be the conjugacy class of 〈x〉, h be a subgroup of g from conjugacy class m, and θ be the permutation character of the action of g on the right cosets of h. then ap,m = θ(x)|m| |g : h| and bp,m = θ(x)|p| |g : h| . 2 for clarity we use the same numbering for the groups l3(4).2 as in [11]. in [7], this group is called l3(4).21. 150 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 proof. let k be the conjugacy class of x. we consider another bipartite graph, with vertex set k∪m and with an edge between y ∈ k and j ∈ m if and only if y ∈ j. any two vertices in k have the same degree α, any two vertices in m have the same degree β, and α|k| = β|m|. by proposition 2.1, β = θ(x)|k| |g:h| , and since y ∈ j if and only if 〈y〉 ≤ j, ap,m = α = β|m| |k| = θ(x)|m| |g:h| . the formula for bp,m then follows from (1). sometimes it is possible to simplify the problem and reduce the number of conjugacy classes of subgroups we must consider. for example, if the members of some conjugacy class of maximal subgroups contain no principal subgroups, then this conjugacy class may be eliminated from further consideration as no minimal cover can contain subgroups from this class. on the other hand, define u to be the set of all maximal subgroups h of g for which there is a principal subgroup c such that h is the unique maximal subgroup of g containing c. then u is a (possibly empty) union of conjugacy classes of maximal subgroups of g and any cover of g consisting of maximal subgroups must contain every member of u. as we only consider covers consisting of maximal subgroups, we must include every member of u. consequently, we need no longer consider the conjugacy classes of subgroups in u, nor the conjugacy classes of principal subgroups covered by the members of u. thus we may delete the rows and columns of a and b corresponding to the classes of subgroups that have been removed from consideration to obtain submatrices a′ and b′, of a and b respectively, which describe the incidence between the remaining classes of subgroups. the third step is to find a small cover which will be a candidate for a minimal cover. this will often, but not always, be a union of conjugacy classes of maximal subgroups of g. one can easily find covers of this type; they consist of the subgroups from u as well as those from the conjugacy classes of maximal subgroups corresponding to a set of columns of a′ whose sum has no zero entries. in any case, once a cover has been found, the number of subgroups in the cover gives an upper bound on σ(g). we establish a lower bound for the covering number by solving a certain integer linear programming (ilp) problem. to make this more explicit, consider a cover c consisting of maximal subgroups of g, and let m1, . . . ,mn be the remaining conjugacy classes of maximal subgroups of g (i.e. those not contained in u). define xj = |mj ∩c| for 1 ≤ j ≤ n. for each remaining conjugacy class p of principal subgroups of g the following inequality must hold: bp,m1x1 + bp,m2x2 + · · · + bp,mnxn ≥ |p|. the sum of |u| and the minimum value of the function x1 +· · ·+xn subject to these linear constraints, as well as the conditions xj ∈ z, and 0 ≤ xj ≤ |mj| for 1 ≤ j ≤ n, is a lower bound for σ(g). if this lower bound is equal to the number of subgroups in a cover c, then necessarily c is minimal and σ(g) = |c|, though in general this bound may be strictly less than the actual covering number of the group. one may be able to work around this difficulty in some instances by deriving additional linear constraints by, for example, considering how the group acts on appropriate combinatorial objects. this is illustrated in the computation of the covering number of the mclaughlin group in section 3. in cases where the lower bound described above is less than the number of subgroups in the best cover we have found, we can formulate a different linear programming problem to find the covering number. let m be the matrix with rows and columns indexed by the remaining principal subgroups and the remaining maximal subgroups of g respectively, such that the entry in the row c and column h is 1 if c ≤ h and 0 otherwise. then a minimal cover of g consisting of maximal subgroups is then of the form u ∪x, where the members of x are the maximal subgroups corresponding to the positions of the 1’s in a (0,1)-vector x whose entries have minimum possible sum subject to the condition that mx ≥~1, where ~1 denotes the column vector of all 1’s of length equal to the number of rows of m. the benefit of this second ilp formulation is that a solution to the linear programming problem always yields the exact covering number of the group, but the trade off is that it generally has far more variables and constraints than the other linear programming problem described above, and is usually much more difficult to solve. however, even if the problem is intractable, we can get upper and lower bounds on the covering number from an incomplete attempt to solve the ilp from the best incumbent solution and best bound found 151 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 by the solver at the time it is interrupted. we take this approach in section 4 when investigating the covering numbers of low degree primitive groups. 3. the mclaughlin group the mclaughlin group, one of the sporadic simple groups, is a subgroup of index two in the full automorphism group of the mclaughlin graph, a strongly regular graph with parameters (275, 112, 30, 56). we note that the independence number of the mclaughlin graph is 22 (see [4]), and that the mclaughlin group acts transitively on the vertices, the edges, and the nonedges of the mclaughlin graph. the maximal subgroups from classes m1, m6, and m7 are the stabilizers of the vertices, edges, and nonedges of the mclaughlin graph respectively. tables 1 and 2 give the conjugacy classes of principal and maximal subgroups of the mclaughlin group, and the matrix a for the mclaughlin group is given in table 3. table 1. conjugacy classes of principal subgroups of the mclaughlin group class order class size p1 5 8981280 p2 6 12474000 p3 8 28066500 p4 9 11088000 p5 11 16329600 p6 12 18711000 p7 14 21384000 p8 30 7484400 table 2. conjugacy classes of maximal subgroups of the mclaughlin group class order class size structure m1 3265920 275 u4(3) m2 443520 2025 m22 m3 443520 2025 m22 m4 126000 7128 u3(5) m5 58320 15400 31+4+ : 2s5 m6 58320 15400 34 : m10 m7 40320 22275 l3(4) : 2 m8 40320 22275 2.a8 m9 40320 22275 24 : a7 m10 40320 22275 24 : a7 m11 7920 113400 m11 m12 3000 299376 51+2+ : 3 : 8 152 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 table 3. matrix a for mcl m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 p1 5 5 5 3 0 5 5 0 5 5 5 1 p2 2 3 3 3 2 1 6 7 6 6 6 0 p3 1 1 1 2 2 2 1 1 1 1 2 4 p4 2 0 0 0 1 1 0 0 0 0 0 0 p5 0 1 1 0 0 0 0 0 0 0 1 0 p6 1 0 0 0 3 2 0 1 0 0 0 2 p7 0 0 0 0 0 0 1 1 1 1 0 0 p8 0 0 0 0 1 0 0 1 0 0 0 1 theorem 3.1. the covering number of the mclaughlin group is 24553. proof. we show that there exists a cover c ⊆ m1 ∪m2 ∪m8 with |c| = 24553 as follows: first, observe that m2∪m8 is sufficient to cover all of the principal elements of mcl except for those of order 9, which generate the cyclic subgroups from p4, and that these are contained in the subgroups from class m1. each element of order 9 fixes two adjacent vertices in the mclaughlin graph, the edge between them, and no nonedges. as the independence number of the mclaughlin graph is 22, we may choose an independent set i consisting of 22 vertices of the mclaughlin graph. let s be the subset of m1 consisting of the stabilizers of the vertices of the mclaughlin graph which are not in i. then |s| = 253, and given any element x of order 9 in mcl, at least one of the two subgroups from m1 which contain x is in s. consequently, c = s ∪m2 ∪m8 is a cover of mcl with |c| = 253 + 2025 + 22275 = 24553. as a result, σ(mcl) ≤ 24553. we now prove that no smaller cover can exist. suppose that c is a minimal cover of mcl consisting of maximal subgroups. for 1 ≤ j ≤ 12, let xj = |c ∩mj|. the cyclic subgroups from p5 of order 11 are contained only in the maximal subgroups from classes m2, m3, and m11. each subgroup h ∈m2∪m3 contains 8064 of these cyclic subgroups and each subgroup h ∈m11 contains 144 of them. since c covers all of the elements of order 11, 8064(x2 + x3) + 144x11 ≥ |p5| = 16329600, from which it follows that x2 + x3 + x11 ≥ 2025. now, the cyclic subgroups of order 14 from p7 are contained only within the maximal subgroups from classes mi for 7 ≤ i ≤ 10, and a subgroup from any of these classes contains exactly 960 members of p7. hence, 960(x7 + x8 + x9 + x10) ≥ |p7| = 21384000, from which we may deduce that x7 + x8 + x9 + x10 ≥ 22275. a similar analysis for the subgroups from p4 will yield the inequality x1 + x5 + x6 ≥ 138, but this is insufficient for our purposes. we consider the action of the principal subgroups of mcl of order 9 on the mclaughlin graph to obtain a stronger inequality. let w be the set of vertices of the mclaughlin graph whose stabilizers are not contained in c. observe that x1 + |w | = 275. now, an cyclic subgroup 〈x〉 of order 9 in mcl is left uncovered by the subgroups from m1 ∩c if and only if the unique edge e of the mclaughlin graph fixed by x is an edge of the subgraph induced by w . the stabilizer in mcl of an edge of the mclaughlin graph contains 720 cyclic subgroups of order 9, so the total number of members of p4 left uncovered by the subgroups from m1 ∩c is 720n, where n is the number of edges in the subgraph of the mclaughlin graph induced by w . aside from the subgroups from class m1, the only maximal subgroups which contain members of p4 are those from m5 ∪m6, each of which contains 720 of them. consequently, at least n subgroups from m5 ∪m6 are needed to cover the remaining cyclic subgroups of order 9. now, the mclaughlin graph has independence number 22, so the subgraph induced by w can have no independent set of more than 22 vertices. therefore n ≥ |w |− 22. thus, x1 + x5 + x6 ≥ x1 + n ≥ x1 + |w |− 22 = 275 − 22 = 253. 153 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 it follows from the inequalities x2 + x3 + x11 ≥ 2025, x7 + x8 + x9 + x10 ≥ 22275, and x1 + x5 + x6 ≥ 253 that σ(mcl) = |c| = ∑12 j=1 xj ≥ 2025 + 22275 + 253 = 24553, which completes the proof. this settles one of two open cases for which upper and lower bounds were given in [13]. the other such case, that of the janko group j1, remains open, though improved bounds were given in [16]. 4. some primitive groups of low degree in this section we take a computational approach to investigating the covering numbers of some of the remaining primitive groups of degree less than 129 from table 2 of [11]. the approach is based on linear programming and follows the method described in section 2. all of the group-theoretic computations were done using magma [1], and the linear programming problems were solved using gurobi optimization software [12]. in many cases we are unable to solve the larger ilp problem in a reasonable amount of time; in this case we can still obtain upper and lower bounds for the covering number from gurobi’s best incumbent solution and best lower bound at the time the computation is interrupted. we summarize our results in tables 4 and 5. previous previous group lower bound upper bound covering number l5(3) 393030144 – 393031475 pσl2(121) 671 794 794 a5 wr 3 216 342 317 (a5 ×a5 ×a5).6 1000 1217 1127 pγl2(125) 7750 7876 7876 l7(2) 184308203520 – 184308218125 table 4. exact covering numbers of some of the low degree primitive groups from the list of open cases in [11]. previous previous new new group lower bound upper bound lower bound upper bound a7 wr 2 447 667 460 667 l3(4).22 138 166 144 166 hs : 2 11859 22375 15127 22375 l2(11) wr 2 570 926 721 817 pgu3(5) 6000 6526 6307 6378 table 5. improved upper and lower bounds for the covering numbers of some of the low degree primitive groups from the list of open cases in [11]. we illustrate our method by presenting two cases in detail. the others are similar, and for these we simply describe a cover of size equal to the covering number given in table 4 or the upper bound given in table 5. 154 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 4.1. l5(3) l3(5) has 17 conjugacy classes of principal subgroups and 8 classes of maximal subgroups. these are as in tables 6 and 7. we note that each member of p17 is contained in a unique maximal subgroup of l5(3), which is from m8, so we must use all 393030144 subgroups from that class in the cover. the matrix a′ is given in table 8. table 6. conjugacy classes of principal subgroups of l5(3) class order class size p1 6 366949440 p2 6 366949440 p3 6 2201696640 p4 8 1857681540 p5 8 928840770 p6 9 489265920 p7 12 1238454360 p8 18 733898880 p9 24 825636240 p10 24 825636240 p11 24 1238454360 p12 24 137606040 p13 26 1524251520 p14 78 508083840 p15 80 743072616 p16 104 381062880 p17 121 393030144 table 7. conjugacy classes of maximal subgroups of l5(3) class order class size structure m1 1965150720 121 3 : (33 : 2).l4(3).2 m2 1965150720 121 3 : (33 : 2).l4(3).2 m3 196515072 1210 36.q8.s3.l3(3) m4 196515072 1210 36.q8.s3.l3(3) m5 51840 4586868 c2(3).2 m6 7920 30023136 m11 m7 7920 30023136 m11 m8 605 393030144 121 : 5 155 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 table 8. matrix a′ for l5(3) m1 m2 m3 m4 m5 m6 m7 p1 5 5 9 9 0 0 0 p2 5 5 8 8 36 0 0 p3 2 2 3 3 0 9 9 p4 1 1 2 2 0 8 8 p5 1 1 2 2 8 0 0 p6 1 1 1 1 9 0 0 p7 2 2 3 3 0 0 0 p8 2 2 2 2 0 0 0 p9 1 1 2 2 0 0 0 p10 1 1 1 1 0 0 0 p11 2 2 3 3 0 0 0 p12 4 4 5 5 0 0 0 p13 2 2 1 1 0 0 0 p14 1 1 1 1 0 0 0 p15 1 1 0 0 0 0 0 p16 0 0 1 1 0 0 0 observe that c = m1 ∪m3 ∪m8 is a cover of size 393031475, and therefore σ(l5(3)) ≤ 393031475. on the other hand, it is necessary to use all 393030144 subgroups from class m8 in order to cover the cyclic subgroups of order 121, and we must use at least 121 subgroups from m1 ∪m2 to cover the members of p15 and at least 1210 subgroups from m3 ∪m4 to cover the members of p16. consequently σ(l5(3)) = 393031475. 4.2. (a5 ×a5 ×a5).6 the group (a5 × a5 × a5).6 has 19 conjugacy classes of principal subgroups and 6 classes of maximal subgroups, described in tables 9 and 10. we note that each member of p17 is contained in a unique maximal subgroup, which is from m5. therefore it is necessary to use all 1000 subgroups from m5 in the cover. likewise, we must use the subgroup from m1 in order to cover the members of p8 ∪p9 ∪p10 ∪p11. the only classes of principal subgroups whose members are not covered by m1 ∪m5 are p12 and p15. we present the matrix a′ in table 11. 156 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 table 9. conjugacy classes of principal subgroups of (a5 ×a5 ×a5).6 class order class size p1 5 2592 p2 6 3000 p3 6 6000 p4 6 36000 p5 10 3240 p6 10 3240 p7 10 4050 p8 12 4500 p9 12 4500 p10 12 9000 p11 12 13500 p12 12 54000 p13 15 2160 p14 15 3600 p15 15 21600 p16 15 2160 p17 18 24000 p18 30 2700 p19 30 2700 table 10. conjugacy classes of maximal subgroups of (a5 ×a5 ×a5).6 class order class size structure m1 432000 1 a5.a5.s5 m2 648000 1 a5 wr 3 m3 10368 125 26.33.6 m4 6000 216 53.a4.4 m5 1296 1000 33.(22 ×a4) m6 360 3600 3 ×s5 table 11. matrix a′ for (a5 ×a5 ×a5).6 m2 m3 m4 m6 p12 0 1 2 1 p15 1 0 1 1 we can cover the remaining principal subgroups with the 126 subgroups from m2 ∪m3. moreover, covering the members of p15 without using the subgroup from m2 requires at least 216 subgroups from 157 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 m4 ∪m6, and will not yield a minimal cover. therefore we must use the subgroup from m2, and we can complete a minimal cover by finding a minimal subset of m3 ∪m4 ∪m6 which covers the elements of p12. therefore we construct the 54000 × 3941 (0,1)-incidence matrix between the members of p12 and those of m3 ∪m4 ∪m6. we use gurobi to solve the corresponding linear programming problem, and find that the minimum number of subgroups required to cover the members of p12 is indeed 125. therefore m1∪m2∪m3∪m5 is a minimal cover of (a5×a5×a5).6, and σ((a5×a5×a5).6) = 1127. 4.3. the remaining groups from tables 4 and 5 for each of the remaining groups in tables 4 and 5 we describe a cover which is either minimal (in the case of groups from table 4) or else has size equal to the upper bound given in table 5. in this section dn denotes the dihedral group of order 2n. pσl2(121): there is a minimal cover c (of size 794) which is a union of three conjugacy classes of maximal subgroups. c consists of the normal subgroup l2(121), the conjugacy class of subgroups isomorphic to 112 : (5 × d12) (122 subgroups), and all 671 subgroups from one of the two classes of subgroups isomorphic to 2 ×l2(11).2 (and no subgroups from the other class). a5 wr 3: there is a minimal cover c (of size 317) consisting of the following subgroups: the normal subgroup a35, all 216 subgroups from the conjugacy class of maximal subgroups isomorphic to d5 wr 3, and 100 of the 125 subgroups (all conjugate) which are isomorphic to 26 : 3 : 31+2+ . pγl2(125): there is a minimal cover c (of size 7876) which is the union of two conjugacy classes of maximal subgroups, namely the class of 7750 subgroups isomorphic to 7 : (2 ×d9 : 3), and the class of 126 subgroups isomorphic to 53 : (4 × 31 : 3). l7(2): there is a minimal cover c (of size 184308218125) which is the union four conjugacy classes of maximal subgroups, namely the class of 184308203520 subgroups isomorphic to 127 : 7, one of the two classes of 127 subgroups isomorphic to 26 : l6(2), one of the two classes of 2667 subgroups isomorphic to 210.s3.l5(2), and one of the two classes of 11811 subgroups isomorphic to 212.a8.l2(7). a7 wr 2: there is a cover of size 667 which is the union of three conjugacy classes of maximal subgroups, namely the class containing the normal subgroup a27, one of the two conjugacy classes of subgroups isomorphic to l2(7) wr 2 (225 subgroups), and the conjugacy class of 441 subgroups isomorphic to s5 wr 2. l3(4).22: there is a cover c of size 166 consisting of the following maximal subgroups: the normal subgroup l3(4), all 105 subgroups isomorphic to 42.s4 (all conjugate), and a total of 60 subgroups isomorphic to so3(7). there are three conjugacy classes of maximal subgroups isomorphic to so3(7) in l3(4).22, each of size 120. the cover c contains 28 subgroups from one of these classes, 32 subgroups from a second class, and no subgroups from the remaining class. hs : 2: there is a cover of size 22375 which is the union of four conjugacy classes of maximal subgroups: the class of 100 subgroups isomorphic to m22.2, the class of 1100 subgroups isomorphic to 2 ×s8, the class of 5775 subgroups isomorphic to 4.25.s5, and the class of 15400 subgroups isomorphic to 22.a6.22. l2(11) wr 2: there is a cover c of size 817 consisting of the normal subgroup l2(11)2, a total of 77 subgroups isomorphic to a5 wr 2 (66 from one conjugacy class of maximal subgroups isomorphic to a5 wr 2 and 11 subgroups from the other), all 144 subgroups from the conjugacy class of maximal subgroups isomorphic to (11 : 5) wr 2, and 595 subgroups from the conjugacy class of 660 maximal subgroups isomorphic to pgl2(11). pgu3(5): there is a cover of size 6378 consisting of the following subgroups: all 6000 maximal subgroups isomorphic to 3 × 7 : 3 (all conjugate), all 126 maximal subgroups isomorphic to 51+2+ : 24 (all conjugate), and 252 maximal subgroups isomorphic to 3 × sl2(5) : 2 (out of 525, all of which are conjugate). 158 m. epstein / j. algebra comb. discrete appl. 9(3) (2022) 149–159 references [1] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symbolic comput. 24(3-4) (1997) 235–265. 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[18] m. j. tomkinson, groups as the union of proper subgroups, math. scand. 81 (1997) 191–198. 159 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1016/j.jcta.2007.07.002 https://doi.org/10.1016/j.jcta.2007.07.002 https://doi.org/10.1016/j.jcta.2010.08.006 https://doi.org/10.1016/j.jcta.2010.08.006 https://doi.org/10.1016/j.jcta.2010.08.006 https://www.win.tue.nl/~aeb/graphs/mcl.html https://www.win.tue.nl/~aeb/graphs/mcl.html https://doi.org/10.1017/s0004972700036364 https://doi.org/10.1017/s0004972700036364 https://doi.org/10.7146/math.scand.a-12501 https://doi.org/10.1017/s001309150002839x https://doi.org/10.1017/s001309150002839x https://www.wiley.com/en-ie/abstract+algebra,+3rd+edition-p-9780471433347 https://www.proquest.com/openview/709599a66045e22e98e56258dcd88a6b/1?pq-origsite=gscholar&cbl=18750&diss=y https://www.proquest.com/openview/709599a66045e22e98e56258dcd88a6b/1?pq-origsite=gscholar&cbl=18750&diss=y https://mathscinet.ams.org/mathscinet-getitem?mr=3676187 https://mathscinet.ams.org/mathscinet-getitem?mr=3676187 https://doi.org/10.1080/10586458.2019.1636425 https://doi.org/10.1080/10586458.2019.1636425 https://doi.org/10.1016%2fj.jcta.2005.09.006 https://doi.org/10.1016%2fj.jalgebra.2009.10.011 https://doi.org/10.1016%2fj.jalgebra.2009.10.011 http://dx.doi.org/10.1090/conm/511 http://dx.doi.org/10.1090/conm/511 https://doi.org/10.1515/gcc-2016-0010 https://doi.org/10.1515/gcc-2016-0010 https://doi.org/10.1016/j.jcta.2004.10.003 https://doi.org/10.1016/j.jcta.2004.10.003 https://doi.org/10.7146/math.scand.a-12873 introduction preliminaries the mclaughlin group some primitive groups of low degree references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1000778 j. algebra comb. discrete appl. 8(3) • 151–160 received: 15 may 2020 accepted: 15 may 2021 journal of algebra combinatorics discrete structures and applications rotated dn-lattices in dimensions power of 3∗ research article agnaldo j. ferrari, grasiele c. jorge, antonio a. de andrade abstract: in this work, we present constructions of families of rotated dn-lattices which may be good for signal transmission over both gaussian and rayleigh fading channels. the lattices are obtained as sublattices of a family of rotated z ⊕ak2 lattices, where ak2 is a direct sum of k = 3 r−1−1 2 copies of the a2-lattice, using free z-modules in z[ζ3r + ζ−13r ]. 2010 msc: 11h06, 11r18, 94b12 keywords: lattices, cyclotomic fields, signal transmission 1. introduction a lattice λ ⊆ rn is a discrete set generated by integer combinations of n linearly independent vectors in rn over r. its packing density ∆(λ) is the proportion of the space rn covered by congruent disjoint spheres of maximum radius [8]. a lattice λ has diversity m ≤ n if m is the maximum number such that for all y = (y1, . . . ,yn) ∈ λ, with y 6= 0, there are at least m non-zero coordinates. given a full diversity lattice λ ⊆ rn, with m = n, the minimum product distance is defined as dmin(λ) = inf{ ∏n i=1 |yi| for all y = (y1, . . . ,yn) ∈ λ, with y 6= 0} [5]. lattices have been considered in different areas, especially in coding theory, and they have been studied in several papers, from different points of view [1–7, 9, 10, 12, 13, 15]. signal constellations having lattice structure have been studied for signal transmission over both gaussian and single-antenna rayleigh fading channel [7]. usually the problem of finding good signal constellations for a gaussian ∗ this work was supported by cnpq (conselho nacional de desenvolvimento científico e tecnológico) under grants no. 432735/2016-0 and 429346/2018-2 and fapesp (fundação de amparo à pesquisa do estado de são paulo) under grant no. 2013/25977-7. agnaldo j. ferrari (corresponding author); department of mathematics, são paulo state university, bauru, sp 17033-360, brazil (email: agnaldo.ferrari@unesp.br). grasiele c. jorge; institute of science and technology, federal university of são paulo, são josé dos campos, sp 12247-014, brazil (email: grasiele.jorge@unifesp.br). antonio a. de andrade; department of mathematics, são paulo state university, são josé do rio preto, sp 15054-000, brazil (email: antonio.andrade@unesp.br). 151 https://orcid.org/0000-0002-1422-1416 https://orcid.org/0000-0002-1474-6001 https://orcid.org/0000-0001-6452-2236 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 channel is associated to the search for lattices with high packing density [8]. on the other hand, for a rayleigh fading channel the efficiency is strongly related to the lattice diversity and minimum product distance [5, 7]. the approach in this work, following [12] and [13] is the use of algebraic number theory to construct rotated dn-lattices with full diversity via free z-modules. in [1, 4, 5] some families of rotated zn-lattices for n = p−1 2 , where p ≥ 5 is a prime number, and n = 2s, for s ≥ 1, with full diversity and good minimum product distance are studied for transmission over rayleigh fading channels. in [12, 13] are studied some families of rotated dn-lattices with full diversity and good minimum product distance for transmission over both gaussian and rayleigh fading channels. in [12] are constructed rotated dn-lattices for n = (p − 1)/2, where p ≥ 7 is a prime and n = 2k, for k ≥ 2 integer, and in [13] families of rotated dn-lattices for n = 2k(p− 1), with k ≥ 0 integer and p ≥ 5 a prime, and n = (p− 1)(q − 1)/4, where p,q ≥ 5 are distinct prime numbers. in this work, we construct families of rotated dn-lattices with full diversity n for n = 3s, s ≥ 1, (propositions 3.4 and 3.5). a dn-lattice has better packing density δ(dn) when compared to zn, i.e., dn has the best lattice packing density for n = 3, 4, 5 and limn−→∞ δ(zn) δ(dn) = 0, and also a very efficient decoding algorithm [8]. 2. algebraic lattices let {v1, . . . ,vm} be a set of linearly independent vectors in rn and λ = { ∑m i=1 aivi; ai ∈ z} the associated lattice. the set {v1, . . . ,vm} is called a basis for λ. a matrix m whose rows are these vectors is said to be a generator matrix for λ while the associated gram matrix is g = mmt = (〈vi,vj〉) m i,j=1 . the determinant of λ is det λ = det g and it is an invariant under change of basis (see [8, p. 4]). two lattices λ1 and λ2 are said to be similar if there is an orthogonal mapping φ : rn → rn and a real positive number c such that cφ(λ1) = λ2. when c = 1 the similar lattices λ1 and λ2 are said to be congruent or isomorphic. in this paper, as in [5, 12], we will say that λ1 is a rotated λ2-lattice if λ1 and λ2 are congruent. let k be a totally real number field of degree n and ok its ring of integes. let σi, for i = 1, . . . ,n, be the n distinct q-homomorphisms from k to r. the canonical embedding σ : k −→ rn is defined by σ(x) = (σ1(x), . . . ,σn(x)) [14, 16]. it can be shown that if i ⊆ ok is a free z-module of rank n with z-basis {w1, . . . ,wn}, then the image λ = σ(i) is a lattice in rn with basis {σ(w1), . . . ,σ(wn)} [16, chapter 8] and it has full diversity [2, 5]. a gram matrix for σ(i) is g = ( trk|q(wiwj) )n i,j=1 , where trk|q(x) = ∑n i=1 σi(x) for any x ∈ k [5]. in what follows let q(ui,uj) = trk|q(uiuj) for any ui,uj ∈ k. in this paper, we focus on the maximal totally real subfields of the cyclotomic fields q(ζ3r ), where ζ3r is a primitive 3r-th root of unity, with r ≥ 3 a positive integer [17]. 3. rotated dn-lattices via k = q(ζ3r + ζ−13r ), where r ≥ 3 and n = 3r−1 in [13, proposition 2.7] it was shown that if k is a totally real galois extension with dk an odd integer, then it is impossible to construct rotated dn-lattices via fractional ideals of ok. in particular, it is impossible to construct rotated dn-lattices via fractional ideals of z[ζ3r +ζ−13r ] since dk = 3 2(r+1)3r−1−3r−1 2 by [11]. thus, in this section, we present some families of rotated dn-lattices using free z-modules in z[ζ3r +ζ−13r ]. our strategy is to construct these lattices as sublattices of a family of rotated z⊕a k 2-lattices, where ak2 is a direct sum of k = 3r−1−1 2 copies of the a2-lattice. in [3] is presented a family of rotated z ⊕ak2-lattices as the image of a twisted embedding [2] applied to z[ζ3r + ζ −1 3r ]. in proposition 3.3, we construct a family of rotated z ⊕ak2-lattices using the canonical embedding, where the lemma 3.1 and proposition 3.2 are support for the proof of proposition 3.3. 152 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 lemma 3.1. [9] consider e0 = 1 and ei = ζi3r + ζ −i 3r , for i = 1, 2, . . . , 3 r−1 − 1. 1. if i = 0, . . . , 3r−1 − 1, then q(ei,ei) = { 3r−1 if i = 0, 2 · 3r−1 otherwise. 2. if i = 1, 2, . . . , 3r−1 − 1, then q(ei,e0) = 0. 3. if i,j = 1, . . . , 3r−1 − 1, with i 6= j, then q(ei,ej) = { −3r−1 if i + j = 3r−1, 0 otherwise. proposition 3.2. consider u0 = e0, u1 = e1 and for i = 2, 3, . . . , 3r−1 − 1 ui = { ei+1 2 if i ≡ 1 (mod 2), e3r−1− i 2 otherwise. 1. if i = 0, . . . , 3r−1 − 1, then q(ui,ui) = { 3r−1 if i = 0, 2 · 3r−1 otherwise. 2. if i = 1, 2, . . . , 3r−1 − 1, then q(ui,u0) = 0. 3. if i,j = 1, . . . , 3r−1 − 1, with i 6= j, then q(ui,uj) = { −3r−1 if i + j ≡ 3 (mod 4) and |i− j| = 1, 0 otherwise. proof. from lemma 3.1, it follows that q(u0,u0) = q(e0,e0) = 3r−1 and for i = 1, 2, . . . , 3r−1 − 1, it follows that q(ui,ui) = 2 · 3r−1 and q(ui,u0) = q(ui,e0) = 0, for ui ∈ {e1,e2, . . . ,e3r−1−1}. if i,j = 1, 2, . . . , 3r−1 − 1, with i 6= j, then q(ui,uj) =   q(ei+1 2 ,ej+1 2 ) if i,j ≡ 1 (mod 2), q(ei+1 2 ,e3r−1−j 2 ) if i ≡ 1 and j ≡ 0 (mod 2), q(e3r−1− i 2 ,ej+1 2 ) if i ≡ 0 and j ≡ 1 (mod 2), q(e3r−1− i 2 ,e3r−1−j 2 ) if i,j ≡ 0 (mod 2). for i,j ≡ 1 (mod 2), it follows that either i + j ≡ 0 (mod 4) or i + j ≡ 2 (mod 4) and i+1 2 + j+1 2 6= 3r−1. otherwise, since i 6= j, it follows that i = j = 3r−1 − 1, which is a contradiction. thus, q(ui,uj) = 0. for i ≡ 1 (mod 2) and j ≡ 0 (mod 2), it follows that i+1 2 + 3r−1 − j 2 = 3r−1 if and only if i = j − 1. for i ≡ 0 (mod 2) and j ≡ 1 (mod 2), it follows that 3r−1 − i 2 + j+1 2 = 3r−1 if and only if j = i− 1. in the last two cases, as i + j is odd, it follows that i + j ≡ 3 (mod 4), because if i + j ≡ 1 (mod 4), with i = j−1 (respectively, j = i−1), it follows that j is odd (respectively, i is odd), which is a contradiction. therefore, q(ui,uj) = −3r−1 if i + j ≡ 3 (mod 4) and |i− j| = 1. for i,j ≡ 0 (mod 2), it follows that either i + j ≡ 0 (mod 4) or i + j ≡ 2 (mod 4) and 3r−1 − i 2 + 3r−1 − j 2 6= 3r−1. otherwise, since i 6= j, it follows that i = j = 3r−1 − 1, which is a contradiction. thus, q(ui,uj) = 0. proposition 3.3. the lattice 1 √ 3r−1 σ(ok) is a rotated version of z⊕ak2, where ak2 is a direct sum of k = 3 r−1−1 2 copies of the a2-lattice. proof. from proposition 3.2, it follows that {u0,u1, . . . ,u3r−1−1} is a z-basis of ok because it is a permutation of the z-basis {e0,e1, . . . ,e3r−1−1}. a generator matrix of the algebraic lattice 1√3r−1 σα(ok) 153 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 is given by m = 1√ 3r−1 n, where n = (σi(uj−1))3 r−1 i,j=1, and the associated gram matrix is given by g = mmt = 1 3r−1 (q(ui,uj)) 3r−1−1 i,j=0 . so, g =   1 2 −1 −1 2 2 −1 −1 2 ... 2 −1 −1 2   . it follows that the matrix g is a gram matrix of z⊕ak2-lattice. in what follows, we split in two cases, i.e., we construct rotated dn-lattices for n = 3r−1, for r even and for r odd. 3.1. rotated dn-lattices for n = 3r−1, where r ≥ 4 is even in this section, we present a construction of rotated dn-lattices using z-modules in the totally real number field k = q(ζ3r + ζ−13r ), where r is even. the dn-lattice is obtained as sublattice of z⊕a k 2 using b = {u0,u1, . . . ,u3r−1−1} a z-basis of ok. proposition 3.4. let i = zω0 ⊕zω1 ⊕ . . .⊕zω3r−1−1 be a free z-module of ok, where 1. ω0 = −4u0 − 2u1 − 2u2; ω1 = −2u1 + 2u2; ω2 = 4u0 − 2u2; ω3 = −2u0 + 2u1 + 2u2 −u5 + u6 −u9 + u10; 2. for j = 1, 2 . . . , 3 r−1−11 8 , ω4j = u8j−3 −u8j−2 −u8j−1 + u8j + u8j+1 −u8j+2 −u8j+3 + u8j+4; ω4j+1 = −u8j−3 + u8j−2 + u8j−1 −u8j + u8j+1 −u8j+2 + u8j+3 −u8j+4; ω4j+2 = u8j−3 −u8j−2 −u8j−1 + u8j −u8j+1 + u8j+2 + u8j+3 −u8j+4; ω4j+3 = u8j−1 −u8j −u8j+3 + u8j+4 −u8j+5 + u8j+6 −u8j+9 + u8j+10; ω3r−1+1−4j = −u8j−3 −u8j−2 −u8j−1 −u8j + 3u8j+1 + 3u8j+2 −u8j+3 −u8j+4; ω3r−1+2−4j = −u8j−3 −u8j−2 + 3u8j−1 + 3u8j −u8j+1 −u8j+2 + u8j+3 + u8j+4; ω3r−1+3−4j = 3u8j−3 + 3u8j−2 −u8j−1 −u8j + u8j+1 + u8j+2 + u8j+3 + u8j+4; if j 6= 1, ω3r−1+4−4j = −u8j−9 −u8j−8 − 2u8j−7 − 2u8j−6 + u8j−5 + u8j−4 −u8j−3 −u8j−2 −u8j+1 −u8j+2 − 2u8j+3 − 2u8j+4; 3. for j = 3 r−1−3 8 , ω4j = u3 −u4 + u8j−3 −u8j−2 −u8j−1 + u8j + u8j+1 −u8j+2; ω4j+1 = −u3 + u4 −u8j−3 + u8j−2 + u8j−1 −u8j + u8j+1 −u8j+2; ω4j+2 = −u3 + u4 + u8j−3 −u8j−2 −u8j−1 + u8j −u8j+1 + u8j+2; ω4j+3 = 2u3 − 2u8j − 2u8j+1 − 2u8j+2; ω3r−1+1−4j = −u3 −u4 −u8j−3 −u8j−2 −u8j−1 −u8j + 3u8j+1 + 3u8j+2; ω3r−1+2−4j = u3 + u4 −u8j−3 −u8j−2 + 3u8j−1 + 3u8j −u8j+1 154 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 −u8j+2; ω3r−1+3−4j = u3 + u4 + 3u8j−3 + 3u8j−2 −u8j−1 −u8j + u8j+1 + u8j+2; ω3r−1+4−4j = −2u3 − 2u4 + u8j−9 + u8j−8 − 2u8j−7 − 2u8j−6 + u8j−5 + u8j−4 −u8j−3 −u8j−2 −u8j+1 −u8j+2. therefore, λ = 1 2 √ 3r σ(i) ⊆ r3 r−1 is a rotated version of the d3r−1 -lattice. proof. from proposition 3.2, it follows that q(ω0,ω0) = trk/q(ω0ω0) = trk/q((−4u0 − 2u1 − 2u2)(−4u0 − 2u1 − 2u2) = trk/q(16u0u0 + 16u0u1 + 16u0u2 + 4u1u1 + 8u1u2 + 4u2u2) = 16q(u0,u0) + 16q(u0,u1) + 16q(u0,u2) + 4q(u1,u1) + 8q(u1,u2) + 4q(u2,u2) = 24 · 3r−1. q(ω1,ω1) = 4q(u1,u1) + 4q(u2,u2) − 8q(u1,u2) = 24 · 3r−1. q(ω2,ω2) = 16q(u0,u0) + 4q(u2,u2) = 24 · 3r−1. q(ω3,ω3) = 4q(u0,u0) + 4q(u1,u1) + 8q(u1,u2) + 4q(u2,u2) +q(u5,u5) − 2q(u5,u6) + q(u6,u6) + q(u9,u9) −2q(u9,u10) + q(u10,u10) = 24 · 3r−1. q(ω0,ω1) = q(ω0,ω3) = q(ω1,ω3) = 0. q(ω0,ω2) = q(ω1,ω2) = q(ω2,ω3) = q(ω3,ω4) = −12 · 3r−1. let j = 1, 2, . . . , 3 r−1−3 8 . since q(ui,uj) 6= 0 if and only if i + j ≡ 3 (mod 4) and |i−j| = 1, it follows that q(ω4j,ω4j) = q(u8j−3,u8j−3) − 2q(u8j−3,u8j−2) + q(u8j−2,u8j−2) + q(u8j−1,u8j−1) − 2q(u8j−1,u8j) + q(u8j,u8j) + q(u8j+1,u8j+1) − 2q(u8j+1,u8j+2) + q(u8j+2,u8j+2) + q(u8j+3,u8j+3) − 2q(u8j+3,u8j+4) + q(u8j+4,u8j+4) = 24 · 3r−1. similarly, q(ω3r−1+1−4j,ω3r−1+1−4j = q(u8j−3,u8j−3) + 2q(u8j−3,u8j−2) + q(u8j−2,u8j−2) + q(u8j−1,u8j−1) + 2q(u8j−1,u8j) + q(u8j,u8j) + 9q(u8j+1,u8j+1) + 18q(u8j+1,u8j+2) + 9q(u8j+2,u8j+2) + q(u8j+3,u8j+3) + 2q(u8j+3,u8j+4) + q(u8j+4,u8j+4) = 24 · 3r−1, q(ω4j+1,ω4j+1) = q(ω4j+2,ω4j+2) = q(ω4j+3,ω4j+3) = = q(ω3r−1+2−4j,ω3r−1+2−4j) = q(ω3r−1+3−4j,ω3r−1+3−4j) = q(ω3r−1+4−4j,ω3r−1+4−4j) = 24 · 3r−1, q(ω4j,ω4j+1) = −q(ω8j−3,ω8j−3) + 2q(ω8j−3,ω8j−2) −q(ω8j−2,ω8j−2) −q(ω8j−1,ω8j−1) + 2q(ω8j−1,ω8j) −q(ω8j,ω8j) + q(ω8j+1,ω8j+1) − 2q(ω8j+1,ω8j+2) + q(ω8j+2,ω8j+2) −q(ω8j+3,ω8j+3) + 2q(ω8j+3,ω8j+4) −q(ω8j+4,ω8j+4) = −12 · 3r−1, q(ω4j+1,ω4j+2) = q(ω4j+2,ω4j+3) = q(ω4j+3,ω4(j+1)) = = q(ω3r−1+1−4j,ω3r−1+2−4j) = q(ω3r−1+2−4j,ω3r−1+3−4j) = q(ω3r−1+3−4j,ω3r−1+4−4j) = −12 · 3r−1. finally, for k,l = 1, 2, . . . , 3r−1 − 2, with l > k + 1, it follows that q(ωk,ωl) = 0. now, c = {ω0,ω1, . . . ,ω3r−1−1} is a basis of a free z-module i. a generator matrix of the algebraic lattice 1 2 √ 3r σ(i) is given by m = 1 2 √ 3r n, where n = (σi(ωj−1))3 r−1 i,j=1, and the associated gram matrix is 155 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 g = mmt = 1 12 · 3r−1 (q(ωi,ωj)) 3r−1−1 i,j=0 = =   2 0 −1 0 0 0 . . . 0 0 2 −1 0 0 0 . . . 0 −1 −1 2 −1 0 0 . . . 0 0 0 −1 2 −1 0 . . . 0 0 0 0 −1 2 −1 . . . 0 ... ... ... ... ... ... ... ... 0 0 0 . . . 0 −1 2 −1 0 0 0 . . . 0 0 −1 2   . therefore, g is the gram matrix of a d3r−1-lattice. 3.2. rotated dn-lattices for n = 3r−1, where r ≥ 3 is odd in this section, we present a construction of rotated dn-lattices using z-modules via the totally real number field k = q(ζ3r + ζ−13r ), where r is odd. the dn-lattice is obtained as sublattice of z⊕a k 2 using b = {u0,u1, . . . ,u3r−1−1} a z-basis of ok. proposition 3.5. let i = zω0 ⊕zω1 ⊕ . . .⊕zω3r−1−1 be a free z-module of ok, where 1. ω0 = −6u0 − 3u1 − 3u3; ω1 = 6u0 − 3u1 − 3u3; ω2 = 6u1; 2. for 3 ≤ j ≤ 3 r−1−3 2 , where j is odd, ωj = −3u2j−5 + 3u2j−3 − 3u2j−1 − 3u2j+1; ωj+1 = 6u2j−1; 3. for j = 3 r−1+1 2 , ωj = −u2j−7 − 2u2j−6 − 5u2j−5 − 4u2j−4 + 4u2j−3 + 2u2j−2; ωj+1 = −u2j−9 − 2u2j−8 −u2j−7 − 2u2j−6 + 3u2j−5 + 6u2j−4 −u2j−3 − 2u2j−2; ωj+2 = −u2j−9 − 2u2j−8 + 3u2j−7 + 6u2j−6 −u2j−5 − 2u2j−4 + u2j−3 + 2u2j−2; ωj+3 = 3u2j−9 + 6u2j−8 −u2j−7 − 2u2j−6 + u2j−5 + 2u2j−4 + u2j−3 + 2u2j−2; 4. for j = 1, 2, . . . , 3 r−1−9 8 , with r > 3, ω3r−1−4j = −u8j−5 − 2u8j−4 − 2u8j−3 − 4u8j−2 + u8j−1 + 2u8j −u8j+1 − 2u8j+2 −u8j+5 − 2u8j+6 − 2u8j+7 − 4u8j+8; ω3r−1+1−4j = −u8j−7 − 2u8j−6 −u8j−5 − 2u8j−4 + 3u8j−3 + 6u8j−2 −u8j−1 − 2u8j; ω3r−1+2−4j = −u8j−7 − 2u8j−6 + 3u8j−5 + 6u8j−4 −u8j−3 − 2u8j−2 + u8j−1 + 2u8j; ω3r−1+3−4j = 3u8j−7 + 6u8j−6 −u8j−5 − 2u8j−4 + u8j−3 + 2u8j−2 + u8j−1 + 2u8j. therefore, λ = 1 6 √ 3r−1 σ(i) ⊆ r3 r−1 is a rotated version of a d3r−1 -lattice. proof. from proposition 3.2, it follows that q(ω0,ω0) = trk/q(ω0ω0) = trk/q((−6u0 − 3u1 − 3u3)(−6u0 − 3u1 − 3u3) = trk/q(36u0u0 + 36u0u1 + 36u0u3 + 9u1u1 + 18u1u3 + 9u3u3) = 36q(u0,u0) + 36q(u0,u1) 156 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 + 36q(u0,u3) + 9q(u1,u1) + 18q(u1,u3) + 9q(u3,u3) = 72 · 3r−1. q(ω1,ω1) = 36q(u0,u0) + 9q(u1,u1) + 9q(u3,u3) = 72 · 3r−1. q(ω2,ω2) = 36q(u1,u1) = 72 · 3r−1. q(ω0,ω1) = q(ω0,ω3) = q(ω1,ω3) = 0. q(ω0,ω2) = q(ω1,ω2) = q(ω2,ω3) = −36 · 3r−1. let 3 ≤ j ≤ 3 r−1−3 2 , with j odd. since q(ui,uj) 6= 0 if and only if i + j ≡ 3 (mod 4) and |i− j| = 1, it follows that q(ωj,ωj) = 9q(u2j−5,u2j−5) + 9q(u2j−3,u2j−3) + 9q(u2j−1,u2j−1) + 9q(u2j+1,u2j+1) = 72 · 3r−1. q(ωj+1,ωj+1) = 36q(u2j−1,u2j−1) = 72 · 3r−1. furthermore, q(ωj,ωj+1) = −18q(u2j−1,u2j−1) = −36 · 3r−1, and for j < 3 r−1−3 2 , q(ωj+1,ωj+2) = q(6u2j−1,−3u2(j+2)−5 + 3u2(j+2)−3 − 3u2(j+2)−1 − 3u2(j+2)+1) = q(6u2j−1,−3u2j−1 + 3u2j+1 − 3u2j+3 − 3u2j+5) = −18q(u2j−1,u2j−1) = = −36 · 3r−1. for j = 3 r−1+1 2 , it follows that q(ωj,ωj) = q(u2j−7,u2j−7) + 4q(u2j−7,u2j−6) + 4q(u2j−6,u2j−6) + 25q(u2j−5,u2j−5) + 40q(u2j−5,u2j−4) + 16q(u2j−4,u2j−4) + 16q(u2j−3,u2j−3) + 16q(u2j−3,u2j−2) + 4q(u2j−2,u2j−2) = 72 · 3r−1. in the same way, it follows that q(ωj+1,ωj+1) = q(ωj+2,ωj+2) = q(ωj+3,ωj+3) = 72 · 3r−1. also, q(ωj,ωj+1) = q(ω2j−7,ω2j−7) + 4q(ω2j−7,ω2j−6) + 4q(ω2j−6,ω2j−6) − 15q(ω2j−5,ω2j−5) − 42q(ω2j−5,ω2j−4) − 24q(ω2j−4,ω2j−4) − 4q(ω2j−3,ω2j−3) − 10q(ω2j−3,ω2j−2) − 4q(ω2j−2,ω2j−2) = −36 · 3r−1. in the same way, it follows that q(ωj+1,ωj+2) = q(ωj+2,ωj+3) = −36 · 3r−1. and for k = 3 r−1−9 8 , q(ωj+3,ω3r−1−4k) = q(ω3r−1+7 2 ,ω3r−1+9 2 ) = −36 · 3r−1. for j = 1, 2, . . . , 3 r−1−9 8 , with r > 3, q(ω3r−1−4j,ω3r−1−4j) = q(ω8j−5,ω8j−5) + 4q(ω8j−5,ω8j−4) + 4q(ω8j−4,ω8j−4) + 4q(ω8j−3,ω8j−3) + 16q(ω8j−3,ω8j−2) + 16q(ω8j−2,ω8j−2) + q(ω8j−1,ω8j−1) + 4q(ω8j−1,ω8j) + 4q(ω8j,ω8j) + q(ω8j+1,ω8j+1) + 4q(ω8j+1,ω8j+2) + 4q(ω8j+2,ω8j+2) + q(ω8j+5,ω8j+5) + 4q(ω8j+5,ω8j+6) + 4q(ω8j+6,ω8j+6) + 4q(ω8j+7,ω8j+7) + 16q(ω8j+7,ω8j+8) + 16q(ω8j+8,ω8j+8) = 72 · 3r−1. in the same way, it follows that q(ω3r−1+1−4j,ω3r−1+1−4j) = q(ω3r−1+2−4j,ω3r−1+2−4j) = q(ω3r−1+3−4j,ω3r−1+3−4j) = 72 · 3r−1. also, q(ω3r−1−4j,ω3r−1+1−4j) = q(u8j−5,u8j−5) + 4q(u8j−5,u8j−4) + 4q(u8j−4,u8j−4) − 6q(u8j−3,u8j−3) − 24q(u8j−3,u8j−2) − 24q(u8j−2,u8j−2) −q(u8j−1,u8j−1) − 4q(u8j−1,u8j) − 4q(u8j,u8j) = −36 · 3r−1. in the same way, it follows that q(ω3r−1+1−4j,ω3r−1+2−4j) = q(ω3r−1+2−4j,ω3r−1+3−4j) = −36 · 3r−1. finally, for k,l = 1, 2, . . . , 3r−1 − 2, with l > k + 1, it follows that q(ωk,ωl) = 0. now, n c = {ω0,ω1, . . . ,ω3r−1−1} is a basis of a free z-module i. a generator matrix of the algebraic 157 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 lattice 1 6 √ 3r−1 σ(i) is given by m = 1 6 √ 3r−1 n, where n = (σi(ωj−1))3 r−1 i,j=1, and the associated gram matrix is g = mmt = 1 36 · 3r−1 (q(ωi,ωj)) 3r−1−1 i,j=0 = =   2 0 −1 0 0 0 . . . 0 0 2 −1 0 0 0 . . . 0 −1 −1 2 −1 0 0 . . . 0 0 0 −1 2 −1 0 . . . 0 0 0 0 −1 2 −1 . . . 0 ... ... ... ... ... ... ... ... 0 0 0 . . . 0 −1 2 −1 0 0 0 . . . 0 0 −1 2   . therefore, g is the gram matrix of a d3r−1-lattice. 4. conclusions in this paper, we construct full diversity rotated versions of d3r−1-lattices via the canonical embedding and two families of z-modules of the ring of the integers z[ζ3r + ζ−13r ], for r ≥ 3 a positive integer, since it is impossible to construct rotated dn-lattices via fractional ideals of z[ζ3r + ζ−13r ] [13]. the lattices obtained here are sublattices of the family of rotated z ⊕ak2-lattices, where ak2 is a direct sum of k = 3 r−1−1 2 copies of the a2-lattice. in [1] and [4] families of rotated z2 r−2 -lattices were obtained via the ring of integers z[ζ2r + ζ−12r ]. in [5] a family of rotated z(p−1)/2-lattices was obtained via the ring of integers z[ζp +ζ−1p ], with p prime. in [9] two families of rotated z3 r−1 -lattices were obtained via free z-modules of z[ζ3r + ζ−13r ], one for r odd and one for r even. in [12] two families of rotated d2r−2-lattices were obtained, one via the ring of integers z[ζ2r +ζ−12r ] and one via a principal ideal of z[ζ2r +ζ −1 2r ]. also in [12] a family of rotated d(p−1)/2-lattices was presented via free z-modules in z[ζp +ζ−1p ], with p prime, that are not ideals. in [13] considering the compositum of q(ζ2r +ζ−12r ) and q(ζp +ζ −1 p ) and the compositum of q(ζp1 +ζ −1 p1 ) and q(ζp2 +ζ −1 p2 ), where p,p1 and p2 are prime numbers with p1 6= p2, were constructed families of rotated dn-lattices via free zmodules of rank n that are not ideals. in table 1, we list the number fields considered in [1, 4, 5, 9, 12, 13] and here for constructing rotated zn and dn-lattices for some values of n. let k1 = q(ζ2r + ζ−12r ), k2 = q(ζp + ζ−1p ), where p is a prime, k3 = q(ζ2r + ζ −1 2r )q(ζp + ζ −1 p ), k4 = q(ζp1 + ζ −1 p1 )q(ζp2 + ζ −1 p2 ), with p1 6= p2, and k5 = q(ζ3r + ζ−13r ). we observe that for r = 14, 21, 25, 26, 28, 29 and 30 there are not p,p1,p2 prime numbers with p1 6= p2 such that the degree of q(ζp + ζ−1p ) and q(ζp1 + ζ−1p1 )q(ζp2 + ζ −1 p2 ) be 3r−2. n zn dn k1 k2 k5 k1 k2 k3 k4 k5 2 r = 3 p = 5 − − − − − − 3 − p = 7 r = 3 − p = 7 − − r = 3 4 r = 4 − − r = 4 − r = 3, p = 5 − − 8 r = 5 p = 17 − r = 5 p = 17 r = 4, p = 5 − − 9 − p = 19 r = 4 − p = 19 − − r = 4 16 r = 6 − − r = 6 − r = p = 5 p1,p2 ∈{5,17} − 27 − − r = 5 − − − p1,p2 ∈{7,19} r = 5 32 r = 7 − − r = 7 − r = 4, p = 17 − − 64 r = 8 − − r = 8 − r = 7, p = 5 − − 81 − p = 163 r = 6 − p = 163 − − r = 6 158 a. j. ferrari et. al. / j. algebra comb. discrete appl. 8(3) (2021) 151–160 n zn dn k1 k2 k5 k1 k2 k3 k4 k5 128 r = 9 p = 257 − r = 9 p = 257 r = 8, p = 5 − − 243 − p = 487 r = 7 − p = 487 − p1,p2 ∈{7,163} r = 7 256 r = 10 − − r = 10 − r = 7, p = 17 p1,p2 ∈{5,257} − 512 r = 11 − − r = 11 − r = 10, p = 5 − − 729 − p = 1459 r = 8 − p = 1459 − p1,p2 ∈{7,487} r = 8 p1,p2 ∈{19,163} 1024 r = 12 − − r = 12 − r = 9, p = 17 p1,p2 ∈{17,257} − 2048 r = 13 − − r = 13 − r = 10, p = 17 − − 2187 − − r = 9 − − − p1,p2 ∈{7,1459} r = 9 p1,p2 ∈{19,487} 4096 r = 14 − − r = 14 − r = 13, p = 5 − − 6561 − − r = 10 − − − p1,p2 ∈{19,1459} r = 10 8192 r = 15 − − r = 15 − r = 12, p = 17 − − 16384 r = 16 − − r = 16 − r = 13, p = 5 − − 19683 − p = 39367 r = 11 − p = 39367 − p1,p2 ∈{163,487} r = 11 32768 r = 17 p = 65537 − r = 17 p = 65537 r = 14, p = 17 − − 59049 − − r = 12 − − − p1,p2 ∈{7,39367} r = 12 p1,p2 ∈{163,1459} 65536 r = 18 − − r = 18 − r = 17, p = 5 − − 131072 r = 19 − − r = 19 − r = 16, p = 17 − − 177147 − − r = 13 − − − p1,p2 ∈{19,39367} r = 13 p1,p2 ∈{487,1459} 262144 r = 20 − − r = 20 − r = 13, p = 257 − − 524288 r = 21 − − r = 21 − r = 20, p = 5 − − 531441 − − r = 14 − − − − r = 14 1048576 r = 22 − − r = 22 − r = 19, p = 17 − − 1594323 − − r = 15 − − − p1,p2 ∈{163,39367} r = 15 table 2. rotated zn and dn-lattices for n powers of 2 and 3. references [1] a. a. andrade, c. alves, t. b. carlos, rotated lattices via th cyclotomic field q(ζ2r ), international journal of applied mathematics 19(3) (2006) 321-331. 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[17] l. c. washington, introduction to cyclotomic fields, springer-verlag, new york (1982). 160 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.13069/jacodesmath.729440 https://doi.org/10.1142/s0219498806001636 https://doi.org/10.1142/s0219498806001636 https://doi.org/10.1016/j.jnt.2012.05.002 https://doi.org/10.1016/j.jnt.2012.05.002 https://doi.org/10.1007/s00013-013-0501-8 https://doi.org/10.1007/s00013-013-0501-8 https://doi.org/10.13069/jacodesmath.75353 https://doi.org/10.13069/jacodesmath.75353 https://doi.org/10.1007/978-1-4684-0133-2 introduction algebraic lattices rotated dn-lattices via k=q(3r+3r-1), where r 3 and n=3r-1 conclusions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.561322 j. algebra comb. discrete appl. 6(2) • 95–103 received: 9 december 2018 accepted: 16 april 2019 journal of algebra combinatorics discrete structures and applications coretractable modules relative to a submodule research article ali reza moniri hamzekolaee, yahya talebi abstract: let r be a ring and m a right r-module. let n be a proper submodule of m. we say that m is n-coretractable (or m is coretractable relative to n) provided that, for every proper submodule k of m containing n, there is a nonzero homomorphism f : m/k → m. we present some conditions that a module m is coretractable if and only if m is coretractable relative to a submodule n. we also provide some examples to illustrate special cases. 2010 msc: 16d10, 16d40, 16d80 keywords: coretractable module, n-coretractable module 1. introduction throughout this paper r will denote an arbitrary associative ring with identity and all modules will be unitary right r-modules unless stated otherwise. let m be an r-module. we use endr(m), annr(m) (in the case m is a right r-module), annl(m) (in the case m is a left r-module) to denote the ring of endomorphisms of m, the right annihilator in r of m and the left annihilator in r of m, respectively. let m be a module and k a submodule of m. then k is essential in m denoted by k ≤e m, if l∩k 6= 0 for every nonzero submodule l of m. dually, k is small in m (k � m), in case m = k + l implies that l = m. a submodule n of m is called supplement, if there is a submodule k of m such that m = n + k and n ∩k � n. a module m is called supplemented if every submodule of m has a supplement in m. for any unexplained terminology we refer to [3], [9] and [11]. khuri in [5] introduced the concept of a retractable module. let m be a module. then m is retractable in case for every nonzero submodule n of m, there is a nonzero homomorphism f : m → n, i.e homr(m,n) 6= 0. in the literature, there are some works about retractable modules (see [6, 12, 14]). amini, ershad and sharif in [2] defined a dual notation namely coretractable modules. a module m is coretractable provided that, homr(m/n,m) 6= 0 for every proper submodule n of m. there are also some papers whose main subject is to study and investigate coretractable modules. we refer readers to [1, 4, 13] for more information about coretractable modules. ali reza moniri hamzekolaee (corresponding author), yahya talebi; department of mathematics, faculty of mathematical sciences, university of mazandaran, babolsar, iran (email: a.monirih@umz.ac.ir, talebi@umz.ac.ir). 95 https://orcid.org/0000-0002-2852-7870 https://orcid.org/0000-0003-2311-4628 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 in [10], the author introduced a generalization of coretractable modules via the cosingular submodule. following [10], a module m is called z(m)-coretractable in case, for every proper submodule n of m containing z(m), there is a nonzero homomorphism f : m/n → m. it is proved in [10, theorem 2.11] that a ring r is z(rr)-coretractable if and only if every finitely generated free right r-module f is z(f)-coretractable. also, a characterization of commutative semiperfect kasch rings is presented via z-coretractablity ([10, corollary 2.14]). inspiring by [10], we are interested to study coretractablity of modules relative to their submodules. if in the definition of a coretractable module m, we fix a submodule n and focus just on nonzero homomorphisms from m/k to m where k 6= m contains n, we have a special generalization of coretractable modules. we may choose special submodules of a module m such as soc(m), rad(m) and some others. we present some necessary conditions to prove that when two concepts coretractable and coretractable relative to a submodule coincide. among them, we show that for a small or a semisimple submodule n of m, m is coretractable if and only if m is n-coretractable. it is also shown that if m is n-coretractable and n is coretractable, then m is coretractable. for a right ideal i of r, we show that rr is i-coretractable if and only if every simple right r-module that is annihilated by i, can be embedded in rr. as a consequence, rr is coretractable if and only if r is right kasch. 2. coretractable modules relative to a submodule in this section we introduce a new generalization of coretractable modules via submodules. recall that a module m is coretractable, in case for every proper submodule n of m, there exists a nonzero homomorphism f : m/n → m. definition 2.1. let m be a module and n a proper submodule of m. we say m is n-coretractable in case for every proper submodule k of m containing n, there is a nonzero homomorphism f : m/k → m. note that a module m is coretractable if and only if m is {0}-coretractable. let m be a module and n a proper submodule of m. it is not hard to verify that m is ncoretractable if and only if for every proper essential submodule k of m containing n, there is a nonzero homomorphism from m/k to m. note that if a module m is n-coretractable, then for every submodule t ⊆ n, there is a nonzero homomorphism g : m/t → m. in fact, if m is n-coretractable, then for every submodule t of m, either contained in n or containing n, there will be a nonzero homomorphism from m/t to m. recall from [7], a ring r is right (left) kasch in case every simple right (left) r-module can be embedded in rr (rr). in [2, theorem 2.14], the authors proved that r is right kasch if and only if rr is coretractable. let r be a right kasch ring which is not left perfect. then by [4, proposition 2.9], there is a right ideal i of r such that r/i is not coretractable while rr is coretractable as r is a kasch ring (see also [4, example 2.10]). lemma 2.2. (1) let n,k,ni < m. let m be n-coretractable. if k ⊇ n, then m is k-coretractable. in particular, if m is ni-coretractable for each i ∈ i, then m is ( ∑ i∈i ni)-coretractable. (2) let m be n-coretractable. if k ≤ n such that k contains no nonzero image of any endomorphism of m, then m/k is n/k-coretractable. in a special case, if m is n-coretractable such that for every f ∈ end(m), imf * n, then m/n is coretractable (see [4, proposition 2.11]). proof. (1) this is straightforward. (2) let t/k be a proper submodule of m/k containing n/k. then n ⊆ t ⊂ m. since m is n-coretractable, there exists a nonzero homomorphism g : m/t → m. now define h : m/k t/k → m/k by h(x + k + t k ) = g(x + t) + k for every x ∈ m. if imh = 0, then img ⊆ k. now, k contains 96 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 the image of the endomorphism goπ of m where π : m → m/t is the natural epimorphism, this gives a contradiction. therefore, m/k is n/k-coretractable. let r be a right noetherian ring and m be a n-coretractable module where n is a finitely generated proper submodule of m. then by lemma 2.2(2), m/n is coretractable (see [4, corollary 2.13]). proposition 2.3. let m be a module and k ≤ n < m. if m/k is n/k-coretractable and m/k can be embedded in m, then m is n-coretractable. in particular, if m = k ⊕k′ and n is any submodule of m such that k′ is (n ∩k′)-coretractable, then m is n-coretractable. proof. let t be a proper submodule of m containing n. then t/k is a proper submodule of m/k containing n/k. by assumption, there is a nonzero homomorphism g : m/k t/k ∼= m/t → m/k. there also exists a monomorphism h : m/k → m. now, the homomorphism hog : m/t → m is the required one. corollary 2.4. let m be a module and n < m such that m/n is coretractable. if m/n can be embedded in m, then m is n-coretractable. in particular, if m is supplemented with rad(m) a direct summand of m, then m is rad(m)-coretractable. proof. this is a special case of proposition 2.3. the last part follows from the fact that for a supplemented module m, the module m/rad(m) is coretractable since m/rad(m) is semisimple. in this case m is rad(m)-coretractable. example 2.5. (1) let m be a coretractable module and n < m. then m is n-coretractable. in particular, every cogenerator m in the category of right r-modules is coretractable relative to every n < m. (2) let m be a module such that for every submodule k of m we have m/k ∼= m. then m is coretractable relative to each n < m. (3) let m be a module and n < m. if every proper submodule of m containing n, is contained in a proper summand of m, then m is n-coretractable. (4) let m be an uniserial module. if m is coretractable relative to a proper submodule n, then m is coretractable. the following introduces a n-coretractable module which is not coretractable. in fact, the class of relative coretractable modules properly contains the class of coretractable modules. example 2.6. let p be the set of all prime numbers and m = ∏ p∈p zp as an z-module. take n = {0}×z3 ×z5 × . . . which is a maximal submodule of m, since m/n ∼= z2. consider g : z2 → m defined by g(x) = (x,0,0, . . .). then g is a nonzero homomorphism indicating that m is n-coretractable. note that by [2, example 2.9], m is not a coretractable z-module. remark 2.7. let m be a module and n < m. if there is not a nonzero homomorphism from m/n to m, then m is not n-coretractable. for example, let m be a nonsingular module and n be a proper submodule of m such that m/n is singular. so there does not exist any nonzero homomorphism from m/n to m. now, m is not n-coretractable (for example, z-modules q and z can not be nz-coretractable). we shall consider some conditions under which the two concepts coretractable and n-coretractable coincide. lemma 2.8. let m be a module and n < m. in each of the following cases m is n-coretractable if and only if m is coretractable. (1) n is a small submodule of m. (2) n is a coretractable module. 97 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 proof. (1) let m be n-coretractable where n � m and k be a proper submodule of m. since n is small in m, we have n + k 6= m. now since m is n-coretractable, then there is a nonzero homomorphism f : m/(n +k) → m. so that homr(m/k,m) 6= 0. it follows that m is coretractable. the converse is clear. (2) let k be a proper submodule of m. then either k + n 6= m or k + n = m. if k + n 6= m, then similarly to (1) we have homr(m/k,m) 6= 0. now suppose that k + n = m. then there is an isomorphism h : m/k → n/(n ∩ k) induced from m = n + k. since n is coretractable, there is a nonzero homomorphism g : n/(n∩k) → n. therefore, jogoh : m/k → m is a nonzero homomorphism where j : n → m is the inclusion. recall that a module m is hollow, provided every proper submodule of m is small in m. corollary 2.9. (1) let m be a hollow module and n < m. then m is n-coretractable if and only if m is coretractable. (2) let m be a finitely generated module. then m is rad(m)-coretractable if and only if m is coretractable. (3) let n be a semisimple submodule of m. then m is n-coretractable if and only if m is coretractable. (4) let m be a module. then m is soc(m)-coretractable if and only if m is coretractable. let m be a module and n a submodule of m. following [15], n is δ-small in m (denoted by n �δ m), in case m = n + k with m/k singular implies that m = k. note that by definitions, every small submodule of m is δ-small in m. the sum of all δ-small submodules of m is denoted by δ(m). also δ(m) is the reject of the class of all simple singular modules in m. proposition 2.10. let m be a module and n be a proper δ-small submodule of m. then m is ncoretractable if and only if m is coretractable. proof. let m be n-coretractable and k be a proper submodule of m. suppose that m 6= n+k. since m is n-coretractable, there is a nonzero homomorphism f : m/(n +k) → m. so that foπ : m/k → m is the required homomorphism where π : m/k → m/(n + k) is natural epimorphism. otherwise, m = n + k. now from [15, lemma 1.2], there is a decomposition m = y ⊕k where y is a semisimple projective submodule of n. therefore, there is a monomorphism from m/k to m since k is a direct summand of m. it follows that m is coretractable. proposition 2.11. let m be a module and n be a proper submodule of m. if m is n-coretractable and m/n has a maximal submodule, then soc(m) 6= 0. in particular, if m is finitely generated and n-coretractable, then soc(m) 6= 0. proof. let k/n be a maximal submodule of m/n. then k is a maximal submodule of m. so there is a nonzero homomorphism h : m/k → m. it follows that imh is a simple submodule of m. this completes the proof. the following is an immediate consequence of last proposition. corollary 2.12. let r be a ring such that every cyclic right r-module is coretractable relative to at least one of its submodules. then r is semi-artinian. let r be a ring. then r is called a right v -ring in case every simple right r-module is injective. as a generalization of v -rings, r is a right generalized v -ring (gv -ring for short), if every simple singular right r-module is injective ([11]). 98 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 proposition 2.13. let r be a ring and m be an indecomposable right r-module with rad(m) 6= m. if each of the following statements holds, then m is rad(m)-coretractable if and only if m is simple. (1) r is a right gv -ring. (2) m is noncosingular. proof. (1) let m be rad(m)-coretractable. then for each maximal submodule k of m there is a monomorphism g : m/k → m. it follows that img is a simple submodule of m. then img is either singular or projective. if img is projective, then k is a direct summand of m and hence k = 0 or k = m. so that k = 0. if img is singular, it will be injective as r is right gv . therefore, img is a summand of m and since g 6= 0 we conclude that img = m. in both cases, m is simple. the converse is obvious. (2) it follows from (1) and the fact that every homomorphic image of m is noncosingular. corollary 2.14. let r be a right v -ring and m an indecomposable right r-module. then m is coretractable if and only if m is simple. following [8], a module m is dual rickart provided that for every f ∈ end(m), imf is a direct summand of m. remark 2.15. let m be an indecomposable dual rickart module with rad(m) 6= m. then m is (rad(m)-)coretractable if and only if m is simple. let k be a maximal submodule of m. then there is a monomorphism g : m/k → m. consider the endomorphism h = goπ : m → m where π : m → m/k is the natural epimorphism. then imh = img is a summand of m. so img = m as m is indecomposable. it follows that m is simple. proposition 2.16. let m be a module and l a proper submodule of m such that l has a supplement k in m. if m is l-coretractable and k is fully invariant in m, then k is coretractable. proof. let k be a supplement of l in m. then m = k + l and k ∩ l � k. let n be a proper submodule of k. then n + l is a proper submodule of m. for if, n + l = m, by modular law n + (k ∩ l) = k, which implies that n = k, a contradiction. since m is l-coretractable, there is a nonzero homomorphism f : m/(n + l) → m. since k is a fully invariant submodule of m, we have foπ(k) ⊆ k where π : m → m/(n + l) is the natural epimorphism. now consider h : k/n → k by h(x + n) = f(x + n + l) for every x ∈ k. it is not hard to verify that h is well-defined. now, there is y ∈ m such that y /∈ n +l and f(y +n +l) 6= 0. now there exists k ∈ k and l ∈ l such that y = k +l. it is easy to see that h(k + n) = f(k + l + n + l) = f(y + l) 6= 0. it follows that h is nonzero. corollary 2.17. ([2, proposition 2.5]) every fully invariant direct summand of a coretractable module is coretractable. let m be a module. then m is called a duo module provided every submodule of m is fully invariant. corollary 2.18. let m be a duo module. if m is coretractable relative to each direct summand of m, then every direct summand of m is coretractable. proposition 2.19. let m = m1 ⊕ . . .⊕mn and n < m. if each mi is n ∩mi-coretractable, then m is n-coretractable. especially a finite direct sum of coretractable modules is coretractable. proof. the proof is exactly similar to proof of [2, proposition 2.6]. proposition 2.20. let r be a right max ring and m = ⊕ i∈i mi be a direct sum of n∩mi-coretractable right r-modules where n < m. then m is n-coretractable. in particular, an arbitrary direct sum of coretractable right r-modules is coretractable. proof. similar to the proof of [2, proposition 2.7]. 99 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 let m be an r-module. a submodule k of m is said to be dense in m if, for any y ∈ m and 0 6= x ∈ m, there exists r ∈ r such that xr 6= 0 and yr ∈ k. obviously, any dense submodule of m is essential in m. from [7, proposition 8.6], k is dense in m if and only if homr(p/k,m) = 0 for every submodule p ⊇ k. remark 2.21. let m be a module and n < m. if n is dense in m, then m is not n-coretractable. in fact for a n-coretractable module m, we have n is not dense in m. this follows from the fact that if m is n-coretractable, then there is a nonzero homomorphism from m/n to m. proposition 2.22. let m be a module and n a proper submodule of m. if m is quasi-injective or every proper submodule of m is contained in a maximal submodule, then m is n-coretractable if and only if every proper submodule of m containing n is not dense in m. proof. (1) let m be a quasi-injective module such that every proper submodule of m containing n is not dense in m. suppose that k is a proper submodule of m containing n. since k is not dense in m, there is a f : p/k → m where p is a submodule of m containing k. it follows that foπ : p → m is a nonzero homomorphism where π : p → p/k is the natural epimorphism. consider the inclusion homomorphism j : p → m. since m is quasi-injective, there exists h : m → m such that hoj = foπ. by defining h : m/k → m with h(m + k) = h(m) we conclude that m is n-coretractable. note that h is nonzero. conversely, if m is n-coretractable and n ⊆ k < m, then there is a homomorphism g : m/k → m which shows that k is not dense in m. (2) suppose that every submodule of m is contained in a maximal submodule of m. let n ⊆ k < m. then there is a maximal submodule l of m such that k ≤ l. since l is not dense in m, there is a nonzero homomorphism h : m/l → m. as f : m/k → m/l with f(x + k) = x + l is a nonzero homomorphism, then hof is nonzero. it follows that m is n-coretractable. the converse is the same as (1). the following presents a characterization of i-coretractable rings. theorem 2.23. let r be a ring and i be a proper right ideal of r. then the following are equivalent: (1) rr is i-coretractable; (2) every n-generated free right r-module is i(n)-coretractable; (3) for every right ideal t ⊇ i, annl(t) 6= 0. proof. (1) ⇔ (2) follows from proposition 2.19. (1) ⇒ (3) let t be a right ideal of r containing i. since rr is i-coretractable, there is a nonzero homomorphism f : r/t → r. consider the endomorphism g = foπ : r → r where π is the natural epimorphism from r to r/t. then there is an element 0 6= a ∈ r such that g(x) = ax. let y ∈ t . then g(y) = ay = 0 as t ⊆ kerg. this shows that 0 6= a ∈ annl(t). (3) ⇒ (1) let t be a right ideal of r containing i. since annl(t) 6= 0, there exists an element of r such as a that at = 0 and a 6= 0. define f : r/t → r by f(x + t) = ax. it is easy to check that f is an r-homomorphism and in particular f 6= 0. remark 2.24. let r be a ring and i ≤ rr with annl(i) = 0. then rr is not i-coretractable. for example, let r = [ k k 0 k ] be the ring of 2 × 2 upper triangular matrices over a field k. let i = [ 0 k 0 k ] which is a right ideal of r. then annl(i) = 0. hence, rr is not i-coretractable. in other words, r/j(r) is coretractable relative to each of its ideals as r/j(r) is a semisimple ring. note that j(r) = [ 0 k 0 0 ] . theorem 2.25. let r be a ring and i be a proper two-sided ideal of r. then the following statements are equivalent: (1) rr is i-coretractable; (2) every simple right r-module that is annihilated by i can be embedded in rr. 100 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 proof. (1) ⇒ (2) let m ∼= r/k be a simple right r-module such that mi = 0. it follows that i ⊆ k. since rr is i-coretractable, there is a nonzero homomorphism f : r/k → r. (2) ⇒ (1) let t be a right ideal of r containing i. now there exists a right maximal ideal k of r such that i ⊆ t ⊆ k. consider the simple right r-module m = r/k. since mi = 0, there is a nonzero homomorphism g : r/k → r by assumption. as t is a submodule of k, there exists f : r/t → r/k defined by f(x + t) = x + k. hence gof is the desired homomorphism. for a ring r, theorem 2.25 implies that rr is (j(r)-)coretractable if and only if r is a right kasch ring. in [2, proposition 4.4], it is shown that if r is a von neumann regular ring then r is right kasch if and only if r is semisimple. in the following we shall investigate a more general version. proposition 2.26. let r be a right gv -ring. then the following are equivalent: (1) r is right kasch; (2) r is semisimple. proof. (1) ⇒ (2) let r be right kasch. so rr is j(r)-coretractable. now suppose that k is an arbitrary maximal right ideal of r. then there is a monomorphism g : r/k → r. it follows that r/k ∼= img is a simple right r-module. so, img is either singular or projective. in first case img should be injective as r is right gv . therefore, img is a direct summand of rr. now img is singular projective which implies that img = 0, a contradiction. so that img and hence every simple right r-module will be projective. this shows that r is semisimple. (2) ⇒ (1) it is obvious. corollary 2.27. let r be a right v -ring. then r is a kasch ring if and only if r is semisimple. example 2.28. (1) let r = [ k k 0 k ] where k is a field. then j(r) = [ 0 k 0 0 ] . it is easy to check that r is a semilocal ring as r/j(r) ∼= k × k which is a semisimple ring. now by [3, exercise 10, page 113], soc(rr) = [ k k 0 0 ] . however, soc(rr) = [ 0 k 0 k ] . set m1 = soc(rr) and m2 = soc(rr). then both m1 and m2 are maximal left and right ideals of r. a quick calculation shows that annl(m1) = m2, annl(m2) = 0, annr(m1) = 0 and annr(m2) = m1. now by theorem 2.23, rr is m1-coretractable while rr is not m2-coretractable. also left version of theorem 2.23, implies that rr is m2-coretractable but it is not m1-coretractable. since the simple right r-module r/m2 can not be embedded in rr and the simple left r-module r/m1 can not be embedded in rr, the ring r is neither right kasch nor left kasch (note that since r is right gv which is not a v -ring, it can not be kasch from proposition 2.26). (2) let k be a division ring and r = {a =   a 0 b c 0 a 0 d 0 0 a 0 0 0 0 e   | a,b,c,d,e ∈ k}. then j(r) = {a ∈ r | a = 0 = e}, soc(rr) = annl(j(r)) = {a ∈ r | a = 0}, soc(rr) = annr(j(r)) = j(r). since r/j(r) ∼= k ×k, r is a semilocal ring. now soc(rr) = {a ∈ r | a = 0} and soc(rr) = j(r). from [7, example 8.29], soc(rr) is a left and right maximal ideal of r. since annr(soc(rr)) = {a ∈ r | a = e = 0} = j(r) 6= 0, it follows from [7, corollary 8.28], r/soc(rr) can be embedded in rr (see also theorem 2.23). therefore, rr is soc(rr)-coretractable while rr is not soc(rr)-coretractable (see also corollary 2.9). now an easy computation shows that annl(soc(rr)) = {a ∈ r | a = c = d = e = 0} 6= 0. so r/soc(rr) can be embedded in rr by [7, corollary 8.28]. as soc(rr) is a maximal right ideal of r, then rr is soc(rr)-coretractable. also from [7, example 8.29], r is a right kasch ring while it is not a left kasch ring. (3) let k be a field and r = ∏∞ i=1 k. it is well-known that r is a von neumann regular v -ring. consider the ideal ti = k×k×. . .×k×0×k×k×. . .. it is clear that ti for each i ∈ n is a maximal 101 a. r. moniri hamzekolaee, y. talebi / j. algebra comb. discrete appl. 6(2) (2019) 95–103 ideal of r. it is easy to see that ann(ti) = 0×0× . . .×0×k×0× . . . which is nonzero. therefore, from theorem 2.23, r is i-coretractable for each i ⊆ ti. now consider the ideal l = ⊕∞ i=1 k of r. then ann(l) = 0 and of course ann(m) = 0 for every maximal ideal m of r containing l. hence the simple r-module r/m can not be embedded in r (see [7, corollary 8.28]). therefore, r is not coretractable relative to l. this means that r is not a kasch ring. proposition 2.29. let r be a ring and i a right ideal of r such that every free right r-module r(a) is (i(a))-coretractable. then for every right r-module m with i ⊆ annr(m), homr(m,r) 6= 0. proof. let m be a right r-module such that i ⊆ annr(m). then there is a free right r-module f and a submodule k of f such that m ∼= f/k. since mi = 0, we have i(a) ⊆ k where a is an indexed set. by assumption, there is a nonzero homomorphism f : f/k → f. then the homomorphism πof : m → r is the required one where π : f → r is the natural epimorphism. proposition 2.30. let r be a ring having a radical right r-module m with mi 6= m where i ≤ rr. if for every right ideal t of r, rad(t) 6= t, then there is a free right r-module r(a) which is not i(a)-coretractable. proof. let rad(m) = m such that mi is a proper submodule of m. there exists a free right r-module f = r(a) and a submodule k of f such that m/mi ∼= f/k. being m radical implies that m/mi is radical. so, homr(m/mi,r) = 0. since (f/k)i = 0, i(a) ⊆ k. it follows that homr(f/k,f) = 0 which implies f is not i(a)-coretractable. proposition 2.31. let r be a right max ring and i ≤ rr such that every cyclic r-module n is nicoretractable. then every right r-module m is mi-coretractable. in particular, if r is a (semiperfect) right perfect ring with all cyclic right r-modules coretractable, then every (finitely generated) right rmodule is coretractable. proof. let m be a right r-module. suppose that k is a proper submodule of m containing mi. since r is a right max ring, k is contained in a maximal submodule l of m. for every x ∈ m \l, we know m/l ∼= xr/(xr∩l) as xr + l = m. note that mi ⊆ l. so that (xr/(xr∩l))i = 0. it follows that (xr)i ⊆ xr ∩ l. being xr a (xr)i-coretractable module implies that homr(xr/(xr ∩ l),xr) 6= 0. hence there is a nonzero homomorphism f : m/l → m. therefore, homr(m/k,m) 6= 0 as k ⊆ l. references [1] a. n. abyzov, a. a. tuganbaev, retractable and coretractable modules, j. math. sci. 213(2) (2016) 132–142. [2] b. amini, m. ershad, h. sharif, coretractable modules, j. aust. math. soc. 86(3) (2009) 289–304. [3] f. w. anderson, k. r. fuller, rings and categories of modules, springer-verlog, new york, 1992. [4] n. o. ertaş, d. k. tütüncü, r. tribak, a variation of coretractable modules, bull. malays. math. sci. soc. 41(3) (2018) 1275–1291. [5] s. m. khuri, endomorphism rings and lattice isomorphisms, j. algebra 56(2) (1979) 401–408. [6] s. m. khuri, nonsingular retractable modules and their endomorphism rings, bull. aust. math. soc. 43(1) (1991) 63–71. [7] t. y. lam, lectures on modules and rings, springer-verlag, new york, 1999. [8] g. lee, s. t. rizvi, c. s. roman, dual rickart modules, comm. algebra 39(11) (2011) 4036-4058. [9] s. h. mohamed, b. j. müller, continuous and discrete modules, london math. soc. lecture notes series 147, cambridge, university press, cambridge, 1990. 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[12] j. m. zelmanowitz, correspondences of closed submodules, proc. amer. math. soc. 124(10) (1996) 2955–2960. [13] j. žemlička, completely coretractable rings, bull. iranian math. 39(3) (2013) 523–528. [14] z. zhengping, a lattice isomorphism theorem for nonsingular retractable modules, canad. math. bull. 37(1) (1994) 140–144. [15] y. zhou, generalizations of perfect, semiperfect, and semiregular rings, algebra colloq. 7(3) (2000) 305–318. 103 https://mathscinet.ams.org/mathscinet-getitem?mr=1144522 https://doi.org/10.1090/s0002-9939-96-03469-7 https://doi.org/10.1090/s0002-9939-96-03469-7 https://mathscinet.ams.org/mathscinet-getitem?mr=3095342 https://doi.org/10.4153/cmb-1994-020-5 https://doi.org/10.4153/cmb-1994-020-5 https://doi.org/10.1007/s10011-000-0305-9 https://doi.org/10.1007/s10011-000-0305-9 introduction coretractable modules relative to a submodule references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(2) • 85-94 received: 31 october 2014; accepted: 21 february 2015 doi 10.13069/jacodesmath.75353 journal of algebra combinatorics discrete structures and applications lattice polytopes in coding theory research article ivan soprunov∗ department of mathematics, cleveland state university, cleveland, oh usa abstract: in this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. we also include a new inductive bound for the minimum distance of generalized toric codes. as an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations. 2010 msc: 14m25, 14g50, 52b20 keywords: toric code, lattice polytope, minkowski length, sparse polynomials 1. introduction toric codes are examples of a large class of evaluation codes studied by goppa, tsfasman, vlǎdut, and others, using methods of algebraic geometry [18]. yet the construction is very explicit: given a lattice polytope p in rm, consider the set of all m-variate polynomials whose exponent vectors lie in p . the code is produced by evaluating these polynomials at the points of (f∗q) m. this makes toric codes a wonderful example of an interconnection between algebraic geometry (toric varieties), geometric combinatorics (lattice polytopes), and coding theory. toric codes were first introduced by j. hansen in [7] for m = 2 and have been actively studied in the last decade. here is a list of some recent papers on the subject: [8–11, 14, 16, 17, 19]. apart from numerous theoretical results, about a dozen new “champion" toric codes and generalized toric codes have been found just recently [3, 4, 12]. a “champion" code is the one that has the largest known minimum distance for a given block length and dimension, as in the table of best known codes [6]. in this paper we concentrate on combinatorial questions about lattice polytopes which arise when one studies the minimum distance of toric codes. in section 3 we relate the minimum distance to a geometric invariant called the minkowski length of p . in particular, we look at the problem of estimating the number of lattice points in polytopes of fixed minkowski length. the results there are not new, although some of them have not been published previously. section 4 is concerned with generalized toric codes. there we prove a general inductive bound for the minimum distance. as an application we generalize previously known formulas for the minimum distance (theorem 3.2) to generalized toric codes as well as provide some examples. ∗ e-mail: i.soprunov@csuohio.edu the author is partially supported by nsa grant h98230-13-1-0279 85 lattice polytopes in coding theory 2. preliminaries 2.1. linear codes to set our notation we start with basic definitions from coding theory. throughout the paper, fq denotes a finite field of q elements and f∗q its multiplicative group of non-zero elements. a subspace c of fnq is called a linear code, and its elements c = (c1, . . . ,cn) are called codewords. the number n is called the block length of c. the weight of c in c is the number of non-zero entries in c. the distance between two codewords a and b in c is the weight of a− b ∈ c. the block length n, the dimension k = dim(c), and the minimum distance d = d(c) are the parameters of c. a code with parameters n, k, and d is referred to as an [n,k,d]q-code. 2.2. newton polytopes let f be a polynomial in m variables over a field k. if we allow negative exponents in the monomials of f we call it a laurent polynomial. the set of the exponent vectors of the monomials appearing in f is called the support of f, denoted by a(f). thus we may write f = ∑ a∈a(f) cat a, where ta = ta11 · · ·t am m , ca ∈ k. the newton polytope p(f) is the convex hull of the support of f. it is a convex lattice polytope in rm. (a polytope is called lattice if its vertices lie in zm ⊂ rm.) for example, the newton polytope of f(t1, t2) = t −1 1 + 2t −1 1 t2 − 3t1t2 is the triangle with vertices (−1, 0), (−1, 1) and (1, 1). notice that it makes sense to evaluate laurent polynomials at points none of whose coordinate is zero, i.e., points in the algebraic torus tm = (k∗)m. laurent polynomials with a prescribed newton polytope are usually called sparse polynomials to emphasize that, compared to a generic polynomial of the same degree, it may have only a few monomials (the ones that correspond to the lattice points in its newton polytope). the newton polytope plays the role of the degree for a sparse polynomial. note that for any two sparse polynomials f,g we have p(fg) = p(f) + p(g), just as for usual degrees. the sum here is the minkowski sum of the polytopes, which is the set of all sums p1 +p2 for all pairs p1 ∈ p(f) and p2 ∈ p(g), and turns out to be again a polytope. therefore, factorizations of a sparse polynomial are related to minkowski sum decompositions of its newton polytope. we will see in section 3 how this relation helps to estimate the number of solutions to f = 0 over a finite field in terms of the newton polytope p(f). here is a bit of terminology. we say a lattice segment in rm is primitive if it contains exactly two lattice points. we say a lattice simplex rm is unimodular if it contains exactly m + 1 lattice points. we say a lattice triangle in r2 is exceptional if it contains exactly three boundary lattice points and one interior lattice point. 3. toric codes let {p1, . . . ,pn} be the set of all points in the algebraic torus tm = (f∗q)m in some linear order. fix a lattice polytope p ⊂ rm and let l(p) be the finite-dimensional space of laurent polynomials over fq whose support is contained in p : l(p) = spanfq{t a | a ∈ p ∩zm}. (1) we have the following evaluation map evtm : l(p) → fnq , f 7→ (f(p1), . . . ,f(pn)). (2) 86 i. soprunov the image of evtm is called the toric code and is denoted by cp . remark 3.1. one may regard toric codes as a multivariate generalization of the reed–solomon codes. indeed, if m = 1 and p is the lattice segment [0,`] the toric code cp coincides with the reed–solomon code with parameters [q − 1,` + 1,q − 1 − `]q. clearly, the block length n of cp equals (q − 1)m, the size of tm. in [14] d. ruano showed that the dimension k of cp equals the number of lattice points of p if no two of them are congruent modulo (zq−1)m. in particular, this is true if we assume that p is contained in the cube kmq = [0,q − 2]m. the main problem we are concerned with is how to compute or estimate the minimum distance d = d(cp ). we will start with some explicit results. j. little and r. schwarz in [11] computed the minimum distance of cp in the case of p = `∆m, the standard m-simplex of side length ` and p = π`1,...,`m, the product of m segments [0,`1] ×···× [0,`m]: d(c`∆m ) = (q − 1) m−1(q − 1 − `), d(cπ`1,...,`m ) = m∏ i=1 (q − 1 − `i). it turned out that this is an instance of a general phenomenon. in the following theorem we describe how the minimum distance behaves under basic operations on lattice polytopes (see [17] for details). theorem 3.2. [17] 1. let p ⊆ km1q and q ⊆ km2q be lattice polytopes. then d(cp×q) = d(cp ) d(cq). 2. let q be a lattice polytope of dim q ≥ 1, and let {kq | 0 ≤ k ≤ n} be a sequence of k-dilates of q, contained in kmq . let p(q) be the pyramid over q, i.e. the convex hull in rm+1 of the set {(x, 0) | x ∈ q}∪{em+1}. then d(ckp(q)) = (q − 1) d(ckq). using this result one can compute the minimum distance explicitly for a large class of polytopes obtained from a lattice segment by taking the direct product or constructing a pyramid and dilating. in particular, umana and velasco [19] used this to compute the minimum distance for toric codes on degree one polytopes. in section 4 we generalize this theorem to generalized toric codes. next we turn to the case of arbitrary polytopes. the situation is far from being understood even in the case of polytopes of small dimension. the first results in this direction were obtained by hansen [7, 8] who used intersection theory on the toric surface defined by the lattice polygon to obtain lower bound for the minimum distance of cp . it turns out that there is a more direct relation between d(cp ) and geometry of lattice polytopes (at least for large q) — the minimum distance d(cp ) can be bounded in terms of what is called the minkowski length of p . here is the definition. definition 3.3. let p be a lattice polytope in rm. the minkowski length of p is the maximum number of lattice polytopes of positive dimension whose minkowski sum is contained in p: l(p) = max{` |q1 + · · · + q` ⊆ p, dim qi > 0}. a minkowski decomposition of q into l(p) summands of positive dimension will be referred to as a maximal decomposition in p and q will be called maximal. it is not hard to see that there are only finitely many lattice polytopes q contained in p and there are only finitely many possible decompositions of q into the minkowski sum of lattice polytopes of positive dimension, so the number l(p) is well-defined. moreover, it is easy to see that in the definition of l(p) one may assume that the qi are lattice segments. 87 lattice polytopes in coding theory recall from section 2 that a factorization of a sparse polynomial corresponds to minkowski sum decomposition of its newton polytope. therefore, the minkowski length is the geometric invariant of p which describes the largest possible number of factors in factorizations of polynomials f ∈l(p). consider the case m = 2. one can use the hasse–weil bound to estimate the number of zeroes in t2 of absolutely irreducible factors of f ∈l(p). little and schenck in [10] used this bound to show that the more factors f has, the more it has zeroes in t2, provided q is large enough. it turns out that if f ∈l(p) has a factorization with the largest number of factors then the newton polytope of each factor is either a primitive segment, or a unimodular triangle, or an exceptional triangle, see [16]. moreover, we have the following lower bound for the minimum distance of cp . theorem 3.4. [16] let p be a lattice polygon of minkowski length l. there is an explicit function α(p) such that for all q ≥ α(p) we have d(cp ) ≥ (q − 1)(q − 1 −l) − (2 √ q − 1). moreover, the term 2 √ q − 1 may be omitted if no maximal decomposition of p contains an exceptional triangle. there is a natural action of the isomorphism group agl(m,z) of the lattice zm on the space of lattice polytopes, under which l(p) is invariant. the group agl(m,z) consists of translations by a lattice vector and integer linear non-degenerate transformations, called unimodular transformations. let p and p ′ be agl(m,z)-equivalent. then the corresponding toric codes cp and cp′ are monomially equivalent [11] (although the opposite is not true, see [13] for a counterexample). this means that for the purpose of coding theory it is enough to consider lattice polytopes up to agl(m,z)-equivalence. returning to definition 3.3, note that each summand in a maximal decomposition has l(qi) = 1. such polytopes are called strongly indecomposable and they play an important role in estimating the minimum distance d(cp ), see [16], as well as [20, chapter 2]. in dimension m = 2 there are exactly three strongly indecomposable polytopes up to agl(m,z)equivalence: the unit segment, the unit triangle, and the exceptional triangle, see figure 1. figure 1. strongly indecomposable polytopes up to agl(2, z)-equivalence. note that the latter has the largest number of lattice points, which is four. the following theorem is a generalization of this fact, which was discovered by i. barnett, b. fulan, c. quinn, and j. soprunova in an reu project at kent state university in 2011. since this result is not written up anywhere we include a short proof here. theorem 3.5. let q ⊂ rm be strongly indecomposable. then the number of lattice points in q is at most 2m. moreover, there exist strongly indecomposable polytopes with exactly 2m lattice points. proof. for the first part, consider the lattice points of q modulo (z/2z)m. if q has more than 2m lattice points then there exists distinct lattice points a,b ∈ q ∩ zm which coincide modulo (z/2z)m. then the lattice segment [a,b] ⊂ q must contain at least one interior lattice point, hence, decomposes into lattice segments. this contradicts the assumption that l(q) = 1. the construction of q for which the bound is attained is by induction on m. we start with the exceptional triangle in r2. after a unimodular transformation we may assume that it contains no horizontal lattice segments, i.e. segments whose direction vector has zero first coordinate. we will call the direction vector of a lattice segment in a polytope p simply a direction vector in p . 88 i. soprunov assume that p ⊂ rm is a strongly indecomposable polytope with 2m lattice points, such that no direction vector in p has zero first coordinate. let k be the largest first coordinate of all direction vectors in p . there is a unimodular transformation α ∈ gl(m,z) such that every direction vector in α(p) has the first coordinate greater than k. for example, we can take α = α2 ⊕ idm−2, where α2 has matrix[ a 1 a− 1 1 ] with large enough a. finally, let p ′ be the convex hull of p ×{0}∪ α(p) ×{1} in rm+1. to show that p ′ is strongly indecomposable it is enough to show that there are no lattice segments of length more than one connecting a point in p and a point in α(p), and there are no lattice parallelograms with two vertices in p and two vertices in α(p). the former is clear since all lattice points in p ′ are distinct modulo (z/2z)m+1. the latter follows from the fact that the first coordinate of every direction vector in α(p) is greater than the first coordinate of any direction vector in p . there has been recent progress in understanding the structure of polytopes with l(p) = 1 in higher dimensions. in particular, new results have been obtained about 3-dimensional lattice polytopes and longest minkowski sum decompositions of their subpolytopes [1]. as for the bounds in theorem 3.4, a similar approach was taken in [20] for 3-dimensional toric codes. the author gives an algorithmic way of obtaining lower bound for the minimum distance, but one still hopes for more explicit bounds than the ones in [20]. classifying polytopes of minkowski length larger than one is not easy even in dimension m = 2. in figure 2 we present 16 classes of lattice polygons of minkowski length two. the proof that these are all of them is not hard, but tedious, so we do not include it here. lattice polygons of minkowski length two up to gl(2, z)-equivalence figure 2. the sixteen polytopes with l(p) = 2 up to gl(2, z)-equivalence. it does not seem feasible to classify polygons with l(p) ≥ 3 by hand. recall that the dimension of a toric code equals the number of lattice points in p . thus, a more important question is the following: given `, what could be the largest number of lattice points in p with l(p) = `? the naive bound |p ∩zm| ≤ (` + 1)m which follows from considering the lattice points of p modulo (z/(` + 1)z)m, as in the proof of theorem 3.5, appears to be too rough. 89 lattice polytopes in coding theory suppose m = 2, so p is a lattice polygon. from figure 2 we see that for ` = 2 the answer is 7. in [5] v. cestaro showed that for ` = 3 the answer is 9. for larger ` the question is open and no better estimate than (` + 1)2 is currently known. 4. generalized toric codes generalized toric codes are a natural extension of toric codes. they first appeared in the work of d. ruano [15] and j. little [12]. the definition is similar to the one of a toric code, except we allow arbitrary configurations of lattice points instead of the lattice points of a lattice polytope. more precisely, let s be a set of lattice points in rm contained in the m-cube kmq . similar to (1) we let l(s) be the vector space over fq of laurent polynomials with support in s: l(s) = spanfq{t a | a ∈ s}. the image of the corresponding evaluation map evtm : l(s) → fnq , f 7→ (f(p1), . . . ,f(pn)). is called the generalized toric code cs. the weight of each nonzero codeword equals the number of points ξ ∈ tm where the corresponding polynomial does not vanish. we denote it by w(f). let z(f) denote the number of zeroes of f in tm. also let zs denote the maximum number of zeroes over all nonzero f ∈l(s). obviously, z(f) = (q − 1)m −w(f) and zs = (q − 1)m −d(s). (3) as before, cs is a linear code of block length n = (q−1)m and dimension dimcs = |s|, the cardinality of s. note that if p is the convex hull of s then dimcs ≤ dimcp and d(cs) ≥ d(cp ). the idea is that by omitting just a few lattice points of p one could, in principle, obtain s for which the minimum distance d(cs) is significantly larger than d(cp ). examples of this phenomenon were provided by j. little [12]. at the same time he gave some evidence that for large q this often does not happen. this prompted a search for generalized toric codes with parameters better than previously known over fields of small size. g. brown and a. kasprzyk [3, 4] used an exhaustive search of lattice polygons and lattice point configurations contained in k2q for q up to 8. they were able to find a new toric code champion and seven new generalized toric code champions. 4.1. two examples below we give two examples of generalized toric code with best known parameters. the corresponding configurations (see figure 3) are agl(2,z)-equivalent to the ones found in [4]. they produce a [49, 13, 27]-code and a [49, 19, 21]-code over f8, respectively. as pointed out by markus grassl (private communication), by omitting the point (1, 2) in s one obtains a subcode with parameters [49, 12, 28]. applying construction x to this pair of codes (see [6]), one obtains a [50, 13, 28]-code over f8, which gave another champion. 4.2. inductive bound we finish with a new general lower bound for the minimum distance of generalized toric codes. the bound is inductive in a sense that it uses the codes from the fibers and the images of a projection of s 90 i. soprunov figure 3. two lattice configurations producing a [49, 13, 27]and [49, 19, 21]-code over f8. onto a coordinate subspace. as a corollary we get a generalization of theorem 3.2 to generalized toric codes. let s ⊆ kmq be a set of lattice points. choose a coordinate subspace y ⊆ rm (defined by setting a subset of coordinates equal zero) and let π : rm → y be the corresponding projection. for every a ∈ π(s) let sa denote the fiber sa = s ∩ π−1(a). theorem 4.1. let s be a set of lattice points in kmq and π : r m → y a projection onto a coordinate subspace. then d(s) ≥ min s′⊆π(s) ( d(s′) max a∈s′ d(sa) ) . proof. we may assume that π : rm → y is the projection onto the last m−k coordinates. furthermore, we use (x,y) = (x1, . . . ,xk,y1, . . . ,ym−k) to denote coordinates in tm = tk ×tm−k. consider an arbitrary nonzero f ∈ l(s) with support a(f), and let s′ denote the projection s′ = π(a(f)). we have a(f) ⊆ ∪a∈s′sa, hence, we can write f as a linear combination of monomials ya for a ∈ s′ with coefficients fa that are nonzero polynomials in l(sa): f(x,y) = ∑ a∈s′ fa(x)y a. (4) given a point ξ = (ξ1, . . . ,ξk) ∈ (f∗q)k let lξ be the coset of the subtorus {1}× (f∗q)m−k containing ξ, i.e. lξ = {(ξ,y) | y ∈ (f∗q) m−k}. here 1 denotes the identity element in (f∗q) k. note that on every lξ where f is identically zero, f has exactly (q− 1)m−k zeroes, and on every lξ where f is not identically zero, it has at most zs′ zeroes, since the (nonzero) polynomial f(ξ,y) lies in l(s′). then the number of zeroes of f in tm is bounded by z(f) ≤ (q − 1)m−kn + zs′ ( (q − 1)k −n ) , (5) where n is the number of the cosets lξ where f is identically zero. substituting zs′ = (q−1)m−k−d(s′) (see (3)) and simplifying we obtain z(f) ≤ (q − 1)m −d(s′) ( (q − 1)k −n ) , 91 lattice polytopes in coding theory or, simply, w(f) ≥ d(s′) ( (q − 1)k −n ) . (6) notice that n is, in fact, the number of common zeroes of the fa in (f∗q) k, and is at most the number of zeroes of each fa. therefore, n ≤ min a∈s′ z(fa) ≤ (q − 1)k − max a∈s′ d(sa). now (6) implies w(f) ≥ d(s′) max a∈s′ d(sa). notice that the right hand side depends only on the projection of the support of f, so it remains to take the minimum over all subsets s′ ⊆ π(s) and the statement of the theorem follows. our first application of the inductive formula is a generalization of theorem 3.2, part (1). corollary 4.2. suppose s = s1 ×s2 ⊂ rm1 ×rm2 for some lattice sets si ⊆ kmiq ∩zmi, i = 1, 2. then d(s) = d(s1)d(s2). proof. consider the projection π : rm1 × rm2 → rm2. then π(s) = s2. as every fiber sa equals a lattice translate of s1, for a ∈ s2, by theorem 4.1 we have d(s) ≥ min s′⊆s2 (d(s′)d(s1)) = d(s1) min s′⊆s2 d(s′). it is clear that if s′ ⊆ s2 then d(s′) ≥ d(s2). therefore, the above minimum equals d(s2). conversely, let fi ∈ l(si) for i = 1, 2 be polynomials with the minimum weight. we have d(si) = w(fi) = (q − 1)mi − z(fi), where z(fi) is the number of zeroes of fi in tmi. then, by the inclusionexclusion principle, the polynomial f = f1f2 has (q − 1)m2z(f1) + (q − 1)m1z(f2) −z(f1)z(f2) zeroes in tm1 ×tm2. this implies that its weight equals w(f) = (q − 1)m1+m2 − (q − 1)m2z(f1) − (q − 1)m1z(f2) + z(f1)z(f2) = w(f1)w(f2). therefore, d(s) ≤ d(s1)d(s2), and we are done. corollary 4.3. let πm : rm → r be the projection to the last coordinate and suppose πm(s) = {0, 1, . . . ,`}. if d(s0) ≤ d(s1) ≤ ···≤ d(s`) then d(s) ≥ min 0≤i≤` (q − 1 − i)d(si). proof. indeed, consider s′ ⊂ πm(s) and let i be the length of the convex hull of s′. on one hand we have d(s′) ≥ (q−1−i). on the other hand, since d(s0) ≤ d(s1) ≤ ···≤ d(s`), when finding the minimum over all s′ it is enough to consider only those s′ that contain 0. in that case maxa∈s′ d(sa) = d(si) and the statement follows from theorem 4.1. to connect this result to the second part of theorem 3.2, we will need an extra assumption on the configuration s. first, we have the following proposition. its proof is similar to the one of [17, proposition 2.2] proposition 4.4. let s,s′ be lattice sets in kmq and t the set of lattice points of a lattice segment. if s + t ⊆ s′ (up to a lattice translation) then (q − 1)d(s′) ≤ (q −|t|)d(s). 92 i. soprunov proof. after a unimodular transformation we may assume that s + t ⊆ s′, and t is the set of lattice points of the segment [0,ke1], where e1 is the first basis vector and k = |t| − 1 is the length of the segment. let g ∈l(s) be a polynomial with z(g) = zs. then for any ξ1, . . . ,ξk ∈ f∗q the polynomial f(x) = g(x) k∏ j=1 (x1 − ξj) belongs to l(s + t) ⊆l(s′). by the inclusion-exclusion formula we have z(f) = z(g) + k(q − 1)m−1 − k∑ j=1 z(g|x1=ξj ). since tm is the union of q−1 subtori given by x1 = ξ, for ξ ∈ f∗q, we have z(g) = ∑ ξ∈f∗q z(g|x1=ξ). choose ξ1, . . . ,ξk ∈ f∗q so that {z(g|x1=ξj ) | j = 1, . . . ,k} are the k smallest integers among the q − 1 integers {z(g|x1=ξ) | ξ ∈ f∗q}. then 1 k k∑ j=1 z(g|x1=ξj ) ≤ z(g) q − 1 . therefore, we obtain zs′ ≥ z(f) ≥ z(g) + k(q − 1)m−1 − k q − 1 z(g) replacing z(g) with zs and using zs = (q − 1)m −d(s) we see that the latter inequality is equivalent to (q − 1)d(s′) ≤ (q −k − 1)d(s), as required. the following is a generalization of theorem 3.2, part (2) to generalized toric codes. theorem 4.5. let s be a lattice set in kmq . let πm : r m → r be the projection to the last coordinate, πm(s) = {0, 1, . . . ,`}, and s0, . . . ,s` the corresponding fibers. suppose there is a primitive lattice segment [a,b] such that for every 1 ≤ i ≤ `, the set si + {a,b} is contained in si−1, up to a lattice translation. then d(s) = (q − 1)d(s0). proof. first, note that in this special situation, the conditions of corollary 4.3 are satisfied. indeed, by proposition 4.4, (q − 1)d(si−1) ≤ (q − 2)d(si), so in particular, d(si−1) ≤ d(si). next, we have si + i{a,b} ⊆ s0 up to a lattice translation. here i{a,b} (which is the minkowski sum of {a,b} with itself i times) is the set of lattice points of a lattice segment of length i. thus, by proposition 4.4, (q − 1)s0 ≤ (q − 1 − i)d(si), for every 0 ≤ i ≤ `. applying corollary 4.3, we obtain d(s) ≥ (q − 1)d(s0). conversely, let g ∈ l(s0) be a polynomial with z(g) = zs0. by definition, g depends only on the first m− 1 variables. therefore, it has (q − 1)zs0 zeroes in tm. this implies that zs ≥ (q − 1)zs0, i.e. d(s) ≤ (q − 1)d(s0). 93 lattice polytopes in coding theory the last result can be applied to constructing a generalized toric code with parameters [(q − 1)n,k′, (q − 1)d] from a given generalized toric [n,k,d]-code. as an example, let s0 be the 13-point configuration in figure 3. for the primitive segment [a,b] we choose a = (0, 0) and b = (1, 1). then by removing the points with the largest sum of coordinates in every line parallel to [a,b] we obtain a 6-point configuration s1 satisfying s1 +{a,b}⊂ s0. a repetition of this process produces a 2-point configuration s2 satisfying s2 + {a,b} ⊂ s1. now define s = ⋃2 i=0 si ×{i}, which is a 21-point configuration in z 3. according to theorem 4.5, the corresponding generalized toric code has parameters [343, 21, 189] over f8. acknowledgment: the first part of this paper is based on a talk given at karatekin mathematics days 2014 international mathematics symposium. i am grateful to the organizers for inviting me and to mesut şahin, pinar celebi demirarslan, and gökhan demirarslan for their hospitality. references [1] o. beckwith, m. grimm, j. soprunova, b. weaver, minkowski length of 3d lattice polytopes, discrete and computational geometry 48, issue 4, 1137-1158, 2012. [2] w. bosma, j. cannon, c. playoust, the magma algebra system. i. the user language, j. symbolic comput., 24, 235-265, 1997. [3] g. brown, a. m. kasprzyk, small polygons and toric codes, journal of symbolic computation, 51, 55-62, april 2013. [4] g. brown, a. m. kasprzyk, seven new champion linear codes, lms journal of computation and mathematics, 16, 109-117, 2013. [5] v. cestaro, parameters of toric codes in small dimension, senior undergraduate project, csu 2011. [6] m. grassl, bounds on the minimum distance of linear codes and quantum codes, online, http:// www.codetables.de/, accessed on october 1, 2013. [7] j. hansen, toric surfaces and error-correcting codes in coding theory, cryptography, and related areas, springer, 132-142, 2000. [8] j. hansen, toric varieties hirzebruch surfaces and error-correcting codes, appl. algebra engrg. comm. comput., 13, 289-300, 2002. [9] d. joyner, toric codes over finite fields, appl. algebra engrg. comm. comput., 15, 63-79, 2004. [10] j. little, h. schenck, toric surface codes and minkowski sums, siam j. discrete math. 20, no. 4, (electronic) 999-1014, 2006. [11] j. little, r. schwarz, on toric codes and multivariate vandermonde matrices, appl. algebra engrg. comm. comput., 18(4), 349-367, 2007. [12] j. little, remarks on generalized toric codes, finite fields and their applications, 24, 1-14, november 2013. [13] x. luo, s. s.-t. yau, m. zhang, h. zuo, on classification of toric surface codes of low dimension, arxiv:1402.0060. [14] d. ruano, on the parameters of r-dimensional toric codes, finite fields and their applications, 13, 962-976, 2007. [15] d. ruano, on the structure of generalized toric codes, journal of symbolic computation, 44(5), 499-506, 2009. [16] i. soprunov, j. soprunova, toric surface codes and minkowski length of polygons, siam j. discrete math., 23(1), 384-400, 2009. [17] i. soprunov, j. soprunova, bringing toric codes to the next dimension, siam j. discrete math., 24(2), 655-665, 2010. [18] m. tsfasman, s. vlǎdut, d. nogin, algebraic geometric codes: basic notions providence, r.i.: american mathematical society, 2007. [19] v. g. umana, m. velasco dual toric codes and polytopes of degree one, preprint arxiv:1404.4063. [20] j. whitney, a bound on the minimum distance of three dimensional toric codes, ph.d. thesis, 2010. 94 introduction preliminaries toric codes generalized toric codes references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(2) • 75-84 received: 29 october 2014; accepted: 2 february 2015 doi 10.13069/jacodesmath.17537 journal of algebra combinatorics discrete structures and applications on a class of repeated-root monomial-like abelian codes research article edgar martínez-moro1∗, hakan özadam2,3∗∗, ferruh özbudak2§, steve szabo3∗∗∗ 1. institute of mathematics and applied mathematics department university of valladolid, castilla, spain 2. department of mathematics and institute of applied mathematics, middle east technical university i̇nönü bulvarı, 06531, ankara, turkey 3. university of massachusetts, medical school worcester, massachusetts 4. department of mathematics and statistics, eastern kentucky university richmond, ky, usa abstract: in this paper we study polycyclic codes of length ps1 × · · · × psn over fpa generated by a single monomial. these codes form a special class of abelian codes. we show that these codes arise from the product of certain single variable codes and we determine their minimum hamming distance. finally we extend the results of massey et. al. in [10] on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables. 2010 msc: 94b15,11t71,11t55 keywords: repeated-root cyclic code, abelian code, weight-retaining property 1. introduction cyclic codes are said to be repeated-root when the codeword length and the characteristic of the alphabet are not coprime. despite that it has been proved that in general they are asymptotically bad in some cases repeated-root cyclic codes are optimal and they have interesting properties. massey et. al. have shown in [10] that cyclic codes of length p over a finite field of characteristic p are optimal. there also exist infinite families of repeated-root cyclic codes in even characteristic according to the results of [14]. also in [10] it has been pointed out that some repeated-root cyclic codes can be decoded using a very simple circuitry. among other studies on repeated-root cyclic codes with several different settings are [1, 2, 7, 8, 11, 12, 14]. ∗ e-mail: edgar@maf.uva.es ∗∗ e-mail: ozhakan@metu.edu.tr § e-mail: ozbudak@metu.edu.tr ∗∗∗ e-mail: steve.szabo@eku.edu 75 monomial-like codes contrary to the simple-root case, there are repeated root cyclic codes of the form ( f(x)i ) where i > 1. specifically, all cyclic codes of length ps over a finite field of characteristic p are generated by a single “monomial” of the form (x− 1)i, where 0 ≤ i ≤ ps (see [2, 11]). in this paper, as a generalisation of these codes to several variables, we study cyclic codes of the form i(i1,...in) = ( (x1 − 1)i1 · · ·(xn − 1)in ) ⊂r = fpa [x1, . . . ,xn]( x ps1 1 − 1, . . . ,x psn n − 1 ), (1) i.e. i(i1,...in) is the ideal of r generated by a single monomial of the form (x1 − 1) i1 · · ·(xn − 1)in. this paper is organised as follows. first we introduce some notation, give some definitions and prove some structural properties of the ambient space of a particular class of abelian codes in section 2. in section 3, we show thatmonomial like codes arise from product codes and we determine their hamming distance. we describe their duals which yields a parity check matrix for these codes. in section 4, we explain the relationship of the hasse derivative with the dual of this type of codes. finally in section 5, we generalise the weight retaining property of monomials in single variable to the multivariable case. 2. the ambient space throughout the paper, we consider the finite ring r = fpa [x1, . . . ,xn]( x ps1 1 − 1, . . . ,x psn n − 1 ) (2) as the ambient space of the codes to be studied unless otherwise stated. it is a well known fact that r is a local ring with maximal ideal (x1 − 1, . . . ,xn − 1). we define l = {(α1,α2, . . . ,αn) | 0 ≤ αj < psj, αj ∈ z for all 1 ≤ j ≤ n}. (3) the elements of r can be identified uniquely with the polynomials of the form f(x1, . . . ,xn) = ∑ (α1,α2,...,αn)∈l f(α1,α2,...,αn)x α1 1 x α2 2 · · ·x αn n , (4) so throughout the paper, we identify the equivalence class f(x1, . . . ,xn) + ( x ps1 1 − 1,x ps2 2 − 1, . . . ,x psn n − 1 ) with the polynomial f(x1, . . . ,xn). we shall consider a repeated-root code as just an ideal c of r. the length of the code is ps1 × ps2 × ···× psn and the support of a codeword f(x1, . . . ,xn) ∈ c is the set supp(f) = {(α1,α2, . . . ,αn) ∈ l | f(α1,α2,...,αn) 6= 0}. the hamming weight of f(x1, . . . ,xn) is defined as w(f(x1, . . . ,xn)) = |supp(f)|, i.e. the number of nonzero coefficients of f(x1, . . . ,xn). the minimum hamming distance of the code c is defined as d(c) = min{w(f(x1, . . . ,xn)) | f(x1, . . . ,xn) ∈c \{0}}. 3. monomial-like codes in this paper we shall study a particular class of the codes over r called monomial-like codes given by an ideal generated by a single monomial of the form c(i1,...,in) = ( (x1 − 1)i1 · (x2 − 1)i2 · · ·(xn − 1)in ) ⊂r. (5) 76 e. martínez-moro et al. note that not all the ideals in r can be generated by a single monomial of this form. in one variable case, the minimum hamming distance of c was computed in [11] and [2]. it turns out that, in multivariate case, c(i1,...,in) can be considered as a product code of single variable codes. this decomposition allows us to express the minimum hamming distance of c(i1,...,in) in terms of the hamming distances of cyclic codes of length psj . definition 3.1. the product of two linear codes c,c′ over fpa is the linear code c⊗c′ whose codewords are all the two dimensional arrays for which each row is a codeword in c and each column is a codeword in c′. the following are some well-known facts about the product codes. 1. if c and c′ are [n,k,d] and [n′,k′,d′] codes respectively, then c ⊗c′ is a [nn′,kk′,dd′] code. 2. if g and g′ are generator matrices of c and c′ respectively, then g⊗g′ is a generator matrix of c ⊗c′, where ⊗ denotes the kronecker product of matrices and the codewords of c ⊗c′ are seen as concatenations of the rows in arrays in c ⊗c′. theorem 3.2. let n1,n2 be positive integers and let r̂ = fpa [x,y] (xn1 − 1,yn2 − 1) ,rx = fpa [x] (xn1 − 1) , ry = fpa [y] (yn2 − 1) . suppose that (x− 1)k1|xn1 − 1 and (y − 1)k2|yn2 − 1. the code c = ( (x− 1)k1 · (y − 1)k2 ) ⊂r̂ is the product of the codes cx = ( (x− 1)k1 ) ⊂rx and cy = ( (y − 1)k2 ) ⊂ry, i.e., c = cx ⊗cy. proof. let g(x) = (x− 1)k1 = gk1x k1 + · · · + g1x + g0, h(y) = (y − 1)k2 = hk2y k2 + · · · + h1y + h0. then gx =   0 . . . 0 0 gk1 . . . g1 g0 0 . . . 0 gk1 . . . g1 g0 0 ... ... gk1 . . . g1 g0 0 . . . 0 0   ,gy =   0 . . . 0 0 hk2 . . . h1 h0 0 . . . 0 hk2 . . . h1 h0 0 ... ... hk2 . . . h1 h0 0 . . . 0 0   are two generator matrices for cx and cy, respectively. we identify the polynomial f(x,y) = ∑ 0≤i<n1,0≤j<n2 cijx iyj ∈ fpa [x,y], with the codeword (cn1−1,n2−2, . . . ,cn1−1,1,cn1−1,0, . . . ,c1,n2−1, . . . ,c1,1,c1,0,c0,n2−1, . . . ,c0,1,c0,0). the elements of c = ( (x− 1)k1 (y − 1)k2 ) ⊂r̂ are exactly all the fpa-linear combinations of the elements of the set β = {xiyj(x− 1)k1 (y − 1)k2 : 0 ≤ i < n−k1, 0 ≤ j < n−k2} now we consider g = gx ⊗gy. using the above identification for the rows of g, we obtain a basis for cx ⊗cy which is equal to β. thus c = cx ⊗cy. 77 monomial-like codes corollary 3.3. let r1, . . . ,rn, i1, . . . , in be positive integers and let r′ = fpa [x1, . . . ,xn] (xr11 − 1, . . . ,x rn n − 1) , rxj = fpa [xj]( x rj j − 1 ). suppose that (xj − 1)ij|x rj j − 1 for all 1 ≤ j ≤ n. the code c(i1,...,in) = ( (x1 − 1)i1 · · ·(xr − 1)ir ) is the product of the codes cxj = ( (xj − 1)ij ) ⊂rxj , i.e., c(i1,...,in) = (· · ·((cx1 ⊗cx2 ) ⊗cx3 ) ⊗···) ⊗cxn = n⊗ i=1 cxi. (6) remark 3.4. 1. note that the tensor product is associative in the sense that there is a natural isomorphism (c ⊗ c′) ⊗c′′ ∼= c ⊗ (c′ ⊗c′′). thus we can remove all the parenthesis in equation 6. 2. the reader can identify in theorem and corollary 3.3 as a polynomial version of the the fact that for g a finite p-group such that g = g1×g2×···×gn and k a field then kg ∼= kg1⊗kg2⊗···⊗kgn where g = g1g2 . . .gn is mapped to g1 ⊗g2 ⊗···⊗gn. the previous construction give us a straightforward result for the minimum distance of our codes as follows. corollary 3.5. let c(i1,...,in) ⊂r then d(c(i1,...,in)) = n∏ j=1 d(c(ij )), (7) where d(c(ij )) is the minimum distance of the code (x ij i − 1) in fpa [x]/(x ij i − 1). note that d(c(ij )) is explicitly given in [2, theorem 6.4] and [11, theorem 1] in terms of p,a and ij. 3.1. weight hierarchy of some two-variable cases in some very special two-variable cases we can go slightly further and compute explicitly the whole weight hierarchy of the code. the r-th generalised hamming weight dr(c) , 1 ≤ r ≤ k, of a fp-linear code c of dimension k is defined as the minimum of the cardinalities of the supports of all the subcodes (linear subspaces) of dimension r of c. we will define d0(c) = 0. the sequence {dr(c)}kr=0 is called the hamming weight hierarchy of c. let r′ = fpa [x] (xp−1) and c(i1) = ( (x− 1)i1 ) ⊂r′. it was shown in [10, theorem 5] that c(i1) is a maximum distance separable (mds) code.the weight hierarchy of a mds code c is completely determined by its length n and dimension k as dr(c) = n − k + r for r = 1, 2, . . . ,k, see for example [6, theorem 7.10.7]. consider now c(i2) = ( (x2 − 1)i2 ) ⊂ fpa [x2] (xp2−1) and let k1,k2 the dimension as fpa-linear spaces of the 78 e. martínez-moro et al. codes c(i1),c(i2) respectively. using [13, theorem 1] and since c(i1) ⊗c(i2) = c(i1,i2) we get dr(c(i1,i2)) = min{ s∑ i=1 (di(c(i1)) −di−1(c(i1)))dti (c(i2)), 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1, s∑ i=1 ti = r} = min{d1(c(i1))(i2 + t1) + s∑ i=2 (i2 + ti), 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1, s∑ i=1 ti = r} = min{(d1(c(i1)) − 1)(i2 + t1) + r + si2, 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1, s∑ i=1 ti = r}. since d1(c1) = i1 + 1 and the minimum value of t1, subject to 1 ≤ ts ≤ ··· ≤ t1 ≤ k2 and ∑s i=1 ti = r, is ⌈ r s ⌉ , we obtain dr(c(i1,i2)) = min{i1(i2 + ⌈r s ⌉ ) + r + si2, s ≤ k1}, r = 1, 2, . . . ,k1 ·k2. (8) 3.2. dual codes note that the elements of the form ∏n k=1(xk − 1) jk with j ∈ nn form a basis of fpa [x1, . . . ,xn] and the elements of this form with jk ≥ psk for some k form a basis of ({xp sk − 1}nk=1). let us consider 0 6= f(x1, . . . ,xn) = ∑ j∈l cj n∏ k=1 (xk − 1)jk. therefore (x1 − 1)i1 · · ·(xn − 1)inf(x1, . . . ,xn) = 0 in r if and only if for every j ∈ l with cj 6= 0 we have psk ≤ jk + ik for some k if and only if f(x1, . . . ,xn) ∈ ({(x− 1)p sk−ik}nk=1). this proves that the annihilator of c(i1,...,in) is ({(x − 1) psk−ik}nk=1) and the dual of an ideal of r is exactly its annihilator. therefore we have proved the following statement. theorem 3.6. c⊥(i1,...,in) = ({(x− 1) psk−ik}nk=1) ⊂r. remark 3.7. note that the above fact does not hold for arbitrary ideals of algebras of type f[x1, . . . ,xn]/({xnii − 1} n i=1) and it relies on the fact that the ni = psi. let us construct an fpa-basis for c⊥. this will provide us a generator matrix for c⊥ and hence a parity check matrix for c. let tk = {(a1, . . . ,an) ∈ nn | psj − ij ≤ aj < psj if j = k, 0 ≤ aj < psj if j 6= k} and t = t1 ∪·· ·∪tn. let s = s1 + s2 + · · · + sn, it is clear that for e1, . . . ,er pairwise distinct |te1 ∩·· ·∩ter| = ie1 · · ·ier ps pse1 · · ·pser . now applying the inclusion-exclusion principle we obtain |t | = n∑ j=1 ij ps psj − ∑ j<k ijik ps psjpsk −··· + (−1)n+1i1 · · ·in = ps − (ps1 − i1) · · ·(psn − in). 79 monomial-like codes let b = {(x1 − 1)a1 · · ·(xn − 1)an | (a1, . . . ,an) ∈ t}. clearly the elements of b are fpa–linearly independent and |b| = |t |. on the other hand, we know, from theorem 3.2, that dim(c(i1,...,in)) = (ps1 −i1) · · ·(psn −in). this implies that dim(c⊥(i1,...,in)) = p s−dim(c(i1,...,in)) which agree the cardinality of b, thus the set b is an fq-basis for c⊥. i.e., if we consider the vector representations of the elements of b, we obtain a generator matrix for c⊥ and a parity check matrix for c. 4. duality and the hasse derivative in this subsection we will show the natural relation between the hasse derivative and the dual of monomial-like of codes. we begin by recalling the hasse derivative which is used in the repeatedroot factor test. for a detailed treatment of the hasse derivative, we refer to [4, chapter 1] and [5, chapter 5]. note that the standard derivative for polynomials over a field of positive characteristic, say p, is inappropriate because from the pth derivative on, the result is always zero. for this reason, it is more convenient to work with the hasse derivative. sometimes the hasse derivative is also called the hyper derivative. throughout this section, we will use the convention that ( a b ) = 0 whenever b > a. let g(x1, . . . ,xn) = ∑ gα1,...,αnx α1 1 · · ·x αn n be a polynomial in fq[x1, . . . ,xn]. the hasse derivative of g(x1, . . . ,xn) in the direction a = (a1, . . . ,an) is defined as d[a](g(x1, . . . ,xn)) = ∑ gα1,...,αn ( α1 a1 ) · · · ( αn an ) xα1−a11 · · ·x αn−an n . (9) we denote the evaluation of d[a](g(x1, . . . ,xn)) at the point (λ1, . . . ,λn) ∈ fnpa by d[a](g)(λ1, . . . ,λn). we can express g(x1, . . . ,xn) as g(x1, . . . ,xn) = ∑ (j1,...,jn)∈s cj1,...,jn (x1 − 1) j1 · · ·(xn − 1)jn where s is a finite nonempty subset of nn. let s = u` tp` where u` = {(j1, . . . ,jn) ∈ s | j` ≥ i`}, p` = {(j1, . . . ,jn) ∈ s | j` < i`}. therefore g(x1, . . . ,xn) = ∑ (j1,...,jn)∈u` cj1,...,jn (x1 − 1) j1 · · ·(xn − 1)jn + ∑ (j1,...,jn)∈p` cj1,...,jn (x1 − 1) j1 · · ·(xn − 1)jn, and the term (x` − 1)i` divides g(x1, . . . ,xn) if and only if cj1,...,jn = 0 for all (j1, . . . ,jn) ∈ p`. now suppose that (x` − 1)i` g(x1, . . . ,xn). then there is a (æ̂1, . . . , æ̂n) ∈ p` such that cæ̂1,...,æ̂n 6= 0. hence d[æ̂](g)(1, . . . , 1) = cæ̂1,...,æ̂n ( æ̂1 æ̂1 ) · · · ( æ̂n æ̂n ) 6= 0. conversely, if (x` − 1)i` divides g(x1, . . . ,xn), then g(x1, . . . ,xn) = ∑ (j1,...,jn)∈u` cj1,...,jn (x1 − 1) j1 · · ·(xn − 1)jn. therefore d[~a](g)(1, . . . , 1) = 0 for all ~a = (a1, . . . ,an) with 0 ≤ a` < i`. this proves the following result. lemma 4.1. let g(x1, . . . ,xn) ∈ fpa [x1, . . . ,xn] and let a` = {~a = (a1, . . . ,an) ∈ nn | 0 ≤ a` < i`}. then (x` − 1)i` divides g(x1, . . . ,xn) if and only if d[~a](g)(1, . . . , 1) = 0 for all ~a ∈ a`. 80 e. martínez-moro et al. as an immediate consequence, we have the following theorem. theorem 4.2. let a` = {~a = (a1, . . . ,an) ∈ nn | 0 ≤ a` < i`} and a = ∪n`=1a`. let g(x1, . . . ,xn) ∈ fpa [x1, . . . ,xn]. we have (x1 −1)i1 · · ·(xn−1)in divides g(x1, . . . ,xn) if and only if d[~a](g)(1, . . . , 1) = 0 for all ~a ∈ a. let r be as in (2) and let our code be c(i1,...in) ⊂ r. we know that the polynomial g(x1, . . . ,xn) is in the code c(i1,...in) if and only if (x1 − 1) i1 · · ·(xn − 1)in divides g(x1, . . . ,xn). note that d[a1,...,an](g)(1, . . . , 1) = 0 if a` ≥ ps` for some 1 ≤ ` ≤ n. together with this fact, theorem 4.2 implies the following result. theorem 4.3. let c(i1,...in) ⊂r, and let us define q = n⋃ `=1 {~a = (a1, . . . ,an) ∈ nn | 0 ≤ a` < i`, 0 ≤ aj < psj for j 6= `}. then g(x1, . . . ,xn) ∈c(i1,...in) if and only if d [~a](g)(1, . . . , 1) = 0 for all ~a ∈ q. now let us fix a monomial order so that x1 > · · · > xn. let ~a = (a1, . . . ,an) ∈ q. consider the vector wa = (( ps1 − 1 a1 ) · · · ( psn − 1 an ) , ( ps1 − 1 a1 ) · · · ( psn−1 − 1 an−1 )( psn − 1 an ) , · · · ( 0 a1 ) , · · · ( 0 an )) . for g(x1, . . . ,xn) ∈r, let ug be the vector representation of the polynomial with respect to the fixed ordering. then the dot product of wa and ug gives us the evaluation of the hasse derivative of g(x1, . . . ,xn) at (1, . . . , 1) in the direction ~a, i.e., wa ·ug = d[~a](g)(1, . . . , 1). if we construct the matrix h whose rows are the vectors wa where ~a ∈ q and q is as in theorem 4.3 then h is an alternative parity check matrix for the code c(i1,...in) by theorem 4.3. 5. a generalisation of the weight retaining property in [10], the so-called weight retaining property of polynomials over finite fields was stated and proved. this property turned out to be very useful for determining the hamming distance of cyclic codes. in this section, we give a generalisation of the weight retaining property to multivariate polynomials. we prove that the hamming weight of any fpa-linear combination of the monomials (x1 −c1)i1 · · ·(xn − cn) in is greater than or equal to the hamming weight of the “minimal” nonzero term, where a “minimal” term is the one that is not divisible by the rest of the nonzero terms of the summation. first, we consider the case in one variable which was studied in [10]. the weight retaining property of (x− c)i is given in the following two theorems. theorem 5.1. [10, theorem 1.1 and theorem 6.1] let l be any nonempty finite subset of non-negative integers with least integer imin and let f(x) = ∑ i∈l bi(x− c)i where c and each bi are nonzero elements of fpa. then w(f(x)) ≥ w((x− c)imin ). it is not hard to see that theorem 5.1 is equivalent to the following theorem. 81 monomial-like codes theorem 5.2. [10, theorem 6.2] for any polynomial q(x) over fpa and c ∈ fpa \{0}, and any nonnegative integer n, w(q(x)(x− c)n ) ≥ w((x− c)n )w(q(c)). the hamming weight of the monomial (x− c)i, which is used above, was also determined in [10]. theorem 5.3. [10, lemma 1] let c ∈ fpa \{0} and let i be an integer with the p-adic expansion i = ι0 + ι1p + · · ·ιm−1pm−1 where 0 ≤ ι` ≤ p− 1 for all 0 ≤ ` ≤ m− 1. then w((x− c)i) = p(i) = m−1∏ j=0 (ιj + 1). the following theorem is a generalisation of the massey’s weight retaining property to n variables. its proof is very similar to the proof of [3, proposition 1.2]. theorem 5.4. let ψ ⊂ nn be a finite set and let (n1,n2, . . . ,nn) ∈ ψ. let f(x1, . . . ,xn) = ∑ β∈ψ cβ(x1 − c1)β1 (x2 − c2)β2 · · ·(xn − cn)βn ∈ fpa [x1, . . . ,xn], where cβ ∈ fpa \{0}, β = (β1, . . . ,βn) and (x1 −c1)n1 (x2 −c2)n2 · · ·(xn−cn)nn divides (x1 −c1)β1 (x2 − c2) β2 · · ·(xn − cn)βn for every β ∈ ψ. then w(f(x1, . . . ,xn)) ≥ n∏ i=1 p(ni). proof. the proof is via induction on n. for n = 1, the claim follows by theorem 5.1. now assume that the claim holds true for n− 1. we can express f(x1, . . . ,xn) as (xn − cn)nn ( ∑ β∈ψ c (0) β (x1 − c1) β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 +(xn − cn) ∑ β∈ψ c (1) β (x1 − c1) β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 ... +(xn − cn)r ∑ β∈ψ c (r) β (x1 − c1) β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 ) for some non-negative integer r and c(`)β ∈ fpa. by the induction step, we have w( ∑ β∈ψ c (0) β (x1 − c1) β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 ) ≥ p(n1) · · ·p(nn−1). if we express each summand ∑ β∈ψ c (u) β (x1 − c1) β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 in the form∑ β∈ψ′ e (u) β x β1 1 x β2 2 · · ·x βn−1 n−1 , we get (xn − cn)nn ( ∑ β∈ψ′ e (0) β x β1 1 x β2 2 · · ·x βn−1 n−1 + (xn − cn) ∑ β∈ψ′ e (1) β x β1 1 x β2 2 · · ·x βn−1 n−1 . . . + (xn − cn)r ∑ β∈ψ′ e (r) β x β1 1 x β2 2 · · ·x βn−1 n−1 ). 82 e. martínez-moro et al. note that we have just shown that there are at least p(n1) · · ·p(nn−1) many nonzero e (0) β ’s. we define hβ(xn) = e (0) β + e (1) β (xn − cn) + · · · + e (r) β (xn − cn) r. so f(x1, . . . ,xn) = (xn − cn)nn ( ∑ β∈ψ hβ(xn)x β1 1 · · ·x βn−1 n−1 ). there are at least p(n1) · · ·p(nn−1) many β’s such that hβ(xn) 6= 0. for every such β = (β1, . . . ,βn), we have w((xn − cn)nnhβ(xn)x β1 1 · · ·x βn−1 n−1 ) ≥ p(nn) because w((xn − cn)nnhβ(xn)) ≥ p(nn) as the claim holds for one variable. hence w(f(x1, . . . ,xn)) ≥ p(n1) · · ·p(nn−1)p(nn). remark 5.5. this result only applies for polynomials f of a special kind, namely those for which the set denoted ψ contains (n1, . . . ,nn). for example, ψ = {(1, 2), (2, 1)} does not have that property. note that the condition (n1,n2) ∈ ψ is necessary, consider the following example f(x1,x2) = (x1 + 1) 4(x2 + 1) 3 + (x1 + 1) 3(x2 + 1) 4 with coefficients in the field of 2 elements. it is easy to check that w(f(x1,x2)) = 14 but p(3) = 4 where p is the polynomial of theorem 5.3. using theorem 5.4, we generalise theorem 5.3 to n variables. corollary 5.6. let q(x1, . . . ,xn) ∈ fpa [x1, . . . ,xn], c1, . . . ,cn ∈ fpa and n1, . . . ,nn ∈ n. we have w[q(x1, . . . ,xn)(x1 − c1)n1 · · ·(xn − cn)nn ] ≥ w[(x1 − c1)n1 · · ·(xn − cn)nn ][q(c1, . . . ,cn)] = p(n1) · · ·p(nn)wh[q(c1, . . . ,cn)]. note that this property roughly states that the hamming weight of a polynomial of a linear combination of polynomials of the form (x1 − 1)i1, . . . (xn − 1)in is at least the hamming weight of a minimal term (in the lexicographical order of exponents). acknowledgment: the research of the first author was partly supported by the spanish grants mtm2007-64704 and mtm2010-21580-c02-02. he was also funded by the vernon wilson endowed chair at eastern kentucky university during his sabbatical leave. references [1] g. castagnoli, j. l. massey, p. a. schoeller, and n. von seemann, on repeated-root cyclic codes, ieee trans. inform. theory, 37(2), 337-342, 1991. [2] h. q. dinh. on the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, finite fields appl., 14(1), 22-40, 2008. [3] v. drensky and p. lakatos, monomial ideals, group algebras and error correcting codes, in applied algebra, algebraic algorithms and error-correcting codes, (rome, 1988), volume 357 of lecture notes in comput. sci., pages 181-188. springer, berlin, 1989. 83 monomial-like codes [4] d. m. goldschmidt, algebraic functions and projective curves, volume 215, graduate texts in mathematics, springer-verlag, new york, 2003. [5] j. w. p. hirschfeld, g. korchmáros, and f. torres, algebraic curves over a finite field, princeton series in applied mathematics. princeton university press, princeton, nj, 2008. [6] w. c. huffman, v. pless, fundamentals of error-correcting codes, cambridge university press, cambridge, 2003. [7] s. r. lópez-permouth, s. szabo, on the hamming weight of repeated root cyclic and negacyclic codes over galois rings, adv. math. commun., 3(4), 409-420, 2009. [8] e. martínez-moro, i. f. rúa, on repeated-root multivariable codes over a finite chain ring, des. codes cryptogr., 45(2), 219-227, 2007. [9] c. martínez-pérez, w. willems, on the weight hierarchy of product codes, designs, codes and cryptography. an international journal, 33(2), 95-108, 2004. [10] j. l. massey, d. j. costello, jørn justesen, polynomial weights and code constructions, ieee trans. information theory, it-19, 101-110, 1973. [11] hakan özadam and ferruh özbudak, a note on negacyclic and cyclic codes of length ps over a finite field of characteristic p, adv. math. commun., 3(3), 265-271, 2009. [12] s. r. lópez-permouth, h. özadam, f. özbudak, s szabo, polycyclic codes over galois rings with applications to repeated-root constacyclic codes, finite fields and their applications, 19(1), 16-38, 2013. [13] h. g. schaathun, the weight hierarchy of product codes, ieee trans. inform. theory, 46(7), 26482651, 2000. [14] j. h. van lint. repeated-root cyclic codes, ieee trans. inform. theory, 37(2), 343-345, 1991. 84 introduction the ambient space monomial-like codes duality and the hasse derivative a generalisation of the weight retaining property references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.645029 j. algebra comb. discrete appl. 7(1) • 73–84 received: 21 june 2019 accepted: 14 october 2019 journal of algebra combinatorics discrete structures and applications constructions of mds convolutional codes using superregular matrices∗ research article julia lieb, raquel pinto abstract: maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. in this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an mds convolutional code with a certain rate and a certain degree. we then present two novel constructions that fulfill these conditions by considering two types of superregular matrices. 2010 msc: 94b10 keywords: convolutional codes, mds codes, superregular matrices 1. introduction the (free) distance of a code measures its capability of detecting and correcting errors introduced during information transmission through a noisy channel. maximum distance separable (mds) block codes of rate k/n are the block codes with distance equal to the singleton bound n−k + 1. the class of mds block codes is very well understood and there exist prominent constructions of mds block codes like the reed-solomon codes [12]. the theory of convolutional codes is more involved than the theory of block codes and there are not many known constructions of mds convolutional codes. these codes have maximum free distance in the class of convolutional codes of a certain rate k/n and a certain degree δ, i.e., are the ones with free distance equal to the singleton bound (n−k) (⌊ δ k + 1 ⌋) + δ + 1 [13]. the first construction of mds convolutional codes was obtained by justesen in [9] for codes of rate 1/n and restricted degrees. in [16] smarandache and rosenthal presented constructions of convolutional codes of rate 1/n and arbitrary ∗ this work was supported by fundação para a ciência e a tecnologia (fct) within project uid/mat/04106/2019 (cidma) and the german research foundation (dfg) within grant li3103/1-1. julia lieb (corresponding author), raquel pinto; department of mathematics, university of aveiro, portugal (email: jlieb@ua.pt, raquel@ua.pt). 73 https://orcid.org/0000-0003-4211-1596 https://orcid.org/0000-0002-8168-4023 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 degree using linear systems representations. however these constructions require a larger field size than the constructions obtained in [9]. gluesing-luerssen and langfeld introduced in [6] a new construction of convolutional codes of rate 1/n that requires the same field sizes as the ones obtained in [9] but also with a restriction on the degree of the code. finally, smarandache, gluesing-luerssen and rosenthal [15] constructed mds convolutional codes for arbitrary parameters. we will define new constructions of convolutional codes of any degree and sufficiently low rate using superregular matrices with a specific property. a similar procedure was done for constructing twodimensional mds convolutional codes [3, 4] but it is not possible to derive 1d convolutional codes from the constructions of these papers. moreover, the proof which we present in this paper that the obtained codes are mds uses different tecniques from the corresponding ones in [3, 4]. we also provide explicit constructions of these codes using cauchy circulant matrices [14] and superregular matrices as defined in [2]. the paper is organized as follows: in the next section we start by introducing some preliminaries on superregular matrices. we give the definition of these matrices and two different types of superregular matrices. then we give some definitions and results on convolutional codes. in section 3, we present a procedure to construct mds convolutional codes using superegular matrices. we show that generator matrices whose coefficients of its entries fulfill certain conditions are generator matrices of an mds convolutional code. in section 4, we give two different constructions of mds convolutional codes of an arbitrary degree and rate smaller than some upper bound. finally, in section 5, we compare the necessary field size and the restrictions on the parameters of our obtained constructions with those of already known constructions. 2. preliminaries 2.1. superregular matrices we denote, as usual, the finite field of order q as fq. definition 2.1 ([14]). a matrix a ∈ fr×`q is said to be superregular if every minor of a is different from zero. the following lemma is easy to see and we will use it several times to derive our conditions for mds convolutional codes. lemma 2.2. (i) let a ∈ fr×`q be superregular. then, each vector which is a linear combination of s columns of a has at most s− 1 zeros. (ii) let a ∈ fr×`q with r ≥ ` be such that all its fullsize minors are nonzero. then, each vector which is a linear combination of ` columns of a has at most `− 1 zeros. there are many examples of superregular matrices. we will present two types of superregular matrices that we will use later for the constructions that we introduce in this paper. the first one will be the cauchy circulant matrices. theorem 2.3. [14] let q be an odd number, let α be an element of order q−1 2 in fq and let b be a nonsquare element in fq. then the (q−12 ) × ( q−1 2 ) matrix c = [ cij ] where cij = 1 1 − bαj−i , for 0 ≤ i,j ≤ q − 3 2 is superregular. the matrix considered in the above theorem is a cauchy circulant matrix. another type of superregular matrix is given in the next theorem. 74 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 theorem 2.4. [2] let p be a prime number and α a primitive element of fpn and b = [νi`] a matrix over fpn with the following properties 1. νi` = αβi` for a positive integer βi`; 2. if ` < `′, then 2βi` ≤ βi`′; 3. if i < i′, then 2βi` ≤ βi′ `. suppose n is greater than any exponent of α appearing as a nontrivial term of any minor of b. then b is superregular. 2.2. convolutional codes let fq[z] denote the ring of the polynomials with coefficients in fq. a convolutional code of rate k/n is an fq[z]-submodule of fq[z]n of rank k. a generator matrix of a convolutional code c of rate k/n is any n×k matrix whose columns constitute a basis of c, i.e., it is a full column rank matrix g(z) such that c = imfq[z]g(z) = {g(z)u(z) : u(z) ∈ fq[z]k}. if g(z) ∈ fq[z]n×k is a generator matrix of a convolutional code c, then all generator matrices of c are of the form g(z)u(z) for some unimodular matrix u(z) ∈ fq[z]k×k, i.e. for a matrix that is invertible in the ring of polynomial matrices. two generator matrices of the same code are said to be equivalent generator matrices. since two equivalent generator matrices differ by right multiplication of a unimodular matrix, they have the same full size minors, up to multiplication by a nonzero constant. the complexity or degree of a convolutional code is defined as the maximum degree of the full size minors of a generator matrix of the code. define the j-th column degree νj of a polynomial matrix g(z) ∈ fq[z]n×k to be the maximum degree of the entries of the j-th column of g(z). obviously, the sum of the column degrees of g(z) is greater or equal than the maximum degree of its full size minors. if the sum of the column degrees of g(z) equals the maximum degree of its full size minors, g(z) is said to be column reduced. a convolutional code always admits column reduced generator matrices and two column reduced generator matrices have the same column degrees up to a column permutation [5, 10]. therefore, column reduced generator matrices are the ones that have minimal sum of the column degrees and thus the sum of its column degrees is equal to the degree of the code. definition 2.5. for g(z) ∈ fq[z]n×k, let [gij]hc denote the coeffcient of zνj in gij(z). then, the highest column degree coeffcient matrix [g]hc ∈ fn×kq is defined as the matrix consisting of the entries [gij]hc. a matrix g(z) ∈ fq[z]n×k is column reduced if and only if [g]hc ∈ fn×kq has full rank. the weight of a vector c ∈ fnq , wt(c), is the number of its nonzero entries and the weight of a polynomial vector v(z) = ∑ i∈n0 viz i ∈ fq[z]n is given by wt(v(z)) = ∑ i∈n0 wt(vi). the free distance of a convolutional code c is the minimum weight of the nonzero codewords of the code, i.e., dfree(c) = {wt(v(z)) : v(z) ∈c\{0}}. 75 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 in [13] smarandache and rosenthal obtained an upper bound for the free distance of a convolutional code c of rate k/n and degree δ given by dfree(c) ≤ (n−k) (⌊ δ k ⌋ + 1 ) + δ + 1. this bound is called the generalized singleton bound. a convolutional code of rate k/n and degree δ with free distance equal to the generalized singleton bound is called maximum distance separable (mds) convolutional code. if c is such a code and g(z) ∈ fq[z]k×n is a column reduced generator matrix of c, its column degrees are equal to ⌊ δ k ⌋ + 1 with multiplicity t := δ −k ⌊ δ k ⌋ and ⌊ δ k ⌋ with multiplicity k − t; see [13]. 3. conditions to obtain mds convolutional codes let c be a convolutional code of rate k/n and degree δ. in this section, we will derive conditions on a column reduced generator matrix g(z) of c that ensure that the code is an mds convolutional code. to this end, we assume that g(z) has non-increasing column degrees with values ⌊ δ k ⌋ + 1 and ⌊ δ k ⌋ . we write g(z) = ∑µ i=0 giz i with gµ 6= 0, i.e. µ = deg g, and ν := ⌊ δ k ⌋ + 1, i.e. ν = µ if k δ and ν = µ + 1 if k | δ. furthermore, we write g(z) = [g1(z) . . .gk(z)] with gr(z) =   ∑ 0≤i≤ν gi,rz i, for r = 1, 2, . . . , t,∑ 0≤i≤ν−1 gi,rz i, for r = t + 1, t + 2, . . . ,k, i.e. gi = [gi,1 · · ·gi,k] for i = 1, . . . ,ν − 1 and gν = [gν,1 · · ·gν,t 0 · · ·0], where t = δ −k ⌊ δ k ⌋ . set g = [ g0,1 · · ·gν,1 · · · g0,t · · ·gν,t g0,t+1 · · ·gν−1,t+1 · · · g0,k · · ·gν−1,k ] ∈ fn×(kν+t)q . (1) write u(z) = ∑ i∈n0 uiz i. if l := deg u ≤ µ, we have v(z) = g(z)u(z) = g0u0 + · · · [gl · · ·g0]  u0... ul  zl + · · · + [gµ · · ·gµ−l]  u0... ul  zµ + · · · + gµulzµ+l (2) for l ≥ µ, one obtains v(z) = g(z)u(z) = g0u0 + · · · + [gµ · · ·g0]  u0... uµ  zµ + · · · + [gµ · · ·g0]  ul−µ... ul  zl + · · · + gµulzµ+l (3) as wt(g(z)u(z)) = wt(g(z)u(z)zr) for r ∈ n, throughout this paper, we assume, without loss of generality that u0 6= 0. theorem 3.1. if the matrix g defined in (1) is superregular, g(z) is the generator matrix of an (n,k,δ) convolutional code. 76 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 proof. since the highest column degree coefficient matrix of g(z) is equal to[ gν,1 gν,2 · · · gν,t gν−1,t+1 · · · gν−1,k ] , it is a submatrix of the superregular matrix g and hence full rank. consequently, g(z) is column reduced. therefore, the degree of the code generated by g(z) is equal to the sum of the column degrees of g(z), which is νt + (ν − 1)(k − t) = δ. the generated code is an mds convolutional code if and only if for each u(z) ∈ fq[z]k \ {0} and v(z) = g(z)u(z), one has wt(v(z)) ≥ (n−k) (⌊ δ k ⌋ + 1 ) + δ + 1 = nν − (k − t) + 1. (4) next, we will show that under certain conditions equation (4) is fulfilled by considering different cases for the value of δ. in any case, one of the conditions will always be the superregularity of g. however, this condition is not necessary to obtain an mds convolutional code as the following example shows. example 3.2. let c be the convolutional code of rate 2/3 and degree 1 with generator matrix g(z) =   1 11 2 0 1  +   1 01 0 2 0  z. the free distance of this code is dfree(c) = 3 and hence it is an mds convolutional code but g =  1 1 11 2 1 0 1 2   is not superregular. 3.1. conditions for the case δ < k in this case, we have to prove that wt(v(z)) ≥ n−k + δ + 1. theorem 3.3. assume that δ < k and let g be superregular. if n ≥ δ + k−1, then g(z) is the generator matrix of an (n,k,δ) mds convolutional code. proof. as δ < k, we have ν = µ = 1 and t = δ. case 1: l = 0 one has v(z) = g0u0 + g1u0z, where g0 and the δ nonzero columns of g1 form superregular matrices. if the first δ components of u0 are zero, i.e. g1u0 = 0, we have wt(v(z)) ≥ n−(k−δ) + 1, since g0u0 is a nonzero linear combination of k−δ columns of g0. if one of the first δ components of u0 is nonzero, one obtains wt(v(z)) = wt(g0u0) + wt(g1u0) ≥ n−k + 1 + n−δ + 1 ≥ n−k +δ + 1 as n ≥ δ +k−1 ≥ 2δ−1. case 2: l ≥ 1 here, one has v(z) = g0u0 +[g1 g0] ( u0 u1 ) z+· · ·+[g1 g0] ( ul−1 ul ) zl+g1ulz l+1. if the first δ components of ul are zero, one has wt(v(z)) ≥ wt(g0u0) + wt ( [g1 g0] ( ul−1 ul )) ≥ n − k + 1 + n − (k + δ − δ) + 1 ≥ n − k + δ + 1, since [g1 g0] ( ul−1 ul ) is a nonzero linear combination of δ columns of g1 and k − δ columns of g0 and n ≥ δ + k − 1. if one of the first δ components of ul is nonzero, one obtains wt(v(z)) ≥ wt(g0u0) + wt(g1ul) ≥ n−k + 1 + n−δ + 1 ≥ n−k + δ + 1 as n ≥ δ + k − 1 ≥ 2δ − 1. 77 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 3.2. conditions for the case δ ≥ k for this subsection, we need the additional definitions g1 =  g0... gν   ∈ f(ν+1)n×kq , g2 =   g0... gν−1   ∈ fνn×kq , ḡ =  g0... gµ   ∈ f(µ+1)n×kq . we have ḡ = g1 for k δ and ḡ = g2 for k | δ and g(z) = [in inz · · ·inzµ]ḡ. (5) theorem 3.4. assume that δ ≥ k and let g defined in (1) be superregular. moreover, assume that all fullsize minors of ḡ are nonzero. if n ≥ 2δ +k−ν, then g(z) is the generator matrix of an (n,k,δ) mds convolutional code. proof. we distinguish several cases. case 1: l = 0 case 1.1: k | δ in this case, the generalized singleton bound is equal to nν − k + 1. if we define v = g2u, we obtain that v is a nonzero linear combination of the columns of a matrix with nonzero fullsize minors and hence wt(v) ≥ nν −k + 1. case 1.2: k δ let us write u0 = [ u (1) 0 u (2) 0 ] , with u(1)0 ∈ f t q and u (2) 0 ∈ f k−t q , and set v = g1u. then v = [ v(1) v(2) ] , with v(1) = g2u ∈ fνnq and v(2) = gνu(1) ∈ fnq . hence, v(1) is a nontrivial linear combination of columns of an nν ×k matrix with nonzero fullsize minors and v(2) is a linear combination of columns of an n× t matrix with nonzero fullsize minors. we distinguish two further subcases. case 1.2.1: u(1) = 0. in this case, one has v = [ v(1) 0 ] where v1 is a nontrivial linear combination of the columns of an nν × (k− t) matrix with nonzero fullsize minors and k− t < nν. applying lemma 2.2, we obtain wt(v) ≥ nν − (k − t) + 1. case 1.2.2: u(1) 6= 0. in this case, v(1) and v(2) are nontrivial linear combinations of the columns of an nν ×k and an n× t matrix with nonzero fullsize minors, respectively. moreover, since nν > k and n > t, it follows from lemma 2.2 that wt(v(1)) ≥ nν −k + 1 and wt(v(2)) ≥ n− t + 1 and thus we get wt(v) = wt(v(1)) + wt(v(2)) ≥ nν + n−k − t + 2 = nν − (k − t) + 1 + n− 2t + 1 ≥ nν − (k − t) + 1 where the last inequality follows from the fact that n ≥ 2δ+k−ν = δ+k−1 +δ−bδ k c≥ δ+k−1 ≥ 2t−1. using equation (5), wt(g(z)u(z)) = wt(v) and the result follows. case 2: 1 ≤ l < µ note that for this case, one has n ≥ 2δ + k −ν ≥ k + δ − 1 = ( δ k − 1 k + 1 ) k ≥ µk ≥ (l + 1)k. (6) case 2.1: k | δ 78 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 using equations (2) and (6), the superregularity of g and that u0 and ul are nonzero, we obtain wt(v(z)) ≥ 2 l∑ i=1 (n− ik + 1) + ( δ k − l + 1)(n− (l + 1)k + 1) = = 2nl + 2l−k(l + 1)l + ( δ k + 1)(n−k) + ( δ k + 1)(−lk + 1) − ln + l(l + 1)k − l = ( δ k + 1)(n−k) + δ + 1 + nl + l + ( δ k + 1)(−lk + 1) − δ − 1. consequently, in order to get wt(v(z)) ≥ ( δ k + 1)(n−k) + δ + 1, one needs n ≥ 1 l ( δ + 1 − l + lδ + lk − δ k − 1 ) = δ + k − 1 + 1 l ( δ − δ k ) . the result follows from δ + k − 1 + 1 l ( δ − δ k ) ≤ 2δ + k −ν. case 2.2: k δ additionally to the previous subcase, here we have to regard that gµul might be zero and that [gµ · · ·gi] for i = µ− l, . . .µ− 1 has k − t = kµ−δ zero columns. therefore, we get wt(v(z)) ≥ 2 l∑ i=1 (n− ik + 1) − (n−k + 1) + (µ− l + 1)(n− (l + 1)k + 1) + (kµ−δ)l = µ(n−k) + δ + 1 + nl + l− lk + µ(−lk + 1) + (kµ−δ)l− δ − 1 = µ(n−k) + δ + 1 + nl + l− lk + µ− δl− δ − 1 hence, one needs n ≥ k + δ− 1 + 1 l (δ + 1 −µ). this is true as k + δ− 1 + 1 l (δ + 1 −µ) ≤ k + 2δ−µ and µ = ν for k δ. case 3: l ≥ µ for this case, we consider equation (3). as it could happen that 2δ + k−ν is smaller than (µ + 1)k, the weight of vi might be zero for i = µ,. . . , l. however, one has µk ≤ ( δ k − 1 k + 1 ) k = δ +k−1 ≤ 2δ +k−ν. case 3.1: k | δ using equation (3) and the superregularity of g, we obtain wt(v(z)) ≥ 2 µ∑ i=1 (n− ik + 1) = 2nµ + 2µ− (µ + 1)µk = (n−k)(µ + 1) + n(µ− 1) + 2µ− (µ + 1)(µ− 1)k + δ + 1 − δ − 1. hence, for µ ≥ 2, one needs n ≥ k(µ + 1) − 2 + δ−1 µ−1 = k + δ − 2 + δ−1 µ−1. this is true for µ ≥ 3 since k+δ−2+ δ−1 µ−1 ≤ k+ 3 2 δ−5 2 ≤ k+2δ−ν for k ≥ 2 and k+δ−2+ δ−1 µ−1 = k+δ−1 for k = 1. it is also true for µ = 2 since k + δ − 2 + δ − 1 = k + 2δ − 3 ≤ k + 2δ −ν. it remains to consider the case µ = 1. if we consider above estimation for the weight for µ = 1, we get the condition δ ≤ 1. hence, for the following consideration, we could assume k = δ ≥ 2. for these parameters one has 2δ + k − ν ≥ k + δ and thus, we could exploit the superregularity of [g1 g0]. doing this, we get wt(v(z)) ≥ 2(n−k + 1) + (n− 2k + 1) = 2(n−k) + δ + 1 −δ + 2 + n− 2k ≥ 2(n−k) + δ + 1 because n ≥ 2δ + k −ν = δ + 2k − 2. case 3.2: k δ if one of the first t components of ul is nonzero, we get wt(v(z)) ≥ 2 µ∑ i=1 (n− ik + 1) = (n−k)µ + nµ + 2µ−µ2k + δ + 1 −δ − 1. 79 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 consequently, one needs n ≥ kµ− 2 + δ+1 µ , which is true as kµ− 2 + δ+1 µ ≤ k + δ − 3 + δ+1 2 ≤ k + 3 2 δ − 5 2 ≤ 2δ + k −ν because k δ implies k ≥ 2. if the first t components of ul are zero, what changes in the previous estimation for the weight of v(z) is that we have to subtract n−k + 1 as gµul = 0 but in turn we could add k − t = kµ− δ to each of the weights of vi for i = l + 1, . . . , l + µ− 1. finally, we obtain wt(v(z)) ≥ (n−k)µ + n(µ− 1) + 2µ− 1 − (µ2 − 1)k + δ + 1 − δ − 1 + (kµ−δ)(µ− 1). therefore, we need n ≥ (µ+ 1)k−2 +δ−kµ+ δ µ+1 = k +δ−2 + δ µ+1 , which is true since k +δ−2 + δ µ+1 ≤ k + 4 3 δ − 2 ≤ 2δ + k −ν because k δ implies k ≥ 2. 4. constructions of mds convolutional codes in this section, we will use the results of the preceding section to obtain two different constructions for mds convolutional codes. 4.1. constructions for δ < k theorem 4.1 (construction 1). assume δ < k, t and ν as defined before and n ≥ k + δ− 1. moreover, let q be an odd number such that q ≥ 2 max{k + δ,n} + 1 and let c = [ cij ] be the cauchy circulant matrix defined in theorem 2.3 over fq. set g =   c0,0 · · · c0,k+δ−1... ... cn−1,0 · · · cn−1,k+δ−1   (7) then, the matrix g is superregular. let us write g = [ g0,1 · · ·gν,1 · · · g0,t · · ·gν,t g0,t+1 · · ·gν−1,t+1 · · · g0,k · · ·gν−1,k ] . then, g(z) = ∑µ i=0 giz i with gi = [gi,1 · · ·gi,k] is the generator matrix of an (n,k,δ) mds convolutional code. proof. the proof of theorem 4.1 follows immediately from theorem 2.3. example 4.2. in this example we use theorem 4.1 to construct a (4, 3, 2) mds convolutional code over f11. to this end, we choose a = 3 and b = 2 in theorem 2.3 and get the superregular matrix c =   10 2 9 6 3 3 10 2 9 6 6 3 10 2 9 9 6 3 10 2 2 9 6 3 10  . thus, considering g = [g0,1 g1,1 g0,2 g1,2 g0,3] =   10 2 9 6 3 3 10 2 9 6 6 3 10 2 9 9 6 3 10 2   it follows from theorem 4.1 that g(z) =   10 9 3 3 2 6 6 10 9 9 3 2   +   2 6 0 10 9 0 3 2 0 6 10 0  z 80 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 is a generator matrix of a (4, 3, 2) mds convolutional code. theorem 4.3 (construction 2). assume that δ < k, n ≥ k + δ − 1 and n ≥ 2n+k+δ−1. let α be a primitive element of a finite field fpn . set g =   α · · · α 2k+δ−1 ... ... α2 n−1 · · · α2 n+k+δ−2   . then, the matrix g is superregular. let us write g = [ g0,1 · · ·gν,1 · · · g0,t · · ·gν,t g0,t+1 · · ·gν−1,t+1 · · · g0,k · · ·gν−1,k ] . then, g(z) = ∑µ i=0 giz i with gi = [gi,1 · · ·gi,k] is the generator matrix of an (n,k,δ) mds convolutional code over fpn . proof. according to theorem 2.4, g is superregular over fpn if n is greater than∑k+δ−1 i=0 2 k+δ+n−2−2i = 2n−k−δ ∑k+δ−1 i=0 2 2i < 2n+δ+k−1. for the last inequality, we used the geometric sum. example 4.4. in this example we construct a (4, 3, 2) mds convolutional code but now over f 22 8 . to this end, we choose a primitive element α of f28 and set g = [g0,1 g1,1 g0,2 g1,2 g0,3] =   α α2 α4 α8 α16 α2 α4 α8 α16 α32 α4 α8 α16 α32 α64 α8 α16 α32 α64 α128   and g(z) =   α α4 α16 α2 α8 α32 α4 α16 α64 α8 α32 α128   +   α2 α8 0 α4 α16 0 α8 α32 0 α16 α64 0  z, which, according to theorem 4.3, is a generator matrix of a (4, 3, 2) mds convolutional code. 4.2. constructions for δ ≥ k theorem 4.5. [construction 1] assume δ ≥ k, t, ν as defined before and n ≥ k + 2δ−ν. moreover, let q be an odd number such that q ≥ 2n(ν + 1) + 1 and let c = [ cij ] be the cauchy circulant matrix defined in theorem 2.3 over fq. set gj,r =   cjn,r−1 cjn+1,r−1 ... c(j+1)n−1,r−1   (8) for (j,r) ∈ ({0, 1, . . . ,ν − 1}×{1, 2, . . . ,k}) ∪ ({ν}×{1, 2, . . . , t}) if t 6= 0 and for (j,r) ∈ ({0, 1, . . . ,ν − 1}×{1, 2, . . . ,k}) if t = 0. then, the matrix g(z) = ∑µ i=0 giz i with gi = [gi,1 · · ·gi,k] is the generator matrix of an (n,k,δ) mds convolutional code. proof. by theorem 2.3, c is a superregular matrix. then the matrix ḡ is superregular because it is a submatrix of c. since α q−1 2 = 1, we obtain cu,v = 1 1 − bαv−u = 1 1 − bα q−1 2 −u+v = c0,q−1 2 −u+v, 81 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 for 0 ≤ u,v ≤ q−3 2 , and, hence, gj,r =   c0,q−1 2 −jn+r−1 c1,q−1 2 −jn+r−1 ... cn−1,q−1 2 −jn+r−1   . consequently, after an appropriate rearrangement of the columns of g, we obtain a submatrix of the cauchy matrix c. therefore, the matrix g is also superregular. theorem 4.6. [construction 2] assume that δ ≥ k and n ≥ k + 2δ −ν and n ≥ 2(b δ k c+1)n+k−1. let α be a primitive element of a finite field fpn . set gj,r =   α 2r−1+nj ... α2 r−1+n(j+1)   for r = 1, . . . ,k and j = 0, . . . ,bδkc and gbδ k c+1,r =   α 2nr−1 ... α2 n(r+1)−2   for r = 1, . . . , t. then, g(z) = ∑µi=0 gizi with gi = [gi,1 · · ·gi,k] is the generator matrix of an (n,k,δ) mds convolutional code over fpn . proof. with the definitions of the above theorem, g consists of k + δ columns of  α · · · α2 k−1+nbδ k c ... ... α2 n−1 · · · α2 k+n−2+nbδ k c   . hence, according to theorem 2.4, it is superregular over fpn if n is greater than∑k+δ−1 i=0 2 k+n−2+nbδ k c−2i = 2n+nb δ k c−k−2δ ∑k+δ−1 i=0 2 2i < 2n+nb δ k c+k−1. for the last inequality, we used the geometric sum. moreover, ḡ is equal to  α · · · α2 k−1 ... ... α2 n−1+nbδ k c · · · α2 k+n−2+nbδ k c   , which, according to theorem 2.4, is superregular over fpn again if n is greater than∑k+δ−1 i=0 2 k+n−2+nbδ k c−2i < 2n+nb δ k c+k−1. example 4.7. using theorem 4.5 it is possible to construct a (4, 2, 2) mds convolutional code over f25 and with theorem 4.6 a (4, 2, 2) mds convolutional code over f 22 9 . 5. comparison of constructions for mds convolutional codes in this section, we want to compare the new constructions for mds convolutional codes in this paper with the already known constructions. the comparison should be in terms of conditions on the parameters n,k and δ and in terms of the necessary field size. throughout this section, we refer to the new constructions of the preceding section as construction 1 and construction 2. the constructions in [9], [16] and [6], which we already mentioned in the introduction, work only for k = 1 but in this case the required field sizes are smaller than the field sizes required for construction 1 and construction 2. 82 j. lieb, r. pinto / j. algebra comb. discrete appl. 7(1) (2020) 73–84 for nearly all parameters with k > 1, the construction of [15] leads to the smallest field size of all known constructions. but this construction has the drawback that it only works for |fq| ≡ 1 mod n. moreover, construction 1 obtained in this paper could improve the necessary field size of [15] in some particular cases, e.g. it leads to smaller field sizes for (17, 2, 1) and (17, 2, 4) convolutional codes. however, also this construction has restrictions, i.e. it works only for odd field sizes and if n is larger than a particular lower bound. maximum distance profile (mdp) convolutional codes are convolutional codes whose so-called column distances increase as rapidly as possible for as long as possible; see [8] or [7] for more explanation. as each (n,k,δ) mdp convolutional code with (n−k) | δ is an mds convolutional code [7], for comparison, one also has to consider constructions for mdp convolutional codes if (n−k) | δ. in [1] and [11, theorem 3.2], one could find such constructions that have no other requirements on the parameters than (n−k) | δ. there, the required field sizes are larger than the field size from [15] but again this construction has the drawback that it only works for |fq| ≡ 1 mod n. theorem 3.2 of [11] provides a construction for mdp convolutional codes where the required field size is smaller than the field size in [1]. however, it only works for very large characteristic of the field, while the construction in [1] and also construction 2 work for arbitrary characteristic. if n is sufficiently large, such that the conditions for construction 2 are fulfilled, it depends on the parameters if it is better than the construction in [1] or not. for example, for an (5, 2, 2) code the construction from [1] is better, and for an (5, 1, 5) code, construction 2 is better. references [1] p. j. almeida, d. napp, r. pinto, a new class of superregular matrices and mdp convolutional codes, linear algebra appl. 439(7) (2013) 2145–2157. 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[16] r. smarandache, j. rosenthal, a state space approach for constructing mds rate 1/n convolutional codes, proceedings of the 1998 ieee information theory workshop on information theory, 116–117. 84 https://doi.org/10.1109/18.930938 https://doi.org/10.1109/18.930938 https://doi.org/10.1109/itw.1998.706461 https://doi.org/10.1109/itw.1998.706461 introduction preliminaries conditions to obtain mds convolutional codes constructions of mds convolutional codes comparison of constructions for mds convolutional codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.784992 j. algebra comb. discrete appl. 7(3) • 247–258 received: 14 april 2020 accepted: 1 may 2020 journal of algebra combinatorics discrete structures and applications generalization of pinching operation to binary matroids research article vahid ghorbani, ghodratollah azadi, habib azanchiler abstract: in this paper, we generalize the pinching operation on two edges of graphs to binary matroids and investigate some of its basic properties. for n ≥ 2, the matroid that is obtained from an n-connected matroid by this operation is a k-connected matroid with k ∈ {2,3,4} or is a disconnected matroid. we find conditions to guarantee this k. moreover, we show that eulerian binary matroids are characterized by this operation and we also provide some interesting applications of this operation. 2010 msc: 05b35 keywords: binary matroid, connectivity, pinching, splitting, splitting off, element splitting 1. introduction the matroid and graph notations and terminology used here will follow [4] and [7]. let g be a graph and s be a subset of the edge set e(g). we denote by gps the graph that is obtained from g by adding a new vertex w and replacing any edge uv ∈ s with two edges uw and wv. the transition from g to gps is called the k-pinching operation [2] where k = |s|. if s is the empty set, then 0-pinching means adding a new single vertex w. if |s| = 1, then applying the pinching operation is equivalent to the subdivision of uv where s = {uv}. if s = e(g), then we have the parallel classes on the new vertex w on gps such that the sequence of the cardinality of them is equal to the degree sequence of the vertices of g. applying the pinching operation on graphs is a useful method for solving 2k-edge-connectivity problems for graphs [2]. for example, suppose that g is a 2k-edge-connected graph. then the graph g′ obtained from g by pinching together any k edges of g is also 2k-edge-connected. raghunathan, shikare, and wapare [5] extended the splitting operation and azadi [1] extended the splitting off and the element splitting operations from graphs to binary matroids. these operations are defined as follows. definition 1.1. let m be a binary matroid on a set e and a be a matrix that represents m over gf(2). consider two elements x and y of e(m). let ax,y be the matrix that is obtained by adjoining an extra vahid ghorbani (corresponding author), ghodratollah azadi, habib azanchiler; department of mathematics, urmia university, iran (email: v.ghorbani@urmia.ac.ir, gh.azadi@urmia.ac.ir, h.azanchiler@urmia.ac.ir) 247 http://orcid.org/0000-0002-7301-6973 http://orcid.org/0000-0002-1807-4732 http://orcid.org/0000-0002-2949-3836 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 row to a whose entries are zero everywhere except in the columns corresponding to x and y where it takes value 1. let mx,y be the matroid that represents by the matrix ax,y. then the transition from m to mx,y is called the splitting operation. definition 1.2. let m be a binary matroid on a set e and a be a matrix that represents m over gf(2). consider two elements x and y of e(m). let a′x,y be the matrix that is obtained by adjoining an extra row to a whose entries are zero everywhere except in the columns corresponding to x and y where it takes value 1 and then adjoining an extra column to the resulting matrix with this column being zero everywhere except in the last row. let m′x,y be the matroid that represents by the matrix a ′ x,y. then the transition from m to m′x,y is called the element splitting operation. definition 1.3. let m be a binary matroid on a set e and a be a matrix that represents m over gf(2). consider two elements x and y of e(m). let axy be the matrix that is obtained by adjoining an extra column to a which is the sum of the columns corresponding to x and y, and then deleting the two columns corresponding to x and y. let mxy be the matroid that represents by the matrix axy. then the transition from m to mxy is called the splitting off operation. splitting off operation on graphs is defined as follows. definition 1.4. let g be a graph. consider two adjacent non-loop edges x = uw and y = vw. the splitting off a pair (x,y) of edges from a vertex w means that we replace the edges x and y by a new edge α = uv. if u = v then the resulting loop is deleted from the graph. 2. 2-pinching operation in 2-connected graphs let g be a 2-connected graph with the edge set e(g) and s ⊆ e(g) where s = {x = u1v1,y = u2v2}. to apply the 2-pinching operation, we first delete x and y and add new vertex w. then we add new edges x′ = u1w, x′′ = wv1, y′ = u2w and y′′ = wv2. when x and y are adjacent, we take v1 = v2 = v. therefore, x′ = u1w, x′′ = wv, y′ = u2w and y′′ = wv (figure 1). we denote by gpxy the graph that is obtained by applying the 2-pinching operation on edges x and y. note that we can retrieve the graph g from gpxy by splitting off the pair (x ′,x′′) and then splitting off the pair (y′,y′′). figure 1. 2-pinching operation on {x,z} and {x,y}. now let g be a 2-connected graph and x,y ∈ e(g). then g has a cycle cn of length n containing both x and y. assume that the cycle cn has the minimum cardinality among all such cycles. after applying the 2-pinching operation on x and y, the cycle cn transforms into two cycles c′k+2 and c ′ n−k where k is the length of shortest path q among endpoints of x and y such that x′′,y′′ ∈ c′k+2 and x ′,y′ ∈ c′n−k. in other words, c ′ k+2 = e(q)∪{x ′′,y′′} 1 and c′n−k = ( cn−(e(q)∪{x,y}) ) ∪{x′,y′}. note that if k = 0, then x and y are adjacent and cn transforms to c′2 = {x′′,y′′} and c′n = ( cn−{x,y} ) ∪{x′,y′}. 1 in this section, for cycle cn of length n from a graph g, we consider just its edge set 248 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 theorem 2.1. let g be a graph and x,y ∈ e(g). suppose that g has a cycle cn with a minimum length containing both x and y. let x′,x′′,y′,y′′,c′k+2 and c ′ n−k be the notations as mentioned above. moreover, let δ = {c′k+2,c ′ n−k} and c be a subset of edges of g p xy. then c forms a cycle of g p xy if and only if c ∈ δ or c satisfies one of the following conditions. (i) c is a cycle of g which contains neither x nor y; (ii) c = ( c′−{x} ) ∪{x′,x′′} where c′ is a cycle of g containing x but not y, or c = ( c′′−{y} ) ∪{y′,y′′} where c′′ is a cycle of g containing y but not x; and (iii) c = c1∆c2 where c1 ∈ δ, and c2 is a cycle of type (i) or (ii) such that c1 ∩c2 6= ∅. proof. it is clear that if c ∈ δ or if c is a cycle of g containing neither x nor y, then c is a cycle of gpxy. now suppose that c = ( c′ −{x} ) ∪{x′,x′′} = ( c′ −{u1v1} ) ∪{u1w,v1w} where c′ is a cycle of g containing x but not y and has length k′ + 1, so c′ = v1xu1e1z1...e ′ k′−1zk′−1ek′v1. after applying the pinching operation on x we delete x and add x′ = u1w and x′′ = v1w. thus (c′ −{x}) ∪{x′,x′′} = u1e1z1...ek′−1zk′−1ek′v1x′′wx′u1. therefore c is a cycle of gpxy. similarly, if c = ( c′′−{y} ) ∪{y′,y′′} = ( c′−{u2v2} ) ∪{u2w,v2w} where c′′ is a cycle of g containing y but not x, then c is a cycle of gpxy. now suppose that c1 is a cycle of type (i) or (ii) and c2 ∈ δ such that c1 ∩c2 6= ∅. to show that c1∆c2 is a cycle of gpxy, we consider the following two possible cases. figure 2. internally disjoint paths. case(1) let c1 be a cycle of type (i). assume that c1 has a non-empty intersection with both c′k+2 and c′n−k. let q1 and q2 be the paths such that c1∩c ′ n−k = e(q1) and c1∩c ′ k+2 = e(q2). moreover, let q3 , q4 be internally disjoint paths which have the same end vertices as of q1 and q2 such that c1 = e(q1)∪e(q2)∪e(q3)∪e(q4) (see figure 2). suppose that c′n−k = e(q1)∪e(q5)∪e(q7) and c′k+2 = e(q2)∪e(q6)∪e(q8) where qi and qj for i,j ∈{1, 2, 5, 6, 7, 8} are internally disjoint paths. then clearly c1∆c′n−k and c1∆c ′ k+2 are cycles of g p xy. similarly, if c1 has a non-empty intersection with exactly one of c′k+2 or c ′ n−k, then c1∆c ′ n−k or c1∆c ′ k+2 is a cycle of g p xy. case(2) let c2 = e(q3)∪e(q5)∪e(q6) and c3 = e(q4)∪e(q7)∪e(q8). clearly, c2 and c3 are cycles of type (ii). it is easy to see that c2∆c′n−k, c2∆c ′ k+2, c3∆c ′ n−k and c3∆c ′ k+2 are cycles of g p xy. now suppose that x and y are adjacent. then v1 = v2 = v and c′k+2 = vy ′′wx′′v. an argument similar to one as given above shows that c1∆c′2 and c1∆c ′ n are cycles of g p xy where c1 is a cycle of type (i) or (ii) such that c1 ∩c′n 6= ∅. 249 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 conversely, let c be a cycle of gpxy. by the definition of g p xy one can see that x,y /∈ e(gpxy) and x′,x′′,y′,y′′ /∈ e(g). let n = {x′,x′′,y′,y′′}. we know that the induced subgraph of n is either the graph (a) or the graph (b) of the following figure. figure 3. two possible induced subgraphs of n. it is straightforward to check that if c is a cycle of gpxy, then c contains a subset n ′ of n with |n′| ∈ {0, 2}. therefore, every cycle of gpxy is a cycle of type (i), (ii) or (iii). 3. 2-pinching operation for binary matroids now, by using theorem 2.1 we extend the notion of the 2-pinching operation from 2-connected graphs to binary matroids. let m be a binary matroid and x,y ∈ e(m). we denote by cp the collection of all the circuits of m which contain both x and y. choose a member of cp and denote it by cxy. partition cxy −{x,y} into sets x1 and x2 where x1 or x2 may be empty. we denote such a partition by p = (x1,x2). clearly, when cp is empty or |cxy| = 2, there is no such partition and we say that p is empty. definition 3.1. let a be a matrix that represents the matroid m over gf(2). for x,y ∈ e(m), let cp 6= ∅ and p = (x1,x2) constructed as above. let apxy be the matrix obtained from a by the following way. (1) adjoin four extra columns to a with the labels α1,α2,α3 and α4 such that: – the corresponding entries of α1 and x are equal; – α2 is a zero column; – α3, α4 are columns whose entries are 1 or 0 such that sum of the corresponding entries of the corresponding columns of x1 ∪{α1,α3} and x2 ∪{α2,α4} are zero. (2) delete the two columns corresponding to x and y (3) adjoin an extra row with this row being zero everywhere except in the columns corresponding to the new elements αk where it takes value 1, for 1 ≤ k ≤ 4. we call a vector matroid obtained in this way a pinching on x and y with respect to p or 2-pinching with respect to p and denoted it by mpxy. definition 3.2. let all hypotheses of definition 3.1 be established except that cp 2 is an empty set. to build a∅xy from a, carry out all the steps of definition 3.1 with the following changes. • the corresponding entries of α1 and x are equal; • the corresponding entries of α3 and y are equal; 2 this means x or y or both are cocircuits of m 250 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 • α2 and α4 are zero columns; remark 3.3. let x and y be two elements of a matroid m. note that, when |cxy| = 2, we can build a∅xy with both definition 3.1 and definition 3.2. so throughout this paper, we may assume that cp 6= ∅ unless in special cases of some proofs. figure 4. {x=6,y=7} and p=({1},{4}). example 3.4. consider the cycle matroid of the graph in figure 4 with the following matrix a over gf(2) that represents it. a =   1 2 3 4 5 6 7 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 1 1 1  . let x = 6 and y = 7. then cp = {{2, 6, 7},{1, 4, 6, 7}}. let cxy = {1, 4, 6, 7}. partition cxy−{x,y} to two sets x1 = {1} and x2 = {4}. so by definition 3.1, after applying the pinching operation on x and y with respect to p1 = ({1},{4}), the matrix a transforms into the following matrix. a p1 xy =   1 2 3 4 5 α1 α2 α3 α4 1 0 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 1  . one can easily check that the vector matroid of ap1xy is isomorphic to m ∗(k3,3). hence the 2-pinching operation on a graphic matroid may not yield a graphic matroid. moreover, m∗(k3,3) is a 3-connected matroid while the original matroid is not. clearly, if cxy = {2, 6, 7}, then p2 = ({2},∅) and mp2xy is a graphic matroid. theorem 3.5. let m = (e,c) be a binary matroid on the set e together with the collection c of circuits. let x,y ∈ e(m), and for a given circuit cxy, let p = (x1, x2) be a partition of cxy −{x,y}. suppose that γ = {α1,α2,α3,α4} such that αi /∈ e(m) for 1 ≤ i ≤ 4. then mpxy = ( (e −{x,y}) ∪ γ,cpxy ) with cpxy = c0 ∪c1 ∪ δ ∪c2 ∪c3 is a binary matroid where (i) c0 = { c ∈c : c contains neither x nor y } ; (ii) c1 = {( c−{x} ) ∪{α1,α2} : c ∈c, and c contains x but not y }⋃{( c−{y} ) ∪{α3,α4} : c ∈c, and c contains y but not x } ; (iii) δ = { x1 ∪{α1,α3},x2 ∪{α2,α4} } ; (iv) c2 = { c1∆c2 : c1 ∈c0 ∪c1 and c2 ∈ δ, and if c1 ∈c0, then c1 ∩c2 6= ∅; and there is no circuit c of c0 and c1 such that c ⊆ c1∆c2 } ; and 251 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 (v) c3 = the set of minimal members of { (c −{x,y}) ∪ γ : x,y ∈ c and c 6= cxy } . proof. it is easy to see that every member of cpxy is a circuit in mpxy. to show that cpxy is the collection of circuits of a binary matroid on ( e −{x,y}∪ γ ) , we must show that if x,y ∈ cpxy, then x * y and y*x, and x∆y contains at least one member of cpxy. so there are fifteen cases to check and checking them is straightforward. let a be a matrix that represents m over gf(2). let c ⊆ e(m). then c is a circuit of m if and only if c is minimal and the mod-2 sum of the columns of a corresponding to c is a zero columns [4]. by using this fact, a consequence of the last theorem is the following partial result. corollary 3.6. under the hypotheses of theorem 3.5, let a be a matrix that represents m over gf(2), then the collection of minimal dependent sets of vector matroid of the matrix apxy is cpxy. proof. depending on how to construct the matrix apxy, the proof is straightforward. corollary 3.7. under the hypotheses of theorem 3.5, let r(m) and r(mpxy) be the ranks of m and m p xy, respectively. then r(mpxy) = r(m) + 1. proof. suppose that b is a basis of m such that cxy is a fundamental circuit with respect to b. then exactly one of the following holds. (i) ( cxy −{x} ) ⊆ b. let b1 = (b −{y}) ∪{α3,α4}. then |b1| = |b| + 1. by theorem 3.5(iii), c′1 = x1 ∪{α1,α3} and c′2 = x2 ∪{α2,α4} are circuits of mpxy. clearly, b1 does not contain either c′1 or c′2, nor any member of c0 ∪c1 ∪c2 ∪c3. so b1 is an independent set. since b1 ∪α1 contains c′1 and b1 ∪α2 contains c′2, the set b1 is a maximal independent set and so b1 is a basis of m p xy. (ii) ( cxy −{y} ) ⊆ b. similar to (i) we can see that if b2 = (b −{x}) ∪{α1,α2}, then b2 is a maximal independent set and |b2| = |b| + 1. (iii) cxy −{z}⊆ b where z /∈{x,y}. suppose that z ∈ x1 and b3 = ( b−{x,y} ) ∪{α1,α2,α3,α4}. consider two sets ( x1−{z} ) ∪{α1,α3} and x2 ∪{α2,α4}. clearly, ( x1 −{z} ) ∪{α1,α3} is an independent set and x2 ∪{α2,α4} is a circuit of mpxy. hence b3 cannot be a basis of m p xy, but b4 = b3 −{z′} where z′ ∈ ( x2 ∪{α2,α4} ) is a basis of mpxy and |b4| = |b| + 1. by (i),(ii) and (iii), we conclude that r(mpxy) = r(m) + 1. 4. connectivity of mpxy in this section, by studying the cardinality of some circuits and cocircuits of mpxy, we explore the relationship between connectivity of a given connected binary matroid m and mpxy. theorem 4.1. let mpxy be the pinching of a connected binary matroid m. if c1 ∪c3 = ∅, then mpxy is a disconnected matroid. otherwise, mpxy is a connected matroid. 252 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 proof. since m is a connected matroid, for every pair (x,y) of two elements of e(m), there exists a circuit of c(m) containing both x and y. now assume c1 ∪c3 = ∅. by theorem 3.5, we conclude that there is no circuit of m which contains x but not y, also there is no circuit of m which contains y but not x. let p = (x1,x2). then δ = { c′1 = x1 ∪{α1,α3},c′2 = x2 ∪{α2,α4} } and c2 = { c′1∆c : c ∈ c0, and c ∩ c′1 6= ∅}∪{c′2∆c : c ∈ c0, and c ∩ c′2 6= ∅ } . hence every member of c2 contains either {α1,α3} or {α2,α4}. since cpxy = c0 ∪ δ ∪c2, there is no member of cpxy containing {α1,α2}, {α3,α4}, {α1,α4} and {α2,α3}. therefore mpxy is a disconnected matroid. similarly, it can be shown that if c1∪c3 is non-empty, then mpxy is a connected matroid. definition 4.2. we say that two elements x and y from given matroid m are equivalent and denoted by x ∼ y, if at least one of x and y is a coloop of m or {x,y} is a cocircuit of m. if a matroid m has at least one pair of equivalent elements, then m is a connected or is a disconnected matroid. our important task of this section is to rely connectivity of obtained matroid by applying the 2pinching operation on a given n-connected matroid. so we investigate the cocircuits of matroid obtained by this operation. it is well known that all circuits and all cocircuits of an n-connected matroid with at least 2(n− 1) elements have at least n elements [4]. for a given binary matroid m with matrix representation a and two elements x,y ∈ e(m), we denote by r(c∗) and r(c∗p) the cocircuit spaces of a and apxy, respectively. in the following lemmas, we use the fact that the intersection of a circuit and a cocircuit of a binary matroid m has even cardinality [4]. lemma 4.3. let mpxy be a pinching of a binary matroid m. then γ = e(m p xy) −e(m) is a cocircuit of mpxy if and only if x is not equivalent to y. proof. suppose that mpxy is such a pinching with p = (x1,x2) and γ = {α1,α2, α3,α4}. let a be the matrix that represent m over gf(2). build apxy for mpxy according to definition 3.1. since there is a row in the row space of apxy which this row is zero everywhere except in the corresponding entries of γ, we get γ ∈r(c∗p). assume, without loss of generality, that x ∼ y. then exactly one of the following holds. (i) {x,y} is a cocircuit of m. then, there is a row in the row space of a such that has the following form [ e(m)−{x,y} x y 0 0 ... 0 ∣∣∣ 1 1 ]. let δxi be the sum of the corresponding entries of xi in this row, for i ∈ {1, 2}. clearly, δx1 = δx2 = 0. thus, after applying the pinching operation on m, we have the following row in the row space of apxy [ e(m)−{x,y} α1 α2 α3 α4 0 0 ... 0 ∣∣∣ 1 0 1 0 ]. we conclude that {α1,α3} and {α2,α4} are in r(c∗p). by theorem 3.5, c1 = x1 ∪{α1,α3} and c2 = x2 ∪{α2,α4} are circuits of mpxy and since |cj ∩{αi}| = 1 for some j ∈ {1, 2} and some i ∈{1, 2, 3, 4}. hence {α1,α3} and {α2,α4} are minimal and so γ is not a cocircuit of mpxy. (ii) {x} is a cocircuit of m. then m is a disconnected matroid and so {α2,α4} is a circuit of m∅xy. we see that in the row space of a∅xy the corresponding entries of α1 and α3 are 1 and 0, respectively and δx1 = δx2 = 0. therefore {α1} is a cocircuit of mpxy and so γ is not. 253 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 to complete the proof, we must show that γ is minimal when x is not equivalent to y. on the contrary, let z be a proper subset of γ which is minimal. clearly, if |z| = 1 or 3, then there is a circuit in mpxy such that its intersection with z or z∆γ is a singleton set and if |z| = 2, we get x ∼ y; a contradiction. hence γ is a cocircuit of mpxy. for a given matroid m, the girth g(m) is the minimum circuit size of m unless m has no circuits, in which case, g(m) = ∞. the cogirth is the girth of the dual of m. by lemma 4.3, we conclude that if m is an n-connected matroid with n ≥ 4, then the cogirth of mpxy is 4. now suppose that the girth of such a matroid m is k and |cxy| = k. then clearly, k ≥ n and by theorem 3.5, there is a matroid mpxy which will decrease the girth of m to w where 2 ≤ w ≤ [ k 2 ] + 1. note that if g(m) = ∞, for a given matroid m, then g(m∅xy) = 2. lemma 4.4. let m be an n-connected binary matroid with n ≥ 4 and |e(m)| ≥ 2(n − 1). let x,y ∈ e(m) and p = (x1,x2) be a partition of cxy −{x,y} with xi 6= ∅ for i ∈{1, 2}. then (γ,e(mpxy)−γ) is a 4-separation of mpxy where γ = e(m p xy) −e(m). proof. suppose that m is an n-connected binary matroid with n ≥ 4 and |e(m)| ≥ 2(n−1). let mpxy be a pinching of m with respect to partition p with non-empty parts. then |e(mpxy)| ≥ 2n. consider the partition (x,y ) of e(mpxy) with x = γ and y = e(m p xy) − γ where γ = e(m) −e(mpxy). clearly, min{|x|, |y |} = 4. suppose that r and r′ be the rank functions of m and mpxy, respectively. the set γ is an independent set and y ⊂ e(m). by using theorem 3.7 r′(x) + r′(y ) −r′(mpxy) ≤ 4 + r(m) −r′(mpxy) ≤ 3. hence, for n ≥ 4, the partition (x,y ) is a 4-separation of mpxy. the last result says that for every n-connected matroid with n ≥ 4, the maroid mpxy is a k-connected matroid for some k in {2, 3, 4}. lemma 4.5. let x and y be two elements of an n-connected binary matroid m, for n ≥ 3, and let mpxy be a pinching of m. let z be a cocircuit of mpxy such that z 6= γ where γ = e(mpxy) − e(m). then |z| ≥ n. proof. assume, without loss of generality, that z′ is an n-element cocircut of m and p = (x1,x2) is a partition of cxy−{x,y}. let a be a matrix that represents m and apxy constructed from a by definition 3.1. suppose that δxi, for i ∈ {1, 2} is the sum of the corresponding entries of xi in the corresponding row of z′ in row space of a. so δxi ∈{0, 1}. then exactly one of the following holds. (i) x,y /∈ z′ and |z′∩x1| and |z′∩x2| are even. then δx1 = δx2 = 0 and therefore all corresponding entries of γ in the corresponding row in row space of apxy are 0. we conclude that z ′ is a cocircuit of mpxy. (ii) x,y /∈ z′ and |z′∩x1| and |z′∩x2| are odd. then δx1 = δx2 = 1 and therefore the corresponding entries of α1 and α2 in apxy are 0 and α3 and α4 are 1. so z ′ ∪{α3,α4} is a cocircuit of mpxy. note that there is no cocircuit z′ of m not containing x and y such that |z′ ∩ x1| is odd and |z′ ∩x2| is even or vice versa. since in this case |z′ ∩cxy| is odd; a contradiction (iii) x is in z′ but y not, then |z′∩x1| is odd and |z′∩x2| is even or vice versa. then (z′−{x})∪{α1} or (z′ −{x}) ∪{α1,α3,α4} is a cocircuit of mpxy. (iv) y is in z′ but x not, then |z′∩x1| is odd and |z′∩x2| is even or vice versa. then (z′−{y})∪{α3} or (z′ −{y}) ∪{α4} is a cocircuit of mpxy. 254 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 (v) x,y ∈ z′ and |z′∩x1| and |z′∩x2| are even. then δx1 = δx2 = 0 and therefore the corresponding entries of α2 and α4 in apxy are 0 and α1 and α3 are 1. so (z ′−{x,y}) ∪{α1,α3} is a cocircuit of mpxy. (vi) x,y ∈ z′ and |z′∩x1| and |z′∩x2| are odd. then δx1 = δx2 = 1 and therefore the corresponding entries of α2 and α3 in apxy are 0 and α1 and α4 are 1. so (z ′−{x,y}) ∪{α1,α4} is a cocircuit of mpxy. clearly, every cocircuit z in (i)-(vi) has at least n elements and x is not equivalent to y. by lemma 4.3 γ is a cocircuit of mpxy, and if z ′′∆γ is a cocircuit of mpxy, then |z′′| ≥ n. consider the following proposition [4], which is straightforward consequence of the fact that all circuits and all cocircuits of an n-connected matroid have at least n elements where the ground set of such matroid has at least 2(n− 1) elements. proposition 4.6. let (x,y) be an n-separation of an n-connected matroid and suppose that |x| = n. then x is either a coindependent circuit or an independent cocircuit. by using this proposition and the last three lemmas, for a given n-connected matroid with n ≥ 4, we have the following result to determine connectivity of pinching matroid with respect to p . theorem 4.7. let m be an n-connected binary matroid with n ≥ 4 and |e(m)| ≥ 2(n−1). then mpx,y with respect to p = (x1,x2) is a k-connected matroid for k ∈{2, 3, 4}, if min{|x1|, |x2|}≥ k − 2. proof. let m be an n-connected binary matroid and let cxy be an n′-element circuit of m where n′ ≥ n. assume that p = (x1,x2) with |x1| = n′ −k and |x2| = k − 2. let x = (x2 ∪{α2,α4}) and y = e(mpxy)−x and let r and r′ be the rank functions of m and mpxy, respectively. since x is a circuit of mpxy, so r ′(x) = k − 1. clearly, r′(y ) ≤ r′(e(mpxy). now by these facts, we have r′(x) + r′(y ) −r′(mpxy) ≤ k − 1. since min{|x|, |y |} = k, the partition (x,y ) is a k-separation of mpxy. to complete the proof, we shall show that, mpxy has no j-separation for j < k. on the contrary, let (x ′,y ′) be a j-separation of mpxy with |x′| = j. then, by proposition 4.6, the set x′ must be a coindependent circuit or independent cocircuit in mpxy, but such circuit or cocircuit by theorem 3.5 and lemma 4.5 does not exist; a contradiction. definition 4.8. let m be an n-connected binary matroid and x,y ∈ e(m) such that x is not equivalent to y. we say that a collection of some subsets of e(mpxy) is a pinched set if all its members have at least (n + 1) elements. note that if we consider c∗(mpxy)−γ where γ = e(mpxy)−e(m), then the cardinality of its members can be characterized by lemma 4.5. theorem 4.9. let m be an n-connected binary matroid for n ∈ {2, 3} and x,y ∈ e(m) such that x is not equivalent to y and cp has at least one member with k-elements where k ≥ 2n and every nelement circuit of m contains exactly one of x and y. let c∗(mpxy)−γis a pinched set. then mpxy is an (n + 1)-connected matroid. proof. suppose that m is an n-connected binary matroid and x,y ∈ e(m). choose cxy from cp with |cxy| ≥ 2n. partition cxy −{x,y} to two sets x1 and x2 with |x1| = n − 1 and |x2| ≥ k − n − 1. let x = x1 ∪{α1,α2} and y = e(mpxy) −x. then, min{|x|, |y |} = n + 1. therefore, by the similar argument in theorem 4.7, one can easily show that (x,y ) is a minimal (n + 1)-separation for mpxy. 255 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 5. applications in this section, we study the relations between the pinching operation and other binary operations such as splitting, the splitting off operation and element splitting. using these relations, we will extend the notions of subdivision on to edges and special case of vertex identification on two vertices from graphs to binary matroids. moreover, we shall show that eulerian binary matroids are characterized in terms of the 2-pinching, splitting and splitting off operations. the next theorem shows the relationship between m and mpxy by combining the notions of minors, the splitting off and element splitting operation. theorem 5.1. let m be a connected binary matroid. let x,y ∈ e(m) and γ = {u,v,u′,v′} such that γ /∈ e(m). consider the matroid mpxy with δ = { x1 ∪{u,u′},x2 ∪{v,v′} } . then (i) mpxy \ γ = m \{x,y}. (ii) ( (mpxy)uv ) u′v′ ∼= ( (mpxy)u′v′ ) uv ∼= m. (iii) if x1 = ∅, then mpxy/{u,u′} = mpxy/{u}\{u′} = mpxy/{u′}\{u} = m where {u,u′} is a 2-circuit of mpxy. moreover, m p xy is a single parallel extension of m ′ x,y, or m p xy\u = m′x,y and mpxy\u′ = m′x,y. proof. the proofs are straightforward. a binary matroid m with ground set e is eulerian if there exist disjoint circuits c1,c2, ...,cp such that e = c1 ∪c2 ∪ ...∪cp and m is called bipartite matroid if every circuit has even cardinality. for a given connected graph g, the matroid m(g) is eulerian if and only if g is an eulerian [6]. a matroid m is eulerian if and only if mxy is eulerian [3]. the next theorem, says that this property is preserved under 2-pinching operation. to prove this theorem, we consider the following lemma and we use the fact that m is a binary matroid if and only if the symmetric difference of any set of circuits is a disjoint union of circuits [4]. lemma 5.2. let mpxy be a pinching of a eulerian binary matroid m. then e(m) can be partitioned into a collection of disjoint circuits of m such that this partition contains cxy. proof. for some k, let π = (c1,c2, ...,ck) be a partition of e(m) such that cxy /∈ π. choose all circuits of this partition that have a non-empty intersection with cxy. let v be the union of these circuits and let v ′ be the union of other circuits of this partition which are not in v . then, there is a circuit c that is not in π and contains x, y or both such that c ∩v ′ = ∅ and v ∆c is a disjoint union of circuits and contains cxy. now e(m) = v ′ ∪ (v ∆c). clearly, this union is a disjoint union of circuits. theorem 5.3. let m be a binary matroid on a set e and x,y ∈ e. then m is eulerian if and only if mpxy is eulerian. proof. suppose that m is a binary eulerian matroid and x,y ∈ e(m). then, for some k, there are disjoint circuits c1,c2, ...,ck of m such that e(m) = ⋃k i=1 ci. let m p xy be a pinching of m. if x ∈ ci and y ∈ cj where i,j ∈{1, 2, ...,k} and i 6= j, then c′ = (ci−{x})∪{α1,α2} and c′′ = (cj−{y})∪{α3,α4} are disjoint circuits of mpxy. clearly, e(m p xy) = c1∪...∪ci−1∪c′∪ci+1∪...∪cj−1∪c′′∪cj+1∪...∪ck and these circuits are pairwise disjoint. if x,y ∈ ci′, for i′ ∈ {1, 2, ...,k}, then we have two following cases. (i) if cxy = ci′ and p = (x1,x2), then e(mpxy) = c1∪...∪ci′−1∪(x1∪{α1,α3})∪(x2∪{α2,α4})∪ ci′+1 ∪ ...∪ck. clearly, any two members of this union are disjoint. 256 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 (ii) if cxy 6= ci′, then we have two following cases: -if ( (ci′ − {x,y}) ∪ {α1,α2,α3,α4} ) ∈ c3, then e(mpxy) = c1 ∪ ... ∪ ci′−1 ∪ (ci′ − {x,y}) ∪ {α1,α2,α3,α4}) ∪ci′+1 ∪ ...∪ck. clearly, any two members of this union are disjoint. -if ( (ci′ −{x,y}) ∪{α1,α2,α3,α4} ) /∈c3, then by lemma 5.2, there is another partition of e(m) containing cxy and by part (i) of this proof we can see that mpxy is an eulerian matroid. we conclude that mpxy is an eulerian matroid. to prove the converse, by theorem 5.1, m = ((mpxy)α1α2 )α3α4. therefore when m p xy is an eulerian matroid, m is eulerian. corollary 5.4. every uniform matroid un,n+1 can be constructed from u1,2 by a sequence of the 2pinching, splitting and splitting off operations. proof. let m = u1,2 and so there is no partition p. then m∅xy ∼= u1,2 ⊕ u1,2. now let {x′,y′} be {α1,α2} or {α3,α4}. then (m∅xy)x′,y′ ∼= u3,4 and ((u3,4)x′,y′ )x′y′ ∼= u2,3. by repeating this process (first applying the 2-pinching operation with respect to partition p with one empty part, then applying splitting and splitting off operations, respectively), we eventually obtain un,n+1 and then un−1,n , respectively. on combining theorem 5.3 with corollary 5, we immediately obtain the following result. theorem 5.5. a binary matroid is eulerian if, by repeated applications of splitting and 2-pinching operations, it can be transformed to a direct sum of copies of u1,2 and u2,3. now we extend the notions of subdivision of edge and vertex identification from graphs to binary matroids by the pinching operation in the following way. let g be a 2-connected graph and x,y ∈ e(g). after applying the pinching operation on x and y, let {x′,y′} and {x′′,y′′} be the disjoint subsets of members of δ. clearly, if we apply the splitting operation using {x′,x′′} or {y′,y′′}, then the obtained graph is isomorphic to the graph resulting by subdivision of g on x and y. let g be a 2-connected graph and u,v,w,z ∈ e(g), such that t1 and t2 are the common vertices of {u,v} and {w,z}, respectively, and there are no other edges on t1 and t2. suppose that α and α′ are the new elements of the obtained graph after applying the splitting off operation on {u,v} and then {w,z}, respectively. now if we implement the pinching operation on {α,α′}, then the obtained graph is isomorphic to the graph obtained by vertex identification of t1 and t2. definition 5.6. let m be a connected binary matroid and x,y ∈ e(m). consider the matroid mpxy with δ = { x1 ∪{u,u′},x2 ∪{v,v′} } . we call n1 as the subdivision of m on x and y if n1 ∼= (mpxy)u,v ∼= (mpxy)u′,v′. definition 5.7. let m be a connected binary matroid and γ ⊆ e(m) such that γ = {u,v,u′,v′}. suppose that {u,v} and {u′,v′} are cocircuits of m and e(muv) −e(m) = {α} and e(mu′v′ ) −e(m) = {α′}. we call n2 is the identification {u,v} with {u′,v′} in m, if n2 ∼= ((muv)u′v′ )pαα′ ∼= ((mu′v′ )uv) p αα′. corollary 5.8. the matroids n1 and n2, defined as in definitions 5.6 and 5.7, are extensions of the notions of subdivision and vertex identification from graphs to connected binary matroids. references [1] g. azadi, generalized splitting operation for binary matroids and related results, ph.d. thesis, university of pune, 2001. 257 v. ghorbani et al. / j. algebra comb. discrete appl. 7(3) (2019) 247–258 [2] a. frank, edge–connection of graphs, digraphs, and hypergraphs, more sets, graphs and numbers 15 (2006) 93–141. [3] m. m. shikare, k. v. dalvi, s. b. dhotre, splitting off operation for binary matroids and its applications, graph and combinatorics 27 (2011) 871–882. [4] j. oxley, matroid theory, oxford university press, 2nd ed. 2011. [5] t. t. raghunathan, m. m. shikare, b. n. waphare, splitting in a binary matroid, discrete math. 184 (1998) 267–271. [6] d. j. a. welsh, euler and bipartite matroids, journal of combinatorial theory 6(4) (1969) 375–377. [7] d. west, introduction to graph theory, prentice–hall, 2nd ed. 2001. 258 https://doi.org/10.1007/978-3-540-32439-3_6 https://doi.org/10.1007/978-3-540-32439-3_6 https://doi.org/10.1007/s00373-010-1005-y https://doi.org/10.1007/s00373-010-1005-y https://doi.org/10.1016/s0012-365x(97)00202-1 https://doi.org/10.1016/s0012-365x(97)00202-1 https://doi.org/10.1016/s0021-9800(69)80033-5 introduction 2-pinching operation in 2-connected graphs 2-pinching operation for binary matroids connectivity of mpxy applications references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.935938 j. algebra comb. discrete appl. 8(2) • 59–71 received: 30 july 2020 accepted: 20 october 2020 journal of algebra combinatorics discrete structures and applications on unit group of finite semisimple group algebras of non-metabelian groups of order 108 research article gaurav mittal, rajendra. k. sharma abstract: in this paper, we characterize the unit groups of semisimple group algebras fqg of non-metabelian groups of order 108, where fq is a field with q = pk elements for some prime p > 3 and positive integer k. upto isomorphism, there are 45 groups of order 108 but only 4 of them are non-metabelian. we consider all the non-metabelian groups of order 108 and find the wedderburn decomposition of their semisimple group algebras. and as a by-product obtain the unit groups. 2010 msc: 16u60, 20c05 keywords: unit group, finite field, wedderburn decomposition 1. introduction let fq denote a finite field with q = pk elements for odd prime p > 3, g be a finite group and fqg be the group algebra. the study of the unit groups of group algebras is a classical problem and has applications in cryptography [4] as well as in coding theory [5] etc. for the exploration of lie properties of group algebras and isomorphism problems, units are very useful see, e.g. [1]. we refer to [11] for elementary definitions and results about the group algebras and [2, 14] for the abelian group algebras and their units. recall that a group g is metabelian if there is a normal subgroup n of g such that both n and g/n are abelian. the unit groups of the finite semisimple group algebras of metabelian groups have been well studied. in this paper, we are concerned about the unit groups of the group algebras of non-metabelian groups. let us first mention the available literature in this direction. from [13], we know that all the groups up to order 23 are metabelian. the only non-metabelian groups of order 24 are s4 and sl(2, 3) and the unit group of their group algebras have been discussed in [7, 9]. further, from [13], we also gaurav mittal (corresponding author); department of mathematics, indian institute of technology roorkee, roorkee, india (email: gmittal@ma.iitr.ac.in). r. k. sharma; department of mathematics, indian institute of technology delhi, new delhi, india (email: rksharma@maths.iitd.ac.in). 59 https://orcid.org/0000-0001-8292-9646 https://orcid.org/0000-0001-5666-4103 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 deduce that there are non-metabelian groups of order of 48, 54, 60, 72, 108 etc. the unit groups of the group algebras of non-metabelian groups up to order 72 have been discussed in [10, 12]. the unit group of the semisimple group algebra of the non-metabelian group sl(2, 5) has been discussed in [15]. the main motive of this paper is to characterize the unit groups of fqg, where g represents a non-metabelian group of order 108. it can be verified that, upto isomorphism, there are 45 groups of order 108 and only 4 of them are non-metabelian. we deduce the wedderburn decomposition of group algebras of all the 4 non-metabelian groups and then characterize the respective unit groups. the rest of the paper is organized in following manner: we recall all the basic definitions and results to be needed in our work in section 2. our main results on the characterization of the unit groups are presented in third section and the last section includes some discussion. 2. preliminaries let e denote the exponent of g, ζ a primitive eth root of unity. on the lines of [3], we define if = {n | ζ 7→ ζn is an automorphism of f(ζ) over f}, where f is an arbitrary finite field. since, the galois group gal ( f(ζ),f ) is a cyclic group, for any τ ∈ gal ( f(ζ),f ) , there exists a positive integer s which is invertible modulo e such that τ(ζ) = ζs. in other words, if is a subgroup of the multiplicative group z∗e (group of integers which are invertible with respect to multiplication modulo e). for any p-regular element g ∈ g, i.e. an element whose order is not divisible by p, let the sum of all conjugates of g be denoted by γg, and the cyclotomic f-class of γg be denoted by s(γg) = {γgn | n ∈ if}. the cardinality of s(γg) and the number of cyclotomic f-classes will be incorporated later on for the characterization of the unit groups. next, we recall two important results from [3]. first one relates the number of cyclotomic f-classes with the number of simple components of fg/j(fg). here j(fg) denotes the jacobson radical of fg. second one is about the cardinality of any cyclotomic f-class in g. theorem 2.1. the number of simple components of fg/j(fg) and the number of cyclotomic f-classes in g are equal. theorem 2.2. let ζ be defined as above and j be the number of cyclotomic f-classes in g. if ki, 1 ≤ i ≤ j, are the simple components of center of fg/j(fg) and si, 1 ≤ i ≤ j, are the cyclotomic f-classes in g, then |si| = [ki : f] for each i after suitable ordering of the indices if required. to determine the structure of unit group u(fg), we need to determine the wedderburn decomposition of the group algebra fg. in other words, we want to determine the simple components of fg. based on the existing literature, we can always claim that f is one of the simple component of decomposition of fg/j(fg). the simple proof is given here for the completeness. lemma 2.3. let a1 and a2 denote the finite dimensional algebras over f. further, let a2 be semisimple and g be an onto homomorphism between a1 and a2, then we must have a1/j(a1) ∼= a3 + a2, where a3 is some semisimple f-algebra. proof. from [6], we have j(a1) ⊆ ker(g). this means there exists f-algebra homomorphism g1 from a1/j(a1) to a2 which is also onto. in other words, we have g1 : a1/j(a1) 7−→ a2 defined by g1(a + j(a1)) = g(a), a ∈ a1. as a1/j(a1) is semisimple, there exists an ideal i of a1/j(a1) such that a1/j(a1) = ker(g1) ⊕ i. 60 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 our claim is that i ∼= a2. for this to prove, note that any element a ∈ a1/j(a1) can be uniquely written as a = a1 + a2 where a1 ∈ ker(g1),a2 ∈ i. so, define g2 : a1/j(a1) 7→ ker(g1) ⊕a2 by g2(a) = (a1,g1(a2)). since, ker(g1) is a semisimple algebra over f and a2 is an isomorphic f-algebra, claim and the result holds. above lemma concludes that f is one of the simple components of fg provided j(fg) = 0. now, we recall a result which characterize the set if defined in the beginning of this section. for proof, see theorem 2.21 in [8]. theorem 2.4. let f be a finite field with prime power order q. if e is such that gcd(e,q) = 1, ζ is the primitive eth root of unity and |q| is the order of q modulo e, then modulo e, we have if = {1,q,q2, . . . ,q|q|−1}. next result is proposition 3.6.11 from [11] and is useful for the determination of commutative simple components of the group algebra fqg. theorem 2.5. if rg is a semisimple group algebra, then rg ∼= r ( g/g′ ) ⊕ ∆(g,g′), where g′ is the commutator subgroup of g, r ( g/g′ ) is the sum of all commutative simple components of rg, and ∆(g,g′) is the sum of all others. we end this section by recalling a proposition 3.6.7 from [11] which is a generalized version of the last result. theorem 2.6. let rg be a semisimple group algebra and h be a normal subgroup of g. then rg ∼= r ( g/h ) ⊕ ∆(g,h), where ∆(g,h) is a left ideal of rg generated by the set {h− 1 : h ∈ h}. 3. unit group of fqg where g is a non-metabelian group of order 108 the main objective of this section is to characterize the unit group of fqg where g is a nonmetabelian group of order 108. upto isomorphism, there are 4 non-metabelian groups of order 108 namely: (1) g1 = ((c3 ×c3) o c3) o c4. (2) g2 = ((c3 ×c3) o c3) o c4. (3) g3 = ((c3 ×c3) o c3) o (c2 ×c2). (4) g4 = c2 × (((c3 ×c3) o c3) o c2). here g1 and g2 are two non-isomorphic groups formed by the semi-direct product of (c3×c3) oc3 and c4 which will be clear once we discuss the presentation of these groups. we consider each of these 4 groups one by one and discuss the unit groups of their respective group algebras along with the wedderburn decompositions in subsequent subsections. throughout this paper, we use the notation [x,y] = x−1y−1xy. 3.1. the group g1 = ((c3 ×c3) o c3) o c4 group g1 has the following presentation: g1 = 〈x,y,z,w,t | x2y−1, [y,x], [z,x]z−1, [w,x]w−1, [t,x], y2, [z,y], [w,y], [t,y], z3, [w,z]t−1, [t,z], w3, [t,w], t3〉. also g1 has 20 conjugacy classes as shown in the table below. 61 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 r e x y z w t xy xt yz yw yt zw t2 xyt xt2 yzw yt2 z2w xyt2 yz2w s 1 9 1 6 6 1 9 9 6 6 1 6 1 9 9 6 1 6 9 6 o 1 4 2 3 3 3 4 12 6 6 6 3 3 12 12 6 6 3 12 6 where r, s and o denote the representative of conjugacy class, size and order of the representative of the conjugacy class, respectively. from the above discussion, it is clear that exponent of g1 is 12. also g′1 ∼= (c3×c3) oc3 with g1/g′1 ∼= c4. next, we discuss the unit group of the group algebra fqg1 when p > 3. theorem 3.1. the unit group of fqg1, for q = pk, p > 3 where fq is a finite field having q = pk elements is as follows: 1. for any p and k even or pk ≡ 1 mod 12 with k odd, we have u(fqg1) ∼= (f∗q) 4 ⊕gl2(fq)8 ⊕gl3(fq)8. 2. for pk ≡ 5 mod 12 with k odd, we have u(fqg1) ∼= (f∗q) 4 ⊕gl2(fq)8 ⊕gl3(fq2 )4. 3. for pk ≡ 7 mod 12 with k odd, we have u(fqg1) ∼= (f∗q) 2 ⊕f∗q2 ⊕gl2(fq) 8 ⊕gl3(fq)4 ⊕gl3(fq2 )2. 4. for pk ≡ 11 mod 12 with k odd, we have u(fqg1) ∼= (f∗q) 2 ⊕f∗q2 ⊕gl2(fq) 8 ⊕gl3(fq2 )4. proof. since fqg1 is semisimple, using lemma 2.3 we get fqg1 ∼= fq ⊕t−1r=1 mnr (fr), for some t ∈ z. (1) first assume that k is even which means for any prime p > 3, we have pk ≡ 1 mod 3 and pk ≡ 1 mod 4. using chinese remainder theorem, we get pk ≡ 1 mod 12. this means |s(γg)| = 1 for each g ∈ g1 as if = {1}. hence, (1), theorems 2.1 and 2.2 imply that fqg1 ∼= fq ⊕19r=1 mnr (fq). (2) incorporating theorem 2.5 with g′1 ∼= (c3 ×c3) o c3 in (2) to obtain fqg1 ∼= f4q ⊕ 16 r=1 mnr (fq), where nr ≥ 2 with 104 = 16∑ r=1 n2r. (3) above equation gives the only possibility (28, 38) for the values of n′rs where a b means (a,a, · · · ,b times) and therefore, (3) implies that fqg1 ∼= f4q ⊕m2(fq) 8 ⊕m3(fq)8. (4) now we consider that k is odd. we shall discuss this possibility in the following four cases: case 1: pk ≡ 1 mod 3 and pk ≡ 1 mod 4. in this case, wedderburn decomposition is given by (4). case 2. pk ≡ 5 mod 12. in this case, we have s(γt) = {γt,γt2}, s(γxt) = {γxt,γxt2}, s(γyt) = {γyt,γyt2}, s(γxyt) = {γxyt,γxyt2}, 62 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore, (1) and theorems 2.1, 2.2 imply that fqg1 ∼= fq ⊕11r=1 mnr (fq) ⊕ 15 r=12 mnr (fq2 ). since g1/g′1 ∼= c4, we have fqc4 ∼= f4q. this with above and theorem 2.5 yields fqg1 ∼= f4q ⊕ 8 r=1 mnr (fq) ⊕ 12 r=9 mnr (fq2 ), nr ≥ 2 with 104 = 8∑ r=1 n2r + 2 12∑ r=9 n2r. (5) further, consider the normal subgroup h1 = 〈t〉 of g1 having order 3 with k1 = g1/h1 ∼= (c3 × c3) o c4. the quotient group k1 has 12 conjugacy classes as shown in the table below. here elements of k1 are cosets, for instance, x ∈ k1 is xh1 but we keep the same notation. r e x y z w xy yz yw zw yzw z2w yz2w s 1 9 1 2 2 9 2 2 2 2 2 2 o 1 4 2 3 3 4 6 6 3 6 3 6 it can be verified that for all the representatives g of k1, |s(γg)| = 1. therefore, from theorems 2.1 and 2.2, we have fqk1 ∼= fq ⊕11r=1 mtr (fq), tr ∈ z. observe that k1/k′1 ∼= c4. so, theorem 2.5 implies that fqk1 ∼= f4q ⊕ 8 r=1 mtr (fq), with 32 = 8∑ r=1 t2r, tr ≥ 2. this gives us the only choice (28) for values of t′rs and therefore, theorem 2.5 and (5) yields fqg1 ∼= f4q ⊕m2(fq) 8 ⊕4r=1 mnr (fq2 ), nr ≥ 2 with 36 = 4∑ r=1 n2r. above leaves us with the only choice (34) for values of n′rs which means the required wedderburn decomposition is fqg1 ∼= f4q ⊕m2(fq) 8 ⊕m3(fq2 )4. case 3. pk ≡ 7 mod 12. in this case, we have s(γx) = {γx,γxy}, s(γxt) = {γxt,γxyt}, s(γxt2 ) = {γxt2,γxyt2}, and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore (1) and theorems 2.1, 2.2 imply that fqg1 ∼= fq ⊕13r=1 mnr (fq) ⊕ 16 r=14 mnr (fq2 ). since g1/g′1 ∼= c4, we have fqc4 ∼= f2q ⊕fq2. this with above and theorem 2.5 yields fqg1 ∼= f2q ⊕fq2 ⊕ 12 r=1 mnr (fq) ⊕ 14 r=13 mnr (fq2 ), nr ≥ 2 with 104 = 12∑ r=1 n2r + 2 14∑ r=13 n2r. (6) 63 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 again consider the normal subgroup h1 of g1. in this case, it can be verified that |s(γg)| = 1 for all the representatives g of k1 except x and xy for which s(γx) = {γx,γxy}. therefore, employing theorems 2.1 and 2.2 to obtain fqk1 ∼= fq ⊕9r=1 mtr (fq) ⊕mt10 (fq2 ), tr ∈ z. since k1/k′1 ∼= c4, above and theorem 2.5 imply that fqk1 ∼= f2q ⊕fq2 ⊕ 8 r=1 mtr (fq), with 32 = 8∑ r=1 t2r, tr ≥ 2. this gives us the only choice (28) for values of t′rs. hence, theorem 2.6 and (6) imply that fqg1 ∼= f2q ⊕fq2 ⊕m2(fq) 8 ⊕4r=1 mnr (fq) ⊕ 6 r=5 mnr (fq2 ), with 72 = 4∑ r=1 n2r + 2 6∑ r=5 n2r. above leaves us with the only choice (36) for values of n′rs which means the required wedderburn decomposition is fqg1 ∼= f2q ⊕fq2 ⊕m2(fq) 8 ⊕m3(fq)4 ⊕m3(fq2 )2. case 4. pk ≡ 11 mod 12. in this case, we have s(γt) = {γt,γt2}, s(γxt) = {γxt,γxyt2}, s(γyt) = {γyt,γyt2}, s(γxyt) = {γxyt,γxt2}, s(γx) = {γx,γxy}, and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore, (1) and theorems 2.1, 2.2 imply that fqg1 ∼= fq ⊕9r=1 mnr (fq) ⊕ 14 r=10 mnr (fq2 ). since g1/g′1 ∼= c4, we have fqc4 ∼= f2q ⊕fq2. this with above and theorem 2.5 yields fqg1 ∼= f2q ⊕fq2 ⊕ 8 r=1 mnr (fq) ⊕ 12 r=9 mnr (fq2 ), nr ≥ 2 with 104 = 8∑ r=1 n2r + 2 12∑ r=9 n2r. (7) in this case, again we have |s(γg)| = 1 for all representatives g of k1 except x and xy which means the wedderburn decomposition of fqk1 is same as obtained in case 3, i.e. fqk1 ∼= f2q ⊕fq2 ⊕m2(fq) 8. now employ theorem 2.6 and (7) to obtain fqg1 ∼= f2q ⊕fq2 ⊕m2(fq) 8 ⊕4r=1 mnr (fq2 ), with 36 = 4∑ r=1 n2r. this leaves us with the only choice (34) for values of n′rs which means the required wedderburn decomposition is fqg1 ∼= f2q ⊕fq2 ⊕m2(fq) 8 ⊕m3(fq2 )4. 64 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 3.2. the group g2 = ((c3 ×c3) o c3) o c4 group g2 has the following presentation: g2 = 〈x,y,z,w,t | x2y−1, [y,x], [z,x]w−2, [w,x]t−1w−1z−2, [t,x], y2, [z,y]z−1, [w,y]t−1w−1, [t,y], z3, [w,z]t−1, [t,z], w3, [t,w], t3〉. further, g2 has 14 conjugacy classes as shown in the table below. r e x y z w t xy xz xt yw yt t2 xyz xyzw s 1 9 9 12 12 1 9 9 9 9 9 1 9 9 o 1 4 2 3 3 3 4 12 12 6 6 3 12 12 from above discussion, clearly the exponent of g2 is 12. also g′2 ∼= (c3 ×c3) o c3 and g2/g′2 ∼= c4. next, we discuss the unit group of fqg2 when p > 3. theorem 3.2. the unit group of fqg2, for q = pk, p > 3 where fq is a finite field having q = pk elements is as follows: 1. for any p and k even or pk ≡ 1 mod 12 with k odd, we have u(fqg2) ∼= (f∗q) 4 ⊕gl3(fq)8 ⊕gl4(fq)2. 2. for pk ≡ 5 mod 12 with k odd, we have u(fqg2) ∼= (f∗q) 4 ⊕gl4(fq)2 ⊕gl3(fq2 )4. 3. for pk ≡ 7 mod 12 with k odd, we have u(fqg2) ∼= (f∗q) 2 ⊕f∗q2 ⊕gl3(fq) 4 ⊕gl4(fq)2 ⊕gl3(fq2 )2. 4. for pk ≡ 11 mod 12 with k odd, we have u(fqg2) ∼= (f∗q) 2 ⊕f∗q2 ⊕gl4(fq) 2 ⊕gl3(fq2 )4. proof. since fqg2 is semisimple, we have fqg2 ∼= fq ⊕t−1r=1 mnr (fr), for some t ∈ z. (8) first assume that k is even which means for any prime p > 3, pk ≡ 1 mod 12. this means |s(γg)| = 1 for each g ∈ g2. hence, (8), theorems 2.1 and 2.2 imply that fqg2 ∼= fq ⊕13r=1 mnr (fq). using theorem 2.5 with g′2 ∼= (c3 ×c3) o c3 to obtain fqg2 ∼= f4q ⊕ 10 r=1 mnr (fq), where nr ≥ 2 with 104 = 10∑ r=1 n2r. (9) above equation gives us four possibilities (28, 62), (25, 32, 4, 52), (24, 34, 4, 6) and (38, 42) for the values of n′rs. further, consider the normal subgroup h2 = 〈t〉 of g2 having order 3 with k2 = g2/h2 ∼= (c3 ×c3) o c4. it can be verified that k2 has 6 conjugacy classes as shown in the table below. r e x y z w xy s 1 9 9 4 4 9 o 1 4 2 3 3 4 65 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 also, for all the representatives g of k2, |s(γg)| = 1 which means by theorems 2.1 and 2.2, we have fqk2 ∼= fq ⊕5r=1 mtr (fq), tr ∈ z. observe that k2/k′2 ∼= c4. this with above and theorem 2.4 imply that fqk2 ∼= f4q ⊕ 2 r=1 mtr (fq), with 32 = 2∑ r=1 t2r, tr ≥ 2. this gives us the only choice (42) for values of t′rs. therefore, theorem 2.6 and (9) imply that (3 8, 42) is the correct choice for values of n′rs and therefore, we have fqg2 ∼= f4q ⊕m3(fq) 8 ⊕m4(fq)2. (10) now we consider that k is odd. we shall discuss this possibility in the following four cases: case 1: pk ≡ 1 mod 12. in this case, wedderburn decomposition is given by (10). case 2. pk ≡ 5 mod 12. in this case, we have s(γt) = {γt,γt2}, s(γxz) = {γxz,γxt}, s(γyw) = {γyw,γyt}, s(γxyz) = {γxyz,γxyzw}, and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore, (8) and theorems 2.1, 2.2 imply that fqg2 ∼= fq ⊕5r=1 mnr (fq) ⊕ 9 r=6 mnr (fq2 ). since g2/g′2 ∼= c4, we have fqc4 ∼= f4q. this with above and theorem 2.5 yields fqg2 ∼= f4q ⊕ 2 r=1 mnr (fq) ⊕ 6 r=3 mnr (fq2 ), nr ≥ 2 with 104 = 2∑ r=1 n2r + 2 6∑ r=3 n2r. (11) further, again consider the normal subgroup h2 = 〈t〉 of g2. it can be verified that for all the representatives g of k2, |s(γg)| = 1 which means (as earlier) fqk2 ∼= f4q ⊕m4(fq) 2. this with theorem 2.6 and (11) imply that fqg2 ∼= f4q ⊕m4(fq) 2 ⊕4r=1 mnr (fq2 ), nr ≥ 2 with 36 = 4∑ r=1 n2r. above leaves us with the only choice (34) for values of n′rs which means the required wedderburn decomposition is fqg2 ∼= f4q ⊕m4(fq) 2 ⊕m3(fq2 )4. case 3. pk ≡ 7 mod 12. in this case, we have s(γx) = {γx,γxy}, s(γxz) = {γxz,γxyzw}, s(γxt) = {γxt,γxyz}, and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore, (8) and theorems 2.1, 2.2 imply that fqg2 ∼= fq ⊕7r=1 mnr (fq) ⊕ 10 r=8 mnr (fq2 ). since g2/g′2 ∼= c4, we have fqc4 ∼= f2q ⊕fq2. this with theorem 2.5 yields fqg2 ∼= f2q ⊕fq2 ⊕ 6 r=1 mnr (fq) ⊕ 8 r=7 mnr (fq2 ), nr ≥ 2 with 104 = 6∑ r=1 n2r + 2 8∑ r=7 n2r. (12) 66 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 further, it can be verified that |s(γg)| = 1 for all the representatives g of k2 except x and xy. for these, we have s(γx) = {γx,γxy} which means by theorems 2.1 and 2.2, we have fqk2 ∼= fq ⊕3r=1 mtr (fq) ⊕mt4 (fq2 ), tr ∈ z. incorporating k2/k′2 ∼= c4 with theorem 2.5 to obtain fqk2 ∼= f2q ⊕fq2 ⊕ 2 r=1 mtr (fq), with 32 = 2∑ r=1 t2r, tr ≥ 2. this gives us only choice (42) for values of t′rs. therefore, theorem 2.6 and (12) yields fqg2 ∼= f2q ⊕fq2 ⊕m4(fq) 2 ⊕4r=1 mnr (fq) ⊕ 6 r=5 mnr (fq2 ), with 72 = 4∑ r=1 n2r + 2 6∑ r=5 n2r. above leaves us with the only choice (36) for values of n′r which means the required wedderburn decomposition is fqg2 ∼= f2q ⊕fq2 ⊕m4(fq) 2 ⊕m3(fq)4 ⊕m3(fq2 )2. case 4. pk ≡ 11 mod 12. in this case, we have s(γt) = {γt,γt2}, s(γx) = {γx,γxy}, s(γxz) = {γxz,γxyz}, s(γxt) = {γxt,γxyzw}, s(γyw) = {γyw,γyt}, and s(γg) = {γg} for the remaining representatives g of conjugacy classes. therefore, (8) and theorems 2.1, 2.2 imply that fqg2 ∼= fq ⊕3r=1 mnr (fq) ⊕ 8 r=4 mnr (fq2 ). since g2/g′2 ∼= c4, we have fqc4 ∼= f2q ⊕fq2. this with above and theorem 2.5 yields fqg2 ∼= f2q ⊕fq2 ⊕ 2 r=1 mnr (fq) ⊕ 6 r=3 mnr (fq2 ), nr ≥ 2 with 104 = 2∑ r=1 n2r + 2 6∑ r=3 n2r. (13) further, it can be verified that |s(γg)| = 1 for all the representatives g of k2 except x and xy which means (as in case 3), fqk2 ∼= f2q ⊕fq2 ⊕m4(fq) 2. now employ theorem 2.6 and (13) to obtain fqg2 ∼= f2q ⊕fq2 ⊕m4(fq) 2 ⊕4r=1 mnr (fq2 ), with 36 = 4∑ r=1 n2r. above leaves us with the only choice (34) for values of n′r which means the required wedderburn decomposition is fqg2 ∼= f2q ⊕fq2 ⊕m4(fq) 2 ⊕m3(fq2 )4. 67 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 3.3. the group g3 = ((c3 ×c3) o c3) o (c2 ×c2) group g3 has the following presentation: g3 = 〈x,y,z,w,t | x2, [y,x], [z,x], [w,x]w−1, [t,x]t−1, y2, [z,y]z−1, [w,y], [t,y]t−1, z3, [w,z]t−1, [t,z], w3, [t,w], t3〉. further, g3 has 11 conjugacy classes as shown in the table below. rep 1 x y z w t xy xz yw zw xyt size of class 1 9 9 6 6 2 9 18 18 12 18 order of rep 1 2 2 3 3 3 2 6 6 3 6 from above discussion, clearly the exponent of g3 is 6. also g′3 ∼= (c3×c3)oc3 with g3/g′3 ∼= c2×c2. next, we discuss the unit group of fqg3 when p > 3. theorem 3.3. the unit group of fqg3, for q = pk, p > 3 where fq is a finite field having q = pk elements is as follows: u(fqg3) ∼= (f∗q) 4 ⊕gl2(fq)4 ⊕gl4(fq) ⊕gl6(fq)2. proof. since fqg3 is semisimple, we have fqg3 ∼= fq ⊕t−1r=1 mnr (fr), for some t ∈ z. (14) first assume that k is even which means for any prime p > 3, pk ≡ 1 mod 6. this means |s(γg)| = 1 for each g ∈ g3. hence, (14), theorems 2.1 and 2.2 imply that fqg3 ∼= fq ⊕10r=1 mnr (fq). incorporating theorem 2.5 with g′3 ∼= (c3 ×c3) o c3 in above to obtain fqg3 ∼= f4q ⊕ 7 r=1 mnr (fq), where nr ≥ 2 with 104 = 7∑ r=1 n2r. (15) above equation gives us four possibilities (24, 4, 62), (23, 32, 5, 7), (2, 32, 42, 52) and (34, 42, 6) for the values of n′rs. further, consider the normal subgroup h3 = 〈t〉 of g3 having order 3 with k3 = g3/h3 ∼= s3 ×s3. it can be verified that k3 has 9 conjugacy classes as shown in the table below. r e x y z w xy xz yw zw s 1 3 3 2 2 9 6 6 4 o 1 2 2 3 3 2 6 6 3 further, for all the representatives g of k3, |s(γg)| = 1 which means by theorems 2.1 and 2.2, we have fqk3 ∼= fq ⊕8r=1 mtr (fq), tr ∈ z. observe that k3/k′3 ∼= c2 ×c2. so, above and theorem 2.5 imply that fqk3 ∼= f4q ⊕ 5 r=1 mtr (fq), with 32 = 5∑ r=1 t2r, tr ≥ 2. 68 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 this gives us the only choice (24, 4) for values of t′rs. therefore, from theorem 2.6 and (15), we conclude that (24, 4, 62) is the correct choice for n′rs which means fqg3 ∼= f4q ⊕m2(fq) 4 ⊕m4(fq) ⊕m6(fq)2. (16) now we consider that k is odd. we shall discuss this possibility in the following two cases: case 1: pk ≡ 1 mod 6. in this case, wedderburn decomposition is given by (16). case 2. pk ≡ 5 mod 6. in this case, we have s(γg) = {γg} for all the representatives g of conjugacy classes. therefore, wedderburn decomposition is again given by (16). 3.4. the group g4 = c2 × (((c3 ×c3) o c3) o c2 group g4 has the following presentation: g4 = 〈x,y,z,w,t | x2, [y,x], [z,x]z−1, [w,x]w−1, [t,x], y2, [z,y], [w,y], [t,y]t−1, z3, [w,z]t−1, [t,z], w3, [t,w], t3〉. further, g4 has 20 conjugacy classes as shown in the table below. r e x y z w t xy xt yz yw yt zw t2 xyt xt2 yzw yt2 z2w xyt2 yz2w s 1 9 1 6 6 1 9 9 6 6 1 6 1 9 9 6 1 6 9 6 o 1 2 2 3 3 3 2 6 6 6 6 3 3 6 6 6 6 3 6 6 from above discussion, clearly the exponent of g4 is 6. also g′4 ∼= (c3×c3)oc3 with g4/g′4 ∼= c2×c2. next, we discuss the unit group of fqg4 when p > 3. theorem 3.4. the unit group of fqg4, for q = pk, p > 3 where fq is a finite field having q = pk elements is as follows: 1. for any p and k even or pk ≡ 1 mod 6 with k odd, we have u(fqg4) ∼= (f∗q) 4 ⊕gl2(fq)8 ⊕gl3(fq)8. 2. for pk ≡ 5 mod 6 with k odd, we have u(fqg4) ∼= (f∗q) 4 ⊕gl2(fq)8 ⊕gl3(fq2 )4. proof. since fqg4 is semisimple, we have fqg4 ∼= fq ⊕t−1r=1 mnr (fr), for some t ∈ z. (17) first assume that k is even which means for any prime p > 3, pk ≡ 1 mod 6. this means |s(γg)| = 1 for each g ∈ g4. hence, (17), theorems 2.1 and 2.2 imply that fqg4 ∼= fq ⊕19r=1 mnr (fq). using theorem 2.5 with g′4 ∼= (c3 ×c3) o c3 in above to obtain fqg4 ∼= f4q ⊕ 16 r=1 mnr (fq), where nr ≥ 2 with 104 = 16∑ r=1 n2r. (18) above equation gives us the only possibility (28, 38) for values of n′rs which means the required wedderburn decomposition is fqg4 ∼= f4q ⊕m2(fq) 8 ⊕m3(fq)8. (19) 69 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 now we consider that k is odd. we shall discuss this possibility in the following two cases: case 1: pk ≡ 1 mod 6. in this case, wedderburn decomposition is given by (19). case 2. pk ≡ 5 mod 6. in this case, we have s(γt) = {γt,γt2}, s(γxt) = {γxt,γxt2}, s(γyt) = {γyt,γyt2}, s(γxyt) = {γxyt, γxyt2}, and s(γg) = {γg} for all the remaining representatives g of conjugacy classes. hence, (17), theorems 2.1 and 2.2 imply that fqg4 ∼= fq ⊕11r=1 mnr (fq) ⊕ 15 r=12 mnr (fq2 ). above with theorem 2.5 yields fqg4 ∼= f4q ⊕ 8 r=1 mnr (fq) ⊕ 12 r=9 mnr (fq2 ), where nr ≥ 2 with 104 = 8∑ r=1 n2r + 2 12∑ r=9 n2r. (20) further, consider the normal subgroup h4 = 〈t〉 of g4 having order 3 with k4 = g4/h4 ∼= c2 × ((c3 × c3) o c2). it can be verified that k4 has 12 conjugacy classes as shown in the table below. r e x y z w xy yz yw zw yzw z2w yz2w s 1 9 1 2 2 9 2 2 2 2 2 2 o 1 2 2 3 3 2 6 6 3 6 3 6 it can be seen that for all the representatives g of k4, |s(γg)| = 1 which means by theorems 2.1 and 2.2, we have fqk4 ∼= fq ⊕11r=1 mtr (fq), tr ∈ z. observe that k4/k′4 ∼= c2 ×c2. this with above and theorem 2.5 imply that fqk4 ∼= f4q ⊕ 8 r=1 mtr (fq), with 32 = 8∑ r=1 t2r, tr ≥ 2. this gives us the only choice (28) for values of t′rs. therefore, theorem 2.6 and (20) yields fqg4 ∼= f4q ⊕m2(fq) 8 ⊕4r=1 mnr (fq2 ), nr ≥ 2 with 36 = 4∑ r=1 n2r. above leaves us with the only choice (34) for values of n′rs which means the required wedderburn decomposition is fqg4 ∼= f4q ⊕m2(fq) 8 ⊕m3(fq2 )4. 4. discussion we have characterized the unit groups of semisimple group algebras of 4 non-metabelian groups having order 108 and the results are verified using gap. clearly, the complexity in the calculation of wedderburn decomposition upsurges with the increase in order of the group and we need to look into the wedderburn decompositions of the quotient groups. the technique used for obtaining the wedderburn decomposition works well provided the group has non-trivial normal subgroups of small order. 70 g. mittal, r. k. sharma / j. algebra comb. discrete appl. 8(2) (2021) 59–71 references [1] a. bovdi, j. kurdics, lie properties of the group algebra and the nilpotency class of the group of units, j. algebra 212 (1999) 28–64. 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[15] r. k. sharma, g. mittal, on the unit group of a semisimple group algebra fqsl(2,z5), math bohemica (2021). 71 https://doi.org/10.14232/actasm-013-510-1 https://doi.org/10.14232/actasm-013-510-1 https://doi.org/10.1080/00927870802103503 https://doi.org/10.1504/ijicot.2009.024047 https://doi.org/10.1504/ijicot.2009.024047 https://doi.org/10.1007/s10474-007-6169-4 https://doi.org/10.1007/s10474-007-6169-4 https://doi.org/10.1017/cbo9781139172769 https://doi.org/10.1017/cbo9781139172769 https://doi.org/10.13069/jacodesmath.83854 https://doi.org/10.13069/jacodesmath.83854 https://doi.org/10.14232/actasm-014-311-2 https://doi.org/10.14232/actasm-014-311-2 https://doi.org/10.21136/mb.2021.0116-19 https://doi.org/10.21136/mb.2021.0116-19 https://doi.org/10.1002/mana.19800950102 https://doi.org/10.21136/mb.2021.0104-20 https://doi.org/10.21136/mb.2021.0104-20 introduction preliminaries unit group of fqg where g is a non-metabelian group of order 108 discussion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.935980 j. algebra comb. discrete appl. 8(2) • 107–118 received: 12 july 2020 accepted: 5 january 2021 journal of algebra combinatorics discrete structures and applications general degree distance of graphs research article tomáš vetrík abstract: we generalize several topological indices and introduce the general degree distance of a connected graph g. for a, b ∈ r, the general degree distance dda,b(g) = ∑ v∈v (g)[degg(v)] asbg(v), where v (g) is the vertex set of g, degg(v) is the degree of a vertex v, sbg(v) = ∑ w∈v (g)\{v}[dg(v, w)] b and dg(v, w) is the distance between v and w in g. we present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices. 2010 msc: 05c35, 05c12, 92e10 keywords: degree distance, chromatic number, vertex connectivity 1. introduction we denote the vertex set and edge set of a graph g by v (g) and e(g), respectively the number of vertices of g is called the order. for v ∈ v (g), the degree of v, degg(v), is the number of vertices adjacent to v. the distance between two vertices v and w in g, denoted by dg(v,w), is the number of edges in a shortest path between v and w. we denote the complete graph and the star of order n by kn and sn, respectively. topological indices are molecular descriptors which have been studied due to their extensive applications. these graph invariants play an important role in engineering, materials science, pharmaceutical sciences and especially in chemistry, since they can be correlated with many chemical and physical properties of molecules. graph theory can be used to characterize these chemical structures. one of the most well-known distance-based topological indices is the degree distance. the degree distance of a connected graph g, dd(g) = ∑ {v,w}⊆v (g) (degg(v) + degg(w))dg(v,w), tomáš vetrík; department of mathematics and applied mathematics, university of the free state, bloemfontein, south africa (email: vetrikt@ufs.ac.za). 107 https://orcid.org/0000-0002-0387-7276 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 was introduced independently by dobrynin and kochetova [5] and gutman [6]. bounds on the degree distance for graphs with given vertex connectivity were obtained in [2], bounds for graphs with given edge connectivity in [1], bounds for graphs of minimum degree in [13], bounds for cacti in [20] and bounds for bicyclic graphs in [3]. the degree distance for unicyclic graphs with prescribed matching number was investigated in [10] and graph products were studied in [19] and [16]. relations between the degree distance and the eccentric distance sum were investigated in [9] and relations between the degree distance and the gutman index in [4]. the reciprocal degree distance rdd(g) = ∑ {v,w}⊆v (g) degg(v) + degg(w) dg(v,w) of a connected graph g has been widely studied too. bounds on the reciprocal degree distance for graphs with cut edges or cut vertices were given in [12], bounds for bipartite graphs and outerplanar graphs in [11]. the reciprocal degree distance of graph products was studied in [15] and the steiner reciprocal degree distance in [17]. the generalized degree distance was first presented in [8] and studied for example in [7] and [14]. for a,b ∈ r, we introduce the general degree distance of a connected graph g as dda,b(g) = ∑ v∈v (g) ( [degg(v)] a ∑ w∈v (g)\{v} [dg(v,w)] b ) = ∑ {v,w}⊆v (g) ([degg(v)] a + [degg(w)] a)[dg(v,w)] b. let sbg(v) = ∑ w∈v (g)\{v}[dg(v,w)] b. then we can write dda,b(g) = ∑ v∈v (g) [degg(v)] asbg(v). if a = 1, then dd1,b(g) is the generalized degree distance. if a = 1 and b = 1, we get the classical degree distance. if a = 1 and b = −1, we get the reciprocal degree distance. if a = 0 and b = 1, then dd0,1(g) = 2w(g), where w(g) is the wiener index. if a = 0 and b = −1, then dd0,−1(g) = 2h(g), where h(g) is the harary index. we present several bounds on the general degree distance of graphs. 2. preliminary results lemmas 2.1 and 2.2 are used in the proofs of some main results. note that (a,b) 6= (0,0) means that not both a and b are 0. lemma 2.1. let g be any connected graph such that u1 and u2 are non-adjacent vertices in g. then for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0), dda,b(g + u1u2) < dda,b(g). proof. let g′ be the graph g + u1u2. then for any two vertices v,w ∈ v (g), we get dg′(v,w) ≤ dg(v,w) and [dg′(v,w)]b ≤ [dg(v,w)]b, where b ≥ 0. therefore, sbg′(v) ≤ s b g(v) for each v ∈ v (g), where b ≥ 0. for v ∈ v (g) \ {u1,u2}, we have degg′(v) = degg(v), thus [degg′(v)]a = [degg(v)]a and [degg′(v)] asbg′(v) ≤ [degg(v)] asbg(v) for a ≤ 0 and b ≥ 0. now we consider the vertices u1 and u2. since 1 = dg′(u1,u2) < dg(u1,u2), we obtain [dg′(u1,u2)] b < [dg(u1,u2)] b for b > 0. thus sbg′(ui) < s b g(ui) for i = 1,2. note that if b = 0, then sbg′(ui) = s b g(ui). 108 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 for the degrees, we have degg′(ui) = degg(ui) + 1. for a < 0, we obtain [degg′(ui)]a < [degg(ui)]a, and for a = 0, we have [degg′(ui)]a = [degg(ui)]a = 1. it follows that [degg′(ui)]a sbg′(ui) < [degg(ui)] asbg(ui) for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0). thus dda,b(g ′) = 2∑ i=1 [degg′(ui)] asbg′(ui) + ∑ v∈v (g′)\{u1,u2} [degg′(v)] asbg′(v) < 2∑ i=1 [degg(ui)] asbg(ui) + ∑ v∈v (g)\{u1,u2} [degg(v)] asbg(v) = dda,b(g ′). lemma 2.2. let g be any connected graph such that u1 and u2 are non-adjacent vertices in g. then for a ≥ 0 and b ≤ 0, where (a,b) 6= (0,0), dda,b(g + u1u2) > dda,b(g). proof. let g′ be the graph g + u1u2. then for any two vertices v,w ∈ v (g), we get dg′(v,w) ≤ dg(v,w) and [dg′(v,w)]b ≥ [dg(v,w)]b, where b ≤ 0. therefore, sbg′(v) ≥ s b g(v) for each v ∈ v (g), where b ≤ 0. for v ∈ v (g) \ {u1,u2}, we have degg′(v) = degg(v), thus [degg′(v)]a = [degg(v)]a and [degg′(v)] asbg′(v) ≥ [degg(v)] asbg(v) for a ≥ 0 and b ≤ 0. now we consider the vertices u1 and u2. since 1 = dg′(u1,u2) < dg(u1,u2), we obtain 1 = [dg′(u1,u2)] b > [dg(u1,u2)] b > 0 for b < 0. thus sbg′(ui) > s b g(ui) for i = 1,2. note that if b = 0, then sbg′(ui) = s b g(ui). for the degrees, we have degg′(ui) = degg(ui) + 1. for a > 0, we obtain [degg′(ui)]a > [degg(ui)]a, and for a = 0, we have [degg′(ui)]a = [degg(ui)]a = 1. it follows that [degg′(ui)]a sbg′(ui) > [degg(ui)] asbg(ui) for a ≥ 0 and b ≤ 0, where (a,b) 6= (0,0). thus dda,b(g ′) = 2∑ i=1 [degg′(ui)] asbg′(ui) + ∑ v∈v (g′)\{u1,u2} [degg′(v)] asbg′(v) > 2∑ i=1 [degg(ui)] asbg(ui) + ∑ v∈v (g)\{u1,u2} [degg(v)] asbg(v) = dda,b(g ′). the proof is complete. lemma 2.3 was presented in [18]. we use it in the next section to compare the dda,b indices of some graphs. lemma 2.3. let 1 ≤ x < y and c > 0. for a > 1 and a < 0, we have (x + c)a −xa < (y + c)a −ya. 3. main results from lemma 2.1, we know that among graphs of order n, the complete graph kn is the graph having the smallest general degree distance for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0). similarly, by lemma 2.2, among graphs of order n, kn is the graph having the largest general degree distance for a ≥ 0 and b ≤ 0, where (a,b) 6= (0,0). 109 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 for an integer k ≥ 2, a k-partite graph is a graph such that we can divide its vertices into k disjoint sets, where any vertices in the same set are non-adjacent. a 2-partite graph is called a bipartite graph. the complete k-partite graph with partite sets of orders n1,n2, . . . ,nk is denoted by kn1,n2,...,nk. any two vertices which are not in the same partite set of kn1,n2,...,nk are adjacent. if ni and nj differ by at most 1 for every i,j, where 1 ≤ i < j ≤ k, then the graph kn1,n2,...,nk of order n is called the turán graph and it is denoted by t(n,k). we show that t(n,k) has the smallest dda,b index among k-partite graphs with n vertices. theorem 3.1. let g be any k-partite graph of order n, where 2 ≤ k ≤ n. then for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0), we have dda,b(g) ≥ dda,b(t(n,k)) with equality only if g is the turán graph t(n,k). proof. let g′ be a graph having the smallest dda,b index among k-partite graphs of order n. by lemma 2.1, any two vertices in different partite sets must be adjacent. thus g′ is kn1,n2,...,nk for some positive integers n1,n2, . . . ,nk. we prove by contradiction that ni and nj differ by at most 1 for every i,j, where 1 ≤ i < j ≤ k. assume that ni and nj differ by at least 2 for some i,j, where 1 ≤ i < j ≤ k. without loss of generality, assume that n1 ≥ n2 + 2. we compare the dda,b indices of g′ = kn1,n2,...,nk and g ′′ = kn1−1,n2+1,...,nk. for every vertex v from the first partite set and v′ from the second partite set of kn1,n2,...,nk, we have degg′(v) = n−n1, sbg′(v) = 1 b(n−n1) + 2b(n1 −1) and degg′(v ′) = n−n2, sbg′(v ′) = 1b(n−n2) + 2b(n2 −1). for every vertex w from the first partite set and w′ from the second partite set of kn1−1,n2+1,...,nk, we have degg′′(w) = n− (n1 −1), sbg′′(w) = 1 b(n− (n1 −1)) + 2b(n1 −2) and degg′′(w ′) = n− (n2 + 1), sbg′′(w ′) = 1b(n− (n2 + 1)) + 2bn2. for any other vertex z, we have degg′(z) = degg′′(z) and sbg′(z) = s b g′′(z). for b > 0, sbg′(v)−s b g′′(w) = 2 b −1 > 0, sbg′′(w)−s b g′′(w ′) = 2b(n1 −n2 −2)−n1 + n2 + 2 = (2b −1)(n1 −n2 −2) ≥ 0, sbg′′(w ′)−sbg′(v ′) = 2b −1 > 0, therefore 0 < sbg′(v ′) < sbg′′(w ′) ≤ sbg′′(w) < s b g′(v). for b = 0, we obtain sbg′(v ′) = sbg′′(w ′) = sbg′′(w) = s b g′(v) = n−1. since n1 −1 ≥ n2 + 1, we have 0 < (n−n2)a < (n−n2 −1)a ≤ (n−n1 + 1)a < (n−n1)a 110 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 for a < 0. obviously, for a = 0, we get (n−n2)a = (n−n2 −1)a = (n−n1 + 1)a = (n−n1)a = 1. thus, for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0), we have dda,b(g ′)−dda,b(g′′) = n1(n−n1)asbg′(v) + n2(n−n2) asbg′(v ′) − (n1 −1)(n−n1 + 1)asbg′′(w)− (n2 + 1)(n−n2 −1) asbg′′(w ′) = (n1 −1)(n−n1)asbg′(v) + (n2 + 1)(n−n2) asbg′(v ′) + (n−n1)asbg′(v)− (n−n2) asbg′(v ′) − (n1 −1)(n−n1 + 1)asbg′′(w)− (n2 + 1)(n−n2 −1) asbg′′(w ′) > (n1 −1)(n−n1)asbg′(v) + (n2 + 1)(n−n2) asbg′(v ′) − (n1 −1)(n−n1 + 1)asbg′′(w)− (n2 + 1)(n−n2 −1) asbg′′(w ′) = (n1 −1)(n−n1)asbg′′(w) + (n2 + 1)(n−n2) asbg′′(w ′) + (n1 −1)(n−n1)a[sbg′(v)−s b g′′(w)] − (n2 + 1)(n−n2)a[sbg′′(w ′)−sbg′(v ′)] − (n1 −1)(n−n1 + 1)asbg′′(w)− (n2 + 1)(n−n2 −1) asbg′′(w ′) ≥ (n1 −1)(n−n1)asbg′′(w) + (n2 + 1)(n−n2) asbg′′(w ′) − (n1 −1)(n−n1 + 1)asbg′′(w)− (n2 + 1)(n−n2 −1) asbg′′(w ′) = (n1 −1)sbg′′(w)[(n−n1) a − (n−n1 + 1)a] + (n2 + 1)s b g′′(w ′)[(n−n2)a − (n−n2 −1)a] ≥ (n2 + 1)sbg′′(w ′)[(n−n1)a − (n−n1 + 1)a] + (n2 + 1)s b g′′(w ′)[(n−n2)a − (n−n2 −1)a] ≥ 0, since for a = 0, we obtain (n−n1)a−(n−n1 + 1)a = 0 and (n−n2)a−(n−n2 −1)a = 0, and for a < 0, by lemma 2.3, (n−n2)a − (n−n2 −1)a > (n−n1 + 1)a − (n−n1)a, thus (n−n2)a − (n−n2 −1)a + (n−n1)a − (n−n1 + 1)a > 0. hence dda,b(g′)−dda,b(g′′) > 0 and dda,b(g′) > dda,b(g′′). so g′ is not a graph with the smallest dda,b index and we have a contradiction. therefore, ni and nj differ by at most 1 which means that g′ is the turán graph t(n,k). we can use k = 2 in theorem 3.1 to get the following corollary for bipartite graphs. corollary 3.2. let g be any bipartite graph of order n, where n ≥ 2. then for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0), we have dda,b(g) ≥ dda,b(kdn 2 e,bn 2 c) with equality only if g is kdn 2 e,bn 2 c. the chromatic number of a graph g is the minimum number of colors needed to color the vertices of g so that no two adjacent vertices have the same color. let us present a bound on the general degree distance for graphs with given chromatic number. theorem 3.3. let g be any connected graph of order n and chromatic number χ, where 2 ≤ χ ≤ n. then for a ≤ 0 and b ≥ 0, where (a,b) 6= (0,0), we have dda,b(g) ≥ dda,b(t(n,χ)) with equality only if g is the turán graph t(n,χ). 111 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 proof. let g′ be any graph having the smallest dda,b index in terms of order n and chromatic number χ. there is no edge between the vertices in the same color class, therefore g′ must be a χ-partite graph. then, by theorem 3.1, g′ is t(n,χ). the union h = h1 ∪ h2 and join f = h1 + h2 of the graphs h1 and h2 have the vertex sets v (h) = v (f) = v (h1) ∪ v (h2). the edge set e(h) = e(h1) ∪ e(h2). the set e(f) contains the edges in e(h) and the edges connecting each vertex in v (h1) and each vertex in v (h2). we show that (kn−κ−1∪k1)+kκ has the extremal dda,b(g) index for graphs of given vertex connectivity, where a ≥ 1 and b ≤ 0. the vertex connectivity of g is the minimum number of vertices whose removal disconnects g. theorem 3.4. let g be any connected graph of order n and vertex connectivity κ, where 1 ≤ κ ≤ n−2. then for a ≥ 1 and b ≤ 0, dda,b(g) ≤ dda,b((kn−κ−1 ∪k1) + kκ) with equality only if g is (kn−κ−1 ∪k1) + kκ. proof. let g′ be any graph with the largest dda,b index with respect to order n and vertex connectivity κ. so there is a set a ⊂ v (g′) of cardinality κ, such that g′−a is disconnected (where g′−a is obtained from g′ be the removal of the vertices in a and the removal of each edge of g′ incident with a vertex in a). we can divide the vertices in v (g′) \a into the sets a1 and a2, where no vertex in a1 is adjacent to a vertex in a2. by lemma 2.2, there is an edge connecting each pair of vertices in a1, each pair of vertices in a2 and the degree of each vertex in a is n−1 in g′. let |a1| = n1 and |a2| = n2. without loss of generality, assume that n1 ≥ n2 ≥ 1. we get n−κ = n1 + n2 and g′ is (kn1 ∪kn2) + kκ. we prove that n2 = 1. assume to the contrary that n2 ≥ 2 (where n1 ≥ n2). we compare the dda,b indices of g′ = (kn1 ∪kn2) + kκ and g′′ = (kn1+1 ∪kn2−1) + kκ. for each z ∈ a, we get degg′(z) = degg′′(z) = n − 1 and sbg′(z) = s b g′′(z) = n − 1. for each v ∈ v (kn1), degg′(v) = κ + n1 −1 and sbg′(v) = 1 b(κ + n1 −1) + 2bn2. for each v′ ∈ v (kn2), we obtain degg′(v ′) = κ + n2 −1 and sbg′(v ′) = 1b(κ + n2 −1) + 2bn1. for each w ∈ v (kn1+1), we have degg′′(w) = κ + n1 and sbg′′(w) = 1 b(κ + n1) + 2 b(n2 −1). for each w′ ∈ v (kn2−1), we get degg′′(w ′) = κ + n2 −2, and sbg′′(w ′) = 1b(κ + n2 −2) + 2b(n1 + 1). for b ≤ 0, sbg′′(w ′)−sbg′(v ′) = 2b −1 ≤ 0, sbg′(v ′)−sbg′(v) = 2 b(n1 −n2)−n1 + n2 = (2b −1)(n1 −n2) ≤ 0, sbg′(v)−s b g′′(w) = 2 b −1 ≤ 0, therefore 0 < sbg′′(w ′) ≤ sbg′(v ′) ≤ sbg′(v) ≤ s b g′′(w). 112 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 note that 0 < (κ + n2 −2)a < (κ + n2 −1)a ≤ (κ + n1 −1)a < (κ + n1)a. for a ≥ 1. thus for a ≥ 1 and b ≤ 0, we have dda,b(g ′)−dda,b(g′′) = n1(κ + n1 −1)asbg′(v) + n2(κ + n2 −1) asbg′(v ′) − (n1 + 1)(κ + n1)asbg′′(w)− (n2 −1)(κ + n2 −2) asbg′′(w ′) = n1(κ + n1 −1)asbg′(v) + n2(κ + n2 −1) asbg′(v ′) − n1(κ + n1)asbg′′(w)−n2(κ + n2 −2) asbg′′(w ′) − (κ + n1)asbg′′(w) + (κ + n2 −2) asbg′′(w ′) < n1(κ + n1 −1)asbg′(v) + n2(κ + n2 −1) asbg′(v ′) − n1(κ + n1)asbg′′(w)−n2(κ + n2 −2) asbg′′(w ′) = n1(κ + n1 −1)asbg′(v) + n2(κ + n2 −1) asbg′(v ′) − n1(κ + n1)asbg′(v)−n2(κ + n2 −2) asbg′(v ′) − n1(κ + n1)a[sbg′′(w)−s b g′(v)] + n2(κ + n2 −2) a[sbg′(v ′)−sbg′′(w ′)] ≤ n1(κ + n1 −1)asbg′(v) + n2(κ + n2 −1) asbg′(v ′) − n1(κ + n1)asbg′(v)−n2(κ + n2 −2) asbg′(v ′) = n1s b g′(v)[(κ + n1 −1) a − (κ + n1)a] + n2s b g′(v ′)[(κ + n2 −1)a − (κ + n2 −2)a] ≤ n1sbg′(v)[(κ + n1 −1) a − (κ + n1)a] + n1s b g′(v)[(κ + n2 −1) a − (κ + n2 −2)a] ≤ 0, since for a = 1, we obtain (κ + n1 − 1)a − (κ + n1)a + (κ + n2 − 1)a − (κ + n2 − 2)a = 0 and for a > 1, by lemma 2.3, (κ + n2 −1)a − (κ + n2 −2)a < (κ + n1)a − (κ + n1 −1)a, thus (κ + n1 −1)a − (κ + n1)a + (κ + n2 −1)a − (κ + n2 −2)a < 0. hence dda,b(g′) −dda,b(g′′) < 0 and dda,b(g′) < dda,b(g′′). so g′ is not a graph with the largest dda,b index and we have a contradiction. we obtain n2 = 1 and consequently, n1 = n−κ−1. thus g′ is (kn−κ−1 ∪k1) + kκ. a pendant vertex is a vertex of degree one. let kn−k ? sk+1 be obtained by joining k pendant vertices to one vertex of kn−k. we study the dda,b index for graphs with pendant vertices and show that kn−k ? sk+1 is the extremal graph. theorem 3.5. let g be any connected graph having order n and k pendant vertices, where 1 ≤ k ≤ n−3. then for a ≥ 1 and b ≤ 0, where (a,b) 6= (1,0), dda,b(g) ≤ dda,b(kn−k ? sk+1) with equality only if g is kn−k ? sk+1. proof. let g′ be a graph with the largest dda,b index with respect to order n and k pendant vertices. let a be the set of pendant vertices of g′. by lemma 2.2, there is an edge connecting each pair of vertices in v (g′) \ a. so g′ contains kn−k as a subgraph. we prove that one vertex of that kn−k is adjacent to all the k pendant vertices in g′. 113 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 suppose to the contrary that kn−k contains two vertices v and w, such that each of them is adjacent to a pendant vertex in g′. let us denote the pendant neighbors of v by vi where i = 1,2, . . . ,n1, and the pendant neighbors of w by wj where j = 1,2, . . . ,n2. clearly, n1,n2 are positive integers and n1 +n2 ≤ k. without loss of generality, assume that n1 ≥ n2. we compare the dda,b indices of the graphs g′ and g′′ having the same vertex sets, where e(g′′) = {vw1,vw2, . . . ,vwn1}∪e(g′)\{ww1,ww2, . . . ,wwn1}. we obtain degg′(z) = degg′′(z) and sbg′(z) = s b g′′(z) if z is not v,w,vi,wj, where i = 1,2, . . . ,n1 and j = 1,2, . . . ,n2. let s = n − k − 1. then s ≥ 2. we have degg′(v) = s + n1, degg′(w) = s + n2, degg′′(v) = s + n1 + n2 and degg′′(w) = s. we obtain sbg′(v) = (s + n1) + 2 b(k −n1), sbg′(w) = (s + n2) + 2 b(k −n2), sbg′′(v) = (s + n1 + n2) + 2 b(k −n1 −n2), sbg′′(w) = s + 2 bk, sbg′(vi) = 1 + 2 b(s + n1 −1) + 3b(k −n1), sbg′(wj) = 1 + 2 b(s + n2 −1) + 3b(k −n2), sbg′′(vi) = s b g′′(wj) = 1 + 2 b(s + n1 + n2 −1) + 3b(k −n1 −n2). for b ≤ 0, sbg′(vi)−s b g′′(vi) = n2(3 b −2b) ≤ 0, sbg′(wj)−s b g′′(wj) = n1(3 b −2b) ≤ 0. for b < 0, sbg′′(w)−s b g′(w) = n2(2 b −1) < 0, sbg′(w)−s b g′(v) = (n2 −n1) + 2 b(n1 −n2) = (2b −1)(n1 −n2) ≤ 0, sbg′(v)−s b g′′(v) = n2(2 b −1) < 0, therefore 0 < sbg′′(w) < s b g′(w) ≤ s b g′(v) < s b g′′(v). if b = 0, obviously 0 < sbg′′(w) = s b g′(w) = s b g′(v) = s b g′′(v). note that 0 < sa < (s + n2) a ≤ (s + n1)a < (s + n1 + n2)a for a ≥ 1. thus, for a ≥ 1 and b ≤ 0, we obtain dda,b(g ′)−dda,b(g′′) = [degg′(v)]asbg′(v)− [degg′′(v)] asbg′′(v) + [degg′(w)] asbg′(w)− [degg′′(w)] asbg′′(w) + n1∑ i=1 ([degg′(vi)] asbg′(vi)− [degg′′(vi)] asbg′′(vi)) + n2∑ j=1 ([degg′(wj)] asbg′(wj)− [degg′′(wj)] asbg′′(wj)) = (s + n1) asbg′(v)− (s + n1 + n2) asbg′′(v) + (s + n2) asbg′(w) − sasbg′′(w) + n1[s b g′(vi)−s b g′′(vi)] + n2[s b g′(wj)−s b g′′(wj)] = (s + n1) asbg′(v)− (s + n1 + n2) asbg′(v) + (s + n2) asbg′(w) − sasbg′(w) + (s + n1 + n2) a[sbg′(v)−s b g′′(v)] + sa[sbg′(w)−s b g′′(w)] + 2n1n2(3 b −2b) ≤ (s + n1)asbg′(v)− (s + n1 + n2) asbg′(v) 114 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 + (s + n2) asbg′(w)−s asbg′(w) = sbg′(v)[(s + n1) a − (s + n1 + n2)a] + sbg′(w)[(s + n2) a −sa] ≤ sbg′(v)[(s + n1) a − (s + n1 + n2)a] + sbg′(v)[(s + n2) a −sa] ≤ 0, since for a = 1, we have (s + n1)a − (s + n1 + n2)a + (s + n2)a −sa = 0 and for a > 1, by lemma 2.3, (s + n2) a −sa < (s + n1 + n2)a − (s + n1)a, thus (s + n1) a − (s + n1 + n2)a + (s + n2)a −sa < 0. so dda,b(g′)−dda,b(g′′) < 0 for a > 1 and b ≤ 0. for a ≥ 1 and b < 0, we have sbg′(v)−s b g′′(v) = s b g′′(w)−s b g′(w) = n2(2 b −1) < 0, thus (s + n1 + n2) a[sbg′(v)−s b g′′(v)] + s a[sbg′(w)−s b g′′(w)] < 0 and we again get dda,b(g′)−dda,b(g′′) < 0. therefore, dda,b(g′) < dda,b(g′′) for a ≥ 1 and b ≤ 0, where (a,b) 6= (1,0). thus g′ is not a graph with the largest dda,b index and we have a contradiction. hence g′ is kn−k ? sk+1. the problem studied in the previous theorem is trivial for a = 1 and b = 0. all graphs g with n vertices and m edges have the same dd1,0(g) index. we obtain dd1,0(g) = ∑ v∈v (g) ( [degg(v)] 1 ∑ w∈v (g)\{v} [dg(v,w)] 0 ) = (n−1) ∑ v∈v (g) degg(v) = 2m(n−1). so, by lemma 2.2, each graph containing kn−k and k pendant vertices is a graph with the largest dd1,0 index with respect to order n and k pendant vertices. finally, we consider connected graphs without cycles called trees. in the following proof, we show that the diameter of the extremal tree t of given order is at most 2, which means that t is a star. note that the distance between any two furthest vertices v and w is called the diameter of t and a shortest path between v and w is a diametral path. theorem 3.6. let t be any tree of order n ≥ 4. then for a ≥ 1 and b ≤ 0, where (a,b) 6= (1,0), we have dda,b(t) ≤ dda,b(sn) with equality only if t is sn. proof. let t ′ be a tree of order n with the largest dda,b index. we prove that t ′ is sn. if n ≤ 3, clearly t is sn, so we study trees for n ≥ 4. assume to the contrary that t ′ is not sn. thus the diameter of t ′ is d ≥ 3. we denote a diametral path of t ′ by u0u1u2 . . .ud, where degt′(u1) = n1 and degt′(u2) = n2. clearly, n1,n2 ≥ 2. so u1 is adjacent to n1 −1 pendant vertices, say v1,v2, . . . ,vn1−1 (one of them is u0). we compare the dda,b indices of the trees t ′ and t ′′ having the same vertex sets, while the edge set e(t ′′) = {u2v1, u2v2, . . . ,u2vn1−1}∪e(t ′)\{u1v1, u1v2, . . . ,u1vn1−1}. we have degt′′(u1) = 1 and degt′′(u2) = n1 + n2 −1. for i = 1,2, . . . ,n1 −1, dt′(u1,vi) = dt′′(u2,vi) = 1 and dt′′(u1,vi) = dt′(u2,vi) = 2, 115 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 thus for b ≤ 0, sbt′(u1)−s b t′′(u1) = (n1 −1)(1 b −2b) ≥ 0, sbt′(u2)−s b t′′(u2) = (n1 −1)(2 b −1b) ≤ 0. since dt′(u1,z) = dt′′(u2,z) + 1 for each z ∈ v (t ′)\{u1,u2,v1,v2, . . . ,vn1−1}, we get [dt′(u1,z)] b < [dt′′(u2,z)] b and sbt′(u1) < s b t′′(u2) for b < 0. obviously, [dt′(u1,z)] b = [dt′′(u2,z)] b and sbt′(u1) = s b t′′(u2) for b = 0. for any z other than u1,u2 and vi, where i = 1,2, . . . ,n1 − 1, we have degt′(z) = degt′′(z) and dt′(z,x) ≥ dt′′(z,x), where x ∈ v (t ′). for b ≤ 0, we obtain [dt′(z,x)]b ≤ [dt′′(z,x)]b and sbt′(z) ≤ s b t′′(z), therefore [degt′(z)] asbt′(z) ≤ [degt′′(z)] asbt′′(z). similarly, [degt′(vi)]asbt′(vi) ≤ [degt′′(vi)] asbt′′(vi). then, for a ≥ 1 and b ≤ 0, dda,b(t ′)−dda,b(t ′′) ≤ [degt′(u1)]asbt′(u1)− [degt′′(u1)] asbt′′(u1) + [degt′(u2)] asbt′(u2)− [degt′′(u2)] asbt′′(u2) = na1s b t′(u1)−1 asbt′′(u1) + n a 2s b t′(u2)− (n1 + n2 −1) asbt′′(u2) = na1s b t′(u1)−s b t′(u1) + n a 2s b t′′(u2)− (n1 + n2 −1) asbt′′(u2) + [sbt′(u1)−s b t′′(u1)] + n a 2[s b t′(u2)−s b t′′(u2)] ≤ na1s b t′(u1)−s b t′(u1) + n a 2s b t′′(u2)− (n1 + n2 −1) asbt′′(u2) = sbt′(u1)[n a 1 −1] + s b t′′(u2)[n a 2 − (n1 + n2 −1) a] ≤ sbt′′(u2)[n a 1 −1] + s b t′′(u2)[n a 2 − (n1 + n2 −1) a] ≤ 0, since for a = 1, we have na1 −1 + na2 − (n1 + n2 −1)a = 0 and for a > 1, by lemma 2.3, na1 −1 a < (n1 + n2 −1)a −na2, thus na1 −1 + n a 2 − (n1 + n2 −1) a < 0. so dda,b(t ′)−dda,b(t ′′) < 0 for a > 1 and b ≤ 0. for a ≥ 1 and b < 0, we have sbt′(u1) < s b t′′(u2), thus sbt′(u1)[n a 1 −1] < s b t′′(u2)[n a 1 −1] and we again get dda,b(t ′)−dda,b(t ′′) < 0. therefore, dda,b(t ′) < dda,b(t ′′) for a ≥ 1 and b ≤ 0, where (a,b) 6= (1,0), which is a contradiction. hence g′ is sn. let us note that if a = 1 and b = 0, then all trees t of order n have the same dd1,0 index, since dd1,0(t) = ∑ v∈v (t) ( [degt (v)] 1 ∑ w∈v (t)\{v} [dt (v,w)] 0 ) = (n−1) ∑ v∈v (t) degt (v) = 2(n−1)2. 116 t. vetrík / j. algebra comb. discrete appl. 8(2) (2021) 107–118 4. conclusion we presented some bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices. there is a huge space for further research, since one can study lower and upper bounds on the dda,b index for general graphs or special classes of graphs for various invariants of graphs. let us state the following problems. problem 4.1. find sharp upper and lower bounds on 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[20] z. zhu, y. hong, minimum degree distance among cacti with perfect matchings, discrete appl. math. 205 (2016) 191–201. 118 http://dx.doi.org/10.13069/jacodesmath.729422 http://dx.doi.org/10.13069/jacodesmath.729422 https://doi.org/10.1142/s1793830920500500 https://doi.org/10.1142/s1793830920500500 https://doi.org/10.1016/j.dam.2020.03.051 https://doi.org/10.1016/j.dam.2020.03.051 https://doi.org/10.1007/s10878-014-9757-6 https://doi.org/10.1007/s10878-014-9757-6 https://doi.org/10.1016/j.dam.2016.01.005 https://doi.org/10.1016/j.dam.2016.01.005 introduction preliminary results main results conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.617239 j. algebra comb. discrete appl. 6(3) • 147–161 received: 10 february 2019 accepted: 19 august 2019 journal of algebra combinatorics discrete structures and applications a generalization of the mignotte’s scheme over euclidean domains and applications to secret image sharing∗ research article ibrahim ozbek, fatih temiz, irfan siap abstract: secret sharing scheme is an efficient method to hide secret key or secret image by partitioning it into parts such that some predetermined subsets of partitions can recover the secret but remaining subsets cannot. in 1979, the pioneer construction on this area was given by shamir and blakley independently. after these initial studies, asmuth-bloom and mignotte have proposed a different (k, n) threshold modular secret sharing scheme by using the chinese remainder theorem. in this study, we explore the generalization of mignotte’s scheme to euclidean domains for which we obtain some promising results. next, we propose new algorithms to construct threshold secret image sharing schemes by using mignotte’s scheme over polynomial rings. finally, we compare our proposed scheme to the existing ones and we show that this new method is more efficient and it has higher security. 2010 msc: 11k31, 94a08, 94a62, 13f07 keywords: mignotte sequences, secret image sharing, secret sharing scheme, euclidean domain 1. introduction the rapid improvement of technology and the increase of the usage of the internet day by day introduces some new challenges. one of the most important challenge is the security of data (message/image). there are several techniques in literature for keeping the data secure. one of them is applying a secret sharing scheme (a.k.a. key safeguarding scheme). secret sharing schemes play an important role in cryptography especially where the secret key is supposed to be distributed in parts to shareholders so that ∗ this research was partially supported by the scientific and technological research council of turkey, project no: 114f388. ibrahim ozbek; yildiz technical university, graduate school of science and engineering, department of mathematics, istanbul, turkey (email: ibrhmozbek@gmail.com). fatih temiz (corresponding author); istanbul gelisim university, department of management information systems, istanbul, turkey (email: ftemiz@gelisim.edu.tr). irfan siap; jacodesmath institute, department of mathematics, istanbul, turkey (email: irfan.siap@gmail.com). 147 https://orcid.org/0000-0001-5477-0463 https://orcid.org/0000-0002-9702-1531 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 some of predetermined shares can recover the key. in order to construct such schemes many methods have been developed over the last 30 years [14]. the first secret sharing scheme is introduced by blakley and shamir independently [2, 12]. they propose a (k,n) threshold secret sharing scheme, i.e. any k out of n participants can reconstruct the secret key but any k−1 or fewer participants cannot reconstruct it. in [1, 9], asmuth-bloom and mignotte propose a different (k,n) threshold modular secret sharing scheme by using chinese remainder theorem. these two methods depend on a particular choice of the ordering and the selection of positive integers that are used as module. mignotte’s method is over integers, on the other hand asmuth-bloom’s modular approach is applicable to not only integers but also to euclidean domains in general. in [5], mignotte’s construction is further generalized to not necessarily relatively coprime integers which are called generalized mignotte sequences and an application to e-voting is presented. also, another generalization of mignotte’s construction over polynomial rings is given in [16]. in [10], naor and shamir propose an interesting method for encrypting visual data which is called visual cryptography. in this method, there is no cryptographic computation to recover secret image, that is, decoding process of the method is based solely on human visual system. after this pioneer construction, to overcome memory space problem, thien and lin propose threshold secret image sharing scheme by using shamir’s scheme [17]. in this scheme, they consider each k pixels of a secret image s as coefficients of polynomials and compute the values of these polynomials to generate the n shadow images. since the scheme’s reconstruction is based on lagrange interpolation, any k out of n shadow images can reconstruct the secret image s but any k− 1 or fewer shadow images cannot reconstruct the secret image. by taking advantage of huffman encoding, wang and su come up with a new secret image sharing scheme that uses smaller shadow images [19]. in [8], meher and patra design a new scheme which is not a threshold by taking advantage of chinese remainder theorem. unlike the previous work, the shadow images of this scheme may have distinct sizes. in [13], shyu and chen give another threshold scheme which is based on mignotte’s secret sharing scheme. in this study, we give a generalization of mignotte’s scheme over euclidean domains. we also construct threshold secret image sharing schemes based on generalization of mignotte’s scheme over polynomial rings. we organize this paper as follows: in section 2, we give basics of secret sharing schemes and recall the construction of mignotte’s scheme. the generalization of mignotte’s scheme over euclidean domains is given in section 3. in section 4, we propose new algorithms to construct threshold secret image sharing schemes and an experimental result is presented in section 5. in section 6, the security analysis of algorithms is also analyzed. in section 7, our proposed secret image sharing scheme is compared with the state of the art methods. finally, conclusions and comparisons to the existing methods are summarized in section 8. 2. preliminaries a secret sharing scheme is a method of sharing a secret s among n participants p = {p1,p2, . . . ,pn} by using a distribution rule f = {f |f : v → p } such that some predetermined subsets of 2p can reconstruct the secret s, where v is the set of shares (shadows). the subsets that can construct the secret s are called access structures and the set of such subsets is denoted by γ, and any subset of participants that is not in γ cannot reconstruct the secret s. if every k out of n participants can determine the secret s and any out of k − 1 or fewer participants cannot determine the secret s, then this scheme is called a (k,n) threshold scheme and the access structure of this scheme is γ = {a| |a| ≥ k,a ⊆ 2p} [12]. also, if k − 1 or fewer participants cannot obtain any helpful information about the secret s, i.e. do not have any advantage in order to reconstruct the key, then this is called a perfect secret sharing scheme [14]. there are various kinds of generalizations of chinese remainder theorem (crt) in literature, we give some of them which are necessary for our constructions in the following sections. the following is called chinese remainder theorem for commutative rings [4]. theorem 2.1. let i1,i2, . . . ,in be ideals of a commutative ring r with identity such that ii + ij = r 148 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 for all i 6= j. then, there exists a ring isomorphism ϕ : r/i1i2 · · ·in → r/i1 ×r/i2 ×···×r/in x 7→ x mod i1,x mod i2, . . . ,x mod in. conversely, given the module values of x, then x is uniquely determined up to congruence modulo the ideal i = i1i2 · · ·in = i1 ∩ i2 ∩·· ·∩ in. the version of this theorem can be given in a similar way for integers and polynomial rings (also euclidean domains) [3, 6]. next, we give a more general version of crt developed by ore such that q1,q2, . . . ,qn do not have to be pairwise relatively prime [11]. theorem 2.2. [11] let q1,q2, . . . ,qn ≥ 2 be a collection of positive integers. for any set of elements ai the system of simultaneous congruences x ≡ a1 mod q1 x ≡ a2 mod q2 ... x ≡ an mod qn has a solution if and only if ai ≡ aj mod (qi,qj) for all i 6= j, 1 ≤ i,j ≤ n , where (qi,qj) is the greatest common divisor of qi and qj. moreover, the system has a unique solution in modulo q = lcm(q1,q2, . . . ,qn), where q is the least common multiple of q1,q2, . . . ,qn and it can be computed as follows x ≡ a1 q q1 b1 + a2 q q2 b2 + · · · + an q qn bn where bi are integers such that b1 q q1 + b2 q q2 + · · · + bn qqn = 1. if q1,q2, . . . ,qn are relatively prime, then we obtain the standard version of crt. furthermore, a general version of crt can be applied to polynomial rings [16]. the chinese remainder theorem has many applications in literature [3]. one of them is on secret sharing. there are two different pioneer constructions to recover the secret s with crt given by asmuthbloom and mignotte [1, 9]. next, we recall the construction of mignotte’s scheme. mignotte’s construction mignotte introduced a (k,n) threshold secret sharing scheme using a special subset of coprime numbers which are called mignotte’s sequences [9]. the trick of this construction is the choice of the secret s in a particular range. now, let us recall the construction given by mignotte based on crt. the system is composed by a dealer and n participants. the dealer constructs the system and gives the shares to the participants as follows: 1. dealer chooses positive integers q1 < q2 < · · · < qn such that (qi,qj) = 1 for all 1 ≤ i < j ≤ n and n∏ i=n−k+2 qi < k∏ i=1 qi. 2. the secret s is a randomly chosen integer in the following interval n∏ i=n−k+2 qi < s < k∏ i=1 qi. 149 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 3. the shadows vi are computed as: v1 ≡ s mod q1 ... vn ≡ s mod qn and distributed to the participants pi = (vi,qi) for all 1 ≤ i ≤ n. 4. given k out of n distinct participants {pi1,pi2, . . . ,pik} , the secret s is recovered uniquely in modulo q = qi1qi2 · · ·qik using the standart crt as follows: s ≡ vi1ri1 q qi1 + · · · + vikrik q qik mod q where rij q qij ≡ 1 mod qij for all 1 ≤ j ≤ k. example 2.3. let us consider a (2, 4) threshold scheme and choose pairwise coprime numbers q1 = 11,q2 = 15,q3 = 17, and q4 = 26. the secret can be chosen inside the interval 26 < s < 165. let s = 124 and the corresponding shadows be v1 = 3, v2 = 4, v3 = 5, v4 = 20. each participant in a system has a pair of information (vi,qi). suppose that p1 and p2 want to find the secret s by using their shadows {(3, 11) , (4, 15)} . first, r1 = 3,r2 = 11 is found and the secret can be computed uniquely modulo 165 as follows: s ≡ (3 · 3 · 15 + 4 · 11 · 11) mod165 ≡ 124. in this example, we see that any two out of four participants can recover the secret by putting their partial information together. 3. generalization of mignotte’s scheme over euclidean domains in this section, we generalize the threshold secret sharing scheme given by mignotte [9] to euclidean domains. let us first recall the definition of euclidean domains. definition 3.1. let d be an integral domain and r = z+ ∪{0} be the set of nonnegative integers. d is called euclidean domain if there is a norm function n : d\{0}→ r with the following two properties: 1. for any a,b ∈ d\{0}, n (a) ≤ n (ab), 2. for all a,b ∈ d with b 6= 0 we can write a = qb + r for some q,r ∈ d such that r = 0 or n (r) < n (b). now, we construct a (k,n)-threshold scheme over a euclidean domain. let d be a euclidean domain. choose the elements q1, . . . ,qn ∈ d such that (qi,qj) = 1 for i,j ∈{1, . . . ,n}, i 6= j and n (qi) < n (qj) for all i < j, where n is a suitable degree function defined on euclidean domain. dealer chooses a secret s ∈ d/〈q1 · · ·qn〉 satisfying the following conditions: 1. n (α) = n (qn−k+2) n (qn−k+3) · · ·n (qn) < n (s) and 2. n (s) < n (β) = n (q1) n (q2) · · ·n (qk). 150 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 the dealer now determines the shares v1, . . . ,vn to be distributed to the n participants in the following way: v1 ≡ s mod q1 v2 ≡ s mod q2 ... vn ≡ s mod qn. since d/〈qj1qj2 . . .qjk〉∼= d/〈qj1〉×d/〈qj2〉×·· ·×d/〈qjk〉 where qj1,qj2, . . . ,qjk ∈{q1,q2, . . . ,qn}, any k out of n participants can reconstruct the secret s by using crt over euclidean domains [6]. for instance, first k participants can reconstruct the secret s in the following way: 1. for 1 ≤ j ≤ k, define µj = k∏ i=1 i6=j qi and ηj = µ −1 j (mod qj) where ηj < qj. 2. then, the secret can be computed by s ≡ v1µ1η1 + v2µ2η2 + · · · + vkµkηk (mod q1q2 · · ·qk). (1) theorem 3.2. for the given construction above, any k− 1 or fewer participants cannot reconstruct the secret s. proof. assume that any k − 1 out of n participants come together to reconstruct the secret s. let these participants pj1,pj2, . . . ,pjk−1 reconstruct r as a secret with their own shares. it is easily seen that n (r) < n (s) and s = r + δqj1qj2 . . .qjk−1 for some δ ∈ d. this means that any k − 1 or fewer participants cannot reconstruct the secret s. it is easily seen that the best probability of finding the secret s is n(qj1qj2 · · ·qjk−1 ) n(β) −n(α) . if the range of norm is large enough, it will be an infeasible problem to determine the secret s. example 3.3. let us construct a (2, 3) threshold secret sharing scheme based on a specific euclidean domain, i.e. gaussian integers z[i]. we choose three coprime gaussian numbers, 11+8i, −3−13i, 7+4i and compute their norms 185, 178, 65 (n (a + bi) = a2 + b2) respectively. dealer chooses the norm of the secret inside the following interval 185 < s < 11570. suppose that dealer chooses the norm of the secret as 424 and a gaussian integer 18−10i. after choosing the secret, dealer computes the shares of participants as follows: 18 − 10i ≡ −1 − 7i (mod 11 + 8i) ⇒ v1 = −1 − 7i, 18 − 10i ≡ 5 − 7i (mod − 3 − 13i) ⇒ v2 = 5 − 7i, 18 − 10i ≡ 3 (mod 7 + 4i) ⇒ v3 = 3. hence a (2, 3) threshold secret sharing scheme is designed such that any 2 out of 3 participants can reconstruct the secret. we pick p1 and p3 and compute the secret in the following way: z[i]/(11 + 8i) ×z[i]/(7 + 4i) → z[i]/(45 + 100i) (−1 − 7i)(7 + 4i)[(7 + 4i)−1 (mod 11 + 8i)] + 3(11 + 8i)[(11 + 8i)−1 (mod 7 + 4i)] after computing the inverse of gaussian integers, (7 + 4i)−1 (mod 11 + 8i) ≡−12 + 2i and (11 + 8i)−1 (mod 7 + 4i) ≡ 7 − 2i, we substitute the values and the secret is computed as follows s = (−1 − 7i)(7 + 4i)(−12 + 2i) + (3)(11 + 8i)(7 − 2i) (mod 45 + 100i) ≡ 18 − 10i. in the next section, we build our methods over two specific euclidean domains. gaussian ring of integers and polynomial rings over fields. 151 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 4. a new algorithm for secret image sharing in this section, we present a new (k,n) threshold secret image sharing scheme such that any k out of n shadow images can reconstruct the secret image s but any k − 1 or fewer shadow images cannot reconstruct s. the idea behind this construction is a generalization of mignotte’s scheme [16] over polynomial rings. a recent study on secret sharing scheme over polynomial rings is presented in [16]. the difference between our scheme and the method introduced in [16] is that the irreducible polynomials in [16] are chosen of the same degrees (remark 3.1, [16]). here in our method we do not impose such a restriction hence this gives us flexibility on construction which leads to a faster applicability and better security. since the gray value of a pixel is between 0 and 255 this fact forces us to consider algorithms for distinct prime numbers p = 251 and p = 257, which are the closest prime numbers smaller and larger than p = 255, or the finite field extension gf(28). for the first algorithm, we must reduce all the gray values between 251 − 255 to 250. in the second algorithm, we can encounter invalid gray value, i.e. 256. to overcome this problem, we must increase the size of shadow images (see algorithm 2, step 3 and step 5). in the third algorithm, since the gray values range from 0 to 255, each gray value has a 2-adic representation a0 + a12 + · · · + a727 for some ai ∈ z2, hence there is a map z256 → z2 [x]/〈f (x)〉, a → (a0, . . . ,a7) where f (x) is an irreducible polynomial of degree 8 over z2. since gf ( 28 ) ∼= z2 [x]/〈f (x)〉 and gf ( 28 )× = 〈α〉, we have a one-to-one and onto map φ : z256 → gf ( 28 ) a → { 0, if a = 0 αia, if a 6= 0 where αia = a0 + a1α + · · · + a7α7. because of the above map φ, unlike the other two algorithms, there is no truncation and invalid gray value in the last algorithm. suppose that we intend to construct a (k,n) threshold secret image sharing for an m×r secret image s. we first choose n polynomials q1,q2, . . . ,qn such that (qi,qj) = 1 for all i,j ∈ {1, . . . ,n}, i 6= j and α = deg (qn−k+2×qn−k+3×···×qn) < β = deg (q1×q2×···×qk). we suppose that the secret image is divided into 1 ×s row vectors such that s divides r. the crucial part of the algorithm is choosing s− 1 degree polynomials si (x) = gi,0 + gi,1x + · · · + gi,s−1xs−1, called secret polynomials corresponding the ith partition of s, such that α < deg (s (x)) < β, where gi,0,gi,1, . . . ,gi,s−1 are the s ordered pixels of ith partition. for the ith partition of s, the ith partition of n shadow images s1,s2, . . . ,sn are obtained in the following way: vi,1 ≡ si(x) mod q1 vi,2 ≡ si(x) mod q2 ... vi,n ≡ si(x) mod qn (2) where the polynomials vi,j are called shadow polynomials. the steps of algorithms for this secret image sharing schemes are illustrated as follows. 4.1. algorithm 1 for p = 251 the share construction 1. if there exists gray values larger than 250, they are set to 250. thus the gray values are now between the range 0−250. here there will be some loose of pixel colors that are not distinguishable by naked eye. 152 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 2. choose n polynomials q1,q2, . . . ,qn such that (qi,qj) = 1 for all i,j ∈ {1, . . . ,n}, i 6= j and α = deg(qn−k+2 ×qn−k+3 ×···×qn) < β = deg(q1 ×q2 ×···×qk). 3. the secret polynomials si(x) corresponding to ith partition of the secret image are obtained by letting the s coefficients be the gray values of s pixels of ith partition such that α < deg(si(x)) = s− 1 < β. 4. using the secret polynomial si(x) for ith partition, generate n shadow polynomials vi,j for all 1 ≤ j ≤ n by using equation 2 and set li = deg(qi) for 1 ≤ i ≤ n. 5. for all partitions of the secret image, apply steps 3 and 4. remark: the degrees of n polynomials q1,q2, . . . ,qn can be chosen arbitrarily as long as the condition deg(qn−k+2×qn−k+3×···×qn) < deg(q1×q2×···×qk) is satisfied. if at least one of the polynomials q1,q2, . . . ,qn have different degree than the others, then the size of one of s1,s2, . . . ,sn may be different from each other. in order to correctly reconstruct the image, we must give the information of number of pixels in partitions for each shadow images to the participants. this means that the shares of participants are (si,qi, li) for all 1 ≤ i ≤ n, where li is the number of pixels in each partition. the reconstruction phase 1. for all 1 ≤ i ≤ n, generate k shadow polynomials using first li pixels of k shadow images. 2. using k shadow polynomials v1,i1,v1,i2, . . . ,v1,ik and equation 1, one can obtain the secret polynomial s1(x), i.e. we first get s pixels of the secret image. 3. for all other pixels of k shadow images, apply step 1 and 2. 4.2. algorithm 2 for p = 257 the share construction 1. choose n polynomials q1,q2, . . . ,qn such that (qi,qj) = 1 for all i,j ∈ {1, . . . ,n}, i 6= j and α = deg (qn−k+2×qn−k+3×···×qn) < β = deg (q1×q2×···×qk) . 2. find the secret polynomial si (x) for the ith partition of the secret image such that α < deg (si (x)) = s− 1 < β. 3. by using equation 2, find the shadow polynomials vi,j for all 1 ≤ j ≤ n and apply the following steps. (a) if the coefficients of vi,j’s are not equal to 255 or 256, the ith partition of sj is generated by using the coefficients of vi,j. (b) if a coefficient vi,jk of vi,j is equal to 255, then consider 255 as a couple 255 and 0 and write 255xk−1 + 0xk as the polynomial vi,j. (c) if a coefficient vi,jk of vi,j is equal to 256, then consider 256 as a couple 255 and 1 and write 255xk−1 + 1xk as the polynomial vi,j. 4. for all other pixels of the secret image, apply steps 2 and 3. 5. extend the shadow images to a rectangular array by padding with redundant pixels such that if the gray value of the last pixel of non-extended shadow image is m then all the redundant pixels are of gray value m + 1. the reconstruction phase 153 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 1. for given k shadow images, find the last pixel value which is different than the consecutive one and reduce them by deleting the redundancy. 2. take the first li pixels of any k shadow images and apply the following steps: (a) if any gray value of the first partition of sj is not 255, the shadow polynomial vi,j is obtained by using first li pixels of sj. (b) if the gray value a1,l of the first partition of sj is equal to 255, then the pixels a1,l and a1,l+1 are converted to (a1,l + a1,l+1) xl−1 in the polynomial v1,j. 3. using the k shadow polynomials v1,i1,v1,i2, . . . ,v1,ik and equation 1, we obtain secret polynomial s1 (x), i.e. we get first s pixels of the secret image. 4. to all pixels of k shadow images, apply step 1 and 2. 4.3. algorithm 3 for gf(28) the share construction 1. find the field elements corresponding to the gray values in the secret image with using the map φ. 2. choose n polynomials q1,q2, . . . ,qn in gf(28) [x] such that (qi,qj) = 1 for all i,j ∈{1, . . . ,n}, i 6= j and α = deg (qn−k+2×qn−k+3×···×qn) < β = deg (q1×q2×···×qk). 3. the secret polynomials si (x) corresponding to ith partition of the secret image are obtained by letting the s coefficients be as the φ-image of the gray values of s pixels of ith partition such that α < deg (si (x)) = s− 1 < β. 4. using the secret polynomial si (x) for ith partition, generate n shadow polynomials vi,j for all 1 ≤ j ≤ n by using equation 2 and set li = deg (qi) for 1 ≤ i ≤ n. 5. for 1 ≤ j ≤ n, calculating φ−1-image of the coefficients of the shadow polynomials vi,j, construct ith partition of shadow images. 6. for all partitions of the secret image, apply steps 3, 4 and 5. remark: for this construction, all operations are performed over the finite field extension gf(28). reconstruction phase 1. for all 1 ≤ i ≤ n, generate k shadow polynomials using first li pixels of k shadow images. 2. calculate φ-image of the coefficients of the shadow polynomials v1,i1,v1,i2, . . . ,v1,ik. for these k polynomials over gf(28), applying an equation 1, determine the secret polynomial s1 (x) over gf(28). 3. calculating φ−1-image of the coefficients of the secret polynomials s1 (x), construct first s pixels of secret images. 4. for all other pixels of k shadow images, apply step 1, 2 and 3. 154 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 5. an experimental result to illustrate algorithm 1, we design a (2, 4) threshold secret image sharing scheme. the secret image s is chosen as a 256 × 256 pixels pepper image shown in figure 1 (a). to construct the shadow images, we choose relatively prime four polynomials q1 = x4 + 85x2 + 4x + 23, q2 = x4 + 54x3 + 99x2 + 27x + 105, q3 = 23x 4 + 81x3 + 201x2 + 153x + 83, q4 = x4 + x3 + 81x2 + 103 where the condition α = 4 < β = 8 is satisfied. we take s = 8 so that the size of the shadow images are 1/2 of the size of the secret image since division of each si(x) corresponding to 8 pixels, by qj(x) gives us a remainder polynomial of degree 3, corresponding to 4 pixels. the shadow images s1,s2,s3,s4 with respect to the secret image s are illustrated in figure 1 (b). figure 1: (a) 256 × 256 secret image s; (b) 128 × 256 shadow images s1,s2,s3,s4 to recover the secret image, a combination of any 2 participants having shadow images is sufficient. this reconstruction is illustrated in figure 2. note that, for (k,n) threshold secret image sharing, if we take the degrees of each polynomial qi as m and degree of secret polynomial km − 1, the size of each shadow image is one-kth of the size of secret image as the construction of thien and lin [17]. further, recently an image sharing method that avoids permutation is introduced in [18]. our method also differs and presents a more applicable nature compared to [18]. in [18], the shares distributed to the holders are each of sizes 256×256. in our method, the sizes are 128 × 256 which gives a storage advantage. we present another illustrative example to show how we can control the size of shares using the same (2, 4) threshold secret image sharing scheme. we can take the irreducible polynomials of degree 5 or 6. by choosing four irreducible polynomials q1 = x5 + 12x4 + 53x3 + 177x2 + 46x+ 91,q2 = x5 + 132x4 + 131x3 + 155 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 figure 2: some examples of reconstruction 117x2 + 114x+ 11,q3 = x 5 + 151x4 + 175x3 + 21x2 + 173x+ 39,q4 = x 5 + 151x4 + 48x3 + 13x2 + 164x+ 145, we obtain the shadow images of size 160 × 256 (figure 3). also, the size of the shadow images would be 196 × 256 if the irreducible polynomials were chosen of degree 6. 6. security analysis in this section, we show that any k−1 or fewer shadow images cannot reconstruct m×r secret image s. without loss of generality, suppose that last k−1 participants come together to reconstruct the secret image s and that assuming they know how to read the pixels of shadow images. assume that they want to reconstruct the first partition of the secret image, without loss of generality, they first generate the shadow polynomials v1,n−k+2,v1,n−k+3, . . . ,v1,n by using the first partitions of the shadow images and the secret polynomial r1 (x) corresponding to these shadow polynomials is computed by using crt such that deg (r1 (x)) < α where α = deg (qn−k+2qn−k+3 . . .qn). assume that r1 (x) has the maximum degree, i.e., deg (r1 (x)) = α− 1 and secret polynomial of the first partition of secret image s1 (x) has minimum degree, i.e., deg (s1 (x)) = α + 1. it is easily seen that s1 (x) = r1 (x) + γ1 (x) (qn−k+2qn−k+3 · · ·qn) (mod p) for some γ1 (x) ∈ gf(p), where p is either 251, 256 or 257. this means that the probability of finding the right polynomial is 1/p2 at the best. since, the secret image s has m × r/s partitions, the probability of reconstruction the right image is ( 1/p2 )m×r/s at the best. for the given example, the probability of obtaining the right image is ( 1 2514 )256×32 ≈ (1 2 )261212 . furthermore, we assume that the 156 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 figure 3: (a) 256 × 256 secret image s; (b) 160 × 256 shadow images s1,s2,s3,s4 person who wants to reconstruct the secret image knows the size of partition of shadow images and secret image. if an unauthorized person does not hold into this information, then the problem becomes more infeasible to solve. it is clear that, as a special case of our construction, if we choose the degrees of polynomials qi as m and the secret polynomial km− 1, then our construction and [17] have the same security level. we present some results of another example where the secret image s is chosen as a 512 × 512 pixels pepper image. the histograms of the original picture and the shares are provided in figure 4. it is observed that the shares have a very uniform histograms. we would like to point out that getting uniformly distributed encrypted images is a problem by itself and some other techniques besides are used in order to be successful ([7, 15]). here, we do not employ additional tools in order to get this achievement. also we have computed the entropy of the original image and the shares s1, s2, s3 and s4 as 7.5937, 7.9700, 7.9700, 7.9701, 7.9702 respectively. the last four evaluations that correspond to the shares are close to 8 which point to a good entropy level. further, the structural similarity index measure (ssim)[20] is applied between the shares that solve the problem (we recall that two of them solve the original image) and the original image and these results are given in table 1. the results in table 1 show that ssim is close to zero which also gives a promising result. 157 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 figure 4: histogram of 512 × 512 original image (oi); histograms of 256 × 512 shadow images s1,s2,s3,s4 respectively share images ssim s1 and s2 −2.6959 × 10−04 s1 and s3 −3.2801 × 10−04 s1 and s4 −7.0751 × 10−05 s2 and s3 2.7211 × 10−04 s2 and s4 5.2297 × 10−04 s3 and s4 5.2318 × 10−04 table 1: ssim of shares and the original image. we present the performance comparison between these three algorithms over the above example graphically (figure 2). the graph shows the elapsed time of the secret sharing process over the various finite fields. (a 2.5 ghz processor and 8.00 gb ram standard computer over matlab r2015b is used.) as seen, the performance over gf(256) is slower than the other fields. this is due to the algebraic manipulations over the extension field. on the other hand, there is no meaningful difference between the 158 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 figure 5: elapsed time of secret sharing process over the various fields table 2: comparison of secret image sharing methods sis entropy size needing s1 s2 s3 s4 sc 7.8942 7.8942 7.8942 7.8942 256 × 512 prn padding tl 7.9080 7.9655 7.9624 7.9696 256 × 512 permutation ots 7.9700 7.9700 7.9701 7.970 256 × 512 no need other two prime fields and algorithms are faster compared to the extension field gf(256). 7. a comparison with other schemes we show the efficiency of our scheme (ots) by comparing it with the threshold secret image sharing (sis) schemes introduced in [13] (sc) and [17] (tl). the entropies and sizes of shadow images are given in table 2. the entropies of shadow images constructed by our scheme are closer to 8 which indicates the reliable security of the system, despite our scheme does not need any permutation or random numbers padding. without applying a permutation, the original image is still perceptible by naked eye after (2,4) secret image sharing using the scheme of [17]. we also derive the same security level with [17] without using a permutation which is also required to be shared as a secret. the sizes of the shadow images are all half of the size of the original image for given applications. however, as distinct from others, the sizes of the shadow images can differ in our scheme. further, by waiving the scheme to be threshold, we can improve the flexibility of the system and discriminate the participants for authorization as shown in the next illustrative example. by choosing four irreducible polynomials q1 = x6 +10x5 +75x4 +23x3 +64x+3, q2 = 5x 2 + 3, q3 = 215x2 + 157 and q4 = x6 + 101x5 + 250x4 + 123x3 + 99x + 231, we obtain the shadow images of sizes 196 × 256, 64 × 256, 64 × 256 and 196 × 256 respectively (figure 6). also, a shadow image would be of size 160 × 256 for an irreducible polynomial of degree 5 or 96 × 256 for an irreducible polynomial of degree 3. the shadow images s2 and s3 cannot reconstruct the secret image s, for instance. on the other hand, the scheme given in [13] is not as flexible as our scheme. for instance, 512×512 pixels gray color image forces us to choose d as a power of 2 and hence qis which are smaller than 255 as given in the example forces us to choose d = 8, in order to design a (2,4) threshold scheme. therefore, the sizes of shadow images will be the same as the original image. besides, the choice of qis are not 159 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 (a) (b) figure 6: secret image s (a) and shadow images s1 of size 196 × 256, s2 of size 64 × 256, s3 of size 64 × 256 and s4 of size 196 × 256 respectively (b) so independent for a secret image sharing since the pixel values of an image are determined in a fixed spectrum. further, the histograms of shadow images may give an information about qis. 8. conclusion in this study, we generalize mignotte’s scheme over euclidean domains and present a new threshold secret image sharing. we show that mignotte’s generalized construction over gaussian integers has higher security level than mignotte’s construction over integers. we also give threshold secret image sharing algorithms for primes p = 251, p = 257 and over galois field gf(256). one of the advantages of these algorithms is that there is no need to apply a permutation which needs to be known by all participants permutation as in [17] or add a random value as in [13]. unlike [17], in our scheme, one can choose arbitrarily the number of pixels s which is the length of each partition that the secret image divided into, providing that s divides r for an m × r secret image to construct (k,n) scheme and so the choice of k which is the minimum number of participants who can reconstruct the secret is flexible. unlike [13] and [17], the sizes of the shadow images can differ and so this increases the applicability and the security. we also show that finding the secret image for unauthorized person is an infeasible problem. furthermore, in [16] there is a restriction on the degree of the irreducible polynomials which narrows the selection choice 160 i. ozbek et al. / j. algebra comb. discrete appl. 6(3) (2019) 147–161 of such polynomials to be applied. here, we do not impose such a restriction which makes the scheme more practical and secure. we finally realize our proposed method over an example which illustrates the new algorithm. references [1] c. asmuth, j. bloom, a modular approach to key safeguarding, ieee trans. inform. theory 29(2) (1983) 208–210. 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[8] p. k. meher, j. c. patra, a new approach to secure distributed storage, sharing and dissemination of digital image, ieee international symposium on circuits and systems (2006) 373-376. [9] m. mignotte, how to share a secret,in: cryptography, eurocrypt 1982, lecture notes in computer science 149 (1983) 371–375. [10] m. naor, a. shamir, visual cryptography, in: advances in cryptology–eurocrypt 1994, lecture notes in computer science 950 (1994) 1–12. [11] o. ore, the general chinese remainder theorem, the american mathematical monthly 59(6) (1952) 365–370. [12] a. shamir, how to share a secret, comm. acm 22(11) (1979) 612–613. [13] s. j. shyu, y. r. chen, threshold secret image sharing by chinese remainder theorem,ieee asia– pacific services computing conference (2008) 1332–1337. [14] d. r. stinson, an explication of secret sharing schemes, des. codes cryptogr. 2(4) (1992) 357–390. [15] s. somaraj, m. a. hussain, performance and security analysis for image encryption using key image, indian journal of science and technology 8(35) (2015). [16] g. tatyana, m. genadii, generalized mignotte’s sequences over polynomial rings, electronic notes in theoretical computer science 186 (2007) 43–48. [17] c. c. thien, j. c. lin, secret image sharing, comput. graph. 26(5) (2002) 765–770. [18] g. ulutas, m. ulutas, v. nabiyev, secret sharing scheme based on mignotte’s scheme, 2011 ieee 19th signal processing and communications applications conference (2011) 291–294. [19] r. z. wang, c. h. su, secret image sharing with smaller shadow images,pattern recognition lett. 27(6) (2006) 551–555. [20] z. wang, a. c. bovik, h. r. sheikh, e. p. simoncelli, image quality assessment: from error visibility to structural similarity, ieee transactions on image processing 13(4) (2004) 600–612. 161 https://doi.org/10.1109/tit.1983.1056651 https://doi.org/10.1109/tit.1983.1056651 https://doi.ieeecomputersociety.org/10.1109/afips.1979.98 https://doi.ieeecomputersociety.org/10.1109/afips.1979.98 https://doi.org/10.1016/j.entcs.2007.01.065 https://doi.org/10.1016/j.entcs.2007.01.065 https://doi.org/10.1109/sibgrapi.2012.17 https://doi.org/10.1109/sibgrapi.2012.17 https://doi.org/10.1109/iscas.2006.1692600 https://doi.org/10.1109/iscas.2006.1692600 https://doi.org/10.1007/3-540-39466-4_27 https://doi.org/10.1007/3-540-39466-4_27 https://doi.org/10.1007/bfb0053419 https://doi.org/10.1007/bfb0053419 https://doi.org/10.1080/00029890.1952.11988142 https://doi.org/10.1080/00029890.1952.11988142 https://doi.org/10.1145/359168.359176 https://doi.org/10.1109/apscc.2008.223 https://doi.org/10.1109/apscc.2008.223 https://doi.org/10.1007/bf00125203 https://doi.org/10.17485/ijst/2015/v8i35/73141 https://doi.org/10.17485/ijst/2015/v8i35/73141 https://doi.org/10.1016/j.entcs.2006.12.044 https://doi.org/10.1016/j.entcs.2006.12.044 https://doi.org/10.1016/s0097-8493(02)00131-0 https://doi.org/10.1109/siu.2011.5929644 https://doi.org/10.1109/siu.2011.5929644 https://doi.org/10.1016/j.patrec.2005.09.021 https://doi.org/10.1016/j.patrec.2005.09.021 https://doi.org/10.1109/tip.2003.819861 https://doi.org/10.1109/tip.2003.819861 introduction preliminaries generalization of mignotte's scheme over euclidean domains a new algorithm for secret image sharing an experimental result security analysis a comparison with other schemes conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.561316 j. algebra comb. discrete appl. 6(2) • 63–74 received: 23 march 2018 accepted: 12 march 2019 0 journal of algebra combinatorics discrete structures and applications fibonacci numbers and resolutions of domino ideals research article rachelle r. bouchat, tricia muldoon brown abstract: this paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a 2 × n tableau. the multi-graded betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. it is well-known that the number of domino tilings of a 2 × n tableau is given by a fibonacci number. using the bijection, this relationship is further expanded to show the relationship between the fibonacci numbers and the graded betti numbers of the corresponding domino ideal. 2010 msc: 05e40, 13a15 keywords: fibonacci numbers, monomial ideals, domino tilings 1. introduction monomial ideals have been studied using mechanisms from several different areas of mathematics, including combinatorics, graph theory, algebra, and topology. given a simplicial complex, there are two common monomial ideals that are studied, the stanley reisner ideal of the complex as well as the facet ideal of the complex (see miller and sturmfels [14] for a comprehensive overview of these results). of particular interest are monomial ideals representing well-known combinatorial objects. for example, conca and de negri [8] introduced the study of edge ideals. these edge ideals are squarefree monomial ideals generated from the edges of a graph. edge ideal results have been extended to the study of path ideals by bouchat, há, and o’keefe [4] and further generalized to facet ideals of simplicial complexes by faridi [10] and to path ideals of hypergraphs by há and van tuyl [12]. furthermore, many other natural combinatorial objects can be used to generate squarefree monomial ideals. in this paper, we consider a class of monomial ideals arising from the set of domino tilings of a 2 ×n tableau. a domino tiling of a 2×n rectangular tableau is a disjoint arrangement of 2×1 tiles placed horizontally or vertically to completely cover the area of the rectangle. domino tilings have been well-studied from rachelle r. bouchat; department of mathematics, indiana university of pennsylvania, usa (email: rbouchat@iup.edu). tricia muldoon brown (corresponding author); department of mathematical sciences, georgia southern university, usa (email: tmbrown@georgiasouthern.edu). 63 https://orcid.org/0000-0003-2286-0805 https://orcid.org/0000-0003-3835-1175 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 a combinatorial viewpoint, and they have many interesting properties. for example, using demoivre’s initial conditions on fibonacci numbers, a classic exercise can show the number of 2 ×n domino tilings is given by the nth fibonacci number. further, a survey paper by ardilla and stanley [2] gives many results for domino and more generalized tilings of the plane. in this paper, we will show the fibonacci numbers also occur naturally when enumerating graded betti numbers related to domino tilings. tilings have been studied in terms of their enumeration, their intersection, and their connection to graph theory. see fisher and temperley [16] and kasteleyn [13] for the first enumerative results; butler, horn, and tressler [7] for intersection results; or benedetto and loehr [3] for graph theoretical results. these results, among others, suggest that interpretation as monomial ideals will also be of interest. here, we extend previous work by the authors [6] to enumerative results concerning the multi-graded and graded betti numbers of the ideal corresponding to the set of all 2 ×n domino tilings. for a field k, to each domino position in the 2 × n tableau, we associate a variable in the ring r = k[x1, . . . ,x2(n−1),y1, . . . ,yn]. the xi for 1 ≤ i ≤ n− 1 are associated with the horizontally placed dominoes covering entries (1, i) and (1, i+ 1), the xi for n ≤ i ≤ 2(n−1) are associated with the dominoes covering entries (2, i− (n− 1)) and (2, i− (n− 1) + 1), and the yi for 1 ≤ i ≤ n are associated with the vertically oriented dominoes covering entries (1, i) and (2, i). definition 1.1. consider a 2 ×n tableau being tiled with 2 × 1 tiles, and denote the set of tilings of the tableau by tn = {τ : τ is a tiling of the 2 ×n tableau}. let zi ∈{x1, . . . ,x2(n−1),y1, . . . ,yn}. 1. the tiling monomial xτ associated to the tiling τ is the monomial xτ = ∏ z δzi (τ) i where δzi (τ) = { 1 , if zi ∈ τ 0 , else 2. associated to a collection of domino tilings {τ1, . . . ,τm} is the domino monomial xτ1,...τm = ∏ z δzi ({τ1,...,τm}) i where δzi ({τ1, . . . ,τm}) = { 1 , if zi ∈ τj for some 1 ≤ j ≤ m, 0 , else. example 1.2. the tiling monomial associated to the following tiling of the 2 × 7 tableau  x1 x3 x5 y7x7 x9 x11   is x1x3x5x7x9x11y7, and the domino monomial associated to the collection of tilings  x1 x3 x5 y7x7 x9 x11 , y1 y2 y3 x4 x6 x10 x12   is x1x3x4x5x6x7x9x10x11x12y1y2y3y7. definition 1.3. the domino ideal corresponding to an 2 ×n tableau is the ideal i := (xτ : xτ is a tiling monomial). we note, the generating set of the domino ideals associated to the set of all 2×n domino tilings can also be viewed as paths of a graph. however, the ideals are not path ideals, as their generating sets do not correspond to all paths of a specified length within a graph, but rather just a subset, as illustrated in example 1.4. 64 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 example 1.4. consider a 2×3 tableau, then the domino ideal i is generated by the domino monomials corresponding to the tilings in the set:  x1 y3x3 , y1 x2 x4 , y1 y2 y3   then i = (x1x3y3,x2x4y1,y1y2y3) ⊂ k[x1,x2,x3,x4,y1,y2,y3]. notice that i is generated by a subcollection of the paths of length two in the graph: x3 x1 y3 y2 y1 x2 x4 before stating our results, we give the necessary background from topology and commutative algebra. 2. background in this paper, we will focus on properties of domino ideals relating to the corresponding minimal free resolutions. since a domino ideal, i, can be viewed as a finitely generated graded r-module. associated to i is a minimal free resolution, which is of the form 0 → ⊕ a r(−a) βp,a(i) δp−→ ⊕ a r(−a) βp−1,a(i) δp−1−→ ··· δ1−→ ⊕ a r(−a) β0,a(i) → i → 0 where the maps δi are exact and where r(−a) denotes the translation of r obtained by shifting the degree of elements of r by a ∈ nn. the numbers βi,a(i) are called the multi-graded betti numbers of i, and they correspond to the number of minimal generators of degree a occurring in the ith-syzygy module of i. the graded betti numbers for a finitely generated ideal i, can be computed using the software system macaulay2 (see [11]) and are displayed in a betti table where: 0 1 · · · i · · · total: – – – 0: 1: ... j : βi,i+j(i) ... example 2.1. consider the domino ideal i = (x1x3y3,x2x4y1,y1y2y3) ⊂ k[x1,x2,x3,x4,y1,y2,y3] corresponding to a 2 × 3 tableau. then the betti table from macaulay2 corresponding to the minimal free resolution of i is: 0 1 2 total : 3 3 1 3 : 3 . . 4 : . 2 . 5 : . 1 1 from this betti table, it is easy to form the graded minimal free resolution of i: 0 −→ r(−7) −→ r2(−5) ⊕ r(−6) −→ r3(−3) −→ i −→ 0. 65 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 throughout this paper, we will be using a result adapted from theorem 2.8 of alilooee and faridi [1] to calculate the multi-graded betti numbers for path ideals of rooted trees. before we provide this result, we need the following definitions from simplicial topology. definition 2.2. 1. an abstract simplicial complex, ∆, on a vertex set x = {x1, . . . ,xn} is a collection of subsets of x satisfying: (a) {xi}∈ ∆ for all i, and (b) f ∈ ∆, g ⊂ f =⇒ g ∈ ∆. the elements of ∆ are called faces, and the maximal faces (under inclusion) are called facets. the simplicial complex ∆ with facets f1, . . . ,fs will be denoted by 〈f1, . . . ,fs〉. 2. for any y ⊆ x, an induced subcollection of ∆ on y, denoted by ∆y, is the simplicial complex whose vertex set is a subset of y and whose facet set is given by {f | f ⊆y and f is a facet of ∆}. 3. if f is a face of ∆ = 〈f1, . . . ,fs〉, the complement of f in ∆ is given by fcx = x \ f, and the complementary complex is then ∆cx = 〈(f1) c x , . . . , (fs) c x〉. note, in the setting of domino monomials, we let xτ represent either a monomial in the ring r = k[x1, . . . ,x2(n−1),y1, . . . ,yn] or a simplex whose vertices are given by the variables. thus given x = xτ1,τ2,...,τm, the complex γx is the simplicial complex 〈xτ1,xτ2, . . . ,xτm〉 and its complementary complex γcx is given by γcx = 〈(xτ1 ) c x, (xτ2 ) c x, . . . , (xτm ) c x〉. the following example illustrates this complex. example 2.3. consider the domino monomial x = x1x2x4x5x6x8y1y2y3 originating from a 2×5 tableau, which is generated from the set of tilings {x1x4x5x8y3,x2x4x6x8y1,x4x8y1y2y3}. we have γcx = 〈x2x6y1y2,x1x5y2y3,x1x2x5x6〉. we often apply a deformation retraction to associate variables that always appear together in the complement, such as xi with xn−1+i, and in doing so we obtain the following complex that is topologically equivalent to γcx: γcx '〈x2y1y2,x1y2y3,x1x2〉' s 1 complementary complexes are important in determining the betti numbers as we see in the following corollary. corollary 2.4 (corollary 2.11 in [5]). let s = k[x1, . . . ,xn] be a polynomial ring over a field k, and let i be a pure, squarefree monomial ideal in s. then the multi-graded betti numbers of i are given by βi,a(i) = dimk h̃i−1 ( γcvert(γ) ) where γ is an induced subcollection of ∆(i) with vert(γ) = {xi | ai = 1} where a = (a1, . . . ,an). concluding the background results, we observe the following: corollary 2.5. let i be a domino ideal. then βi,a(i) ∈{0, 1}. as we can find a path ideal containing each domino ideal and because path ideals have this property (see erey and faridi [9]), the result follows immediately. 66 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 3. statistics in order to associate the domino monomials with the appropriate multi-graded betti numbers, we first introduce three statistics on the set of domino monomials. definition 3.1. given a domino monomial x corresponding to the 2 ×n tableau: 1. set d(x) = d1d2 · · ·dn ∈ {1, 2, 3}n where di = #{z ∈ {xi−1,xi,yi} : z|x} for all 1 ≤ i ≤ n− 1 and dn = #{z ∈{xn−1,yn} : z|x}. we say d(x) is the depth sequence of x. 2. set v(x) = #{yiyi+1 : yiyi+1|x,xixn−1+i x, and di = di+1 = 2} to be the number of pairs of vertical dominos in x with a 2 in the depth sequence at these positions such that the corresponding pair of horizontal dominos is not in x. the depth sequence is so named because if all dominos of the monomial x were minimally stacked in their respective positions on the 2 ×n tableau, the height at position i would correspond to the value of the depth sequence di. both of these statistics will be utilized to classify sets of domino monomials by their the multi-graded betti numbers βi,a(in). to simplify notation we let 1k represent the depth sequence given by the string of k ones 11 · · ·1, and similarly let 2k = 22 · · ·2 and 3k = 33 · · ·3 describe the depth sequences of strings of k twos and threes, respectively. thus, for example, the depth sequence 2221233211 can be written as 231232212. before defining the third statistic, we need to introduce two operations on depth sequences. definition 3.2. let d(x) = d1d2 · · ·dn ∈{1, 2, 3}n be a given depth sequence. o1. we may double the sequence by taking any subsequence of 1k , for 1 ≤ k ≤ n, and replacing it with 2k. o2. we may triple the sequence by taking any single subsequence of 23 and replacing the middle two with a three. to illustrate the above definitions for the two operations, we provide the following examples. example 3.3. consider the depth sequence d(x) = 1111222211. then we can double d(x) to obtain the sequence 2221222211 or 1111222222, among others. we could also triple d(x) to obtain 1111232211 or 1111223211. we can now introduce the next statistic on domino monomials. definition 3.4. the starting column of a domino monomial x is given by the minimum number of operations, doubles or triples, needed to obtain the depth sequence d(x) from the depth sequence 1n. we write sc(x) to represent this quantity. when considering the formation of a domino monomial from the underlying optimal stacking of domino tilings, it becomes clear that the starting column statistic is well-defined. for instance, replacing a subsequence of 1k with 2k could correspond to taking a domino tiling with horizontal tiles at particular entries and combining it with another domino tiling having vertical tiles on those same entries, among other options. similarly, replacing a subsequence of 23 with 232 could correspond to having both pairs of horizontal tiles as well as the two end vertical tiles and then adding in a third tiling which includes the central vertical tile. example 3.5. consider the domino monomial x = x1x3x4x5x6x7x9x10x11x12y1y2y3y7 presented in example 1.2. the depth sequence of x is d(x) = 27, and as 27 is minimally created from 17 by one operation of changing a sequence of seven ones into seven twos, the starting column is sc(x) = 1. 67 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 we intend to use the starting column statistic, sc(x) to identify the minimum label i such that βi,a(in) = 1 where a is the (3n − 2)-tuple such that ai = δzi ({τ1, . . . ,τm}) for x{τ1,...,τm}. we note, monomials with the same depth sequence may appear in different columns of the betti table. thus, the starting column is needed to determine the column i for a given monomial x{τ1,...,τm}. proposition 3.6. let γcx be the complementary complex of a domino monomial x such that xixn−1+i | x, yi x, and yi+1 x for some 1 < i < n− 1. further, let x′ be the domino monomial x′ = x yiyi+1 xixn−1+i . if γcx is homotopic to the sphere s j, then γcx′ is homotopic to the sphere s j+1. proof. by corollary 2.5, if γcx is not contractible we may assume it is homotopic to a sphere s j for some j ≥ 0. now, when transitioning from the domino monomial x to x′ by replacing the variables xixn−1+i with yiyi+1, the vertex set of γx also changes to the vertex set of γx′ in the same way. thus every facet in the complementary complex γcx can be mapped to a facet of γ c x′ through this replacement. that is, a tiling in γx containing xixn−1+i is mapped to a tiling in γx′ containing yiyi+1, and the complements of these tilings within their respective complementary complexes are the same. further, a tiling in γx which does not contain xixn−1+i is mapped to itself in γx′, and thus the complement of that tiling in γcx which contains xixn−1+i is mapped to the facet of γcx′ containing yiyi+1 using the replacement. first, we note by our hypothesis the image of γcx in γ c x′ is homotopic to s j. second, these facets may be grouped into two disjoint sets; that is, into the set of facets which contain the vertices yiyi+1 and the set of facets which contain xi−1xi+1. now, the remaining facets of γcx′ must be complements of tilings that contained yi or yi+1, but not both. therefore, these facets can also be put into two disjoint sets, namely the set of complementary facets which contain the vertices xi+1yi or the set of complementary facets which contain the vertices xi−1yi+1. thus, in the complex γcx′, we have disjoint sets of facets containing xi−1xi+1 or yiyi+1, forming a sphere; and moreover, both of these sets are connected to those facets containing xi−1yi+1, creating a pyramid over the sphere sj. on the other hand these disjoint sets of facets which form the sphere are also connected to xi+1yi, creating another pyramid over this sphere. as there is no connection between the apex of either of these two pyramids, we see the complex is actually a bipyramid over a sphere, sj, which is homotopic to the sphere sj+1, and we have proven the claim. note, although the action in proposition 3.6 of replacing two interior stacked horizontal tiles with their respective pair of vertical tiles affects the dimension of the complementary complex it does not change the depth sequence, that is d(x) = d(x′). recall, in the case that the depth sequence of a domino monomial is a string of ones, we know the domino monomial corresponds to a single tiling, and as a generator of the ideal we have β0,a(in) = 1. thus, if i = 0, the multi-graded betti number, β0,a(in) = 1 if and only if x = xτ. the next proposition describes the multi-graded betti numbers for i ≥ 1. proposition 3.7. given a domino monomial x = x{τ1,...,τm}, let a = (a1,a2, . . . ,a3n−2) where ai = δzi ({τ1, . . . ,τm}). then for i ≥ 1, the multi-graded betti number βi,a(in) = 1 if and only if i = sc(x) + v(x). proof. let x be a domino monomial. we first consider the domino monomials where v(x) = 0; that is, the monomials whose corresponding multi-graded betti numbers are non-zero when i = sc(x). since there is always a deformation retraction that takes xixn−1+i to xi, we will use the variable xi for 1 ≤ i ≤ n−1 to represent the monomial xixn−1+i. case: d(x) ∈ {1, 2}n \ 1n. assume the depth sequence consists of m strings of consecutive twos separated by strings of ones. for example: 222211122221112211 m = 3 122122122221122122 m = 5 for each string of twos in columns ci,ci + 1, . . . ,ci + ki where 1 ≤ ci ≤ n − 1 indexes the first column of the ith string of twos for 1 ≤ i ≤ m and ki ≥ 1, consider a partial domino monomial xcixci+1 · · ·xci+kiyciyci+ki which corresponds to the pair of tilings 68 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 si = { {xcixci+2 · · ·xci+ki−1,xci+1xci+3 · · ·xci+ki−2yciyci+ki} if n is even {xcixci+2 · · ·xci+ki−2yci+ki,xci+1xci+3 · · ·xci+ki−1yci} if n is odd (note, a sequence of only one 2 is impossible so ki > 0.) the maximal set of tilings in the equivalence class of x, sx, consists of the concatenation of one element from the pair of tilings in si for all 1 ≤ i ≤ m, along with fixed choices for the variables representing the strings of ones. in the complementary complex, every variable corresponding to a one in the depth sequence appears in each generator, and thus does not appear in any facets of the complementary complex. furthermore, note that the two sub-tilings in si are complementary to each other, so we may represent si as {ti, t̄i}. with this notation: γcx = 〈t1t2 · · ·tm, t̄1t2 · · ·tm, t1t̄2 · · ·tm, . . . , t1t2 · · · ¯tm, t̄1t̄2 · · ·tm, . . . , t̄1t2 · · · ¯tm, . . . , t1t̄2 · · · ¯tm, . . . , t̄1t̄2 · · · ¯tm〉 is homotopic to the complex with m vertices and m complementary vertices, such that in each of the 2m m-dimensional facets no element is in the same facet as its complement, but is in every other facet that does not contain its complement. as this complex is the cross-product of m zero-dimensional spheres, it is homotopic to a sphere of dimension m− 1. thus we have βm,a(in) = dimkh̃m−1(sm−1) = 1 and i = sc(x) + v(x) = m. case: d(x) ∈ {1, 2, 3}n \{1, 2}n. first assume that x can be written x = x̂ ·yc for some column c for 1 < c < n− 1 and some x̂ where d(x̂) ∈ {1, 2}n \ 1n, and thus the depth sequence of x has exactly one three. we compare the complementary complex γcx with the complementary complex γx̂. from the case above, we know γcx̂ is homotopic to a sphere in dimension m− 1. further every tiling in γx̂ is also a tiling in γx, and thus every facet in γcx̂ appears as a part of a facet of γ c x by the simplicial join with yc. therefore, the image of γcx̂ ∗yc in γ c x is a cone over an (m− 1)-dimensional sphere. the rest of γcx is generated by the complements of tilings which contained the vertical tile yc. recall, that the sphere γcx̂ can be described as strings of barred and unbarred vertices where, without loss of generality, the unbarred vertex corresponds to the tiling with odd indexed x variables and the barred vertex corresponds to the tiling with even indexed x variables. any tiling containing horizontal tiles and yc must contain both odd and even indexed x variables, and hence the complementary facet also contains odd and even indexed x vertices while also avoiding yc. therefore these complements contain tiles in ti and t̄i and so fill the void in the sphere described by γcx̂. thus γ c x is the cone by yc over the boundary of an m-dimensional ball and thus is homotopic to a sphere sm. for depth sequences with more than one three, this process can be iterated so the result follows from the previous case. finally, we need to consider domino monomials with a given depth sequence d(x) where v(x) > 0. consider the set of pairs {yiyi+1 : yiyi+1|x,xixn−1+i x, and di = di+1 = 2} note, if we replace any subset of non-overlapping pairs of yiyi+1 in the monomial x with the corresponding xixn−1+i, the depth sequence remains unchanged. applying proposition 3.6, we see each of these replacements of yiyi+1 with xixn−1+i decreased the dimension of the sphere by one, as well as decreased the statistic v(x) by one. by reversing this process we have proven the claim. example 3.8. consider the domino monomial x = x1x3x4x5x6x7x9x10x11x12y1y2y3y7 from example 1.2, and observe that x contains two adjacent pairs of vertical dominos. however, for only one of these pairs, y2y3, is the corresponding pair of horizontal dominos absent, so we have i = sc(x) + v(x) = 2 + 1 = 3. thus, the equivalence class of tilings x corresponds to the multi-graded betti number β3,a(i7) = 1 where a = (a1 . . . ,a3n−2) is such that am = 1 if and only if m|x and am = 0 otherwise. we may now state our main result. 69 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 theorem 3.9. let tn be the set of domino tilings of a 2 ×n tableau, and let in be the monomial ideal generated by the tilings τi ∈ tn. then, the set of non-zero multi-graded betti numbers βi,a(in) for i ≥ 0 are in bijection with the set of domino monomials generated from tn. proof. given a domino monomial, proposition 3.7 shows the appropriate multi-graded betti number is non-zero as the complementary complex corresponds to a sphere. by definition, non-zero multi-graded betti numbers can be described by a set of domino tilings, so the result follows. 4. enumeration as the correspondence between domino monomials and non-zero multi-graded betti numbers of the domino ideal is well understood, we now wish to utilize this result to enumerate graded betti numbers of the domino ideal. we will see that the nature of the tilings as counted by fibonacci numbers is carried through to the sets of tilings. as stated, the number of 2 ×n domino tilings is given by the nth fibonacci number fn where the fibonacci number are defined by the recursion with demoivre’s initial conditions, fn+2 = fn+1 + fn; f0 = 1,f1 = 1. the fibonacci numbers are one of the most ubiquitous sequences in combinatorics. one can see the entry a000045 in oeis[15] to find numerous references to sets of combinatorial objects enumerated by the fibonacci numbers. we will use products of fibonacci numbers and binomial coefficient expansions of fibonacci numbers to count sets of tilings and consequently to determine the graded betti numbers. in section 3, we defined a depth sequence for a monomials associated with sets of domino tilings. now, we begin by considering specific subsequences which we will call fundamental. using the fundamental sequences, we then build all possible depth sequences corresponding to tiling monomials. the fundamental subsequences are: 1k, 2k, 12k3, 32k1, 32k3, and 3k the first class of fundamental tilings is already well-understood, and thus the number of domino tilings of depth 1k is given by fk for 0 ≤ k ≤ n. further, a subsequence of k ones in the depth sequence of a domino monomial implies that every 2 ×n tiling in the set of tilings which comprise the monomial x must contain the same set of k tiles chosen in fk ways. because there can be no overlap between tiles chosen for this subsequence and those in other positions of the depth sequence, when counting we may choose this set of dominos independent of the number of ways to choose sets of dominoes to fill in the rest of the 2 × n rectangle. thus the tilings with a given depth sequence may be enumerated by the product of appropriate fibonacci numbers for each string of one’s multiplied by the number of ways to choose dominos for the sequences of two’s and three’s. so next, consider the fundamental depth sequences of only two’s or three’s. for a given depth sequence d(x) and a given column in the betti table i ≥ 1, we wish to understand how many distinct domino monomials x have this depth sequence and correspond to a multi-graded betti number in column i. proposition 4.1. given a domino monomial x = x{τ1,...,τm}, let a = (a1,a2, . . . ,a3k−2) where ai = δzi ({τ1, . . . ,τm}). for k ≥ 2, the number of domino monomials x with non-zero multi-graded betti number βi,a(in) = 1 and depth sequence 2k is ( k−i−1 i−1 ) . proof. if the depth of a domino monomial corresponding to a set of 2 ×k tilings is 2k, the first and last columns must contain the vertical dominos y1, yk and the horizontal dominos x1xk and xk−1x2(k−1). therefore there are k − 2 remaining columns in which a choice of dominos can be made. each of these 70 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 columns must be covered with one pair of horizontal dominos and either another pair of horizontal dominos or a vertical domino. we see if d(x) = 2k then sc(x) = 1 and so proposition 3.7 gives if i = 1, then v(x) = 0 which corresponds to the one domino monomial x = x1x2 · · ·x2k−2y1yk containing all the horizontal dominos. thus we satisfy the claim with ( k 0 ) = 1. for i > 1, choose i−1 pairs of non-intersecting horizontal dominos of the form xjxk−1+j to be replaced in the monomial by the pair yj,yj+1. these selections must be non-intersecting because removing both xjxk−1+j and xj+1xk+j leaves a depth of only 1 in column j + 1. further, proposition 3.6 notes that each of these replacements increases the dimension of the complementary complex by one and as there are ( k−2−(i−1) i−1 ) ways to choose i− 1 non-consecutive integers from the set [k − 2], we have the ( k−i−1 i−1 ) ways to choose the non-intersecting pairs of horizontal domino tiles enumerates the domino monomials corresponding to a non-zero ith multi-graded betti number. in the proof of proposition 4.1, we assumed 2k was a depth sequence corresponding to a domino monomial of a 2 × k rectangle. however, by adjusting indices the domino monomials can easily be modified to describe a factor of a 2 ×n domino monomial whose depth sequence contains a subsequence of k consecutive twos adjacent to a subsequence or subsequences of ones. now consider fundamental subsequences of length k + 1 and k + 2 containing two’s or three’s and contained inside a larger length n depth sequence. first, a subsequence of threes, 3k, in columns j,j + 1, . . . ,j + k implies that all dominos, both horizontal and vertical, in those columns must be included in the monomial. thus there is only one way to chose factors for the monomial in those positions. we know that subsequence of threes must be followed and preceded by a string of twos, so we need to understand how these threes affect the choice for an adjacent subsequence of twos. proposition 4.2. given a domino monomial x = x{τ1,...,τm}, let a = (a1,a2, . . . ,a3n−2) where ai = δzi ({τ1, . . . ,τm}). for k ≥ 2, suppose the partial depth sequence djdj+1 · · ·dj+k+1 is 32k1 or 12k3, respectively. then, the number of factors f of x containing dominos from the set {xj, . . .xj+k,xj+n−1, . . . ,xj+k+n−1,yj, . . . ,yj+k} or {xj+1, . . .xj+k+1,xj+n, . . . ,xj+k+n,yj+1, . . . ,yj+k+1}, respectively, such that v(f) = i is ( k−i i−1 ) . similarly, suppose the partial depth sequence djdj+1 · · ·dj+k+1 is 32k3. then, the number of factors f of x containing dominos from the set {xj, . . .xj+k,xj+n, . . . ,xj+k+n,yj, . . . ,yj+k+1} such that v(f) = i is ( k−i+1 i−1 ) . proof. similar to the proof of proposition 4.1, we observe that in the case the subsequence of twos is preceded or followed by a three we are allowed one more position in which xjxj+n−1 may be replaced with yjyj+1. thus we have k−1 or k positions to choose i−1 non-intersection pairs of horizontal dominos to be replaced by pairs of vertical dominos and the result follows. these results now allow us to determine graded betti numbers by listing all depth sequence created from a string of n ones using operations o1 and o2, separating these sequences into their fundamental subsequences, enumerating each subsequence using the results above, and finally multiplying. example 4.3 illustrates this process. example 4.3. letting n = 7, suppose we wish to calculate βi,13. we write all possible combinations of 1’s, 2’s, and 3’s whose sum is 13 and were created from the sequence 17 using the operations o1 and o2 as 71 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 d(x) multiplicity count sc(x) i = 1 i = 2 i = 3 2222221 2 [( 4 0 ) + ( 3 1 ) + ( 2 2 )] · f1 1 2 6 2 2222122 2 [( 2 0 ) + ( 1 1 )] · f1 · ( 0 0 ) 2 2 2 2221222 1 ( 1 0 ) · f1 · ( 1 0 ) 2 1 2322211 2 ( 0 0 ) · [( 2 0 ) + ( 1 1 )] · f2 2 4 4 1232221 2 f1 · ( 0 0 ) · [( 2 0 ) + ( 1 1 )] · f1 2 2 2 1123222 2 f2 · ( 0 0 ) · [( 2 0 ) + ( 1 1 )] 2 4 4 2232211 2 ( 1 0 ) · ( 1 0 ) · f2 2 4 1223221 1 f1 · ( 1 0 ) · ( 1 0 ) · f1 2 1 233211 2 ( 0 0 ) · ( 0 0 ) · f3 3 6 1233211 2 f1 · ( 0 0 ) · ( 0 0 ) · f2 3 4 2321221 2 ( 0 0 ) · ( 0 0 ) · f1 · ( 0 0 ) · f1 3 2 2321122 2 ( 0 0 ) · ( 0 0 ) · f2 · ( 0 0 ) 3 4 1232122 2 f1 · ( 0 0 ) · ( 0 0 ) · f1 · ( 0 0 ) 3 2 2 24 32 we have β1,13(i7) = 2, β2,13(i7) = 24, and β3,13(i7) = 32. figure 1. possible depth sequences whose entries sum to 13 used to compute βi,13(i7) shown in the first column of the table in figure 1. keeping track of the starting column, we enumerate the subsequences by propositions 4.1 and 4.2 and multiply. note, the multiplicity column counts whether or not the reverse of the sequence is distinct from the original sequence. thus β1,13(i7) = 2, β2,13(i7) = 24, and β3,13(i7) = 32. if we wish to disregard the column in the betti table, we may use fibonacci numbers to enumerate all domino monomials with a given depth sequence. corollary 4.4. the number of domino monomials x with the depth sequence d(x) is given by the product of counts for its fundamental subsequences as given in the table below. 72 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 6(2) (2019) 63–74 sequence count 1k fk 2k fk−2 12k3 or 32k1 fk−1 32k3 fk 3k 1 the above corollary is a result of summing over all 0 ≤ i ≤b(k − 2)/2c and the identity fk = bk/2c∑ j=0 ( k − j j ) , found in sequence a000045 of oeis [15]. example 4.5. given the depth sequence d(x) = 2222333321112211232, in order to find the number of domino monomials x with this depth sequence we separate the monomial into its seven fundamental subsequences as follows: d(x) = 22223|33|32|111|22|11|232 thus #{x|d(x) = 24342132212232} = f3 · 1 ·f0 ·f3 ·f0 ·f2 ·f0 ·f0 = 18. we make a final observation on the second column of the betti table associated to the domino ideal. while the i = 0 column of the betti table is given by fibonacci numbers, the i = 1 column also has a nice description in terms of fibonacci numbers. all domino monomials counted by multi-graded betti numbers where i = 1 must come from one action of o1, that is, switching a consecutive sequence of ones into twos. thus, sc(x) = 1 and consequently v(x) = 0 for any such monomial so there is only one choice of all the horizontal dominos for the tilings in the columns whose depth sequence is labeled with twos. because we now only need to count the number of ways to tile the remaining region(s) labeled with ones, we have the following corollary. corollary 4.6. in the minimal free resolution 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10.13069/jacodesmath.39202 journal of algebra combinatorics discrete structures and applications graphical sequences of some family of induced subgraphs research article s. pirzada∗, bilal a. chat∗∗, farooq a. dar department of mathematics, university of kashmir, hazratbal srinagar-190006, india abstract: the subdivision graph s(g) of a graph g is the graph obtained by inserting a new vertex into every edge of g. the svertex or sver join of the graph g1 with the graph g2, denoted by g1∨̇g2, is obtained from s(g1) and g2 by joining all vertices of g1 with all vertices of g2. the sedge or sed join of g1 and g2, denoted by g1∨̄g2, is obtained from s(g1) and g2 by joining all vertices of s(g1) corresponding to the edges of g1 with all vertices of g2. in this paper, we obtain graphical sequences of the family of induced subgraphs of sj = g1 ∨ g2, sver = g1∨̇g2 and sed = g1∨̄g2. also we prove that the graphic sequence of sed is potentially k4 −e-graphical. 2010 msc: 05c07 keywords: graphical sequences, subdivision graph, join of graphs, split graph 1. introduction let g = (v (g),e(g)) be a simple graph with n vertices and m edges having vertex set v (g) = {v1,v2, · · · ,vn}. the set of all non-increasing non-negative integer sequences π = (d1,d2, · · · ,dn) is denoted by nsn. a sequence π ∈ nsn is said to be graphic if it is the degree sequence of a simple graph g on n vertices, and such a graph g is called a realization of π. the set of all graphic sequences in nsn is denoted by gsn. there are several famous results, havel and hakimi [6–8] and erdös and gallai [2] which give necessary and sufficient conditions for a non-negative sequence π = (d1,d2, · · · ,dn) to be the degree sequence of a simple graph g. a graphical sequence π is potentially h-graphical if there is a realization of π containing h as a subgraph, while π is forcibly h graphical if every realization of π contains h as a subgraph. if π has a realization in which the r + 1 vertices of largest degree induce a clique, then π is said to be potentially ar+1-graphic. we know that a graphic sequence π is potentially kk+1-graphic if and only if π is potentially ak+1-graphic [10, 11]. the disjoint union of the graphs g1 and g2 is defined by g1 ⋃ g2. if g1 = g2 = g, we abbreviate g1 ⋃ g2 as 2g. let kk, ck and pk respectively denote a complete graph on k vertices, a cycle on k vertices and a path on k + 1 vertices. ∗ e-mail: pirzadasd@kashmiruniversity.ac.in ∗∗ e-mail: bilalchat99@gmail.com 95 graphical sequences a sequence π = (d1,d2, · · · ,dn) is said to be potentially kr+1graphic if there is a realization g of π containing kr+1 as a subgraph. if π is a graphic sequence with a realization g containing h as a subgraph, then in [4], it is shown that there is a realization g of π containing h with the vertices of h having |v (h)| largest degree of π. in 2014 [1], bu, yan, x. zhou and j. zhou obtained resistance distance in the subdivision vertex join and edge join type of graphs. also conditions for r-graphic sequences to be potentially k(r)m+1-graphic can be seen in [12]. 2. definitions and preliminary results in the simple graph g, let di be the degree of vi for 1 ≤ i ≤ n. then π = (d1,d2, · · · ,dn) is the degree sequence of g. we note that the vertices have been labelled so that π is in increasing order. the degree sequence π = (d1,d2, · · · ,dn) is said to be potentially ar+1-graphic if it has a realization h = (v (h),e(h)), where v (h) = {u1,u2, · · · ,un} and the degree of ui is di for 1 ≤ i ≤ n, such that the subgraph induced by {u1,u2, · · · ,ur+1} is kr+1. for π = (d1,d2, · · · ,dn) ∈ nsn and 1 ≤ k ≤ n, let π′ = (d1 − 1, · · · ,dk−1 − 1, · · · ,dk + 1 − 1,dk + 2, · · · ,dn), if dk ≥ k, = (d1 − 1, · · · ,dk − 1, · · · ,dk + 1, · · · ,dk−1,dk+1,dn), if dk < k. denote π′k = (d i′ 1 ,d i′ 2 , · · · ,di ′ n−1) where 1 ≤ i′ ≤ n and di ′ 1 ,d i′ 2 , · · · ,di ′ n−1 is a rearrangement of the n− 1 terms of π′. then π′ is called the residual sequence obtained by laying off dk from π. gould, jacobson and lehel [4] obtained the following result. theorem 2.1. if π = (d1,d2, · · · ,dn) is the graphic sequence with a realization g containing h as a subgraph, then there exists a realization g′ of π containing h as a subgraph so that the vertices of h have the largest degrees of π. throughout this paper, we take π1 = (d11,d 1 2, · · · ,d1m) and π2 = (d21,d22, · · · ,d2n) respectively to be the graphical sequence of the graphs g1 and g2. let σ(π) = d1 + d2 + · · · + dn and dt in the graphic sequence π means d occurs t-times. the following definitions will be required for obtaining the main results. definition 2.2. the join of g1 and g2 is a graph sj = g1 ∨ g2 with vertex set v (g1) ∪ v (g2) and an edge set consisting of all edges of g1 and g2, together with the edges joining each vertex of g1 with every vertex of g2. definition 2.3. the subdivision graph s(g) of a graph g is the graph obtained by inserting a new vertex into every edge of g. equivalently, each edge of g is replaced by a path of length 2. figure 1 shows the subdivision graph s(g) of the graph g. the vertices inserted are denoted by open circles. d d d d d d d d d d @ @ @ @ @ @ @ @ @@ � � � � �� u u u u u u u u e e e ee e g s(g) figure 1. 96 s. pirzada, b. a. chat, f. a. dar definition 2.4. the subdivision vertex join of g1 onto g2, denoted by sver = g1∨̇g2, is the graph obtained from s(g1) ∪ g2 by joining every vertex of v (g1) to every vertex of v (g2). figure 2 gives sver = k4∨̇k4. @ @ @ @ u u u u u u u ue e e e e e �� �� �� �� �� � sver = k4∨̇k4 figure 2. definition 2.5. let g1 and g2 be two graphs, let s(g1) be the subdivision graph of g1 and let i(g1) be the set of new inserted vertices of s(g1). the subdivision edge join of g1 and g2 denoted by sed = g1∨̄g2, is the graph obtained from s(g1)∪g2 by joining every vertex of i(g1) to every vertex of v (g2). figure 3 below illustrates this operation by taking sed = k4∨̄k4. @ @ @ @ u u u u u u u u sed = k4∨̄k4 e e e e e e figure 3. definition 2.6. if the vertex set of a graph can be partitioned into a clique and an independent set, then it is called a split graph [5]. let kr and ks be complete graphs on r and s vertices. clearly kr ∨ks is one type of split graph on r + s vertices and is denoted by sr,s. pirzada and bilal [9] defined new types of graphical operations and obtained graphical sequences of some derived graphs. definition 2.7. let kr and ks be any two graphs. let kṙ be the subdivision graph of kr and ks be the complement of ks. the graphs ( bṙ,s ) = kṙ∨̇ks is called the r-vertex sub-division-sr,s-graph and the graph ( br,s ) = kṙ∨ks is called the r-edge sub-division-sr,s-graph. these are illustrated in figures 4, 5, 6 and 7 below by taking the graphs k4 and k2. 97 graphical sequences @ @ @ @ @ @@ u u u u u u k4 k2 figure 4. u u u ue e eee u u k4̇ k2 e figure 5. in 2014, pirzada and bilal [9] proved the following assertion. theorem 2.8. if g1 is a realization of π1 = (d11, . . . ,d 1 m) containing kp as a subgraph and g2 is a realization of π2 = (d21, . . . ,d 2 n) containing kq as a subgraph, then the degree sequence π = (d1, . . . ,dm+n) of the join of g1 and g2 is kp+q-graphic. the purpose of this paper is to find the graphic sequence of the family of induced subgraphs of sj, sver and sed. we also give the characterization for a graphic sequence of sed to be potentially k4 −e-graphical. now we have the following observations. remark 2.9. let lr = ka1,a2,··· ,ar and mr = kb1,b2,··· ,br respectively be the r-partite graphs on r∑ i=1 ai and r∑ i=1 bi vertices. let l1 = r∑ i=1 ai and l′1 = r∑ i=1 bi and define l = r∑ i=1 ( ai + bi ) , m = r∑ i=1 ( a2i + b 2 i ) . clearly, the number of edges in ka1,a2,··· ,ar = |elr| = (r2)∑ i,j=1,i6=j aiaj and the number of edges in kb1,b2,··· ,br = |emr| = (r2)∑ i,j=1,i6=j bibj. remark 2.10. let cl2m be the circular ladder with 2m vertices, m ≥ 3. the circular ladder is the graph formed by taking two copies of the cycle cm with corresponding vertices from each copy of cm being adjacent. a ladder graph on 8 vertices is shown in figure 8. let (k2m − cl2m) be the graph obtained from k2m by removing the edges of cl2m. for m ≥ 3, it can be easily seen that (k2m −cl2m) is a 2m−4 regular graph on 2m vertices and the number of edges in this graph being 2m(m− 2). 98 s. pirzada, b. a. chat, f. a. dar u u u u u u e e e e e e (b4̇,2) = k4̇∨̇k2 figure 6. u u u u ee e e e e u u (b4̄,2) = k4̇∨̄k2 figure 7. remark 2.11. in the split graph sr,s, the number of vertices is r + s and number of edges is r(r−1) 2 + rs. that is, |v (sr,s)| = r + s and |e(sr,s)| = r(r−1) 2 + rs. figure 9 ilustrates s3,2 and s4,1. remark 2.12. if π1 and π2 respectively are the graphic sequences of g1 and g2, the graphic sequence of sver = g1∨̇g2 is π = (d11 + n,d12 + n, · · · ,d1m + n,d21 + m, · · · ,d2n + m, 2|e1|) and the graphic sequence of sed = g1∨̄g2 is π = (d11,d12, · · · ,d1m,d21 + |e1|, · · · ,d2n + |e1|, (2 + n)|e1|). 3. main results in the following result, we find the graphic sequence of the induced subgraph (k2m′−cl2m′)∨̇(k2n′− cl2n′) of the graph sver = g1∨̇g2. theorem 3.1. if π1 and π2 respectively are potentially (k2m′ − cl2m′) and (k2n′ − cl2n′)-graphic sequences, m′ ≥ 3, n′ ≥ 3, m ≥ 2m′, n ≥ 2n′, then the graphic sequence of (k2m′ −cl2m′)∨̇(k2n′ −cl2n′) is ( (2(m′ + n′ − 2))2(m ′+n′), 22m ′(m′−2) ) . proof. by remark 2.12 and theorem 2.1, the graphic sequence of sver is π = (d11 +n,d 1 2 +n, · · · ,d1m + n,d21 + m, · · · ,d2n + m, 2|e1|). now let π∗ be the graphic sequence of the induced subgraph (k2m′ − cl2m′)∨̇(k2n′ − cl2n′) of sver. by taking |e(k2m′ − cl2m′)| = 2m′(m′ − 2), we have π∗ = ( d1 ′ 1 + 2n ′,d1 ′ 2 + 2n ′, · · · ,d1 ′ 2m′ + 2n ′,d2 ′ 1 + 2m ′,d2 ′ 2 + 2m ′, 99 graphical sequences s s s s c4 r r r s c4 s s s s @ @ # ## � � s s s s cl8 figure 8. � � � � � s s s s s s s s s s v2 v3 v2 v3 s3,2 s4,1 v1 v1 v4 u1 u2 u1 figure 9. · · · ,d2 ′ 2n′ + 2m ′, 22m ′(m′−2)) = (2m′ − 4 + 2n′, 2m′ − 4 + 2n′, · · · , 2m′ − 4 + 2n′, 2n′ − 4 + 2m′, · · · , 2n′ − 4 + 2m′, 22m ′(m′−2)) = ( (2(m′ + n′ − 2))2 (m′+n′) , 22m ′(m′−2)). corollary 3.2. if π1 and π2 are potentially (k2m′ − cl2m′)-graphic sequences, where m′ ≥ 3, m ≥ 2m′, n ≥ 2m′, then the graphic sequence of the induced subgraph (k2m′ −cl2m′)∨̇(k2m′ −cl2m′) of sver is ( (4(m′ − 1))4m ′ , 22m ′(m′−2) ) and σ(π∗) = 4m′(5m′ − 6). proof. put m′ = n′ in theorem 3.1, we get π∗ = (2m′ − 4 + 2m′, 2m′ − 4 + 2m′, · · · , 2m′ − 4 + 2m′, 2m′ − 4 + 2m′, · · · , 2m′ − 4 + 2m′, 22m ′(m′−2)) = ( (4(m′ − 1))4m ′ , 22m ′(m′−2)) also σ(π∗) = 4m′(4(m′ − 1)) + 2m′(m′ − 2)2 = 4m′(5m′ − 6). the following result shows that the graphic sequence π of sed = g1∨̄g2 is potentially k4 − egraphical. theorem 3.3. if π1 and π2 respectively are potentially kp1 and kp2-graphic sequences, where m ≥ 3, n ≥ 2, p1 ≤ m and p2 ≤ n, then the graphical sequence π of sed is potentially k4 −e-graphical. proof. let π1 and π2 respectively be potentially kp1 and kp2-graphic sequences, where m ≥ 3, n ≥ 2, p1 ≤ m and p2 ≤ n. let sed = g1∨̄g2 and π be its graphic sequence. then π = ( d11,d 1 2, · · · ,d1m,d21 + 100 s. pirzada, b. a. chat, f. a. dar |e1|,d22 + |e1|, · · · ,d2n + |e1|). clearly there are at-least three vertices and at least two edges in g1 and there are at least two vertices and at least one edge in g2, since g1 and g2 are connected. let vi,vj and vk be any three vertices in g1 and ui and uj be any two vertices in g2. since there are at least two edges in g1 and at least one edge in g2, without loss of generality we take vivj,vjvk ∈ e(g1) and uiuj ∈ e(g2). by construction, it can easily be seen that the graph g formed from g1 and g2 contains a subgraph on u′i,u ′ j,ui and uj vertices (where u ′ i and u ′ j are the two inserted vertices in vivj and vjvk of g1) which is k4 −e. thus π is potentially k4 −e graphical. now we obtain the graphic sequence of the induced subgraph sr1,s1∨̄sr2,s2 of sed = g1∨̄g2. theorem 3.4. if π1 and π2 respectively are potentially sr1,s1 and sr2,s2-graphic, then the graphic sequence of the induced subgraph sr1,s1∨̄sr2,s2 of sed is π∗ = (( 2(r2 + s2 − 1) + r1(2s1 + r1 − 1) 2 )r2 , ( 2r2 + r1(2s1 + r1 − 1) 2 )s2 , (2 + r2 + s2) r1(2s1+r1−1) 2 , (r1 + s1 − 1)r1,rs11 ) . proof. let π∗ be the graphic sequence of the induced subgraph sr1,s1∨̄sr2,s2 of sed. by remark 2.11 and theorem 2.1, we have π∗ = ( d1 ′ 1 ,d 1′ 2 , · · · ,d 1′ r1 ,d1 ′ r1+1 ,d1 ′ r1+2 , · · · ,d1 ′ r1+s1 ,d2 ′ 1 ,d 2′ 2 , · · · ,d 2′ r2+s2 , (2 + r2 + s2) |e(sr1,s1 )| ) = ( r1 + s1 − 1,r1 + s1 − 1, · · · ,r1 + s1 − 1,r1,r1, · · · ,r1,r2 + s2 − 1 + r1(r1 − 1) 2 + (r1s1), · · · ,r2 + s2 − 1 + r1(r1 − 1) 2 + (r1s1), (r2 + |e(sr1,s1 )|), · · · , (r2 + |e(sr1,s1 )|), (2 + r2 + s2) |e(sr1,s1 )| ) = ( (r1 + s1 − 1)r1,rs11 , ( 2(r2 + s2 − 1) + r1(r1 − 1) + 2r1s1 2 )r2 ,( 2r2 + r1(r1 − 1) + 2r1s1 2 )s2 , (2 + r2 + s2) r1(r1−1)+2r1s1 2 ) = (( 2(r2 + s2 − 1) + r1(2s1 + r1 − 1) 2 )r2 , ( 2r2 + r1(2s1 + r1 − 1) 2 )s2 , (2 + r2 + s2) r1(2s1+r1−1) 2 , (r1 + s1 − 1)r1,rs11 ) . next we obtain the graphic sequence of the induced subgraph kp1∨̄kp2 and sr2,s2∨̄sr1,s1 of sed = g1∨̄g2. theorem 3.5. if π1 and π2 respectively are potentially kp1 and kp2-graphic, then the graphic sequence of the induced subgraph kp1∨̄kp2 of sed is π∗ = ( (p1 − 1)p1, ( p1(p1 − 1) + 2(p2 − 1) 2 )p2 , (2 + p2) p1(p1−1) 2 ) , where p1 ≥ 2, p2 ≥ 1. proof. by theorem 2.1, in the graphic sequence of the induced subgraph kp1∨̄kp1 of sed, we have d1 ′ 1 = p1 − 1,d1 ′ 2 = p1 − 1, · · · ,d1 ′ p1 = p1 − 1,d2 ′ 1 = p2 − 1, · · · ,d2 ′ p2 = p2 − 1, |e(kp1 )| = p1(p1−1) 2 and n = p2. thus the graphic sequence π∗ of the induced subgraph kp1∨̄kp1 of sed is π∗ = ( p1 − 1, · · · ,p1 − 1,p2 − 1 + p1(p1 − 1) 2 , · · · ,p2 − 1 + p1(p1 − 1) 2 , (2 + p2) p1(p1−1) 2 ) 101 graphical sequences = ( (p1 − 1)p1, ( p1(p1 − 1) + 2(p2 − 1) 2 )p2 , (2 + p2) p1(p1−1) 2 ) . theorem 3.6. if π1 and π2 respectively are potentially (k2m′ −cl2m′) and (k2n′ −cl2n′)-graphic, where m′,n′ ≥ 3, then the graphic sequence of the induced subgraph (k2m′−cl2m′)∨̄(k2n′−cl2n′) of sed = g1∨̄g2 is π∗ = (( 2(m′ − 2) )2m′ , ( 2(n′ + m′ 2 ) − 4(n′ + 1) )2n′ , (2(1 + n′))2m ′(m′−2) ) . proof. let π∗ is the graphic sequence of (k2m′−cl2m′)∨̄(k2n′−cl2n′). then by theorem 2.1, we have d1 ′ 1 = 2m ′−4 = d1 ′ 2 = d 1′ 2m′,d 2′ 1 + |e((k2m′ −cl2m′))| = d2 ′ 2 + |e((k2m′ −cl2m′))|, · · · ,d2 ′ 2n′ + |e((k2m′ − cl2m′))| = 2n′−4 + (2m′−4)m′. thus the graphic sequence π∗ of the required induced subgraph (k2m′− cl2m′)∨̄(k2n′−cl2n′) of sed becomes (( 2(m′−2) )2m′ , ( 2(n′+m′ 2 )−4(n′+1) )2n′ , (2(1+n′))2m ′(m′−2) ) . theorem 3.7. if π1 and π2 respectively are potentially lr = ka1,a2,··· ,ar and mr = kb1,b2,··· ,br graphic, then (a) the graphic sequence of induced subgraph lr∨̇mr of sver = g1∨̇g2 is π∗ = (( l−a1 )a1 , ( l−a2 )a2 , · · · , ( l−ar )ar , ( l− b1 )b1 , · · · , ( l− br )br , 2( r 2) ) , where ( r 2 ) is the number of combinations of a1,a2, · · · ,ar taken two at a time. (b) σ(π∗) = ∑r i=1 ( ai(l−ai) + bi(l− bi) ) + 2 ( r 2 ) . proof. let π1 and π2 respectively be potentially lr = ka1,a2,··· ,ar and mr = kb1,b2,··· ,br graphic. so clearly the graphs g1 and g2 contain respectively lr and mr as a subgraph. let sver = g1∨̇g2 be the graph obtained by sub-division vertex join of graphs and let π be the graphic sequence of sver. we have π = ( d11 + n,d 1 2 + n, · · · ,d 1 m + n,d 2 1 + m, · · · ,d 2 n + m, 2 |e1| ) (1) where |e1| is the size of g1. let π∗ be the graphic sequence of the induced subgraph lr∨̇mr of sver. to prove (a) we use induction on r. for r = 1, the result is obvious. for r = 2, we have g′2 = ka1,a2∨̇kb1,b2. let π′2 be the graphic sequence of g′2. therefore, by remark 2.12, we have π′2 = (( a2 + b1 + b2 )a1 , ( a1 + b1 + b2 )a2 , ( b2 + a1 + a2 )b1 , ( b1 + a1 + a2 )b2 , 2 (22)∑ i,j=1,i 6=j aiaj ) = (( 2∑ i=1 (ai + bi) −a1 )a1 , ( 2∑ i=1 (ai + bi) −a2 )a2 , ( 2∑ i=1 (ai + bi) − b1 )b1 , ( 2∑ i=1 (ai + bi) − b2 )b2 , 2 (22)∑ i,j=1,i 6=j aiaj ) = (( 2∑ i=1 (ai + bi) −ai )ai , ( 2∑ i=1 (ai + bi) − bi )bi , 2a1a2 ) . 102 s. pirzada, b. a. chat, f. a. dar this proves the result for r = 2. assume that the result is true for r = k − 1. therefore, we have g′k−1 = ka1,a2,··· ,ak−1∨̇kb1,b2,··· ,bk−1 and let π′k−1 be the graphic sequence of g ′ k−1. then we have π′k−1 = ((k−1∑ i=1 (ai + bi) −a1 )a1 , · · · , (k−1∑ i=1 (ai + bi) −ak−1 )ak−1 , (k−1∑ i=1 (ai + bi) − b1 )b1 , · · · , (k−1∑ i=1 (ai + bi) − bk−1 )bk−1 , 2 (k−12 )∑ i,j,i6=j aiaj ) . (2) now, for r = k, we have g′k = ka1,a2,··· ,ak−1,ak∨̇kb1,b2,··· ,bk−1,bk = kr,ak∨̇ks,bk, where r = a1,a2, · · · ,ak−1 and s = b1,b2, · · · ,bk−1. since the result is proved for all r = k − 1 and using the fact that the result is proved for each pair and since the result is already proved for r = 2, it follows by induction hypothesis that the result holds for r = k also. that is, π∗ = π′k = (( ak + bk + k−1∑ i=1 (ai + bi) −a1 )a1 , · · · , ( ak + bk + k−1∑ i=1 (ai + bi) −ak−1 )ak−1 , ( ak + bk + k∑ i=1 (ai + bi) −ak )ak , ( ak + bk + k−1∑ i=1 (ai + bi) − b1 )b1 , · · · , ( ak + bk + k−1∑ i=1 (ai + bi) − bk−1 )bk−1 , ( ak + bk + k−1∑ i=1 (ai + bi) − bk )bk , 2 ( (k−12 )∑ i,j,i6=j aiaj + k−1∑ i=1 akai )) = (( l−a1 )a1 , ( l−a2 )a2 , · · · , ( l−ar )ar , ( l− b1 )b1 , · · · , ( l− br )br , 2( r 2) ) . this proves part (a). now we have σ(π∗) = a1(l−a1) + · · · + ar(l−ar) + b1(l− b1) + · · · + br(l− br) + 2 ( (k2)∑ i,j=1,i 6=j aiaj ) = r∑ i=1 ( ai(l−ai) + bi(l− bi) ) + 2 ( r 2 ) . theorem 3.8. if π1 and π2 respectively are potentially lr = ka1,a2,··· ,ar and mr = kb1,b2,··· ,br -graphic, then the graphic sequence of the induced subgraph l∨̄m of sed is π∗ = (( (l1 −ai)ai, (l′1 + |el|− bi )bi)r i=1 , (2 + l′1) |el| ) and σ(π∗) = l21 + l ′ 1 2 −m + 2(1 + l′1)|el|, where l1 = r∑ i=1 ai and l′1 = r∑ i=1 bi. 103 graphical sequences proof. let π1 and π2 respectively be potentially lr and mr graphic. then the graphs g1 and g2 contain lr and mr as a subgraph. let sed = g1∨̄g2 be the graph obtained by sub-division edge join of graphs and let π be the graphic sequence of sed. then, we have π = ( d11,d 1 2, · · · ,d 1 m,d 2 1 + |e1|, · · · ,d 2 n + |e1|, (2 + n) |e1| ) (3) where |e1| is the size of g1. let π∗ be the graphic sequence of the induced subgraph lr∨̄mr of sed. to prove the result we use induction on r. for r = 1, the result follows by theorem 3.5. for r = 2, we have g′2 = ka1,a2∨̄kb1,b2. let π′2 be the graphic sequence of g′2. therefore, by remark 2.12, we have π′2 = ( aa12 ,a a2 1 , ( a1a2 + b2 )b1 , ( a1a2 + b1 )b2 , ( 2 + b1 + b2 )a1a2) = (( 2∑ i=1 ai −a1 )a1 , ( 2∑ i=1 ai −a2 )a2 , ( (22)∑ i,j,i 6=j aiaj + b2 )b1 , ( 2c2∑ i,j,i 6=j aiaj + b1 )b2 , ( 2 + 2∑ i=1 bi ) (22)∑ i,j,i6=j aiaj ) = (( l∗1 −a1 )a1 , ( l∗1 −a2 )a2 , ( |e(l2)| + b2 )b1 , ( |e(l2)| + b1 )b2 , ( 2 + l′1 )|e(l2)|) = ((( l∗1 −ai )ai , ( |e(l2)| + l′1 − bi )bi)2 i=1 , ( 2 + l′1 )|e(l2)|) where l∗1 = 2∑ i=1 ai and |e(l2)| = |e(ka1,a2 )| = a1a2. this proves the result for r = 2. assume that the result is true for r = k − 1, therefore, we have g′k−1 = ka1,a2,··· ,ak−1∨̄kb1,b2,··· ,bk−1. let π′k−1 be the graphic sequence of g ′ k−1, then we have π′k−1 = ((( l∗∗1 −ai ) , ( |e(lk−1)| + l′1 − bi )bi)k−1 i=1 , ( 2 + l′1 )|e(lk−1)|) where l∗∗1 = k−1∑ i=1 ai. now we show that the result holds for r = k. we have g′k = ka1,a2,··· ,ak−1,ak∨̄kb1,b2,··· ,bk−1,bk = kr,ak∨̄ks,bk where r = a1,a2, · · · ,ak−1 and s = b1,b2, · · · ,bk−1. since the result is proved for every r = k − 1 and using the fact that the result is proved for each pair and since the result is already proved for r = 2, it follows by induction hypothesis that the result holds for r = k also. that is, π∗ = π′k = (( ak + k−1∑ i=1 ai −a1 )a1 , ( ak + k−1∑ i=1 ai −a2 )a2 , · · · , ( ak + k−1∑ i=1 ai −ak )ak , 104 s. pirzada, b. a. chat, f. a. dar ( aka1 + aka2 + · · · + akak−1 + (k−12 )∑ i,j,i 6=j aiaj + b2 )b1 , ( aka1 + aka2 + · · · + akak−1 + (k−12 )∑ i,j,i 6=j aiaj + b2 )b2 , · · · , ( aka1 + aka2 + · · · + akak−1 + (k−12 )∑ i,j,i 6=j aiaj + bk )bk , ( 2 + bk + k−1∑ i=1 bi )( (k−12 )∑ i,j,i6=j aiaj + k−1∑ i=1 akai )) = ((( l1 −ai )ai , ( |elr| + l ′ 1 − b1 )bi)k i=1 , ( 2 + l′1 )|elr|) also, we have σ(π∗) = a1(l1 −a1) + · · · + ar(l1 −ar) + b1(l′1 + |el|− b1) + · · · + br(l′1 + |el|− br) + kc2∑ i,j=1,i6=j aiaj(2 + 2l ′ 1) = l21 + l ′ 1 2 −m + 2(1 + l′1)|el|. this completes the proof. let g1 and g2 be any two graphs. let sj = g1 ∨ g2 and let sj∗ = ( bṁ1,n1 ) ∨ ( bṁ2,n2 ) be the induced subgraph of sj and let π∗ be the graphic sequence of s∗j. theorem 3.9. if π1 and π2 respectively be potentially bṁ1,n1 and bṁ2,n2, then (a) the graphic sequence π∗ of induced subgraph ( bṁ1,n1 ) ∨ ( bṁ2,n2 ) of sj is π∗ = (( a + |e(km2 )|− 1 )m1 , ( a + |e(km1 )|− 1 )m2 , ( a + |e(km2 )| + 2 − (m1 + n1) )|e(km1 )| , ( a + |e(km1 )| + 2 − (m2 + n2) )|e(km2 )| , ( a + |e(km2 )|−n1) )n1 , ( a + |e(km1 )|−n2) )n2) and (b) σ(π∗) = a ( a + 2∑ i=1 |e(kmi )| ) + 2∏ i,j=1,i6=j ( mi + ni ) |e(kmj )| + 2 ( |e(km1 )||e(km2 )| + 2∑ i |e(kmi )| ) − 2∑ i=1 ( mi + ( mi + ni ) |e(kmi )| ) − 2∑ i=1 n2i . where a = 2∑ i=1 (mi + ni) and |e(kmi )| = mi(mi−1) 2 . proof. the graphic sequence of bṁ1,n1 and bṁ2,n2 respectively are π′1 = (( m1 + n1 − 1 )m1 , 2 m1(m1−1) 2 ,mn11 ) (4) π′2 = (( m2 + n2 − 1 )m2 , 2 m2(m2−1) 2 ,mn22 ) (5) 105 graphical sequences clearly from (4) and (5), the graphic sequence of s∗j is π∗ = (( m1 + m2 + n1 + n2 − 1 + m2(m2 − 1) 2 )m1 , ( m1 + m2 + n1 + n2 − 1 + m1(m1 − 1) 2 )m2 , ( 2 + m2 + n2 + m2(m2 − 1) 2 )m1(m1−1) 2 , ( 2 + m1 + n1 + m1(m1 − 1) 2 )m2(m2−1) 2 ( m1 + m2 + n2 + m2(m2 − 1) 2 )n1 , ( m1 + m2 + n1 + m1(m1 − 1) 2 )n2) = (( a + |e(km2 )|− 1 )m1 , ( a + |e(km1 )|− 1 )m2 , ( a + |e(km2 )| + 2 − (m1 + n1) )|e(km1 )| , ( a + |e(km1 )| + 2 − (m2 + n2) )|e(km2 )| , ( a + |e(km2 )|−n1) )n1 , ( a + |e(km1 )|−n2) )n2) this proves (a). further σ(π∗) = m1(a + |e(km2 )|− 1) + m2(a + |e(km1|− 1) + |e(km1 )| ( a + |e(km2 )| + 2 −m1 −n1 ) + |e(km2 )| ( a + |e(km1 )| + 2 −m2 −n2 ) + n1 ( a + |e(km2|−n1 ) + n2 ( a + |e(km1 )|−n2 ) = m1a + m1|e(km2 )|−m1 + |e(km2 )|a + m2|e(km1 )|−m2 + |e(km1 )|a + |e(km1 )||e(km2 )| + 2|e(km1 )|− |e(km1 )|m1 −n1|e(km1 )| + a|e(km2 )| + |e(km2 )||e(km1 )| + 2|e(km2 )|−m2|e(km2 )|−n2|e(km2 )| + n1a + n1|e(km2 )|−n 2 1 + n2a + n2|e(km1 )|−n 2 2 = (m1 + n1 + m2 + n2)a + ( |e(km1 )| + |e(km2 )| ) a + m1|e(km2 )| + m2|e(km1 )| + n2|e(km1 )| + n1|e(km2 )|−m1|e(km1 )|−n1|e(km1 )| −m2|e(km2 )|−n2|e(km2 )|− (m1 + m2) + 2(|e(km1 )||e(km2 )|) + 2 ( |e(km1 )| + |e(km1 )| ) − (n21 + n 2 2) = a2 + a 2∑ i=1 |e(kmi )| + 2∏ i,j=1,i6=j mi|e(kmj )| + 2∏ i,j=1,i6=j ni|e(kmj )| − 2∑ i=1 (mi + ni)|e(kmi )|− 2∑ i=1 mi + 2 ( |e(km1 )||e(km2 )| + 2∑ i=1 |e(kmi )| ) − 2∑ i=1 n2i = a ( a + 2∑ i=1 |e(kmi )| ) + 2∏ i,j=1,i6=j ( mi + ni ) |e(kmj )| + 2 ( |e(km1 )||e(km2 )| + 2∑ i |e(kmi )| ) − 2∑ i=1 ( mi + ( mi + ni ) |e(kmi )| ) − 2∑ i=1 n2i . which proves (b). let g1 and g2 be two graphs. let sj = g1 ∨g2 and let s∗∗j = (bm̄1,n1 ) ∨ (bm̄2,n2 ) be the induced subgraph of sj and let π∗∗ be the graphic sequence of s∗∗j . 106 s. pirzada, b. a. chat, f. a. dar theorem 3.10. if π1 and π2 respectively are potentially ( bm̄1,n1 ) and ( bm̄2,n2 ) , then the graphic sequence of induced subgraph ( bm̄1,n1 ) ∨ ( bm̄1,n1 ) of sj is π∗∗ = (( a + |e(km2 )|− (n1 + 1) )m1 , ( a + |e(km2| + 2 −m1 )|e(km1| , ( a + 2∑ i=1 |e(kmi )|− (m1 + n1) )n1 , ( a + |e(km1|− (n2 + 1) )m2 ( a + |e(km1| + 2 −m2 )|(km2| , ( a + 2∑ i=1 |e(kmi )|− (m2 + n2) )n2) . and σ(π∗∗) =a ( a + 2∑ i=1 |e(kmi )| ) + 2∏ i,j=1,i6=j (mi + ni)|e(kmj )| + 2∏ i,j=1,i6=j |e(kmi )||e(kmj )| + 2∑ i=1 (2 + ni)|e(kmi )|− 2∑ i=1 ( 2ni + 1 + |e(kmi )| ) mi − 2∑ i=1 n2i . proof. the graphic sequence of bm̄1,n1 and bm̄2,n2 respectively are π′1 = (( m1 − 1 )m1 , (m1(m1 − 1) 2 )n1 , (2 + n1) m1(m1−1) 2 ) (6) π′2 = (( m2 − 1 )m2 , (m2(m2 − 1) 2 )n2 , (2 + n2) m2(m2−1) 2 ) . (7) then by (6), (7) and by definition 2.2, we have π∗∗ = (( m1 − 1 + m2 + n2 + m2(m2 − 1) 2 )m1 , ( m2 − 1 + m1 + n1 + m1(m1 − 1) 2 )m2 , (m1(m1 − 1) 2 + m2 + n2 + m2(m2 − 1) 2 )n1 , (m2(m2 − 1) 2 + m1 + n1 + m1(m1 − 1) 2 )n2 , ( 2 + n1 + m2 + n2 + m2(m2 − 1) 2 )m1(m1−1) 2 , ( 2 + n2 + m1 + n1 + m1(m1 − 1) 2 )m2(m2−1) 2 ) = (( a + |e(km2 )|− (n1 + 1) )m1 , ( a + |e(km2| + 2 −m1) )|e(km1| , ( a + 2∑ i=1 |e(kmi )|− (m1 + n1) )n1 , ( a + |e(km1|− (n2 + 1) )m2 ( a + |e(km1| + 2 −m2 )|(km2| , ( a + 2∑ i=1 |e(kmi )|− (m2 + n2) )n2) . further σ(π∗∗) = (( a + |e(km2 )|− (n1 + 1) )m1 + ( a + |e(km2|2 −m1 )|e(km1| + + ( a + 2∑ i=1 |e(kmi )|− (m1 + n1) )n1 + ( a + |e(km1|− (n2 + 1) )m2 107 graphical sequences ( a + |e(km1| + 2 −m2 )|(km2| + ( a + 2∑ i=1 |e(kmi )|− (m2 + n2) )n2) . = m1 ( a + |e(km2 )|−n1 − 1 ) + |e(km1 )| ( a + |e(km2 )| + 2 −m1 ) + n1 ( a + |e(km1 )| + |e(km2 )|−m1 −n1 ) + m2 ( a + |e(km1 )|−n2 − 1 ) + |e(km2 )| ( a + |e(km1 )| + 2 −m2 ) + n2 ( a + |e(km1 )| + |e(km2 )|−m2 −n2 ) = m1a + m1|e(km2 )|−n1m1 −m1 + |e(km1 )|a + |e(km1 )||e(km2 )| + 2|e(km1 )| − |e(km1 )|m1 + n1a + n1|e(km1 )| + n1|e(km2 )|−n1m1 −n 2 1 + m2a + m2|e(km1 )| −m2n2 −m2 + |e(km2 )|a + |e(km2 )||e(km1 )| + 2|e(km2 )|− |e(km2 )|m2 + n2a + n2|e(km2 )|−n2m2 −n 2 2 + n2|e(km1 )| = (m1 + n1 + m2 + n2)a + ( |e(km1 )| + |e(km2 )| ) a + m1|e(km2 )| + m2|e(km1 )| + n1|e(km1 )| + n2|e(km2 )| + |e(km1 )||e(km2 )| + |e(km2 )||e(km1 )| + 2(|e(km1 )| + |e(km2 )|) + n1|e(km2 )| + n2|e(km1 )|− 2n1m1 − 2n2m2 − (m1 + m2) − ( |e(km1 )|m1 + |e(km2 )|m2 ) = a2 + a 2∑ i=1 |e(kmi )| + 2∏ i,j=1,i6=j mi|e(kmj )| + 2∏ i,j=1,i6=j ni|e(kmj )| + 2∏ i,j=1,i6=j |e(kmi )||e(kmj )| + 2 2∑ i=1 |e(kmi )| + 2∑ i=1 ni|e(kmi )| − 2 2∑ i=1 nimi − 2∑ i=1 mi − 2∑ i=1 |e(kmi )|mi − 2∑ i=1 n2i = a ( a + 2∑ i=1 |e(kmi )| ) + 2∏ i,j=1,i6=j (mi + ni)|e(kmj )| + 2∑ i=1 (2 + ni)|e(kmi )| − 2∑ i=1 ( 2ni + 1 + |e(kmi )| ) mi − 2∑ i=1 n2i . this completes the proof. acknowledgements: the authors are grateful to the anonymous referee for his useful comments and suggestions which improved the presentation of the paper. references [1] c. bu, b. yan, x. zhou, j. zhou, resistance distance in subdivision-vertex join and subdivision-edge join of graphs, linear algebria and its applications, 458, 454-462, 2014. [2] p. erdős, t. gallai, graphs with prescribed degrees, (in hungarian) matemoutiki lapor, 11, 264-274, 1960. [3] d. r. fulkerson, a. j. hoffman, m. h. mcandrew, some properties of graphs with multiple edges, canad. j. math., 17, 166-177, 1965. 108 s. pirzada, b. a. chat, f. a. dar [4] r. j. gould, m. s. jacobson, j. lehel, potentially g-graphical degree sequences, in combinatorics, graph theory and algorithms, vol. 2, (y. alavi et al., eds.), new issues press, kalamazoo mi, 451-460, 1999. [5] j. l. gross, j. yellen, p. zhang, handbook of graph theory, crc press, boca raton, fl, 2013. [6] s. l. hakimi, on the realizability of a set of integers as degrees of the vertices of a graph, j. siam appl. math., 10, 496-506, 1962. [7] v. havel, a remark on the existance of finite graphs, (czech) casopis pest. mat. 80, 477-480, 1955. [8] s. pirzada, an introduction to graph theory, universities press, orient blackswan, india, 2012. [9] s. pirzada, b. a. chat, potentially graphic sequences of split graphs, kragujevac j. math 38(1), 73-81, 2014. [10] a. r. rao, an erdos-gallai type result on the clique number of a realization of a degree sequence, preprint. [11] a. r. rao, the clique number of a graph with a given degree sequence, proc. symposium on graph theory (ed. a. r. rao), macmillan and co. india ltd, i.s.i. lecture notes series, 4, 251-267, 1979. [12] j. h. yin, conditions for r-graphic sequences to be potentially k(r)m+1-graphic, disc. math., 309, 6271-6276, 2009. 109 introduction definitions and preliminary results main results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.617232 j. algebra comb. discrete appl. 6(3) • 123–134 received: 4 july 2018 accepted: 23 april 2019 journal of algebra combinatorics discrete structures and applications bijective s-boxes of different sizes obtained from quasi-cyclic codes∗ research article dusan bikov, iliya bouyukliev, stefka bouyuklieva abstract: the aim of this paper is to construct s-boxes of different sizes with good cryptographic properties. an algebraic construction for bijective s-boxes is described. it uses quasi-cyclic representations of the binary simplex code. good s-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18 are obtained. 2010 msc: 94a60, 11t71, 06e30, 94b05 keywords: s-box, simplex code, quasi-cyclic codes 1. introduction s-boxes are among the most common and essential components of the block ciphers. they provide block ciphers with resistance to known and potential cryptanalytic attacks. therefore significant research effort has been made in developing methods for constructing s-boxes with optimal parameters and desirable cryptographic properties. there are well studied criteria that a good s-box has to fulfill to make the cipher resistant against differential and linear cryptanalyses. however, the construction of a cryptographically secure s-box is still a problem. for many years, properties as well as various techniques and methods for constructing good s-boxes have been investigated. the popular techniques for constructing s-boxes can be classified into three categories: algebraic structures, pseudo-random generation and different heuristic approaches. the aim of this paper is the constructions of bijective s-boxes of different sizes with good cryptographic properties. to do this, we use binary quasi-cyclic codes. we need a method to construct ∗ this work was supported by bulgarian science fund under contract dn-02-2/13.12.2016. dusan bikov; faculty of computer science, goce delchev university, shtip, macedonia (email: dusan.bikov@ugd.edu.mk). iliya bouyukliev; institute of mathematics and informatics, bulgarian academy of sciences, p.o.box 323, 5000 veliko tarnovo, bulgaria (email: iliyab@math.bas.bg). stefka bouyuklieva (corresponding author); faculty of mathematics and informatics, st. cyril and st. methodius university of veliko tarnovo, bulgaria (email: stefka@ts.uni-vt.bg). 123 https://orcid.org/0000-0002-5145-5297 https://orcid.org/0000-0002-6730-1129 https://orcid.org/0000-0002-9557-4749 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 the codes, then effective algorithms to compute the parameters of the corresponding s-boxes, and fast programs that implement these algorithms. increasing the size of input data leads to complicated computations that become more difficult and use too many objects. on the other hand, some of the algorithms are suitable for parallel implementation, which makes it possible to scan the intermediate objects at the same time, and thus allows the study of s-boxes of relatively large sizes. therefore we have used the parallel library boolsplg [1] to realize the presented constructions and to calculate the parameters of the obtained s-boxes. for the programs we have used gpu computing model with cuda. this allows us to interact directly with the gpus and run programs on them, thus effectively utilizing the advantages of parallelization (many details about the parallel algorithms and programs are given in [2]). for this research, we have used a graphic card nvidia geforce titan x pascal which we have as a donation from nvidia corporation. the bijective s-boxes of size n = 4 have been extensively studied, classifications have been made, criteria of optimality are defined [12, 16, 17]. a general classification of all optimal s-boxes of size n = 4 is given in the work of leander and poschmann [12] in 2007, where the authors make a comprehensive analysis and find all classes of affine equivalent s-boxes for which they explore linearity, differential uniformity, and algebraic degree. saarinen [16] in 2011 has extended the work of leander and poschmann, making a comprehensive analysis and finding all classes of linear equivalence. in the article [17] from 2015, a new classification of s-boxes of size n = 4 is made, dividing them into 183 different categories. almost everything about the s-boxes with size n = 4 is clear, but the situation for larger sizes is radically different. s-boxes with size n = 8 are of particular interest due to the fact that these cryptographic primitives are embedded in the widely used cryptographic standards. it is still unclear whether there are s-boxes for n = 8 with nonlinearity greater than 112 or other better cryptographic properties. therefore, a construction of s-boxes with better "optimal" cryptographic properties is a very topical issue. we construct bijective s-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18. the paper is organized as follows. section 2 provides the necessary definitions and assertions required for our constructions and for the investigation of the obtained s-boxes. in section 3 we describe the constructions that we use. the results are presented in section 4. 2. preliminaries s-boxes are also called multi-output boolean functions or vectorial boolean functions because of their connection with boolean functions. they are defined as functions from fn2 to f m 2 (also called (n,m)-functions or (n,m) s-boxes) where n and m are positive integers. an s-box can be represented by the vector (f1,f2, . . . ,fm), where fi are boolean functions of n variables, called its coordinate functions, i = 1,2, . . . ,m. then the m×2n matrix bs =   tt(f1)... tt(fm)   also represents the considered s-box, where tt(fi) is the truth table of the boolean function fi, i = 1, . . . ,m [5]. an s-box is bijective (or invertible), if n = m and s is an invertible function. if an s-box is represented by the matrix bs, then it is bijective if and only if n = m and the columns of bs are all binary vectors of length n. to connect the codes with s-boxes, we consider the binary simplex codes. an [n,k] linear code c over the binary field f2 is a k-dimensional linear subspace of fn2 . the hamming weight of a vector in fn2 is equal to the number of its nonzero coordinates, and the hamming distance between two vectors is the number of positions in which they differ. we call c an [n,k,d] code if d is the minimum distance of the code, d = min{d(x,y), x,y ∈ c,x 6= y}. two binary codes of length n are equivalent if there is a permutation σ ∈ sn which maps one code to the other. let gs be an n× (2n −1) binary matrix whose columns are all nonzero binary vectors of length n. 124 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 the code generated by gs is called a binary simplex code and we denote it by sn. this is a constant weight code with nonzero weight 2n−1 and its dual code is the [2n−1,2n−1−n,3] binary hamming code (for more information see [14]). the simplex code sn can be considered as an irreducible cyclic code. moreover, if h(x) is any primitive polynomial (a minimal polynomial of a primitive element α of the field f2n over f2), the code with check polynomial h(x) is equivalent to the simplex code sn [14]. the columns of gs can be considered as the binary representations of the integers 1, . . . ,2n − 1 which we denote by 1, . . . ,2n −1, respectively. suppose that the columns of gs are ordered as gs = (1 t · · · 2n −1t ). let gs = (0 t 1 t · · · 2n −1t ). obviously, gs =   tt(x1)... tt(xn)   and so s n = 〈tt(x1), . . . ,tt(xn)〉, where s n is the extended simplex code (extended with a zero coordinate). this proves the following theorem that is very important in our research. theorem 2.1. an s-box is invertible if and only if n = m and the matrix bs generates a [2n,n,2n−1] code equivalent to the extended simplex code s n. in order to study the cryptographic properties of an s-box related to the linearity, we need to consider all non-zero linear combinations of its coordinate functions, namely sb = b ·s = b1f1⊕···⊕bmfm, where b = (b1, . . . ,bm) ∈ fm2 , b 6= 0. these are the component functions of the considered s-box. the truth tables of the component functions are all nonzero linear combinations of the rows of matrix bs and so they coincide with the nonzero codewords of the linear code generated by bs. in the case of bijective s-box, instead of a generator matrix we can consider a (2n − 1) × 2n matrix whose rows are all nonzero codewords of the given extended simplex code (which are the component functions of the corresponding s-box). there are a few different definitions for equivalence of s-boxes but for all of them equivalent but different linear codes can lead to nonequivalent s-boxes with different characteristics. therefore we consider different codes all of which are equivalent to s n, and these codes produce s-boxes with different cryptographic properties. since the building blocks of an s-box are boolean functions, we define in the beginning some of their parameters which are important for cryptography. let f : fn2 → f2 be a boolean function of n variables. the functions of the form fa(x) = a1x1 ⊕a2x2 ⊕···⊕anxn = a ·x are linear, and fa(x)⊕b = a1x1 ⊕a2x2 ⊕···⊕anxn⊕b are affine functions, a = (a1, . . . ,an) ∈ fn2 , b ∈ f2, x = (x1,x2, . . . ,xn). the walsh coefficients of the boolean function f are defined as fw (a) = ∑ x∈fn2 (−1)f(x)⊕fa(x). the 2n-tuple (fw (0),fw (1), . . . ,fw (2n −1) is called the walsh spectrum of the function f, the set of all walsh coefficients is its walsh distribution, and the maximum absolute value of an walsh coefficient of f is its linearity lin(f) = max{|fw (a)| | a ∈ fn2}. another important parameter which is closely connected with the linearity is the nonlinearity. nonlinearity nl(f) of the boolean function f is the minimum hamming distance from f to the nearest affine function: nl(f) = min{dh(f,g) | g − affine function}. the relation between the linearity and nonlinearity is given by the equality lin(f) = 2n − 2nl(f) [4]. obviously, the minimum linearity corresponds to maximum nonlinearity. the parseval’s equality∑ a∈fn2 (fw (a))2 = 22n gives that lin(f) ≥ 2n/2 [4]. functions attaining this lower bound are called bent functions. 125 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 the walsh spectrum of s is defined as the collection of all walsh spectra of its component functions. the linearity and nonlinearity of s are defined as lin(s) = max b∈fm2 \{0} lin(b ·s), nl(s) = min b∈fm2 \{0} nl(b ·s). the nonlinearity and the walsh spectrum of a boolean function can be calculated using linear codes. it is well known that the set of truth tables of all affine boolean functions coincides with the set of codewords of the reed-muller code of first order rm(1,n), which is a linear [2n,n + 1,2n−1] code with a generator matrix g(rm(1,n)) =   tt(1) tt(x1) ... tt(xn)   . the code rm(1,n) is obtained from the extended simplex code by adding the all ones vector 1 to its generator matrix. this means that rm(1,n) consist of the codewords of s n and their complements, or rm(1,n) = s n ∪ (1 + s n). the nonlinearity of the boolean function f is the hamming distance from tt(f) to the reed-muller code rm(1,n), or nl(f) = dh(tt(f),rm(1,n)). this means that we can use algorithms for calculating the distance from a vector to a code (or for minimum distance of a linear code) to find the nonlinearity and linearity of a boolean function without having the whole walsh spectrum. if f is an affine function then nl(f) = 0, otherwise nl(f) is equal to the minimum distance of the linear code with a generator matrix gf = ( g(rm(1,n)) tt(f) ) . this helps us to calculate the nonlinearity of an s-box as the minimum distance of the linear code generated by the matrix bs = ( g(rm(1,n)) bs ) . let us recall that if there is a coordinate function sb which is affine then nl(s) = 0. other important parameters of an s-box are the algebraic degree deg(s), the differential uniformity δ and the autocorrelation ac(s). any boolean function f can be represented uniquely as a binary polynomial of n variables whose monomials have the form xi1xi2 · · ·xik, 1 ≤ i1 < i2 < · · · < ik ≤ n, 0 ≤ k ≤ n, which is called the algebraic normal form anf of f. the degree of this polynomial is the algebraic degree of the boolean function denoted by deg(f). the algebraic degree of an m×n s-box is equal to the minimum algebraic degree of the component functions of s, deg(s) = min{deg(b · s),b ∈ fm2 \{0}}. autocorrelation of the boolean function f is defined by ac(f) = max{| ∑ x∈fn2 (−1)f(x)⊕f(x⊕w)| | w ∈ fn2}. the autocorrelation of an s-box is the maximal autocorrelation of its components functions, ac(s) = max{ac(b ·s),b ∈ fm2 \{0}}. the differential uniformity of an (n×n) s-box s is defined by: δ = max α,β∈fn2 ,α6=0 |{x ∈ fn2 |s(x)⊕s(x⊕α) = β}|. an s-box should have a differential uniformity as low as possible. the smallest possible value of δ in the case of bijective s-boxes is 2. summarized results for good s-boxes are presented in [9, 10]. 126 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 3. the considered constructions in this section we present the main definitions and properties of the linear codes that are important for our constructions. a linear code of length n is quasi-cyclic (qc) code if a cyclic shift of a codeword by m positions results in another codeword. obviously, m must divide the length n of the code, and n/m is called index of the considered qc code. if m = 1 the code is cyclic so qc codes are a generalization of cyclic codes. many quasi-cyclic codes have the largest minimum distance among the linear codes with given length and dimension. there are different methods to construct quasi-cyclic codes. a qc code of length lm and index l is generated by a block matrix such that each block is an m×m circulant. the structure of qc codes is studied in [6, 11, 13]. to connect the codes with s-boxes, we consider the binary simplex codes as quasi-cyclic codes. as we defined in section 2, gs is the n× (2n −1) generator matrix of sn such that the columns of gs are ordered as gs = (1 · · · 2n −1). we use two more constructions for the codes equivalent to sn. let k = f2n be a finite field, α be its primitive element, 2n − 1 = mr, 1 < m,r < 2n − 1, and β = αr. if g = 〈β〉 < k∗ then g is a cyclic group of order m and g,αg,α2g,.. . ,αr−1g are all different cosets of g in k∗. for our constructions, we use left circulant matrices and the trace map. the trace map tr : k → f2 is defined as tr(ξ) = ξ + ξ2 + ξ4 + · · ·+ ξ2 n−1 , ξ ∈ k. we present two constructions of quasi-cyclic codes. for the first construction, we need the left circulant m×m matrices ca = (tr(αmaβi+j))0≤i,j≤m−1, a = 0,1, . . . ,r−1, and the left circulant block matrix m1 =   c0 c1 . . . cr−1 c1 c2 . . . c0 ... ... ... ... cr−1 c0 . . . cr−2   . (1) the second construction is similar to the first one but the constructed block matrix is not circulant. we use again left circulant matrices but these are da = (tr(αaβi+j))0≤i,j≤m−1, a = 0,1, . . . ,2r−2. we construct a block matrix m2 in the following way: m2 =   d0 d1 . . . dr−1 d1 d2 . . . dr ... ... ... ... dr−1 dr . . . d2r−2   . (2) there are two main differences between these two constructions. first, in the circulants ca and da we multiply the powers of β by αma and αa, respectively. second, in the block matrix m1 we take the indexes of the circulants modulo r. the following theorem is important for our research. both codes generated by m1 and m2 are quasi-cyclic. theorem 3.1. the code whose nonzero codewords are the rows of the matrix m2 is equivalent to the simplex [2n−1 = mr,n,2n−1] code. if m and r are coprime, the same is true for the code whose nonzero codewords are the rows of the matrix m1. proof. the proof consists of two parts. first, we will prove that the hamming weight of each row of both matrices is 2n−1. second, the sum of any two rows of m1 (respectively m2) is a row in the same matrix. this proves that the rows in any of both matrices are the nonzero codewords of a linear code of length 2n − 1 and dimension n. moreover, all nonzero codewords of the corresponding code have the same weight 2n−1 and therefore it is a linear constant-weight [2n − 1,n,2n−1] code. up to equivalence, the simplex code sn is the unique code with these parameters for a given positive integer n. consider the elements m1[i,j] and m2[i,j], 0 ≤ i,j ≤ 2n − 2. if i = mi1 + i2, j = mj1 + j2, 127 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 0 ≤ i1,j1 ≤ r −1, 0 ≤ i2,j2 ≤ m−1, then m2[i,j] = di1+j1[i2,j2] = tr(α r(i2+j2)+i1+j1) = tr(α(i1+ri2)+(j1+rj2)), m1[i,j] = c(i1+j1)r [i2,j2] = tr(α r(i2+j2)+m(i1+j1)) = tr(α(mi1+ri2)+(mj1+rj2)), where (i1 + j1)r = (i1 + j1) mod r. if m and r are coprime, for a fixed i the exponents (mi1+ri2)+(mj′1+rj ′ 2) and (mi1+ri2)+(mj ′′ 1 +rj ′′ 2 ) are different for j′ = mj′1 + j ′ 2 6= j′′ = mj′′1 + j′′2 , where j′,j′′ ∈{0,1, . . . ,mr −1}. hence the i-th row of m1 consists of the traces of all nonzero elements of the field k, and this holds for all i = 0,1, . . . ,mr−1. since exactly half of the elements of the field have trace 1 and the other half (including 0) have trace 0, the hamming weight of each row is 2n−1. for the matrix m2 the integers m and r are not necessarily coprime. it is easy to see that if j′1 + rj ′ 2 = j ′′ 1 + rj ′′ 2 , 0 ≤ j′1,j′′1 ≤ r −1, 0 ≤ j′2,j′′2 ≤ m−1, then j′1 − j ′′ 1 = r(j ′′ 2 − j ′ 2) ⇒ r | j ′ 1 − j ′′ 1 ⇒ j ′ 1 = j ′′ 1 ⇒ j ′ 2 = j ′′ 2 . hence the i-th row of m2 consists of the traces of all nonzero elements of the field, and this holds for all i = 0,1, . . . ,mr −1. take the s-th and the t-th rows of the matrix m1 (respectively m2), and s = ms1 +s2, t = mt1 +t2, 0 ≤ s1, t1 ≤ r −1, 0 ≤ s2, t2 ≤ m−1. then m1[s] + m1[t] = (tr(α (ms1+rs2)+(mj1+rj2)) + tr(α(mt1+rt2)+(mj1+rj2)))j=0,...,mr−1 = (tr(α(ms1+rs2)+(mj1+rj2) + α(mt1+rt2)+(mj1+rj2)))j=0,...,mr−1 = (tr((α(ms1+rs2) + α(mt1+rt2))α(mj1+rj2)))j=0,...,mr−1. since ms1 + rs2 6≡ mt1 + rt2 (mod mr), then α(ms1+rs2) + α(mt1+rt2) 6= 0 and so α(ms1+rs2) + α(mt1+rt2) = αc for some c ∈{0,1, . . . ,mr − 1}. if m and r are coprime, c = mc1 + rc2, 0 ≤ c1 ≤ r − 1, 0 ≤ c2 ≤ m−1. it follows that m1[s] + m1[t] = (tr(α cαmj1+rj2))j=0,...,mr−1 = (tr(α mc1+rc2αmj1+rj2))j=0,...,mr−1 = (tr(αm(c1+j1)αr(c2+j2)))j=0,...,mr−1 = (tr(α m(c1+j1)βc2+j2))j=0,...,mr−1 ⇒ m1[s] + m1[t] = (cc1[c2],cc1+1[c2], . . . ,cc1−1[c2]). hence m1[s] + m1[t] is equal to the c-th row of m1. in the similar way we obtain that m2[s] + m2[t] is equal to the c-th row of m2, where αs1+rs2 + αt1+rt2 = αc, 0 ≤ c ≤ mr −1. so we proved that the rows of the matrix mi, i = 1,2, are all nonzero codewords in a linear code, equivalent to sn (for m1 we need m and r to be coprime). denote by c(m1) and c(m2) the codes generated by m1 and m2, respectively. the above theorem says that c(m1) ∼= c(m2) ∼= sn. let m1 be the matrix m1 extended with one zero column in the beginning, and c(m1) be the code whose codewords are the rows of m1 (the same for m2). then any generator matrix of c(m1) can be considered as an invertible s-box. since all these s-boxes generate the same code c(m1) and have the same component boolean functions, they have the same linearity, nonlinearity, degree, autocorrelation and differential uniformity. therefore it doesn’t matter which generator matrix of c(m1) (or c(m2)) we consider, so we take the matrices g(mi) whose rows are the first n linearly independent rows of mi, i = 1,2. 128 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 we use the described constructions of quasi-cyclic codes to obtain s-boxes in two different ways. the constructed s-boxes are called here qcs-boxes. moreover, we take a permutation π ∈ sr which permutes the block-columns of m1 (or m2). unfortunately, the qcs-boxes g(m1π) and g(m2π) do not have good nonlinearity. this construction is natural but looking for better results we transform the matrices m1 and m2. (c1) first construction: we describe this construction for the matrix m1, but we use it in the same way for m2. let σ ∈ s2n permutes the columns of the matrix g(m1) into the vectors 0,1, . . . ,2n −1. now we consider the qcs-box, represented by the matrix σ(g(m1π)). the nonlinearity of this qcs-box is equal to the minimum distance d of the code generated by the matrix g (π) 1 =   1 11 . . .10 g(m1) 0 g(m1π)   . this follows from the fact that σ maps the above matrix into  111 . . .1σ(g(m1)) σ(g(m1π))   = ( g(rm(1,n)) σ(g(m1π)) ) . the quasi-cyclic structure of the matrices provides a faster algorithm for calculating linearity of the obtained qcs-boxes. the code generated by g(π)1 is invariant under the action of the cyclic group 〈τ〉 where τ ∈ s2n is presented as a product of independent cycles in the following way τ = (1,2, . . . ,m)(m + 1, . . . ,2m) . . .(mr −r + 1, . . . ,mr). the group 〈τ〉 defines a relation of equivalence in the considered code (two codewords u and v are equivalent if u = τs(v), 0 ≤ s ≤ m− 1). to calculate the minimum distance d of the above code we need only one codeword from each equivalence class. we present this observation in the next proposition proposition 3.2. let a = (a0,a1, . . . ,ar−1) and b = (b0,b1, . . . ,br−1) be block matrices, where ai and bi are m × m circulants, i = 0,1, . . . ,r − 1. if a0,a1, . . . ,am−1 are the rows of a, and b0,b1, . . . ,bm−1 are the rows of b, then d(ai,bj) = d(ai+1,bj+1) for 0 ≤ i,j ≤ m−1 (i+ 1 and j + 1 are taken modulo m). since fw (a) = 2n−2d(f,fa) [4], proposition 3.2 shows that the walsh distributions of all boolean function in one equivalence class are the same. this allows us to calculate the linearity (the same for the other parameters) listing only r of the component functions of the considered qcs-box. (c2) second construction: for each of the circulants c0,c1, . . . ,cr−1 we reorder the columns in the following way: first we take the last column, then the previous one, and in the end the first one. in this way we obtain the circulants c′0, c ′ 1, . . . , c ′ r−1, which define the matrix g(m ′ 1). if ca =   c (a) 1 c (a) 2 · · · c (a) m−1 c (a) m c (a) 2 c (a) 3 · · · c (a) m c (a) 1 ... ... ... ... ... c (a) m−1 c (a) m · · · c (a) m−3 c (a) m−2 c (a) m c (a) 1 · · · c (a) m−2 c (a) m−1   , then c ′ a =   c (a) m c (a) m−1 · · · c (a) 2 c (a) 1 c (a) 1 c (a) m · · · c (a) 3 c (a) 2 ... ... ... ... ... c (a) m−2 c (a) m−3 · · · c (a) m c (a) m−1 c (a) m−1 c (a) m−2 · · · c (a) 1 c (a) m   . 129 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 hence c′a is a right circulant matrix, and since ca = (tr(α maβi+j))0≤i,j≤m−1, then c′a = (tr(αmaβi−j−1))0≤i,j≤m−1, a = 0,1, . . . ,r − 1. so m′1 is a left circulant block matrix whose blocks are right circulants. we use g(m′1π), where π ∈ sr is a permutation on the block-columns of g(m′1), but now we compute the minimum distance d of the code generated by the matrix g (π) 2 =   1 11 . . .10 g(m1) 0 g(m′1π)   . if σ is defined as in (c1) then d is the nonlinearity of the qcs-box represented by the matrix σ(g(m′1π)). this construction extends the first construction c1. using such constructions we can easily compute the walsh distributions of the considered qcs-boxes and take only those that have large nonlinearity. we apply these constructions to obtain n×n bijective s-boxes with sizes 4 ≤ n ≤ 18 such that 2n −1 is not prime. remark 3.3. in our paper [3] we considered only the first construction technique and its application only in the case of 8×8 s-boxes. now we extend our study to more construction methods with applications to bijective s-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18. 4. constructed qcs-boxes in this section we present the constructed qcs-boxes, that have linearity close to the parseval bound, small differential uniformity δ (δ ≥ 2), high algebraic degree and small autocorrelation ac(s), and compare them with the best known s-boxes. we use a fixed finite field with 2n elements, generated by a primitive binary polynomial g(x) of degree n, and take α = x. if we change the generator polynomial of the field, we can get different results, but we have not studied how the selected field affects the parameters of the constructed s-boxes. this is still an open problem as well as the equivalence of the constructed qcs-boxes with respect to an equivalence relation (affine, ccz, or another equivalence relation). the obtained results are presented in tables. the first column shows the used construction (c1 or c2), the used matrix (m1 or m2) and the integers m and r, the next columns contain the values of the computed cryptographic parameters, and the last column gives the number of the constructed qcs-boxes in each of the cases. for r ≤ 15 we study the s-boxes for all permutations π ∈ sr used as it is described in c1 and c2, otherwise we consider only a part of the permutations (in this cases the study of the s-boxes constructed from the matrices m1 and m2 using construction methods c1 and c2 is not completed we put ∗ in the last column of the corresponding row in the table). in the tables we list only those s-boxes which have linearity lin ≤ 2n/2+1. for example, if m = 5, r = 3, we have 3! = 6 permutations π but only three of the s-boxes constructed in the combination (c1, m1) have linearity 8 (the other three s-boxes have bigger linearity). all qcs-boxes with minimum linearity among the constructed s-boxes of even sizes have linearity lin(s) = 2n/2+1 (respectively nonlinearity nl(s) = 2n−1 − 2n/2). this is the best known linearity for s-box of the corresponding sizes but it is far from the parseval bound lin(s) ≥ 2n/2. 1. n=4: a definition for optimal 4×4 s-boxes is given in [12]. using our constructions and the field generated by the polynomial 1 + x + x4, we obtain many optimal s-boxes presented in table 1. all of them have the same parameters lin(s) = 8, nl(s) = 4, deg(s) = 3, ac(s) = 8, and δ = 4. 2. n=6: the used generator polynomial is g(x) = 1+x+x3 +x4 +x6. the constructed bijective 6×6 qcs-boxes with best cryptographic properties are given in table 2. they have linearity 16, most of them have algebraic degree 5, all but one have differential uniformity 4, and their autocorrelations are 16, 32, 64. 130 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 table 1. bijective 4 × 4 qcs-boxes qcs-boxes lin nl δ deg(s) ac(s) number c1, m1, m = 5,r = 3 8 4 4 3 8 3 c1, m1, m = 3,r = 5 8 4 4 3 8 60 c1, m2, m = 5,r = 3 8 4 4 3 8 3 c1, m2, m = 3,r = 5 8 4 4 3 8 28 c2, m1, m = 5,r = 3 8 4 4 3 8 3 c2, m1, m = 3,r = 5 8 4 4 3 8 60 c2, m2, m = 5,r = 3 8 4 4 3 8 6 c2, m2, m = 3,r = 5 8 4 4 3 8 28 3. n=8: in this case g(x) = 1+x+x3 +x5 +x8. many of the constructed bijective 8×8 qcs-boxes for different values of m and r have parameters close or equal to the best known nonlinearity for this size, namely nl(s) = 112. this is the nonlinearity of the s-box of the most popular block cipher aes [8]. our results are described in table 3. table 2. bijective 6 × 6 qcs-boxes qcs-boxes lin nl δ deg(s) ac(s) number c1, m1, m = 9,r = 7 16 24 4 5 16 7 16 24 4 3 32 7 16 24 4 2 64 7 c1, m2, m = 7,r = 9 16 24 8 4 24 1 c2, m1, m = 9,r = 7 16 24 4 5 16 7 c2, m1, m = 7,r = 9 16 24 4 5 16 18 c2, m2, m = 21,r = 3 16 24 4 5 16 1 c2, m2, m = 9,r = 7 16 24 4 5 16 1 c2, m2, m = 7,r = 9 16 24 4 5 16 1 4. n ≥ 10: we consider the sizes n = 10, n = 12, n = 14, n = 16 and n = 18. for these sizes we obtain qcs-boxes with nonlinearity nl(s) = 2n−1 −2n/2. the aes s-box have the same nonlinearity for n = 8 but these values are not so close to the parseval bound nl(s) =≤ 2n−1 − 2n/2−1. the used generator polynomials of the considered fields, as well as the cryptographic parameters of the constructed s-boxes in this case are presented in table 4. 5. n odd: for the odd values of n, we apply all our constructions for sizes n = 9, n = 11 and n = 15 but only the second construction gives bijective s-boxes with good cryptographic properties. we list them in table 5. the used generator polynomials are 1+x5+x9, 1+x+x2+x3+x6+x7+x9+x10+x11 and 1 + x2 + x4 + x5 + x15, respectively. we use the procedures from the parallel library boolsplg [1] to design algorithms realizing the presented constructions. boolsplg is a cuda library that includes algorithms for calculation of some cryptographic parameters and characteristics of boolean and vectorial boolean functions (s-boxes) (see 131 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 table 3. bijective 8 × 8 qcs-boxes s-boxes lin nl δ deg(s) ac(s) number aes s-box [8] 32 112 4 7 32 / c1, m1, m = 17,r = 15 32 112 4 7 32 15 c1, m1, m = 15,r = 17 32 112 4 5 48 4* 32 112 4 5 56 4* c2, m1, m = 85,r = 3 32 112 4 7 32 3 c2, m1, m = 51,r = 5 32 112 4 7 32 5 c2, m1, m = 17,r = 15 32 112 4 7 32 15 c2, m1, m = 15,r = 17 32 112 4 7 32 1* c2, m2, m = 85,r = 3 32 112 4 7 32 1 c2, m2, m = 51,r = 5 32 112 4 7 32 1 c2, m2, m = 17,r = 15 32 112 4 7 32 1 [7] for more information about cuda parallel computing platform). the advantage of using parallel algorithms is essential for bigger values of n (especially for n ≥ 14). for our calculations, we used a server with intel xeon e5-2640 processor that contains two graphics cards. the first graphics card is nvidia geforce gtx titan [15], which has 2688 cores running at 837 mhz and 288.4 gb/sec memory bandwidth. the second graphics card is nvidia titan x pascal [15], which has 3584 cores running at 1.5 ghz and 549 gb/sec memory bandwidth. we have used cuda toolkit 8.0 and developed environment ms visual studio 2012. all constructed qcs-boxes are available at the web page of the second author: http://www.moi.math.bas.bg/~iliya/. each s-box is represented as a sequence of hexadecimal numbers, representing the corresponding columns in the matrix bs. we give two examples. 1. the sequence (0,b,2,7,4,5,f,3,d,9,a,1,e,8,c,6) represents the s-box with bs =   0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 0 0   . we give also the values of n, m and r, the permutation π in word representation, and the parameters lin(s), nl(s), ac(s) and δ. in this case n = 4, m = 5, r = 3, π = (1, 0, 2), lin(s) = 8, nl(s) = 4, deg(s) = 3, ac(s) = 8, and δ = 4. 2. one of the obtained 6×6 s-boxes is 0,2f,17,25,2b,1c,32,f,3a,15,21,2e,1b,19,34,7,3d,33,2a,2c,30,9,37,2, 29,d,1d,c,5,1a,20,23,1e,a,39,1f,35,3,36,28,27,18,12,4,13,3b,b,1,14, 3f,6,11,e,24,26,16,3e,22,8,2d,3c,10,38,31. its parameters are n = 6, m = 9, r = 7, π = (6, 5, 4, 3, 2, 1, 0), lin(s) = 16, nl(s) = 24, deg(s) = 5, ac(s) = 16, δ = 4. acknowledgment: we gratefully acknowledge the support of nvidia corporation with the donation of the titan x pascal gpu used for this research. we also thank the anonymous reviewer for his/her valuable comments. 132 d. bikov et al. / j. algebra comb. discrete appl. 6(3) (2019) 123–134 table 4. bijective qcs-boxes for n = 10, n = 12, n = 14, n = 16 and n = 18 qcs-boxes lin nl δ deg(s) ac(s) number n = 10 g(x) = 1 + x3 + x4 + x8 + x10 c1, m1, m = 33,r = 31 64 480 4 9 64 1* c2, m1, m = 341,r = 3 64 480 4 9 64 3 c2, m1, m = 93,r = 11 64 480 4 9 64 11 c2, m1, m = 33,r = 31 64 480 4 9 64 1* c2, m2, m = 341,r = 3 64 480 4 9 64 1 c2, m2, m = 93,r = 11 64 480 4 9 64 1 n = 12 g(x) = 1 + x + x5 + x8 + x10 + x11 + x12 c1, m1, m = 65,r = 63 128 1984 4 11 128 1* c2, m1, m = 819,r = 5 128 1984 4 11 128 5 c2, m1, m = 585,r = 7 128 1984 4 11 128 7 c2, m1, m = 455,r = 9 128 1984 4 11 128 9 c2, m1, m = 315,r = 13 128 1984 4 11 128 13 c2, m2, m = 1365,r = 3 128 1984 4 11 128 1 c2, m2, m = 819,r = 5 128 1984 4 11 128 1 c2, m2, m = 585,r = 7 128 1984 4 11 128 1 c2, m2, m = 455,r = 9 128 1984 4 11 128 1 c2, m2, m = 315,r = 13 128 1984 4 11 128 1 n = 14 g(x) = 1 + x + x2 + x3 + x10 + x12 + x14 c1, m1, m = 129,r = 127 256 8064 4 13 256 1* c2, m1, m = 5461,r = 3 256 8064 4 13 256 3 c2, m2, m = 5461,r = 3 256 8064 4 13 256 1 n = 16 g(x) = 1 + x + x2 + x4 + x5 + x9 + x10 + x12 + x16 c1, m1, m = 257,r = 255 512 32512 4 15 512 1* c2, m1, m = 21845,r = 3 512 32512 4 15 512 3 c2, m1, m = 13107,r = 5 512 32512 4 15 512 5 c2, m2, m = 21845,r = 3 512 32512 4 15 512 1 c2, m2, m = 13107,r = 5 512 32512 4 15 512 1 n = 18 g(x) = 1 + x2 + x4 + x5 + x6 + x9 + x10 + x11 + x15 + x17 + x18 c1, m1, m = 513,r = 511 1024 130560 4 17 1024 1* table 5. bijective qcs-boxes for n = 9, n = 11 and n = 15 qcs-boxes lin nl δ deg(s) ac(s) number c2, m1, n = 9, m = 73,r = 7 44 234 2 8 48 7 c2, m2, n = 9, m = 73,r = 7 44 234 2 8 48 1 c2, m1, n = 11, m = 89,r = 23 88 980 2 10 88 1* c2, m1, n = 15, m = 4681,r = 7 360 16204 2 14 360 7 c2, m2, n = 15, m = 4681,r = 7 360 16204 2 14 360 1 133 d. bikov et al. / j. algebra 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(eds) fast software encryption. lecture notes in computer science, vol 9054. springer, berlin, heidelberg (2015) 494– 515. 134 http://www.moi.math.bas.bg/moiuser/~data/results/crypto/boolspl.html http://www.moi.math.bas.bg/moiuser/~data/results/crypto/boolspl.html https://www.doi.org/10.2478/cait-2018-0018 https://www.doi.org/10.2478/cait-2018-0018 https://www.doi.org/10.2478/cait-2018-0018 https://doi.org/10.1016/j.endm.2017.02.012 https://doi.org/10.1016/j.endm.2017.02.012 https://doi.org/10.1109/tit.2006.890727 https://doi.org/10.1109/tit.2006.890727 https://developer.nvidia.com/cuda-zone/ https://mathscinet.ams.org/mathscinet-getitem?mr=1986943 https://mathscinet.ams.org/mathscinet-getitem?mr=1986943 https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.390.7159 https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.390.7159 https://doi.org/10.1007/s12095-015-0170-5 https://doi.org/10.1007/s12095-015-0170-5 https://doi.org/10.1016/s0166-218x(00)00350-4 https://doi.org/10.1016/s0166-218x(00)00350-4 https://doi.org/10.1007/978-3-540-73074-3_13 https://doi.org/10.1007/978-3-540-73074-3_13 https://doi.org/10.1007/978-3-540-73074-3_13 https://doi.org/10.1109/18.959257 https://doi.org/10.1109/18.959257 https://mathscinet.ams.org/mathscinet-getitem?mr=465509 https://mathscinet.ams.org/mathscinet-getitem?mr=465509 https://www.nvidia.com/en-us/data-center/ https://www.springer.com/gp/book/9783642284953 https://www.springer.com/gp/book/9783642284953 https://www.springer.com/gp/book/9783642284953 https://doi.org/10.1007/978-3-662-48116-5_24 https://doi.org/10.1007/978-3-662-48116-5_24 https://doi.org/10.1007/978-3-662-48116-5_24 https://doi.org/10.1007/978-3-662-48116-5_24 introduction preliminaries the considered constructions constructed qcs-boxes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729402 j. algebra comb. discrete appl. 7(2) • 103–119 received: 19 february 2019 accepted: 22 september 2019 0 journal of algebra combinatorics discrete structures and applications a modified bordered construction for self-dual codes from group rings research article joe gildea, abidin kaya, alexander tylyshchak, bahattin yildiz abstract: we describe a bordered construction for self-dual codes coming from group rings. we apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. in particular we find a new extremal binary self-dual code of length 78. 2010 msc: 94b05 keywords: group rings, self-dual codes, codes over rings, bordered constructions 1. introduction the standard form of the generator matrix of any binary self-dual code of length 2n is of the form (in|a) where a is an n × n matrix satisfying aat = −in. when searching for self-dual codes, some special structure is imposed on the matrix a to make the search field more feasible. taking a to be a circulant or a block circulant matrix is one of the methods that has been utilized in the literature. sometimes, from a special such generator matrix, an extension can be achieved by modifying the matrix to get a new self-dual code of higher lengths. while these are generally known as extension methods in the literature, we can also view them as new construction methods for self-dual codes. one such example of a matrix, that we will extend in our constructions is defined in [13] as  1 0 x1 · · · xn 1 · · · 1 y1 y1 ... ... in a yn yn   , (1) joe gildea; university of chester, uk (email: j.gildea@chester.ac.uk). abidin kaya; sampoerna university, indonesia (email: nabidin@gmail.com). alexander tylyshchak; uzhgorod state university, ukraine (email: alxtlk@gmail.com). bahattin yildiz (corresponding author); northern arizona university, usa (email: bahattin.yildiz@nau.edu). 103 https://orcid.org/0000-0003-0175-1909 https://orcid.org/0000-0001-8106-3123 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 where yi = xi + 1 and (in|a) is the generator matrix of a self-dual code of length 2n, possibly coming from a special construction method described above. in this work, we shall extend the above construction, using matrices a that arise from group rings. group rings have been used to construct self-dual codes on many occasions. in [2], certain ideals of f2s4 were used to construct the extended binary golay code. in [15], an isomorphism from a group ring to a certain subring of the n × n matrices was described. this was used to construct self-dual codes in [16, 18, 19]. in [5], it is shown that zero divisors can’t be used to construct the putative [72, 36, 16] code. in [11], it is shown that unitary units can be used to construct self-dual codes under a certain construction, and using such units, many new extremal binary self-dual codes were obtained. in the same work, groups of different orders were used to describe many new construction methods for self-dual codes. in our work, we will extend the structure of the matrix given in (1) with the matrices that we get from group ring elements to find new methods for constructing self-dual codes. we will apply the constructions coming from groups of order 2p with p an odd number (using the cyclic group c2p and the dihedral group d2p) over the binary field f2, f4 and the rings rk and f4 + uf4 to obtain many extremal and best known binary self-dual codes of various lengths: 14, 28, 56, 44, 30, 38, 46, 54, 62, 70 and 78. in particular, we obtain a new extremal binary self-dual code of length 78. many of these lengths are not well-known like the oft-studied lengths of 64, 66 and 68. the rest of the paper is organized as follows: in section 2, we give some definitionas and notations that will be used in subsequent sections. in section 3 we give the construction together with a special case when it produces self-dual codes. in section 4 we give the computational results. we finish the paper with some concluding remarks and directions for possible future research. 2. definitions and notations 2.1. codes in this paper, all rings are assumed to be commutative, finite, frobenius rings with a multiplicative identity. a code over a finite commutative ring r is said to be any subset c of rn. when the code is a submodule of the ambient space then the code is said to be linear. to the ambient space, we attach the usual inner-product, specifically [v,w] = ∑ viwi. the orthogonal with respect to this inner-product is defined as c⊥ = {w | w ∈ rn, [v,w] = 0,∀v ∈ c}. since the ring is frobenius we have that for all linear codes over r, |c||c⊥| = |r|n. if a code satisfies c = c⊥ then the code c is said to be self-dual. if c ⊆ c⊥ then the code is said to be self-orthogonal. for binary codes, a self-dual code where all weights are congruent to 0 (mod 4) is said to be type ii and the code is said to be type i otherwise. an upper bound on the minimum hamming distance of a binary self-dual code finalized in [20]. theorem 2.1. ([20]) let di(n) and dii(n) be the minimum distance of a type i and type ii binary code of length n, respectively. then dii(n) ≤ 4b n 24 c + 4 and di(n) ≤ { 4b n 24 c + 4 if n 6≡ 22 (mod 24) 4b n 24 c + 6 if n ≡ 22 (mod 24). self-dual codes meeting these bounds are called extremal. a best known self-dual code of a given 104 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 length is a self-dual code that has the highest possible known minimum distance, for a length for which the existence of an extremal code is not currently known. throughout the paper, we will be constructing extremal or best known binary self-dual codes of different lengths. 2.2. group rings and special matrices we shall use group rings in our construction so we give the standard definition of a group ring. let g be a finite group or order n, then the group ring rg consists of ∑n i=1 αgigi, αgi ∈ r, gi ∈ g. addition in the group ring is done by coordinate addition, namely n∑ i=1 αgigi + n∑ i=1 βgigi = n∑ i=1 (αgi + βgi )gi. (2) the product of two elements in a group ring is given by( n∑ i=1 αgigi ) n∑ j=1 βgjgj   = ∑ i,j αgiβgjgigj. (3) it follows that the coefficient of gi in the product is ∑ gigj=gk αgiβgj . we restrict ourselves to finite groups since we are mainly concerned with using these to construct codes whose lengths will be determined by the order of the group. the following construction of a matrix was first given for codes over fields by hurley in [15] and extended to rings in [5]. let r be a finite commutative frobenius ring and let g = {g1,g2, . . . ,gn} be a group of order n. let v = αg1g1 + αg2g2 + · · · + αgngn ∈ rg. define the matrix σ(v) ∈ mn(r) to be σ(v) =   αg−11 g1 αg−11 g2 αg−11 g3 . . . αg−11 gn αg−12 g1 αg−12 g2 αg−12 g3 . . . αg−12 gn ... ... ... ... ... αg−1n g1 αg−1n g2 αg−1n g3 . . . αg−1n gn   . (4) we note that the elements g−11 ,g −1 2 , . . . ,g −1 n are the elements of the group g in some given order. lemma 2.2. if v = ∑n i=1 αgigi is unitary unit of rg and µ = ∑n i=1 αgi then µ 2 = 1. proof. the map ∗ : rg → rg defined by  ∑ g∈g agg  ∗ = ∑ g∈g agg −1 is an antiautomorphism of rg of order 2. an element v of v (kg) satisfying vv∗ = 1 is called unitary. the homomorphism ε : rg → r given by ε ( n∑ i=1 αgigi ) = n∑ i=1 αgi is called the augmentation mapping of rg. let v = ∑n i=1 αgigi, then v∗ = ∑n i=1 αgig −1 i and ε(v) = ε(v ∗) = ∑n i=1 αgi = µ. therefore ε(vv ∗) = ε(v)ε(v∗) = µ2 = 1. 2.2.1. σ(v) for d2p and c2p in what follows, circ(a1,a2, . . . ,am) denotes the m×m criculant matrix whose first row is given by (a1,a2, . . . ,am). 105 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 let g = c2p = 〈x |x2p = 1〉. if α = p−1∑ i=0 1∑ j=0 a1+i+pjx 2i+j ∈ rc2p, then σ(α) = ( a b b′ a ) . where a = circ(a1, . . . ,ap), b = circ(ap+1, . . . ,a2p) and b′ = circ(a2p,ap+1, . . . ,a2p−1). let g = d2p = 〈x,y |xp = y2 = 1, xy = x−1〉. if α = p−1∑ i=0 1∑ j=0 a1+i+pjx iyj ∈ rd2p, then σ(α) = ( a b bt at ) . where a = circ(a1, . . . ,ap) and b = circ(ap+1, . . . ,a2p), and at represents the transpose of a. 2.3. rings we shall use several alphabets in our constructions, including the binary field f2, the quaternary field f4 and rings rk and f4 + uf4. 2.3.1. the ring family rk the ring family rk were defined in [8] and [9]. we briefly give the descriptions of these rings. for k ≥ 1, define rk = f2[u1,u2, . . . ,uk]/〈u2i = 0,uiuj = ujui〉. (5) for k = 1, we denote the rings by f2 +uf2, and when k = 2, we denote them by f2 +uf2 +vf2 +uvf2, both of which have been considered in coding theory quite extensively. the rings can also be defined recursively as: rk = rk−1[uk]/〈u2k = 0,ukuj = ujuk〉 = rk−1 + ukrk−1. (6) for any subset a ⊆{1, 2, . . . ,k} we will fix ua := ∏ i∈a ui (7) with the convention that u∅ = 1. then any element of rk can be represented as∑ a⊆{1,...,k} caua, ca ∈ f2. (8) it is shown in [8] that the ring rk is a commutative ring with |rk| = 2(2 k). it is also shown that ∀a ∈ rk a2 = { 1 if a is a unit 0 otherwise. (9) 106 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 we shall now recall the gray map from rk to f2 k 2 . for r1 we have the following map: φ1(a + bu) = (b,a + b). then let c ∈ rk, c can be written as c = a + buk−1,a,b ∈ rk−1. then φk(c) = (φk−1(b),φk−1(a + b)). (10) the map φk is a distance preserving map and the following is shown in [9]. theorem 2.3. let c be a self-dual code over rk, then φk(rk) is a binary self-dual code of length 2kn. 2.3.2. the ring f4 + uf4 let f4 = f2 (ω) be the quadratic field extension of f2, where ω2 + ω + 1 = 0. the ring f4 + uf4 is defined via u2 = 0. note that f4 + uf4 can be viewed as an extension of r1 = f2 + uf2 and so we can describe any element of f4 + uf4 in the form ωa + ω̄b uniquely, where a,b ∈ f2 + uf2. a linear code c of length n over f4 + uf4 is an (f4 + uf4)-submodule of (f4 + uf4) n. in [10] and [6] the following gray maps were introduced; ψf4 : (f4) n → (f2) 2n ϕf2+uf2 : (f2 + uf2) n → f2n2 aω + bω 7→ (a,b) , a,b ∈ fn2 a + bu 7→ (b,a + b) , a,b ∈ fn2 . those were generalized to the following maps in [17]; ψf4+uf4 : (f4 + uf4) n → (f2 + uf2) 2n ϕf4+uf4 : (f4 + uf4) n → f2n4 aω + bω 7→ (a,b) , a,b ∈ (f2 + uf2) n a + bu 7→ (b,a + b) , a,b ∈ fn4 these maps preserve orthogonality in the corresponding alphabets. the binary images ϕf2+uf2 ◦ ψf4+uf4 (c) and ψf4 ◦ ϕf4+uf4 (c) are equivalent. the lee weight of an element is defined to be the hamming weight of its binary image. proposition 2.4. ([17]) let c be a code over f4 + uf4. if c is self-orthogonal, so are ψf4+uf4 (c) and ϕf4+uf4 (c). c is a type i (resp. type ii) code over f4 +uf4 if and only if ϕf4+uf4 (c) is a type i (resp. type ii) f4-code, if and only if ψf4+uf4 (c) is a type i (resp. type ii) f2 + uf2-code. furthermore, the minimum lee weight of c is the same as the minimum lee weight of ψf4+uf4 (c) and ϕf4+uf4 (c). corollary 2.5. suppose that c is a self-dual code over f4 +uf4 of length n and minimum lee distance d. then ϕf2+uf2 ◦ψf4+uf4 (c) is a binary [4n, 2n,d] self-dual code. moreover, c and ϕf2+uf2 ◦ψf4+uf4 (c) have the same weight enumerator. if c is type i (type ii), then so is ϕf2+uf2 ◦ψf4+uf4 (c). 3. the construction let v ∈ rg where r is a finite frobenius ring of characteristic 2 and g is a finite group of order 2p and p is odd. define the following matrix: α1 0 α2 + 1 α2 + 1 ... ... α2 + 1 α2 + 1 α3 + 1 α3 + 1 ... ... α3 + 1 α3 + 1 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 σ(v) ip 0p 0p ip m(σ) = . 107 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 let cσ be a code that is generated by the matrix m(σ) and µ = ∑n i=1 αgi. let a1 = (α1, 0) ∈ r 2, a2 = (α2, . . . ,α2,α3, . . . ,α3) ∈ r2p, a3 = (α4, . . . ,α4) ∈ r2p and b1 =   α2+1 α2+1 ... ... α2+1 α2+1 α3+1 α3+1 ... ... α3+1 α3+1  . then, m(σ)(m(σ))t = ( a1a t 1 + a2a t 2 + a3a t 3 a1b t 1 + a2 + a3σ(v) t b1a t 1 + a t 2 + σ(v)a t 3 b1b t 1 + i + σ(v)σ(v) t ) where • a1at1 + a2at2 + a3at3 = α21 + pα22 + pα23 + 2pα24 = α21 + p(α2 + α3)2, • a1bt1 + a2 + a3σ(v)t = (α1(α2 + 1), . . . ,α1(α2 + 1),α1(α3 + 1), . . . ,α1(α3 + 1)) + (α2, . . . ,α2,α3, . . . ,α3)+(α4, . . . ,α4)σ(v) = (α1(α2+1)+α2+µα4, . . . ,α1(α2+1)+α2+µα4,α1(α3+ 1) + α3 + µα4, . . . ,α1(α3 + 1) + α3 + µα4) and • b1bt1 + i + σ(v)σ(v)t = i + σ(v)(σ(v)t = i + σ(vv∗). theorem 3.1. let r be a finite commutative frobenius ring of characteristic 2, g be a finite group of order 2p where p is odd and µ = ∑2p i=1 αgi. if α 2 1 + p(α2 + α3) 2 = 0, α1(α2 + 1) + α2 + µα4 = 0, α1(α3 + 1) + α3 + µα4 = 0 vv ∗ = 1 and (α1,p(α2(α2 + 1) + α3(α3 + 1)),α2 + 1,α3 + 1) has free rank 1 then cσ is a self-dual code of length 4p + 2. proof. if vv∗ = 1, then i +σ(vv∗) = 0. additionally, let α21 +p(α2 +α3) 2 = 0, α1(α2 +1)+α2 +µα4 = 0 and α1(α3 + 1) + α3 + µα4 = 0, therefore cσ is self-orthogonal. it remains to show that the m(σ) has free rank 2p + 1. rank(m(σ)) = rank   α1 0 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 α2 + 1 α2 + 1 ... ... ip 0p α2 + 1 α2 + 1 σ(v) α3 + 1 α3 + 1 ... ... 0p ip α3 + 1 α3 + 1   = rank   α1 0 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 α2 + 1 ... ... ip 0p 0 α2 + 1 σ(v) 0 α3 + 1 ... ... 0p ip 0 α3 + 1   108 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 = rank   α1 γ1 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... ip 0p 0 0 σ(v) 0 α3 + 1 ... ... 0p ip 0 α3 + 1   = rank   α1 γ2 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... ip 0p 0 0 σ(v) 0 0 ... ... 0p ip 0 0   = rank     α1 γ2 α2 · · · α2 α3 · · · α3 α4 · · · α4 α4 · · · α4 0 0 ... ... ip 0p 0 0 σ(v) 0 0 ... ... 0p ip 0 0    i2 0 00 ip 0 0 σ(v)t ip     = rank   α1 γ2 γ3 · · · γ3 γ4 · · · γ4 α4 · · · α4 α4 · · · α4 0 0 ... ... 0 0 i + σ(v)σ(v∗) σ(v) 0 0 ... ... 0 0   = rank   α1 γ2 γ3 · · · γ3 γ4 · · · γ4 α4 · · · α4 α4 · · · α4 0 0 ... ... 0p 0p 0 0 σ(v) 0 0 ... ... 0p 0p 0 0   109 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 where γ1 = pα2(α2 + 1), γ2 = pα2(α2 + 1) +pα3(α3 + 1), γ3 = α2 +µα4 = α2 + 1 and γ4 = α3 +µα4 = α3 + 1. finally, rank(m(σ)) =   α1 γ2 α2 + 1 · · · α2 + 1 α3 + 1 · · · α3 + 1 γ5 · · · γ5 γ5 · · · γ5 0 0 ... ... 0p 0p 0 0 σ(v) 0 0 ... ... 0p 0p 0 0   where γ5 = α4 +µ2α4 = α4 + (1)α4 = 0 by lemma 2.2. therefore cσ is self if (α1,p(α2(α2 + 1) +α3(α3 + 1)),α2 + 1,α3 + 1) has free rank 1. the family of rings rk is particularly well suited for this construction. corollary 3.2. let r = rk and let g be a finite group of order 2p where p is odd. let v ∈ rg be a unitary unit. then if α2 + α3 is any unit then cσ is a self-dual code of length 4p + 2. proof. we will show that this case satisfies the hypotheses of theorem 3.1. if v is a unitary unit then vv∗ = 1. if α2 + α3 is a unit then (α2 + α3)2 = 1 by (9). then 1 + p(α2 + α3)2 = 1 + p(1) = 0 and vv∗ = 1. 4. computational results in this section, we apply the constructions discussed in the previous section over particular groups and rings that have been described before. the sizes of the groups and the alphabets used lead to particular lengths for self-dual codes. in all the subsequent subsections, we tabulate the extremal binary self-dual codes or the best-known (if the existence of extremal self-dual codes is not known for that length) self-dual codes of the certain lengths. 4.1. constructions from groups of order 6 we first apply construction d6 over the binary field and f4. table 1. extremal binary self-dual code of length 14 from d6 (α1,α2,α3,α4) (a1, . . . ,a6) |aut(c)| (1,0,1,1) (0,1,1,1,1,1) 27 ·32 ·72 table 2. extremal self-dual code of length 28 coming from applying d6 over f4 (α1,α2,α3,α4) (a1, . . . ,a6) |aut(c)| (1,ω,ω + 1,1) (0,0,1,1,ω,ω + 1) 25 ·3 ·7 110 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 in [14], the possible weight enumerators for a self-dual type i [56, 28, 10] code were obtained in two forms as: w56,1 = 1 + (308 + 4α) y 10 + (4246 − 8α) y12 + (40852 − 28α) y14 + · · · w56,2 = 1 + (308 + 4α) y 10 + (3990 − 8α) y12 + (42900 − 28α) y14 + · · · where α is an integer. applying the constructions over the ring f4 + uf4, we will be able to get binary self-dual codes of length 56. for brevity of notation, we need a brief notation for the elements of f4 + uf4. 0 ↔ 0000, 1 ↔ 0001, 2 ↔ 0010, 3 ↔ 0011, 4 ↔ 0100, 5 ↔ 0101, 6 ↔ 0110, 7 ↔ 0111, 8 ↔ 1000, 9 ↔ 1001, a ↔ 1010, b ↔ 1011, c ↔ 1100, d ↔ 1101, e ↔ 1110, f ↔ 1111. we use the ordered basis {uω,ω,u, 1} to express the elements of f4 + uf4. for instance, 1 + uω corresponds to 1001, which is represented by the hexadecimal 9. in the following tables α = mi denotes that the weight enumerator of the code has parameter α = m in wi, i = 1, 2. table 3. [56,28,10] codes over f4 + uf4 from d6 where (α1,α2,α3,α4) = (1,6,f,1) (a1, . . . ,a6) |aut(c)| type (a1, . . . ,a6) |aut(c)| type (a,a,1,1,4,5) 23 ·3 α = −322 (a,a,1,3,c,f) 22 ·3 α = −82 (a,a,1,b,6,d) 24 ·3 ·7 α = −562 (a,8,3,1,6,7) 2 ·3 α = −262 (a,8,3,3,e,d) 2 ·3 α = −202 (8,8,1,1,4,5) 24 ·3 α = −82 (8,8,1,3,c,f) 22 ·3 α = −322 table 4. [56,28,10] codes over f4 + uf4 from d6 where |aut(c)| = 2 ·3 (α1, . . . ,α4) (a1, . . . ,a6) type (a1, . . . ,a6) (α1, . . . ,α4) type (1,4,7,1) (0,0,9,9,c,d) α = −82 (1,4,7,1) (0,8,3,1,4,f) α = −242 (1,4,7,1) (0,0,b,1,c,7) α = −182 (1,4,7,1) (0,0,b,3,4,d) α = −122 (1,4,7,1) (2,0,b,3,c,7) α = −202 (1,4,7,1) (2,0,b,b,6,5) α = −142 (1,4,7,1) (2,a,1,3,6,d) α = −262 (1,4,7,9) (a,8,3,3,c,7) α = −62 (1,4,7,9) (8,2,9,3,e,7) α = −382 (1,4,7,9) (8,2,b,3,4,f) α = −302 (1,4,d,1) (0,8,1,1,e,7) α = −322 (1,4,d,1) (a,0,9,9,e,5) α = −321 (1,4,d,1) (a,2,b,b,4,d) α = −261 (1,4,d,1) (a,a,3,3,6,7) α = −141 (1,4,d,9) (0,0,9,1,4,5) α = −261 (1,4,d,9) (a,0,3,9,4,d) α = −381 (1,4,f,1) (2,8,9,b,6,f) α = −281 (1,4,f,1) (a,8,1,1,c,f) α = −221 (1,4,f,9) (0,0,9,3,c,f) α = −161 (1,4,f,9) (0,8,1,b,e,5) α = −281 (1,4,f,9) (8,2,3,9,4,d) α = −401 (1,4,f,9) (8,2,3,b,c,7) α = −461 we can also construct self-dual codes of length 56 from applying the construction d6 over the ring f2 + uf2 + vf2 + uvf2 as well. this is a ring of size 16, so in the same way as was done for f4 + uf4, we can use hexadecimals to shorten the notations. the correspondence between the binary 4 tuples and 111 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 the hexadecimals is as follows: 0 ↔ 0000, 1 ↔ 0001, 2 ↔ 0010, 3 ↔ 0011, 4 ↔ 0100, 5 ↔ 0101, 6 ↔ 0110, 7 ↔ 0111, 8 ↔ 1000, 9 ↔ 1001, a ↔ 1010, b ↔ 1011, c ↔ 1100, d ↔ 1101, e ↔ 1110, f ↔ 1111. the ordered basis {uv,v,u, 1} is used to express elements of f2 +uf2 +vf2 +uvf2 for instance, 1 +u+v is represented as 0111 which corresponds to hexadecimal 7. table 5. [56,28,10] codes over f2 + uf2 + vf2 + uvf2 from d6 where (α1,α2,α3,α4) = (1,6,b,1) (a1, . . . ,a6) |aut(c)| type (a1, . . . ,a6) |aut(c)| type (4,1,9,1,7,b) 23 ·3 α = −282 (4,5,d,1,f,3) 23 ·3 α = −522 (4,1,9,1,b,7) 23 ·3 α = −42 (4,5,d,9,7,3) 22 ·3 α = −42 (4,5,d,9,3,7) 22 ·3 α = −162 (4,7,f,1,3,f) 22 ·3 α = −282 (4,7,f,9,f,b) 22 ·3 α = −402 table 6. [56,28,10] codes over f2 + uf2 + vf2 + uvf2 from d6 where (α1,α2,α3,α4) = (1,2,5,1) (a1, . . . ,a6) |aut(c)| type (a1, . . . ,a6) |aut(c)| type (6,1,9,1,b,5) 22 ·3 α = −141 (6,1,9,1,d,3) 22 ·3 α = −261 (0,1,9,5,3,f) 22 ·3 α = −181 (0,1,9,5,7,b) 22 ·3 α = −421 (0,1,9,7,d,3) 22 ·3 α = −301 table 7. [56,28,10] codes over f2 + uf2 + vf2 + uvf2 from d6 where (α1,α2,α3,α4) = (1,2,7,1) (a1, . . . ,a6) |aut(c)| type (a1, . . . ,a6) |aut(c)| type (c,7,f,9,7,b) 22 ·3 α = −381 (c,7,f,9,b,7) 22 ·3 ·13 α = −381 4.2. constructions from groups of order 10 table 8. extremal binary self-dual code of length 22 from d10 (α1,α2,α3,α4) (a1, . . . ,a10) |aut(c)| (1,0,1,1) (0,0,0,1,1,0,1,0,1,1) 28 ·32 ·5 ·7 ·11 in [7], the possible weight enumerators for a self-dual type i [44, 22, 8] code were obtained in two forms as: w44,1 = 1 + (44 + 4β)y 8 + (976 − 8β)y10 + · · · where 10 ≤ β ≤ 122 and 112 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 w44,2 = 1 + (44 + 4β)y 8 + (1232 − 8β)y10 + · · · where 10 ≤ β ≤ 154. table 9. extremal self-dual code of length 44 over f4 from d10 (α1,α2,α3,α4) (a1, . . . ,a6) |aut(c)| type (1,ω,ω + 1,1) (0,0,0,1,1,1,ω,ω,ω + 1,ω + 1) 24 ·5 w44,2 (β = 14) (1,ω,ω + 1,1) (0,0,0,ω,ω + 1,0,ω,ω + 1,ω,ω + 1) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,0,0,ω,ω + 1,1,1,ω,1,ω + 1) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,0,1,1,1,0,ω,ω,ω + 1,ω + 1) 23 ·5 w44,2 (β = 34) (1,ω,ω + 1,1) (0,0,1,1,1,1,1,1,ω,ω + 1) 23 ·5 w44,2 (β = 34) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,0,1,1,ω + 1,ω) 5 w44,2 (β = 14) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,0,1,ω,1,ω + 1) 5 w44,2 (β = 9) (1,ω,ω + 1,1) (0,0,1,ω,ω + 1,1,ω,ω,ω,ω) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω,0,1,1,ω + 1,ω + 1) 22 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω,1,ω,ω,ω,ω + 1) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,0,ω,1,ω + 1,1,1,1,1,1) 216 ·32 ·52 w44,2 (β = 74) (1,ω,ω + 1,1) (0,0,1,ω,ω,1,0,ω,ω,ω + 1,ω) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (0,ω,ω,ω + 1,ω + 1,1,ω,ω + 1,ω,ω + 1) 2 ·5 w44,2 (β = 4) (1,ω,ω + 1,1) (1,1,1,ω,ω + 1,1,ω,ω + 1,ω,ω + 1) 22 ·5 w44,2 (β = 14) table 10. extremal self-dual code of length 44 over f2 + uf2 from c10 (α1,α2,α3,α4) (a1, . . . ,a6) |aut(c)| type (1,u,1,1) (u,u,u,1,1,1,1,0,u + 1,0) 22 ·5 w44,1 (β = 32) (1,u,1,1) (u,u,u,1,1,u + 1,u + 1,0,u + 1,0) 215 ·34 ·52 ·72 w44,1 (β = 122) (1,u,1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 w44,2 (β = 10) (1,u,1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 w44,2 (β = 30) (1,u,1,1) (0,u,0,1,1,u + 1,u + 1,u,u + 1,u) 23 ·5 w44,1 (β = 12) (1,u,1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·32 ·52 w44,2 (β = 90) (1,u,u + 1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 w44,2 (β = 14) (1,u,u + 1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 w44,2 (β = 34) (1,u,u + 1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·32 ·52 w44,2 (β = 74) (1,0,u + 1,1) (u,0,u,1,1,1,1,0,1,0) 24 ·5 w44,2 (β = 14) (1,0,u + 1,1) (u,0,u,1,1,u + 1,u + 1,0,1,0) 23 ·5 w44,2 (β = 34) (1,0,u + 1,1) (0,0,0,1,1,1,1,u,1,u) 216 ·34 ·52 ·72 ·112 w44,2 (β = 154) 113 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 4.3. constructions from groups of order 14 table 11. extremal binary self-dual code of length 30 from d14 (α1,α2,α3,α4) (a1, . . . ,a14) |aut(c)| (1,0,1,1) (0,0,0,0,0,0,1,0,0,1,0,1,1,1) 211 ·3 ·7 (1,0,1,1) (0,0,0,0,0,1,1,0,0,1,0,0,1,1) 28 ·7 (1,0,1,1) (0,0,0,1,0,1,1,0,1,1,1,1,1,1) 27 ·32 ·5 ·7 table 12. extremal binary self-dual code of length 30 from c14 (α1,α2,α3,α4) (a1, . . . ,a14) |aut(c)| (1,0,1,1) (0,0,0,0,0,1,1,0,1,1,0,0,1,0) 28 ·7 table 13. extremal binary self-dual code of length 60 over f4 from d14 (α1,α2,α3,α4) (a1, . . . ,a14) |aut(c)| type (1,ω,ω + 1,1) (0,0,1,ω,1,ω,1,0,1,ω + 1,0,ω,ω + 1,ω + 1) 22 ·7 w60,1 (β = 0) 4.4. constructions from groups of order 18 table 14. extremal binary self-dual code of length 38 from d18 (α1,α2,α3,α4) (a1, . . . ,a18) |aut(c)| (1,0,1,1) (0,0,0,0,0,0,1,1,1,0,0,1,1,0,1,1,1,1) 2 ·32 (1,0,1,1) (0,0,0,0,1,0,1,1,1,0,0,1,1,0,1,0,1,1) 2 ·32 ·19 4.5. constructions from groups of order 22 table 15. [46,22,8] codes from d22 (α1,α2,α3,α4) (a1, . . . ,a22) |aut(c)| (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,1,1,1,1,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,0,1,0,1,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,1,1,0,1,1) 11 (1,0,1,1) (0,0,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1) 215 ·34 ·52 ·72 ·112 ·232 114 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 table 16. [46,22,8] codes from c22 (α1,α2,α3,α4) (a1, . . . ,a22) |aut(c)| (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,0,1,1,0,0,0,1) 2 ·11 (1,0,1,1) (0,0,0,0,0,0,1,0,1,1,1,0,0,1,1,0,1,1,0,1,0,0) 2 ·11 (1,0,1,1) (0,0,0,0,1,0,1,1,1,1,1,0,1,1,0,1,1,1,1,0,1,0) 2 ·11 (1,0,1,1) (0,0,0,1,0,0,1,1,1,1,1,1,1,0,1,1,0,1,1,0,0,1) 2 ·11 (1,0,1,1) (0,0,0,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1) 215 ·34 ·52 ·72 ·112 ·232 4.6. constructions from groups of order 26 from [3], it is known that the weight enumerator of a [54, 27, 10] self-dual code can be of the follwing form: w54,1 = 1 + (351 − 8β)y10 + (5031 + 24β)y12 + . . . w54,2 = 1 + (351 − 8β)y10 + (5543 + 24β)y12 + . . . in the following tables, we consturct inequivalent self-dual codes of parameters [54, 27, 10] from d26 and c26. table 17. inequivalent [54,27,10] codes from d26 (α1,α2,α3,α4) (a1, . . . ,a26) |aut(c)| type (1,0,1,1) (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,1,1,1,0,1) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,1,0,1,0,1,1,1) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0,1,1) 13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,0,0,1,1,1,1,0,0,1) 2 ·3 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,1,1,1) 13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,1,0,0,1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1) 2 ·34 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0,1,1,0,1,1,1,1) 2 ·3 ·13 w54,1 (β = 0) table 18. inequivalent [54,27,10] codes from c26 (α1,α2,α3,α4) (a1, . . . ,a26) |aut(c)| type (1,0,1,1) (0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,1,0) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,0,0,1,1,0,1,1,1,0) 2 ·3 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,0,1,0,0,1,1,1,1,1,1) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,0,1,1,0,1,1,1,1,0,1,0) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,1,0,1,0,1,1,1,1,1,0,0,1,0,1,1,1,1,0,1,0) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,0,0,1,0,1,1,0,1) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,0,1,1,1,1,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,0,0,1,1,0,1,1,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,0) 2 ·13 w54,1 (β = 0) (1,0,1,1) (0,0,0,1,0,0,1,1,0,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,1,0) 2 ·3 ·13 w54,1 (β = 0) 115 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 4.7. constructions from groups of order 30 there are two possibilities for the weight enumerators of extremal singly-even [62, 31, 12]2 codes ([3]): w62,1 = 1 + 2308y 12 + 23767y14 + · · · w62,2 = 1 + (1860 + 32β) y 12 + (28055 − 160β) y14 + · · · , : 0 ≤ β ≤ 93. only codes with weight enumerator for β = 0, 2, 9, 10, 15, 16 in w62,2 known to exist. table 19. [62,31,12] codes from d30 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a30) |aut(c)| type (0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,1) 2 ·3 ·5 w62,2 (β = 10) (0,0,0,0,0,0,0,0,1,0,1,1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1) 2 ·3 ·5 w62,2 (β = 10) (0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,0,1,1,1,0,1,1,1,0,1,1,1) 22 ·3 ·5 w62,2 (β = 0) (0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,0,0,1,1,1,1,0,0,1,1,1,1,1,1,1) 2 ·3 ·5 w62,2 (β = 0) (0,0,0,0,0,1,0,0,1,0,0,1,1,1,1,0,1,0,1,1,0,1,1,1,1,1,1,0,1,1) 2 ·3 ·5 w62,2 (β = 10) (0,0,0,0,0,1,1,0,0,1,0,1,1,1,1,0,0,1,1,0,1,1,1,0,1,1,0,1,1,1) 3 ·5 w62,2 (β = 0) (0,0,0,0,0,1,1,0,1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1,1,1) 2 ·3 ·5 w62,2(β = 0) (0,0,0,0,0,1,1,1,0,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1) 2 ·3 ·5 w62,2(β = 0) (0,0,0,0,1,1,0,0,1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1) 2 ·3 ·5 w62,2(β = 0) 4.8. constructions from groups of order 34 the weight enumerator of a self-dual [70, 35, 12]2 code is in one of the following forms ([13]): w70,1 = 1 + 2β + (11730 − 2β − 128γ) y14 + (150535 − 22β + 896γ) y16 + · · · w70,2 = 1 + 2β + (9682 − 2β) y14 + (173063 − 22β) y16 + · · · the code with weight enumerator for γ = 1,β = 416 is constructed in [13]. together with the results from [4] and [12], the existence of codes with weight enumerators γ = 0 in w70,1 is known for many β values. in the following tables we tabulate the [70, 35, 12] self-dual codes from d34 and c34 together with their β values and automorphism groups. note that the automorphism groups all have an element of order 17 in them. naturally, these have the same parameters as the ones obtained in [12]. however, here we have given an alternative construction to those codes. 116 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 table 20. [70,35,12] codes from d34 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a34) |aut(c)| w70,1 (γ = 0) (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,0,1,1,0,1,0,1,1,0,1,0,1,1) 2 ·17 β = 102 (0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,0,0,0,0,0,1,0,1,1,0,0,1,1,0,1,1,1) 2 ·17 β = 136 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,1,0,1,1,0,1,1) 17 β = 170 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,0,0,1,0,1,0,0,1,1,0,0,1,1,1) 17 β = 204 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,0,1,0,1,1,0,1,1,0,1,0,1,1) 17 β = 238 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,1,0,1,1,0,1,0,1,0,1,1,0,0,1,1) 17 β = 272 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,0,1,0,0,0,1,1,0,1,1,0,1,1,1,1) 17 β = 306 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,0,0,0,0,1,0,1,0,1,0,0,1,1,0,0,1,1,1) 17 β = 340 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,1,1) 17 β = 374 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,1,1) 17 β = 408 (0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,0,1,0,1,0,1,1,0,0,0,1,0,1,1,1) 17 β = 442 (0,0,0,0,0,0,0,0,1,1,0,1,1,1,1,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,1) 2 ·17 β = 476 (0,0,0,0,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,1,1,1,1) 17 β = 510 table 21. extremal [70,35,12] codes from c34 where (α1,α2,α3,α4) = (1,0,1,1) (a1, . . . ,a34) |aut(c)| w70,1 (γ = 0) (0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0,1,1,0) 2 ·17 β = 102 (0,0,0,0,0,0,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,1,1,1,1) 2 ·17 β = 136 (0,0,0,0,0,0,0,0,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0,1,1,0,0,0,1,1) 2 ·17 β = 238 (0,0,0,0,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,1,1,1) 2 ·17 β = 272 (0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,1,0,1,0,1) 2 ·17 β = 306 (0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1) 2 ·17 β = 374 (0,0,0,0,0,0,1,0,0,0,1,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,1,1,1) 2 ·17 β = 442 4.9. constructions from groups of order 38 the possible weight enumerators for self-dual codes of parameters [78, 39, 14] are given as follows ([7]): w78,1 = 1 + (3705 + 8β)y 14 + (62244 + 512α− 24β)y16 + . . . , 0 ≤ α ≤ −1 16 β ≤ 28, w78,2 = 1 + (3750 + 8α)y 14 + (71460 − 24α)y16 + . . . ,−486 ≤ α ≤−135. for many of these parameters, the existence of a code with that weight enumerator is not known. together with the ones that were recently found in [1, 21, 22], the existence of codes which have w78,1 with α = 0 and β = 0,−13,−19 − 26,−38,−39,−52,−65,−78,−104,−117 and which have α = −135 in w78,2. in the following table we construct [78, 39, 14] self-dual codes from d38. the one with β = −57 is a new code. 117 j. gildea et al. / j. algebra comb. discrete appl. 7(2) (2020) 103–119 table 22. extremal [78,39,14] codes from d38 where (α1,α2,α3,α4) = (1,0,1,1) and a1 = a2 = a3 = a4 = 0 (a5, . . . ,a34) |aut(c)| w78,1 (α = 0) (1,1,1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0,1,1,1,1,1,1,0,1,1,1,1,1) 2 ·19 β = 0 (1,1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,1,1,1,1,0,1,1,1,1,1) 19 β = 0 (1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,1,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,0,1,1) 19 β = −19 (1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,1,1) 19 β = −38 (1,1,1,1,0,0,1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,1,1,0,1,1) 19 β = −57 5. conclusion we have integrated a modified bordered construction with the matrices corresponding to a group ring element in rc2p and rd2p where r is a commutative frobenius ring of characteristic 2 and as a result we have been able to obtain many extremal binary self-dual codes. the structure of the groups has allowed us to look at such lengths as 62, 70, 78, etc. that are different than the oft-studied lengths of 64, 66 and 68. in addition to giving an alternative construction to many extremal or best-known self-dual codes we were able to obtain a new code of length 78, with α = 0, β = −57 in w78,1. the results we have obtained demonstrate the relevance of our constructions and may lead to more such results when considered over different rings and groups. acknowledgment: the authors would like to thank the anonymous referees for their useful comments that helped improve the paper. references [1] d. anev, n. yankov, self–dual codes of length 78 with an automorphism of order 13, in xvth international workshop on optimal codes and related topics, sofia, bulgaria, july 2017. 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[22] t. zhang, j. michel, t. feng, g. ge, on the existence of certain optimal self–dual codes with lengths between 74 and 116, the electronic journal of combinatorics 22(4) (2015) 1–25. 119 https://doi.org/10.1016/j.ffa.2018.01.002 https://doi.org/10.1016/j.ffa.2018.01.002 https://hrcak.srce.hr/157710 https://hrcak.srce.hr/157710 https://doi.org/10.1006/ffta.1996.0174 https://doi.org/10.1006/ffta.1996.0174 https://arxiv.org/abs/0711.3983 https://arxiv.org/abs/0711.3983 https://doi.org/10.1006/eujc.2001.0509 https://doi.org/10.1007/s10623-011-9530-0 https://doi.org/10.1007/s10623-011-9530-0 https://doi.org/10.1109/tit.2008.928260 https://doi.org/10.1109/tit.2008.928260 https://doi.org/10.1109/18.651000 http://dx.doi.org/10.3934/amc.2017047 http://dx.doi.org/10.3934/amc.2017047 https://doi.org/10.37236/5213 https://doi.org/10.37236/5213 introduction definitions and notations the construction computational results conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.645021 j. algebra comb. discrete appl. 7(1) • 35–53 received: 13 june 2019 accepted: 17 august 2019 journal of algebra combinatorics discrete structures and applications locally recoverable codes from planar graphs research article kathryn haymaker, justin o’pella abstract: in this paper we apply kadhe and calderbank’s definition of lrcs from convex polyhedra and planar graphs [4] to analyze the codes resulting from 3-connected regular and almost regular planar graphs. the resulting edge codes are locally recoverable with availability two. we prove that the minimum distance of planar graph lrcs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes. constructions of regular and almost regular planar graphs are given, and their associated code parameters are determined. in certain cases, the code families meet the rate bound. 2010 msc: 94b05, 94b25, 94b65 keywords: error-correction, local recovery, planar graphs, availability, rate bound 1. introduction in the classical view of error-correcting codes (eccs), the central question involves recovering an original transmitted message from an entire received word, even in the presence of errors or erasures. the recent rise of distributed storage applications has inspired research on the local erasure-correcting capabilities of eccs. a code c is a locally recoverable code (lrc) with locality r if for all codewords c ∈ c, any erased symbol of c can be recovered by accessing at most r other symbols from c. in [2], gopalan, et al. describe the need for efficient erasure coding with small locality to address node failure in distributed storage networks. a code c has availability t if for any codeword position i, there are t disjoint recovery sets of sizes r1, r2, · · · , rt, respectively for position i. kadhe and calderbank present a construction of lrcs with availability two from convex polyhedra, specifically demonstrating the construction with the five examples of the platonic solids. the authors prove that lrcs from convex polyhedra have length and dimension given by the number of edges and the number of faces minus one, respectively, of the polyhedra, and they provide the weight distribution of the codes from the platonic solids [4]. the pre-print [5] is an kathryn haymaker (corresponding author); department of mathematics and statistics, villanova university, united states (email: kathryn.haymaker@villanova.edu). justin o’pella; thomas jefferson university, united states (email: justin.opella@jefferson.edu). 35 https://orcid.org/0000-0001-5965-4197 https://orcid.org/0000-0002-1381-4172 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 extended version of the conference paper [4] and also contains results on the rate of optimal binary lrcs with small availability and locality. we direct the reader to [5] and the references therein for additional background on binary lrcs. the goal of this paper is to build upon the results in [4] on lrcs generated by convex polyhedra to classify and construct infinite families of lrcs generated by simple, 3-connected planar graphs. we prove a relationship between the girth of planar graphs and the minimum distance d of the code. we also prove a bound on the parameters n, k, and d of an lrc generated by a planar graph using classical facts about planar graphs. in many cases, the bound is tighter than previously discovered bounds for lrcs with availability two. we discuss how constructions of j-regular planar graphs (j = 3, 4, 5) yield planar graph lrcs with recovery set size j − 1. finally, we present families of almost regular planar graphs. a graph is a (j, j + 1) almost regular graph if its degree sequence contains only j and j + 1, for some positive integer j. we determine the code parameters from these graphs and classify the types of graphs that yield codes meeting the rate bound. 2. preliminaries we begin with the necessary definitions and notation. a planar graph is a graph that can be drawn in the plane with no edges crossing. let vi represent the number of vertices with degree i. the maximum degree of a graph is denoted by ∆. a wheel graph is a graph that contains a cycle where every vertex on the cycle is connected to a universal vertex (see the black edges of figure 8). the girth of a graph is the minimum length of a cycle contained in the graph. the degree of a face f of a planar graph is the number of edges bordering f. figure 1 shows an almost regular (2, 3) graph, with three faces. the outer or infinite face is degree four. the other two faces are each of degree three. the girth of the graph is three and ∆ = 3. a 3-connected graph is a graph in which the removal of any collection of two or fewer vertices does not disconnect the graph. figure 1. a planar type (2, 3) graph with v2 = 2, v3 = 2. this graph is not 3-connected, since the deletion of the two vertices of degree 3 disconnects the graph. the following bound on the rate of lrcs with availability two is proven in [8] for sequential recovery codes and in [4] for non-sequential recovery codes. the rate r of a locally recoverable code with availability two and recovery set sizes at most r satisfies: r ≤ r r + 2 . (1) in [4], kadhe and calderbank introduced codes generated by convex polyhedra. definition 2.1 (def. 2, [4]). consider a convex polyhedron γ with v vertices, e edges, and f faces. fix an arbitrary labeling of its edges 1 through e. let c be a subset of fe2 such that for a vector c ∈ c, every collection of entries corresponding to edges that meet at a vertex sum to zero over f2. we say that the code is generated by γ, and denote it as c(γ). kadhe and calderbank use properties of convex polyhedra to prove that the dimension of a code c(γ) is f − 1. 36 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 lemma 2.2 (kadhe and calderbank, [4]). for a convex polyhedron γ with v vertices, e edges and f faces, the code c(γ) generated by γ is an [e, f − 1] code. in this paper we restrict our work to 3-connected planar graphs because there is a correspondence between the embeddings of convex polyhedra and 3-connected planar graphs [10]. moreover, it is important to guarantee that every edge is involved in a cycle that encloses a face so that we can unambiguously quantify the degree of a face. throughout, we will use the notion of an lrc generated by a planar graph introduced in [4], with the additional understanding that all planar graphs considered in this paper are 3-connected. definition 2.3 (def. 7, [4]). consider a planar graph γ with v vertices and e edges. fix an arbitrary labeling of its edges from 1 through e. let c be a subset of fe2 such that for every vector c ∈ c , the entries of c corresponding to edges that meet at a vertex sum to zero over f2. we say that the code c is generated by γ, and denote it as c(γ). we call a code generated by a planar graph in this way an edge code. remark 2.4. edge codes have availability t = 2 since every edge is incident with two vertices that provide parity checks on the edge. the sizes of the recovery sets for edge {vi, vj} are deg(vi)-1 and deg(vj)-1, respectively. that is, r ≤ ∆ − 1. throughout the paper we make use of euler’s formula for planar graphs: a planar graph with v vertices, e edges, and f faces satisfies v −e + f = 2. in the proof of lemma 2.2, the fact that γ is a convex polyhedron is used to invoke euler’s formula. since euler’s formula also applies to 3-connected planar graphs, the proof of lemma 2.2 implies the following corollary. corollary 2.5. for a 3-connected planar graph γ with v vertices, e edges and f faces, the code c(γ) generated by γ is an [e, f − 1] code. example 2.6. a tetrahedron embedded into the plane contains 4 vertices of degree 3, 6 edges, and 4 faces. the code generated by the tetrahedron is a [6, 3] code with locality r = 2 and availability t = 2. the tetrahedron embedded in the plane and one example codeword are shown in figure 2. the examples constructed in [4] include lrcs from the platonic solids. parameters of these codes are summarized in table 1. figure 2. tetrahedron with example codeword bits in black, and index positions in red. the codeword shown is (1, 1, 1, 0, 0, 0). 37 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 table 1. parameters of codes associated with the platonic solids [4]. platonic solid (n, k) r t tetrahedron (6, 3) 2 2 cube (12, 5) 2 2 octahedron (12, 7) 3 2 dodecahedron (30, 11) 2 2 icosahedron (30, 19) 4 2 3. parameters of edge codes we begin by proving that the minimum distance of an edge code is equal to the girth of the planar graph, which is the same as the smallest degree of a face. proposition 3.1. the minimum distance of an edge code generated by a planar graph is equal to the smallest degree of a face. proof. let c represent the set of edges in a planar graph g corresponding to a smallest face of g. then the indicator vector of c, xc, is a codeword, since every vertex in the cycle c has exactly two incident edges labeled 1, while every vertex outside of the cycle has all incident edges labeled 0. next we show that a minimum-weight codeword must contain a cycle from g. suppose that x is a nonzero minimum-weight codeword in a code generated by the edges of a 3-connected planar graph g. let gx be the subgraph induced by the edges that correspond to the support of x. consider an edge e corresponding to a position in the support of the codeword x. seeking a contradiction, suppose that the connected component of gx containing e is a tree. then there must be at least two leaf vertices, each of which has exactly one neighboring edge labeled 1, and therefore these leaf vertices represent unsatisfied parity checks. this contradicts the assumption that x is a codeword. next we use facts about planar graphs to prove the following rate bound. theorem 3.2. an [n, k, d] edge code generated by a 3-connected planar graph g with v ≥ 3 satisfies the following bound: k n ≤ 2 d − 1 n . (2) proof. note that for any simple planar graph, the sum of the degrees of all faces is equal to twice the number of edges. let d be the smallest degree of a face in g. more formally, ∑ i fi = 2e, and therefore: 3f ≤ ∑ i fi = 2e. since d is the smallest degree of a face in g, df can replace 3f in the inequality as follows: df ≤ 2e. therefore we obtain f − 1 e ≤ 2 d − 1 e . applying corollary 2.5, we have established that r ≤ 2 d − 1 n . 38 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 remark 3.3. a graph in which every face has degree d would result in equalities in the steps of the proof, producing a code whose rate meets the bound. in section 5.1, we use this fact to classify almost regular planar graphs whose corresponding codes meet the rate bound. notice that the smallest degree of a face in g is also the girth of the graph, so bound 2 can also be stated in terms of the girth. the cases in this paper include many codes with d = 3. figure 3 shows a comparison of bound 1 and bound 2 for different recovery set sizes and d = 3. for cases where r ≥ 4, bound 2 is tighter than bound 1. when r = 3, bound 2 is tighter than bound 1 for n < 15. in the case where r = 2, bound 2 is tighter than bound 1 for n < 6. for codes generated by planar graphs with d > 3, bound 2 is tighter than bound 1. figure 3. bound comparison for edge codes: bound 1 and bound 2, for d = 3. we will repeatedly make use of the following well-known fact about planar graphs, so we include a brief proof. fact 1. the average degree of a planar graph is strictly less than 6. proof. considering that 3f ≤ 2e for a planar graph and substituting this into euler’s formula gives: ∑ i di ≤ 6v − 12. therefore the average vertex degree is at most 6 − 12 v . 4. j-regular planar graphs regular planar graphs of degree 3, 4, or 5 can yield lrcs with availability two, where every recovery set has the same size, ∆ − 1. relaxing the regularity condition can still yield codes that meet bound 2, so we also explore the expanded class of almost regular planar graph constructions in section 5. 39 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 using the website [7], with information from [6], we determine the parameters of lrcs from small 3-regular planar graphs of girth at least 5 in table 2. the graph with v = 20 is the dodecahedron edge code that was presented in [4, 5]. table 2. 3-regular planar graphs of girth at least 5 and their edge code parameters. all codes have minimum distance at least 5. bound 3 assumes d = 5. rate and bound 3 rounded to 4 decimal places. there are three non-isomorphic graphs with 28 vertices. v e f code parameters rate bound 3 20 30 12 [30, 11] .3667 .3667 24 36 14 [36, 13] .3611 .3722 26 39 15 [39, 14] .3590 .3744 28 42 16 [42, 15] .3591 .3762 we now summarize some constructions of infinite families of 3, 4, and 5-regular planar graphs and give their corresponding edge code parameters. there are no j-regular planar graphs for j > 5 by fact 1. we concentrate on girth 3, 4, and 5 to obtain lrcs with minimum distance at most 5. bound 2 shows that larger minimum distance results in low code rates (r < 1 3 ), and therefore we do not consider d > 5 in this paper. the proof of bound 2 shows that a planar graph with all faces of degree g for a planar graph of girth g yields an lrc with rate meeting bound 2. the graphs in which every face has the same degree are called triangulations (g = 3), quadrangulations (g = 4), and pentangulations (g = 5), and the j-regular versions of these are precisely the platonic solids (see [4]). case 1: 3-regular planar graphs define the operation of splitting a face of a planar graph as adding a vertex to each of two distinct edges of the face and joining the two new vertices by an edge. steinetz (edited by rademacher) showed that 3-regular planar graphs can be generated by “adding edges" to the tetrahedron [10], including by splitting faces. this process is also called adding handles or adding ears. the face splitting operation increases the number of vertices in the graph by two, increases the number of edges by three, and increases the number of faces by one. the degree of all vertices remains three. therefore, starting with the tetrahedron with v = 4, e = 6, f = 4, the splitting process at iteration i results in a graph with v = 4+2i, e = 6+3i, f = 4+i. the resulting family of edge codes has parameters: [6 + 3i, 3 +i, 3], with rate ri = 3+i6+3i, which approaches 1 3 from above as i increases. the rate can be improved by adding edges between existing vertices, but this process destroys the 3-regularity of the graph, which would also negatively impact the small locality of the resulting lrc. face splitting is the only edge-addition operation that preserves 3-regularity, and the cost is that every operation that splits a face adds at least one additional non-triangular face to the graph. case 2: 4-regular planar graphs broersma, et al. detailed a process of generating all 3-connected 4-regular planar graphs from the octahedron [1]. inserting a new triangle into an existing triangular face of the octahedron is one method that results in an infinite family of 4-regular planar graphs. see the yellow edges being inserted into the red triangle in figure 15 for an example. on iteration i of this process, the resulting graph has v = 6 + 3i, e = 12 + 6i, f = 8 + 3i. the resulting family of edge codes has parameters [12 + 6i, 7 + 3i, 3], with rate ri = 7+3i12+6i, which approaches 1 2 from above as i increases. as in case 1, adding edges could increase the rate at the cost of the graph regularity and the small locality of the edge code. case 3: 5-regular planar graphs families of 5-regular simple planar graphs are generated in [3]. an infinite family d1, d2, . . . is given in [3] where d1 is the icosahedron planar embedding. d2 is formed by splitting an edge of d1, duplicating the graph and gluing the split edges back together. see [3], page 420 for the first few examples in this infinite family. the graph di has v = 12i, e = 30i, f = 18i + 2. the resulting edge code family has parameters [30i, 18i + 1, 3] and rate ri = 18i+130i . 40 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 section 5.3 contains another approach to constructing an infinite family of 5-regular planar graphs. dropping the strict regularity condition and considering almost regular graphs in the next section in some cases yields improved code rates and more flexible constructions, without a large difference in the local erasure correction capabilities of the codes. 5. almost regular type (j, j + 1) planar graphs almost regular type (j, j + 1) planar graphs will be considered for j ∈ {3, 4, 5}. values of j ∈{1, 2} are not considered due to the resulting small locality. since the average degree of a planar graph is strictly less than 6 (fact 1), j cannot be greater than 5. a type (j, j + 1) graph has vj + vj+1 vertices. the number of edges e can be calculated as follows: e = jvj + (j + 1)vj+1 2 . (3) we use euler’s formula to find the number of faces f and the code rate for all codes in this section. 5.1. almost regular graphs attaining the bound first we consider graphs that yield codes that meet bound 2, with minimum distance d = 3. that is, we assume the graphs are triangulations since every face must have degree 3 to meet bound 2 with d = 3. theorem 5.1. there are no infinite families of planar graphs with vertices of degree 3, 4, or 5 that are triangulations. the finite list of such graphs is given in table 3. proof. the number of vertices v in an almost regular graph with only vertices of degree 3, 4, and 5 can be restated as v = v3 + v4 + v5. the number of edges is e = 32v3 + 2v4 + 5 2 v5. note a graph that generates a code with d = 3 will attain bound 3 when every face of the graph is degree 3. more formally, the bound is attained if 3f = 2e. substituting this information into euler’s formula allows for an algebraic solution. v −e + f = 2 3v − 3e + 3f = 6 3v −e = 6 3v3 + 3v4 + 3v5 − 3 2 v3 − 2v4 − 5 2 v5 = 6 3 2 v3 + v4 + 1 2 v5 = 6. there are a finite number of combinations of non-negative integers that satisfy the equation above. in [9], schmeichel and hakimi determine which almost regular degree sequences are planar graphical. table 3 contains the list of 11 combinations of v3, v4, v5 that are planar graphical. notice that some of the cases listed in table 3 are regular graphs, some are almost regular graphs, and some contain other combinations of vertex degrees (such as only degree 3 and degree 5 vertices, for example). figure 6 shows all type (4, 5) planar graphs whose corresponding codes are rate optimal according to bound 2. next we consider the cases d = 4 and d = 5. we first present two iterative constructions of almost regular (3, 4) planar graphs whose codes meet bound 2. both constructions begin with one of the platonic solids. 41 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 table 3. graphs that generate codes attaining bound 3. code rate rounded to 4 decimal places. codes possess minimum distance d = 3. v3 v4 v5 code rate 0 0 12 .6333 0 2 8 .6250 0 3 6 .6190 0 4 4 .6111 0 5 2 .6000 0 6 0 .5833 1 3 3 .6000 2 0 6 .6111 2 2 2 .5833 2 3 0 .5556 4 0 0 .5000 construction 1. the starting point of this construction is the cube graph—a 3-regular planar graph with faces of degree 4, shown on the left in figure 4. the first iteration in the construction is to insert a copy of the black edges and vertices into any face in the graph that has all vertices of degree 3. in figure 4, the new vertices and edges are inserted into the center face. there will be at least one face incident with all degree-3 vertices on every iteration because the inner-most face being inserted always has this property. on iteration i, the graph has v3 = 8, v4 = 4i, with 6 + 4i faces, and all faces of degree 4. the code parameters at the ith iteration are [12 + 8i, 5 + 4i, 4], and the rate meets bound 2. figure 4. the cube and the first iteration of construction 1. construction 2. this construction begins with the planar embedding of the dodecahedron—a 3-regular planar graph with faces of degree 5, shown in figure 5. the first iteration of the construction takes a copy of the edges and vertices in black in the figure, and inserts them into any face that is incident with only vertices of degree 3. like in construction 1, there will always be a face incident with vertices of degree 3, since there are several such faces being inserted at each iteration. at iteration i, the graph has v3 = 20 + 10i, v4 = 5i, with 12 + 10i faces, each of degree 5. the code at iteration i has parameters [30 + 25i, 11 + 10i, 5], and the rate meets bound 2. the same argument as theorem 5.1 can be applied with 4f = 2e and 5f = 2e to gain insight into the possible parameters of rate optimal codes with d = 4 and d = 5, respectively. the resulting equations are v3 −v5 = 8 for d = 4 and v32 −v4 − 5v5 2 = 10 for d = 5. for d = 4, construction 1 gives an infinite family of (3, 4) almost regular planar graphs in which every face has degree 4 and whose resulting code rates meet bound 2. there are no type (4, 5) almost regular graphs satisfying the d = 4 equation. furthermore, the resulting equation gives a necessary condition for a rate optimal type (3, 5) graph to exist. 42 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 5. the dodecahedron with black edges and vertices denoting the portion of the graph that should be copied and inserted into a face bordered by vertices of degree 3 for each iteration of construction 2. the results for d = 5 do not provide a finite list of cases that meet the bound but, rather, show necessary conditions for a rate optimal code to exist. these observations are in table 4. table 4. necessary existence conditions for number of vertices needed for a rate optimal code to exist with d = 5. graph degrees (type) necessary conditions (3, 4) v3 ≥ 22, v4 ≥ 1 (4, 5) does not exist (3, 5) v3 ≥ 30, v5 ≥ 2 (3, 4, 5) v3 2 −v4 − 5v52 = 10 5.2. type (3, 4) planar graphs with girth 3 since theorem 5.1 establishes that there are no infinite families of (3, 4) almost regular planar graphs that are triangulations, in this section we present alternative graph constructions which contain some faces of degree four. algorithm 5.2. this algorithm subdivides the edges of a triangle embedded into a plane with new vertices to form a planar graph with v2 = 2, v3 = 2, v4 = k, and f = k + 3 for any non-negative integer k. see figure 7 for an example. begin with a planar 2-regular graph with 3 vertices labeled 1, 2, and 3. 1. subdivide edge v1v2 by inserting vertex i = 4. 2. insert edge v3v4. 3. if this is the desired number of v2, v3 and v4, stop. else, continue to step 4. 4. set i = i + 1. 5. subdivide edge vi−2v2 by inserting vertex i. 6. insert edge vi−1vi. return to step 3. in steps 1-3, an initial graph with v2 = 2, v3 = 2, v4 = 0, and f = 3 is constructed. for each iteration of steps 3-6, a vertex of degree 4 is added to the degree sequence and another face is bounded. 43 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 (a) (b) (c) (d) figure 6. all rate optimal type (4, 5) planar graphs. degree 5 vertices in black. (a) [5, 5, 4, 4, 4, 4, 4] (b) [5, 5, 5, 5, 4, 4, 4, 4]; (c) [5, 5, 5, 5, 5, 5, 4, 4, 4]; and, (d) [5, 5, 5, 5, 5, 5, 5, 5, 4, 4]. figure 7. a graph with v2 = 2, v3 = 2, v4 = k = 1, and f = k + 3 = 4. vertices and edges added through the steps of algorithm 5.2 in red. algorithm 5.2 can be generalized to cases where a triangle is present in a larger graph so the initial degree of vertices 1, 2, and 3 depends on the total edges incident with those vertices. construction 3. this construction (figure 8) generates an infinite family of type (3, 4) planar graphs with v3 = 4, v4 = k and f = k + 4 for all k ∈ z+. begin with a wheel graph with 5 vertices. select a triangle in the graph and implement algorithm 5.2. note that the universal vertex cannot be labeled vertex 3. the rate of the code in construction 3 simplifies to 1 2 for any value of k, thus any combination of v3 and v4 that can be attained in construction 3 yields the same code rate. 44 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 8. a graph with v3 = 4, v4 = k = 5, and f = k + 4 = 9. vertices and edges added through the steps of algorithm 5.2 in red. construction 4. this construction generates an infinite family of type (3, 4) planar graphs with v3 = 2, v4 = k and f = k + 3 for k ∈ z+, k ≥ 3. consider the type (3, 4) graph generated by construction 3. insert an edge between vertex 2 and another vertex of degree 3 on the original “wheel” that is not already neighbors with vertex 2 (see figure 9). return to step 3 of algorithm 5.2. figure 9. a graph with v3 = 2, v4 = k = 7, and f = k + 3 = 10. as degree 4 vertices are added in construction 4, the code rate decreases. the degree of the infinite face increases and since 3 is the minimum number of edges needed to form a closed face, it is an expensive face with respect to edges. similarly, the degree of the face with vertices 1 and 2 that is incident with the face where algorithm 5.2 takes place increases and becomes an expensive face with respect to edges. as the number of degree 4 vertices becomes larger, the constants in the code rate become less significant and the code rate approaches 1 2 . algorithm 5.3. algorithm 5.3 subdivides the edges of a rectangle with new vertices to form a new planar graph with v2 = 4, v3 = 2k, and f = k + 2 for any k ∈ z+. see figure 10. begin with a planar 2-regular graph with 4 vertices. label the vertices 1, 2, 3 and 4 with edges v1v2, v1v3, v2v4 and v3v4. 1. subdivide edge v1v3 and v2v4 by inserting vertex i = 5 and i + 1 = 6, respectively. 2. insert edge v5v6. 3. if this is the desired number of v2 and v3, stop. else, continue to step 4. 4. set i = i + 2. 45 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 10. a graph with v2 = 4, v3 = 2k = 2(3) = 6, and f = 3 + 2 = 5. figure 11. a graph with v3 = 2(9) = 18, v4 = 2, and f = 9 + 4 = 13 5. subdivide edges vi−2v3 and vi−1v4 with vertex i and i + 1, respectively. 6. insert edge vivi+1. return to step 3. algorithm 5.3 can be applied to cases where a rectangle is present in a larger graph so the initial degree of vertices 1, 2, 3 and 4 depends on the total edges incident with those vertices. construction 5. this construction generates an infinite family of type (3, 4) planar graph with v vertices and v3 = 2l, v4 = 2 and f = l + 4 for any l ∈ z+, l ≥ 3. see figure 11. • embed a rectangle into the plane with vertices a, b, c, and d and edges vavb, vavd, vbvc, vcvd. • add two vertices, u and v, to the infinite face and edges vavu, vbvu, vcvv, vdvd. • subdivide vavb and vcvd with w and z, respectively, and add edges vuvw, vvvz, and vwvz. • set va = 1, vw = 2, vd = 3, and vz = 4 and implement algorithm 5.3 throughout. construction 5 utilizes algorithm 5.3 to add edges, vertices, and faces to a graph. each iteration of algorithm 5.3 adds a face of degree 4. the rate of the code generated by this graph decreases as more degree 3 vertices are added since the degree of each new face is 4. as the number of degree 3 vertices becomes larger, the constants in the code rate become less significant and the code rate approaches 1 3 . 46 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 12. a graph with v3 = 2(9) = 18, v4 = m = 6, and f = 9 + 6 + 2 = 17 construction 6. this construction generates an infinite family of type (3, 4) planar graph with v3 = 2l and f = l + 4 for any l ∈ z+, l ≥ 3 can have v4 = m for any m ∈ z+, m ≥ 2. the number of faces can be restated as f = l + m + 2. see figure 12. • construct a graph using construction 5. • select a triangle and implement algorithm 5.2. note: a vertex of degree 4 cannot be labeled 3 in algorithm 5.2. the rate of the code generated by the graph in construction 6 increases as degree 4 vertices are added before reaching a limit that does not attain the bound shown earlier in the paper. while fixing the number of degree 3 vertices, as the number of degree 4 vertices added becomes larger, the code rate approaches 1 2 . for larger values of l, the code rate approaches 1 2 at a slower rate since the constants have a larger impact in the calculation. taking constructions 3, 4, 5, and 6 it is possible to construct a type (3, 4) planar graph with v3 = 2a and v4 = b such that v3 + v4 ≥ 5 for any a, b ∈ z+. table 5 includes a summary of code rates for these constructions. all of the code families in table 5 have minimum distance 3. the code rates for type (3, 4) graphs constructed in section 5.2 are shown in figures 13 and 14. 5.3. type (4, 5) planar graphs with girth 3 next we construct infinite families of type (4, 5) almost regular graphs. the following operations will be used sequentially in the construction and are demonstrated in figure 15. definition 5.4 (operation a). let operation a be defined as an operation performed on a planar graph with a face f, where the degree of f is 3 and each e ∈ f is incident with another face of degree 3 such that every pair of vertices of f does not share a common neighbor. 1. label the vertices of f as v1, v2, and v3 2. subdivide v1v2, v1v3, and v2v3 with v4, v5, v6, respectively. 3. add edges v4v5, v4v6, and v5v6 definition 5.5 (operation b). let operation b be defined as an operation performed after operation a has been applied. the labeling of the vertices for operation b follows from operation a. label the vertices adjacent to v1 and v2, v1 and v3, and v2 and v3 as v7, v8, and v9, respectively. 47 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 table 5. performance of families of codes generated by graph constructions. the constructions with an (*) meet rate bound 2. all codes have availability t = 2. graph type code rate rec. set size(s) min. dist. d case 1 (3-reg) 3+i 6+3i , i ∈ z+ 2 3 case 2 (4-reg) 7+3i 12+6i , i ∈ z+ 3 3 case 3 (5-reg) 18i+1 30i , i ∈ z+ 4 3 construction 1∗ (type (3,4)) 5+4i 12+8i , i ∈ z+ {2, 3} 4 construction 2∗ (type (3,4)) 11+10i 30+25i , i ∈ z+ {2, 3} 5 construction 3 (type (3,4)) k+3 2(k+3) = 1 2 {2, 3} 3 construction 4 (type (3,4)) k+2 2k+3 for k ∈ z+, k ≥ 3 {2, 3} 3 construction 5 (type (3,4)) l+3 3l+4 for l ∈ z+, l ≥ 3 {2, 3} 3 construction 6 (type (3,4)) l+m−1 3l+2m , for l, m ∈ z+, l ≥ 3, m ≥ 2 {2, 3} 3 construction 7 (type (4,5)) 9p+3m+1 3(2m+5p) for m ∈{1, 2}, p ∈ z+ {3, 4} 3 construction 8∗ (type (5,6)) 19+6k 3(10+3k) for k ∈ z+ {4, 5} 3 figure 13. code rates generated by type (3, 4) graphs in constructions 3, 4, and 5 compared to bound 2. 1. add edges v4v7, v5v8, and v6v9. construction 7. consider the graph g created by embedding the octahedron into the plane so that v4 = 6 and there are 8 faces. 1. select any face of degree 3 that is not incident with the infinite face to implement a series of operations a and b. perform operation a followed by operation b. if this is the desired number of v4 and v5, stop. else, continue to step 2. 2. perform operation a on degree 3 face added in previous iteration of operation a then perform operation b. 48 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 14. code rate generated by graph from construction 6 compared to bound 2 for varying levels of l. 3. perform operation a on degree 3 face added in previous iteration of operation a. if this is the desired number of v4 and v5, stop. else, continue to step 4. 4. perform operation a on degree 3 face added in previous iteration of operation a. if this is the desired number of v4 and v5, stop. else, continue to step 5. 5. perform operation a on degree 3 face added in previous iteration of operation a then perform operation b. if this is the desired number of v4 and v5, stop. else, return to step 2. the octahedron is a 4-regular graph with 6 vertices. each application of operation a increases v4 by 3. each application of operation b decreases v4 by 6 and increases v5 by 6. letting a and b represent the number of applications of operations a and b, respectively, v4 = 6 + 3a−6b and v5 = 6b. applying operations a and b in construction 5 leads to v4 = 3m for m ∈{1, 2} and v5 = 6p for p ∈ z+. figure 16 shows the code rate as the number of degree 5 vertices increases. the code rate is highest when v4 = 3 and v5 = 6 after step 1 in construction 7. this is the only instance when every face of the graph has degree 3 and the bound is attained. all future steps involve at least one face of degree 4. while fixing m at either 1 or 2, as the number of degree 5 vertices are added, the constants in the code rate become less significant and the code rate approaches 3 5 . remark 5.6. it is possible to generate an infinite family of 5-regular planar graphs with v5 = 12i for i ∈ z+ using the steps in construction 7 by removing the stopping criterion from steps 1, 3-5 and adding an option to stop after step 2. the 5-regular graph generated by this construction produces a code with rate 18i+1 30i , matching the code rate generated by the 5-regular graph is case 3 of section 4. 5.4. type (5, 6) planar graphs with girth 3 the final type of almost regular planar graph to consider is type (5, 6). we construct infinite families of almost regular (5, 6) planar graphs and determine the code parameters of the resulting edge codes. the following construction results in graphs that are triangulations. 49 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 15. operations a and b on f. f in red. operation a in gold. operation b in blue. algorithm 5.7. for a graph g with face f where the degree of f is 3 and each edge e ∈ f is incident with another face of degree 3 such that every pair of vertices in f do not share a common neighbor, the number of vertices, faces, and edges of a graph g can be increased by 3, 6, and 9, respectively, for i iterations, where i ∈ z+. the following steps are shown in figure 17. begin with a 2-regular planar graph with 3 vertices v1, v2, and v3. add three disconnected vertices to the outside face labeled v4, v5, and v6. add the following edges to the graph: v1v4, v1v6, v2v4, v2v5, v3v5, v3v6. note that the graph constructed is in the appropriate form required for the algorithm. 1. subdivide edges v1v2, v1v3, and v2v3 with new vertices v7, v8, and v9, respectively, then add edges v7v8, v7v9, and v8v9. 2. insert edges v4v7, v5v9, and v6v8. 3. if this is the desired graph, stop. else, relabel the vertices as follows: vertices v4, v5, and v6 become unlabeled, and the rest are relabeled v1 → v6, v2 → v4, v3 → v5, v7 → v1, v8 → v3, v9 → v2. return to step 1. construction 8. this construction generates an infinite family of type (5, 6) planar graphs with v5 = 12 and v6 = 3k for all k ∈ z+. consider the graph g created by embedding the icosahedron into the plane so that there are 12 vertices of degree 5 and 20 faces. the structure of g allows for algorithm 5.7 to be used. any face of degree 3 may be chosen to initialize vertices 1, 2, and 3 for algorithm 5.7. see figure 18, for example. the code generated by the graph in construction 8 is rate optimal by bound 2. all faces of the graph are degree 3 at any iteration of algorithm 5.7. the code approaches the graph theoretic bound of 2 3 (for d = 3) as the number of iterations of algorithm 5.7 go to infinity. 50 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 16. type (4, 5) construction 7 code rate figure 17. an iteration of algorithm 5.7. new edges added in red. face f in yellow. 6. conclusion an edge code generated by a planar graph has minimum distance d equal to the smallest degree of a face, and availability t = 2. the recovery set size for a type (j, j + 1) planar graph is at most r = j. in this paper we proved that the minimum distance of an edge code coincides with the girth of the planar graph, and that the code rate satisfies a bound that depends on the length and minimum distance of the code. we applied expansion constructions of j-regular planar graphs to compute the parameters of infinite families of edge codes. we classified the type (3, 4) almost regular graphs whose code rates meet bound 2 and proved that there are no infinite families of this type that achieve the bound. for d = 3, 4, 5 we presented constructions of codes that meet the rate bound (see table 5). we also presented constructions of type (3, 4) almost regular planar graphs with v3 = 2a and v4 = b for any a, b ∈ z+ such that v3 + v4 ≥ 5. operations a and b were presented to construct infinite families of type (4, 5) almost regular planar graphs. finally, type (5, 6) planar graphs that generate rate-optimal lrcs were presented. the family of (5, 6) almost regular edge codes approach the graph theoretic bound of r = 2 3 51 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 figure 18. algorithm 5.7 implemented on the icosahedron. edges added in the first iteration of the algorithm in red. edges added in the second iteration of the algorithm in green. vertices of degree 6 are shown in white. as the number of iterations of operations goes to infinity. potential future work includes codes generated by different families of almost regular planar graphs, or of 2-connected planar graphs. graphs that are “almost” almost regular could also be explored, although a larger difference in recovery set sizes might negatively impact the code rates. codes generated by graphs embedded on other surfaces may also lead to additional constructions of lrcs from graphs. acknowledgment: the authors would like to thank the referees for their helpful suggestions. references [1] h. j. broersma, a. j. w. duijvestijn, f. göbel, generating all 3-connected 4-regular planar graphs from the octahedron graph, j. graph theor. 17(5) (1993) 613–620. [2] p. gopalan, c. huang, h. simitci, s. yekhanin, on the locality of codeword symbols, ieee trans. inform. theory 58(11) (2012) 6925–6934. [3] m. hasheminezhad, b. d. mckay, t. reeves, recursive generation of simple planar 5-regular graphs and pentangulations, journal of graph algorithms and applications 15(3) (2011) 417–436. [4] s. kadhe, r. calderbank, rate optimal binary linear locally repairable codes with small availability, in 2017 ieee international symposium on information theory (isit) (2017) 166–170. [5] s. kadhe, r. calderbank, rate optimal binary linear locally repairable codes with small availability, arxiv preprint, arxiv:1701.02456, 2017. [6] m. meringer, fast generation of regular graphs and construction of cages, j. graph theor. 30(2) (1999) 137–146. [7] m. meringer, regular planar graphs, available online at http://www.mathe2.uni-bayreuth.de/ markus/reggraphs.html, accessed 2009. [8] n. prakash, v. lalitha, p. vijay kumar, codes with locality for two erasures, in 2014 ieee international symposium on information theory (2014) 1962–1966. [9] e. f. schmeichel, s. l. hakimi, on planar graphical degree sequences, siam j. appl. math. 32(3) 52 https://doi.org/10.1002/jgt.3190170508 https://doi.org/10.1002/jgt.3190170508 https://doi.org/10.1109/tit.2012.2208937 https://doi.org/10.1109/tit.2012.2208937 http://dx.doi.org/10.7155/jgaa.00232 http://dx.doi.org/10.7155/jgaa.00232 https://doi.org/10.1109/isit.2017.8006511 https://doi.org/10.1109/isit.2017.8006511 https://doi.org/10.1002/(sici)1097-0118(199902)30:2%3c137::aid-jgt7%3e3.0.co;2-g https://doi.org/10.1002/(sici)1097-0118(199902)30:2%3c137::aid-jgt7%3e3.0.co;2-g http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html https://doi.org/10.1109/isit.2014.6875176 https://doi.org/10.1109/isit.2014.6875176 https://doi.org/10.1137/0132048 https://doi.org/10.1137/0132048 k. haymaker, j. o’pella / j. algebra comb. discrete appl. 7(1) (2020) 35–53 (1977) 598–609. [10] e. steinitz, vorlesungen über die theorie der polyeder: unter einschluß der elemente der topologie, volume 41, springer-verlag, 2013. 53 https://doi.org/10.1137/0132048 https://doi.org/10.1137/0132048 introduction preliminaries parameters of edge codes j-regular planar graphs almost regular type (j, j+1) planar graphs conclusion references issn 2148-838x j. algebra comb. discrete appl. -(-) • 1–13 received: 11 february 2021 accepted: 15 october 2021 ar ti cl e in pr es s journal of algebra combinatorics discrete structures and applications cordiality of digraphs research article leroy b. beasley, manuel a. santana, jonathan mousley, david e. brown abstract: a (0,1)-labelling of a set is said to be friendly if approximately one half the elements of the set are labelled 0 and one half labelled 1. let g be a labelling of the edge set of a graph that is induced by a labelling f of the vertex set. if both g and f are friendly then g is said to be a cordial labelling of the graph. we extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. we investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called (2,3)-cordiality. a directed graph is (2,3)-cordial if there is a friendly labelling f of the vertex set which induces a (1,−1,0)-labelling of the arc set g such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. in particular we determine which tournaments are (2,3)-cordial, which orientations of the n-wheel are (2,3)-cordial, and which orientations of the n−fan are (2,3)-cordial. 2010 msc: 05c20, 05c38, 05c78 keywords: tournament, wheel graph, fan graph, (2, 3)-cordial 1. introduction the study of cordial graphs began in 1987 with an article by i. cahit [2]: “cordial graphs: a weaker version of graceful and harmonious graphs”. in 1991, hovey [3] generalized this concept to a-cordial graphs where a is an abelian group. a further generalization, one that included cordiality of directed graphs, appeared in 2012 with an article by pechenik and wise [4], where the a was allowed to be any quasi group, not necessarily abelian. we modify this concept to (a,a)-cordial digraphs where a is a subset of the quasigroup a. let g = (v,e) be an undirected graph with vertex set v and edge set e. a (0, 1)-labelling of the vertex set is a mapping f : v →{0, 1} and is said to be friendly if approximately one half of the vertices are labelled 0 and the others labelled 1. an induced labelling of the edge set is a mapping g : e →{0, 1} where for an edge uv,g(uv) = ĝ(f(u),f(v)) for some ĝ : {0, 1}×{0, 1}→{0, 1} and is said to be cordial leroy b. beasley (corresponding author), manuel a. santana, jonathan mousley, david e. brown; department of mathematics and statistics, utah state university, logan, utah 84322-3900, u.s.a (email: leroy_beas@aol.com, manuelarturosantana@gmail.com, jonathanmousley@gmail.com, david.e.brown@usu.edu). 1 http://orcid.org/0000-0002-0575-7200 http://orcid.org/0000-0003-1055-3824 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 if f is friendly and about one half the edges of g are labelled 0 and the others labelled 1. a graph, g, is called cordial if there exists a cordial induced labelling of the edge set of g. in this article, as in [1], we define a cordial labelling of directed graphs that is not merely a cordial labelling of the underlying undirected graph. a specific type of (a,a)-cordial digraph is a (2, 3)-cordial digraph defined by beasley in [1]. let d = (v,a) be a directed graph with vertex set v and arc set a. let f : v → {0, 1} be a friendly vertex labelling and let g be the induced labelling of the arc set, g : a → {0, 1,−1} where for an arc −→uv,g(−→uv) = f(v) − f(u). the labellings f and g are (2, 3)-cordial if f is friendly and about one third the arcs of d are labelled 1, one third are labelled -1 and one third labelled 0. a digraph, d, is called (2, 3)-cordial if there exist (2, 3)-cordial labellings f of the vertex set and g of the arc set of d. note that here and what follows, the term “about” when talking about fractions of a quantity we shall mean as close is possible in integral arithmetic, so about half of 9 is either 4 or 5, but not 3 or 6. 2. preliminaries definition 2.1. a quasigroup is a set q with binary operation ◦ such that given any a,b ∈q there are x,y ∈q such that a◦x = b and y ◦a = b. fact: all two element quasigroups are abelian. proof. suppose that q = {a,b} is a quasi group with binary operation ◦. then, there are x,y ∈ q such that a◦x = a and y ◦a = a. if x = y = b then q is abelian. otherwise, we must have a◦a = a. similarly either q is abelian or b◦ b = b. now, suppose that a◦a = a and b◦b = b. then there are c,d ∈q such that a◦d = b and c◦a = b. since a◦a = a., we must have that both c = b and d = b. that is q is abelian. we now formalize the terms mentioned in the introduction. we let zk denote the set of integers {0, 1, . . . ,k} with arithmetic modulo k. further let z−k denote the set zk with binary operation “−”, subtraction modulo k. clearly, for k ≥ 3, z−k is a nonabelian quasigroup. definition 2.2. a zk-labelling (or simply a k-labelling) of a finite set, x, is a mapping f : x → zk and is said to be friendly if the labelling is evenly distributed over zk, that is, given any i,j ∈ zk, −1 ≤ |f−1(i)|− |f−1(j)| ≤ 1 where |x| denotes the cardinality of the set x. definition 2.3. let g = (v,e) be an undirected graph with vertex set v and edge set e, and let f be a friendly (0, 1)-labelling of the vertex set v . given this friendly vertex labelling f, an induced (0, 1)labelling of the edge set is a mapping g : e →{0, 1} where for an edge uv, g(uv) = ĝ(f(u),f(v)) for some ĝ : {0, 1}×{0, 1} → {0, 1} and is said to be cordial if g is also friendly, that is about one half the edges of g are labelled 0 and the others are labelled 1, or −1 ≤ |g−1(0)|− |g−1(1)| ≤ 1. a graph, g, is called cordial if there exists a induced cordial labelling of the edge set of g. the induced labelling g in a cordial graph is usually g(u,v) = ĝ(f(u),f(v)) = |f(v) − f(u)| [2], g(u,v) = ĝ(f(u),f(v)) = f(v)+f(u) (in z2) [3], or g(u,v) = ĝ(f(u),f(v)) = f(v)f(u) (product cordiality) [5]. in [3], hovey introduced a-friendly labellings where a is an abelian group. a labelling f : v (g) →a is said to be a-friendly if given any a,b ∈ a, −1 ≤ |f−1(b)| − |f−1(a)| ≤ 1. if g is an induced edge labelling and f and g are both a-friendly then g is said to be an a-cordial labelling and g is said to be a-cordial. when a = zk we say that g is k-cordial. we shall use this concept with digraphs. given an undirected graph or a digraph, g, let ĝ denote the subgraph (or subdigraph) of g induced by its nonisolated vertices. so ĝ never has an isolated vertex. the need for this will become apparent in example 3.4. 2 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 in this article, we will be concerned mainly with digraphs. we let dn denote the set of all simple directed graphs on the vertex set v = {v1,v2, . . . ,vn}. note that the arc set of members of dn may contain digons, a pair of arcs between two vertices each directed opposite from the other. we shall let tn denote the set of all subdigraphs of a tournament digraph. so the members of tn contain no digons. let d ∈dn, d = (v,a) where a is the arc set of d. then d has no loops, and no multiple arcs. an arc in d directed from vertex u to vertex v will be denoted −→uv,←−vu or by the ordered pair (u,v). we also let gn denote the set of all simple undirected graphs on the vertex set v = {v1,v2, . . . ,vn}. so all members of tn are orientations of graphs in gn. in [4], pechenik and wise introduced quasigroup cordiality. when the quasigroup is nonabelian, this type of cordiality is quite suitable for studying labellings of directed graphs. in fact, if q is a quasigroup with any binary operation ◦ with the property that for any a,b ∈q a◦ b = b◦a if and only if a = b, we have the best situation for directed graphs. now the set zk with binary operation ◦ where for a,b ∈ zk, a ◦ b = (b − a) mod k is such a quasigroup. in our investigations we make one further restriction: we will label our vertices with only a subset of q, not necessarily the whole set q: definition 2.4. let q be a quasigroup with binary operation ◦ and let q be a subset of q. let d = (v,a) be a directed graph with vertex set v and arc set a. let f : v → q be a friendly q-labelling of v and let g : a → q be an induced arc labelling where for −→uv ∈ a, g(−→uv) = ĝ(f(u),f(v)) for some ĝ : q × q → q. the mapping g is said to be (q,q)-cordial if g is also friendly, that is, given any a,b ∈ q, −1 ≤ |g−1(a)| − |g−1(b)| ≤ 1. a directed graph, d, is called (q,q)-cordial if there exists a induced (q,q)-cordial labelling of the arc set of d. we now shall restrict our attention to the smallest case of (q,q)-cordiality that is appropriate for directed graphs, (z2,z−3 )-cordiality, that defined by beasley in [1], (2, 3)-cordiality. 3. (2, 3)-orientable digraphs let d = (v,a) be a directed graph with vertex set v and arc set a. let f : v → {0, 1} be a friendly labelling of the vertices of d. as for undirected graphs, an induced labelling of the arc set is a mapping g : a → x for some set x where for an arc (u,v) = −→uv,g(u,v) = ĝ(f(u),f(v)) for some ĝ : {0, 1}×{0, 1} → x . as we are dealing with directed graphs, it would be desirable for the induced labelling to distinguish between the label of the arc (u,v) and the label of the arc (v,u), otherwise, the labelling would be an induced labelling of the underlying undirected graph. if we let x = {−1, 0, 1} and ĝ(f(u),f(v)) = f(v) −f(u) using real arithmetic, or arithmetic in z3, we have an asymmetric labelling. in this case, if about one third of the arcs are labelled 0, about one third of the arcs are labelled 1 and about one third of the arcs are labelled -1 we say that the labelling is (2, 3)-cordial. formally: definition 3.1. let d ∈tn, d = (v,a), be a digraph without isolated vertices. let f : v →{0, 1} be a friendly labelling of the vertex set v of d. let g : a →{1, 0,−1} be an induced labelling of the arcs of d such that for any i,j ∈{1, 0,−1}, −1 ≤ |g−1(i)|− |g−1(j)| ≤ 1. such a labelling is called a (2, 3)-cordial labelling. a digraph d ∈tn whose subgraph ĝ can possess a (2, 3)-cordial labelling will be called a (2, 3)-cordial digraph. an undirected graph g is said to be (2, 3)-orientable if there exists an orientation of g that is (2, 3)cordial. in [1] the concept of (2, 3)-cordial digraphs was introduced and paths and cycles were investigated. in [6] one can find further investigation of orientations of paths and trees as well as finding the maximum number of arcs possible in a (2, 3)-cordial digraph. in this article we continue this investigation, showing 3 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 which tournaments, which orientations of the wheel graphs, and which orientations of the fan graphs are (2, 3)-cordial. definition 3.2. let d = (v,a) be a digraph with vertex labelling f : v → {0, 1} and with induced arc labelling g : a → {0, 1,−1}. define λf,g : dn → n3 by λf,g(d) = (α,β,γ) where α = |g−1(1)|,β = |g−1(−1)|, and γ = |g−1(0)|. let d ∈tn and let dr be the digraph such that every arc of d is reversed, so that −→uv is an arc in dr if and only if −→vu is an arc in d. let f be a (0, 1)-labelling of the vertices of d and let g(−→uv) = f(v)−f(u) so that g is a (1,−1, 0)-labelling of the arcs of d. let f be the complementary (0, 1)-labelling of the vertices of d, so that f(v) = 0 if and only if f(v) = 1. let g be the corresponding induced arc labelling of d, g(−→uv) = f(v) −f(u). lemma 3.3. let d ∈ tn with vertex labelling f and induced arc labelling g. let λf,g(d) = (α,β,γ). then 1. λf,g(dr) = (β,α,γ). 2. λf,g(d) = (β,α,γ), and 3. λf,g(d r) = λf,g(d). proof. if an arc is labelled 1,−1, 0 respectively then reversing the labelling of the incident vertices gives a labelling of −1, 1, 0 respectively. if an arc −→uv is labelled 1,−1, 0 respectively, then −→vu would be labelled −1, 1, 0 respectively. example 3.4. now, consider a graph, xn in gn consisting of three parallel edges and n-6 isolated vertices. is xn (2, 3)-orientable? if n = 6, the answer is no, since any friendly labelling of the six vertices would have either no arcs labelled 0 or two arcs labelled 0. in either case, the orientation would never be (2, 3)-cordial. that is x6 is not (2, 3)-orientable, however with additional vertices like x7 the graph is (2, 3)-orientable. thus, for our investigation here, we will use the convention that a graph, g, is (2, 3)-orientable/(2, 3)cordial if and only if the subgraph of g induced by its nonisolated vertices, ĝ, is (2, 3)-orientable/(2, 3)cordial. 3.1. (2, 3)-orientations of a complete graph-tournaments it is an easy exercise to show that every 3-tournament is (2, 3)-cordial and that two of the four non isomorphic 4-tournaments are (2, 3)-cordial. see figures 2 and 3. note that the 4-tournaments that are not (2, 3)-cordial may require more than a cursory glance to verify that they are not (2, 3)-cordial. lemma 3.5. every 5-tournament is (2, 3)-cordial. proof. let t ∈d5 be a tournament. then there are two vertices, without loss of generality, v1 and v2, whose total out degree is four. (and hence the total in-degree of v1 and v2 is also four.) let f be the vertex labelling and let f(v1) = f(v2) = 1 and f(v3) = f(v4) = f(v5) = 0. let g be the arc labelling g(−−→vivj) = f(vj) − f(vi). then, the arc between v1 and v2 is labelled 0, as are the three arcs between v3,v4 and v5. thus there are four arcs labelled 0. the three arcs from v1 or v2 to vertices v3,v4 or v5 are labelled 1 and the three arcs from v3,v4 or v5 to vertices v1 or v2 are labelled -1. in figure 1 is an example of the labelling described above. thus λf,g(t) = (3, 3, 4). that is t is (2, 3)-cordial. lemma 3.6. if n ≥ 6 and t ∈dn is a tournament on n vertices then t is not (2, 3)-cordial. 4 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 0 0 11 0 0 0 0 −1 1 −1 1 −11 0 v1v2 v3 v4 v5 figure 1. a (2,3)-cordial labelling of a 5-tournament proof. we divide the proof into two cases: case 1. n is even. let n = 2k. we shall show that there must be more arcs labelled 0 than is allowed in any (2, 3)-cordial digraph with n(n−1) 2 arcs. for any vertex labelled 0, there are k − 1 other vertices also labelled 0 so that there are k − 1 arcs labelled 0 that either begin or terminate at that vertex. also there are k such vertices so there are k(k − 1)/2 arcs between pairs of vertices each labelled 0. (note, since each arc is adjacent to two vertices we have divided the total number by 2 to get the number of distinct arcs labelled 0.) there are also k(k − 1)/2 arcs between pairs of vertices each labelled 1. thus we must have k(k − 1) arcs labelled 0. now, there must be at most one third the number of arcs labelled 0, so we must have 3k(k − 1) ≤ n(n−1) 2 + 2 = 4k 2−2k+4 2 . that is, we must have k2 − 2k − 2 ≤ 0. but that only happens if k ≤ 2. so if k ≥ 3 or n ≥ 6, t is not (2, 3)-cordial. case 2. n is odd. let n = 2k + 1. without loss of generality, we may assume that there are k vertices labelled 0 and k + 1 vertices labelled 1. thus there are 1 2 k(k−1) arcs labelled 0 that connect two vertices labelled 0 and 1 2 (k + 1)k arcs labelled 0 that connect two vertices labelled 1. thus there are at least k2 arcs labelled 0. to be (2, 3)-cordial we must have that 3k2 ≤ n(n−1) 2 + 2, or k2−k−2 ≤ 0. that happens only if k ≤ 2. but, since n is odd, n ≥ 7 so k ≥ 3. thus, t is not (2, 3)-cordial. lemma 3.7. the tournaments t4,3 and t4,4 of figure 3 are not (2, 3)-cordial. proof. since t4,4 is the reversal of t4,3, by lemma 3.3 we only need show that t4,3 is not (2, 3)-cordial. further, by lemma 3.3 we may assume that the upper left vertex of t4,3 in figure 3 is labelled 0. since any permutation of the other three vertices results in an isomorphic graph we may assume that the upper 5 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 0 0 1 and 0 0 1 1 0 −1 0 1 −1 figure 2. (2,3)-cordial labellings of 3tournaments t4,3 : (2,2,2,0) not (2,3)-cordial t4,4 : (3,1,1,1) not (2,3)-cordial 0 1 1 0 1 −1 −1 0 0 1 t4,1 : (2,2,1,1) 0 1 0 1 1 0 0 1 −1 −1 t4,2 : (3,2,1,0) figure 3. (2,3)-cordial labellings of two 4-tournaments and two non (2,3) cordial 4-tournaments with their out-degree sequences. right vertex is labelled 0 and the bottom two are labelled 1. this results in one arc labelled 1, three arcs labelled −1 and two arcs labelled 0. thus t4,3 is not (2, 3)-cordial. theorem 3.8. let t be an n-tournament. then t is (2, 3)-cordial if and only if n ≤ 5 and t is not isomorphic to t4,3 or t4,4. proof. lemmas 3.7, 3.5 and 3.6 together with figures 2 and 3 establish the theorem. 6 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 we end this section with a couple of observations we label as corollaries: corollary 3.9. the property of being (or not being) (2, 3)-cordial is not closed under vertex deletion. proof. every tournament on k vertices is a vertex deletion of a tournament on k + 1 vertices. thus, t4,3, which is not (2, 3)-cordial, is a vertex deletion of a tournament on 5 vertices, which is (2, 3)-cordial, and this tournament is a vertex deletion of a tournament on 6 vertices, which is not (2, 3)-cordial. corollary 3.10. the property of being (or not being) (2, 3)-cordial is not closed under arc contraction. proof. as in the above corollary, every tournament on k vertices is an arc contraction of a tournament on k + 1 vertices. thus, t4,3, which is not (2, 3)-cordial, is an arc contraction of a tournament on 5 vertices, which is (2, 3)-cordial, and this tournament is an arc contraction of a tournament on 6 vertices, which is not (2, 3)-cordial. 3.2. (2, 3)-orientations of wheel graphs a wheel graph on n vertices consists of an (n− 1)-star together with edges joining the non central vertices in a cycle. a 6-wheel is shown in figure 4. since we are not concerned with digraphs that contain digons, we shall assume that n ≥ 4 in this section. an orientation of the wheel graph with the central vertex being a source/sink is called an out/inwheel. if the outer cycle of the wheel is oriented in a directed cycle the wheel is called a cycle-wheel. if the n-wheel is oriented such that it is both an out-wheel and a cycle-wheel it is called an n-cycle-out-wheel. see figure 5. v1 v2 v3v4 v5 v6 figure 4. a 6-wheel graph. definition 3.11. let wn = (v,a) be an n-wheel digraph with central vertex vn and with vertex labelling f : v → {0, 1}. let g be the induced arc labelling g : a → {0, 1,−1} where g(−→uv) = f(v) − f(u). let 7 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 v1 v2 v3v4 v5 v6 figure 5. a 6-cycle-out-wheel graph. s be the set of arcs incident with the central vertex and let t be the set of arcs not incident with the central vertex. define λsf,g to be the real triple λ s f,g(d) = (αs,βs,γs) where αs = |g −1(1) ∩ s|,βs = |g−1(−1) ∩s|, and γs = |g−1(0) ∩s| and λtf,g to be the real triple λ t f,g(d) = (αt ,βt ,γt ) where αt = |g−1(1) ∩ t |,βt = |g−1(−1) ∩ t |, and γt = |g−1(0) ∩ t |. since s ∪ t = a, the set of all arcs of d, αs + αt = α,βs + βt = β, and γs + γt = γ, where λf,g(d) = (α,β,γ). theorem 3.12. let −→ wn be an n-cycle-out-wheel graph with central vertex vn. then −→ wn is not (2, 3)cordial. proof. we proceed with two cases, the case that n is even then the case that n is odd. let −→ wn = (v,a) and let f : v → {0, 1} be a vertex labelling and g : a → {0, 1,−1} be the induced arc labelling, g(−→uv = f(v) −f(u). suppose that f and g is a (2, 3)-cordial labelling. case 1: n = 2k. without loss of generality, we may assume that f(vn) = 0. thus, αs = k,βs = 0 and γs = k − 1. further, since the orientation is cyclic, αt = βt . since α − 1 ≤ β ≤ α + 1 we have αt + k− 1 = αt + αs − 1 = α− 1 ≤ β = βt + βs = βt = αt . thus k− 1 ≤ 0 or k ≤ 1, a contradiction since n ≥ 4. case 2: n = 2k + 1. without loss of generality we may assume that f(vn) = 0. since either k or k + 1 of the non central vertices must be labelled 0, we have two possibilities: αs = k or αs = k + 1. subcase 1. αs = k here we have γs = k and βs = 0. further, αt = βt since −→ w is cyclic. since the labelling is (2, 3)-cordial, α−1 ≤ β ≤ α+ 1. thus k+αt −1 = αs = αt −1 = α−1 ≤ β = βs +βt = αt . that is k − 1 ≤ 0 or k ≤ 1, a contradiction since n ≥ 4. subcase 2. αs = k + 1 here we have γs = k−1 and βs = 0. further, αt = βt since −→ w is cyclic. since the labelling is (2, 3)-cordial, α−1 ≤ β ≤ α + 1. thus k + αt = k + 1 + αt −1 = αs + αt −1 = α−1 ≤ β = βs + βt = αt . that is k ≤ 0, a contradiction since n ≥ 4. 8 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 in all cases we have arrived at a contradiction thus we must have that −→ w is not (2, 3)-cordial. lemma 3.13. let c be an undirected cycle with a (0, 1)-vertex labelling. then, there is an even number of edges in c whose incident vertices are labelled differently. proof. we may assume that the vertex v1 is labelled 0. going around the cycle, the labelling goes from 0 to 1 then back again to zero. this two step change must happen a fixed number of times then return to vertex v1. thus there are an equal number of changes in labelling from 0 to 1 and from 1 to 0. that is, the total number of changes is an even number. theorem 3.14. let wn be the undirected wheel graph on n vertices. then, wn is not (2, 3)-orientable if and only if n = 2k for some integer k, 4 does not divide n, and 2n− 2 = 3z for some integer z. proof. let −→ wn be an orientation of the wheel graph on n vertices with central vertex vn. let ah be the set of arcs incident with vn, and let ar be the arcs not incident with vn. then a = ah ∪ar. let f be a friendly vertex labelling and g the induced arc labelling of −→ wn. define , λf,h(x) = |g−1(x) ∩ah|, and λf,r(x) = |g−1(x) ∩ar|, define λf (x) = λf,h(x) + λf,r(x), that is λf (x) = |g−1(x)|. we begin by showing for n = 2k for some integer k, k = 2` + 1 for some integer `, and 2n− 2 = 3z for some integer z that wn is not (2, 3)-orientable. in this case, we may assume that f(vn) = 0 and λf,h(1) +λf,h(−1) = k, an odd integer. by lemma 3.13 the number of arcs that are not incident with vn and labelled either 1 or −1 is even. thus λf (1) + λf (−1) = (λf,h(1) + λf,h(−1)) + (λf,r(1) + λf,r(−1)) is the sum of an even integer and an odd integer, so that λf (1) + λf (−1) is an odd integer. but since the total number of arcs is 2n− 2 = 3z, if −→ wn is (2, 3)-cordial, we must have λf (1) + λf (−1) = 2z, an even integer, a contradiction. thus, in this case wn is not (2, 3)-orientable. we now show that for all other cases wn is (2, 3)-orientable. we divide the proof into three cases, those being whether the total number of edges in wn is a multiple of three, one more than a multiple of three, or two more than a multiple of three. case 1. 2n− 2 = 3z for some integer z. subcase 1.1. n = 2k and k = 2`. in this case, let f be the labelling such that the labelling of the cycle has 2(z − k 2 ) edges incident with vertices labelled differently. orient all arcs not incident with vn clockwise around the cycle. orient half the arcs incident with vn that are labelled 1 away from vn, and half toward vn. in this case, λf,h(0) = k − 1, and λf,h(1) = λf,h(−1) = k2 = `. further, λf,r(1) = λf,r(−1) = z− k2 . thus, λf (1) = λf (−1) = λf,h(1) + λf,r(1) = ` + z− k 2 = z. thus, we must also have λf (0) = z, and that wn is (2, 3)-orientable. subcase 1.2. n = 2k + 1. in this case proceed as in subcase 1.1, labelling the vertices not incident with vn with an even number of 1’s (either k or k + 1). let ` be half of this even number. then, we produce a (2, 3)-cordial orientation of wn the same as in subcase 1.1. case 2. 2n− 2 = 3z + 1 for some integer z. subcase 2.1. n = 2k, k = 2`. in this case, let f be the labelling such that the labelling of the cycle has 2(z−k 2 ) edges incident with vertices labelled differently. orient all arcs not incident with vn clockwise around the cycle. orient half the arcs incident with vn that are labelled 1 away from vn, and half toward vn. in this case, λf,h(0) = k − 1,λf,h(1) = λf,h(−1) = k2 = `. further, λf,r(1) = λf,r(−1) = z − k 2 . thus, λf (1) = λf (−1) = λf,h(1) + λf,r(1) = ` + z− k2 = z. thus, we must also have λf (0) = z + 1, and that wn is (2, 3)-orientable. subcase 2.2. n = 2k, k = 2`+1 in this case, let f be the labelling such that the labelling of the cycle has 2(z− k−1 2 ) edges incident with vertices labelled differently. orient ` of the arcs incident with vn that are labelled 1 away from vn, and ` + 1 of those arcs toward vn. in this case, λf,h(0) = k−1,λf,h(1) = ` and λf,h(−1) = ` + 1. further, λf,r(1) = λf,r(−1) = z− k2 = z = `. thus, λf (1) = λf,h(1) + λf,r(1) = ` + z − ` = z, and λf (−1) = λf,h(−1) + λf,r(−1) = ` + 1 + z − ` = z + 1. thus, we must also have λf (0) = 2n− 2 − (z) − (z + 1) = z, and thus wn is (2, 3)-orientable. 9 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 subcase 2.3. n = 2k + 1. in this case proceed as in subcase 2.1 labelling the vertices not incident with vn with an even number of 1’s (either k or k + 1 depending upon whether k is even or odd). let ` be half of this even number. then, we produce a (2, 3)-cordial orientation of wn the same as in subcase 2.1. case 3. 2n− 2 = 3z + 2 for some integer z. subcase 3.1. n = 2k, k = 2`. in this case, let f be the labelling such that the labelling of the cycle has 2(z−k 2 ) edges incident with vertices labelled differently. orient all arcs not incident with vn clockwise around the cycle. orient half the arcs incident with vn that are labelled 1 away from vn, and half toward vn. in this case, λf,h(0) = k − 1,λf,h(1) = λf,h(−1) = k2 = `. further, λf,r(1) = λf,r(−1) = z − k 2 . thus, λf (1) = λf (−1) = λf,h(1) + λf,r(1) = ` + z− k2 = z. thus, we must also have λf (0) = z + 1, and that wn is (2, 3)-orientable. subcase 3.2. n = 2k, k = 2` + 1 in this case, let f be the labelling such that the labelling of the cycle has 2(z − k−1 2 ) edges incident with vertices labelled differently. orient ` of the arcs incident with vn that are labelled 1 away from vn, and ` + 1 toward vn. in this case, λf,h(0) = k− 1,λf,h(1) = ` and λf,h(−1) = ` + 1. further, λf,r(1) = λf,r(−1) = z − k2 = z = `. thus, λf (1) = λf,h(1) + λf,r(1) = ` + z − ` = z, and λf (−1) = λf,h(−1) + λf,r(−1) = ` + 1 + z − ` = z + 1. thus, we must also have λf (0) = z, and that wn is (2, 3)-orientable. subcase 3.3. n = 2k + 1. in this case proceed as in subcase 3.1 labelling the vertices not incident with vn with an even number of 1’s (either k or k + 1 depending upon whether k is even or odd. let ` be half of this even number. then, we produce a (2, 3)-cordial orientation of wn the same as in subcase 3.1. we have now established the theorem. 3.3. (2, 3)-orientations of fan graphs a fan graph is isomorphic to a wheel graph with one edge of the cycle deleted. thus, by deleting one properly chosen arc from the cycle of a (2, 3)-oriented n-wheel graph we obtain an orientation of the n-fan graph that is (2, 3)-cordial. note that if there are at least as many arcs labelled x (x = 1,−1 or 0) as any other labelling, the properly chosen arc would be in the set of arcs labelled x. thus there is only one case to consider, the case where 2n− 2 = 3z,n = 2k and k = 2` + 1 for some z,k, and `. theorem 3.15. let n ≥ 5 and let fn be the n-fan graph with central vertex v1, that is the edges not on the cycle are all incident to v1. let −→ f be a cyclic-out orientation of fn. then −→ f is not (2, 3)-cordial. proof. as for wheel graphs, the number of arcs labelled 1 on the cycle is equal to the number of arcs labelled −1 and there are at least two arcs labelled 1 on the interior of the cycle. thus, the number of arcs labelled 1 in −→ f is at least two more that the arcs labelled −1 in −→ f . that is −→ f is not (2, 3)-cordial. theorem 3.16. let fn be the fan graph on n vertices, 2n−3 = 3z + 2, n = 2k and k = 2` + 1 for some integers k,`, and z. then there is an orientation of fn that is (2, 3)-cordial. proof. let α = z−`+ 1, and define f : v →{0, 1} by f(v2i−1) = 0, i = 1, . . . ,α, f(v2i) = 1, i = 1, . . . ,α, f(v2α+i) = 1, i = 1, . . . ,k−α, and f(vk+α+i) = 0, i = 1, . . . ,k−α, note that (k−α) + (k + α) = 2k = n, so all vertices are labelled. orient the cycle clockwise, so that the oriented cycle is −−→v1v2, −−→v2v3, . . . , −−−−→vn−1vn,−−→vnv1. see figure 6 where the vertex labellings are outside the cycle. now, orient ` of the inner arcs from v1 to arcs labelled 1 (except for v2 which is not an inner arc) away from v1 and the remaining ` such arcs inward so that we get −→ fn = (v,a). let g : a →{0, 1,−1} be the induced labelling, g(−→uv) = f(v)−f(u). then there are α arcs labelled 1 on the cycle, α arcs labelled −1 on the cycle, ` of the inner arcs are labelled −1 and ` of the inner arcs are labelled 1. thus, in all of −→ fn there are α + ` = z + 1 arcs labelled −1, α + ` = z + 1 arcs labelled 1 and (hence) z arcs labelled 0. that is, this orientation of fn is (2, 3)-cordial. 10 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 v1 v2 v3 v4 v2α−1 v2α v2α+1 vs−1 vsvs+1vn−1vn . . . · · · ... s = k + α 0 01 1 0 1 1 1 1 000 figure 6. fn with oriented cycle arcs. 3.4. extremes of (2, 3)-cordiality as seen in section 3, complete graphs are not (2, 3)-orientable if n ≥ 6. so the question arises: how large can a (2, 3)-orientable graph be (how many edges)? or: how large can a (2, 3)-cordial digraph be? that question was fully answered by m. a. santana in [6] for completeness we shall include the proofs of his results. theorem 3.17. [6, theorem 3.1] every simple directed graph is (2, 3)-cordial if and only if there exists a friendly vertex labelling such that about 1 3 of the edges are connected by vertices of the same label. proof. let g be a graph such that there exists a friendly labelling on g such that about 1 3 of the edges are connected by vertices of the same label. this would mean about 2 3 of the edges are connected by vertices of different labels, and therefore arcs may be assigned such that g is cordial. now let h be a graph such that there does not exist a friendly labelling on h such that about 1 3 of the edges are connected by vertices of the same label then there will be no way h can be cordial since only then could about one third of the edges be labelled 0. santana’s application of theorem 3.17 is theorem 3.18. [6, theorem 4.2] given a directed graph g = (v,e) with vertex set v and n = |v | with n ≥ 6, and edge set e, the maximum size of e such that g is cordial for any given n is |e|max = ( n 2 ) −z + ⌈ 1 2 ((n 2 ) −z )⌉ z = ( dn 2 e 2 ) + ( bn 2 c 2 ) . (1) 11 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 00 0 1 1 1 figure 7. a complete graph. dashed lines represent edges labelled zero regardless of arc orientation f. rom section 3 we have that for any tournament with n ≤ 5 there exists a cordial labelling, save for the case when n = 4 thus we begin with a complete graph with n ≥ 6. recall that the number of edges on a complete graph is ( n 2 ) . due to our cordial labelling the number of edges with an induced labelling of 0 will be our z. this is because it will be the number of edges connected by two vertices of the same label, as shown in figure 7. if n is even that will mean that z = 2 (n 2 2 ) , i.e., it will be the number of edges on two complete graphs each on n 2 vertices represented by the labellings of ones and zeros. the floor and ceiling function in (1) simply account for the odd case. for every tournament with n ≥ 6 vertices, z > 1 3 ( n 2 ) . therefore some of the arcs labelled zero will need to be removed to get a cordial graph. how many arcs need to be removed is going to be equal to how much greater z is than the number of half the number of arcs not labelled zero. by the definition of a directed cordial graph we know that z can be larger than α or β and we can still have a cordial graph, hence the ceiling function. as mentioned in the introduction, the smallest non (2, 3)-cordial digraph is an orientation of xn, three parallel arcs. a question may be asked: what is the minimum number of arcs in a non (2, 3)-cordial digraph that has no isolated vertices? 4. conclusions in this article, we have shown that the only tournaments that are (2, 3)-cordial are when n ≤ 5 and then not for two 4-tournaments. except for one case when n is even, the n-wheel graph has an orientation that is (2, 3)-cordial and that at least one orientation of any wheel graph is not (2, 3)-cordial. further, we show that every fan graph has a (2, 3)-cordial orientation, and there is always an orientation of the n-fan that is not (2, 3)-cordial. references [1] l. b. beasley, cordial digraphs, arxiv:2212.05142. 12 https://arxiv.org/abs/2212.05142 ar ti cl e in pr es s l. b. beasley et. al. / j. algebra comb. discrete appl. -(-) (2023) 1–13 [2] i. cahit, cordial graphs: a weaker version of graceful and harmonious graphs, ars comb. 23 (1987) 201–207. [3] m. hovey, a-cordial graphs, discrete math. 93(2-3) (1991) 183–194. [4] o. pechenik, j. wise, generalized graph cordiality, discuss. math. graph theory 32(3) (2012) 557– 567. [5] e. salehi, pc-labelling of a graph and its pc-set, bull. ins.t comb. appl. 58 (2010) 112–121. [6] m. a. santana, j. m. mousley, d. e. brown, l. b. beasley, (2, 3) cordial trees and paths, arxiv:2012.10591. 13 https://mathscinet.ams.org/mathscinet-getitem?mr=886954 https://mathscinet.ams.org/mathscinet-getitem?mr=886954 https://doi.org/10.1016/0012-365x(91)90254-y https://doi.org/10.7151/dmgt.1626 https://doi.org/10.7151/dmgt.1626 http://salehi.faculty.unlv.edu/pcl1.pdf https://doi.org/10.48550/arxiv.2012.10591 https://doi.org/10.48550/arxiv.2012.10591 introduction preliminaries (2,3)-orientable digraphs conclusions references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1000959 j. algebra comb. discrete appl. 8(3) • 219–231 received: 07 december 2020 accepted: 29 april 2021 journal of algebra combinatorics discrete structures and applications cyclic dna codes over the ring z4 + uz4 + u2z4 research article karthick gowdhaman, somi gupta, cruz mohan, kenza guenda, durairajan chinnapillai abstract: in this work, we have investigated the one generator cyclic dna codes with reverse and reverse complement constraints over the ring r = z4+uz4+u2z4 with u3 = 0. skew cyclic codes with reverse complement constraint are constructed over r. we have also determined a one-to-one correspondence between the elements of the ring r and dna codons satisfying the watson-crick complement. finally, we have established some examples which satisfy the given constraints. 2010 msc: 94b05, 94b15 keywords: dna codes, skew codes, reversible codes 1. introduction dna is shortened word of deoxyribonucleic acid. it is a molecule made of two chains that curl around one another to frame a two-fold helix conveying the hereditary guidelines utilized in the development of all creatures. this two-fold helix is assembled by blending the four fundamental structure units a(adenine), c-(cytosine), g-(guanine) and t-(thymine) which are called the nucleotides held by hydrogen ties. the dna strand is held by an important feature called complementary base pairing which connects the watson-crick complementary bases with each other denoted by a = t,g = c,c = g,t = a. in [4] adleman performed a successful experiment for dna computing. the basic idea of his work was to use dna which is an ideal source of computing due to its dense and self-replicating property to solve a mathematical problem. after this successful experiment, the area of dna computing was flooded with different approaches such as dna tile assembly, the building of dna nanostructures, dna-based data storage system and study of error-correcting properties of dna. karthick gowdhaman, cruz mohan, durairajan chinnapillai; department of mathematics, bharathidasan university, tiruchirapalli, tamil nadu, india (email: karthimath123@bdu.ac.in, cruzmohan@gmail.com, cdurai66@rediffmail.com). somi gupta; department of mathematics and applications "renato caccioppoli", university of napoli federico ii, naples, italy (email: gupta7somi200@gmail.com). kenza guenda; laboratory of algebra and number theory, usthb, algiers (email: ken.guenda@gmail.com). 219 https://orcid.org/0000-0003-0104-2993 https://orcid.org/0000-0002-1482-7565 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 cyclic codes over rings have been studied by many authors (see for example [6] and [2]). later dna cyclic codes have gained interest of many researchers for their applications (see [16], [11], [18], [12], [14], [19], [5]). abhay et al. [10] have studied dna cyclic codes over the ring z4 + uz4 with u2 = 0. cyclic codes over the ring r = z4 + uz4 + u2z4 with u3 = 0 have been studied by ozen et al. in [15]. cyclic code over skew polynomial ring have been constructed by [7]. boucher and ulmer [8] have found a link between arithmetic structure of skew polynomials and existence of such codes. in [17] siap et al. studied skew cyclic code of arbitrary length and established a strong connection with well known codes. most recently, the dna codes over ring of order 256 has been studied in [9]. they have obtained dna skew cyclic codes over the ring f2 + uf2 + vf2 + wf2 + uvf2 + uwf2 + vwf2 + uvwf2, where, u2 = 0,v2 = v,w2 = w,uv = vu,uw = wu,vw = wv, addressing reversibility problem. in this work, we have investigated the one generator cyclic dna codes with reverse and reverse complement constraint over the ring r = z4 + uz4 + u2z4 with u3 = 0. skew cyclic codes with reverse complement constraint are constructed over r. we have also determined a one-to-one correspondence between elements of the ring r and dna codons satisfying watson-crick complement. finally, we have established some examples which satisfy the given constraints. beside the theoretical results concerning the reverse and the reverse complement codes over the ring r and the one-to-one correspondence with codes, our other motivation in choosing this ring is the fact that it contains an additive subgroup of order 16. then the idea of the additive stem distance characterizing the hybridization energy given in [12] can be extended to this subgroup and then to the ring. this paper is structured in the following way: section 2 contains basic definitions of cyclic dna codes. we have established a one-to-one correspondence between elements of the ring r = z4+uz4+u2z4 with u3 = 0, elements of z34 and 64 codons. in section 3, we have studied the reversibility condition of one generator cyclic codes over the ring r and have determined reverse complement codes. binary image of the ring and skew cyclic codes are studied in sections 4 and 5 respectively. in section 6, we have obtained some examples. finally, section 7 concludes the paper with some establishments of future work that can be done using this work. 2. preliminaries ozen et al. in [15] have determined cyclic codes over the ring r = z4 + uz4 + u2z4, where u3 = 0. they obtained that the total number of cyclic codes over r is 13r. they also found the general form of generator of cyclic codes over r and one generator cyclic codes (cyclic codes generated by one element) over the ring r. let r = z4 +uz4 +u2z4 = {a+ub+u2c | a,b,c ∈ z4}, where u3 = 0. the ring r has 11 nontrivial ideals given by a2u2 = {0,2u2} au2 = {0,u2,2u2,3u2} a2u = {0,2u,2u2,2u + 2u2} a2u+u2 = {0,2u2,2u + u2,2u + 3u2} a2u,u2 = {0,u2,2u2,3u2,2u,2u + u2,2u + 2u2,2u + 3u2} a2 = {0,2,2u,2u2,2 + 2u,2 + 2u2,2u + 2u2,2 + 2u + 2u2} a2+u2 = {0,2u,2u2,2u + 2u2,2 + u2,2 + 3u2,2 + 2u + u2,2 + 2u + 3u2} a2,u2 = {0,2,2u,u2,2u2,3u2,2 + 2u,2 + u2, 2 + 2u2,2 + 3u2,2u + u2,2u + 2u2, 2u + 3u2,2 + 2u + u2,2 + 2u + 2u2,2 + 2u + 3u2} 220 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 au = {0,u,2u,3u,u2,2u2,3u2,u + u2,u + 2u2,u + 3u2,2u + u2,2u + 2u2, 2u + 3u2,3u + u2,3u + 2u2,3u + 3u2} a2+u = {0,2u,u2,2u2,3u2,2 + u,2 + 3u,2u + u2,2u + 2u2,2u + 3u2,2 + u + u2, 2 + u + 2u2,2 + u + 3u2,2 + 3u + u2,2 + 3u + 2u2,2 + 3u + 3u2} a2,u = {0,2,u,2u,3u,u2,2u2,3u2,2 + u,2 + 2u,2 + 3u,2 + u2,2 + 2u2,2 + 3u2, u + u2,u + 2u2,u + 3u2,2u + u2,2u + 2u2,2u + 3u2,3u + u2,3u + 2u2, 3u + 3u2,2 + u + u2,2 + u + 2u2,2 + u + 3u,2 + 2u + u2,2 + 2u + 2u2, 2 + 2u + 3u2,2 + 3u + u2,2 + 3u + 2u2,2 + 3u + 3u2}. the ring r is a finite local ring with a2,u as its unique maximal ideal (as it contains all the non-zero divisors of r). the residue field of r is k = r/a2,u = {0 + a2, u,1 + a2, u}∼= z2. let p(x) be a monic basic irreducible polynomial of degree m in r[x], then the galois ring extension over r is defined by the residue class ring qm = r[x]/(p(x)) having 64m elements. the triplet of nucleotides called codons is the basic coding unit for amino acids during the protein synthesis in living organisms. since the ring r has 64 elements, then it is convenient to describe a one-to-one correspondence between the elements of r and the codons. in the following table, we give a one-to-one correspondence between the elements of r, the elements of z34 and the codons. the oneto-one correspondence, called gray map between the rings r and z34 is defined as ψ : r → z34 with (a + ub + u2c) 7→ (a,b,c), where a,b,c ∈ z4. another important one-to-one correspondence between the elements of z34 and d 3 is denoted by φ : z34 → d3, where d denotes the set of all nucleotides. the full names of codons have been taken from http://www.hgmd.cf.ac.uk/docs/cd_amino.html. table 1. correspondence between elements of r, z34 and codons a ∈ r ψ(a) ∈ z34 φ(a) ∈ d3 full name of codons 0 (0, 0, 0) gtg valine u (0, 1, 0) ccg alanine u+u2 (0, 1, 1) cga arginine u+2u2 (0, 1, 2) caa glutamine u+3u2 (0, 1 ,3) tag termination (amber) 2u+u2 (0,2 , 1) cat histidine 2u+2u2 (0, 2, 2) ctc serine 2u+3u2 (0, 2, 3) ata isoleucine 3u+u2 (0, 3, 1) gcg alanine 3u+2u2 (0, 3, 2) tcg serine 3u+3u2 (0, 3, 3) taa termination (ochre) 1 (1, 0, 0) ggc arginine 1+u (1, 1, 0) tga termination (opal or umber) 1+u+u2 (1, 1, 1) ggg glycine 1+ u+2u2 (1, 1, 2) gca alanine 1+u+3u2 (1, 1, 3) gaa glutamate 1+2u (1, 2, 0) cca proline 1+ 2u+u2 (1, 2, 1) ttt phenylalanine 1+2u+2u2 (1, 2, 2) cct proline 1+2u+3u2 (1, 2, 3) agg arginine 1+3u (1, 3, 0) tgg trytophan 1+3u+u2 (1, 3, 1) tta leucine 1+3u+2u2 (1, 3, 2) tac tyrosine 1+3u+3u2 (1, 3, 3) tgc cysteine 2 (2, 0, 0) gag arginine 2u (0, 2,0) aca threonine 2+u (2, 1, 0) agc serine 2+2u (2, 2, 0) aga glutamate 221 http://www.hgmd.cf.ac.uk/docs/cd_amino.html k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 2+ u+u2 (2, 1, 1) att isoleucine 2+ u+2u2 ( 2, 1, 2) gat aspartate 2+u+3u2 (2, 1, 3) cgc arginine 2+2u+u2 (2, 2, 1) cag glutamine 2+2u+2u2 (2, 2, 2) cac histidine 2+2 u+3u2 (2, 2, 3) gac aspartate 2+3u (2, 3, 0) gtt valine 2+3 u+u2 (2, 3, 1) atc isoleucine 2+3u+2u2 (2, 3, 2) ggc glycine 2+3 u+3u2 (2, 3, 3) gct valine 3 (3, 0, 0) gga glycine 3u (0, 3, 0) cta leucine 3+u (3, 1, 0) atg methionine 3+u+u2 (3, 1, 1) acg threonine 3+u+2u2 (3, 1, 2) acc threonine 3+u+3u2 (3, 1, 3) aat asparagine 3+2u (3, 2, 0) tca serine 3+2u+u2 (3, 2, 1) aag lysine 3+2u+2u2 (3, 2, 2) ccg alanine 3+2u+3u2 (3, 2, 3) ttg leucine 3+3u (3, 3, 0) cgt arginine 3+3u+u2 (3, 3, 1) ctt leucine 3+3u+2u2 (3, 3, 2) act threonine 3+3u+3u2 (3 ,3 ,3) ccc proline 1+u2 (1, 0, 1) aac asparagine 1+2u2 (1, 0, 2) agt serine 1+3u2 (1, 0, 3) ttc phenylalanine 2+u2 (2, 0 , 1) tat tyrosine 2+2u2 (2, 0, 2) tgt cysteine 2+3u2 (2, 0, 3) gta valine 3+u2 (3, 0, 1) tcc serine 3+2u2 (3, 0, 2) ggt glycine 3+3u2 (3, 0,3) aaa lysine u2 (0, 0, 1) ctg leucine 2u2 (0, 0, 2) tct leucine 3u2 (0, 0, 3) gtc valine definition 2.1. let a,b ∈ z4, then we define a distance called gray distance by dg(a,b) = dh(φ(a),φ(b)), where φ(a),φ(b) ∈ d3. note that it can be extended upto length n. hence the map φ is a distance preserving map from (rn,dg) to (d3n,dh). example 2.2. let a = 1 + u + u2 and b = 2 + 2u + 2u2, then dg(a,b) = dh(ggg,cac) = 3. definition 2.3. if c is invariant under the cyclic shift operator δ : rn → rn given by δ(c1,c2, · · · ,cn) = (cn,c1, · · · ,cn−1), then the code c is called a cyclic code of length n. definition 2.4. let c be a code of arbitrary length n over any finite set a. then c is called reversible code if it remains invariant under the reversal of each codeword, i.e., if c = (c1,c2, . . . ,cn) ∈ c, then cr = (cn,cn−1, . . . ,c1) ∈c. while working with cyclic and reversible codes we need to deal with a polynomial called selfreciprocal polynomial defined in the following definition. definition 2.5. let p(x) = a0 + a1x + · · · + anxn be a polynomial of degree n. the polynomial g(x) = an + an−1x + · · · + a0xn is called reciprocal polynomial of p(x). a polynomial p(x) over r is said to be self-reciprocal polynomial if p(x) = p∗(x), where p∗(x) = xnp ( 1 x ) . 222 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 we follow the definition of a dna code from [12]. a code c is called a dna code if it satisfies some of the following properties: (i) the hamming distance constraint is defined as dh(a, b) ≥ d for all a, b ∈ c and a 6= b for some predefined distance d. (ii) the reverse constraint is defined as dh(ar, b) ≥ d for all a,b ∈c and a 6= b for some predefined distance d and where ar denotes the reverse sequence of alphabets in a. (iii) the reverse complement constraint is defined as dh(ar,b) ≥ d for all a,b ∈ c,a 6= b for some predefined distance d and where b denotes a word in which each alphabet of b is replaced by its watson-crick complement. dna code satisfying reverse complement constraint is called the reverse complement dna code. (iv) the fixed gc-content constraint specifies that each codeword must have fixed number of g’s and c’s. generally this fixed number is bn 2 c where n denotes the length of codewords. definition 2.6. let c be a code of arbitrary length n over d. then c is called reverse complement code if it contains both the reverse and the complement of every codeword in c, that is, if (c1,c2, . . . ,cn) ∈ c, then (cn,cn−1, . . . ,c1) ∈c. it is interesting to observe that one-to-one correspondence we define is a customized gray map between the dna codons and elements of the ring r. the elements of ideal a2 correspond only to self reversible dna codons. this map take care of gc content for the ideals a2u2 and a2u. the gc content is 30 to 50 percentage in these ideals. further, the map satisfies the following property. lemma 2.7. let a ∈ r, then a = a + 2(1 + u + u2). 3. cyclic dna codes over r in this section, we study the reverse and the reverse complement cyclic codes over r. for this, we will use the following lemmas. lemma 3.1. [1] let p(x) and q(x) be polynomials over z4 with deg p(x) ≥ deg q(x). then (i) [p(x)q(x)]∗ = p∗(x)q∗(x) (ii) [p(x) + q(x)]∗ = p∗(x) + x(deg p(x)−deg q(x))q∗(x). lemma 3.2. [3] let c = 〈p(x)〉 be a cyclic code of odd length n over z4; where p(x) is a monic polynomial of degree r over z4. then c is reversible if and only if p(x) is self reciprocal polynomial. now, we are able to prove the following result which will be useful thereafter. lemma 3.3. let p1(x),p2(x), . . . ,pr(x) be polynomials over z4 with deg pi(x) = ki and kr ≤ kr−1 ≤ ···≤ k1, then (i) [p1(x)p2(x) · · ·pr(x)]∗ = p∗1(x)p∗2(x) · · ·p∗r(x) and (ii) [p1(x) + p2(x) + · · ·+ pr(x)]∗ = p∗1(x) + xk1−k2p∗2(x) + · · ·+ xk1−kip∗i (x) + · · ·+ x k1−krp∗r(x). proof. we prove this by induction on r. by lemma 3.1, this is true for r = 2. let us assume that this is true for less than or equal to l, that is, [p1(x)p2(x) · · ·pl(x)]∗ = p∗1(x)p ∗ 2(x) · · ·p ∗ l (x), [p1(x) + p2(x) + · · ·+ pl(x)]∗ = p∗1(x) + x k1−k2p∗2(x) + · · ·+ x k1−klp∗l (x). 223 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 now, we will check whether the hypothesis is true for r = l + 1. by using hypothesis for r = 2 and r = l consecutively, [p1(x)p2(x) · · ·pl(x)pl+1(x)]∗ = [p1(x)p2(x) · · ·pl(x)]∗p∗l+1(x) = p∗1(x)p ∗ 2(x) · · ·p ∗ l+1(x). repeat the same process to prove the second identity. [p1(x) + p2(x) + · · ·+ pl+1(x)]∗ = [p1(x) + p2(x) + · · ·+ pl(x)]∗ + x(k1−kl+1)p∗l+1(x) = p∗1(x) + x k1−k2p∗2(x) + · · ·+ x k1−klp∗l (x) + x (k1−kl+1)p∗l+1(x). hence the induction hypothesis is true for r = l+1. therefore, by mathematical induction, the statement is true for all positive integer r. proposition 3.4. let pi(x) and qi(x) be in z4[x] for i = 1,2 and 3. if the following equality holds p1(x) + up2(x) + u 2p3(x) = q1(x) + uq2(x) + u 2q3(x), (1) then pi(x) = qi(x) for i = 1,2 and 3. proof. assume that equation (1) is true. multiplying it by u2 and using u3 = 0, we have u2p1(x) = u2q1(x). therefore p1(x) = q1(x), because p1(x),q1(x) ∈ z4[x]. again multiplying (1) by u and substituting p1(x) = q1(x), we get u2p2(x) = u2q2(x) and hence p2(x) = q2(x). similarly, we get p3(x) = q3(x). now, we study the reverse constraint of one generator cyclic code of odd length n over r. for this, first we need the following result which states one generator cyclic codes of odd length n over the ring r. this result with its proof can be found in [15]. theorem 3.5. [15] let c be a cyclic code of odd length n over r. if c = 〈h1(x) + ug1(x) + u2b1(x),uh2(x) + u 2b2(x),u 2h3(x)〉 and h1(x) and h3(x) are equal, then c = 〈h1(x) + ug1(x) + u2b1(x)〉. definition 3.6. let r be a ring and p(x) ∈ r[x] is called regular polynomial if it is not a zero divisor in r[x]. theorem 3.7. let c be a cyclic code of odd length n over the ring r. let p(x) be a regular polynomial in z4[x] and c = 〈 p(x) + uq(x) + u2h(x) 〉 where p(x),q(x) and h(x) are in z4[x]. if deg p(x) = r, deg q(x) = s and deg h(x) = t with r ≥ s ≥ t, then c is reversible if and only if (a) p(x) is a self reciprocal polynomial. (b) (i) xr−sq∗(x) = q(x) and xr−th∗(x) = h(x), or (ii) xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = q(x) + h(x), or (iii) xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = p(x) + q(x) + h(x), or (iv) xr−sq∗(x) = q(x) and xr−th∗(x) = p(x) + h(x). proof. let us assume that c is a reversible code over r. then c mod u = 〈p(x)〉 is a reversible code over z4. therefore, by lemma 3.2 we have that p(x) is a self reciprocal polynomial in z4[x]. now, by lemma 3.3 we have [p(x) + uq(x) + u2h(x)]∗ =p∗(x) + uxr−sq∗(x) + u2xr−th∗(x) =p(x) + uxr−sq∗(x) + u2xr−th∗(x) =(p(x) + uq(x) + u2h(x))k(x) ∈c. 224 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 since the degrees of both sides are the same, we have k(x) is a constant in r, i.e., k = k1 + uk2 + u2k3 and hence p(x) + uxr−sq∗(x) + u2xr−th∗(x) =(p(x) + uq(x) + u2h(x))k =(p(x) + uq(x) + u2h(x))(k1 + uk2 + u 2k3) =p(x)k1 + u(p(x)k2 + q(x)k1) + u 2(p(x)k3 + q(x)k2 + h(x)k1) (2) now by proposition 3.4, we get k1 = 1 and each k2 and k3 have 4 possibilities. therefore, we have 16 possible cases, i.e., conditions for k = 1,1 + u,1 + 2u,1 + 3u,1 + u + u2,1 + u + 2u2,1 + u + 3u2,1 + 2u + u2,1 + 2u + 2u2,1 + 2u + 3u2,1 + 3u + u2,1 + 3u + 2u2,1 + 3u + 3u2,1 + u2,1 + 2u2,1 + 3u2. we have investigated all these cases by substituting values of k in (2). then we get the following summarized results: constant k conditions obtained k = 1 xr−sq∗(x) = q(x) and xr−th∗(x) = h(x) k = 1 + u xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = q(x) + h(x) k = 1 + 2u 2xr−sq∗(x) = 2q(x) and 2xr−th∗(x) = 2h(x) k = 1 + 3u 2xr−sq∗(x) = 2p(x) + 2q(x) and 2xr−th∗(x) = 2q(x) + 2h(x) k = 1 + u + u2 xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = p(x) + q(x) + h(x) k = 1 + u + 2u2 2xr−sq∗(x) = 2p(x) + 2q(x) and 2xr−th∗(x) = 2q(x) + 2h(x) k = 1 + u + 3u2 xr−sq∗(x) = p(x) + q(x) and 2xr−th∗(x) = 2p(x) + 2q(x) + 2h(x) k = 1 + 2u + u2 2xr−sq∗(x) = 2q(x) and 2xr−th∗(x) = 2p(x) + 2h(x) k = 1 + 2u + 2u2 2xr−sq∗(x) = 2q(x) and 2xr−th∗(x) = 2h(x) k = 1 + 2u + 3u2 2xr−sq∗(x) = 2q(x) and 2xr−th∗(x) = 2p(x) + 2h(x) k = 1 + 3u + u2 2xr−sq∗(x) = 2p(x) + 2q(x) and 2xr−th∗(x) = 2p(x) + 2q(x) + 2h(x) k = 1 + 3u + 2u2 2xr−sq∗(x) = 2p(x) + 2q(x) and 2xr−th∗(x) = 2q(x) + 2h(x) k = 1 + 3u + 3u2 2xr−sq∗(x) = 2p(x) + 2q(x) and 2xr−th∗(x) = 2p(x) + 2q(x) + 2h(x) k = 1 + u2 xr−sq∗(x) = q(x) and xr−th∗(x) = p(x) + h(x) k = 1 + 2u2 xr−sq∗(x) = q(x) and 2xr−th∗(x) = 2h(x) k = 1 + 3u2 xr−sq∗(x) = q(x) and 2xr−th∗(x) = 2p(x) + 2h(x) from the above table, we can observe that there are 12 similar cases. therefore, we can reduce above conditions into following four cases. (i) xr−sq∗(x) = q(x) and xr−th∗(x) = h(x), or (ii) xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = q(x) + h(x), or (iii) xr−sq∗(x) = p(x) + q(x) and xr−th∗(x) = p(x) + q(x) + h(x), or (iv) xr−sq∗(x) = q(x) and xr−th∗(x) = p(x) + h(x). for the converse part, let us assume that the hypothesis is true. since c is a cyclic code over r, it is enough to show that [p(x) + uq(x) + u2h(x)]∗ belong to c. [p(x) + uq(x) + u2h(x)]∗ =p∗(x) + uxr−sq∗(x) + u2xr−th∗(x) =p(x) + uxr−sq∗(x) + u2xr−th∗(x). now using conditions in the hypothesis, we have [p(x) + uq(x) + u2h(x)]∗ = (p(x) + uq(x) + u2h(x))k ∈c where the constant k ∈{1,1 + u,1 + 2u,1 + 3u,1 + u + u2,1 + u + 2u2,1 + u + 3u2,1 + 2u + u2,1 + 2u + 2u2,1 + 2u + 3u2,1 + 3u + u2,1 + 3u + 2u2,1 + 3u + 3u2,1 + u2,1 + 2u2,1 + 3u2}⊆ r. therefore, c is a reversible cyclic code in r[x]. 225 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 by using table 1 and lemma 2.7, we have the following result. theorem 3.8. let c = 〈 p(x) + uq(x) + u2h(x) 〉 be a cyclic code over r where p(x), q(x),h(x) ∈ z4[x]. if deg p(x) = r, deg q(x) = s and deg h(x) = t with r ≥ s ≥ t, then c is a reverse complement code if and only if 1. the element 2(1+u+u 2)(xn−1) (x−1) ∈c and 2. the cyclic code c is reversible. proof. let c be the code satisfying the hypothesis. let c(x) = c0 + c1x + · · · + ckxk ∈ c. since c is reversible, c∗(x) = ck + ck−1x + · · · + c0xk ∈ c. as c is an ideal of r[x] 〈xn−1〉, so x n−k−1c∗(x) = ckx n−k−1 + ck−1x n−k + · · · + c0xn−1 belongs to c. by hypothesis 2(1 + u + u2) (xn−1) (x−1) ∈ c, therefore 2(1 + u + u2) (xn−1) (x−1) + x n−k−1c∗(x) ∈c. then 2(1 + u + u2) (xn −1) (x−1) + xn−k−1c∗(x) =2(1 + u + u2)(1 + x + x2 + · · ·+ xn−k−2) + (ck + 2(1 + u + u 2))xn−k−1 + · · · + (c0 + 2(1 + u + u 2))xn−1 =2(1 + u + u2)(1 + x + x2 + · · ·+ xn−k−2) + ckx n−k−1 + ck−1x n−k + · · ·+ c0xn−1 2(1 + u + u2) (xn −1) (x−1) + xn−k−1c∗(x) =c∗(x). (using lemma(2.7)) hence, we conclude that c is a reverse complement code. conversely, we assume that c is a reverse complement code, i.e., if c(x) ∈ c, then c∗(x) ∈ c. first we observe that since c is linear this implies that the element a(x) = 0 ∈c and therefore a(x) = 2(1 + u + u2)(1 + x + · · ·+ xn−1) = 2(1 + u + u2) (xn −1) (x−1) ∈c. now, let c(x) = c0 + c1x + · · ·+ ckxk ∈c, then c∗(x) =2(1 + u + u2)(1 + x + · · ·+ xn−k−2) + xn−k−1ck + · · ·+ c0 =2(1 + u + u2)(1 + x + · · ·+ xn−k−2) + xn−k−1(ck + 2(1 + u + u2))+ · · ·+ (c0 + 2(1 + u + u2))xn−1 adding 2(1 + u + u2)(x n−1) (x−1) to the above equation, we get c∗(x) + 2(1 + u + u2) (xn −1) (x−1) =ckx n−k−1 + ck−1x n−k + · · ·+ c0xn−1 =xn−k−1(ck + ck−1x + · · ·+ c0xk) multiplying both side by xk+1, we get xk+1(c∗(x) + 2(1 + u + u2) (xn −1) (x−1) ) = ck + ck−1x + · · ·+ c0xk = c∗(x). thus, we have c is reversible code. 226 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 theorem 3.9. let c be a cyclic code of odd length n which satisfies the conditions of theorem 3.8, then c is a dna code. proof. combining the proof of theorem 3.7 and theorem 3.8, we get the result. example 3.10. let x = (2u+ 2u2,2 + 2u2,2 + 2u) and y = (2 + 2u,2 + 2u2,2u+ 2u2). we define a code c consisting of all cyclic shifts and linear combinations of the vectors x and y over r. thus we obtain corresponding dna code of parameters (9,32,4) as follows, ctctgtaga gagcacgag cacgaggag ctcctcgtg gtggtggtg agatgtctc tctacagag caccaccac gtgctcctc gaggagcac ctcgtgctc gagacatct acagagtct ctcagatgt agagtgaga cactcttct tcttctcac tgtgtgtgt tctgagaca gtgtgttgt tgtagactc acaacacac tgttgtgag acatctgag cacacaaca agactctgt acacacaca agaagagtg gtgagaaga tctcactct gagtctaca tgtctcaga example 3.11. let x = (2,2 + 2u,2u,2 + 2u + 2u2), then we define a code generated by a generator matrix consisting of the cyclic shifts of the vector x over r. thus, we have a dna code of parameters (12,16,6) as follows, tgtgtgacagtg tgttgtacaaca gtggtggtggtg gtgtgtgtgaca tgtacaacatgt tgtcacacacac gtgacagtgtgt gtgcacgtgcac cacgtgcacgtg cactgtcacaca acagtgtgtgtg acatgttgtaca cacacacactgt caccaccaccac acaacatgttgt acacactgtcac 4. binary image of elements in r in this section, we define binary images of elements of the ring r which will be useful for dna computing. an element in the ring r is of the form a + ub + u2c; where a,b,c ∈ z4. now, we can define a map between r and z2. a one-to-one correspondence between r and z34 is defined as ψ : r → z34 with (a + ub + u2c) 7→ (a,b,c) where a,b,c ∈ z4. we define a gray map ϕ : z4 → z22 using 2-adic expansion of elements in z4 which are as follows: c π(c) ρ(c) υ(c) 0 0 0 0 1 1 0 1 2 0 1 1 3 1 1 0 we have ϕ(c) = (ρ(c),υ(c)) for all c ∈ z4 (as in [13]). therefore, 0 → 00,1 → 01,2 → 11,3 → 10. for any v ∈ z4, the lee weight wl(v) is defined as min(v,4 − v). this lee weight can be extended to the ring r as follow: for x = a + ub + u2c lee weight of x is defined as wl(x) = wl(a,b,c). the hamming distance dh(c1,c2) between two codewords c1 and c2 is the hamming weight of the codeword wh(c1 −c2) that is the number of non zero element in (c1 −c2). define χ : r → z62 by χ(a+ub+u2c) = (ρ(a),υ(a),ρ(b),υ(b),ρ(c),υ(c)). the map χ is clearly a linear map. lemma 4.1. the gray map from rn to z6n2 is a distance preserving map. 227 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 proof. let x1,x2 be two elements of rn. then dl(x1,x2) = wl(x1 −x2). since the map is bijective, we have wl(x1 − x2) = wh(χ(x1 − x2)) = wh(χ(x1) − χ(x2)). therefore, the gray map χ is distance preserving. given an element c of length n in a cyclic code c. if the cyclic shift δ(c) ∈ c then ψ(δ(c)) will be quasi cyclic code of length 3n with index 3 over z3n4 . then it can be easily seen that ϕ(ψ(δ(c))) satisfies quasi cyclic shift of length 6n with index 6 over z6n2 . thus we have the following theorem. theorem 4.2. if c is a cyclic code of length n, then its image is a quasi cyclic code of length 6n and with index 6 over z2. 5. skew cyclic codes over r let θ be a non-trivial automorphism defined by θ : r → r such that (a + ub + u2c) 7→ a−ub + u2c. the order of θ is 2 that is θ(θ(a + ub + u2c)) = a + ub + u2c. the ring r[x,θ] = {a0 + a1x + a2x2 + · · ·+ an−1x n−1 : ai ∈ r,n ∈ n}, a non commutative ring with usual addition and multiplication defined as axi · bxj = aθi(b)xi+j is called skew polynomial ring. definition 5.1. a set c of codewords over rn is skew cyclic code if it satisfies the following (i) c is a submodule over r. (ii) whenever (c0,c1, . . . ,cn−1) ∈c then (θ(cn−1),θ(c0), . . . ,θ(cn−2)) ∈c. let p(x) + 〈xn − 1〉 ∈ rθ = r[x,θ]/〈xn − 1〉 and r(x) ∈ r[x,θ], then define multiplication as r(x)(p(x) + 〈xn −1〉) = r(x)p(x) + 〈xn −1〉, for any r(x) ∈ r[x,θ]. clearly rθ is left r[x,θ]−module. theorem 5.2. a code c over r is a skew cyclic code of length n if and only if c is left ideal of r[x,θ]module rnθ . proof. the proof is same as the proof of [7, theorem 1]. theorem 5.3. let c be a skew cyclic code of length n over r. if c contains a monic polynomial of minimal degree p(x), then c = 〈p(x)〉, where p(x) is a right divisor of xn −1. proof. the proof is same as the proof of [7, lemma 1]]. now, we will introduce reverse complement skew cyclic codes over r. for this we have to notice that the multiplication over r[x,θ] is not commutative therefore we need to see things differently. let c = (c0,c1, . . . ,cn−1) be in r[x,θ] then reversal of c denoted by cr is given by cr = (cn−1,cn−2, · · · ,c0). c(x−1) ·xn−1 = (c0 + c1x−1 + · · ·+ cn−1x−n+1) ·xn−1 = c0 ·xn−1 + c1x−1 ·xn−1 + · · ·+ cn−1x−n+1 ·xn−1 = c0θ 0(1)xn−1 + c1θ −1xn−1−1 + · · ·+ cn−1θ−n+1x−n+1+n−1 = c0x n−1 + c1x n−2 + · · ·+ cn−1 = cr as θ is an automorphism we have θ(1) = 1 and θr(1) = 1 for all r ∈ z. note that the reciprocal polynomial c∗(x) and cr(x) are different due to the operations on them. also see (p(x) · q(x))r 6= pr(x) · qr(x) as in lemma 3.3. if pr(x) coincide with (rpr(x)), then p(x) is called self reciprocal polynomial where r is a constant in r. theorem 5.4. let c = 〈p(x)〉 be a skew cyclic code of length n, where p(x) is a monic polynomial. then if c is reverse complement cyclic code then p(x) is a self reciprocal polynomial and 2(1 + u + u2)(xn − 1)/(x−1) ∈c. 228 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 proof. let c = 〈p(x)〉 be a reverse complement cyclic code. since c is linear (0, · · · ,0) ∈c, therefore 2(1 + u + u2)(xn −1)/(x−1) ∈c. let p(x) = 1 + p1x + p2x2 + · · ·+ xk. then, p(x) r = (p(x−1) ·xk−1) = 2(1 + u + u2)(1 + x + · · ·+ xn−k−2) + 1xn−k−1 + pk−1xn−k + · · ·+ 1xn−1 = 2(1 + u + u2)(1 + x + · · ·+ xn−k−2) + (3 + 2(1 + u + u2))xn−k−1+ (3pk−1 + 1 + u + u 2)xn−k + · · ·+ (3 + 2(1 + u + u2))xn−1 now as the c is a reverse complement code therefore p(x) r ∈c. using linearity of c we have 2(1 + u + u2)(xn −1)/(x−1) + p(x) r = 3xn−k−1 + 3pk−1x n−k + · · ·+ 3xn−1 ∈c. multiplying xk+1−n both side and making use of c is a left ideal of r[x,θ]-module rnθ we get, (2(1 + u + u2)(xn −1)/(x−1) + p(x) r ) ·xk+1−n = 3(θn−k−1(1) + pk−1θn−k(1)x + · · ·+ θn−1(1)xk) = 3(1 + pk−1x + · · · + xk) = 3pr(x) ∈ c. since c = 〈p(x)〉 implies 3pr(x) = h(x)p(x) for some h(x) ∈ r[x,θ] degree of p(x) and pr(x) is same implies h(x) is constant. hence p(x) is self reciprocal. example 5.5. let p(x) = a+bx+cx2+dx3, where a = 1+u+u2,b = 2+2u+2u2 = c and d = 1+3u+u2. we define a code c obtained by using g =   p(x) xp(x) x2p(x) x3p(x)   as a generator matrix. then c correspond to dna code which is reverse complement skew cyclic code of length 21 and |c| = 48 ·22. 6. examples example 6.1. for n = 9, x9−1 = (x+3)(x2 +x+1)(x6 +x3 +1), let c = 〈g(x)+up(x)+u2h(x)〉, where g(x) = p(x) = h(x) = (x2 + x + 1)(x6 + x3 + 1). clearly, this code satisfies g(x) = g∗(x), p(x) = xip∗(x), h(x) = xjh∗(x), where i,j = 0, gives that c is a reverse complement cyclic dna code of length n = 27 with minimum distance d = 9 and cardinality | c |= 64. the codewords in c are as follows. atcatcatcatcatcatcatcatcatc gcggcggcggcggcggcggcggcggcg gtagtagtagtagtagtagtagtagta ggaggaggaggaggaggaggaggagga tcatcatcatcatcatcatcatcatca tcgtcgtcgtcgtcgtcgtcgtcgtcg aaaaaaaaaaaaaaaaaaaaaaaaaaa gcagcagcagcagcagcagcagcagca ctactactactactactactactacta acgacgacgacgacgacgacgacgacg ttttttttttttttttttttttttttt gatgatgatgatgatgatgatgatgat ataataataataataataataataata agaagaagaagaagaagaagaagaaga ccccccccccccccccccccccccccc gaggaggaggaggaggaggaggaggag acaacaacaacaacaacaacaacaaca aataataataataataataataataat aagaagaagaagaagaagaagaagaag gacgacgacgacgacgacgacgacgac aacaacaacaacaacaacaacaacaac ttattattattattattattattatta tgatgatgatgatgatgatgatgatga gttgttgttgttgttgttgttgttgtt tattattattattattattattattat tagtagtagtagtagtagtagtagtag tactactactactactactactactac gtggtggtggtggtggtggtggtggtg ttgttgttgttgttgttgttgttgttg ttcttcttcttcttcttcttcttcttc tgttgttgttgttgttgttgttgttgt gtcgtcgtcgtcgtcgtcgtcgtcgtc tggtggtggtggtggtggtggtggtgg tgctgctgctgctgctgctgctgctgc tcttcttcttcttcttcttcttcttct ggtggtggtggtggtggtggtggtggt tcctcctcctcctcctcctcctcctcc taataataataataataataataataa attattattattattattattattatt ggcggcggcggcggcggcggcggcggc atgatgatgatgatgatgatgatgatg agtagtagtagtagtagtagtagtagt aggaggaggaggaggaggaggaggagg gctgctgctgctgctgctgctgctgct 229 k. gowdhaman et. al. / j. algebra comb. discrete appl. 8(3) (2021) 219–231 agcagcagcagcagcagcagcagcagc actactactactactactactactact accaccaccaccaccaccaccaccacc gccgccgccgccgccgccgccgccgcc gaagaagaagaagaagaagaagaagaa caacaacaacaacaacaacaacaacaa cgacgacgacgacgacgacgacgacga ccaccaccaccaccaccaccaccacca catcatcatcatcatcatcatcatcat cagcagcagcagcagcagcagcagcag caccaccaccaccaccaccaccaccac cttcttcttcttcttcttcttcttctt ctgctgctgctgctgctgctgctgctg ctcctcctcctcctcctcctcctcctc cgtcgtcgtcgtcgtcgtcgtcgtcgt cgtcgtcgtcgtcgtcgtcgtcgtcgt cgccgccgccgccgccgccgccgccgc cctcctcctcctcctcctcctcctcct ccgccgccgccgccgccgccgccgccg ggggggggggggggggggggggggggg example 6.2. for n = 5, x5 −1 = (x+ 3)(x4 +x3 +x2 +x+ 1), let c = 〈g(x) +up(x) +u2h(x)〉, where g(x) = p(x) = h(x) = (x2 + x + 1)(x6 + x3 + 1). clearly, this code satisfies g(x) = g∗(x), p(x) = xip∗(x), h(x) = xjh∗(x) and hence this is a reverse complement dna cyclic code of length n = 15 with minimum distance d = 5 and cardinality | c |= 64. the code c contains the following codewords. atcatcatcatcatc gcggcggcggcggcg gtagtagtagtagta ggaggaggaggagga tcatcatcatcatca tcgtcgtcgtcgtcg aaaaaaaaaaaaaaa gcagcagcagcagca ctactactactacta acgacgacgacgacg ttttttttttttttt gatgatgatgatgat ataataataataata agaagaagaagaaga ccccccccccccccc gaggaggaggaggag acaacaacaacaaca aataataataataat aagaagaagaagaag gacgacgacgacgac aacaacaacaacaac ttattattattatta tgatgatgatgatga gttgttgttgttgtt tattattattattat tagtagtagtagtag tactactactactac gtggtggtggtggtg ttgttgttgttgttg ttcttcttcttcttc tgttgttgttgttgt gtcgtcgtcgtcgtc tggtggtggtggtgg tgctgctgctgctgc tcttcttcttcttct ggtggtggtggtggt tcctcctcctcctcc taataataataataa attattattattatt ggcggcggcggcggc atgatgatgatgatg agtagtagtagtagt aggaggaggaggagg gctgctgctgctgct agcagcagcagcagc actactactactact accaccaccaccacc gccgccgccgccgcc gaagaagaagaagaa caacaacaacaacaa cgacgacgacgacga ccaccaccaccacca catcatcatcatcat cagcagcagcagcag caccaccaccaccac cttcttcttcttctt ctgctgctgctgctg ctcctcctcctcctc cgtcgtcgtcgtcgt cgtcgtcgtcgtcgt cgccgccgccgccgc cctcctcctcctcct ccgccgccgccgccg ggggggggggggggg 7. conclusion in this paper, we have investigated one generator cyclic codes over the ring r = z4 + uz4 + u2z4, where u3 = 0 with reverse and reverse complement constraint. we have 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preliminaries cyclic dna codes over r binary image of elements in r skew cyclic codes over r examples conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729477 j. algebra comb. discrete appl. 7(2) • 195–207 received: 10 september 2019 accepted: 18 april 2020 journal of algebra combinatorics discrete structures and applications generalization of the ball-collision algorithm research article carmelo interlando, karan khathuria, nicole rohrer, joachim rosenthal, violetta weger abstract: in this paper we generalize the ball-collision algorithm by bernstein, lange, peters from the binary field to a general finite field. we also provide a complexity analysis and compare the asymptotic complexity to other generalized information set decoding algorithms. 2010 msc: 94b35, 94a60 keywords: coding theory, isd, ball-collision 1. introduction since 1978 it has been known that decoding a random linear code is an np-complete problem, this was shown in [7] by berlekamp, mceliece and van tilborg. therefore the interesting task arises of finding the complexity of decoding a random linear code using the best algorithms available. until today two main methods for decoding have been proposed: information set decoding (isd) and the generalized birthday algorithm (gba). the isd is more efficient if the decoding problem has only a small number of solutions, whereas gba is efficient when there are many solutions. also other ideas such as statistical decoding [1], gradient decoding [2] and supercode decoding [5] have been proposed but fail to outperform isd algorithms. the input of an isd algorithm is a corrupted codeword and it outputs the error vector. isd algorithms are often formulated via the parity check matrix, since it is enough to find a vector of a certain weight which has the same syndrome as the corrupted codeword, this problem is also referred to as the syndrome decoding problem. isd algorithms are based on a decoding algorithm proposed by prange [28] in 1962 and their structures do not change much from the original: as a first step one chooses an information set, then gaussian elimination brings the parity check matrix in a standard form and assuming that the errors are outside of the information set, these row operations on the syndrome will exploit the error vector, if the weight does not exceed the given error correction capacity. carmelo interlando; department of mathematics and statistics, san diego state university, san diego, ca 92182-7720 (email: carmelo.interlando@sdsu.edu). karan khathuria (corresponding author), nicole rohrer, joachim rosenthal, violetta weger; institute of mathematics, university of zurich, winterthurerstrasse 190, 8057 zurich, switzerland (email: karan.khathuria@math.uzh.ch, nicole.rohrer@uzh.ch, rosenthal@math.uzh.ch, violetta.weger@math.uzh.ch). 195 https://orcid.org/0000-0003-4928-043x https://orcid.org/0000-0002-9886-2770 https://orcid.org/0000-0003-4545-3559 https://orcid.org/0000-0001-9186-2885 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 the problem of decoding random linear codes has recently been receiving prominence with the proposal of using code-based public key cryptosystems for an upcoming post-quantum cryptographic public key standard. the idea of using linear codes in public key cryptography was first formulated by robert mceliece [24]. since the publication of mceliece a large amount of research has been done and the interested reader will find more information in a recent survey [9]. if the secret code is hidden well enough an adversary who wants to break a code-based cryptosystem encounters the decoding problem of a random linear code. it is therefore of crucial importance to understand the complexity of the best algorithms capable of decoding a general linear code. the isd algorithms were often considered when proposing a variant of the mceliece cryptosystem, to find the key size needed for a given security level. isd algorithms hence do not break a code-based cryptosystem but they determine the choice of secure parameters. since some of the new proposals (for example [3, 4, 18]) involve codes over general finite fields, having efficient isd algorithms generalized to fq is an essential problem. bernstein, lange and peters found a clever improvement of the isd algorithm which they called ball-collision decoding [8]. the algorithm of bernstein et. al. was presented for random binary linear codes. the main contribution of our paper is a generalization of the ball-collision decoding algorithm to arbitrary finite fields. the paper is structured as follows: in section 2 we discuss the previous work on isd algorithms focusing on those which have been generalized to an arbitrary finite field. in section 3 we describe the ball-collision algorithm over the binary field and the notations and concepts involved in the algorithm. in section 4 we present the ball-collision algorithm over fq and in section 5 we perform the complexity analysis of our algorithm including numerical parameter optimization and asymptotic analysis. 2. related work eventhough the cost of one iteration of prange’s original isd algorithm was very low, the algorithm was still coming with a huge complexity due to the number of iterations needed. many improvements have been suggested to prange’s simplest form of isd (see for example [11–13, 19, 21, 31]), they all focus on a more elaborate and more likely weight distribution of the error vector, which results in a higher cost of one iteration, but less iterations have to be performed. the improvements were splitting from an early time on into two directions: the first direction is following the splitting of lee and brickell [20] into the information set and the redundant set, i.e. they ask for v errors in the information set and t−v outside. the second direction is dumer’s splitting approach [13], which is asking for v errors in k + ` bits, which are containing an information set, and t−v in the remaining n−k − ` bits. apart from improvements regarding the success probability, one can also improve the cost of one iteration: canteaut and chabaud [10] have provided a speed up for finding information sets. they show that the information set should not be taken at random after one unsuccessful iteration, but rather a part of the previous information set should be reused and therefore a part of the gaussian elimination step is already performed. the second direction has resulted in many improvements, for example in 2009 finiasz and sendrier [14] have built two intersecting subsets of the k + ` bits, which contain an information set, and ask for v disjoint errors in both sets and t − 2v in the remaining n − k − ` bits. niebuhr, persichetti, cayrel, bulygin and buchmann [26] in 2010 improved the performance of isd algorithms over fq based on the idea of finiasz and sendrier [14]. in 2011 may, meurer and thomae [22] proposed an improvement using the representation technique introduced by howgrave-graham and joux [17]. to this algorithm becker, joux, may and meurer [6] (bjmm) in 2012 introduced further improvements. in the same year meurer in his dissertation [25] proposed a new generalized isd algorithm based on these two papers. in 2015, may-ozerov [23] used the nearest neighbor algorithm to improve the bjmm version of isd. later in 2017, the nearest neighbor algorithm over fq was applied to generalized bjmm algorithm by gueye, klamti and hirose [15]. now we focus on the first direction of improvements, which were first proposed for codes over the 196 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 binary field and then later generalized over an arbitrary finite field. in 1988, the same year as lee and brickell proposed their algorithm, leon [21] introduced a zero window inside the redundant set of size `, where no error are allowed. in 1993 stern [30] kept this zero window and proposed to partition the information set in to two sets and asks for v errors in each part and t−2v errors outside the information set. the generalization of both lee-brickell and stern’s algorithm to a general finite field fq were performed by peters [27] in 2010. in 2011 bernstein, lange and peters proposed the ball-collision algorithm [8], where they keep the partitioning of the information set but they reintroduce errors in the zero window, in fact they partition the zero window into two sets and ask for w errors in both and hence for t−2v−2w errors outside. in 2016, hirose [16] generalized the nearest neighbor algorithm over fq and applied it to the generalized stern algorithm. it is important to remark (see [25]) that the bjmm algorithm, even if having the smallest complexity, comes with a different cost: memory. in order to achieve a complexity of 128 bits, bjmm needs about 109 terabytes of memory. in fact, meurer observed, that if one restricts the memory to 240, bjmm and the ball-collision algorithm are performing almost the same. in this paper we provide the missing generalization of the ball-collision algorithm. the order of the complexities of isd algorithms over f2 is consistent also with their generalizations over fq. 3. preliminaries 3.1. notation we first want to fix some notation: let q be a prime power and let n,k,t ∈ n be positive integers, such that k,t < n. we will denote by 1n the n×n identity matrix. for an m×n matrix a and a set s ⊆{1, ...,n} of size k, we denote by as the m×k matrix consisting of the columns of a indexed by s. for a set s ⊆ {1, ...,n} of size k, we denote by fnq (s) the vectors in fnq having support in s. the projection of x ∈ fnq (s) to fkq is then canonical and denoted by πs(x). on the other hand we denote by σs the canonical embedding of a vector x ∈ fkq into fnq (s), where s ⊆{1, ...,n} is again of size k. for an [n,k] linear code c over fq we denote by h a parity check matrix of size (n−k) ×n and by g a generator matrix of size k ×n. we denote the hamming weight of a vector x ∈ fnq , by w(x). the corrupted codeword c ∈ fnq is given by c = mg + e, where m ∈ fkq is the message and e ∈ fnq is the error vector. the syndrome of c is then defined as s = hc> and coincides with the syndrome of the error vector, since hc> = h(mg + e)> = hg>m> + he> = he>. 3.2. ball-collision algorithm over the binary field in what follows we describe the ball-collision algorithm over the binary proposed in [8] by bernstein, lange and peters. 3.3. concepts there are a few concepts for computing the complexity of the ball-collision algorithm introduced in [8] that we will present and generalize to arbitrary finite fields beforehand. in general the complexity of an isd attack consists of the cost of one iteration times the expected number of iterations. the cost in the following refers to operations, i.e. additions or multiplications, over the given field. 197 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 algorithm 1 ball-collision over the binary input: the (n−k)×n parity check matrix h, the syndrome s ∈ fn−k2 and the positive integers v1, v2, w1, w2, k1, k2, `1, `2 ∈ z, such that k = k1 + k2, vi ≤ ki, wi ≤ `i and t−v1 −v2 −w1 −w2 ≤ n−k − `1 − `2. output: e ∈ fn2 with he> = s and w(e) = t. 1: choose i ⊆{1, ...,n} a uniform random information set of size k. 2: choose a uniform random partition of i into disjoint sets x1 and x2 of size k1 and k2 = k−k1 respectively. 3: choose uniform random partition of y = {1, ...,n} \ i into disjoint sets y1,y2 and y3 of sizes `1,`2 and `3 = n−k − `1 − `2. 4: find an invertible matrix u ∈ f(n−k)×(n−k)2 such that (uh)y = 1n−k and (uh)i = ( a1 a2 ) , where a1 ∈ f(`1+`2)×k2 and a2 ∈ f `3×k 2 . 5: compute us = ( s1 s2 ) with s1 ∈ f`1+`22 and s2 ∈ f `3 2 . 6: compute the set s consisting of all triples (a1(πi(x1)) + πy1∪y2(y1),x1,y1), where x1 ∈ f n 2 (x1), w(x1) = v1 and y1 ∈ fn2 (y1), w(y1) = w1. 7: compute the set t consisting of all triples (a1πi(x2)+πy1∪y2(y2)+s1,x2,y2), where x2 ∈ f n 2 (x2), w(x2) = v2 and y2 ∈ fn2 (y2), w(y2) = w2 . 8: for each (a,x1,y1) ∈ s do 9: for each (a,x2,y2) ∈ t do 10: if w(a2(πi(x1 + x2)) + s2) = t−v1 −v2 −w1 −w2: then output: e = x1 + y1 + x2 + y2 + σy3(a2(πi(x1 + x2)) + s2). 11: else start over with step 1 and a new selection of i. i) the success probability over the binary is usually given by having chosen the correct weight distribution of the error vector. for example, if one assumes that the error vector has weight v in the information set and t−v outside the information set, the success probability results in( k v )( n−k t−v )( n t )−1 . this will not change over fq, since the scalar multiples will cross out:( k v ) (q − 1)v ( n−k t−v ) (q − 1)t−v( n t ) (q − 1)t = ( k v )( n−k t−v )( n t )−1 . ii) the concept of intermediate sums is important whenever one wants to compute something for all vectors in a certain space. for example we are given a k×n matrix a and want to compute ax> for all x ∈ fn2 , of weight t. this would usually cost k times t−1 additions and t multiplications, for each x ∈ fn2 . but if we first compute ax>, where x has weight one, this only outputs the corresponding column of a and has no cost. from there we can compute the sums of two columns of a, there are ( n 2 ) many of these sums and each one costs k additions. from there we can compute all sums of three columns of a, which are ( n 3 ) many and using the sums of two columns we have already computed, means we only need to add one more column costing k additions. proceeding in this way, until one reaches the weight t, to compute ax> for all x ∈ fn2 , of weight t costs k ·(l(n,t)−n) additions, where l(n,t) = t∑ i=1 ( n i ) . this changes slightly over a general finite field. as a first step one computes ax> for all x ∈ fnq , of weight 1. hence this step is no longer for free, but rather means computing aλ for all λ ∈ f?q, 198 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 costing (q − 1)kn multiplications. from there on one computes the sum of two multiples of the columns, there are ( n 2 ) (q−1)2 many and each sum costs k additions. hence proceeding in the same manner the cost turns out to be (q−1)kn multiplications and (l̄(n,t)−n(q−1))k additions, where l̄(n,t) = t∑ i=1 ( n i ) (q − 1)i. iii) the next concept called early abort is also important whenever a computation is done while checking the weight of the result. for example one wants to compute x + y, where x,y ∈ fn2 , which usually costs n additions, but we only proceed in the algorithm if w(x + y) = t. hence we compute and check the weight simultaneously and if the weight of the partial solution exceeds t one does not need to continue. over the binary one expects a randomly chosen bit to have weight 1 with probability 1 2 , hence after 2t we should reach the wanted weight t, and after 2(t + 1) we should exceed the weight t. hence on average we expect to compute only 2(t + 1) many bits of the solution, before we can abort. over fq, we expect a randomly chosen element to have weight 1 with probability q−1q , therefore we need to compute q q−1 (t + 1) many entries of the solution before we can abort. iv) an important step in the ball-collision algorithm is to check for a collision, i.e. if ax> = by> one continues, where again a,b ∈ fk×n2 and x,y are living in some sets s and t respectively. there are | s | · | t | many choices for (x,y), assuming that they are distributed uniformly over fn2 , then on average one expects the number of collisions to be | s | · | t | 2−n. similarly over fq the number of collisions will be | s | · | t | q−n. 4. generalization of the ball-collision algorithm in this section we generalize the ball-collision algorithm from the binary case [8] to a general finite field. again, as in the binary case, the idea of the algorithm is to solve uhe> = us instead of he> = s, where an invertible u is chosen such that (up to permutation) uh = ( a 1n−k ) and us = ( s1 s2 ) with s1 ∈ f`1+`2q , s2 ∈ f`3q . we are therefore looking for a vector e ∈ fnq fulfilling uhe> = ( a1 1`1+`2 0 a2 0 1`3 )e1e2 e3   = (s1 s2 ) , with e1 ∈ fkq, e2 ∈ f`1+`2q , e3 ∈ f`3q . this leads to the following system of equations: a1e1 + e2 = s1, a2e1 + e3 = s2. the algorithm solves the above by finding e1 = πi (x1 + x2), e2 = πy1∪y2 (y1 + y2), e3 = −a2(πi (x1 + x2)) + s2, such that a1(πi (x1)) + πy1∪y2 (y1) = s1 −a1(πi (x2)) −πy1∪y2 (y2). 199 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 algorithm 2 ball-collision over fq input: the (n−k)×n parity check matrix h, the syndrome s ∈ fn−kq and the positive integers v1, v2, w1, w2, k1, k2, `1, `2 ∈ z, such that k = k1 + k2, vi ≤ ki, wi ≤ `i and t−v1 −v2 −w1 −w2 ≤ n−k − `1 − `2. output: e ∈ fnq with he> = s and w(e) = t. 1: choose an information set i ⊆{1, ...,n} of h of size k. 2: partition i into two disjoint subsets x1 and x2 of size k1 and k2 = k −k1 respectively. 3: partition y = {1, ...,n}\i into disjoint subsets y1 of size `1, y2 of size `2 and y3 of size `3 = n−k − `1 − `2. 4: find an invertible matrix u ∈ f(n−k)×(n−k)q , such that (uh)y = 1n−k and (uh)i = ( a1 a2 ) , where a1 ∈ f(`1+`2)×kq and a2 ∈ f`3×kq . 5: compute us = ( s1 s2 ) , where s1 ∈ f`1+`2q and s2 ∈ f`3q . 6: compute the following set: s ={(a1(πi(x1)) + πy1∪y2(y1),x1,y1) | x1 ∈ fnq (x1),w(x1) = v1,y1 ∈ f n q (y1),w(y1) = w1} 7: compute the following set: t ={(−a1(πi(x2)) + s1 −πy1∪y2(y2),x2,y2) | x2 ∈ fnq (x2),w(x2) = v2,y2 ∈ f n q (y2),w(y2) = w2} 8: for (a,x1,y1) ∈ s do 9: for (a,x2,y2) ∈ t do 10: if w(−a2(πi(x1 + x2)) + s2) = t−v1 −v2 −w1 −w2 then output: e = x1 + x2 + y1 + y2 + σy3(−a2(πi(x1 + x2)) + s2) 11: else go to step 1 and choose new information set i. this last condition is fulfilled by the collision between s and t in step 9. observe that for q = 2 the above algorithm is equivalent to the one proposed over the binary. we hence did not change it in its substantial form. we now want to prove that the ball-collision algorithm over fq works, i.e. that it returns any vector e of the desired form, if it exists. for this we follow the idea of [8]. theorem 4.1. the ball-collision algorithm over fq finds any vector e that fulfills uhe> = us and is of the desired form, i.e., e has v1,v2,w1,w2 and t−v1 −v2 −w1 −w2 nonzero entries in x1,x2,y1,y2 and y3 respectively. proof. clearly the output e is of the desired form: • x1 is of weight v1 and in fnq (x1), • x2 is of weight v2 and in fnq (x2), • y1 is of weight w1 and in fnq (y1), • y2 is of weight w2 and in fnq (y2), • w(−a2(πi (x1 + x2)) + s2) = w(a2(πi (x1 + x2))−s2) = t−v1 −v2 −w1 −w2 and it lies in fnq (y3). as the above subspaces do not intersect, w(e) can be calculated by adding up the weights of each of them. hence w(e) = t and each of the subspaces has the desired weight distribution by definition. 200 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 it remains to prove that uhe> = us. let us write each of the subspaces fnq (i),f n q (y1 ∪ y2) and fnq (y3) separately. uhe> = ( a1 1`1+`2 0 a2 0 1`3 ) πi (x1 + x2)πy1∪y2 (y1 + y2) −a2(πi (x1 + x2)) + s2   = ( a1(πi (x1 + x2)) + πy1∪y2 (y1 + y2) a2(πi (x1 + x2)) −a2(πi (x1 + x2)) + s2 ) = ( a1(πi (x1 + x2)) + πy1∪y2 (y1 + y2) s2 ) . and we know that a1(πi (x1 + x2)) + πy1∪y2 (y1 + y2) = s1 by the collision of s and t in step 9. we now want to prove that the algorithm returns each of the above vectors such that he> = s under the assumption, that we worked with a correct partitioning into x1,x2,y1,y2,y3. we do that by checking whether the algorithm considers all possible combinations and does not exclude any possible solution. u is invertible and hence does not exclude any solution when multiplied by h and s. in step 6, where we build the sets s and t, we go through all possible error vectors x1,x2,y1 and y2, which have the desired weight distribution. there are only two steps in the algorithm, where we exclude certain vectors: 1. when we only keep the collisions between s and t in step 9. but this is justified as a1e1 +e2 = s1, i.e. a1(πi (x1)) + πy1∪y2 (y1) = −a1(πi (x2)) + s1 −πy1∪y2 (y2) needs to be satisfied. 2. when we check whether w(−a2(πi (x1 + x2)) + s2) = t−v1−v2−w1−w2. but also this is justified as e3 ∈ f`3q needs this weight to complete the weight of e to be t. hence we consider all possible error vectors that are of the given weight distribution and satisfy uhe> = us. 5. complexity analysis in this section we want to analyze the complexity of the extended ball-collision algorithm over fq. since the cost will be given in operations over fq, we will denote by m the multiplications needed and by a the amount of additions. note that one addition over fq costs log2(q) bit operations and one multiplication over fq costs log2(q) 2 bit operations, observe that one could also use the following speed up [29] such that multiplication costs log2(q) log2(log2(q)) log2(log2(log2(q))) bit operations. success probability of one iteration we have the same success probability over fq as over f2, as observed in section 3.3, hence one iteration succeeds with a probability of ( n t )−1( `3 t−v1 −v2 −w1 −w2 )( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ) . (1) 201 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 cost of one iteration 1. in step 4 of the algorithm, one uses gaussian elimination to find an invertible matrix u, bringing h into systematic form, since we will also need to compute us we will directly perform gaussian elimination on the matrix ( h | s ) , where we adjoined the vector s as a column to h. a broad estimate of the cost for this step is (n−k)2(n + 1)(a + m). 2. to build the set s we want to use the concept of intermediate sums over fq described before. hence to compute a1(πi (x1)), for all x1 ∈ fnq (x1) we need (q − 1)(`1 + `2)k1 multiplications and (l̄(k1,v1) − k1(q − 1))(`1 + `2) additions. after having computed a1(πi (x1)), we add πy1∪y2 (y1) using intermediate sums. this costs l̄(`1,w1) additions for each of the x1 ∈ fnq (x1), which are( k1 v1 ) (q − 1)v1 many. hence resulting in a total cost of (q − 1)(`1 + `2)k1m + (l̄(k1,v1) −k1(q − 1))(`1 + `2)a + ( k1 v1 ) l̄(`1,w1)(q − 1)v1a. 3. to build the set t we proceed similarly, the only difference being that s1 needs to be added to the first step of the intermediate sums over fq, hence adding a cost of (`1 + `2)(q−1)k2 additions. the total cost of this step is hence given by (q − 1)(`1 + `2)k2(m + a) + (l̄(k2,v2 −k2(q − 1)))(`1 + `2)a + ( k2 v2 ) l̄(`2,w2)(q − 1)v2a. 4. in step 9, when checking for collisions between s and t, we want to calculate the number of collisions we can expect on average. the elements in s and t are all of length `1 + `2 and hence there is a total of q`1+`2 possible elements. s has ( k1 v1 )( `1 w1 ) (q − 1)v1+w1 many elements and t has( k2 v2 )( `2 w2 ) (q − 1)v2+w2 many elements, we therefore get that the expected number of collisions is( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ) (q − 1)v1+v2+w1+w2 q`1+`2 . 5. for each collision we have, we check whether w(−a2(πi (x1 + x2)) + s2) = t − v1 − v2 − w1 − w2 is satisfied. for this we will use the method of early abort: to compute one bit of the result costs (v1 + v2 + 1) additions and (v1 + v2) multiplications, hence this step costs on average q q − 1 (t−v1 −v2 −w1 −w2 + 1) ((v1 + v2 + 1)a + (v1 + v2)m) . hence the total cost of one iteration is given by (n−k)2(n + 1)(a + m) + (`1 + `2)[(q − 1)((k1 + k2)m + k2a) + (l̄(k1,v1) −k1(q − 1))a + (l̄(k2,v2) −k2(q − 1))a] + ( k1 v1 ) l̄(`1,w1)(q − 1)v1a + ( k2 v2 ) l̄(`2,w2)(q − 1)v2a (2) + ( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ) (q − 1)v1+v2+w1+w2q−(`1+`2) · q q − 1 (t−v1 −v2 −w1 −w2 + 1) ((v1 + v2 + 1)a + (v1 + v2)m) . 202 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 overall cost combining the result from (1) and (2) the overall cost of the ball-collision algorithm over fq then amounts to ( n t )(( `3 t−v1 −v2 −w1 −w2 )( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ))−1 · [(n−k)2(n + 1)(a + m) + (`1 + `2)[(q − 1)((k1 + k2)m + k2a) + (l̄(k1,v1) −k1(q − 1))a + (l̄(k2,v2) −k2(q − 1))a] + ( k1 v1 ) l̄(`1,w1)(q − 1)v1a + ( k2 v2 ) l̄(`2,w2)(q − 1)v2a + ( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ) (q − 1)v1+v2+w1+w2q−(`1+`2) · q q − 1 (t−v1 −v2 −w1 −w2 + 1) ((v1 + v2 + 1)a + (v1 + v2)m)]. 5.1. asymptotic complexity in this subsection we want to find the asymptotic complexity of the ball-collision algorithm over fq. fix real numbers 0 < t < 1/2 and r, with −t logq(t) − (1 −t) logq(1 −t) ≤ 1 −r < 1. we consider codes of large length n, we fix functions k,t : n → n which satisfy limn→∞ t(n)/n = t and limn→∞k(n)/n = r. we fix real numbers v,w,l with 0 ≤ v ≤ r/2, 0 ≤ w ≤ l and 0 ≤ t − 2v − 2w ≤ 1 −r− 2l. we fix the parameters v1,v2,w1,w2,`1,`2,k1,k2 of the ball-collision algorithm over fq such that i) limn→∞ vin = v, ii) limn→∞ win = w, iii) limn→∞ kin = r/2, iv) limn→∞ `in = l, for i ∈ {1, 2}. we use the convention that x logq(x) = 0, for x = 0. in what follows we will use the following asymptotic formula for binomial coefficients: lim n→∞ 1 n logq ( α + o(1)n β + o(1)n ) = α logq(α) −β logq(β) − (α−β) logq(α−β). with this formula we get the following: i) limn→∞ 1n logq ( n t ) = −t logq(t) − (1 −t) logq(1 −t), ii) limn→∞ 1n logq ( ki vi ) = r/2 logq(r/2) −v logq(v ) − (r/2 −v ) logq(r/2 −v ), 203 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 iii) limn→∞ 1n logq ( `i wi ) = l logq(l) −w logq(w) − (l−w) logq(l−w), iv) limn→∞ 1n logq ( n−k−`1−`2 t−v1−v2−w1−w2 ) = (1 −r− 2l) logq(1 −r− 2l) − (t − 2v − 2w) logq(t − 2v − 2w) − (1 −r− 2l−t + 2v + 2w) logq(1 −r− 2l−t + 2v + 2w). success probability we will denote by s(v,w,l) the asymptotic exponent of the success probability: s(v,w,l) = lim n→∞ 1 n logq (( n t )−1( n−k − `1 − `2 t−v1 −v2 −w1 −w2 )( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 )) = t logq(t) + (1 −t) logq(1 −t) + (1 −r− 2l) logq(1 −r− 2l) −(t − 2v − 2w) logq(t − 2v − 2w) −(1 −r− 2l−t + 2v + 2w) logq(1 −r− 2l−t + 2v + 2w) +r logq(r/2) − 2v logq(v ) − (r− 2v ) logq(r/2 −v ) + 2l logq(l) −2w logq(w) − 2(l−w) logq(l−w). cost of one iteration we will denote by c(v,w,l) the asymptotic exponent of the cost of one iteration. c(v,w,l) = lim n→∞ 1 n logq (( k1 v1 ) (q − 1)v1 + ( k2 v2 ) (q − 1)v2 + ( k1 v1 )( `1 w1 ) (q − 1)v1+w1 + ( k2 v2 )( `2 w2 ) (q − 1)v2+w2 + ( k1 v1 )( k2 v2 )( `1 w1 )( `2 w2 ) (q − 1)v1+v2+w1+w2q−`1−`2 ) = max { logq(q − 1)v + r/2 logq(r/2) −v logq(v ) − (r/2 −v ) logq(r/2 −v ), logq(q − 1)(v + w) + r/2 logq(r/2) −v logq(v ) − (r/2 −v ) logq(r/2 −v ) +l logq(l) −w logq(w) − (l−w) logq(l−w), logq(q − 1)(2v + 2w) − 2l + r logq(r/2) − 2v logq(v ) −(r− 2v ) logq(r/2 −v ) + 2l logq(l) − 2w logq(w) − (2l− 2w) logq(l−w) } . overall cost the overall asymptotic cost exponent of the ball-collision algorithm over fq is given by the difference of c(v,w,l) and s(v,w,l): d(v,w,l) = max { logq(q − 1)v −r/2 logq(r/2) + v logq(v ) + (r/2 −v ) logq(r/2 −v ) −2l logq(l) + 2w logq(w) + 2(l−w) logq(l−w), logq(q − 1)(v + w) −r/2 logq(r/2) + v logq(v ) + (r/2 −v ) logq(r/2 −v ) −l logq(l) + w logq(w) + (l−w) logq(l−w), logq(q − 1)(2v + 2w) − 2l } −t logq(t) − (1 −t) logq(1 −t) (1 −r− 2l) logq(1 −r− 2l) + (t − 2v − 2w) logq(t − 2v − 2w) +(1 −r− 2l−t + 2v + 2w) logq(1 −r− 2l−t + 2v + 2w). the asymptotic complexity is then given by qd(v,w,l)n+o(n). asymptotically, we assume that the code attains the gilbert-varshamov bound, i.e. the code rate 204 c. interlando et al. / j. algebra comb. discrete appl. 7(2) (2020) 195–207 r = k/n and the distance d = d/n relate via: r = 1 + d logq(d) + (1 −d) logq(1 −d) −d logq(q − 1). (3) in order to compute the asymptotic complexity of half-distance decoding (i.e. t = d/2) for a fixed rate r, we performed a numerical optimization of the parameters v,w and l such that the overall cost d(v,w,l) is minimized subject to the following constraints: 0 ≤ v ≤ r/2, 0 ≤ w ≤ l and 0 ≤ t − 2v − 2w ≤ 1 −r− 2l. let f(q,r) be the exponent of the optimized asymptotic complexity. the asymptotic complexity of half-distance decoding at rate r over fq is then given by qf (q,r)n+o(n). in table 1, the values refer to the exponent of the worst-case complexity of distinct algorithms, i.e. f(q,rw) where rw = argmax0<r<1 (f(q,r)). it compares peter’s generalization of stern’s algorithm to fq, hirose ’s generalization of stern’s algorithm using may-ozerov’s nearest neighbor algorithm (mo) to fq, gueye et al. generalization of the algorithm of bjmm using mo to fq and the generalization of the ball-collision algorithm to fq. table 1: comparison of the asymptotic complexities over fq. the values q−stern and q−stern-mo are from [16, table 1], and q−bjmm-mo are from [15, table3]. q q−stern q−stern-mo q−bjmm-mo q−ball-collision 2 0.05563 0.05498 0.04730 0.055573 3 0.05217 0.05242 0.04427 0.052145 4 0.04987 0.05032 0.04294 0.049846 5 0.04815 0.04864 0.03955 0.048140 7 0.04571 0.04614 0.03706 0.045697 8 0.04478 0.04519 0.03593 0.044770 11 0.04266 0.04299 0.03335 0.042656 we can observe that the ball-collision algorithm over fq outperforms peter’s generalization of stern’s algorithm to fq and hirose’s isd algorithm over fq, for all q ≥ 3. like in the binary case, the ball-collision algorithm does not outperform the generalization of gueye et al. of the bjmm algorithm using mo to fq. acknowledgment: the fourth author is thankful to the swiss national science foundation grant number 169510. references [1] a. al jabri, a statistical decoding algorithm for general linear block codes, in ima international conference on cryptography and coding, pages 1–8, springer, 2001. 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γ-lie structures in γ-prime gamma rings with derivations research article okan arslan1∗, hatice kandamar1∗∗ 1. adnan menderes university, faculty of arts and sciences, department of mathematics, aydın, turkey abstract: let m be a γ-prime weak nobusawa γ-ring and d 6= 0 be a k-derivation of m such that k (γ) = 0 and u be a γ-lie ideal of m. in this paper, we introduce definitions of γ-subring, γ-ideal, γ-prime γ-ring and γ-lie ideal of m and prove that if u * cγ, charm 6= 2 and d3 6= 0, then the γ-subring generated by d(u) contains a nonzero ideal of m. we also prove that if [u,d(u)]γ ∈ cγ for all u ∈ u, then u is contained in the γ-center of m when charm 6= 2 or 3. and if [u,d(u)]γ ∈ cγ for all u ∈ u and u is also a γ-subring, then u is γ-commutative when charm = 2. 2010 msc: 16n60, 16w25, 16y99 keywords: gamma ring, γ-prime gamma ring, γ-lie ideal, k-derivation, γ-commutativity 1. preliminaries let m and γ be additive abelian groups. m is said to be a γ-ring in the sense of barnes[2] if there exists a mapping m × γ ×m → m satisfying these two conditions for all a,b,c ∈ m, α,β ∈ γ: (1) (a + b) αc = aαc + bαc a(α + β)c = aαc + aβc aα (b + c) = aαb + aαc (2) (aαb) βc = aα (bβc) in addition, if there exists a mapping γ × m × γ → γ such that the following axioms hold for all a,b,c ∈ m, α,β ∈ γ: (3) (aαb) βc = a (αbβ) c ∗ e-mail: oarslan@outlook.com.tr ∗∗ e-mail: hkandamar@adu.edu.tr 25 γ-lie structures with derivations (4) aαb = 0 for all a,b ∈ m implies α = 0, where α ∈ γ then m is called a γ-ring in the sense of nobusawa[10]. if a γ-ring m in the sense of barnes satisfies only the condition (3), then it is called weak nobusawa γ-ring[9]. let m be a γ-ring in the sense of barnes. then m is said to be a prime gamma ring if aγmγb = 0 with a,b ∈ m implies either a = 0 or b = 0[2]. it is also defined in [2] that m is a completely prime gamma ring if aγb = 0 with a,b ∈ m implies either a = 0 or b = 0. for a subset u of m and γ ∈ γ, the set cγ (u) = {a ∈ m | aγu = uγa, ∀u ∈ u} and the set cγ = {a ∈ m | aγm = mγa, ∀m ∈ m} are called γ-center of the subset u and γ-center of m respectively. in 2000, kandamar[7] firstly introduced the notion of a k-derivation for a gamma ring in the sense of barnes and proved some of its properties and commutativity conditions for nobusawa gamma rings. commutativity conditions with derivations for a gamma ring has been investigated by a number of authors. in [8], khan, chaudhry and javaid proved that if m is a prime gamma ring (in the sense of barnes) of characteristic not 2, i is a nonzero ideal of m and f is a generalized derivation on m, then m is a commutative gamma ring. in [12], suliman and majeed showed a nonzero lie ideal of a 2-torsion-free prime γ-ring m with a nonzero derivation d is central if d(u) is contained in the center of m. in this paper, we define γ-lie ideal for a weak nobusawa gamma ring and show that if u * cγ, charm 6= 2 and d3 6= 0, then the γ-subring generated by d(u) contains a nonzero ideal of m. we also prove that if [u,d(u)]γ ∈ cγ for all u ∈ u, then u ⊆ cγ when charm 6= 2 or 3. and if [u,d(u)]γ ∈ cγ for all u ∈ u and u is also a γ-subring, then u is γ-commutative when charm = 2. 2. γ-lie ideals and derivations now we give some new definitions and make some preliminary remarks we need later. let m be a weak nobusawa γ-ring and 0 6= γ ∈ γ. a subgroup i of m is said to be a γ-subring if xγy ∈ i for all x,y ∈ i. a subgroup a of m is said to be a γ-left ideal(resp. γ-right ideal) if mγa ∈ a(resp. aγm ∈ a) for all m ∈ m, a ∈ a. if a is both γ-left and γ-right ideal then a is called a γ-ideal of m. m is called γ-commutative gamma ring if xγy = yγx for all x,y ∈ m. we say that the additive subgroup u of m is said to be a γ-lie ideal of m if [u,m]γ ⊆ u. we also say that if there exists a γ ∈ γ such that aγmγb = 0 with a,b ∈ m implies either a = 0 or b = 0 then m is called a γ-prime gamma ring. an element a of m is called γ-nilpotent if there exists a positive integer n such that anγ := (aγ) na = 0. in what follows, let m be a γ-prime weak nobusawa γ-ring of characteristic not 2, d 6= 0 be a k-derivation of m such that k (γ) = 0 and u be a γ-lie ideal of m unless otherwise specified. lemma 2.1. if a ∈ m γ-commutes with [a,x]γ for all x ∈ m, then a is in the γ-center of m. proof. let x,y ∈ m. therefore, we get [a,x]γ γ [a,y]γ = 0 by hypothesis. replacing y by mγx with m ∈ m, we obtain [a,x]γ γmγ [a,x]γ = 0. hence, a is in the γ-center of m since m is γ-prime. lemma 2.2. suppose that u 6= (0) is both a γ-subring and a γ-lie ideal of m. then either u ⊆ cγ or u contains a nonzero ideal of m. proof. first, suppose that the γ-subring u is not γ-commutative. then, there exists x,y ∈ u such that [x,y]γ 6= 0. since u is a γ-lie ideal, [x,y]γ γm ⊆ u. hence, [ [x,y]γ γa,b ] γ ∈ u for all a,b ∈ m. expanding this, we get bγ [x,y]γ γa ∈ u leading to mγ [x,y]γ γm ⊆ u. moreover, mγ [x,y]γ γm 6= 0. we have shown that the result is correct if the γ-subring u is not commutative. 26 o. arslan, h. kandamar now suppose that u is γ-commutative. then [ a, [a,x]γ ] γ = 0 for a ∈ u and x ∈ m. therefore, we have u ⊆ cγ by lemma 2.1. lemma 2.3. let u be a γ-lie ideal of m and u * cγ. then there exists a nonzero ideal k of m such that [k,m]γ ⊂ u but [k,m]γ * cγ. proof. since u * cγ, it follows from lemma 2.1 that [u,u]γ 6= 0. let k = mγ [u,u]γ γm. then it is clear that k is a nonzero ideal of m. let t (u) = { x ∈ m : [x,m]γ ⊆ u } . then, it can be shown that u ⊆ t (u) and t (u) is both a γ-subring and a γ-lie ideal of m. let u,v ∈ u such that [u,v]γ 6= 0. replacing v by vγm with m ∈ m, we obtain [u,v]γ γm ⊆ t (u). hence, [ [u,v]γ γm,n ] γ ∈ t (u) for all m,n ∈ m. expanding this, we get k ⊆ t (u). therefore, we have shown that [k,m]γ ⊆ u. suppose that [k,m]γ ⊆ cγ. then, [ x, [x,m]γ ] γ = 0 for all x ∈ k, m ∈ m. let y ∈ m. since k ⊆ cγ by lemma 2.1, we have [y,nγkγm]γ = 0 for all m,n ∈ m,k ∈ k which leads to y ∈ cγ. but this contradicts with u * cγ. lemma 2.4. let u ∈ m. if a ∈ cγ and aγu ∈ cγ, then a = 0 or u ∈ cγ. proof. suppose that a 6= 0. since [aγu,m]γ = 0 for all m ∈ m, we get aγ [u,m]γ = 0. replacing m by mγn with n ∈ m, we obtain [u,n]γ = 0 for all n ∈ m. this gives that u ∈ cγ. lemma 2.5. if u is a γ-lie ideal of m and u * cγ, then cγ(u) = cγ. proof. it is clear that cγ (u) is both a γ-subring and γ-lie ideal of m. we claim that cγ (u) cannot contain a nonzero ideal of m. suppose k is a nonzero ideal of m which is contained in cγ (u). then, it is clear that [u,kγm]γ = 0 for all u ∈ u, k ∈ k and m ∈ m. expanding this, we get kγ [u,m]γ = 0. replacing k by kγm with m ∈ m, we obtain u ∈ cγ which leads to a contradiction. hence, cγ (u) ⊆ cγ by lemma 2.2. lemma 2.6. if u is a γ-lie ideal of m, then cγ ( [u,u]γ ) = cγ (u). proof. first, suppose that [u,u]γ * cγ. since [u,u]γ is a γ-lie ideal of m, we have cγ ( [u,u]γ ) = cγ by lemma 2.5. now, suppose that [u,u]γ ⊆ cγ. let a = [ u, [u,x]γ ] γ for u ∈ u and x ∈ m. since a ∈ cγ and aγu ∈ cγ, we write a = 0 or u ∈ cγ by lemma 2.4. if a = 0, we have u ∈ cγ by lemma 2.1. hence, we get u ⊆ cγ. thus, cγ ( [u,u]γ ) = cγ (u). lemma 2.7. let u be a γ-lie ideal of m and u * cγ. if aγuγb = 0 for a,b ∈ m, then a = 0 or b = 0. proof. there exists a nonzero ideal k of m such that [k,m]γ ⊂ u but [k,m]γ * cγ by lemma 2.3. thus, aγ [kγaγu,m]γ γb = 0 for u ∈ u, k ∈ k and m ∈ m by hypothesis. expanding this, we get aγkγaγmγuγb = 0. since m is γ-prime, we obtain aγkγa = 0 or uγb = 0. let aγkγa = 0. if a 6= 0, then we have k = 0 which is a contradiction. now, let uγb = 0. therefore, [u,m]γ γb = 0 for all u ∈ u and m ∈ m. hence, we get uγmγb = 0 which means b = 0. lemma 2.8. if d is a k-derivation of m such that k (γ) = 0 and d2 = 0, then d = 0. proof. since d2 (xγy) = 0 for all x,y ∈ m, we have d (x) γd (y) = 0. replacing y by mγx with m ∈ m, we get d (x) γmγd (x) = 0 for all x ∈ m. thus, d = 0 since m is γ-prime. 27 γ-lie structures with derivations lemma 2.9. if d 6= 0 is a k-derivation of m such that k (γ) = 0, then cγ (d (m)) = cγ. proof. let a ∈ cγ (d (m)) and suppose a /∈ cγ. thus, [a,d (xγy)]γ = 0 for all x,y ∈ m. expanding this, we get d (x) γ [a,y]γ + [a,x]γ γd (y) = 0. if y ∈ m γ-commutes with a, then [a,y]γ = 0. so the last equation reduces to [a,x]γ γd (y) = 0 for all x ∈ m. then, d (y) = 0 since a /∈ cγ. indeed, if d (y) 6= 0, we get a ∈ cγ. but this is a contradiction. therefore, d (y) = 0 for all y ∈ cγ (a). thus, d = 0 by lemma 2.8 which contradicts with the assumption. lemma 2.10. let d 6= 0 be a k-derivation of m such that k (γ) = 0 and u be a γ-lie ideal of m. (i) if d (u) = 0, then u ⊆ cγ. (ii) if d (u) ⊆ cγ then u ⊆ cγ. proof. (i) let u ∈ u and x ∈ m. since d (u) = d ( [u,x]γ ) = 0 by hypothesis, we get [u,d (x)]γ = 0 for all x ∈ m. therefore, u centralizes d (m). then, we get u ⊆ cγ by lemma 2.9. (ii) suppose that u * cγ. then, v = [u,u]γ * cγ by proof of lemma 2.6. since d ( [u,v]γ ) = 0 for all u,v ∈ u, we get d (v ) = 0. it follows that v ⊆ cγ by (i). but this is a contradiction. lemma 2.11. let d be a k-derivation of m such that k (γ) = 0 and u be a γ-lie ideal of m such that u * cγ. if tγd (u) = 0 (or d (u) γt = 0) for t ∈ m, then t = 0. proof. let u ∈ u and x ∈ m. using the fact [u,x]γ γu = [u,xγu]γ ∈ u, we have tγd ( [u,x]γ γu ) = 0. expanding this, we get tγ [u,x]γ γd (u) = 0 for all x ∈ m andu ∈ u. replacing x by d (v) γy with v ∈ u, y ∈ m, we obtain tγuγd (v) = 0 for all v,u ∈ u since tγd (u) = 0 and m is γ-prime. hence, t = 0 by lemma 2.7. theorem 2.12. let d 6= 0 be a k-derivation of m such that k (γ) = 0. if u is a γ-lie ideal of m such that d2 (u) = 0, then u ⊆ cγ. proof. suppose that u * cγ. by proof of lemma 2.6, we have v = [u,u]γ * cγ. there exists a nonzero ideal k of m such that [k,m]γ ⊂ u but [k,m]γ * cγ by lemma 2.3. let y ∈ m, t ∈ [k,m]γ and u ∈ v . if w := d (u), then d (w) = 0. by hypothesis, d2 ( [tγw,y]γ ) = 0. expanding this, we have d (t) γd ( [w,y]γ ) = 0 for all t ∈ [k,m]γ , y ∈ m, w ∈ d (v ). since [k,m]γ is a γ-lie ideal of m and [k,m]γ * cγ, we have d ( [d (v ) ,m]γ ) = 0 by lemma 2.11. expanding last equation, we conclude that [d (u) ,d (x)]γ = 0 for all x ∈ m,u ∈ v which means d (v ) ⊆ cγ (d (m)). therefore, we have v ⊆ cγ by lemma 2.9 and by lemma 2.10. but this is a contradiction. theorem 2.13. let d 6= 0 be a k-derivation of m such that k (γ) = 0. if u is a γ-lie ideal of m such that u * cγ, then cγ (d (u)) = cγ. proof. let a ∈ cγ (d (u)) and suppose that a /∈ cγ. we have v = [u,u]γ * cγ by proof of lemma 2.6. since d (v ) ⊆ u and a ∈ cγ (d (u)) we get aγd2 (u) = d2 (u) γa and aγd (u) = d (u) γa. now, applying given derivation d to last equation gives d (a) ∈ cγ (d (v )). since a ∈ cγ (d (u)), u ∈ v and v is a γ-lie ideal, we have [d (a) ,u]γ = d ( [a,u]γ ) ∈ d (v ). it follows that [d (a) ,v ]γ = 0 which means d (a) ∈ cγ (v ). therefore, d (a) ∈ cγ (v ) = cγ by lemma 2.5. using same process for the element aγa gives d (aγa) = 2aγd (a) ∈ cγ since aγa ∈ cγ (d (u)). thus, d (a) = 0 by lemma 2.4. therefore, if d (b) 6= 0 for any b ∈ cγ (d (u)) we have b ∈ cγ. so we get 28 o. arslan, h. kandamar a + b ∈ cγ since d (a + b) = d (b) 6= 0. then we have a ∈ cγ. but this is a contradiction. consequently, when we suppose cγ (d (u)) * cγ, we are forced to d (a) = 0 for all a ∈ cγ (d (u)). let w = {x ∈ m | d (x) = 0}. then we have cγ (d (u)) ⊆ w . moreover, d ( [a,u]γ ) = 0 for any a ∈ cγ (d (u)) and u ∈ u. there exists a nonzero ideal k of m such that [k,m]γ ⊂ u but [k,m]γ * cγ by lemma 2.3. if t ∈ [k,m]γ ⊂ u ∩ k, then tγa ∈ k. thus, [ a,d ( [tγa,u]γ )] γ = 0. expanding this, we get d (t) γ [ a, [a,u]γ ] γ = 0 for all t ∈ [k,m]γ , u ∈ u. hence, [ a, [a,u]γ ] γ = 0 by lemma 2.11. since u * cγ, we have a ∈ cγ (u). therefore, a ∈ cγ by lemma 2.5. but this is a contradiction. lemma 2.14. if d3 6= 0 and u * cγ, then d (m) the γ-subring generated by d (m) contains a nonzero γ-ideal of m. proof. since d2 (d (m)) 6= 0, we have y ∈ d (m) ⊆ d (m) such that d2 (y) 6= 0. thus, we get mγd (y) ⊆ d (m) since d (xγy) and d (x) γy in d (m) for all x ∈ m. similarly, d (y) γm ⊆ d (m). if we act d to the element aγd (y) γb for a,b ∈ m, we get aγd2 (y) γb ∈ d (m) by above, that is to say mγd2 (y) γm ⊆ d (m). we also have mγd2 (y) ⊆ d (m) and d2 (y) γm ⊆ d (m) by above. therefore, the γ-ideal of m generated by d2 (y) 6= 0 contained in d (m). lemma 2.15. let d3 6= 0, u * cγ and v = [u,u]γ. if d (v ) contains a nonzero left ideal λ of m and a nonzero right ideal ρ of m, then d (u) contains a nonzero ideal of m. proof. since d (v ) ⊆ u, we have d (d (v )) ⊆ d (u). let a ∈ λ ⊆ d (v ) and x ∈ m. thus, d (xγa) ∈ d (u). expanding this, we get xγd (a) ∈ d (u) for all x ∈ m and a ∈ λ. therefore, we have mγd (λ) ⊆ d (u). similarly, d (ρ) γm ⊆ d (u). let a ∈ λ and u ∈ v . if we act d to the element [a,u]γ, we get d (a) γu ∈ d (u) by above, that is to say d (λ) γv ⊆ d (u). similarly, v γd (ρ) ⊆ d (u). let i = λγv γρ. then by lemma 2.7, i is a nonzero ideal of m. moreover, d (i) ⊆ d (u). by lemma 2.14, d (i) contains a nonzero γ-ideal k of i since d3 6= 0 and i is γ-prime. let s := λγkγρ. then s is an ideal of m which is contained in d (u). lemma 2.16. let 0 6= i < m and u * cγ. if d (u) does not contain a nonzero right ideal(or left ideal) of m and [c,i]γ ⊆ d (u) then c ∈ cγ. proof. let t ∈ d (u) and i ∈ i. then [c,tγi]γ ∈ d (u) by hypothesis. expanding this, we get [c,d (u)]γ γi ⊆ d (u). but, since d (u) does not contain a nonzero right ideal of m, we get [c,d (u)]γ γi = 0. thus, [c,d (u)]γ = 0 since 0 6= i < m and m is a γ-prime gamma ring. then by theorem 2.13, we get c ∈ cγ (d (u)) = cγ. lemma 2.17. let u * cγ, v = [u,u]γ and w = [v,v ]γ. if d 2 (u) γd2 (u) = 0 then d3 (w) = 0. proof. by proof of lemma 2.6, v and w are not contained in cγ since u * cγ. also, we have d (w) ⊆ v and d2 (w) ⊆ d (v ) ⊆ u. if u ∈ u, v ∈ v and w ∈ w , then we have d2 (u) γd2 ([ d (v) ,d2 (w) γt ] γ ) = 0 for any t ∈ u. expanding this, we get d2 (u) γd (v) γ ( d4 (w) γt + 2d3 (w) γd (t) ) = 0 (1) by hypothesis. in (1), if we choose t ∈ d (v ) ⊆ u, it follows d2 (u) γd (v) γd4 (w) γt = 0 for such t. thus, we have d2 (u) γd (v) γd4 (w) = 0 by lemma 2.11. then we get from (1) that d2 (u) γd (v) γd3 (w) γd (t) = 29 γ-lie structures with derivations 0 for all t ∈ u. by lemma 2.11, we conclude that d2 (u) γd (v) γd3 (w) = 0 for all u ∈ u, v ∈ v and w ∈ w . similarly, we have d3 (w) γd (v) γd2 (u) = 0 by reversing sides. by hypothesis, d2 (d (t)) γd2 ( [v,d (w)]γ ) = 0 for w,t ∈ w and v ∈ v . expanding this, we get d3 (t) γvγd3 (w) = 0 that is to say d3 (t) γv γd3 (w) = 0 for all w,t ∈ w . it follows that d3 (w) = 0 by lemma 2.7. lemma 2.18. if u * cγ and d3 (u) = 0 then d3 = 0. proof. let u ∈ u and m ∈ m. then we have d3 ( [u,m]γ ) = 0. expanding this, we get 3 [ d2 (u) ,d (m) ] γ + 3 [ d (u) ,d2 (m) ] γ + [ u,d3 (m) ] γ = 0. (2) let v = [u,u]γ and w = [v,v ]γ. in (2), replacing u by d 2 (w) with w ∈ w, we get [ d2 (w) ,d3 (m) ] γ = 0 by hypothesis. again replacing u by d (w) and m by d (m) in (2), we obtain [ d (w) ,d4 (m) ] γ = 0. by proof of lemma 2.6, w * cγ. thus, by theorem 2.13, cγ (d (w)) = cγ. since [ d (w) ,d4 (m) ] γ = 0 for all w ∈ w and m ∈ m, it follows d4 (m) ⊆ cγ. by hypothesis, d4 ( [u,m]γ ) = 0 for u ∈ u and m ∈ m. expanding this, we get 6 [ d2 (u) ,d2 (m) ] γ + 4 [ d (u) ,d3 (m) ] γ = 0 (3) since d4 (m) ∈ cγ. similarly, expanding the equation d3 ( [u,d (m)]γ ) = 0, we get 3 [ d2 (u) ,d2 (m) ] γ + 3 [ d (u) ,d3 (m) ] γ = 0. (4) combining the equation (4) and the equation (3) we get [ d (u) ,d3 (m) ] γ = 0 for all u ∈ u and m ∈ m. therefore, by theorem 2.13, d3 (m) ⊆ cγ (d (u)) = cγ. hence, we get d3 (m) γd2 (u) ∈ cγ that is to say d3 (m) γd2 (u) ⊆ cγ. suppose that d3 (m) 6= 0. by lemma 2.4, we have d2 (u) ⊆ cγ. since d4 (mγd (u)) ∈ cγ it follows d4 (m) γd (u) ⊆ cγ. since d (u) cannot be contained in cγ by lemma 2.10, we get d4 (m) = 0 by lemma 2.4. so d4 (mγd (u)) = 4d3 (m) γd2 (u) = 0. hence, d3 (m) γd2 (u) = 0. on the other hand, we have d2 (u) 6= 0 by theorem 2.12. since d2 (u) ⊆ cγ and m is γ-prime gamma ring, d3 (m) = 0, that is to say d3 = 0. lemma 2.19. if [u,u]γ ⊆ cγ then u ⊆ cγ. proof. let [u,u]γ = 0. then, we get [ u, [u,x]γ ] γ = 0 for all u ∈ u and x ∈ m by hypothesis. therefore, u ∈ cγ by lemma 2.1. now, let [u,u]γ 6= 0. then, there exist u,v ∈ u such that [u,v]γ 6= 0. let d(x) = [x,v]γ for x ∈ m. then d2(x) = [d(x),v]γ ∈ cγ for all x ∈ m by hypothesis. let a = d(u) and b = d 2(x). therefore, d2(uγx) = 2aγd(x) + bγu ∈ cγ. then we have [u, 2aγd(x) + bγu]γ = 0. expanding this, we get 2aγ [u,d(x)]γ = 0 for all x ∈ m. replacing x by uγv in the last equation, we obtain a 3 γ = 0. therefore, we have a nonzero γ-nilpotent element a in the γ-center of the γ-prime gamma ring m. but this is a contradiction. lemma 2.20. let m be a gamma ring of characteristic not 2 and u be a γ-lie ideal of m. if [u,d(u)]γ ∈ cγ and u2 ∈ u for all u ∈ u, then [u,d(u)]γ = 0. 30 o. arslan, h. kandamar proof. we know that [ u + u2,d(u + u2) ] γ ∈ cγ for all u ∈ u by hypothesis. expanding this, we get 4 [u,d(u)]γ γu ∈ cγ. hence, [u,d(u)]γ γ [u,m]γ = 0 for all u ∈ u and m ∈ m. replacing m by mγx with x ∈ m, we obtain [u,d(u)]γ γmγ [u,x]γ = 0 that leads to [u,d(u)]γ = 0 or [u,x]γ = 0 for all x ∈ m since m is γ-prime gamma ring. lemma 2.21. let u be a γ-lie ideal of m and [u,d(u)]γ ∈ cγ for all u ∈ u. then [ [d(m),u]γ ,u ] γ ∈ cγ for all u ∈ u and m ∈ m. moreover, if [u,d(u)]γ = 0 for all u ∈ u, then [ [d(m),u]γ ,u ] γ = 0 for all u ∈ u and m ∈ m. proof. let u ∈ u and m ∈ m. by hypothesis, [ u + [u,m]γ ,d(u + [u,m]γ) ] γ ∈ cγ. expanding this, we get [ [u,m]γ ,d(u) ] γ + [ u, [d(u),m]γ ] γ + [ u, [u,d(m)]γ ] γ ∈ cγ. but for any u ∈ u and m ∈ m one can show that[ [u,m]γ ,d(u) ] γ + [ u, [d(u),m]γ ] γ = [ m, [d(u),u]γ ] γ = 0. therefore, we get desired result. similary, the other statement can easily be shown. lemma 2.22. let a ∈ m. if aγd(x) = 0 for all x ∈ m, then a = 0 or d = 0. proof. by hypothesis, aγd(xγy) = 0 for all x,y ∈ m. expanding this, we get aγxγd(y) = 0 for all x,y ∈ m. since m is γ-prime gamma ring we have the desired result. 3. main results theorem 3.1. let m be a γ-prime weak nobusawa γ-ring of characteristic not 2 and d be a k-derivation of m such that k (γ) = 0 and d3 6= 0. if u is a γ-lie ideal of m such that u * cγ then d (u) contains a nonzero ideal of m. proof. let v = [u,u]γ and w = [v,v ]γ. according to lemma 2.15, it is enough to show that the γ-subring d (v ) contains a nonzero left ideal of m and a nonzero right ideal of m. suppose that d (v ) does not contain a nonzero right ideal of m. let w ∈ [w,w]γ and a := d (w). since aγ [a,x]γ ∈ w , we have d ( aγ [a,x]γ ) ∈ d (w). expanding this, we get d (a) γ [a,x]γ ∈ d (v ), ∀a ∈ d ( [w,w]γ ) , x ∈ m. (5) on the other hand, since d ( [a,u]γ ) ∈ d (v ) and [a,d (u)]γ ∈ d (v ) for u ∈ v , we have [d (a) ,v ]γ ⊆ d (v ), ∀a ∈ d ( [w,w]γ ) . (6) for m ∈ m we also have d (a) γ [d (a) ,m]γ + d (a) γ [a,d (m)]γ ∈ d (v ) 31 γ-lie structures with derivations since d (a) γd ( [a,m]γ ) ∈ d (v ). hence, by (5) d (a) γ [d (a) ,m]γ ∈ d (v ), ∀a ∈ d ( [w,w]γ ) , m ∈ m. (7) in (7) replacing a by a + b with a, b ∈ d ( [w,w]γ ) we obtain s := d (a) γ [d (b) ,m]γ + d (b) γ [d (a) ,m]γ ∈ d (v ), ∀a,b ∈ d ( [w,w]γ ) . if t := [d (a) γd (b) ,m]γ = d (a) γ [d (b) ,m]γ + [d (a) ,m]γ γd (b) then s− t = d (b) γ [d (a) ,m]γ − [d (a) ,m]γ γd (b) = [ d (b) , [d (a) ,m]γ ] γ . by (6), s − t ∈ d (v ). thus, we get t ∈ d (v ), that is [d (a) γd (b) ,m]γ ⊆ d (v ). since d (v ) does not contain a nonzero right ideal of m, d (a) γd (b) ∈ cγ for all a,b ∈ d ( [w,w]γ ) by lemma 2.16. let n := d (a) γd (b). by (5), d (b) γ [b,x]γ ∈ d (v ). it follows nγ [b,x]γ = d (a) γd (b) γ [b,x]γ ∈ d (v ) since d (a) ∈ d (v ). on the other hand, since nγ [b,x]γ = [b,nγx]γ ∈ d (v ), we have [b,nγm]γ ⊆ d (v ). let i = nγm. if i 6= 0, then b ∈ cγ for all b ∈ d ( [w,w]γ ) by lemma 2.16. thus, by lemma 2.10 we get [w,w]γ ⊆ cγ that is to say u ⊆ cγ by lemma 2.19. but this is a contradiction. therefore, i = nγm = 0. so we get n = d (a) γd (b) = 0 for all a,b ∈ d ( [w,w]γ ) since m is γ-prime gamma ring. that is, d2 ( [w,w]γ ) γd2 ( [w,w]γ ) = 0. hence, we conclude that the contradiction d3 = 0 by lemma 2.17 and lemma 2.18. theorem 3.2. let m be a gamma ring of characteristic not 2 or 3 and u be a γ-lie ideal of m. if [u,d(u)]γ ∈ cγ for all u ∈ u, then u ⊂ cγ. proof. by lemma 2.21 we have[ [d(m),u]γ ,u ] γ γu = uγ [ [d(m),u]γ ,u ] γ . expanding this, we get 3u2γd(m)γu + d(m)γu3 = 3uγd(m)γu2 + u3γd(m). (8) let d(m) = m′. replacing m by u in (8), we obtain u3γu′ −u′γu3 = 3(uγu′ −u′γu)γu2 (9) for all u ∈ u. since [u,u′]γ γu = uγ [u,u ′]γ we have 2γ(uγu′ −u′γu)γu = u2γu′ −u′γu2. (10) again replacing m by uγm′ in (8), we obtain 3uγu′γm′γu2 + u3γu′γm′ − 3u2γu′γm′γu−u′γm′γu3 = 0 (11) for all u ∈ u and m ∈ m. multiplying (8) with u′ gives 3u′γuγm′γu2 + u′γu3γm′ − 3u′γu2γm′γu−u′γm′γu3 = 0. substracting the last equation from (11) we get 3(uγu′ −u′γu)γm′γu2 + (u3γu′ −u′γu3)γm′ − 3(u2γu′ −u′γu2)γm′γu = 0. 32 o. arslan, h. kandamar using the equations (9) and (10) we conclude (uγu′ −u′γu)γ(m′γu2 + u2γm′ − 2uγm′γu) = 0 for all u ∈ u and m ∈ m. if uγu′ −u′γu 6= 0 for some u, then m′γu2 + u2γm′ − 2uγm′γu = 0 (12) for that u. replacing m by uγm, we obtain (u′γm + uγm′)u2 + u2γ(u′γm + uγm′) − 2uγ(u′γm + uγm′) = 0. expanding last equation, we have u′γmγu2 + u2γu′γm− 2uγu′γmγu = 0 (13) for all m ∈ m. if we replace m by u in (12) and multiply by m on the right, then we get u′γu2γm + u2γu′γm− 2uγu′γuγm = 0. (14) substracting (14) from (13) gives u′γ(mγu2 −u2γm) − 2uγu′γ(mγu−uγm) = 0. (15) replacing m by uγm in (15), we obtain u′γuγ(mγu2 −u2γm) − 2uγu′γuγ(mγu−uγm) = 0. (16) mulyiplying (15) by u we get uγu′γ(mγu2 −u2γm) − 2u2γu′γ(mγu−uγm) = 0. (17) substracting (16) from (17) gives (uγu′ −u′γu)γ(mγu2 −u2γm− 2uγ(mγu−uγm)) = 0 for all m ∈ m. since m is γ-prime gamma ring we have mγu2 −u2γm− 2uγ(mγu−uγm) = 0. now think the inner iγu-derivation iuγ on m. from the last equation we write i2uγ = 0 that leads to iuγ = 0 by lemma 2.8. hence, u ∈ cγ. so far we proved that if [u,u′]γ 6= 0 for u ∈ u, then u ∈ cγ. now assume that [u,u ′]γ = 0 for all u ∈ u. by lemma 2.21, [ [d(m),u]γ ,u ] γ = 0 for all m ∈ m and u ∈ u. replacing u by u + w with w ∈ u, we obtain [ [d(m),u]γ ,w ] γ + [ [d(m),w]γ ,u ] γ = 0. (18) choose v,w ∈ u such that wγv ∈ u. replacing w by wγv in (18), we obtain [w,u]γ γ [d(m),v]γ + [d(m),w]γ γ [v,u]γ = 0. for any t ∈ m and w ∈ u the element v = [t,w]γ ensures the condition wγv ∈ u. so by above we have [w,u]γ γ [ d(m), [t,w]γ ] γ + [d(m),w]γ γ [ [t,w]γ ,u ] γ = 0 (19) 33 γ-lie structures with derivations for all t,m ∈ m and u,w ∈ u. replacing u by w, we obtain [d(m),w]γ γ [ [t,w]γ ,w ] γ = 0. (20) replacing t by tγd(a) with a ∈ m, we obtain [d(m),w]γ γ [t,w]γ γ [d(a),w]γ = 0 (21) for all m,t,a ∈ m and w ∈ u. replacing u by [t,w]γ in (19), we get [ [t,w]γ ,w ] γ γ [ [t,w]γ ,d(m) ] γ = 0. if we replace t by t + d(a) we get[ [t,w]γ ,w ] γ γ [ [d(a),w]γ ,d(m) ] γ = 0 (22) for all m,t,a ∈ m and w ∈ u. replacing t by d(t)γs with s ∈ m in (22), we obtain [d(t),w]γ γ [s,w]γ γd(m)γ [d(a),w]γ = 0 for all m,t,a,s ∈ m and w ∈ u. replacing t by tγd(s) in (21), we conclude [d(m),w]γ γmγ [d(s),w]γ γ [d(a),w]γ = 0 for all m,a,s ∈ m and w ∈ u. since m is γ-prime gamma ring we get [d(m),w]γ = 0 or [d(s),w]γ γ [d(a),w]γ = 0 for all m,a,s ∈ m and w ∈ u. if [d(m),w]γ = 0 for all m ∈ m and w ∈ u, then w ∈ cγ by lemma 2.9, so we are done. suppose there is a pair m ∈ m and w ∈ u such that [d(m),w]γ 6= 0. hence, w /∈ cγ and [d(s),w]γ γ [d(a),w]γ = 0 (23) for all a,s ∈ m. replacing a by bγc with b,c ∈ m in (23), we get [d(s),w]γ γd(b)γ [c,w]γ = 0. if we replace b by [t,w]γ in this equation we have [d(s),w]γ γ [t,d(w)]γ γ [w,c]γ = 0 for all c,t,s ∈ m and w ∈ u. replacing c by cγm1 with m1 ∈ m, we obtain [d(s),w]γ γ [t,d(w)]γ = 0. hence, replacing t by tγk with k ∈ m in the last equation, we get d(w) ∈ cγ. now suppose u ∈ u ∩cγ. then d([u,a]γ) = 0 for all a ∈ m. therefore, we have d(u) ∈ cγ. hence, d(u) ∈ cγ for all u ∈ u and then we get d([w,a]γ) ∈ cγ for all a ∈ m. expanding this, we obtain [w,d(a)]γ ∈ cγ and replacing a by aγw, we have [w,d(a)]γ γw + [w,a]γ γd(w) ∈ cγ. (24) therefore, commuting this element by w we get[ w, [w,a]γ ] γ γd(w) = 0 for all a ∈ m. since m is γ-prime gamma ring we have [ w, [w,a]γ ] γ = 0 or d(w) = 0 for all a ∈ m. if [ w, [w,a]γ ] γ = 0, then w ∈ cγ by lemma 2.1. but this is a contradiction. hence, d(w) = 0 and [w,d(a)]γ γw ∈ cγ for all a ∈ m by (24). it follows that [d(a),w]γ γ [w,b]γ = 0 for a,b ∈ m. replacing b by bγc with c, we obtain [d(a),w]γ = 0 or [w,b]γ = 0. so in both cases we have w ∈ cγ which is a contradiction. 34 o. arslan, h. kandamar corollary 3.3. let m be a gamma ring of characteristic 3 and u be a γ-lie ideal of m. if [u,d(u)]γ ∈ cγ and u2 ∈ u for all u ∈ u, then u ⊂ cγ. theorem 3.4. let m be a gamma ring of characteristic 2 and u be a γ-lie ideal and γ-subring of m. if [u,d(u)]γ ∈ cγ for all u ∈ u, then u is γ-commutative. proof. by lemma 2.21, [ [d(m),u]γ ,u ] γ ∈ cγ for all u ∈ u and m ∈ m. hence, d(m)γu2 + u2γd(m) ∈ cγ (25) for all u ∈ u and m ∈ m. then [ d(m),d(m)γu2 + u2γd(m) ] γ = 0 and [ u2,d(m)γu2 + u2γd(m) ] γ = 0. expanding these equations, we get u2γ(d(m))2 = (d(m))2γu2 (26) and u4γd(m) = d(m)γu4 (27) respectively. since d(u2) = uγd(u) + d(u)γu ∈ cγ for u ∈ u, replacing m by v + u2γv with v ∈ u, we obtain( u2γd(v) + d(v)γu2 )2 = 0 for all u,v ∈ u. using γ-primeness of m we have u2γd(v) = d(v)γu2 (28) for all u,v ∈ u by (25). replacing u by u + w with w ∈ u in (28), we get (uγw + wγu) γd(v) = d(v)γ (uγw + wγu) . replacing w by wγu, we get (uγw + wγu) γ (uγd(v) + d(v)γu) = 0 for all u,v,w ∈ u. we conclude( u21γw + wγu 2 1 ) γ (uγd(u) + d(u)γu) = 0, ∀u,u1,w ∈ u replacing u by u + u21 with u1 ∈ u and taking v = u. hence, if [d(u),u]γ 6= 0 for some u ∈ u, then u 2 1γw = wγu 2 1 for all u1,w ∈ u. then, we have u2γ (wγm + mγw) = (wγm + mγw) γu2 for all m ∈ m and u,w ∈ u. expanding this, and replacing m by mγu, we obtain ( u2γm + mγu2 ) γ (wγu + uγw) = 0 35 γ-lie structures with derivations for all u,w ∈ u and m ∈ m. replacing w by [u,t]γ with t ∈ m, we get( u2γm + mγu2 ) γ ( u2γt + tγu2 ) = 0 for all u ∈ u and m,t ∈ m. again replacing t by tγp with p ∈ p , we conclude u2 ∈ cγ for all u ∈ u. assume that [d(u),u]γ = 0 for all u ∈ u. then, by lemma 2.21 [ [d(m),u]γ ,u ] γ = 0 for all u ∈ u and m ∈ m. expanding this, we get u2γd(m) = d(m)γu2. replacing m by mγa with a ∈ m, we obtain d(m)γ ( u2γa + aγu2 ) + ( u2γm + mγu2 ) γd(a) = 0. replacing a by v2, we get d(m)γ ( u2γv2 + v2γu2 ) = 0 for all u,v ∈ u,m ∈ m since d(v2) = vγd(v) + d(v)γv = 0 for v ∈ u. therefore, u2γv2 = v2γu2 for all u,v ∈ u by lemma 2.22. hence, u2γ (vγw + wγv) = (vγw + wγv) γu2 for all u,v,w ∈ u. replacing v by vγw in the last equation, we have (vγw + wγv) γ ( u2γw + wγu2 ) = 0. again replacing v by [w,m]γ with m ∈ m, we obtain( w2γm + mγw2 ) γ ( u2γw + wγu2 ) = 0. that is, iw2γ(m)γ ( u2γw + wγu2 ) = 0 for the inner iγw2derivation iw2γ on m. so by lemma 2.22, if w2 /∈ cγ for some w ∈ u, then u2γw = wγu2 for that w. therefore, [ [u,v]γ ,w ] γ = 0 for all u,v ∈ u. expanding last equation and replacing v by vγw, we obtain [v,w]γ γ [w,u]γ = 0 for all u,v ∈ u. replacing v by [w,r]γ with m,t ∈ m and u by [w,t]γ, we get ( w2γm + mγw2 )( w2γt + tγw2 ) = 0 for all m,t ∈ m. again replacing t by tγp with p ∈ m, we conclude ( w2γm + mγw2 ) γtγ ( w2γp + pγw2 ) = 0 for all p,t ∈ m. since m is γ-prime gamma ring we get w2 ∈ cγ from the last equation. but this is a contradiction. so far we conclude u2 ∈ cγ for all u ∈ u. hence, uγv + vγu ∈ cγ and (uγv + vγu) γu ∈ cγ for all u,v ∈ u. therefore, we have u ∈ cγ or uγv + vγu = 0. then, u is γ-commutative. if we assume that u is only γ-lie ideal of m or only γ-subring of m, then u may not be γcommutative. moreover, according to the assumptions of the theorem, the result u ⊆ cγ cannot be obtained. example 3.5. let r be a noncommutative prime ring with identity. if m is the set of all matrices over r of the form ( a b a c d c ) , γ = m3×2(r) and γ =   1 00 1 0 0   ∈ γ, then m is a γ-prime gamma ring. it can be shown that the subset u = {( a 0 a 0 b 0 ) | a,b ∈ r } of m is a γ-subring but it is not a γ-lie ideal of m. define the inner iγn-derivation inγ on m with n = ( 1 0 1 0 1 0 ) ∈ m. then it is easily verified that [u,inγ(u)]γ ∈ cγ for all u ∈ u but u is not γ-commutative. example 3.6. let m = {( a b a c d c ) | a,b,c,d ∈ z2 } , γ = m3×2(z2) and γ =   1 00 1 0 0   ∈ γ. then m is a γ-prime gamma ring. let u = {( a b a c a c ) | a,b,c ∈ z2 } . it is easily seen that u is a γ-lie ideal but it is not a γ-subring of m. define the inner iγn-derivation inγ on m with n = ( 1 0 1 0 1 0 ) ∈ m. then it is easily verified that [u,inγ(u)]γ ∈ cγ for all u ∈ u but u is not γ-commutative. example 3.7. let m = {( a b a c d c ) | a,b,c,d ∈ z2 } , γ = m3×2(z2) and γ =   1 00 1 0 0   ∈ γ. then m is a γ-prime gamma ring. let u = {( a b a b a b ) | a,b ∈ z2 } . it is easily seen that u is a γ-lie 36 o. arslan, h. kandamar ideal and a γ-subring of m but it is not a γ-ideal. define the inner iγn-derivation inγ on m with n = ( 1 0 1 0 1 0 ) ∈ m. then it is easily verified that [u,inγ(u)]γ ∈ cγ for all u ∈ u. hence, u is γ-commutative but cannot be contained in the γ-center of m. acknowledgment: the authors would like to thank the referees for their valuable suggestions and comments. references [1] r. awtar, lie and jordan structure in prime rings with derivations, proc. amer. math. soc., 41, 67-74, 1973. [2] w. e. barnes, on the γ-rings of nobusawa, pacific j. math., 18, 411-422, 1966. [3] j. bergen, j.w. kerr, i.n. herstein, lie ideals and derivations of prime rings, j. algebra, 71, 259-267, 1981. [4] i. n. herstein, a note on derivations, canad. math. bull., 21(3), 369-370, 1978. [5] i. n. herstein, a note on derivations ii, canad. math. bull., 22(4), 509-511, 1979. [6] i. n. herstein, topics in ring theory, the univ. of chicago press, 1969. [7] h. kandamar, the k-derivation of a gamma-ring, turk. j. math., 23(3), 221-229, 2000. [8] a. r. khan, m. a. chaudhry, i. javaid, generalized derivations on prime γ-rings, world appl. sci. j., 23(12), 59-64, 2013. [9] s. kyuno, gamma rings, hadronic press, 1991. [10] n. nobusawa, on a generalization of the ring theory, osaka j. math., 1, 81-89, 1964. [11] e. c. posner, derivations in prime rings, proc. amer. math. soc., 8, 1093-1100, 1957. [12] n. n. suliman, a. h. majeed, lie ideals in prime γ-rings with derivations, discussiones mathematicae, 33, 49-56, 2013. 37 preliminaries -lie ideals and derivations main results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729422 j. algebra comb. discrete appl. 7(2) • 121–140 received: 10 august 2017 accepted: 15 august 2019 journal of algebra combinatorics discrete structures and applications degree distance and gutman index of two graph products research article shaban sedghi, nabi shobe abstract: the degree distance was introduced by dobrynin, kochetova and gutman as a weighted version of the wiener index. in this paper, we investigate the degree distance and gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph. 2010 msc: 05c07, 05c90 keywords: degree distance, adjacency matrix, distance matrix, complete product, strong product 1. introduction all graphs in this paper are assumed to be undirected, finite and simple. we refer to [2] for graph theoretical notation and terminology not specified here. for a graph g, let v (g), e(g) and g denote the set of vertices, the set of edges and the complement of g, respectively. if g is a connected graph and u,v ∈ v (g), then the distance d(u,v) between u and v is the length of a shortest path connecting u and v. if v is a vertex of a connected graph g, then the eccentricity e(v) of v is defined by e(v) = max{d(u,v) |u ∈ v (g)}. furthermore, the diameter diam(g) of g is defined by diam(g) = max{e(v) |v ∈ v (g)}. let g be a finite, simple, connected, undirected graph with p vertices and q edges. in what follows, we say that g is an (p,q)-graph. let v (g) = {v1,v2, . . . ,vp} and e(g) = {e1,e2, . . . ,eq} be the vertex set and edge set of g, respectively. the adjacency matrix of g is the p×p matrix a = a(g) whose (i,j) entry, denoted by aij, is defined by aij = { 1 if vi and vj are adjacent 0 otherwise. shaban sedghi(corresponding author); department of mathematics, qaemshahr branch, islamic azad university, qaemshahr, iran (email: sedghi_gh@yahoo.com, sedghi.gh@qaemiau.ac.ir). nabi shobe; department of mathematics, babol branch, islamic azad university, babol, iran (email: nabi_shobe@yahoo.com). 121 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 the distance matrix of g is the p×p matrix dg whose (i,j) entry, denoted by dij, is defined by dij = { dg(vi,vj) if vi 6= vj 0 otherwise, where dg(vi,vj) is the length of a shortest directed path in g from vi to vj. the vertex u is said to be a neighbor of v if they are adjacent. the neighborhood of a vertex v, denoted by ng(v), is the set of all neighbors of v. the degree of a vertex v in a graph g, denoted by dv = dg(v), is the number of vertices in its neighborhood, that is, dg(v) = |n(v)|. the common neighborhood graph con(g) (in short congraph) of a graph g is defined as the graph with v (con(g)) = v (g) and two vertices in con(g) are adjacent if they have a common neighbor in g. for every x,y ∈ v (g), xy ∈ e(con(g)) if and only if ng(x) ∩ng(y) 6= ∅. some basic properties of congraphs have been established; see [1, 3]. the oldest and most studied degree-based structure descriptors are the first and second zagreb indices [15], defined as m1(g) = ∑ v∈v (g) (dg(v)) 2 and m2(g) = ∑ uv∈e(g) (dg(u)) (dg(v)) . it has been shown that the first zagreb index obeys the identity [10] m1(g) = ∑ uv∈e(g) (dg(u) + dg(v)) . the first investigation of the sum of distance between all pairs of vertices of a (connected) graph was done by harold wiener in 1947, who realized that there exists a correlation between the boiling points of paraffins and this sum [20]. eventually, the distance–based graph invariant, w(g) = ∑ {u,v}⊆v (g) d(u,v). for more details, we refer to [8, 11, 13, 19]. the degree distance was introduced by dobrynin and kochetova [9] and gutman [14] as a weighted version of the wiener index. the degree distance dd(g) of a graph g is defined as dd(g) = ∑ {u,v}⊆v (g) dg(u,v)[dg(u) + dg(v)] = 1 2 ∑ u,v∈v (g) dg(u,v)[dg(u) + dg(v)] with the summation runs over all pairs of vertices of g. the degree distance is also known as the schultz index in chemical literature; see [21]. in [14], gutman showed that if g is a tree on n vertices, then dd(g) = 4w(g) −n(n− 1); see [5, 6] and [9]. in [7], gutman index gut(g) of a graph g is defined as gut(g) = ∑ {u,v}⊆v (g) dg(u)dg(v)d(u,v). for more details on gutman index, we refer to [4, 7, 12]. the relations between the degree distance, gutman index and wiener index are shown in the following table 1. the join and strong products are defined as follows. the join or complete product g∨h of two disjoint graphs g and h, is the graph with vertex set v (g) ∪v (h) and edge set e(g) ∪e(h) ∪{uv |u ∈ v (g),v ∈ v (h)}. 122 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 table 1. three distance parameters wiener index w(g) = ∑ {u,v}⊆v (g) dg(u, v) degree distance dd(g) = ∑ {u,v}⊆v (g) dg(u, v)[dg(u) + dg(v)] gutman index gut(g) = ∑ {u,v}⊆v (g) dg(u, v)dg(u)dg(v) the strong product g�h of graphs g and h has the vertex set v (g)×v (h). two vertices (u,v) and (u′,v′) are adjacent whenever uu′ ∈ e(g) and v = v′, or u = u′ and vv′ ∈ e(h), or uu′ ∈ e(g) and vv′ ∈ e(h). paulraja and agnes [16] studied the degree distance of cartesian and lexicographic products. later, they [17] investigated the gutman index of cartesian and lexicographic products. in this paper, we investigate the degree distance and gutman index of strong and complete product graphs. 2. preliminary we define, n1(g) = ∑ v∈v (g) dg(v) dcon(g)(v) and n2(g) = ∑ uv∈e(con(g)) dg(u) dg(v). definition 2.1. let a = [aij]m×n. then, we define s(a) = ∑ 1≤i≤m, 1≤j≤n aij. the following lemma is immediate. lemma 2.2. let a = [aij]n×n and b = [bij]n×n. then (1) s(at ) = s(a) and s(αa) = αs(a) for every α ∈ r; (2) s(a + b) = s(a) + s(b). lemma 2.3. let g be a (p,q)-graph, and let con(g) be a (p,q′)-graph. let a,b,k be the adjacency matrices of g,con(g),kp, respectively. then (1) s(a) = 2q; (2) s(a2) = m1(g); (3) s(ab) = n1(g); (4) s(a3) = 2m2(g); (5) s(ak) = 2q(p− 1). (6) s(ab a) = 2n2(g). proof. for (1), we have s(a) = ∑ 1≤i,j≤p aij = p∑ i=1 p∑ j=1 aij = p∑ i=1 dvi = 2q. 123 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 for (2), we have s(a2) = ∑ 1≤i,j≤p a (2) ij = ∑ 1≤i,j≤p p∑ k=1 aik akj = p∑ k=1 p∑ i=1 aik p∑ j=1 akj = p∑ k=1 dvk dvk = m1(g). for (3), we have s(ab) = ∑ 1≤i,j≤p p∑ k=1 aik bkj = p∑ k=1 p∑ i=1 aik p∑ j=1 bkj = p∑ k=1 dvk dcongvk = n1(g). for (4), we have m2(g) = ∑ vivj∈e(g) dvi dvj = 1 2 ∑ 1≤i,j≤p dvi dvj aij = 1 2 p∑ i=1 p∑ j=1 ( p∑ k=1 aki )( p∑ s=1 ajs ) aij = 1 2 p∑ k=1 p∑ j=1 p∑ s=1 ajs p∑ i=1 akiaij. since ∑p i=1 akiaij is the entry tkj of matrix a 2, it follows that m2(g) = 1 2 p∑ k=1 p∑ j=1 p∑ s=1 ajs p∑ i=1 akiaij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 tkjajs = 1 2 p∑ k=1 p∑ s=1 a (3) ks = 1 2 s(a3). for (5), we have s(ak) = ∑ 1≤i,j≤p p∑ r=1 air krj = p∑ r=1 p∑ i=1 air p∑ j=1 krj = p∑ r=1 dvr (p− 1) = 2q(p− 1). for (6), we have n2(g) = ∑ vivj∈e(cong) dvi dvj = 1 2 ∑ 1≤i,j≤p dvi dvj bij = 1 2 p∑ i=1 p∑ j=1 ( p∑ k=1 aki )( p∑ s=1 asj ) bij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 asj p∑ i=1 akibij. 124 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 since, ∑p i=1 akibij is the entry tkj of matrix ab, hence n2(g) = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 asj p∑ i=1 akibij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 tkjajs = 1 2 p∑ k=1 p∑ s=1 fks = 1 2 s(ab a), where ∑p i=1 tkjajs is the entry fks of matrix ab a. the following result for classical distance are from the book [18]. lemma 2.4. [18] let (u,v) and (u′,v′) be two vertices of g1 � g2. then dg1�g2 ((u,v), (u ′,v′)) = max{dg1 (u,u ′),dg2 (v,v ′)}. for g2 = kp, the following result is immediate. corollary 2.5. let kp be a complete graph, and let (u,v) and (u′,v′) be two vertices of g � kp. then dg�kp ((u,v), (u ′,v′)) =   dg(u,u ′) if u 6= u′, 1 if u = u′ and v 6= v′, 0 if u = u′ and v = v′. 3. main results in this section, we give our main results and their proofs. 3.1. relation between degree distance and gutman index we first define a matrix, which will be used later. definition 3.1. let g(v,e) be a graph with order n and m edges. for k = 1, 2, · · · ,α where α denotes the diameter of graph g, we define ak = [a k ij]n×n, where akij = { 1 d(vi,vj) = k 0 otherwise. the following results are easily seen. observation 3.1. let a and dg be the adjacency matrix and the distance matrix of a graph g, respectively. then (1) a1 = a; (2) dg = a1 + 2a2 + · · · + αaα; (3) a1 + a2 + · · · + aα = k, where k is the adjacency matrix complete graph kn; (4) if diam(g) = 2 then dg = a + 2a. 125 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 lemma 3.2. let g be a graph containing no triangles, and let a,b,k be the adjacency matrix of g,con(g),kn, respectively. then (1) for every u,v ∈ v (g), dg(u,v) = 2 if and only if uv ∈ e(con(g)); (2) if diam(g) = 3 then dg = 3k − 2a−b. proof. (1) suppose dg(u,v) = 2. then there exists a vertex x ∈ v (g) such that x /∈ {u,v} and ux,xv ∈ e(g), and hence n(u)∩n(v) 6= ∅. therefore, we have uv ∈ e(con(g)). conversely, we suppose uv ∈ e(con(g)). then n(u) ∩n(v) 6= ∅, and hence there exists a vertex x ∈ n(u) ∩n(v). note that ux,xv ∈ e(g). therefore, d(u,v) ≤ 2. if d(u,v) = 1, then we have a triangle, a contradiction. so dg(u,v) = 2, as desired. (2) from observation 3.1, we have dg = a1 + 2a2 + 3a3. note that a1 = a, a2 = b and a + b + a3 = k. therefore, dg = 3k − 2a−b. lemma 3.3. let g(p,q) be a graph, and let a,dg be the adjacency matrix and the distance matrix of a graph g, respectively. then (1) s(adg) = dd(g); (2) if diam(g) = 2 then dd(g) = 4(p− 1)q −m1(g); (3) if diam(g) = 3 and g has no triangles, then dd(g) = 6q(p− 1) − 2m1(g) −n1(g). proof. (1) since s(adg) = p∑ i=1 p∑ j=1 p∑ k=1 aikdkj = ∑ 1≤j,k≤p p∑ i=1 aik d(vk,vj) = ∑ 1≤j,k≤p d(vk) d(vk,vj) and s(dg a) = p∑ i=1 p∑ j=1 p∑ k=1 dikakj = ∑ 1≤i,k≤p d(vi,vk) p∑ j=1 akj = ∑ 1≤i,k≤p d(vi,vk) d(vk) = ∑ 1≤j,k≤p d(vk,vj) d(vj), it follows that 2s(adg) = s(adg) + s((adg) t ) = s(adg) + s(dg a) = ∑ 1≤j,k≤p d(vk,vj)[d(vk) + d(vj)] = 2 ∑ {vk,vj}⊆v (g) d(vk,vj)[d(vk) + d(vj)] = 2dd(g). for (2), we have dd(g) = s(adg) = s(a(a + 2a)) = 2s(a(a + a)) −s(a2) = 2s(ak) −s(a2) = 4(p− 1)q −m1(g). 126 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 for (3), we have dd(g) = s(adg) = s(a(3k − 2a−b)) = 3s(ak) − 2s(a2) −s(ab) = 6q(p− 1) − 2m1(g) −n1(g). lemma 3.4. let g be a (p,q)-graph and a be the adjacency matrix of g. then gut(g) = 1 2 s(adg a) proof. s(adg a) = p∑ i=1 p∑ j=1 p∑ k=1 aik p∑ s=1 dksasj = p∑ k=1 p∑ s=1 p∑ i=1 aik p∑ j=1 asjdks = p∑ k=1 p∑ s=1 dg(vk) ·dg(vs) ·d(vk,vs) = 2 ∑ {vk,vs}⊆v dg(vk)dg(vs)d(vk,vs) = 2gut(g). corollary 3.5. let g(p,q) be a graph, then δ 2 ≤ gut(g) dd(g) ≤ ∆ 2 . proof. since, s(adg) = dd(g) and gut(g) = 12s(adg a), hence 2gut(g) −dd(g) = s(adg a) −s(adg) = s(adg(a− i)) = ∑ 1≤i,j≤p p∑ k=1 tik(akj − 1kj) = ∑ 1≤i,k≤p tik(dg(vk) − 1). therefore, (δ − 1) ∑ 1≤i,k≤p tik ≤ 2gut(g) −dd(g) ≤ (∆ − 1) ∑ 1≤i,k≤p tik. hence, (δ − 1)s(adg) ≤ 2gut(g) −dd(g) ≤ (∆ − 1)s(adg). thus, δdd(g) ≤ 2gut(g) ≤ ∆dd(g), that is δ 2 ≤ gut(g) dd(g) ≤ ∆ 2 . 127 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 3.2. for degree distance in this subsection, we study the degree distance of strong product graphs. we first begin with an easy case. theorem 3.6. let g be a connected graph with p1 vertices and q1 edges, and kp be a complete graph with order p. then dd(g � kp) = p 3dd(g) + 2p2(p− 1)[w(g) + q1] + p1p(p− 1)2. proof. let v (g) = v1 and v (kp) = v2. from the definition of strong product and corollary 2.5, we have dd(g � kp) = ∑ {(a,b),(c,d)}⊆v1×v2 [dg�kp (a,b) + dg�kp (c,d)]dg�kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg(a) + dkp (b) + dg(a)dkp (b) + dg(c) + dkp (d) + dg(c)dkp (d)] = ∑ {(a,b),(c,d)}⊆v1×v2,a 6=c [dg(a) + p− 1 + dg(a)(p− 1) + dg(c) + p− 1 + dg(c)(p− 1)] ·dg(a,c) + ∑ {(a,b),(c,d)}⊆v1×v2,a=c [dg(a) + p− 1 + dg(a)(p− 1) + dg(c) + p− 1 + dg(c)(p− 1)] · 1 = ∑ {(a,b),(c,d)}⊆v1×v2,a 6=c [p(dg(a) + dg(c)) + 2(p− 1)]dg(a,c) + ∑ {(a,b),(c,d)}⊆v1×v2,a=c [2pdg(a) + 2(p− 1)] = p ∑ {(a,b),(c,d)}⊆v1×v2,a 6=c [dg(a) + dg(c)]dg(a,c) + 2(p− 1) ∑ {(a,b),(c,d)}⊆v1×v2,a6=c dg(a,c) +2p ∑ {(a,b),(c,d)}⊆v1×v2,a=c dg(a) + 2(p− 1) ∑ {(a,b),(c,d)}⊆v1×v2,a=c 1 = p3dd(g) + 2p2(p− 1)w(g) + 2p · p(p− 1) 2 · 2q1 + 2p1(p− 1) · p(p− 1) 2 = p3dd(g) + 2p2(p− 1)[w(g) + q1] + p1p(p− 1)2. for the strong product of two general graphs, we have the following. theorem 3.7. let g1 be a connected graph with p1 vertices and q1 edges, and g2 be a connected graph 128 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 with p2 vertices and q2 edges. then max { dd(g1)[2p2q2 + p 2 2] + 4p2q2w(g1) + 2p2(p2 − 1)q1w(g2) + dd(g2)(2q1 + p1), dd(g2)[2p1q1 + p 2 1] + 4p1q1w(g2) + 2p1(p1 − 1)q2w(g1) + dd(g1)(2q2 + p2) } ≤ dd(g1 � g2) ≤ (2q2 + p2)(p2 + 1)dd(g1) + q2(4p2 + 2p21 − 2p1)w(g1) +(2q1 + p1)(p1 + 1)dd(g2) + q1(4p1 + 2p 2 2 − 2p2)w(g2). moreover, the lower bound is sharp. in particular, if g be a connected graph with p vertices and q edges, then (2q + p)(p + 1)dd(g) + 2pq(p + 1)w(g) ≤ dd(g � g) ≤ 2 { (2q + p)(p + 1)dd(g) + 2pq(p + 1)w(g) } . proof. from lemma 2.4 and the definition of degree distance, we have dd(g1 � g2) = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1�g2 (a,b) + dg1�g2 (c,d)]dg1�g2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)] · max{dg1 (a,c),dg2 (b,d)} ≥ max { ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d), ∑ {(a,b),(c,d)}⊆v1×v2 b6=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 b=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) } 129 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 = max { ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg1 (a) + dg1 (c)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg2 (b) + dg2 (d)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(a,d)}⊆v1×v2 [2dg1 (a) + (dg1 (a) + 1)(dg2 (b) + dg2 (d))]dg2 (b,d), ∑ {(a,b),(c,d)}⊆v1×v2 b6=d [dg1 (a) + dg1 (c)]dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 b 6=d [dg2 (b) + dg2 (d)]dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 b6=d [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg2 (b,d) + ∑ {(a,b),(c,b)}⊆v1×v2 [2dg2 (b) + (dg2 (b) + 1)(dg1 (a) + dg1 (c))]dg1 (a,c) } = max { p22dd(g1) + 4p2q2w(g1) + ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg1 (a,c) +2p2(p2 − 1)q1w(g2) + dd(g2)(2q1 + p1), p21dd(g2) + 4p1q1w(g2) + ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg2 (b,d) +2p1(p1 − 1)q2w(g1) + dd(g1)(2q2 + p2) } since ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg1 (a,c) = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a)dg2 (b) ·dg1 (a,c)] + ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (c)dg2 (d) ·dg1 (a,c)] 130 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 = ∑ {a,c}⊆v1 dg1 (a)dg1 (a,c) · ∑ {b,d}⊆v2 dg2 (b) + ∑ {a,c}⊆v1 dg1 (c)dg1 (a,c) · ∑ {b,d}⊆v2 dg2 (d) = 2p2q2 · ∑ {a,c}⊆v1 dg1 (a)dg1 (a,c) + 2p2q2 · ∑ {a,c}⊆v1 dg1 (c)dg1 (a,c) = 2p2q2 ( ∑ {a,c}⊆v1 [dg1 (a) + dg1 (c)]dg1 (a,c) ) = 2p2q2 ·dd(g1) and similarly ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a)dg2 (b) + dg1 (c)dg2 (d)]dg2 (b,d) = 2p1q1 ( ∑ {b,d}⊆v2 [dg2 (b) + dg2 (d)]dg2 (b,d) ) = 2p1q1dd(g2), it follows that dd(g1 � g2) ≥ max { dd(g1)[2p2q2 + p 2 2] + 4p2q2w(g1) + 2p2(p2 − 1)q1w(g2) + dd(g2)(2q1 + p1), dd(g2)[2p1q1 + p 2 1] + 4p1q1w(g2) + 2p1(p1 − 1)q2w(g1) + dd(g1)(2q2 + p2) } . also, we have dd(g1 � g2) = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1�g2 (a,b) + dg1�g2 (c,d)]dg1�g2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)] · max{dg1 (a,c),dg2 (b,d)} ≤ ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b) + dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d) = (2q2 + p2)(p2 + 1)dd(g1) + q2(4p2 + 2p 2 1 − 2p1)w(g1) +(2q1 + p1)(p1 + 1)dd(g2) + q1(4p1 + 2p 2 2 − 2p2)w(g2). to show the sharpness of the lower bounds of theorem 3.7, we consider the following example. example 1. let g be a complete graph of order n. if n = 2, then g = k2 and g � g = k4, and hence dd(g � g) = 36 = (2q + p)(p + 1)dd(g) + 2pq(p + 1)w(g). if n = 3, then g = k3 and g � g = k9, and hence dd(g � g) = 576 = (2q + p)(p + 1)dd(g) + 2pq(p + 1)w(g). from the proof of theorem 3.7, one can check that kn � kn is an sharp example of the lower bound. 131 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 3.3. for gutman index in this subsection, we study the gutman index of strong product graphs. we first begin with an easy case. theorem 3.8. let g be a connected graph with p1 vertices and q1 edges, and kp be a complete graph with p vertices. then gut(g � kp) = p 4gut(g) + p3(p− 1)dd(g) + p2(p− 1)2 ·w(g) + p3(p− 1) 2 m1(g) + p(p− 1)3 2 p1 + 2p 2(p− 1)2q1. proof. let v (g) = v1 and v (kp) = v2. from the definition of strong product and corollary 2.5, we have gut(g � kp) = ∑ {(a,b),(c,d)}⊆v1×v2 dg�kp (a,b) ·dg�kp (c,d) ·dg�kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg(a) + dkp (b) + dg(a)dkp (b)] · [dg(c) + dkp (d) + dg(c)dkp (d)] ·dg�kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2,a 6=c [pdg(a) + p− 1] · [pdg(c) + p− 1] ·dg(a,c) + ∑ {(a,b),(c,d)}⊆v1×v2,a=c [pdg(a) + p− 1] · [pdg(a) + (p− 1)] · 1 = p2 ∑ {(a,b),(c,d)}⊆v1×v2,a6=c dg(a)dg(c)dg(a,c) +p(p− 1) ∑ {(a,b),(c,d)}⊆v1×v2,a 6=c [dg(a) + dg(c)]dg(a,c) +(p− 1)2 ∑ {(a,b),(c,d)}⊆v1×v2,a6=c dg(a,c) + p 2 ∑ {(a,b),(c,d)}⊆v1×v2,a=c d2g(a) + ∑ {(a,b),(c,d)}⊆v1×v2,a=c (p− 1)2 + 2p(p− 1) ∑ {(a,b),(c,d)}⊆v1×v2,a=c dg(a) = p4gut(g) + p3(p− 1)dd(g) + p2(p− 1)2 ·w(g) + p3(p− 1) 2 m1(g) + p(p− 1)3 2 p1 + 2p 2(p− 1)2q1. for the strong product of two general graphs, we have the following. 132 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 theorem 3.9. let g1 be a connected graph with p1 vertices and q1 edges, and g2 be a connected graph with p2 vertices and q1 edges. then max { gut(g1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)dd(g1) + 4q 2 2w(g1) + m1(g1)w(g2) + [2q1 + m1(g1)]dd(g2) + [p1 + 4q1 + m1(g1)]gut(g2), gut(g2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)dd(g2) + 4q 2 1w(g2) +m2(g2)w(g1) + [2q2 + m2(g2)]dd(g1) + [p2 + 4q2 + m2(g2)]gut(g1) } ≤ gut(g1 � g2) ≤ gut(g1)[p22 + 4p2q2 + 4q 2 2 + p2 + 4q2 + m2(g2)] + [2p2q2 + 4q 2 2 + 2q2 + m2(g2)]dd(g1) +[4q22 + m2(g2)]w(g1) + gut(g2)[p 2 1 + 4p1q1 + 4q 2 1 + p1 + 4q1 + m1(g1)] +[2p1q1 + 4q 2 1 + 2q1 + m1(g1)]dd(g2) + [4q 2 1 + m1(g1)]w(g2). in particular, if g be a connected graph with p vertices and q edges, then gut(g)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + m1(g)]dd(g) + [4q 2 + m1(g)]w(g) ≤ gut(g � g) ≤ 2 { gut(g)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + m1(g)]dd(g) + [4q 2 + m1(g)]w(g) } proof. let v (g1) = v1 and v (g2) = v2. from the definition of strong product and lemma 2.4, we have gut(g1 � g2) = ∑ {(a,b),(c,d)}⊆v1×v2 dg1�g2 (a,b) ·dg1�g2 (c,d) ·dg1�g2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)] · max{dg1 (a,c),dg2 (b,d)} ≥ max { ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d), ∑ {(a,b),(c,d)}⊆v1×v2 b 6=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 b=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) } 133 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 = max { ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(a,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (a) + dg2 (d) + dg1 (a)dg2 (d)]dg2 (b,d), ∑ {(a,b),(c,d)}⊆v1×v2 b6=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d) + ∑ {(a,b),(c,b)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (b) + dg1 (c)dg2 (b)]dg1 (a,c) } = max{x1 + x2,y1 + y2}, where x1 = ∑ {(a,b),(c,d)}⊆v1×v2 a6=c [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c), x2 = ∑ {(a,b),(a,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (a) + dg2 (d) + dg1 (a)dg2 (d)]dg2 (b,d), y1 = ∑ {(a,b),(c,d)}⊆v1×v2 b6=d [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d), and y2 = ∑ {(a,b),(c,b)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (b) + dg1 (c)dg2 (b)]dg1 (a,c). note that x1 = ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg1 (a) ·dg1 (c) ·dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a 6=c dg1 (a) ·dg2 (d) ·dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg1 (a)dg1 (c)dg2 (d)dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg2 (b)dg1 (c)dg1 (a,c) 134 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg2 (b)dg2 (d)dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg2 (b)dg1 (c)dg2 (d)dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg1 (a)dg2 (b)dg1 (c)dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg1 (a)dg2 (b)dg2 (d)dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 a6=c dg1 (a)dg2 (b)dg1 (c)dg2 (d)dg1 (a,c) = p22gut(g1) + 4p2q2gut(g1) + 4q 2 2gut(g1) + 2p2q2dd(g1) + 4q 2 2dd(g1) + 4q 2 2w(g1) = gut(g1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)dd(g1) + 4q 2 2w(g1) and x2 = ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a) 2 ·dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a) ·dg2 (d) ·dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a) 2dg2 (d)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg2 (b)dg1 (a)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg2 (b)dg2 (d)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg2 (b)dg1 (a)dg2 (d)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a) 2dg2 (b)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a)dg2 (b)dg2 (d)dg2 (b,d) + ∑ {(a,b),(c,d)}⊆v1×v2 dg1 (a) 2dg2 (b)dg2 (d)dg2 (b,d) = m1(g1)w(g2) + 2q1dd(g2) + m1(g1)dd(g2) + p1gut(g2) + 4q1gut(g2) +m1(g1)gut(g2) = m1(g1)w(g2) + [2q1 + m1(g1)]dd(g2) + [p1 + 4q1 + m1(g1)]gut(g2). similarly, we have y1 = gut(g2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)dd(g2) + 4q 2 1w(g2) and y2 = m2(g2)w(g1) + [2q2 + m2(g2)]dd(g1) + [p2 + 4q2 + m2(g2)]gut(g1). then gut(g1 � g2) ≥ max { gut(g1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)dd(g1) + 4q 2 2w(g1) +m1(g1)w(g2) + [2q1 + m1(g1)]dd(g2) + [p1 + 4q1 + m1(g1)]gut(g2), gut(g2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)dd(g2) + 4q 2 1w(g2) +m2(g2)w(g1) + [2q2 + m2(g2)]dd(g1) + [p2 + 4q2 + m2(g2)]gut(g1) } 135 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 and gut(g1 � g2) = ∑ {(a,b),(c,d)}⊆v1×v2 dg1�g2 (a,b) ·dg1�g2 (c,d) ·dg1�g2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)] · max{dg1 (a,c),dg2 (b,d)} ≤ ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg1 (a,c) + ∑ {(a,b),(c,d)}⊆v1×v2 [dg1 (a) + dg2 (b) + dg1 (a)dg2 (b)] · [dg1 (c) + dg2 (d) + dg1 (c)dg2 (d)]dg2 (b,d) = x1 + x2 + y1 + y2 ≤ gut(g1)[p22 + 4p2q2 + 4q 2 2 + p2 + 4q2 + m2(g2)] + [2p2q2 + 4q 2 2 + 2q2 + m2(g2)]dd(g1) +[4q22 + m2(g2)]w(g1) + gut(g2)[p 2 1 + 4p1q1 + 4q 2 1 + p1 + 4q1 + m1(g1)] +[2p1q1 + 4q 2 1 + 2q1 + m1(g1)]dd(g2) + [4q 2 1 + m1(g1)]w(g2). to show the sharpness of the lower bounds of theorem 3.9, we consider the following example. example 1. let g be a complete graph of order n. if n = 2, then g = k2 and g � g = k4, and hence gut(g�g) = 54 = gut(g)[p2 +4pq+4q2 +p+4q]+[2pq+4q2 +2q+m1(g)]dd(g)+[4q 2 +m1(g)]w(g). if n = 3, then g = k3 and g � g = k9, and hence gut(g � g) = 2304 = gut(g)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + m1(g)]dd(g) + [4q 2 + m1(g)]w(g). from the proof of theorem 3.9, one can check that kn � kn is an sharp example of the lower bound. 3.4. for complete product we first give the following lemma. lemma 3.10. (1) if a = [aij]n×m be any matrix and i = [1]p×n, then s(i a) = ps(a); (2) if a = [aij]m×n and i = [1]n×p, then s(ai) = ps(a); (3) if a = [aij]p×m, i = [1]m×n and b = [bij]n×q, then s(ai b) = s(a) · s(b). in particular, if a = [aij]n×n then s(ai a) = s(a)2. proof. for (1), we have s(i a) = p∑ i=1 m∑ j=1 n∑ k=1 1ikakj = p∑ i=1 n∑ k=1 m∑ j=1 akj = p∑ i=1 s(a) = ps(a). 136 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 for (2), we have s(ai) = m∑ i=1 p∑ j=1 n∑ k=1 aik1kj = p∑ j=1 m∑ i=1 n∑ k=1 aik = p∑ j=1 s(a) = ps(a). for (3), we have s(ai b) = p∑ i=1 q∑ j=1 m∑ k=1 aik n∑ s=1 1ksbsj = p∑ i=1 m∑ k=1 aik q∑ j=1 n∑ s=1 bsj = s(a) ·s(b). corollary 3.11. let g be a (p,q)-graph and a and k be the adjacency matrix of g and kp respectively. let i = [1]p×p and ip be the identity matrix. then s(aka) = 4q2 −m1(g). proof. by lemma 3.10, we have s(ak a) = s(a(i − ip)a) = s(ai a) −s(a2) = s(a) 2 −s(a2) = 4q2 −m1(g). theorem 3.12. let g be a (p,q)-graph, then (1) if diam(g) = 2, then gut(g) = 4q2 −m1(g) −m2(g). (2) if diam(g) = 3 and g has no cycles of size 3 then gut(g) = 6q2 − 3 2 m1(g) − 2m2(g) −n2(g). proof. (1) by lemma 3.4 and observation 3.1, we have: 2gut(g) = s(adg a) = s(a(a + 2a)a) = s(a 3) + 2s(aaa) = 2s(a(a + a)a) −s(a3) = 2s(ak a) −s(a3) = 8q2 − 2m1(g) − 2m2(g). (2) by lemma 3.4 and observation 3.1, we have: 2gut(g) = s(adg a) = s(a(3k − 2a−b)a) = 3s(ak a) − 2s(a3) −s(ab a) = 12q2 − 3m1(g) − 4m2(g) − 2n2(g). 137 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 remark 3.13. let a1 = [aij]n1×n1 and a2 = [bij]n2×n2 be the adjacency matrix of g1 and g2, respectively. let dg be distance matrix of graph g = g1 ∨g2. let i1 = [1]n1×n1, i2 = [1]n2×n2, i ′ 1 = [1]n1×n2, i ′ 2 = [1]n2×n1 and in be identity matrices. then dg = ( 2i1 −a1 − 2in1, i ′ 1 i ′ 2, 2i2 −a2 − 2in2 ) is distance matrix of g1 ∨g2. theorem 3.14. let g1 be a graph with order n1 and m1 edges and g2 be a graph with order n2 and m2 edges. then dd(g1 ∨g2) = 4(n1 + n2 − 1)(m1 + m2 + n1n2) −m(g1 ∨g2). proof. since diam(g1 ∨g2) = 2, it follows from lemma 3.3 that dd(g1 ∨g2) = 4(n1 + n2 − 1)(m1 + m2 + n1n2) −m1(g1 ∨g2). for computing m1(g1 ∨g2), let a be the adjacency matrix of graph g = g1 ∨g2. then m1(g1 ∨g2) = s(a2) = s    a1, i′1 i ′ 2, a2    a1, i′1 i ′ 2, a2     = s   a12 + i′1i′2, a1i′1 + i′1a2 i ′ 2a1 + a2i ′ 2, i ′ 2i ′ 1 + a2 2   = s(a1 2) + s(i ′ 1i ′ 2) + s(a1i ′ 1) + s(i ′ 1a2) + s(i ′ 2a1) + s(a2i ′ 2) + s(i ′ 2i ′ 1) + s(a2 2) = m1(g1) + n 2 1n2 + 4n2m1 + 4n1m2 + n 2 2n1 + m1(g2). theorem 3.15. let g1 be an (n1,m1)-graph and let g2 be an (n2,m2)-graph. then gut(g1 ∨g2) = 4(m1 + m2 + n1n2)2 −m1(g1 ∨g2) −m2(g1 ∨g2). proof. let a1 = [aij]n1×n1 and a2 = [bij]n2×n2 be the adjacency matrix of g1 and of g2 respectively. let dg be distance matrix of graph g = g1 ∨g2. if we set i1 = [1]n1×n1, i2 = [1]n2×n2, i ′ 1 = [1]n1×n2, i ′ 2 = [1]n2×n1 and in be identity matrix, then it follows from theorem 3.12 that gut(g1 ∨g2) = 4(m1 + m2 + n1n2)2 −m1(g1 ∨g2) −m2(g1 ∨g2), since diam(g1 ∨g2) = 2. for computing m2(g1 ∨g2), let a be the adjacency matrix of graph g = g1 ∨g2. then 138 s. sedghi, n. shobe / j. algebra comb. discrete appl. 7(2) (2020) 121–140 2m2(g1 ∨g2) = s(a3) = s    a1, i′1 i ′ 2, a2    a1, i′1 i ′ 2, a2    a1, i′1 i ′ 2, a2     = s     a12 + i′1i′2 a1i′1 + i′1a2 i ′ 2a1 + a2i ′ 2 i ′ 2i ′ 1 + a2 2    a1, i′1 i ′ 2, a2     = s     a13 + i′1i′2a1 + a1i′1i′2 + i′1a2i′2 a12i′1 + i′1i′2i′1 + a1i′1a2 + i′1a22 i ′ 2a1 2 + a2i ′ 2a1 + i ′ 2i ′ 1i ′ 2 + a2 2i ′ 2 i ′ 2a1i ′ 1 + a2i ′ 2i ′ 1 + i ′ 2i ′ 1a2 + a2 3     = s(a1 3) + s(i ′ 1i ′ 2a1) + s(a1i ′ 1i ′ 2) + s(i ′ 1a2i ′ 2) + s(a1 2i ′ 1) + s(i ′ 1i ′ 2i ′ 1) + s(a1i ′ 1a2) + s(i ′ 1a2 2) + s(i ′ 2a1 2) + s(a2i ′ 2a1) + s(i ′ 2i ′ 1i ′ 2) + s(a2 2i ′ 2) + s(i ′ 2a1i ′ 1) + s(a2i ′ 2i ′ 1) + s(i ′ 2i ′ 1a2) + s(a2 3). from lemma 3.10, we have 2m2(g1 ∨g2) = 2m2(g1) + 4n1n2m1 + 2n2m1(g1) + 2n21n 2 2 + 8m1m2 + 2n21m2 + 2n1m1(g2) + 2n 2 2m1 + 4n1n2m2 + 2m2(g2), and hence m2(g1 ∨g2) = m2(g1) + m2(g2) + n2m1(g1) + n1m1(g2) + 2n1n2m2 + 2n1n2m1 + n 2 1n 2 2 + 4m1m2 + n 2 1m2 + n 2 2m1 = m2(g1) + m2(g2) + n2m1(g1) + n1m1(g2) + (n1n2 + 2m2)(n1n2 + 2m1) + n 2 1m2 + n 2 2m1. references [1] a. alwardi, b. arsić, i. gutman, n. d. soner, the common neighborhood graph and its energy, iran. j. math. sci. inf. 7 (2012) 1–8. 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accepted: 31 may 2015 doi 10.13069/jacodesmath.80607 journal of algebra combinatorics discrete structures and applications the rainbow vertex-index of complementary graphs∗ research article fengnan yanling1∗∗, zhao wang1∗∗∗, chengfu ye1§, shumin zhang1§§ 1. qinghai normal university abstract: a vertex-colored graph g is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. the rainbow vertex-connection number of a connected graph g, denoted by rvc(g), is the smallest number of colors that are needed in order to make g rainbow vertex-connected. if for every pair u, v of distinct vertices, g contains a vertex-rainbow u−v geodesic, then g is strongly rainbow vertex-connected. the minimum k for which there exists a k-coloring of g that results in a strongly rainbow-vertex-connected graph is called the strong rainbow vertex number srvc(g) of g. thus rvc(g) ≤ srvc(g) for every nontrivial connected graph g. a tree t in g is called a rainbow vertex tree if the internal vertices of t receive different colors. for a graph g = (v, e) and a set s ⊆ v of at least two vertices, an s-steiner tree or a steiner tree connecting s (or simply, an s-tree) is a such subgraph t = (v ′, e′) of g that is a tree with s ⊆ v ′. for s ⊆ v (g) and |s| ≥ 2, an s-steiner tree t is said to be a rainbow vertex s-tree if the internal vertices of t receive distinct colors. the minimum number of colors that are needed in a vertex-coloring of g such that there is a rainbow vertex s-tree for every k-set s of v (g) is called the k-rainbow vertex-index of g, denoted by rvxk(g). in this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. the k-rainbow vertex-index of complementary graphs are also studied. 2010 msc: 05c15, 05c40 keywords: strong rainbow vertex-connection number, complementary graph, rainbow vertex s-tree, k-rainbow vertex-index 1. introduction the graphs considered in this paper are finite undirected and simple graphs. we follow the notation of bondy and murty [1], unless otherwise stated. for a graph g, let v (g), e(g), n(g), m(g), and g, respectively, be the set of vertices, the set of edges, the order, the size, and the complement graph of g. ∗ supported by the national science foundation of china (no. 11161037, 11101232, 11461054) and the science found of qinghai province (no. 2014-zj-907). ∗∗ e-mail: fengnanyanlin@yahoo.com (corresponding author) ∗∗∗ e-mail: wangzhao380@yahoo.com § e-mail: yechf@qhnu.edu.cn §§ e-mail: zhsm_0926@sina.com 157 the rainbow vertex-index of complementary graphs let g be a nontrivial connected graph on which an edge-coloring c : e(g) → {1, 2, · · · , n}, n ∈ n, is defined, where adjacent edges may be colored the same. a path is rainbow if no two edges of it are colored the same. an edge-coloring graph g is rainbow connected if any two vertices are connected by a rainbow path. clearly, if a graph is rainbow connected, it must be connected, whereas any connected graph has a trivial edge-coloring that makes it rainbow connected; just color each edge with a distinct color. thus, in [4] l. chen, x. li, h. lian defined the rainbow connection number of a connected graph g, denoted by rc(g), as the smallest number of colors that are needed in order to make g rainbow connected. they showed that rc(g) ≥ diam(g) where diam(g) denotes the diameter of g. for more results on the rainbow connection, we refer to the survey paper [2],[3],[4] and [12], and a new book [10] of li and sun. in [8], krivelevich and yuster proposed the concept of rainbow vertex-connection. a vertex-colored graph g is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. the rainbow vertex-connection number of a connected graph g, denoted by rvc(g), is the smallest number of colors that are needed in order to make g rainbow vertex-connected. for more results on the rainbow vertex-connection, we refer to the survey paper [5] and [9]. an easy observation is that if g is of order n, then rvc(g) ≤ n− 2 and rvc(g) = 0 if and only if g is a complete graph. notice that rvc(g) ≥ diam(g) − 1 with equality if the diameter is 1 or 2. if for every pair u, v of distinct vertices, g contains a vertex-rainbow u − v geodesic, then g is strong rainbow vertex-connected. the definition of strongly rainbow vertex-connected was defined by li et al. in [11]. the minimum k for which there exists a k-coloring of g that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number srvc(g) of g. thus rc(g) ≤ srvc(g) for every nontrivial connected graph g. if g is a nontrivial connected graph of order n whose diameter is diam(g), then diam(g) − 1 ≤ rvc(g) ≤ srvc(g) ≤ n−s, (1) where s denote the number of pendent vertices in g. proposition 1.1. let g be a nontrivial connected graph of order n. then (a) srvc(g) = 0 if and only if g is a complete graph; (b) srvc(g) = 1 if and only if diam(g) = 2 if and only if rvc(g) = 1. a tree t in g is called a rainbow vertex tree if the internal vertices of t receive different colors. for a graph g = (v, e) and a set s ⊆ v of at least two vertices, an s-steiner tree or a steiner tree connecting s (or simply, an s-tree) is a such subgraph t = (v ′, e′) of g that is a tree with s ⊆ v ′. for more problems on s-steiner tree, we refer to [6] and [7]. for s ⊆ v (g) and |s| ≥ 2, an s-steiner tree t is said to be a rainbow vertex s-tree if the internal vertices of t receive distinct colors. the minimum number of colors that are needed in an vertex-coloring of g such that there is a rainbow vertex s-tree for every k-set s of v (g) is called the k-rainbow vertexindex of g, denoted by rvxk(g). the vertex-rainbow index of a graph was first defined by yaping mao in [13]. 2. the strong rainbow vertex-connection of complementary graphs in this section, we investigate the rainbow vertex-connection number of a graph g according to some constraints to its complement g. we give some conditions to guarantee that srvc(g) is bounded by a constant. we investigate the rainbow vertex-connection number of connected complement graphs of graphs with diameter at least 3. 158 fengnan yanling et al. theorem 2.1. if g is a connected graph with diam(g) ≥ 3, then srvc(g) = { 1, if diam(g) ≥ 4; 2, if diam(g) = 3. proof. we choose a vertex x with eccg(x) = diam(g) = d ≥ 3. let nig(x) = {v : dg(x, v) = i} where 0 ≤ i ≤ d. so n0g(x) = {x}, n 1 g(x) = ng(x) as usual. then ⋃ 0≤i≤d n i g(x) is a vertex partition of v (g) with |nig(x)| = ni. let a = ⋃ i is even n i g(x), b = ⋃ i is odd n i g(x). for example, see figure 1, a graph with diam(g) = 5. so, if d = 2k(k ≥ 2), then a = ⋃ 0≤i≤d is even n i g(x), b = ⋃ 1≤i≤d−1 is odd n i g(x); if d = 2k + 1(k ≥ 2) then a = ⋃ 0≤i≤d−1 is even n i g(x), b = ⋃ 1≤i≤d is odd n i g(x). then by the definition of complement graphs, we know that g[a] (g[b]) contains a spanning complete k1-partite subgraph(complete k2-partite subgraph) where k1 = dd+12 e (k2 = d d 2 e). for example, see figure 1, g[a] contains a spanning complete tripartite subgraph kn0,n2,n4, g[b] contains a spanning complete tripartite subgraph kn1,n3,n5. figure 1. graphs for the proof of theorem 2. first of all, we see that g must be connected, since otherwise, diam(g) ≤ 2, contradicting the condition diam(g) ≥ 3. case 1. d ≥ 5. in this case, k1, k2 ≥ 3. we will show that diam(ḡ) ≤ 2 in this case. for u, v ∈ v (g), we consider the following cases: subcase 1.1. u, v ∈ a or u, v ∈ b. if u, v ∈ a, then u, v is contained in the spanning complete k1-partite subgraph of g[a]. thus dg(u, v) ≤ 2. the same is true for u, v ∈ b. subcase 1.2. u ∈ a and v ∈ b. if u = x, v ∈ b, then u is adjacent to all vertices in g[b] \ n1g(x). so dg(u, v) = 1 for v ∈ g[b] \n1g(x). for v ∈ n 1 g(x), let p = ux3v, where x3 ∈ n 3 g(x). clearly, dg(u, v) = 2. if u 6= x, without loss of generality, we assume that u ∈ n2g(x) and v ∈ n 1 g(x). let q = ux5v, where x5 ∈ n5g(x). thus dg(u, v) = 2. from the above, we conclude that diam(g) ≤ 2. so, by proposition 1(b), we have srvc(g) = 1. 159 the rainbow vertex-index of complementary graphs case 2. d = 4. it is obvious that a = n0g(x) ∪ n 2 g(x) ∪ n 4 g(x), b = n 1 g(x) ∪ n 3 g(x). so g[a](g[b]) contains a spanning complete 3-partite subgraph kn0,n2,n4 (complete bipartite subgraph kn1,n3). so, we will show that diam(g) ≤ 2. subcase 2.1. u, v ∈ a or u, v ∈ b. if u, v ∈ a, then u, v is contained in the spanning complete k1-partite subgraph of g[a]. thus dg(u, v) ≤ 2. if u, v ∈ b, then u, v is contained in the spanning complete bipartite subgraph of g[b]. also we have dg(u, v) ≤ 2. subcase 2.2. u ∈ a and v ∈ b. if u = x, v ∈ b, then u is adjacent to all vertices in n3g(x). for v ∈ n 1 g(x), let p = ux3v, where x3 ∈ n3g(x). clearly, dg(u, v) = 2. so dg(u, v) ≤ 2. if u 6= x, then we assume that u ∈ n2g(x) and v ∈ n 1 g(x). let q = ux4v, where x4 ∈ n 4 g(x). thus dg(u, v) = 2. suppose u ∈ n 4 g(x) and v ∈ n 3 g(x). let r = ux1v, where x1 ∈ n 1 g(x). thus dg(u, v) = 2. if u ∈ n2g(x) and v ∈ n 3 g(x), then s = uxv is a path of length 2. then diam(g) ≤ 2. so, by proposition 1, we have srvc(g) = 1. case 3. d = 3. in this case, a = n0g(x) ∪ n 2 g(x), b = n 1 g(x) ∪ n 3 g(x). so g[a] contains a spanning complete bipartite subgraph kn0,n2. so, we give g a vertex-coloring as follows: color vertex x with 1 and color all vertices of n3g(x) with 2. it is easy to see that for any u ∈ n 2 g(x), v ∈ n 1 g(x), there is a rainbow {1, 2} path connecting them in g. so srvc(g) = 2 in this case. for the case of diam(g) = 2, srvc(g) can be very large since diam(g) may be very large. for example, let g = kn \ e(cn), where cn is a cycle of length n in kn. then g = cn and srvc(g) ≥ diam(g) − 1 = dn 2 e− 1 by (1). 3. the k-rainbow vertex-index of complete multipartite graphs theorem 3.1. let kn1,n2,···nl be a complete multipartite graph. if k < 2`, then rvxk = 1; if k ≥ 2`, then rvxk = 2. where s = {v1, v2, · · ·vk} (that is the rainbow s-tree we choose) and vni, (1 ≤ i ≤ l) are the vertices of the partition of kn1,n2,···n`. proof. if k < 2`, then we can find a partition vni, (1 ≤ i ≤ l) of kn1,n2,···nl with vni ∩ s ≤ 1. if vni ∩ s = ∅, then we can choose a vertex v ∈ vni as the root vertex of the rainbow s tree and all the other vertices are leaves. so rvxk(kn1,n2,···n` ) = 1. if vni ∩s = 1, then we choose the vertex v ∈ vni as the root vertex of the rainbow s tree, and all the other vertices are leaves. so rvxk(kn1,n2,···n` ) = 1. if k ≥ 2` and there exists vni such that |s ∩ vni| ≤ 1, then we can choose the vertex v in vni as the root of the rainbow tree and all the other vertices are the leaves the same as when k < 2`. so rvxk(kn1,n2,···nl ) = 1. suppose k ≥ 2` and |s ∩vni| ≥ 2 for any vni. now we give a rainbow vertex-coloring as follows. c(vni ) = { 1, if 1 ≤ i ≤ `− 1; 2, if i = `. next we prove it is a k-rainbow vertex-coloring. choose one vertex v in vn` as the root vertex of the rainbow tree. obviously v is adjacent to all the vertices in vn1 ∩s, vn2 ∩s, . . . vn`−1 ∩s. then choose a vertex in v′ ∈ vn1. since v′ is adjacent to all the remaining vertices in vn` ∩ s, one can prove that the tree is rainbow s-tree. 160 fengnan yanling et al. 4. the k-rainbow vertex-index of complementary graphs theorem 4.1. if g is a connected graph with diam(g) ≥ 3, then rvxk(ḡ) ≤ 2 and the bound is tight. proof. we choose a vertex x with eccg(x) = diam(g) = d ≥ 3 as figure 1. then g[a](g[b]) contains a spanning complete k1-partite subgraph (complete k2-partite subgraph). if the rainbow s-tree contains in g[a](g[b]), then rvxk(ḡ) ≤ 2 by theorem 3.1. now we consider the rainbow s-tree does not contain in g[a] or g[b]. if s ∩ n1g(g) = ∅, then we choose x as the root vertex, and all the other vertices are the leaves. so one can prove that there is a rainbow s-tree. suppose s ∩ n1g(g) 6= ∅. now we give a rainbow vertex-coloring as follows. { c(x) = 1, c(v) = 2, v ∈ v (g)\x. we choose the vertex x as the root of the rainbow tree. we know x is adjacent to all the vertices in n j g(x) ∩ s, (j ∈{2, 3, 4, · · ·}), and there must be a v ∈ n j g(x), (j ∈{2m + 1 and m ≥ 1}) such that v is adjacent to n1g(x) ∩s. one can prove that the tree is rainbow s-tree. let g is a connected graph of diam(g) = 3. we have rvxk(ḡ) = 2, so the bound is tight. acknowledgement: the authors are very grateful to the referees’ valuable comments and suggestions, which helped greatly to improve the presentation of this paper. references [1] j. a. bondy and u. s. r. murty, graph theory, gtm 244, springer, 2008. [2] g. chartrand, g. l. johns, k. a. mckeon and p. zhang, rainbow connection in graphs, math. bohem., 133, 85–98, 2008. [3] g. chartrand, g. l. johns, k. a. mckeon and p. zhang, the rainbow connectivity of a graph, networks, 54(2), 75–81, 2009. [4] l. chen, x. li and h. lian, nordhaus-gaddum-type theorem for rainbow connection number of graphs, graphs combin., 29(5), 1235–1247, 2013. [5] l. chen, x. li and m. liu, nordhaus-gaddum-type theorem for the rainbow vertex connection number of a graph, utilitas math., 86, 335–340, 2011. [6] x. cheng and d. du, steiner trees in industry, kluwer academic publisher, dordrecht, 2001. [7] d. du and x. hu, steiner tree problems in computer communication networks, world scientific, 2008. [8] m. krivelevich and r. yuster, the rainbow connection of a graph is (at most) reciprocal to its minimum degree three, j. graph theory, 63(3), 185-191, 2010. [9] x. li and y. shi, on the rainbow vertex-connection, discuss. math. graph theory, 33(2), 307–313, 2013. [10] x. li and y. sun, rainbow connections of graphs, springerbriefs in math., springer, new york, 2012. [11] x. li, y. mao and y. shi, the strong rainbow vertex-connection of graphs, utilitas math., 93, 213– 223, 2014. [12] x. li, y. shi and y. sun, rainbow connections of graphs–a survey, graphs combin., 29(1), 1-38, 2013. [13] y. mao, the vertex-rainbow index of a graph, arxiv:1502.00151v1 [math.co], 31 jan 2015. [14] e. a. nordhaus and j. w. gaddum, on complementary graphs, amer. math. monthly, 63, 175–177, 1956. 161 introduction the strong rainbow vertex-connection of complementary graphs the k-rainbow vertex-index of complete multipartite graphs the k-rainbow vertex-index of complementary graphs references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.935951 j. algebra comb. discrete appl. 8(2) • 91–105 received: 20 may 2020 accepted: 8 december 2020 journal of algebra combinatorics discrete structures and applications composite g-codes over formal power series rings and finite chain rings research article adrian korban abstract: in this paper, we extend the work done on g-codes over formal power series rings and finite chain rings fq[t]/(ti), to composite g-codes over the same alphabets. we define composite g-codes over the infinite ring r∞ as ideals in the group ring r∞g. we show that the dual of a composite g-code is again a composite g-code in this setting. we extend the known results on projections and lifts of g-codes over the finite chain rings and over the formal power series rings to composite g-codes. additionally, we extend some known results on γ-adic g-codes over r∞ to composite g-codes and study these codes over principal ideal rings. 2010 msc: 94b05, 16s34 keywords: composite g-codes, group rings, finite chain rings, formal power series rings, p-adic integers 1. introduction in [11], t. hurley introduced a map σ which sends the group ring element v ∈ rg to a matrix σ(v) over the ring r. the author also used this map to construct and study codes over fields. the feature of this map is that for different finite groups in the group ring element v, the map σ(v) will produce different matrices over the ring r. for example in [10], the authors show that if v ∈ rd2n then the generator matrix of the form [in | σ(v)] produces the well-known four circulant construction used in coding theory. in [7], the authors apply the above map and study codes generated by 〈σ(v)〉 over the frobenius rings. they define g-codes which are ideals in the group ring rg, where r is a finite commutative frobenius ring and g is a finite group. in [4], the authors study g-codes over formal power series rings and finite chain rings. they extend many well known results on codes over ri and r∞ to g-codes over the same alphabets. the authors also study γ-adic g-codes over r∞ and g-codes over principal ideal rings. adrian korban; department of mathematical and physical sciences, university of chester, thornton science park, pool ln, chester ch2 4nu, england (email: adrian3@windowslive.com). 91 https://orcid.org/0000-0001-5206-6480 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 recently in [3], the authors extended the map σ introduced by t. hurley in [11], so that the group ring element v gets sent to more complex matrices over the ring r. the authors denote this map ω and call the matrices ω(v) the composite matricessee [3] for details. in [6], the authors introduce and study composite g-codes which are defined by taking the row space of the composite matrix ω(v), i.e., 〈ω(v)〉. they also extend many results from [4] on g-codes to composite g-codes. in this work, we generalize the results on g-codes over formal power series rings and finite chain rings fq[t]/(ti) from [4] and some results from [8] to composite g-codes over the same alphabets. we study the projections and lifts of composite g-codes over the finite chain rings and over the formal power series rings respectively. we also extend the results on γ-adic g-codes over r∞ to composite g-codes and some results on g-codes over principal ideal rings to composite g-codes. in many parts of this work, the results we present are a simple generalization or a consequence of the results proven in [4] and [8]. the rest of the work is organized as follows. in section 2, we give preliminary definitions and results on codes, finite chain rings, formal power series and composite g-codes. in section 3, we show that the composite g-codes are ideals in the group ring r∞g. in section 4, we study the projections and lifts of the composite g-codes with a given type. in sections 5 and 6, we extend the results from [4]; we study self-dual γ-adic composite g-codes and composite g-codes over principal ideal rings. we finish with concluding remarks and directions for possible future research. 2. preliminaries 2.1. codes we shall give the definitions for codes over rings. for a complete description of algebraic coding theory in this setting, see [2]. let r be a commutative ring. a code of length n over r is a subset of rn and a code is linear if it is a submodule of the ambient space rn. we assume that all finite rings we use as alphabets are frobenius, where a frobenius ring is characterized by the following. let r̂ be the character module of the ring r. for a finite ring r the following are equivalent: • r is a frobenius ring. • as a left module, r̂ ∼= rr. • as a right module, r̂ ∼= rr. the hamming weight of a vector is the number of non-zero coordinates in that vector and the minimum weight of a code is the smallest weight of all non-zero vectors in the code. we define the standard inner-product on the ambient space, namely [v,w] = ∑ viwi. we define the orthogonal with respect to this inner-product as: c⊥ = {v ∈ rn | [v,w] = 0,∀w ∈c}. the code c⊥ is linear, whether or not c is. if r is a finite frobenius ring, then we have that (c⊥)⊥ = c for all linear codes c over r. however, if r is infinite this is not always true. definition 2.1. a linear code c over an infinite ring r is called basic if c = (c⊥)⊥. 2.2. finite chain rings and formal power series rings we recall the definitions and properties of a finite chain ring r and the formal power series ring r∞. we refer the reader to [8] and [9] for details and further explanations. in this paper, we assume that all 92 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 rings have a multiplicative identity and that all rings are commutative. we also stress that the results we present in this work are given only for finite chain rings fq[t]/(ti). 2.2.1. finite chain rings a ring is called a chain ring if its ideals are linearly ordered by inclusion. in particular, this means that any finite chain ring has a unique maximal ideal. let r be a finite chain ring. denote the unique maximal ideal of r by m, and let γ̃ be the generator of the unique maximal ideal m. this gives that m = 〈γ̃〉 = rγ̃, where rγ̃ = 〈γ̃〉 = {βγ̃ | β ∈ r}. we have the following chain of ideals: r = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃i〉⊇ ·· · . (1) the chain in (1) can not be infinite, since r is finite. therefore, there exists i such that 〈γ̃i〉 = {0}. let e be the minimal number such that 〈γ̃e〉 = {0}. the number e is called the nilpotency index of γ̃. this gives that for a finite chain ring we have the following: r = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃e〉. (2) if the ring r is infinite then the chain in equation 1 is also infinite. let r× denote the multiplicative group of all units in the ring r. let f = r/m = r/〈γ̃〉 be the residue field with characteristic p, where p is a prime number, then |f| = q = pr for some integers q and r. we know that |f×| = pr −1. we now state two well-known lemmas for which the proofs can be found in [12]. lemma 2.2. for any 0 6= r ∈ r there is a unique integer i, 0 ≤ i < e such that r = µγ̃i, with ν a unit. the unit µ is unique modulo γ̃e−i. lemma 2.3. let r be a finite chain ring with maximal ideal m = 〈γ̃〉, where γ̃ is a generator of m with nilpotency index e. let v ⊆ r be a set of representatives for the equivalence classes of r under congruence modulo γ̃. then (i) for all r ∈ r there are unique r0, · · · ,re−1 ∈ v such that r = ∑e−1 i=0 riγ̃ i; (ii) |v | = |f|; (iii) |〈γ̃j〉| = |f|r−j for 0 ≤ j ≤ e− 1. from lemma 2.3, we know that any element ã of r can be written uniquely as ã = a0 + a1γ̃ + · · · + ae−1γ̃e−1, where the ai can be viewed as elements in the field f. it is well-known that the generator matrix for a code c over a finite chain ring ri, where i < ∞ is permutation equivalent to a matrix of the following form: g =   ik0 a0,1 a0,2 a0,3 a0,e γik1 γa1,2 γa1,3 γa1,e γ2ik2 γ 2a2,3 γ 2a2,e ... ... ... ... γe−1ike−1 γ e−1ae−1,e   , (3) where e is the nilpotency index of γ. this matrix g is called the standard generator matrix form for the code c. in this case, the code c is said to have type 1k0γk1 (γ2)k2 . . . (γe−1)ke−1. (4) 93 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 2.2.2. formal power series rings in the next definitions, which can be found in [8], γ will indicate the generator of the ideal of a chain ring, not necessarily the maximal ideal. definition 2.4. the ring r∞ is defined as a formal power series ring: r∞ = f[[γ]] = { ∞∑ l=0 alγ l|al ∈ f}. let i be an arbitrary positive integer. the rings ri are defined as follows: ri = {a0 + a1γ + · · · + ai−1γi−1|ai ∈ f}, where γi−1 6= 0, but γi = 0 in ri. if i is finite or infinite then the operations over ri are defined as follows: i−1∑ l=0 alγ l + i−1∑ l=0 blγ l = i−1∑ l=0 (al + bl)γ l (5) i−1∑ l=0 alγ l · i−1∑ l′=0 bl′γ l′ = i−1∑ s=0 ( ∑ l+l′=s albl′)γ s. (6) the following results can be found in [8]. 1. the ring ri is a chain ring with the maximal ideal 〈γ〉 for all i < ∞. 2. the multiplicative group r×∞ = { ∑∞ j=0 ajγ j|a0 6= 0}. 3. the ring r∞ is a principal ideal domain. let c be a finitely generated linear code over r∞. then the generator matrix of code c is permutation equivalent to the following standard form generator matrix. let c be a finitely generated, nonzero linear code over r∞ of length n, then any generator matrix of c is permutation equivalent to a matrix of the following form: g =   γm0ik0 γ m0a0,1 γ m0a0,2 γ m0a0,3 γ m0a0,r γm1ik1 γ m1a1,2 γ m1a1,3 γ m1a1,r γm2ik2 γ m2a2,3 γ m2a2,r ... ... ... ... γmr−1ikr−1 γ mr−1ar−1,r   , (7) where 0 ≤ m0 < m1 < · · · < mr−1 for some integer r. the column blocks have sizes k0,k1, . . . ,kr and ki are nonnegative integers adding to n. definition 2.5. a code c with generator matrix of the form given in equation 7 is said to be of type (γm0 )k0 (γm1 )k1 . . . (γmr−1 )kr−1, where k = k0 + k1 + · · · + kr−1 is called its rank and kr = n−k. a code c of length n with rank k over r∞ is called a γ-adic [n,k] code. we call k the dimension of c and we write by dim c = k. 94 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 let i,j be two integers with i ≤ j, we define a map ψ j i : rj → ri, (8) j−1∑ l=0 alγ l 7→ i−1∑ l=0 alγ l. (9) if we replace rj with r∞ then we obtain a map ψ∞i . for convenience, we denote it by ψi. it is easy to get that ψji and ψi are ring homomorphisms. let a,b be two arbitrary elements in rj. it is easy to get that ψ j i (a + b) = ψ j i (a) + ψ j i (b), ψ j i (ab) = ψ j i (a)ψ j i (b). (10) if a,b ∈ r∞, we have that ψi(a + b) = ψi(a) + ψi(b), ψi(ab) = ψi(a)ψi(b). (11) note that the map ψji and ψi can be extended naturally from r n j to r n i and r n ∞ to r n i . the construction method above gives a chain of rings where ri is a finite ring for all finite i and r∞ is an infinite principal ideal domain. this gives the following diagram: r f ‖ ‖ r∞ → ··· → re → re−1 → ··· → r1 2.3. composite g-codes in this section, we define a circulant matrix, give the definitions for group rings and introduce composite gcodes. a circulant matrix is one where each row is shifted one element to the right relative to the preceding row. we label the circulant matrix as a = circ(α1,α2, . . . ,αn), where αi are ring elements. we shall now give the necessary definitions for group rings. let g be a finite group of order n and let r be a ring, then the group ring rg consists of ∑n i=1 αigi, αi ∈ r, gi ∈ g. addition in the group ring is done by coordinate addition, namely n∑ i=1 αigi + n∑ i=1 βigi = n∑ i=1 (αi + βi)gi. (12) the product of two elements in a group ring is given by ( n∑ i=1 αigi)( n∑ j=1 βjgj) = ∑ i,j αiβjgigj. (13) it follows that the coefficient of gk in the product is ∑ gigj=gk αiβj. the following matrix construction was first introduced in [3]. in [6], the authors have shown that the same construction produces codes in rn from elements in the group ring rg. let {g1,g2, . . . ,gn} be a fixed listing of the elements of g. let {h1,h2, . . . ,hr} be a fixed listing of the elements of h, where h is a group of order r. here, let r be a factor of n with n > r and n,r 6= 1. also, let gr be a subset of g containing r distinct elements of g. define the map: 95 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 φ : h 7→ gr h1 φ−→ g1 h2 φ−→ g2 ... ... ... hr φ−→ gr. next, let v = αg1g1 + αg2g2 + · · · + αgngn ∈ rg. define the matrix ω(v) ∈ mn(r) to be ω(v) =   a1 a2 a3 . . . an r an r +1 an r +2 an r +3 . . . a2n r ... ... ... ... ... a(r−1)n r +1 a(r−1)n r +2 a(r−1)n r +3 . . . an2 r2   , (14) where at least one block has the following form: a′l =   αg−1 j gk αg−1 j gk+1 . . . αg−1 j gk+(r−1) αφl((hl)−12 (hl)1) αφl((hl)−12 (hl)2) . . . αφl((hl)−12 (hl)r) αφl((hl)−13 (hl)1) αφl((hl)−13 (hl)2) . . . αφl((hl)−13 (hl)r) ... ... ... ... αφl((hl)−1r (hl)1) αφl((hl)−1r (hl)2) . . . αφl((hl)−1r (hl)r)   , and the other blocks are of the form: al =   αg−1 j gk αg−1 j gk+1 . . . αg−1 j gk+(r−1) αg−1 j+1 gk αg−1 j+1 gk+1 . . . αg−1 j+1 gk+(r−1) αg−1 j+2 gk αg−1 j+2 gk+1 . . . αg−1 j+2 gk+(r−1) ... ... ... ... αg−1 j+r−1gk αg−1 j+r−1gk+1 . . . αg−1 j+r−1gk+(r−1)   , where l = {1, 2, 3, . . . , n 2 r2 } and where: φl : hi 7→ gr (hi)1 φl−→ g−1j gk (hi)2 φl−→ g−1j gk+1 ... ... ... (hi)r φl−→ g−1j gk+(r−1). . here we notice that when when l = 1 then j = 1,k = 1, when l = 2 then j = 1,k = r + 1, when l = 3 then j = 1,k = 2r + 1, . . . when l = n r then j = 1,k = n − r + 1. when l = n r + 1 then j = r + 1,k = 1, when l = n r + 2 then j = r + 1,k = r + 1, when l = n r + 3 then j = r + 1,k = 2r + 1, . . . when l = 2n r then j = r + 1,k = n−r + 1, . . . , and so on. in [6], it is shown that the matrix ω(v) can be written as: ω(v) =   αg−111 g1 αg−112 g2 αg−113 g3 . . . αg−11n gn αg−121 g1 αg−122 g2 αg−123 g3 . . . αg−12n gn ... ... ... ... ... αg−1n1 g1 αg−1n2 g2 αg−1n3 g3 . . . αg−1nngn   , 96 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 where g−1ji are simply the elements of the group g. these elements are determined by how the matrix has been partitioned, what groups hi of order r have been employed and how the maps φl have been defined to form the composite matrix. this representation of the composite matrix ω(v) will make it easier to prove the upcoming results. for a given element v ∈ rg and some groups hl of order r, we define the following code over the ring r : c(v) = 〈ω(v)〉. (15) the code is formed by taking the row space of ω(v) over the ring r. the code c(v) is a linear code over the ring r, since it is the row space of a generator matrix. it is not possible to determine the size of the code immediately from the matrix. in [6], it is shown that such codes are ideals in the group ring rg, and are held invariant by the action of the elements of g. such codes are referred to as composite g-codes. we note that the matrix ω(v) is an extension of the matrix σ(v) defined in [11]. also, in [6], the authors show when the matrices ω(v) are inequivalent to the matrices obtained from σ(v). this is one reason to study codes constructed from ω(v)this technique can produce codes which can not be obtained from codes constructed from σ(v) or other classical techniques. for example, please see [5] where many new binary self-dual codes are constructed via the composite matrices. 3. composite g-codes and ideals in the group ring r∞g in this section, we show that the composite gcodes are ideals in the group ring r∞g and that the dual of the composite gcode is also a composite gcode in this setting. these two results are a simple generalization of theorem 3.1 and theorem 3.2 from [4]. we use the same arguments as in [4] to prove our results. for simplicity, we write each non-zero element in r∞ in the form γia where a = a0 + a1γ + · · ·+ · · · with a0 6= 0 and i ≥ 0, which means that a is a unit in r∞. we note that if v = γlg1 ag1g1 +γ lg2 ag2g2 +· · ·+γlgn agngn ∈ r∞g, then each row of ω(v) corresponds to an element in r∞g of the following form: v∗j = n∑ i=1 γ lgji gi agjigigjigi, (16) where γlgji gi agjigi ∈ r∞, gi,gji ∈ g and j is the jth row of the matrix ω(v). in other words, we can define the composite matrix ω(v) as: ω(v) =   γ lg11 g1 ag11g1 γ lg12 g2 ag12g2 γ lg13 g3 ag13g3 . . . γ lg1n gn ag1ngn γ lg21 g1 ag21g1 γ lg22 g2 ag22g2 γ lg23 g3 ag23g3 . . . γ lg2n gn ag2ngn ... ... ... ... ... γ lgn1 g1 agn1g1 γ lgn2 g2 agn2g2 γ lgn3 g3 agn3g3 . . . γ lgnn gn agnngn   , (17) where the elements gji are simply the group elements g. which elements of g these are, depends how the composite matrix is defined, i.e., what groups we employ and how we define the φl map in individual blocks. then we take the row space of the matrix ω(v) over r∞ to get the corresponding composite g-code, namely c(v). 97 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 theorem 3.1. let r∞ be the formal power series ring and g a finite group of order n. let hi be finite groups of order r such that r is a factor of n with n > r and n,r 6= 1. also, let v ∈ r∞g and let c(v) = 〈ω(v)〉 be the corresponding code in rn∞. let i(v) be the set of elements of r∞g such that∑ γliaigi ∈ i(v) if and only if (γl1a1,γl2a2, . . . ,γlnan) ∈c(v). then i(v) is a left ideal in r∞g. proof. we saw above that the rows of ω(v) consist precisely of the vectors that correspond to the elements of the form v∗j = ∑n i=1 γ lgji gi agjigigjigi in r∞g, where γ lgji gi agjigi ∈ r∞, gi,gji ∈ g and j is the jth row of the matrix ω(v). let a = ∑ γliaigi and b = ∑ γljbjgi be two elements in i(v), then a + b = ∑ (γliai + γ ljbj)gi, which corresponds to the sum of the corresponding elements in c(v). this implies that i(v) is closed under addition. let w1 = ∑ γlibigi ∈ r∞g. then if w2 corresponds to a vector in c(v), it is of the form ∑ (γljαj)v ∗ j . then w1w2 = ∑ γlibigi ∑ (γljαj)v ∗ j = ∑ γlibiγ ljαjgiv ∗ j which corresponds to an element in c(v) and gives that the element is in i(v). therefore i(v) is a left ideal of r∞g. next we show that the dual of a composite g-code is also a composite g-code. let i be an ideal in a group ring r∞g. define r(c) = {w | vw = 0, ∀v ∈ i}. it follows that r(i) is an ideal of r∞g. let v = γlg1 ag1g1 + γ lg2 ag2g2 + · · · + γlgn agngn ∈ r∞g and c(v) be the corresponding code. let ω : r∞g → rn∞ be the canonical map that sends γlg1 ag1g1 + γlg2 ag2g2 + · · · + γlgn agngn to (γlg1 ag1,γ lg2 ag2, · · · ,γlgn agn ). let i be the ideal ω−1(c). let w = (w1,w2, . . . ,wn) ∈ c⊥. then the operator of product between any row of ω(v) and w is zero: [(γ lgj1 g1 agj1g1,γ lgj2 g1 agj2g1, . . . ,γ lgjn g1 agjng1 ), (w1,w2, . . . ,wn)] = 0, ∀j. (18) which gives n∑ i=1 γ lgji gi agjigiwi = 0, ∀j. (19) let w = ω−1(w) = ∑ γkgi wgigi and define w ∈ r∞g to be w = γkg1 bg1g1 + γkg2 bg2g2 + · · · + γkgn bgngn, where γkgi bgi = γ k g −1 i wg−1 i . (20) then n∑ i=1 γ lgji gi agjigiwi = 0 =⇒ n∑ i=1 γ lgji gi agjigiγ k g −1 i bg−1 i = 0. (21) here, gjigig −1 i = gji, thus this is the coefficient of gji in the product of w and v ∗ j , where v ∗ j is any row of the matrix ω(v). this gives that w ∈r(i) if and only if w ∈c⊥. let φ : rn∞ → r∞g by φ(w) = w, then this map is a bijection between c⊥ and r(ω−1(c)) = r(i). theorem 3.2. let c = c(v) be a code in r∞g formed from the vector v ∈ r∞g. then ω−1(c⊥) is an ideal of r∞g. proof. the composite mapping ω(φ(c⊥)) is permutation equivalent to c⊥ and φ(c⊥) is an ideal of r∞g. we know that φ is a bijection between c⊥ and r(ω−1(c)), and we also know that ω−1(c) is an ideal of r∞g as well. this proves that the dual of a composite g-code is also a composite g-code over the formal power series ring. 98 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 4. projections and lifts of composite g-codes in this section, we extend more results from [4]. in fact, many of the results presented in this section are a consequence of the results proven in [8] and a simple generalization of the results proven in [4]. we first show that if v ∈ r∞g then ω(v) is permutation equivalent to the matrix defined in equation 7. for simplicity, we write each non-zero element in r∞ in the form γia where a = a0 +a1γ+· · ·+· · · with a0 6= 0 and i ≥ 0, which means that a is a unit in r∞. theorem 4.1. let v = γlgi ag1g1 + γ lg2 ag2g2 + · · · + γlgn agngn ∈ r∞g, where agi are units in r∞. let c be a finitely generated code over r∞. then ω(v) =   γ lg11 g1 ag11g1 γ lg12 g2 ag12g2 γ lg13 g3 ag13g3 . . . γ lg1n gn ag1ngn γ lg21 g1 ag21g1 γ lg22 g2 ag22g2 γ lg23 g3 ag23g3 . . . γ lg2n gn ag2ngn ... ... ... ... ... γ lgn1 g1 agn1g1 γ lgn2 g2 agn2g2 γ lgn3 g3 agn3g3 . . . γ lgnn gn agnngn   , is permutation equivalent to the standard generator matrix given in equation 7. proof. take one non-zero element of the form γm0agi, where m0 is the minimal non-negative integer. by applying column and row permutations and by dividing a row by a unit, the element that corresponds to the first row and column of ω(v) can be replaced by γm0. the elements in the first column of matrix ω(v) have the form γlgj agj with lgj ≥ m0 and agj a unit, thus, these can be replaced by zero when they are added to the first row multiplied by −γlgj−m0 (agj )−1. continuing the process using elementary operations, we obtain the standard generator matrix of the code c given in equation 7. example 4.2. let g = 〈x,y | x4 = 1,y2 = x2,yxy−1 = x−1〉∼= q8. let v = ∑3 i=0 ( αi+1x i + αi+5x iy ) ∈ r∞q8, where αi = αgi ∈ r∞. let h1 = 〈a,b | a2 = b2 = 1,ab = ba〉 ∼= c2 × c2. we now define the composite matrix as: ω(v) = ( a′1 a2 a3 a ′ 4 ) =   αg−11 g1 αg−11 g2 αg−11 g3 αg−11 g4 αg−11 g5 αg−11 g6 αg−11 g7 αg−11 g8 αφ1((h1)−12 (h1)1) αφ1((h1)−12 (h1)2) αφ1((h1)−12 (h1)3) αφ1((h1)−12 (h1)4) αg−12 g5 αg−12 g6 αg−12 g7 αg−12 g8 αφ1((h1)−13 (h1)1) αφ1((h1)−13 (h1)2) αφ1((h1)−13 (h1)3) αφ1((h1)−13 (h1)4) αg−13 g5 αg−13 g6 αg−13 g7 αg−13 g8 αφ1((h1)−14 (h1)1) αφ1((h1)−14 (h1)2) αφ1((h1)−14 (h1)3) αφ1((h1)−14 (h1)4) αg−14 g5 αg−14 g6 αg−14 g7 αg−14 g8 αg−15 g1 αg−15 g2 αg−15 g3 αg−15 g4 αg−15 g5 αg−15 g6 αg−15 g7 αg−15 g8 αg−16 g1 αg−16 g2 αg−16 g3 αg−16 g4 αφ4((h1)−12 (h1)1) αφ4((h1)−12 (h1)2) αφ4((h1)−12 (h1)3) αφ4((h1)−12 (h1)4) αg−17 g1 αg−17 g2 αg−17 g3 αg−17 g4 αφ4((h1)−13 (h1)1) αφ4((h1)−13 (h1)2) αφ4((h1)−13 (h1)3) αφ4((h1)−13 (h1)4) αg−18 g1 αg−18 g2 αg−18 g3 αg−18 g4 αφ4((h1)−14 (h1)1) αφ4((h1)−14 (h1)2) αφ4((h1)−14 (h1)3) αφ4((h1)−14 (h1)4)   , where: φ1 : (h1)i φ1−→ g−11 gi φ4 : (h1)i φ4−→ g−15 gj for i = {1,2,3,4} for when {i = 1, . . . ,4 and j = i + 4}, in a′1 and a ′ 4 respectively. this results in a composite matrix over r∞ of the following form: 99 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 ω(v) =   x1 y1 x2 y1 x1 x3 x4 y4 y4 x4   =   α1 α2 α3 α4 α5 α6 α7 α8 α2 α1 α4 α3 α8 α5 α6 α7 α3 α4 α1 α2 α7 α8 α5 α6 α4 α3 α2 α1 α6 α7 α8 α5 α7 α6 α5 α8 α1 α4 α3 α2 α8 α7 α6 α5 α4 α1 α2 α3 α5 α8 α7 α6 α3 α2 α1 α4 α6 α5 α8 α7 α2 α3 α4 α1   . if we let v = γ2x3 + γ2(1 + γ)xy + γ2(1 + γ + γ2)x2y + γ2x3y ∈ r∞q8, where 〈x,y〉∼= q8, then c(v) = 〈ω(v)〉 =   0 0 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 0 γ2 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) 0 γ2 0 0 γ2(1 + γ + γ2) γ2 0 γ2(1 + γ) γ2 0 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 γ2(1 + γ + γ2) γ2(1 + γ) 0 γ2 0 γ2 0 0 γ2 γ2(1 + γ + γ2) γ2(1 + γ) 0 γ2 0 0 0 0 γ2 γ2(1 + γ + γ2) γ2(1 + γ) 0 0 0 γ2 γ2(1 + γ) 0 γ2 γ2(1 + γ + γ2) 0 0 γ2 0   , and c(v) is equivalent to  γ2 0 0 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 γ2 0 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) 0 0 γ2 0 γ2(1 + γ + γ2) γ2 0 γ2(1 + γ) 0 0 0 γ2 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0   . clearly c(v) = 〈ω(v)〉 is the [8, 4, 4] extended hamming code. we now generalize the results from [4] on the projection of codes with a given type. proposition 4.3. let c be a composite g-code over r∞ of type {(γm0 )k0, (γm1 )k1, . . . , (γmr−1 )kr−1} with generator matrix ω(v). the code generated by ψi(ω(v)) is a code over ri of type {(γm0 )k0, (γm1 )k1, . . . , (γms−1 )ks−1} where ms is the largest mi that is less than e. also, the code generated by ψi(ω(v)) is equal to {(ψi(c1), ψi(c2), . . . , ψi(cn)) | (c1,c2, . . . ,cn) ∈c}. (22) proof. if mi > e − 1 then ψi sends γmim′, where m′ is a matrix, to a zero matrix which gives the first part. the code c is formed by taking the row space of ω(v) over the ring r∞, i.e. γl1a1v1 +γl2a2v2 +· · ·+ γlnanvn where γliai ∈ r∞ and vi are the rows of ω(v). if w = γljajvj, then ψi(w) = ψi(γliai)ψi(vi) by the equation given in (11) where ψi(vi) applies the map coordinate-wise. this gives the second part. since a composite gcode over r∞ is a linear code, the following results are a direct consequence of some results proven in [8]. we omit the proofs. lemma 4.4. let c be a composite g-code of length n over r∞, then, 100 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 (1) c⊥ has type 1m for some m, (2) c = (c⊥)⊥ if and only if c has type 1k for some k, (3) if c has a standard generator matrix g as in equation (7), then we have (i) the dual code c⊥ of c has a generator matrix h = ( b0,r b0,r−1 . . . b0,2 b0,1 ikr ) , (23) where b0,j = − ∑j−1 l=1 b0,la t r−j,r−l −a t r−j,r for all 1 ≤ j ≤ r; (ii) rank(c)+rank(c⊥)=n. example 4.5. if we take the generator matrix g of a code c from example 1, we can see that g =  γ2   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1   γ2   0 1 + γ 1 + γ + γ2 1 1 0 1 + γ 1 + γ + γ2 1 + γ + γ2 1 0 1 + γ 1 + γ 1 + γ + γ2 1 0     , which is the standard generator matrixhere, a0,1 =   0 1 + γ 1 + γ + γ2 1 1 0 1 + γ 1 + γ + γ2 1 + γ + γ2 1 0 1 + γ 1 + γ 1 + γ + γ2 1 0   . in this case the generator matrix of the dual code c⊥ of c has the form: h = ( b0,1 ik1 ) . now, b0,1 = −at0,1, thus h =   0 −(1 + γ) −(1 + γ + γ2) −1 1 0 0 0 −1 0 −(1 + γ) −(1 + γ + γ2) 0 1 0 0 −(1 + γ + γ2) −1 0 −(1 + γ) 0 0 1 0 −(1 + γ) −(1 + γ + γ2) −1 0 0 0 0 1   . we also have rank(c) + rank(c⊥) = 4 + 4 = 8 = n. proposition 4.6. let c be a self-orthogonal composite g-code over r∞. then the code ψi(c) is a selforthogonal composite g-code over ri for all i < ∞. proof. we first show that ψi(c) is self-orthogonal. let v ∈ r∞g and 〈ω(v)〉 = c(v) be the corresponding self-orthogonal composite g-code. this implies that [v,w] = 0 for all v,w ∈ 〈ω(v)〉 = c(v). this gives that n∑ l=1 vlwl ≡ n∑ l=1 ψi(vl)ψi(wl)(mod γi) ≡ ψi([v,w])(mod γi) ≡ 0 (mod γi). hence ψi(c) is a self-orthogonal code over ri. to show that ψi(c) is also a g-code, we notice that when taking ψi(c) = ψi(〈ω(v)〉), it corresponds to ψi(v) = ψi(γlg1 ag1 )g1 + ψi(γlg2 ag2 )g2 +· · ·+ ψi(γlgn agn )gn, then ψi(c) ∈ rig. thus ψi(c) is also a composite g-code. 101 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 definition 4.7. let i,j be two integers such that 1 ≤ i ≤ j < ∞. we say that an [n,k] code c1 over ri lifts to an [n,k] code c2 over rj, denoted by c1 � c2, if c2 has a generator matrix g2 such that ψ j i (g2) is a generator matrix of c1. we also denote c1 by ψ j i (c2). if c is a [n,k] γ-adic code, then for any i < ∞, we call ψi(c) a projection of c. we denote ψi(c) by ci. lemma 4.8. let c be a composite g-code over r∞ with type 1k. if ω(v) is a standard form of c, then for any positive integer, i, ψi(ω(v)) is a standard form of ψi(c). proof. we know from theorem 4.1 that ω(v) is permutation equivalent to a standard form matrix defined in equation 7. we also have that c has type 1k, hence ψi(c) has type 1k. the rest of the proof is the same as in [8]. in the following, to avoid confusion, we let v∞ and v be elements of the group rings r∞g and rig respectively. let v∞ = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ r∞g, and c(v∞) = 〈ω(v∞)〉 be the corresponding composite g-code. define the following map: ω1 : r∞g →c(v∞), (γlg1 ag1g1 + γ lg2 ag2g2 + · · · + γ lgn agngn) 7→ m(r∞g,v∞). we define a projection of composite g-codes over r∞g to rig. let ψi : r∞g → rig (24) γia 7→ ψ(γia). (25) the projection is a homomorphism which means that if i is an ideal of r∞g, then ψi(i) is an ideal of rig. we have the following commutative diagram: rn∞g ω1−→ c(v∞) ψi ↓ ↓ ψi rni g −→ ω1 c(v) . this gives that ψiω1 = ω1ψi, which gives the following theorem. theorem 4.9. if c is a composite g-code over r∞, then ψi(c) is a composite g-code over ri for all i < ∞. proof. let v∞ ∈ r∞g and c(v∞) be the corresponding composite g-code over r∞. then ω1(v∞) = c(v∞) is an ideal of r∞g. by the homomorphism in equation 24 and the commutative diagram above, we know that ψi(ω1(v∞)) = ω1(ψi(v∞)) is an ideal of the group ring rig. this implies that ψi(c) is a composite g-code over ri for all i < ∞. theorem 4.10. let c be a composite g-code over ri, then the lift of c, c̃ over rj, where j > i, is also a composite g-code. proof. let v1 = αg1g1 + αg2g2 + · · · + αgngn ∈ rig and c = 〈ω(v1)〉 be the corresponding composite g-code. let v2 = βg1g1 + βg2g2 + · · · + βgngn ∈ rjg and c̃ = 〈ω(v2)〉 be the corresponding composite g-code. we can say that v1 and v2 act as generators of c and c̃ respectively. we can clearly see that we can have ψji (v2) = ψ j i (βg1 )g1 + ψ j i (βg2 )g2 + · · · + ψ j i (βgn )gn = αg1g1 + αg2g2 + · · · + αgngn ∈ rig, thus ψji (v2) is a generator matrix of c. this implies that the composite g-code c(v1) over ri lifts to a composite g-code over rj, for all j > i. 102 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 the following results consider composite g-codes over chain rings that are projections of γ-adic codes. the results are just a simple consequence of the results proven in [8]. for details on notation and proofs, please refer to [8] and [4]. lemma 4.11. let c be a [n,k] composite g-code of type 1k, and g,h be a generator and parity-check matrices of c. let gi = ψi(g) and hi = ψi(h). then gi and hi are generator and parity check matrices of ci respectively. let i < j < ∞ be two positive integers, then (i) γj−igi ≡ γj−igj (mod γj); (ii) γj−ihi ≡ γj−ihj (mod γj). (iii) γj−1ci ⊆cj; (iv) v = γiv0 ∈cj if and only if v0 ∈cj−i; (v) ker(ψji)=γ icj−i. theorem 4.12. let c be a composite g-code over r∞. then the following two results hold. (i) the minimum hamming distance dh(ci) of ci is equal to d = dh(c1) for all i < ∞; (ii) the minimum hamming distance d∞ = dh(c) of c is at least d = dh(c1). the final two results we present in this section are a simple extension of the two results from [8] on mds and mdr codes over r∞. we omit the proofs since a composite gcode over r∞ is a linear code and for that fact, the proofs are the same as in [8]. theorem 4.13. let c be a composite g-code over r∞. if c is an mdr or mds code then c⊥ is an mds code. theorem 4.14. let c be a composite g-code over ri, and c̃ be a lift of c over rj, where j > i. if c is an mds code over ri then the code c̃ is an mds code over rj. 5. self-dual γ-adic composite g-codes in this section, we extend some results for self-dual γ-adic codes to composite g-codes over r∞. as in previous sections, the results presented here are just a simple generalization of the results proven in [8] and [4]. fix the ring r∞ with r∞ →···→ ri →···→ r2 → r1 and r1 = fq where q = pr for some prime p and nonnegative integer r. the field fq is said to be the underlying field of the rings. we now generalize four theorems from [8]. the first two consider self-dual codes over ri with a specific type and projections of self-dual codes over r∞ respectively. the third one considers a method for constructing self-dual codes over f from a self-dual code over ri. we extend these to self-dual composite g-codes over ri and r∞ respectively. theorem 5.1. let i be odd and c be a composite g-code over ri with type 1k0 (γ)k1 (γ2)k2 . . . (γi−1)ki−1. then c is a self-dual code if and only if c is self-orthogonal and kj = ki−j for all j. proof. it is enough to show that ω(v) where v ∈ rig and g is a finite group, is permutation equivalent to the matrix (3). the rest of the proof is the same as in [8]. 103 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 theorem 5.2. if c is a self-dual composite g-code of length n over r∞ then ψi(c) is a self-dual composite g-code of length n over ri for all i < ∞. proof. this is a direct consequence of theorem 3.4 in [8] and proposition 4.4 of this work. theorem 5.3. let i be odd. a self-dual composite g-code of length n over ri induces a self-dual composite g-code of length n over fq. proof. the first part of the proof is identical to the one of theorem 5.5 from [4]. secondly, when the map ψi1(g̃) is used in [8], we notice that in our case the map will correspond to ψ i 1(g̃) = ψ i 1(v) = ψi1(γ lg1 ag1 )g1 + ψ i 1(γ lg2 ag2 )g2 + · · · + ψi1(γlgn agn )gn, assuming that g̃ is the generator matrix of a composite g-code and v ∈ rig. then ψi1(g̃) is the generator matrix of a composite g-code over fq. theorem 5.4. let r = re be a finite chain ring, f = r/〈γ〉, where |f| = q = pr, 2 6= p is a prime. then any self-dual composite g-code c over f can be lifted to a self-dual composite g-code over r∞. proof. from theorem 4.10 we know that a composite g-code over ri can be lifted to a composite gcode over rj, where j > i. to show that a self-dual composite g-code over f lifts to a self-dual composite g-code over r∞, it is enough to follow the proof in [8]. 6. composite g-codes over principal ideal rings in this section, we study composite g-codes over principal ideal rings. we study codes over this class of rings by the generalized chinese remainder theorem. please see [2] for more details on the notation and definitions of the principal ideal rings. let r1e1,r 2 e2 , . . . ,rses be chain rings, where r j ej has unique maximal ideal 〈γj〉 and the nilpotency index of γj is ej. let fj = rjej/〈γj〉. let a = crt(r1e1, . . . ,r j ej , . . . ,rses ). we know that a is a principal ideal ring. for any 1 ≤ i < ∞, let a j i = crt(r 1 e1 , . . . ,r j i , . . . ,r s es ). this gives that all the rings aji are principal ideal rings. in particular, a j ej = a. we denote crt(r1e1 . . . ,r j ∞, . . . ,r s es ) by aj∞. for 1 ≤ i < ∞, let cji be a code over r j i . let cji = crt(c 1 e1 , . . . ,cji , . . . ,c s es ) be the associated code over aji. let cj∞ = crt(c 1 e1 , . . . ,cj∞, . . . ,c s es ) be associated code over aj∞. we can now prove the following. theorem 6.1. let cjej be a composite g-code over the chain ring r j ej that is cjej is an ideal in rejg. then cj∞ =crt(c1e1, . . . , c j ∞, . . . ,cses ) is a composite g-code over a j ∞. proof. let vj ∈cjej. we know that v ∗ j also belongs to c j ej where v∗j has the form defined in (16). let v ∈ cj∞. now if v = crt(v1,v2, . . . ,vs), then v∗ = crt(v∗1,v∗2, . . . ,v∗s) and so v∗ ∈ cj∞ giving that cj∞ is an ideal in aj∞g, and thus giving that cj∞ is a composite g-code over aj∞. 104 a. korban / j. algebra comb. discrete appl. 8(2) (2021) 91–105 7. conclusion in this work, we generalized the known results on g-codes over the formal power series rings and finite chain rings fq[t]/(ti) to composite g-codes over the same alphabets. we showed that the dual of a composite g-code is also a composite g-code and we studied the projections and lifts of the composite g-codes with a given type in this setting. we extended many theoretical results on γ-adic g-codes and g-codes over principal ideal rings to composite γ-adic g-codes and composite g-codes over principal ideal rings. since the results presented in this paper and in [4] only consider the finite chain rings fq[t]/(ti), it is suggested that for future research, these families of codes; g codes and composite g codes, are studied over a more general finite chain rings as it was done using a unified treatment in [1]. references [1] r. l. bouzara, k. guenda, e. martinez-moro, lifted codes and lattices from codes over finite chain rings, arxiv:2007.05871. [2] s. t. dougherty, algebraic coding theory over finite commutative rings, springerbriefs in mathematics springer (2017). [3] s. t. dougherty, j. gildea, a. korban, extending an established isomorphism between group rings and a subring of the n×n matrices, international journal of algebra and computation, published: 25 february 2021. [4] s. t. dougherty, j. gildea, a. korban, gcodes over formal power series rings and finite chain rings, j. algebra comb. discrete appl. 7 (2020) 55–71. [5] s. t. dougherty, j. gildea, a. korban, a. kaya, composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, advances in mathematics of communications 14(4) (2020) 677–702. [6] s. t. dougherty, j. gildea, a. korban, a. kaya, composite matrices from group rings, composite g-codes and constructions of self-dual codes, arxiv:2002.11614. 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[12] b. r. mcdonald, finite rings with identity, new york: marcel dekker (1974). 105 https://arxiv.org/abs/2007.05871 https://arxiv.org/abs/2007.05871 https://doi.org/10.1007/978-3-319-59806-2 https://doi.org/10.1007/978-3-319-59806-2 https://doi.org/10.1142/s0218196721500223 https://doi.org/10.1142/s0218196721500223 https://doi.org/10.1142/s0218196721500223 https://doi.org/10.13069/jacodesmath.645026 https://doi.org/10.13069/jacodesmath.645026 https://doi.org/10.3934/amc.2020037 https://doi.org/10.3934/amc.2020037 https://doi.org/10.3934/amc.2020037 https://arxiv.org/abs/2002.11614 https://arxiv.org/abs/2002.11614 https://doi.org/10.1007/s10623-017-0440-7 https://doi.org/10.1007/s10623-017-0440-7 https://doi.org/10.1016/s0252-9602(11)60233-6 https://doi.org/10.1016/s0252-9602(11)60233-6 https://doi.org/10.1016/j.ffa.2018.01.002 https://doi.org/10.1016/j.ffa.2018.01.002 introduction preliminaries composite g-codes and ideals in the group ring rg projections and lifts of composite g-codes self-dual -adic composite g-codes composite g-codes over principal ideal rings conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056492 j. algebra comb. discrete appl. 9(1) • 9–15 received: 25 may 2020 accepted: 30 september 2021 journal of algebra combinatorics discrete structures and applications on commuting probabilities in finite groups and rings research article martin juráš, mihail ursul abstract: we show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class ≤ 2. these two sets are equal when restricted to groups and rings with odd number of elements. 2010 msc: 16u80, 05c25, 20p05, 16n40, 20d15 keywords: finite group, finite ring, commuting probability, annihilating probability, nilpotent group, nilpotent ring 1. introduction and preliminaries in 1940, philip hall [17] introduced the notion of the commuting probability in groups. feit and fine [12], derived a combinatorial formula and a generating function for commuting probability in matrix rings over finite fields. in the second half of 1960’s, the series of papers [8], [9], [10], [11] by erdös and turán, gave birth to the statistical group theory. in the fourth paper, among other results, the authors derived a lower bound for commuting probability in a finite group of order n, and showed that the commuting probability in the symmetric group sn is asymptotically equal to 1n· a number of research and expository papers on commuting probability in groups appeared during late sixties and the seventies: joseph [19], [20], galagher [13], gustafson [16], machale [22], and rusin [27], to name a few1. rusin [27], characterized all finite groups with commuting probability > 11 32 · there has also been interest in the study of commuting probability of other algebraic structures, [20]. machale [23], investigated the notion of commuting probability in rings. in 1995 lescot [21], rederived classification of groups with commuting probability > 1 2 , using the notion of isoclinism in groups introduced by hall [17]. recently, the commuting probability in semigroups has been studied in [14], [24], [26] and [29]. martin juráš (corresponding author); scad, savannah, ga 31401, usa (email: mjuras@scad.edu, martinjuras@gmail.com). mihail ursul; department of mathematics, png university of technology, lae, png (email: mihail.ursul@ gmail.com). 1 dixon, provides an extensive list of publications on statistical group theory in the references of his paper [7], up to the year 2002. 9 https://orcid.org/0000-0003-4752-7734 https://orcid.org/0000-0003-4744-0890 m. juráš, m. ursul / j. algebra comb. discrete appl. 9(1) (2022) 9–15 since the dawn of the twenty-first century we have seen an escalation of interest in the study of the commuting probability in groups, and commuting and other types of probabilities in rings, such as anticommuting and annihilating probability. publications [28], [7], [15], [6] and [18] deal with commuting probability in groups. in papers [3] and [1], buckley et. al. classified all rings with commuting probability ≥ 11 32 and anticommuting probability ≥ 15 32 , respectively. throughout this paper, |a| denotes cardinality of the set a. z(g) denotes the center of a group g. for a, b ∈ g, [a, b] = a−1b−1ab denotes the commutator of a and b, and [g, g] denotes the derived subgroup of g generated by all commutators in g. recall that g is nilpotent of class n, if its lower central series (of normal subgroups) terminate in the trivial subgroup after n steps, i.e. g = g0 . g1 . · · ·. gn = {eg}, where gi = [gi−1, g] for i = 1, 2, . . . , n, and gn−1 6= {eg}. commuting probability2 in a group g is defined to be the number prc(g) = |{(a, b) ∈ g×g : ab = ba}| |g|2 . for a class g of finite groups, the set sc(g) = {prc(g) : g ∈g} is called the commuting spectrum of g. rings are not assumed to be associative or unitary. by r(+) we denote the additive group of r. recall that a ring r is called antisymmetric if for all a, b ∈ r, ab = −ba. r is called strongly antisymmetric if the dinipotent condition, a2 = 0, is satisfied for all a ∈ r. strong antisymmetry implies antisymmetry. a ring r is said to be of nilpotent class ≤ n if the product of any n elements with any correct distribution of brackets is zero. for a prime p, r is called a p-ring if |r| = pn for some positive integer n. the symbol [·, ·] denotes the commutator in both a group g and a ring r (for rings, [a, b] = ab−ba). whenever needed, we will write [·, ·]g and [·, ·]r to distinguish between the two cases. buckley [2], introduced the following generalization of the notion of commuting probability in rings. let f(x, y ) = axy + by x be a formal "non-commutative polynomial" with integer coefficients. for any ring r define a function fr : r ×r → r, (x, y) = axy + byx. let prf (r) = |{(x, y) ∈ r ×r : fr(x, y) = 0}| |r|2 · for a class r of finite rings, the set sf (r) = {prf (r) : r ∈r} is called the f-spectrum of r. here, we are going to be mostly concerned with the commuting spectrum, sc(r) and the annihilating spectrum, sann(r), with the associated formal "non-commutative polynomials" f(x, y ) = xy − y x and f(x, y ) = xy , respectively. the commuting probability and the annihilating probability in a ring r are denoted by prc(r) and prann(r), respectively. we will use the following classes of groups and rings: g the class of finite groups; gnil the class of finite nilpotent groups; g (2) nil the class of finite nilpotent groups of class ≤ 2; r the class of finite rings; r (2) nil the class of finite nilpotent rings of class ≤ 3; 2 some publications use the term commuting degree in place of the commuting probability. 10 m. juráš, m. ursul / j. algebra comb. discrete appl. 9(1) (2022) 9–15 rsa the class of finite strongly antisymmetric rings; rp the class of p-rings; for the class c of finite sets, denote odd(c) = {a ∈c : |a| is odd}. recall the following well know construction. for a given ring r, we construct the ring n(r) in the following way: the additive group of n(r) is (r × r, +) with multiplication (a, x)(b, y) = (0, ab). the following lemma is immediate. lemma 1.1. let r be a ring. then n(r) is a nilpotent ring of class at most 3. furthermore, if f(x, y ) = axy + by x is a formal non-commutative polynomial with integer coefficients and r is finite, then prf (r) = prf (n(r)). in particular, the lemma implies sf (r) = sf (r (2) nil). (1) ever since it was discovered that there are no finite groups with commuting probability in the open interval 5 8 ), 1), there has been an interest to understand the structure of the commuting spectrum of groups, and later, the structure of the commuting spectrum of rings and semigroups. the commuting spectrum for semigroups turned out to be the simplest to understand. givens [14] showed that the commuting spectrum for semigroups is dense in the interval [0, 1]. later ponomarenko and selinski [26] proved that for any rational number in r ∈ (0, 1], there is a finite semigroup s such that the commuting probability in s is equal to r. soule [29] found a single family of semigroups that has this property. these semigroups are defined as follows. let x = {x1, x2, . . . , xm}∈ {1, 2, . . . , n} and x1 < x2 < x3 < . . . < xm. set x0 = 0. define a semigroup s(n, x) as follows: for any a, b ∈{1, 2, . . . , n} a ? b = { xi if xi−1 < a ≤ xi and b ≥ xm, max{a, b} if a > xm or b > xm. the author than shows that any rational commuting probability can be achieved by an appropriate choice of parameters. contrastingly, for groups, hegarty [18] showed that for any limit point3 l of sc(g), l ∈ (29, 1], there is no increasing sequence of numbers {an}⊂ sc(g), such that l = limn→∞ an. recently, buckley and machale investigated relations between the commuting spectra of finite groups and rings. comparing the structure of these two spectra for large probabilities, the authors formulated two conjectures, [4], page 9: conjecture 1. sc(r) ⊂ sc(g). conjecture 2. sc(r) = sc(gnil) or sc(r) = sc(g (2) nil). this paper positively resolves the first conjecture and partially resolves the second one.4 2. main results theorem 2.1. sc(r) ⊆ sc(g (2) nil) ⊆ sann(rsa ∩r (2) nil). 3 x is a limit point of a set s if every neighborhood of x contains at least one point of s different from x itself. 4 the authors would like to thank victor bovdi for his interest in this paper. 11 m. juráš, m. ursul / j. algebra comb. discrete appl. 9(1) (2022) 9–15 in [5], the authors determined all values in sc(r) that are ≥ 1132. these are 1, 7 16 , 11 27 , 25 64 , 11 32 , and 22k + 1 22k+1 for k = 1, 2, 3, . . . . thus, 1 2 6∈ sc(r). but, 12 ∈ sc(g), ([27], page 246), and so sc(r) 6= sc(g). in particular, prc(s3) = 1 2 (see [20]); s3 denotes the symmetric group of order 3. this, together with the first inclusion of theorem 2.1, positively resolves conjecture 1. as for conjecture 2, the theorem states sc(r) ⊆ sc(g (2) nil). now that we know sc(r) is a subset of the potentially smaller one of the two sets, sc(g (2) nil) and sc(gnil) (it is unknown whether or not sc(g (2) nil) = sc(gnil)), we ask the following question: does sc(r) = sc(g (2) nil) (2) hold true? we don’t know. but, equation (2) does hold true, when restricted to finite groups and finite rings with odd number of elements. in fact, we prove the following: theorem 2.2. sc(odd(r)) = sc(odd(g (2) nil)) = sann(odd(rsa ∩r (2) nil)). next, we would like to formulate a condition, purely in terms of probabilities in rings, that would imply equation (2). using theorem 2.1, one obvious choice could be sann(rsa ∩ r (2) nil) ⊆ sc(r). we can do slightly better. because things are working smoothly when restricted to rings with odd number of elements, it is sufficient to focus on the "trouble makers" which are the 2-rings. proposition 2.3. if sann(rsa ∩r (2) nil ∩r2) ⊆ sc(r), then equation (2) holds true. the condition of proposition 2.3 implies a stronger statement: if sann(rsa ∩ r (2) nil ∩ r2) ⊆ sc(r), then both inclusions in theorem 2.1 can be replaced by equal signs. note that if there is a counterexample to the condition above, i.e. if there exists a ring r such that r ∈ rsa∩r (2) nil∩r2 and prann(r) 6∈ sc(r), then prann(r) < 1132. we conjecture that sc(r) = sann(rsa). 3. proofs let n be an associative nilpotent ring of class n. then n, endowed with "circular multiplication", a ◦ b = a + b + ab, is a group which we will denote by gn.5 0 is the unit element in gn and a−1 = −a + a2 −a3 + · · ·+ (−1)n−1an−1 is the inverse of a in gn, a◦a−1 = a−1 ◦a = 0. since, ab = ba if and only if a◦ b = b◦a, then, if n is finite, prc(n) = prc(gn), (3) lemma 3.1. let n be a nilpotent ring of class at most 3 (hence, also an associative ring). let a, b, c ∈ n. then (i) [a, b]gn = [a, b]n, (ii) [a, b]gn ◦ [c, d]gn = [a, b]n + [c, d]n, (iii) gn is a nilpotent group of class ≤ 2. 5 another way to associate a group to a ring such that their commuting probabilities equate can be obtained by modifying a construction of mal’cev [25]. for an arbitrary ring r, define a binary operation on r × r by (a, b) · (c, d) = (a + c, ac + b + d). this operation is associative, has unit (0, 0) and (a, b)−1 = (−a, a2 − b). g = (r × r, · ) is a nilpotent group of class at most 2 and prc(r) = prc(g). note that, unlike the construction of gn, the ring r is not required to be nilpotent or associative! 12 m. juráš, m. ursul / j. algebra comb. discrete appl. 9(1) (2022) 9–15 proof. (i) follows by direct computation. (ii). by (i), [a, b]gn ◦ [c, d]gn = [a, b]n ◦ [c, d]n = [a, b]n + [c, d]n + [a, b]n[c, d]n = [a, b]n + [c, d]n. (iii). by (i), [[a, b]gn , c]gn = [[a, b]n, c]n = 0. let g be a nilpotent group of class ≤ 2 and let z = z(g) be the center of g. then g/z is abelian. by rg, denote the ring with the additive group g/z ⊕z, and the multiplication defined by (az, x) · (bz, y) = (z, [a, b]), (4) where [a, b] = a−1b−1ab is the commutator in g. explicitly, the addition in rg is given by (az, x) + (bz, y) = (abz, xy). (z, eg) is the zero element and (a−1z, x−1) is the additive inverse of (az, x). to verify that rg is indeed a ring, the distributive laws have to be satisfied. let a, b, c ∈ g and x, y, z ∈ z. we have (cz, z) · ((az, x) + (bz, y)) = (cz, z) · (abz, xy) = (z, [c, ab]). on the other hand, (cz, z) · (az, x) + (cz, z) · (bz, y)) = (z, [c, a]) + (z, [c, b]) = (z, [c, a][c, b]). using [g, g] ⊆ z, we deduce [c, a][c, b] = c−1a−1cac−1b−1cb = c−1[a, c−1]b−1cb = c−1b−1[a, c−1]cb = c−1b−1a−1cac−1cb = c−1b−1a−1cab = c−1(ab)−1c(ab) = [c, ab]. hence, the left distributive law is satisfied. the proof of the right distributive law is similar.6 lemma 3.2. let g be a nilpotent group of class at most 2. then rg is a strongly antisymmetric nilpotent ring of class at most 3. if g is finite, then |rg| = |g| and prc(g) = prann(rg). (5) proof. |rg| = |g/z||z| = |g|. r3g = 0 and strong antisymmetry of rg follows immediately from the multiplication formula (4). to prove (5), it suffices to note that (az, x) · (bz, y) = (z, [a, b]) = (z, eg) if and only if [a, b] = eg. but, this is exactly when ab = ba. proof of theorem 2.1. we first show that sc(r) ⊆ sc(g (2) nil). let r ∈ sc(r) by lemma 1.1, there is a nilpotent ring n of class at most 3 such that r = prc(n). by lemma 3.1(iii), gn is a nilpotent group of class at most 2 and by equation (3), prc(gn) = prc(n). we conclude that r ∈ sc(g (2) nil). to prove the second inclusion, consider g ∈ g(2)nil. by lemma 3.2 rg ∈ rsa ∩r (2) nil and prann(rg) = prc(g). 6 proposition 3 [4], states that the condition [c, a][c, b] = [c, ab] for all a, b, c ∈ g is equivalent to g being nilpotent of class ≤ 2. 13 m. juráš, m. ursul / j. algebra comb. discrete appl. 9(1) (2022) 9–15 lemma 3.3. let r be a finite antisymmetric ring and with odd number of elements. then prc(r) = prann(r). proof. in an antisymmetric ring, ab = −ba. hence, ab = ba iff 2ab = 0. since |r| is odd, 2ab = 0 iff ab = 0. proof of theorem 2.2. to prove sc(odd(r)) ⊆ sc(odd(g (2) nil)) ⊆ sann(odd(rsa ∩r (2) nil)), we follow the proof of theorem 2.1 and note that |n| = |gn| and |g| = |rg|. to conclude the proof of theorem 2.2, it suffices to show sann(odd(rsa ∩r (2) nil)) ⊆ sc(odd(r)). let r ∈ sann(odd(rsa ∩r (2) nil)) and let r ∈ odd(rsa ∩r (2) nil) such that r = prann(r). by lemma 3.3, r = prann(r) = prc(r) ∈ sc(odd(r)). for a noncommutative formal polynomial f(x, y ) = axy + by x and rings r1 and r2, prf (r1 ×r2) = prf (r1) prf (r2). (6) let p be a prime number and let c be a class of finite rings. denote cp = c ∩ rp. assume c is closed under cartesian products. then cp is closed under cartesian products and both sf (c) and sf (cp) are multiplicative monoids. furthermore, values in sf (c) are finite products of values taken from the set⋃ p sf (cp), where p runs over all prime numbers. we say that a class c of finite rings is hereditary, if any subring of a ring in c is also in r. if a class c is hereditary, then cp is also hereditary. proof of proposition 2.3. it is easy to see that the class c = rsa ∩ r (2) nil is hereditary and closed under cartesian products. assume sann(c ∩ r2) ⊆ sc(r). by theorem 2.2, sann(odd(c)) = sc(odd(r)) and so sann(c ∩ rp) ⊆ sc(r) for a prime p 6= 2. hence, for all primes p, the monoids sann(c ∩ rp) ⊆ sc(r) and so sann(c) ⊆ sc(r). by theorem 2.1, the reverse inclusion is satisfied, thus the proposition follows. references [1] s. m. buckley, d. machale, y. zelenyuk, finite rings with large anticommuting probability, appl. math. inf. sci. 8(1) (2014) 13–25. 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[19] k. s. joseph, commutativity in non-abelian groups, ph.d. thesis, university of california, los angeles (1969). [20] k. s. joseph, several conjectures on commutativity in algebraic structures, amer. math. monthly 84(7) (1977) 550–551. [21] p. lescot, isoclinism classes and commutativity degrees of finite groups, j. algebra 177(3) (1995) 847–869. [22] d. machale, how commutative can a non-commutative group be?, math. gaz. 58 (1974) 199–202. [23] d. machale, commutativity in finite rings, amer. math. monthly 84(1) (1976) 30-32. [24] d. machale, probability in fnite semigroups, irish math. soc. bull. 25 (1990) 64–68. [25] a. i. mal’cev, on a correspondence between rings and groups, in fifteen papers on algebra, ams translation american mathematical soc. (1965) 221–232. [26] v. ponomarenko, n. selinski, two semigroup elements can commute with any positive rational probability, college math. j. 43(4) (2012) 334–336. 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[29] m. soule, a single family of semigroups with every positive rational commuting probability, college math. j. 45(2) (2014) 136–139. 15 https://doi.org/10.1007/bf02280290 https://doi.org/10.1007/bf02280290 https://doi.org/10.1007/bf01894517 https://doi.org/10.1007/bf01894517 https://doi.org/10.1215/s0012-7094-60-02709-5 https://doi.org/10.1215/s0012-7094-60-02709-5 https://doi.org/10.1007/bf01113339 http://www.jstor.org/stable/27646688 http://www.jstor.org/stable/27646688 https://doi.org/10.1016/j.jalgebra.2005.09.044 https://doi.org/10.1016/j.jalgebra.2005.09.044 https://doi.org/10.2307/2318778 https://doi.org/10.2307/2318778 http://eudml.org/doc/150084 https://doi.org/10.1515/jgt-2012-0040 https://doi.org/10.1515/jgt-2012-0040 https://doi.org/10.2307/2320020 https://doi.org/10.2307/2320020 https://doi.org/10.1006/jabr.1995.1331 https://doi.org/10.1006/jabr.1995.1331 https://doi.org/10.2307/3615961 https://doi.org/10.2307/2318829 http://www.irishmathsoc.org/bull25/bull25_64-68.pdf https://www.ams.org/books/trans2/045/ https://www.ams.org/books/trans2/045/ https://doi.org/10.4169/college.math.j.43.4.334 https://doi.org/10.4169/college.math.j.43.4.334 https://mathscinet.ams.org/mathscinet-getitem?mr=549847 https://mathscinet.ams.org/mathscinet-getitem?mr=549847 https://doi.org/10.4169/college.math.j.45.2.136 https://doi.org/10.4169/college.math.j.45.2.136 introduction and preliminaries main results proofs references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.784982 j. algebra comb. discrete appl. 7(3) • 209–227 received: 7 september 2019 accepted: 6 may 2020 journal of algebra combinatorics discrete structures and applications self-dual codes over fq + ufq + u2fq and applications∗ research article parinyawat choosuwan, somphong jitman abstract: self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. recently, characterization and enumeration of euclidean self-dual linear codes over the ring fq + ufq + u2fq with u3 = 0 have been established. in this paper, hermitian self-dual linear codes over fq + ufq + u2fq are studied for all square prime powers q. complete characterization and enumeration of such codes are given. subsequently, algebraic characterization of h-quasi-abelian codes in fq[g] is studied, where h ≤ g are finite abelian groups and fq[h] is a principal ideal group algebra. general characterization and enumeration of h-quasi-abelian codes and self-dual h-quasi-abelian codes in fq[g] are given. for the special case where the field characteristic is 3, an explicit formula for the number of self-dual a× z3-quasi-abelian codes in f3m[a× z3 ×b] is determined for all finite abelian groups a and b such that 3 |a| as well as their construction. precisely, such codes can be represented in terms of linear codes and self-dual linear codes over f3m + uf3m + u2f3m. some illustrative examples are provided as well. 2010 msc: 94b15, 94b05, 94b60 keywords: hermitian self-dual linear codes, quasi-abelian codes, finite chain rings, group algebras 1. introduction self-dual linear codes over finite fields form an interesting class of linear codes that have been extensively studied due to their nice algebraic structures and wide applications (see [8], [11], [12], [22] and references therein). codes over finite rings have been of interest after it was shown that some binary ∗ p. choosuwan was partially supported by the faculty of science and technology, rajamangala university of technology thanyaburi (rmutt). s. jitman was supported by the thailand research fund and silpakorn university under research grant rsa6280042. parinyawat choosuwan; department of mathematics and computer science, faculty of science and technology, rajamangala university of technology thanyaburi (rmutt), pathum thani 12110, thailand (email: parinyawat_c@rmutt.ac.th). somphong jitman (corresponding author); department of mathematics, faculty of science, silpakorn university, nakhon pathom 73000, thailand (email: sjitman@gmail.com). 209 https://orcid.org/0000-0003-0817-282x https://orcid.org/0000-0003-1076-0866 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 non-linear codes such as the kerdock, preparata and goethal codes are the gray images of linear codes over z4 in [7]. in general, families of linear codes and self-dual linear codes over finite chain rings are now become of interest. in [16], the mass formula for euclidean self-dual linear codes over zp3 has been studied. characterization and enumeration of euclidean self-dual linear codes over the ring fq + ufq + u2fq with u3 = 0 have been given in [3]. algebraically structured codes over finite fields such as cyclic codes, abelian codes and quasi-abelian codes are another important family of linear codes that have been extensively studied for both theoretical and practical reasons (see [2], [8], [10], [11], [12] and references therein). in [10], h-quasi-abelian codes and self-dual h-quasi-abelian codes in fq[g] have been studied in the case where fq[h] is semisimple to the best of our knowledge, hermitian self-dual linear codes over fq + ufq + u2fq and nonsemisimple h-quasi-abelian codes in fq[g] have not been well studied. the goals of this paper are to investigate the following families of linear codes and their links. 1) hermitian self-dual linear codes over fq +ufq +u2fq where q is a square prime power. 2) h-quasi-abelian codes and self-dual h-quasi-abelian codes in group algebras fq[g], where h ≤ g are finite abelian groups and fq[h] is a principal ideal group algebra. the paper is organized as follows. in section 2, some results on linear codes and euclidean selfdual linear codes over fq + ufq + u2fq are recalled. in section 3, characterization and enumeration hermitian self-dual linear codes of length n over fq + ufq + u2fq are established for all square prime powers q together with an algorithm to determine all hermitian self-dual codes and illustrative examples. in section 4, the study of h-quasi-abelian codes in fq[g] is given, where fq[h] is a principal ideal group algebra. in the special case where the field characteristic is 3, the characterization and enumeration of a×z3-quasi-abelian codes and self-dual a×z3-quasi-abelian codes in f3m [a×z3 ×b] are completely determined in terms of linear and self-dual linear codes over f3m + uf3m + u2f3m obtained in section 3 for all finite abelian groups a and b such that 3 |a|. summary and remarks are given in section 5. 2. preliminaries in this section, basic results on linear codes and euclidean self-dual linear codes over rings are recalled. 2.1. linear codes over fq + ufq + · · ·+ ue−1fq for a prime power q, denote by fq the finite field of order q. let fq + ufq + · · · + ue−1fq := {a0 +ua1 +· · ·+ue−1ae−1 | ai ∈ fq for all 0 ≤ i < e} be a ring, where the addition and multiplication are defined as in the usual polynomial ring over fq with indeterminate u together with the condition ue = 0. it is easily seen that fq +ufq +· · ·+ue−1fq is isomorphic to fq[u]/〈ue〉 as rings. the galois extension of fq +ufq +· · ·+ue−1fq of degree m is defined to be the quotient ring (fq +ufq +· · ·+ue−1fq)[x]/〈f(x)〉, where f(x) is an irreducible polynomial of degree m over fq. it is not difficult to see that the galois extension of fq + ufq + · · ·+ ue−1fq of degree m is isomorphic to fqm + ufqm + · · ·+ ue−1fqm. the ring fq + ufq + · · ·+ ue−1fq is a finite chain ring with maximal ideal 〈u〉, nilpotency index e and residue field fq. in addition, if q is a square, the mapping ¯ : fq → fq defined by a 7→ a √ q is a field automorphism on fq of order 2. extend ¯ to be a ring automorphism of order 2 on fq + ufq + · · · + ue−1fq of the form a0 + ua1 + · · · + ue−1ae−1 = a0 + ua1 + · · · + ue−1ae−1. let n be a positive integer and let r be a finite ring. the euclidean inner product of u = (u0,u1, . . . ,un−1) and v = (v0,v1, . . . ,vn−1) in rn is defined to be 〈u,v〉e := n−1∑ i=0 uivi. in the case where q is a square and r ∈{fq,fq + ufq + · · ·+ ue−1fq}, the hermitian inner product of u 210 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 and v in rn is defined to be 〈u,v〉h := n−1∑ i=0 uivi. a linear code c of length n over the ring r is defined to be an r-submodule of the r-module rn. a linear code over r is said to be free if it is a free r-module. denote by wt(v) the hamming weight of an element v ∈ rn. precisely, wt(v) is the number of non-zero components in v. for a linear code c over r, let wt(c) = min{wt(c) | c ∈c} be the minimum hamming weight of c. if r = fq, a linear code c of length n and dimension k over r with wt(c) = d is referred as an [n,k,d]q code. the parameters of a linear code c of length n over r satisfies the singleton bond [14], i.e., wt(c) ≤ n−log|r|(|c|) + 1. a linear code c is called a maximum distance separable (mds) code if the equality in the singleton bound holds. a matrix g over r is called a generator matrix for c if the rows of g generate all the elements of c and none of the rows can be written as a linear combination of the others. linear codes c1 and c2 over r are said to be equivalent if there exists a monomial matrix p such that c2 = c1p := {cp | c ∈ c1}. denote by c⊥e = {v ∈ rn | 〈u,v〉e = 0} and c⊥h = {v ∈ rn | 〈u,v〉h = 0} the euclidean and hermitian duals of c, respectively. a linear code c is said to be euclidean (resp., hermitian) self-orthogonal if c ⊆ c⊥e (resp., c ⊆c⊥h). it is called euclidean (resp., hermitian) self-dual if c = c⊥e (resp., c = c⊥h). in section 3 and the remaining parts of this section, we focus on linear and self-dual linear codes over fq + ufq + u2fq. in [19], it has been shown that every linear code of length n over fq + ufq + u2fq is permutation equivalent to a code c with generator matrix g =  ik a2 a3 a40 uil ub3 ub4 0 0 u2im u 2c4   =   a′ub′ u2c   , (1) where ir is the identity matrix of order r, a3 = a30 + ua31,b4 = b40 + ub41,a4 = a40 + ua41 + u2a42, and a2,b3,c4, aij and bij are matrices of appropriate sizes over fq. in this case, the code c is said to be of type {k,l,m} and it contains q3k+2l+m codewords. for each linear code c of length n over fq + ufq + u2fq and i ∈ {0, 1, 2}, the ith torsion code of c is a linear code of length n over fq defined to be tori(c) = { v(modu) | v ∈ ( fq + ufq + u2fq )n and uiv ∈c } . the code tor0(c) is sometime called the residue code of c and denoted it by res(c). from the definitions, it is obvious that res(c) = tor0(c) ⊆ tor1(c) ⊆ tor2(c). for a linear code c of length n over fq +ufq +u2fq with generator matrix g given in (1), the residue code res(c) has dimension k and generator matrix g = [ ik a2 a30 a40 ] , (2) the first torsion code tor1(c) has dimension k + l and generator matrix[ a b ] = [ ik a2 a30 a40 0 il b3 b40 ] , (3) and the second torsion code tor2(c) has dimension k + l + m and generator matrix ab c   =  ik a2 a30 a400 il b3 b40 0 0 im c4   . (4) for 0 ≤ k ≤ n, the gaussian coefficient is defined to be[ n k ] q = (qn − 1)(qn −q) · · ·(qn −qk−1) (qk − 1)(qk −q) · · ·(qk −qk−1) . 211 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 let ne(q,n) denote the number of distinct linear codes of length n over fq + ufq + · · · + ue−1fq. the number ne(q,n) has been studied and summarized in [4]. for e = 3, we have the following result. proposition 2.1 ([4, lemma 2.2]). let q be a prime power and let n be a positive integer. then n3(q,n) = 1 + 3∑ t=1 ∑ n≥h1≥h2≥···≥ht>ht+1=0 t∏ j=1 [ n−hj+1 hj −hj+1 ] q qhj+1(n−hj). 2.2. euclidean self-dual linear codes over fq + ufq + u2fq let c be a linear code of length n and type {k,l,m} over fq +ufq +u2fq and let h = n−(k +l +m). in [3], it has been shown that the euclidean dual c⊥e of c is of type {h,m,l} and it contains q3h+2m+l codewords. therefore, k = h and l = m whenever c is euclidean self-dual. consequently, every euclidean self-dual code of type {k,l,m} over fq + ufq + u2fq has even length n = 2(k + l). characterization of euclidean self-dual linear codes of even length n over fq + ufq + u2fq has been established in [3]. proposition 2.2 ([3, proposition 1]). let q be a prime power and let c be a linear code of length n and type {k,l,m} over fq + ufq + u2fq with generator matrix g in the form of (1). then c is euclidean self-dual if and only if k = h,l = m and the following conditions hold: a′a′ t ≡ 0 (mod u3), (5) a′b′ t ≡ 0 (mod u2), (6) b′b′ t ≡ 0 (mod u), (7) a′ct ≡ 0 (mod u). (8) for a positive integer n and a prime power q, let σe(q,n) denote the number of euclidean self-dual linear codes of length n over fq. further, if q is a square prime power, let σh(q,n) denote the number of hermitian self-dual linear codes of length n over fq. the following results in [21] and [22] are useful in the enumeration of self-dual linear codes over fq + ufq + u2fq. lemma 2.3 ([21] and [22]). let q be a prime power and let n be a positive integer. then σe(q, l) =   n 2 −1∏ i=1 (qi + 1) if q and n are even, 2 n 2 −1∏ i=1 (qi + 1) if q ≡ 1 (mod 4) and 2 | n, 2 n 2 −1∏ i=1 (qi + 1) if q ≡ 3 (mod 4) and 4 | n, 0 otherwise. (9) if q is square, then σh(q,n) =   n 2 −1∏ i=0 (qi+ 1 2 + 1) if n is even, 0 otherwise. (10) 212 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 the empty product is regarded as 1. let nee(q,n) denote the number of distinct euclidean self-dual linear codes of length n over fq + ufq + · · · + ue−1fq. the value of ne3(q,n) has been completely determined in [3]. theorem 2.4 ([3, theorem 1]). let q be a prime power and let n be a positive integer. then ne3(q,n) =   σe(q,n) n/2∑ k=0 [n 2 k ] q qkn/2 if q is even and n is even, σe(q,n) n/2∑ k=0 [n 2 k ] q qk(n/2−1) if q is odd and n is even, 0 otherwise. 3. hermitian self-dual linear codes over fq + ufq + u2fq in this section, we focus on characterization and enumeration of hermitian self-dual linear codes of length n over fq + ufq + u2fq. throughout this section, we assume that q is a square prime power. for each positive integer n, let nhe(q,n) denote the number of distinct hermitian self-dual linear codes of length n over fq + ufq + · · ·+ ue−1fq. by extending techniques introduced in [3], the characterization and the number nh3(q,n) of hermitian self-dual linear codes of length n over fq + ufq + u2fq are established. let c be a linear code of length n over fq + ufq + u2fq of type {k,l,m} and let h = n−(k + l + m). using argument similar to those in section 2 of [3], it can be deduced that the hermitian dual c⊥h of c is of type {h,m,l} and it contains q3h+2m+l codewords. it follows that k = h and l = m if c is hermitian self-dual. hence, every hermitian self-dual code of type {k,l,m} over fq + ufq + u2fq has even length n = 2(k + l). for a matrix a = [aij]s×t over fq + ufq + u2fq, let a := [aij]s×t and a † := a t . characterization of hermitian self-dual linear codes of even length n over fq +ufq +u2fq is given in the following proposition. proposition 3.1. let q be a square prime power and let c be a linear code of even length n and type {k,l,m} over fq +ufq +u2fq with generator matrix g in the form of (1). then c is hermitian self-dual if and only if k = h, l = m and the following hold: a′a′ † ≡ 0 (mod u3), (11) a′b′ † ≡ 0 (mod u2), (12) b′b′ † ≡ 0 (mod u), (13) a′c† ≡ 0 (mod u). (14) proof. assume that c is hermitian self-dual. by the above discussion, we have k = h, l = m and  a′ub′ u2c     a′ub′ u2c  † ≡ [0] (mod u3) which are equivalent to the conditions (11)–(14). 213 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 conversely, assume that c is a linear code such that k = h,l = m and the conditions (11)–(14) hold. from (11)–(14), it is not difficult to see that c is hermitian self-orthogonal. equivalently, c ⊆c⊥h. since k = h and l = m, we have |c| = |c⊥h| which implies that c = c⊥h. therefore, c is hermitian self-dual as desired. corollary 3.2. let c be a hermitian self-dual linear code of length n over fq + ufq + u2fq. then the following statements holds. 1) tor1(c) is a hermitian self-dual code of length n over fq. 2) tor2(c) = res(c)⊥h. proof. assume that c is of type {k,l,m} over fq + ufq + u2fq. then from (11)–(13), it follows that tor1(c) is hermitian self-orthogonal. since dim(tor1(c)) = k + l = n2 = dim((tor1(c)) ⊥h ), tor1(c) is hermitian self-dual. from (11)–(14), we have tor2(c) ⊆ res(c)⊥h. since dim(tor2(c)) = k + 2l = n−k = dim((res(c))⊥h ), we have tor2(c) = res(c)⊥h. since res(c) = tor0(c) ⊆ tor1(c) ⊆ tor2(c), it can be concluded further that res(c) is hermitian self-orthogonal for all hermitian self-dual linear codes c over fq + ufq + u2fq. from corollary 3.2, a hermitian self-dual code c of length n over fq + ufq + u2fq can be induced by hermitian self-dual linear codes of length n over fq. for a given hermitian self-dual code c1 of length n over fq, we first aim to determine the number of hermitian self-dual linear codes c of length n over fq + ufq + u2fq such that tor1(c) = c1. proposition 3.3. let q be a square prime power and let n be an even positive integer. let c1 be a hermitian self-dual linear code of length n over fq. then, for each 0 ≤ k ≤ n2 , there are q kn 2 hermitian self-dual linear codes of length n over fq + ufq + u2fq corresponding to each subspace of c1 of dimension k. proof. let c1 be a hermitian self-dual linear code of length n over fq with dimension n2 = k + l and generator matrix [ a b ] = [ ik a2 a30 a40 0 il b3 b40 ] , (15) where the columns are grouped into blocks of sizes k,l, l and k. since c1 is hermitian self-dual, we have ik + a2a † 2 + a30a † 30 + a40a † 40 = 0, (16) a2 + a30b † 3 + a40b † 40 = 0, (17) il + b3b † 3 + b40b † 40 = 0. (18) let h = [ a30 a40 b3 b40 ] . from (16)–(18), it can be deduced that h(−h†) = −hh† = −hh t = [ −a30at30 −a40at40 −a30bt3 −a40bt40( −a30bt3 −a40bt40 )t −b3bt3 −b40bt40 ] = [ ik + a2a t 2 a2 at2 il ] . 214 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 let j = [ ik −a2 −at2 il + at2 a2 ] . then h(−h†)j = [ ik 0 0 il ] which implies that h is invertible. let c0 be a k-dimensional fq-subspace of c1 with generator matrix a. since c1 is hermitian selfdual, it follows that c0 is hermitian self-orthogonal. up to permutation of the last (k + l) columns (if necessary), its follows that c⊥h0 has a generator matrix of the form ik a2 a30 a400 il b3 b40 0 0 il c4   . (19) then a30 = −a40c † 4 which implies that h = [ −a40ct4 a40 b3 b40 ] . since h is invertible, it follows that a40 is invertible. next, we determined the matrices over fq + ufq + u2fq satisfying conditions (11)–(14) which are equivalent to ik + a2a † 2 + a3a † 3 + a4a † 4 ≡ 0 (mod u 3) (20) a2 + a3b † 3 + a4b † 4 ≡ 0 (mod u 2) (21) il + b3b † 3 + b4b † 4 ≡ 0 (mod u) (22) a3 + a4c † 4 ≡ 0 (mod u). (23) the matrices a2,b3 and c4 are considered modulo u, i.e. all the entries in a2,b3 and c4 are in fq. the matrices a3 and b4 are considered modulo u2 while a4 is considered modulo u3. from these fact, let a3 = a30 + ua31,b4 = b40 + ub41 and a4 = a40 + ua41 + u2a42, where a31,b41,a41 and a42 are matrices of appropriate sizes over fq. therefore, we can write (20) as( ik + a2a † 2 + a30a † 30 + a40a † 40 ) + u ( ã30a † 31 + ã40a † 41 ) +u2 ( a31a † 31 + a41a † 41 + ã40a † 42 ) ≡ 0 (mod u3), where x̃ := x + x†. we can also rewrite (21) as( a2 + a30b † 3 + a40b † 40 ) + u ( a31b † 3 + a40b † 41 + a41b † 40 ) ≡ 0 (mod u2) by substituting (18) into (21), we obtain that b † 41 = −a −1 40 ( a31b † 3 + a41b † 40 ) , from (23), c4 is uniquely determined as c4 = ( −a−140 a30 )† . it is sufficient to focus on (20) because (18) is the same as (22). from (16), we have to determined the matrices satisfying the following: ã30a † 31 + ã40a † 41 = 0, (24) a31a † 31 + a41a † 41 + ã40a † 42 = 0. (25) hence, the code c with generator matrix of the form (1) is a hermitian self-dual linear code if and only if conditions (24) and (25) are satisfied. 215 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 therefore, the number of hermitian self-dual linear codes of length n over fq + ufq + u2fq whose the 1st torsion is c1 is equal to the number of solutions of the system of matrix equations (24) and (25). we take an arbitrary matrix a31 ∈ mk×l(fq) and put [gij] = ã30a † 31 and [xij] = a40a † 41. then condition (24) is equivalent to gij + xij + xji = 0. then −gii = xii + xii = tr(xii) ∈ f√q for each 1 ≤ i ≤ k, where tr : fq → f√q is the trace map defined by α 7→ α + α for all α ∈ fq. note that |tr−1(a)| = √ q for all a ∈ f√q. then we have xii ∈ tr−1(−gii) for all 1 ≤ i ≤ k, xji ∈ fq and xij = −gij −xji for each 1 ≤ i < j ≤ k. therefore, a41 = (a −1 40 [xij]) †. thus we have qkl possible choices for a31 and q k(k−1) 2 + k 2 = q k2 2 for a41. for fixed matrices a31 and a41, let [hij] = a31a † 31 + a41a † 41 and [yij] = a40a † 42. then (25) is equivalent to hij + yij + yji = 0. using a similar argument as above, we have q k2 2 possible choices for a42 = (a −1 40 [yij]) †. therefore, we have qkl ×q k2 2 ×q k2 2 = qk 2+kl = qk(k+l) = q kn 2 possible choices for the matrices a31,a41 and a42 over fq. therefore, the desired result follows immediately. the number of distinct hermitian self-dual linear codes of even length n over fq + ufq + u2fq can be summarized in the following theorem. theorem 3.4. let q be a square prime power and let n be a positive integer. then the number of distinct hermitian self-dual linear codes of length n over fq + ufq + u2fq is nh3(q,n) =   σh(q,n) n/2∑ k=0 [n 2 k ] q qkn/2 if n is even, 0 otherwise. from the proof of theorem 3.3, we obtain not only the number of hermitian self-dual linear codes of length n over fq + ufq + u2fq but also a construction of such hermitian self-dual linear codes. the construction of hermitian self-dual linear codes of length n over fq + ufq + u2fq induced by a hermitian self-dual linear code of length n over fq in the proof of theorem 3.3 is summarized in algorithm 1. based on algorithm 1, an illustrative example of a hermitian self-dual linear code of length 6 over f9 +uf9 +u2f9 constructed from a hermitian self-dual linear code of length 6 over f9 is given as follows. example 3.5. let f9 = f3[α] be the finite field of order 9, where α is a root of x2 + x + 2 over f3. let c0 and c1 be linear codes of length 6 over f9 with generator matrices [ 1 0 1 α2 α5 α2 0 1 0 α 1 α ] and   1 0 1 α2 α5 α20 1 0 α 1 α 0 0 1 α5 α 1   , 216 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 algorithm 1. construction of hermitian self-dual linear codes of length n over fq + ufq + u2fq for a given hermitian self-dual linear code c1 of length n over fq and its linear subcode c0 of dimension 0 ≤ k ≤ n 2 , do the following steps. 1. define l = n 2 −k. 2. construct a generator matrix a = [ ik a2 a30 a40 ] for c0, where the columns are grouped into blocks of sizes k,l, l and k. 3. extend a to be a generator matrix [ ik a2 a30 a40 0 il b3 b40 ] for c1. 4. set c4 = − ( a−140 a30 )† . 5. set a31 to be a k × l matrix over fq. 6. define [gij] = ã30a † 31 and set [xij] to be a k×k matrix over fq such that the strictly lower triangular elements are arbitrary in fq, xii ∈ tr−1(−gii), and xij = −gij −xji for all i < j. (if k = n2 , set [gij] to be the k ×k zero matrix over fq.) 7. set a41 = (a−140 [xij]) †. 8. set b41 = − ( a−140 ( a31b † 3 + a41b † 40 ))† . 9. define [hij] = a31a † 31 + a41a † 41 and set [yij] to be a k × k matrix over fq such that the strictly lower triangular elements are arbitrary in fq, yii ∈ tr−1(−hii), and yij = −gij −xji for all i < j. (if k = n 2 , set [hij] = a41a † 41.) 10. set a42 = (a−140 [yij]) †. 11. define c to be a linear code of length n over fq + ufq + u2fq with generator matrix ik a2 a30 + ua31 a40 + ua41 + u2a420 uil ub3 ub40 + u2b41 0 0 u2il u 2c4   . the c is hermitian self-dual by theorem 3.3. 12. repeat 5.−11. with different choices of a31, a41, and a42. the hermitian self-dual linear codes of length n over fq + ufq + u2fq determined by c0 ⊆c1 are obtained. respectively. then c1 is hermitian self-dual and c0 ⊆ c1. based on algorithm 1, we have k = 2, l = 1, a2 = [ 1 0 ] , a30 = [ α2 α ] , a40 = [ α5 α2 1 α ] , b3 = [ α5 ] , and b40 = [ α 1 ] . then we have c4 = − ( a−140 a30 )† = −  [α5 α2 1 α ]−1 [ α2 α ]† = [0 2]. by choosing a31 = [ 1 1 ] , we have [gij] = ã30a † 31 = ˜[ α2 α ][ 1 1 ] = [ 0 α α3 2 ] . we choose x11 = 0 ∈ {0,α2,α6} = tr−1(0) = tr−1(−g11), x22 = 2 ∈ {2,α5,α7} = tr−1(1) = tr−1(−g22), x21 = 1, and x12 = −g12 − x21 = −α − 1 = α3. it follows that [xij] = [ 0 α3 1 2 ] and a41 = (a −1 40 [xij]) † = [ 2 α α 1 ] . consequently, b41 = − ( a−140 ( a31b † 3 + a41b † 40 ))† = [ α 0 ] . 217 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 let [hij] = a31a † 31 + a41a † 41 = [ 1 1 ][ 1 1 ] + [ α4 α α 1 ][ α4 α3 α3 1 ] = [ 1 α3 α 1 ] . we choose y11 = 1 ∈ {1,α,α3} = tr−1(2) = tr−1(−h11), y22 = 1 ∈ {1,α,α3} = tr−1(2) = tr−1(−h22), y21 = 1, and y12 = −h12 −y21 = −α3 − 1 = α. then [yij] = [ 1 a 1 1 ] and a42 = (a −1 40 [yij]) † = [ α3 α3 0 α5 ] . from algorithm 1, the matrix  ik a2 a30 + ua31 a40 + ua41 + u 2a42 0 uil ub3 ub40 + u 2b41 0 0 u2il u 2c4   =   1 0 1 α2 + u α5 + 2u + α3u2 α2 + αu + α3u2 0 1 0 α + u 1 + αu α + u + α5u2 0 0 u α5u αu + αu2 u 0 0 u2 0 2u2   is a generator matrix for a hermitian self-dual code of length 6 over f9 + uf9 + u2f9 with type {2, 1, 1}. when k = n 2 in theorem 3.3 (equivalently, in algorithm 1), we have the following extension on the parameters of hermitian self-dual linear codes over fq + ufq + u2fq. let n be an even positive integer and let c1 = c0 be a hermitian self-dual code of length n over fq with parameters [n,k = n2 ,d]q and generator matrix a = [ in 2 a40 ] , (26) where a40 is a k×k invertible matrix over fq. based on algorithm 1, the linear code c of length n and type {n 2 , 0, 0} over fq + ufq + u2fq with generator matrix g = [ ik a40 + ua41 + u 2a42 ] (27) is hermitian self-dual. since c is a free code, wt(c) = wt(c1) = d by [18, corollary 4.3]. hence, the following two theorems can be derived directly. theorem 3.6. let q be a prime power and let n be an even positive integer. if there exists an [n, n 2 , n 2 +1]q mds hermitian self-dual code over fq, then an mds hermitian self-dual code of length n over fq +ufq + u2fq of type {n2 , 0, 0} can be constructed with minimum hamming weight n 2 + 1. proof. assume that there exists an [n, n 2 , n 2 + 1]q mds hermitian self-dual code c1 over fq with generator matrix of the form (26). by algorithm 1, a linear code c of length n and type {n 2 , 0, 0} over fq + ufq + u2fq with generator matrix of the form (27) is hermitian self-dual. since c is free, we have wt(c) = wt(c1) = n2 + 1 and logq3 (|c|) = n 2 = dim(c1) by the discussion above. hence, wt(c) = n 2 + 1 = n− logq3 (|c|) + 1 which implies that c is mds. using the above theorem, numerous mds hermitian self-dual codes over fq + ufq + u2fq can be construct based on known mds hermitian self-dual codes over fq (see, for example, [13], [17], [24]). 4. self-dual quasi-abelian codes over principal ideal group algebras in this section, the study of quasi-abelian codes over principal ideal group algebras is given. in the special case where the field characteristic is 3 and the sylow 3-subgroup of the underlying finite abelian group has order 3, complete characterization and enumeration of quasi-abelian codes and self-dual quasiabelian codes are presented in terms linear codes and self-dual linear codes over f3m + uf3m + u2f3m obtained in [3], [4] and section 3. 218 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 4.1. group rings and quasi-abelian codes let r be a finite commutative ring with nonzero identity and let g be a finite abelian group. then r[g] =  ∑ g∈g αgy g | αg ∈ r,g ∈ g   is a commutative ring under the addition and multiplication given for the usual polynomial ring over r with indeterminate y , where the indices are computed additively in g. the ring r[g] is called a group ring of g over r. in the case where r is the finite field fpm, the group ring fpm [g] is called a group algebra of g over fpm and it is called a principal ideal group algebra (piga) if every ideal in fpm [g] is principal. the readers may refer to [15] for more details on group rings. a linear code of length |g| over r can be viewed as an embedded r-submodule of the r-module in r[g] by indexing the |g|-tuples by the elements in g. given a subgroup h of g with index n = [g : h], a linear code c of length |g| viewed as an r-submodule of r[g] is called an h-quasi-abelian code (specifically, an h-quasi-abelian code of index n) in r[g] if c is an r[h]-module, i.e., c is closed under the multiplication by the elements in r[h]. such a code will be called a quasi-abelian code if h is not specified or where it is clear in the context. let {g1,g2, . . . ,gn} be a fixed set of representatives of the cosets of h in g. let r := fq[h]. define φ : fq[g] →rn by φ (∑ h∈h n∑ i=1 αh+giy h+gi ) = (α1(y ),α2(y ), . . . ,αn(y )), where αi(y ) = ∑ h∈h αh+giy h ∈ r for all i = 1, 2, . . . ,n. it is well known that φ is an r-module isomorphism interpreted as follows. lemma 4.1 ([10, lemma 2.1]). the map φ induces a one-to-one correspondence between h-quasi-abelian codes in fq[g] and linear codes of length n over r. we note that a group algebra fpm [h] is semisimple if and only if the sylow p-subgroup of h is trivial (see [20, chapter 2: theorem 4.2]), and it is a piga if and only if he sylow p-subgroup of h is cyclic (see [6]). in [10], complete characterization and enumeration of h-quasi-abelian codes in fpm [g] have been established in the case where fpm [h] is semisimple. here, we focus on a more general case where fpm [h] is a piga, or equivalently, the sylow p-subgroup of h is cyclic. precisely, h ∼= a × zpmi and g ∼= a×zps ×b, where s is a non-negative integer, a and b are finite abelian groups such that p |a|. general characterization is given in subsection 4.2. in the special case where p = 3 and s = 1, complete characterization and enumeration of a×z3-quasi-abelian codes and self-dual a×z3-quasi-abelian codes in f3m [a×z3 ×b] are given in subsection 4.3. 4.2. a×zps-quasi-abelian codes in fpm[a×zps ×b] we focus on h-quasi-abelian codes in fpm [g], where fpm [h] is a piga. equivalently, h ∼= a×zps and g ∼= a× zps ×b, where s is a positive integer, a and b are finite abelian groups such that p |a| (see [6] and [12]). note that the group algebra fpm [a] is semisimple [2] and it can be decomposed using the discrete fourier transform in [23] (see [12] and [11] for more details). for completeness, the decomposition used in this paper are summarized as follows. for co-prime positive integers i and j, denote by ordj(i) the multiplicative order of i modulo j. for each a ∈ a, denote by ord(a) the additive order of a in a and the pm-cyclotomic class of a containing 219 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 a ∈ a is defined to be the set spm (a) := {pmi ·a | i = 0, 1, . . .} = {pmi ·a | 0 ≤ i < ordord(a)(pm)}, where pki · a := pmi∑ j=1 a in a. a subset {a1,a2, . . . ,at} of a is called a complete set of representatives of pm-cyclotomic classes of a if spm (a1),spm (a2), . . . ,spm (at) are distinct and t⋃ i=1 spm (ai) = a. an idempotent in fpm [a] is a nonzero element e such that e2 = e. it is called primitive if for every other idempotent f, either ef = e or ef = 0. the existence of primitive idempotent elements in fpm [a] is proved in [5]. they are induced by the pm-cyclotomic classes of a (see [5, proposition ii.4]). consequently, fpm [a] can be viewed as a direct sum of principal ideals generated by these primitive idempotent elements. proposition 4.2 ([5, corollary iii.6]). let {a1,a2, . . . ,at} be a complete set of representatives of pmcyclotomic classes of a finite abelian group a where p |a| and let ei be the primitive idempotent induced by spm (ai) for all 1 ≤ i ≤ t. then fpm [a] = t⊕ i=1 fpm [a]ei ∼= t∏ i=1 fpmi , where mi = m · ordord(ai)(p m). a piga fpm [a×zps ] can be decomposed in the following theorem. theorem 4.3. let s be a positive integer. let {a1,a2, . . . ,at} be a complete set of representatives of pm-cyclotomic classes of a finite abelian group a where p |a|. then fpm [a×zps ] ∼= t∏ i=1 ( fpmi + ufpmi + · · · + up s−1fpmi ) where mi = m · ordord(ai)(p m) for all 1 ≤ i ≤ t. proof. for each 1 ≤ i ≤ t, let ei be the primitive idempotent induced by spm (ai). from proposition 4.2, we have fpm [a] ∼= fpm [a×zps ]ei ∼= t∏ i=1 fpmi , (28) and hence, fpm [a×zps ] ∼= (fpm [a])[zps ] ∼= t∏ i=1 fpmi [zps ]. (29) under the ring isomorphism that fixes the elements in fpmi and y 1 7→ u + 1, it is not difficult to see that fpmi [zps ] ∼= fpmi + ufpmi + · · · + up s−1fpmi (30) as rings. therefore, fpm [a×zps ] ∼= t∏ i=1 ( fpmi + ufpmi + · · · + up s−1fpmi ) (31) as desired. 220 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 for each finite abelian group b of order n, every a × zps-quasi-abelian code in fpm [a × zps × b] can be viewed as a linear code of length n over fpm [a× zps ] by lemma 4.1. the next corollary follows directly from theorem 4.3. corollary 4.4. let s and m be positive integers. let a and b be finite abelian groups such that |b| = n and p |a|. then every a×zps-quasi-abelian code c in fpm [a×zps ×b] can be viewed as c ∼= t∏ i=1 ci, where ci is a linear code of length n over fpmi + ufpmi + · · · + up s−1fpmi for all i = 1, 2, . . . , t. the enumeration of a×zps-quasi-abelian code in fpm [a×zps ×b] is given as follows. theorem 4.5. let s and m be positive integers. let a and b be finite abelian groups such that |b| = n and the exponent of a is m and p m. then the number of a×zps-quasi-abelian codes in fpm [a×zps×b] is ∏ d|m ( nps (p m·ordd(pm),n) ) na(d) ordd(p m) , where na(d) is the number of elements of order d in a determined in [1] and nps (pm·ordd(p m),n) is the number of linear codes of length n over fpm·ordd(pm) + ufpm·ordd(pm) + · · · + u ps−1fpm·ordd(pm) determined in [4, lemma 2.2]. proof. from theorem 4.3, it suffices to determine the number of linear codes of length n over the ring fpmi + ufpmi + · · · + up s−1fpmi for all i = 1, 2, . . . , t. for each divisor d of m, each pm-cyclotomic class containing an element of order d has ordd(pm) elements and the number of such pm-cyclotomic classes is na(d) ordd(pm) . by theorem 4.3, it follows that the number of linear codes of length n over fpm·ordd(pm) +ufpm·ordd(pm) +· · ·+u ps−1fpm·ordd(pm) corresponding to d is ( nps (p m·ordd(pm),n) ) na(d) ordd(p m) . by taking the summation over all the divisors d of m, the desired result follows. example 4.6. let a ≤ h ≤ g be finite abelian groups such that a ∼= z2 × z4, h ∼= a × z3, and g ∼= h × z4. then the 3-cyclotomic classes of a are s3((0, 0)) = {(0, 0)}, s3((0, 2)) = {(0, 2)}, s3((1, 0)) = {(1, 0)}, s3((1, 2)) = {(1, 2)}, s3((0, 1)) = {(0, 1), (0, 3)}, and s3((1, 1)) = {(1, 1), (1, 3)}. it follows that ordord((0,0))(3) = ordord((0,2))(3) = ordord((1,0))(3) = ordord((1,2))(3) = 1 and ordord((0,1))(3) = ordord((1,1))(3) = 2. by proposition 4.2, f3[a] has 4 primitive idempotents ei such that f3[a]ei ∼= f3 and 2 primitive idempotents ej such that f3[a]ej ∼= f9. such primitive idempotents are induced by the above 6 cyclotomic classes while their explicit forms can be determined using [5, proposition ii.4]. consequently, f3[a] ∼= f3 ×f3 ×f3 ×f3 ×f9 ×f9 and f3[h] = f3[a×z3] ∼= f3[z3] ×f3[z3] ×f3[z3] ×f3[z3] ×f9[z3] ×f9[z3]. by proposition 4.3, we have f3[z3] ∼= f3 + uf3 + u2f3 and f9[z3] ∼= f9 + uf9 + u2f9, where u3 = 0. hence, f3[h] ∼= 4∏ i=1 (f3 + uf3 + u2f3) × 2∏ j=1 (f9 + uf9 + u2f9). (32) 221 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 using corollary 4.4, every h-quasi-abelian code in f3[g] is isomorphic to a code of the form c1 ×c2 ×c3 ×c4 ×c5 ×c6, where c1, c2, c3, and c4 are linear codes of length |z4| = 4 over f3 + uf3 + u2f3, and c5 and c6 are linear codes of length 4 over f9 + uf9 + u2f9. in the next subsections, we focus on self-dual a×zps-quasi-abelian codes in fpm [a×zps ×b] with respect to both the euclidean and hermitian inner products. 4.3. euclidean self-dual a×zps-quasi-abelian codes in fpm[a×zps ×b] euclidean self-dual a×zps-quasi-abelian codes in fpm [a×zps×b] is studied in terms of the following types of pm-cyclotomic classes. a pm-cyclotomic class spm (a) is said to be of type i if a = −a (in this case, spm (a) = spm (−a)), type ii if spm (a) = spm (−a) and a 6= −a, or type iii if spm (−a) 6= spm (a). the primitive idempotent e induced by spm (a) is said to be of type λ ∈{i,ii,iii} if spm (a) is a pm-cyclotomic class of type λ. rearrange the terms in the decomposition in theorem 4.3 based on the pm-cyclotomic classes of types i, ii and iii, we have the next theorem. theorem 4.7. let m and s be positive integers and let a be a finite abelian group such that p |a|. then fpm [a×zps ] ∼= ( ri∏ i=1 ri ) ×   rii∏ j=1 sj  ×  (riii)/2∏ l=1 (tl ×tl)   , where ri,rii and riii are the numbers of elements in a complete set of representatives of pm-cyclotomic classes of a of types i,ii, and iii, respectively, ri = fpm + ufpm + · · · + up s−1fpm for all i = 1, 2, . . . ,ri, sj = fpmri+j + ufpmri+j + · · ·+ u ps−1fpmri+j for all j = 1, 2, . . . ,rii, and tl = fpmri+rii+l + ufpmri+rii+l + . . .up s−1fpmri+rii+l for all l = 1, 2, . . . , (riii)/2. using theorem 4.7 and the analysis similar to those in [11, section ii.d], a a × zps-quasi-abelian code c in fpm [a×zps ×b] and its euclidean dual are given. proposition 4.8. let s and m be positive integers. let a and b be finite abelian groups such that |b| = n and p |a|. then an a×zps-quasi-abelian code in fpm [a×zps ×b] can be viewed as c ∼= ( ri∏ i=1 bi ) ×   rii∏ j=1 cj  ×  (riii)/2∏ l=1 (dl ×d′l)   , (33) where bi, cj, dl and d′l are linear codes of length n over ri, sj, tl and tl, respectively, for all i = 1, 2, . . . ,ri, j = 1, 2, . . . ,rii and l = 1, 2, . . . , (riii)/2. furthermore, the euclidean dual of c in (33) is of the form c⊥e ∼= ( ri∏ i=1 b⊥ei ) ×   rii∏ j=1 c⊥hj  ×  (riii)/2∏ l=1 ( (d′l) ⊥e ×d⊥el ) . the characterization of euclidean self-dual a × zps-quasi-abelian codes in fpm [a × zps × b] is established in terms of a product of linear codes, euclidean self-dual linear codes, and hermitian selfdual linear codes over galois extensions of the ring fpm + ufpm + · · · + up s−1fpm. 222 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 corollary 4.9. let s and m be positive integers. let a and b be finite abelian groups such that |b| = n and p |a|. then a a×zps-quasi-abelian code c in fpm [a×zps ×b] is euclidean self-dual if and only if in the decomposition (33), i) bi is a euclidean self-dual linear code of length n over ri for all i = 1, 2, . . . ,ri, ii) cj is a hermitian self-dual linear code of length n over sj for all j = 1, 2, . . . ,rii, and iii) d′l = d ⊥e l is a linear code of length n over tl for all l = 1, 2, . . . , (riii)/2. from corollary 4.9, the euclidean self-duality of a×zps-quasi-abelian code c in fpm [a×zps ×b] depends only on the structure of a×zps and the index n = |b| but not the structure of b itself. given positive integers m and j, the pair (j,pm) is said to be good if j divides pmt + 1 for some positive integer t, and bad otherwise. this notion have been introduced in [8] and [11] for the enumeration of self-dual cyclic codes and self-dual abelian codes over finite fields and it is completely determined in [9]. let χ be a function defined on pairs (j,pm) as follows. χ(j,pm) = { 0 if (j,pm) is good, 1 otherwise. (34) the number of euclidean self-dual a×zps-quasi-abelian code c in fpm [a×zps×b] can be determined as follows. theorem 4.10. let s and m be positive integers. let a and b be finite abelian groups such that |b| = n is even and the exponent of a is m and p m. then the number of euclidean self-dual a× zps-quasiabelian codes in fpm [a×zps ×b] is (neps (p m,n)) ∑ d|m,ordd(p m)=1 (1−χ(d,pm))na(d) × ∏ d|m ordd(p m) 6=1 ( nhps (p m·ordd(pm),n) )(1−χ(d,pm)) na(d) ordd(p m) × ∏ d|m ( nps (p m·ordd(pm),n) )χ(d,pm) na(d) 2ordd(p m) , where na(d) denotes the number of elements in a of order d determined in [1]. proof. from corollary 4.9, it suffices to determine the numbers of linear codes bi’s, cj’s, and dl’s such that bi and cj are euclidean and hermitian self-dual, respectively. from [12, remark 2.5], the elements in a of the same order are partitioned into pm-cyclotomic classes of the same type. for each divisor d of m, a pm-cyclotomic class containing an element of order d has cardinality ordd(pm) and the number of such pm-cyclotomic classes is na(d) ordd(pm) . we consider the following 3 cases. case 1: χ(d,pm) = 0 and ordd(3k) = 1. by [11, remark 2.6], every 3k-cyclotomic class of a containing an element of order d is of type i. since there are na(d) ordd(pm) such pm-cyclotomic classes, the number of euclidean self-dual linear codes bi’s of length n corresponding to d is (neps (p m,n)) na(d) ordd(p m) = (neps (p m,n)) (1−χ(d,pm))na(d) . case 2: χ(d,pm) = 0 and ordd(pm) 6= 1. by [11, remark 2.6], every pm-cyclotomic class of a containing an element of order d is of type ii and of even cardinality ordd(pm). hence, the number of hermitian self-dual linear codes cj’s of length n corresponding to d is( nhps (p m·ordd(pm),n) ) na(d) ordd(p m) = ( nhps (p m·ordd(pm),n) )(1−χ(d,pm)) na(d) ordd(p m) . 223 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 case 3: χ(d,pm) = 1. by [11, lemma 4.5], every pm-cyclotomic class of a containing an element of order d is of type iii. then the number of linear codes dl’s of length n corresponding to d is ( nps (p m·ordd(pm),n) ) na(d) 2ordd(p m) = ( nps (p m·ordd(pm),n) )χ(d,pm) na(d) 2ordd(p m) . the formula for the number of euclidean self-dual a×zps-quasi-abelian codes in fpm [a×zps×b] follows since d runs over all divisors of m. remark 4.11. in general, the numbers neps (pm,n) and nhps (pm,n) in theorem 4.10 have not been well studied. in the case where the field characteristic is 3, we have the following conclusions. 1. the numbers n3(3m,n), ne3(3m,n) and nh3(3m,n) have been determined in proposition 2.1, [3, theorem 1] and theorem 3.4. by theorem 4.10, the enumeration for euclidean self-dual a× z3quasi-abelian codes in f3m [a×z3 ×b] is completed. . 2. the construction/characterization of linear, euclidean self-dual and hermitian self-dual codes of length n over f3m + uf3m + u2f3m have been given in [3], [4] and in the proof of proposition 3.3. hence, the construction/characterization of euclidean self-dual a × z3-quasi-abelian code in f3m [a×z3 ×b] can be obtained from corollary 4.9. 3. note that, if n is odd, there are no hermitian self-dual linear codes of length n over f3m + uf3m + u2f3m by theorem 3.4. hence, there are no euclidean self-dual a × z3-quasi-abelian codes in f3m [a×z3 ×b] for all abelian groups b of odd order. example 4.12. let a ≤ h ≤ g be finite abelian groups such that a ∼= z2 × z4, h ∼= a × z3, and g ∼= h × z4. form example 4.6, it is easily seen that the 3-cyclotomic classes s3((0, 0)) = {(0, 0)}, s3((0, 2)) = {(0, 2)}, s3((1, 0)) = {(1, 0)}, s3((1, 2)) = {(1, 2)} of a ∼= z2 × z4 are of type i, the 3cyclotomic classes s3((0, 1)) = {(0, 1), (0, 3)} and s3((1, 1)) = {(1, 1), (1, 3)} are of type ii, and there are no 3-cyclotomic classes of type iii. then ri = 4, rii = 2, and riii = 0. in view of theorem 4.7, (32) is recalled as f3[h] ∼= 4∏ i=1 (f3 + uf3 + u2f3) × 2∏ j=1 (f9 + uf9 + u2f9). hence, by corollary 4.9, each euclidean self-dual h-quasi-abelian code in f3[g] is isomorphic to a code of the form c1 ×c2 ×c3 ×c4 ×c5 ×c6, where c1, c2, c3, and c4 are euclidean self-dual linear codes of length 4 over f3 + uf3 + u2f3, and c5 and c6 are hermitian self-dual linear codes of length 4 over f9 + uf9 + u2f9. 4.4. hermitian self-dual a×zps-quasi-abelian codes in fpm[a×zps ×b] in this subsection, we focus on the case where m is even and study hermitian self-dual a × zpsquasi-abelian codes in fpm [a×zps ×b]. the characterization and enumeration of hermitian self-dual a×zps-quasi-abelian codes in fpm [a× zps ×b] are given based on the decomposition of a group algebra fpm [a×zps ] in terms of the following types of pm-cyclotomic classes of a. a pm-cyclotomic class spm (a) is said to be of type i ′ if spm (a) = spm (−p m 2 a) or type ii′ if spm (a) 6= spm (−p m 2 a). the primitive idempotent e induced by spm (a) is said to be of type λ ∈{i′,ii′} if spm (a) is a pm-cyclotomic class of type λ. rearrange the terms in the decomposition in theorem 4.3 based on the pm-cyclotomic classes of types i′ and ii′, the next theorem follows. 224 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 theorem 4.13. let m be an even positive integer and let a be a finite abelian group such that p |a|. fpm [a×zps ] ∼=   ri′∏ j=1 s  ×  (rii′)/2∏ l=1 (tl ×tl)   , where r′i and rii′ are the numbers of elements in a complete set of representatives of p m-cyclotomic classes of a of types i′ and ii′, respectively, sj = fpmj + ufpmj + · · · + up s−1fpmj for all j = 1, 2, . . . ,ri′ and tl = fpmri′+l + ufpkri′+l + · · · + u ps−1f p mri′+l for all l = 1, 2, . . . , (rii′)/2. using theorem 4.13 and the analysis similar to those in [12, section ii.d], the a×zps-quasi-abelian code c in fpm [a×zps ×b] and its hermitian dual are given. proposition 4.14. let s and m be positive integers such that m is even. let a and b be finite abelian groups such that |b| = n and p |a|. then an a× zps-quasi-abelian code in fpm [a× zps ×b] can be viewed as c ∼=   ri′∏ j=1 cj  ×  (rii′)/2∏ l=1 (dl ×d′l)   , (35) where cj, dl and d′l are linear codes of length n over sj, tl and tl, respectively, for all j = 1, 2, . . . ,ri′ and l = 1, 2, . . . , (rii′)/2. furthermore, the hermitian dual of c in (35) is of the form c⊥h ∼=   ri′∏ j=1 c⊥hj  ×  (rii′)/2∏ l=1 ( (d′l) ⊥e ×d⊥el ) . the characterization of hermitian self-dual a × zps-quasi-abelian codes in fpm [a × zps × b] in term of a product of linear codes, and hermitian self-dual linear codes over galois extensions of the ring fpm + ufpm + · · · + up s−1fpm is established. corollary 4.15. let s and m be positive integers such that m is even. let a and b be finite abelian groups such that |b| = n and p |a|. then an a × zps-quasi-abelian code in fpm [a × zps × b] is hermitian self-dual if and only if in the decomposition (35), i) cj is a hermitian self-dual linear code of length n over sj for all j = 1, 2, . . . ,ri′, and ii) d′l = d ⊥e l is a linear code of length n over tl for all l = 1, 2, . . . , (rii′)/2. from corollary 4.15, it follows that the hermitian self-duality of a × zps-quasi-abelian codes in fpm [a×zps ×b] depends only on the structure of a×zps and the index n = |b| but not the structure of b itself. given a positive integer m and a positive integer j, the pair (j,pm) is said to be oddly good if j divides pmt + 1 for some odd positive integer t. this notion has been introduced in [12] for characterizing the hermitian self-dual abelian codes in principal ideal group algebra and completely determined in [9]. let λ be a function defined on the pair (j,pm) as λ(j,pm) = { 0 if (j,pm) is oddly good, 1 otherwise. (36) the number of hermitian self-dual a×zps-quasi-abelian codes in fpm [a×zps×b] can be determined as follows. 225 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 theorem 4.16. let s and m be positive integers such that m is even. let a and b be finite abelian groups such that |b| = n is even and the exponent of a is m and p m. then the number of euclidean self-dual a×zps-quasi-abelian codes in fpm [a×zps ×b] is ∏ d|m ( nhps (p m·ordd(pm),ps) )(1−λ(d,p m2 )) na(d) ordd(p m) × ∏ d|m ( nps (p m·ordd(pm),ps) )λ(d,p m2 ) na(d) 2ordd(p m) , where na(d) denotes the number of elements of order d in a determined in [1]. proof. by corollary 4.15, it is enough to determine the numbers linear codes cj’s and dl’s of length n in (35) such that cj is hermitian self-dual. the result can be deduced using arguments similar to those in the proof of theorem 4.10, where [12, lemma 3.5] is applied instead of [11, lemma 4.5]. remark 4.17. in general, the number nhps (pm,n) of hermitian self-dual linear codes of length n over fpm + ufpm + · · · + up s−1fpm in theorem 4.16 has not been well studied. in the case where the field characteristic is 3, we have the following results. 1. the numbers n3(pm,n) and nh3(3m,n) have been determined in proposition 2.1 and theorem 3.4. hence, the complete enumeration of hermitian self-dual a×z3-quasi-abelian codes in f3m [a×z3× b] follows. 2. the construction/characterization of linear and hermitian self-dual dual linear codes of length n over f3m + uf3m + u2f3m have been given in [4] and in the proof of proposition 3.3. hence, the construction/characterization of hermitian self-dual a×z3-quasi-abelian code in f3m [a×z3 ×b] can be obtained from corollary 4.15. 3. note that, if n is odd, there are no hermitian self-dual linear codes of length n over f3m + uf3m + u2f3m by theorem 3.4. hence, there are no hermitian self-dual a × z3-quasi-abelian codes in f3m [a×z3 ×b] for all abelian groups b of odd order. 5. conclusion and remarks by extending the technique used in the study of euclidean self-dual linear codes over fq +ufq +u2fq in [3], complete characterization and enumeration of hermitian self-dual linear codes over fq +ufq +u2fq have been established for all square prime powers q. an algorithm for constructions of such self-dual codes has veen provided as well as an illustrative example. subsequently, algebraic characterization of h-quasi-abelian codes in fpm [g] has been studied, where h ≤ g are finite abelian groups and the sylow p-subgroup of h is cyclic, or equivalently, fpm [h] is a principal ideal group algebra. in the special case where h ∼= a × z3 with 3 |a|, characterization and enumeration of h-quasi-abelian codes and self-dual h-quasi-abelian codes in f3m [h × b] have been completely determined for all finite abelian group b. as applications, characterization and enumeration of self-dual a × z3-quasi-abelian codes in f3m [a × z3 × b] can be presented in terms of linear codes and self-dual linear codes over some extensions of f3m + uf3m + u2f3m determined in [3], [4] and section 3. in general, it would be interesting to studied a×p-quasi-abelian codes and self-dual a×p-quasiabelian codes in fpm [a×p×b] for all primes p and finite abelian p-groups p. for e ≥ 4, characterization and enumeration of self-dual linear codes over fpm +ufpm +· · ·+ue−1fpm are other interesting problems. acknowledgment: the authors would like to thank the anonymous referees for their helpful comments. 226 p. choosuwan, s. jitman / j. algebra comb. discrete appl. 7(3) (2020) 209–227 references [1] s. benson, students ask the darnedest things: a result in elementary group theory, math. mag. 70 (1997) 207–211. 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[24] h. tong, x. wang, new mds euclidean and hermitian self-dual codes over finite fields, advances in pure mathematics 7 (2017) 325–333. 227 https://doi.org/10.1080/0025570x.1997.11996535 https://doi.org/10.1080/0025570x.1997.11996535 https://doi.org/10.1007/s12190-017-1117-0 https://doi.org/10.1007/s12190-017-1117-0 https://doi.org/10.1016/j.ffa.2014.10.003 https://doi.org/10.1109/18.825811 https://doi.org/10.1080/00927877608822109 https://doi.org/10.1109/18.312154 https://doi.org/10.1109/18.312154 https://doi.org/10.1109/tit.2010.2092415 https://doi.org/10.1109/tit.2010.2092415 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s10623-013-9878-4 https://doi.org/10.1109/tit.2012.2236383 https://doi.org/10.1109/tit.2012.2236383 https://doi.org/10.1109/tit.2013.2296495 https://doi.org/10.1109/tit.2013.2296495 https://doi.org/10.1016/j.jcta.2003.10.003 https://doi.org/10.1016/j.jcta.2003.10.003 https://doi.org/10.1007/s10623-008-9232-4 https://doi.org/10.1007/s10623-008-9232-4 https://doi.org/10.1109/lcomm.2019.2908640 https://doi.org/10.1109/lcomm.2019.2908640 https://doi.org/10.1109/18.841186 https://doi.org/10.1109/18.841186 https://doi.org/10.1007/pl00012382 https://doi.org/10.1007/pl00012382 https://doi.org/10.1016/s0021-9800(68)80067-5 https://doi.org/10.1109/18.165458 https://doi.org/10.1109/18.165458 https://doi.org/10.4236/apm.2017.75019 https://doi.org/10.4236/apm.2017.75019 introduction preliminaries hermitian self-dual linear codes over fq+ufq+u2fq self-dual quasi-abelian codes over principal ideal group algebras conclusion and remarks references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.790751 j. algebra comb. discrete appl. 7(3) • 259–267 received: 5 september 2019 accepted: 18 may 2020 journal of algebra combinatorics discrete structures and applications some results on relative dual baer property research article tayyebeh amouzegar, rachid tribak abstract: let r be a ring. in this article, we introduce and study relative dual baer property. we characterize r-modules m which are rr-dual baer, where r is a commutative principal ideal domain. it is shown that over a right noetherian right hereditary ring r, an r-module m is n-dual baer for all r-modules n if and only if m is an injective r-module. it is also shown that for r-modules m1, m2, . . ., mn such that mi is mj-projective for all i > j ∈ {1, 2, . . . , n}, an r-module n is ⊕n i=1 mi-dual baer if and only if n is mi-dual baer for all i ∈ {1, 2, . . . , n}. we prove that an r-module m is dual baer if and only if s = endr(m) is a baer ring and im = rm (ls(im)) for every right ideal i of s. 2010 msc: 16d10, 16d80 keywords: baer rings, dual baer modules, relative dual baer property, homomorphisms of modules 1. introduction throughout this paper, r will denote an associative ring with identity, and all modules are unitary right r-modules. let m be an r-module. we will use the notation n � m to indicate that n is small in m (i.e., l + n 6= m for every proper submodule l of m). by e(m) and endr(m), we denote the injective hull of m and the endomorphism ring of m, respectively. by q, z, and n we denote the set of rational numbers, integers and natural numbers, respectively. for a prime number p, z(p∞) denotes the prüfer p-group. the concept of baer rings was first introduced in [6] by kaplansky. since then, many authors have studied this kind of rings (see, e.g., [2] and [3]). a ring r is called baer if the right annihilator of any nonempty subset of r is generated by an idempotent. in 2004, rizvi and roman extended the notion of baer rings to a module theoretic version [10]. according to [10], a module m is called a baer module if for every left ideal i of endr(m), ∩φ∈ikerφ is a direct summand of m. this notion was recently dualized by keskin tütüncü-tribak in [14]. a module m is said to be dual baer if for every right ideal tayyebeh amouzegar; department of mathematics, quchan university of advanced technology, quchan, iran (email: t.amoozegar@yahoo.com). rachid tribak (corresponding author); centre régional des métiers de l’education et de la formation (crmeftth)-tanger, avenue my abdelaziz, souani, b.p. 3117, tangier, morocco (email: tribak12@yahoo.com). 259 https://orcid.org/0000-0002-0600-5326 https://orcid.org/0000-0001-8400-4321 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 i of s = endr(m), ∑ φ∈i imφ is a direct summand of m. equivalently, for every nonempty subset a of s, ∑ φ∈a imφ is a direct summand of m (see [14, theorem 2.1]). a module m is said to be rickart if for any ϕ ∈ endr(m), kerϕ is a direct summand of m (see [7]). the notion of dual rickart modules was studied recently in [8] by lee-rizvi-roman. a module m is said to be dual rickart if for every ϕ ∈ endr(m), imϕ is a direct summand of m. in [8], it was introduced the notion of relative dual rickart property which was used in the study of direct sums of dual rickart modules. let n be an r-module. an r-module m is called n-dual rickart if for every homomorphism ϕ : m → n, imϕ is a direct summand of n (see [8]). similarly, we introduce in this paper the concept of relative dual baer property. a module m is called n-dual baer if for every subset a of hom r(m,n),∑ f∈a imf is a direct summand of n. it is clear that if m is n-dual baer, then m is n-dual rickart. we determine the structure of modules m which are rr-dual baer for a commutative principal ideal domain r (proposition 2.7). then we show that for an r-module m, rr is m-dual baer if and only if m is a semisimple module (proposition 2.9). it is shown that over a right noetherian right hereditary ring r, an r-module m is n-dual baer for all r-modules n if and only if m is an injective r-module (corollary 2.17). we prove that if {mi}i is a family of r-modules, then for each j ∈ i, ⊕ i∈i mi is mj-dual baer if and only if mi is mj-dual baer for all i ∈ i (corollary 2.24). it is also shown that for r-modules m1, m2, . . ., mn such that mi is mj-projective for all i > j ∈{1, 2, . . . ,n}, an r-module n is ⊕n i=1 mi-dual baer if and only if n is mi-dual baer for all i ∈{1, 2, . . . ,n} (theorem 2.25). we conclude this paper by showing that an r-module m is dual baer if and only if s = endr(m) is a baer ring and im = rm (ls(i)) for every right ideal i of s, where ls(i) = {ϕ ∈ s | ϕi = 0}, rm (ls(i)) = {m ∈ m | ls(i)m = 0} and im = ∑ f∈i imf (theorem 2.31). 2. main results definition 2.1. let n be an r-module. an r-module m is called n-dual baer if, for every subset a of hom r(m,n), ∑ f∈a imf is a direct summand of n. obviously, an r-module m is dual baer if and only if m is m-dual baer. example 2.2. (1) let n be a semisimple r-module. then for every r-module m, m is n-dual baer. (2) if m and n are r-modules such that homr(m,n) = 0, then m is n-dual baer. it follows that for any couple of different maximal ideals m1 and m2 of a commutative noetherian ring r, e(r/m1) is e(r/m2)-dual baer (see [12, proposition 4.21]). (3) let p be a prime number. note that z/pz and z(p∞) are dual baer z-modules. on the other hand, it is clear that z(p∞) is z/pz-dual baer but z/pz is not z(p∞)-dual baer. recall that a module m is said to have the strong summand sum property, denoted briefly by sssp, if the sum of any family of direct summands of m is a direct summand of m. following [8, definition 2.14], a module m is called n-d-rickart if, for every homomorphism ϕ : m → n, imϕ is a direct summand of n. proposition 2.3. let m and n be two r-modules. if m is n-dual baer, then m is n-d-rickart. the converse holds when n has the sssp. proof. this follows from the definitions of “m is n-d-rickart” and “m is n-dual baer”. the next example shows that the assumption “n has the sssp” is not superfluous in proposition 2.3. example 2.4. let r be a von neumann regular ring which is not semisimple (e.g., r = ∏∞ i=1 z/2z). by [8, proposition 2.26], the r-module rr does not have the sssp. on the other hand, rr is rr-d-rickart, but it is not rr-dual baer (see [14, corollary 2.9] and [8, remark 2.2]). 260 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 proposition 2.5. let n be an indecomposable r-module. then the following conditions are equivalent for an r-module m. (i) m is n-dual baer; (ii) m is n-d-rickart; (iii) every nonzero ϕ ∈ homr(m,n) is an epimorphism. proof. (i) ⇒ (ii) and (iii) ⇒ (i) are clear. (ii) ⇒ (iii) let 0 6= ϕ ∈ homr(m,n). by assumption, imϕ is a direct summand of n. but n is indecomposable. then imϕ = n. this completes the proof. proposition 2.6. let m and n be modules such that homr(m,n) 6= 0 (e.g., n is m-generated). then the following conditions are equivalent: (i) m is n-dual baer and n is indecomposable; (ii) every nonzero homomorphism ϕ ∈ hom r(m,n) is an epimorphism. proof. (i) ⇒ (ii) this follows from proposition 2.5. (ii) ⇒ (i) it is clear that m is n-dual baer. now let k be a nonzero direct summand of n. let k′ be a submodule of n such that n = k ⊕ k′. since homr(m,n) 6= 0, there exists a nonzero homomorphism ϕ ∈ homr(m,n). let π′ : n → k′ be the projection map and let i′ : k′ → n be the inclusion map. then i′π′ϕ ∈ homr(m,n). assume that i′π′ϕ 6= 0. by hypothesis, imi′π′ϕ = n. so k′ = n. thus k = 0, a contradiction. therefore i′π′ϕ = 0. hence k′ = 0 and k = n. it follows that n is indecomposable. the following result describes the structure of r-modules which are rr-dual baer, where r is a commutative principal ideal domain which is not a field. proposition 2.7. let r be a commutative principal ideal domain which is not a field. then the following conditions are equivalent for an r-module m: (i) m is rr-dual baer; (ii) m is rr-d-rickart; (iii) m has no nonzero cyclic torsion-free direct summands; (iv) homr(m,rr) = 0. proof. (i) ⇒ (ii) this is clear. (ii) ⇒ (iii) assume that m has an element x such that xr is a direct summand of m and rr ∼= xr. let π : m → xr be the projection map and let f : xr → rr be an isomorphism. then fπ : m → rr is an epimorphism. let α be a nonzero element of r which is not invertible. consider the homomorphism g : rr → rr defined by g(r) = αr for all r ∈ r. then gfπ ∈ homr(m,rr) and imgfπ = αr. it is clear that αr 6= 0 and αr 6= r. thus αr is not a direct summand of r. so m is not rr-d-rickart, a contradiction. (iii) ⇒ (iv) assume that homr(m,rr) 6= 0. so there exists a nonzero homomorphism f : m → rr. thus imf = ar for some nonzero a ∈ r since r is a principal ideal domain. then m/kerf ∼= ar ∼= rr is a projective r-module. it follows that kerf is a direct summand of m. let y be a submodule of m such that m = kerf ⊕y . therefore y ∼= rr. this contradicts our assumption. hence homr(m,rr) = 0. (iv) ⇒ (i) this is immediate. example 2.8. consider a z-module m = q(i) ⊕ t, where t is a torsion z-module and i is an index set. suppose that m is not z-dual baer. by proposition 2.7, there exists a cyclic submodule l of m such that l ∼= z and l is a direct summand of m. let n be a submodule of m such that m = l⊕n. 261 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 since t is the torsion submodule of m, we have t ⊆ n. hence t is a direct summand of n. let k be a submodule of n such that n = k⊕t . thus m = l⊕k⊕t . therefore l⊕k ∼= q(i). so l is injective, a contradiction. it follows that m is z-dual baer. on the other hand, note that if t ∼= z(2∞) ⊕ z/8z, then m is not a dual baer module (see [14, corollary 3.5]. in proposition 2.7, we studied when an r-module m is rr-dual baer. next, we investigate when rr is m-dual baer for an r-module m. proposition 2.9. the following conditions are equivalent for an r-module m: (i) the r-module rr is m-dual baer; (ii) m is a semisimple module. proof. (i) ⇒ (ii) let x ∈ m. consider the r-homomorphism ϕ : r → m defined by ϕ(r) = xr for all r ∈ r. then imϕ = xr. since rr is m-dual baer, it follows that for any submodule l of m, l = ∑ x∈l xr is a direct summand of m. therefore m is semisimple. (ii) ⇒ (i) is obvious. corollary 2.10. the following conditions are equivalent for a ring r: (i) the r-module rr is dual baer; (ii) the r-module rr is e(r)-dual baer; (iii) r is a semisimple ring. proof. (i) ⇔ (iii) by [14, corollary 2.9]. (ii) ⇔ (iii) this follows from proposition 2.9. remark 2.11. if k is a submodule of an r-module m such that k is m-dual baer, then k is a direct summand of m. in particular, if the r-module m is e(m)-dual baer, then m is an injective module. the next example shows that even if a module m is injective, the module m need not be m-dual baer. example 2.12. let r be a self injective ring which is not semisimple (e.g., r = ∏∞ n=1 z/2z). then e(rr) = rr. by [14, corollary 2.9], the r-module rr is not rr-dual baer. next, we will be concerned with the modules m which are n-dual baer for all modules n. we begin with the following proposition which provides a class of rings r whose semisimple modules are n-dual baer for any r-module n. proposition 2.13. let r be a right noetherian right v-ring and let m be a semisimple r-module. then m is n-dual baer for every r-module n. proof. let n be an r-module. it is clear that for any ϕ ∈ homr(m,n), imϕ is semisimple. let a be a subset of homr(m,n). then ∑ f∈a imf is a semisimple submodule of n. since r is a right noetherian right v-ring, ∑ f∈a imf is injective by [4, proposition 1]. therefore ∑ f∈a imf is a direct summand of n. so m is n-dual baer. the next example shows that the condition “r is a right noetherian ring” in the hypothesis of proposition 2.13 is not superfluous. example 2.14. let f be a field and let r = ∏ n∈n fn such that fn = f for all n ∈ n. then r is a commutative v-ring which is not noetherian. note that soc(r) = ⊕n∈nfn is an essential proper ideal of r. in particular, soc(r) is not a direct summand of r. so soc(r) is not rr-dual baer. 262 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 following [13], a module m is called noncosingular if for every nonzero module n and every nonzero homomorphism f : m → n, imf is not a small submodule of n. proposition 2.15. let m be a module. assume that m is n-dual baer for every r-module n. then every factor module of m is injective. in particular, m is a noncosingular module. proof. let l be a submodule of m. let π : m → m/l be the natural epimorphism and let µ : m/l → e(m/l) be the inclusion map. then µπ ∈ homr(m,e(m/l)) and imµπ = m/l. since m is e(m/l)-dual baer, m/l is a direct summand of e(m/l). so m/l is injective. this completes the proof. proposition 2.16. let r be a right noetherian ring. then the following conditions are equivalent for an r-module m: (i) m is n-dual baer for all r-modules n; (ii) every factor module of m is an injective r-module. proof. (i) ⇒ (ii) by proposition 2.15. (ii) ⇒ (i) let n be an r-module. it is clear that imϕ is injective for every ϕ ∈ homr(m,n). since the ring r is right noetherian, ∑ f∈a imf is injective for every subset a of hom r(m,n) by [1, proposition 18.13]. therefore ∑ f∈a imf is a direct summand of n. this proves the proposition. recall that a ring r is called right hereditary if each of its right ideals is projective. it is well known that a ring r is right hereditary if and only if every factor module of an injective right r-module is injective (see, for example [16, 39.16]). the next result is a direct consequence of proposition 2.16. it determines the structure of r-modules m which are n-dual baer for all r-modules n, where r is a right noetherian right hereditary ring. corollary 2.17. let r be a right noetherian right hereditary ring (e.g., r is a dedekind domain). then the following conditions are equivalent for an r-module m: (i) m is n-dual baer for any r-module n; (ii) m is an injective r-module. example 2.18. let m be a z-module. it is easily seen from corollary 2.17 that m is n-dual baer for any z-module n if and only if m is a direct sum of z-modules each isomorphic to the additive group of rational numbers q or to z(p∞) (for various primes p). combining corollary 2.17 and [8, corollary 2.30], we obtain the following result. corollary 2.19. the following conditions are equivalent for a ring r: (i) every injective r-module is dual baer; (ii) every injective module is n-dual baer for every r-module n; (iii) r is a right noetherian right hereditary ring. the next characterization extends [14, corollary 2.5]. theorem 2.20. let m and n be two r-modules. then m is n-dual baer if and only if for any direct summand m′ of m and any submodule n′ of n, m′ is n′-dual baer. proof. let m′ = em for some e2 = e ∈ endr(m) and let n′ be a submodule of n. let {ϕi}i be a family of homomorphisms in hom r(m′,n′). since ϕie(m) = ϕi(m′) ⊆ n′ ⊆ n for every i ∈ i and m is n-dual baer, ∑ i∈i ϕie(m) is a direct summand of n. therefore ∑ i∈i ϕi(m ′) is a direct summand of n′. it follows that m′ is n′-dual baer. the converse is obvious. 263 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 corollary 2.21. the following conditions are equivalent for a module m: (i) m is a dual baer module; (ii) for any direct summand k of m and any submodule n of m, k is n-dual baer. from [14, example 3.1 and theorem 3.4], it follows that a direct sum of dual baer modules is not dual baer, in general. next, we focus on when a direct sum of n-dual baer modules is also n-dual baer for some module n. proposition 2.22. let n be a module having the sssp and let {mi}i be a family of modules. then⊕ i∈i mi is n-dual baer if and only if mi is n-dual baer for all i ∈ i. proof. suppose that ⊕ i∈i mi is n-dual baer. by theorem 2.20, mi is n-dual baer for all i ∈ i. conversely, assume that mi is n-dual baer for all i ∈ i. let {ϕλ}λ be a family of homomorphisms in hom r( ⊕ i∈i mi,n). for each i ∈ i, let µi : mi → ⊕ i∈i mi denote the inclusion map. then for every i ∈ i and every λ ∈ λ, ϕλµi ∈ homr(mi,n). since mi is n-dual baer for every i ∈ i, it follows that im(ϕλµi) is a direct summand of n for every (i,λ) ∈ i × λ. note that for each λ ∈ λ, imϕλ = ∑ i∈i im(ϕλµi). as n has the sssp, ∑ λ∈λ imϕλ = ∑ λ∈λ ∑ i∈i im(ϕλµi) is a direct summand of n. therefore ⊕ i∈i mi is n-dual baer. the following result is taken from [14, theorem 2.1]. theorem 2.23. the following conditions are equivalent for a module m and s = endr(m): (i) m is a dual baer module; (ii) for every nonempty subset a of s, ∑ f∈a imf = e(m) for some idempotent e ∈ s; (iii) m has the sssp and for every ϕ : m → m, imϕ is a direct summand of m. corollary 2.24. let {mi}i be a family of modules and let j ∈ i. then ⊕ i∈i mi is mj-dual baer if and only if mi is mj-dual baer for all i ∈ i. proof. the necessity follows from theorem 2.20. conversely, by assumption, we have mj is mj-dual baer. then mj is a dual baer module. by theorem 2.23, mj has the sssp. applying proposition 2.22,⊕ i∈i mi is mj-dual baer. in the following result, we present conditions under which a module n is ⊕n i=1 mi-dual baer for some modules mi (1 ≤ i ≤ n). theorem 2.25. let m1, . . . , mn be r-modules, where n ∈ n. assume that mi is mj-projective for all i > j ∈ {1, 2, . . . ,n}. then for any r-module n, n is ⊕n i=1 mi-dual baer if and only if n is mi-dual baer for all i ∈{1, 2, . . . ,n}. proof. the necessity follows from theorem 2.20. conversely, suppose that n is mi-dual baer for all i ∈{1, 2, . . . ,n}. we will show that n is ⊕n i=1 mi-dual baer. by induction on n and taking into account [9, proposition 4.33], it is sufficient to prove this for the case n = 2. assume that n is mi-dual baer for i = 1, 2 and m2 is m1-projective. let {φλ}λ be a family of homomorphisms in hom r(n,m1 ⊕m2). let π2 : m1 ⊕ m2 → m2 be the projection of m1 ⊕ m2 on m2 along m1. we want to prove that∑ λ∈λ imφλ is a direct summand of m1 ⊕ m2. since n is m2-dual baer, ∑ λ∈λ π2φλ(n) is a direct summand of m2. so ∑ λ∈λ π2φλ(n) is m1-projective by [9, proposition 4.32]. as m1 + (∑ λ∈λ imφλ ) = m1 ⊕ (∑ λ∈λ π2φλ(n) ) is a direct summand of m1 ⊕m2, there exists a submodule l ≤ ∑ λ∈λ imφλ such that m1 + (∑ λ∈λ imφλ ) = m1⊕l by [9, lemma 4.47]. thus ∑ λ∈λ imφλ = ( m1 ∩ (∑ λ∈λ imφλ )) ⊕l by modularity. it is easily seen that ∑ λ∈λ π2φλ(n) is a direct summand of m2. let k2 be a submodule of m2 such that m2 = k2⊕ (∑ λ∈λ π2φλ(n) ) . therefore m1⊕m2 = m1⊕l⊕k2. let π1 : m1⊕(l⊕k) → m1 264 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 be the projection of m1⊕m2 on m1 along l⊕k. then π1φλ ∈ hom r(n,m1) for every λ ∈ λ. moreover, we have ∑ λ∈λ π1φλ(n) = π1 (∑ λ∈λ imφλ ) = ((∑ λ∈λ imφλ ) + (l⊕k) ) ∩m1. but ∑ λ∈λ imφλ = ( m1 ∩ (∑ λ∈λ imφλ )) ⊕l. then, ∑ λ∈λ π1φλ(n) = (( m1 ∩ (∑ λ∈λ imφλ )) ⊕l⊕k ) ∩m1 = m1 ∩ (∑ λ∈λ imφλ ) . since n is m1-dual baer, ∑ λ∈λ π1φλ(n) = m1 ∩ (∑ λ∈λ imφλ ) is a direct summand of m1. it follows that ( m1 ∩ (∑ λ∈λ imφλ )) ⊕l is a direct summand of m1⊕l⊕k2. so ∑ λ∈λ imφλ is a direct summand of m1 ⊕m2. consequently, n is m1 ⊕m2-dual baer. this completes the proof. corollary 2.26. let m1, . . . , mn be r-modules, where n ∈ n. assume that mi is mj-projective for all i > j ∈ {1, 2, . . . ,n}. then m = ⊕n i=1 mi is a dual baer module if and only if mi is mj-dual baer for all i,j ∈{1, 2, . . . ,n}. proof. the necessity follows from theorem 2.20. conversely, suppose that mi is mj-dual baer for all i,j ∈ {1, 2, . . . ,n}. by corollary 2.24, m is mj-dual baer for all j ∈ {1, 2, . . . ,n}. since mi is mj-projective for all i > j ∈{1, 2, . . . ,n}, m is ⊕n i=1 mi-dual baer by theorem 2.25. thus m is a dual baer module. note that the sufficiency in corollary 2.26 can be proved by using [14, theorem 3.10]. following [8, definition 5.7], a module m is called n-d2 (or relatively d2 to n) if for any submodule m′ of m, m/m′ is isomorphic to a direct summand of n implies that m′ is a direct summand of m. proposition 2.27. let m1, . . . , mn be r-modules, where n ∈ n. assume that mi is mj-d2 for all i,j ∈ {1, 2, . . . ,n}. then ⊕n i=1 mi is a dual baer module if and only if mi is mj-dual baer for all i,j ∈{1, 2, . . . ,n} and m has the sssp. proof. (⇒) by [8, theorem 5.11], mi is mj-d-rickart for all i,j ∈ {1, 2, . . . ,n}. note that mi has the sssp for every i ∈{1, 2, . . . ,n} (see theorem 2.23). applying proposition 2.3, it follows that mi is mj-dual baer for all i,j ∈{1, 2, . . . ,n}. (⇐) this follows easily from [8, theorem 5.11], proposition 2.3 and theorem 2.23. theorem 2.28. let m = ⊕ i∈i mi be the direct sum of fully invariant submodules mi. then m is a dual baer module if and only if mi is a dual baer module for all i ∈ i. proof. the necessity follows from [14, corollary 2.5]. conversely, let s = endr(m) and let {ϕλ}λ be a family of homomorphisms in s. for each i ∈ i, let πi : m → mi be the projection map and let µi : mi → m be the inclusion map. note that for each λ ∈ λ, ϕλ(m) = ∑ i∈i ϕλµi(mi). since each mi (i ∈ i) is fully invariant in m, it follows that ϕλ(m) = ∑ i∈i πiϕλµi(mi) for all λ ∈ λ. for every i ∈ i and every λ ∈ λ, let ni,λ = πiϕλµi(mi). therefore, ∑ λ∈λ ϕλ(m) = ∑ λ∈λ ∑ i∈i πiϕλµi(mi) = ∑ λ∈λ (∑ i∈i ni,λ ) = ⊕ i∈i (∑ λ∈λ ni,λ ) . since each mi (i ∈ i) is dual baer, each mi (i ∈ i) has the sssp by theorem 2.23. thus ∑ λ∈λ ni,λ is a direct summand of mi for every i ∈ i. so ∑ λ∈λ ϕλ(m) is a direct summand of m. consequently, m is a dual baer module. 265 t. amouzegar, r. tribak / j. algebra comb. discrete appl. 7(3) (2020) 259–267 we conclude this paper by showing a new characterization of dual baer modules. let m be an r-module with s = endr(m). then for every nonempty subset a of s, we denote ls(a) = {ϕ ∈ s | ϕa = 0} and rm (a) = {m ∈ m | am = 0}. we also denote ls(n) = {ϕ ∈ s | ϕ(n) = 0} for any submodule n of m. recall that a ring r is called a baer ring if for every nonempty subset i ⊆ r, there exists an idempotent e ∈ r such that ls(i) = re. proposition 2.29. ([5, proposition 2.3]) for an r-module m, s = endr(m) is a baer ring if and only if rm (ls( ∑ ϕ∈a imϕ)) is a direct summand of m for all nonempty subsets a of s. the next example shows that if m is a module such that s = endr(m) is a baer ring, then m is not a dual baer module, in general. example 2.30. consider the z-module m = z. then s = endz(m) ∼= z. clearly, z is a baer ring. on the other hand, it is easily seen that m is not a dual baer module. note that if m is an r-module with s = endr(m), then for any nonempty subset a of s, ls(a) = ls(am), where am = ∑ f∈a imf. the next result can be considered as an analogue of [8, theorem 3.5]. theorem 2.31. the following are equivalent for an r-module m and s = endr(m): (i) m is a dual baer module; (ii) s is a baer ring and am = rm (ls(am)) for every nonempty subset a of s; (iii) s is a baer ring and im = rm (ls(im)) for every right ideal i of s. proof. (i) ⇒ (ii) from [15, theorem 3.6], it follows that s is a baer ring. moreover, we have rm (ls(am)) = rm (ls(a)) = rm (s(1 −e)) = e(m) = am for all nonempty subsets a of s. (ii) ⇒ (iii) this is obvious. (iii) ⇒ (i) let i be a right ideal of s. since s is a baer ring, rm (ls(im)) is a direct summand of m by proposition 2.29. but im = rm (ls(im)). then im is a direct summand of m. by theorem 2.23, it follows that m is a dual baer module. combining theorem 2.31 and [10, theorem 4.1], we get the following result. corollary 2.32. let m be an r-module such that im = rm (ls(im)) for every right ideal i of s = endr(m). if m is a baer module, then m is a dual baer module. references [1] f. w. anderson, k. r. fuller, rings and categories of modules, vol. 13, springer–verlag, new york 1992. [2] e. p. armendariz, a note on extensions of baer and p.p.–rings, j. austral. math. soc. 18(4) (1974) 470–473. 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[16] r. wisbauer, foundations of module and ring theory, gordon and breach science publishers, philadelphia 1991. 267 https://doi.org/10.1080/00927872.2010.507232 https://doi.org/10.1080/00927872.2010.515639 https://doi.org/10.1081/agb-120027854 https://doi.org/10.1142/9789812701671_0021 https://doi.org/10.1142/9789812701671_0021 https://doi.org/10.1080/00927870209342390 https://doi.org/10.1080/00927870209342390 https://doi.org/10.1017/s0017089509990334 https://dx.doi.org/10.1090/conm/609/12081 https://dx.doi.org/10.1090/conm/609/12081 introduction main results references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.935947 j. algebra comb. discrete appl. 8(2) • 73–90 received: 23 april 2020 accepted: 24 november 2020 journal of algebra combinatorics discrete structures and applications on optimal linear codes of dimension 4∗ research article nanami bono, maya fujii, tatsuya maruta abstract: in coding theory, the problem of finding the shortest linear codes for a fixed set of parameters is central. given the dimension k, the minimum weight d, and the order q of the finite field fq over which the code is defined, the function nq(k, d) specifies the smallest length n for which an [n, k, d]q code exists. the problem of determining the values of this function is known as the problem of optimal linear codes. using the geometric methods through projective geometry, we determine nq(4, d) for some values of d by constructing new codes and by proving the nonexistence of linear codes with certain parameters. 2010 msc: 94b05, 94b27, 51e20 keywords: optimal linear codes, griesmer bound, geometric method 1. introduction we denote by fq the field of q elements. let fnq be the vector space of n-tuples over fq. an [n,k,d]q code c is a k-dimensional subspace of fnq with minimum weight d = min{wt(c) | c ∈ c,c 6= (0, . . . , 0)}, where wt(c) is the number of non-zero entries in the vector c. the weight distribution of c is the list of integers ai where ai is the number of codewords of weight i, 0 ≤ i ≤ n. the weight distribution with (a0,ad, . . .) = (1,α, . . .) is also expressed as 01dα · · · . a fundamental problem in coding theory is to find nq(k,d), the minimum length n for which an [n,k,d]q code exists [10, 11]. an [n,k,d]q code satisfies the inequality called the griesmer bound [8, 10]: n ≥ gq(k,d) = k−1∑ i=0 ⌈ d/qi ⌉ , where dxe denotes the smallest integer greater than or equal to x. the values of nq(k,d) are determined for all d only for some small values of q and k. for k = 3, nq(3,d) is known for all d for q ≤ 9 [1]. see ∗ this research was partially supported by jsps kakenhi grant number 20k03722. nanami bono, maya fujii, tatsuya maruta (corresponding author); department of mathematical sciences, osaka prefecture university, sakai, osaka 599-8531, japan (email: bononanami@gmail.com, ddddy.maya0802@gmail.com, maruta@mi.s.osakafu-u.ac.jp). 73 https://orcid.org/0000-0001-7858-0787 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 [26] for the updated table of nq(k,d) for some small q and k. the following theorems give some known values of nq(4,d). theorem 1.1 ([21, 25]). nq(4,d) = gq(4,d) for 1 ≤ d ≤ q−2, q2−2q + 1 ≤ d ≤ q2−q, q3−2q2 + 1 ≤ d ≤ q3−2q2 +q, q3−q2−q + 1 ≤ d ≤ q3 +q2−q, 2q3−5q2 + 1 ≤ d ≤ 2q3−5q2 + 3q and any d ≥ 2q3−3q2 + 1 for all q. theorem 1.2 ([18, 21, 25]). nq(4,d) = gq(4,d) + 1 for the following d and q: (a) q2 −q + 1 ≤ d ≤ q2 − 1 with q ≥ 3, (b) q3 − 2q2 −q + 1 ≤ d ≤ q3 − 2q2 −b(q + 1)/2c with q ≥ 7, (c) 2q3 − 3q2 −q + 1 ≤ d ≤ 2q3 − 3q2 with q ≥ 4, (d) 2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 −q with q ≥ 5. our main results are the following theorems. theorem 1.3. nq(4,d) = gq(4,d) for 2q3 − 4q2 + 1 ≤ d ≤ 2q3 − 4q2 + 2q for all q. theorem 1.4. nq(4,d) = gq(4,d) + 1 for the following d and q: (a) 2q3 − 3q2 − 3q + 1 ≤ d ≤ 2q3 − 3q2 − 2q with q ≥ 7, (b) 2q3 − 4q2 − 3q + 1 ≤ d ≤ 2q3 − 4q2 with q ≥ 7, (c) 2q3 − 5q2 −q + 1 ≤ d ≤ 2q3 − 5q2 with q ≥ 7. we also tackle the problem to determine n8(4,d) for all d as a continuation of [14, 16, 24]. the problem to determine n8(4,d) for all d has been still open for the 447 values of d, see [26]. we determine n8(4,d) for 32 values of d and give new lower or upper bounds of n8(4,d) for 12 values of d as follows. theorem 1.5. (a) n8(4,d) = g8(4,d) + 1 for d = 381-384, 574, 633-638, 690-701, 745-749, 809-812. (b) n8(4,d) ≤ g8(4,d) + 1 for d = 133, 134, 145, 194. (c) g8(4,d) + 1 ≤ n8(4,d) ≤ g8(4,d) + 2 for d = 173-176, 178, 179, 247, 248. remark 1.6. (a) from theorem 1.4 (a), the problem to determine nq(4,d) for d = 2q3 − 3q2 − 3q + 1 is still open only for q = 5, see [26]. (b) the nonexistence of a [gq(4,d), 4,d]q code for d = 2q3 −rq2 −q + 1 for 3 ≤ r ≤ q−q/p, q = ph with p prime, is proved in [19]. we conjecture that a [gq(4,d), 4,d]q code for d = 2q3 −rq2 − q + 1 with r = q −q/p− 1 does not exist for non-prime q ≥ 8, which is valid for q = 8, 9 by theorem 1.5 and [17]. (c) we conjecture that nq(4,d) = gq(4,d) + 1 for q3 − 2q2 −q + 1 ≤ d ≤ q3 − 2q2 for all q ≥ 3. to prove this, we need to show the existence of a [gq(4,d) + 1, 4,d]q code for d = q3 − 2q2 by theorem 1.2 (b). this is already known for q = 3, 4, 5 and is also valid for q = 8 by theorem 1.5. we recall geometric methods through projective geometry and preliminary results in section 2. we prove theorem 1.3 and some upper bounds on nq(4,d) in theorems 1.4 and 1.5 in section 3. the proofs of theorems 1.4 and 1.5 are completed by the nonexistence of some griesmer codes, which are given in section 4. 74 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 2. geometric methods in this section, we give geometric methods to construct new codes or to prove the nonexistence of codes with certain parameters. we denote by pg(r,q) the projective geometry of dimension r over fq. a j-flat is a projective subspace of dimension j in pg(r,q). the 0-flats, 1-flats, 2-flats, (r − 2)-flats and (r − 1)-flats are called points, lines, planes, secundums and hyperplanes, respectively. we denote by θj the number of points in a j-flat, i.e., θj = (qj+1 − 1)/(q − 1). let c be an [n,k,d]q code having no coordinate which is identically zero. the columns of a generator matrix of c can be considered as a multiset of n points in σ = pg(k − 1,q) denoted by mc. we see linear codes from this geometrical point of view. a point p in σ is called an i-point if it has multiplicity mc(p) = i in mc. denote by γ0 the maximum multiplicity of a point from σ in mc and let ci be the set of i-points in σ, 0 ≤ i ≤ γ0. we denote by ∆1 + · · · + ∆s the multiset consisting of the s sets ∆1, ..., ∆s in σ. we write s∆ for ∆1 + · · · + ∆s when ∆1 = · · · = ∆s. then, mc = ∑γ0 i=1 ici. for any subset s of σ, we denote by mc(s) the multiset {mc(p)p | p ∈ s}. the multiplicity of s with respect to c, denoted by mc(s), is defined as the cardinality of mc(s), i.e., mc(s) = ∑ p∈s mc(p) = γ0∑ i=1 i·|s∩ci|, where |t| denotes the number of elements in a set t. then we obtain the partition σ = ⋃γ0 i=0 ci such that n = mc(σ) and n−d = max{mc(π) | π ∈fk−2}, where fj denotes the set of j-flats in σ. such a partition of σ is called an (n,n−d)-arc of σ. conversely an (n,n−d)-arc of σ gives an [n,k,d]q code in the natural manner. a line l with t = mc(l) is called a t-line. a t-plane, a t-hyperplane and so on are defined similarly. for an m-flat π in σ we define γj(π) = max{mc(∆) | ∆ ⊂ π, ∆ ∈fj}, 0 ≤ j ≤ m. let λs(π) be the number of s-points in π. we denote simply by γj and by λs instead of γj(σ) and λs(σ), respectively. it holds that γk−2 = n−d, γk−1 = n. when c is griesmer, the values γ0,γ1, ...,γk−3 are also uniquely determined ([22]) as follows: γj = j∑ u=0 ⌈ d qk−1−u ⌉ for 0 ≤ j ≤ k − 1. (1) when γ0 = 2, we obtain λ2 = λ0 + n−θk−1 (2) from λ0 + λ1 + λ2 = θk−1 and λ1 + 2λ2 = n. denote by ai the number of i-hyperplanes in σ. note that ai = an−i/(q − 1) for 0 ≤ i ≤ n−d. (3) the list of ai’s is called the spectrum of c. we usually use τj’s for the spectrum of a hyperplane of σ to distinguish from the spectrum of c. simple counting arguments yield the following: γk−2∑ i=0 ai = θk−1, (4) γk−2∑ i=1 iai = nθk−2, (5) 75 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 γk−2∑ i=2 i(i− 1)ai = n(n− 1)θk−3 + qk−2 γ0∑ s=2 s(s− 1)λs. (6) when γ0 ≤ 2, we get the following from (4)-(6): n−d−2∑ i=0 ( n−d− i 2 ) ai = ( n−d 2 ) θk−1 −n(n−d− 1)θk−2 + ( n 2 ) θk−3 + q k−2λ2. (7) lemma 2.1 ([17, 31]). put � = (n−d)q−n and t0 = b(w + �)/qc, where bxc denotes the largest integer less than or equal to x. let π be a w-hyperplane through a t-secundum δ. then t ≤ (w + �)/q and the following hold. (a) aw = 0 if an [w,k − 1,d0]q code with d0 ≥ w − t0 does not exist. (b) γk−3(π) = t0 if an [w,k − 1,d1]q code with d1 ≥ w − t0 + 1 does not exist. (c) let cj be the number of j-hyperplanes through δ other than π. then ∑ j cj = q and∑ j (γk−2 − j)cj = w + �−qt. (8) (d) a γk−2-hyperplane with spectrum (τ0, . . . ,τγk−3 ) satisfies τt > 0 if w + �−qt < q. (e) if any γk−2-hyperplane has no t0-secundum, then mc(π) ≤ t0 − 1. an [n,k,d]q code is called m-divisible if all codewords have weights divisible by an integer m > 1. lemma 2.2 ([31]). let c be an m-divisible [n,k,d]q code with q = ph, p prime, whose spectrum is (an−d−(w−1)m,an−d−(w−2)m, . . . ,an−d−m,an−d) = (αw−1,αw−2, . . . ,α1,α0), where m = pr for some 1 ≤ r < h(k − 2) satisfying λ0 > 0 and⋂ h∈fk−2, mc(h)<n−d h = ∅. (9) then there exists a t-divisible [n∗,k,d∗]q code c∗ with t = qk−2/m, n∗ = ∑w−1 j=0 jαj = ntq − d m θk−1, d∗ = ((n−d)q −n)t whose spectrum is (an∗−d∗−γ0t,an∗−d∗−(γ0−1)t, . . . ,an∗−d∗−t,an∗−d∗) = (λγ0,λγ0−1, . . . ,λ1,λ0). c∗ is called a projective dual of c, see also [4] and [11]. the condition (9) is needed to guarantee that c∗ has dimension k although it was missing in lemma 5.1 of [31]. note that a generator matrix for c∗ is given by considering (n−d− jm)-hyperplanes as j-points in the dual space σ∗ of σ for 0 ≤ j ≤ w − 1 [31]. example 2.3. let c be a [16, 5, 9]3 code with generator matrix g =   1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 2 2 2 0 0 1 1 0 0 0 1 0 0 2 0 0 1 2 2 2 0 1 2 1 0 0 0 1 0 2 2 0 0 1 0 2 2 2 1 1 0 0 0 0 1 2 0 2 2 1 0 1 1 0 2 1  . 76 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 then, the weight distribution of c is 019116121141512, and c is 3-divisible. hence, from (3), c has spectrum (a1,a4,a7) = (6, 57, 58). in this case, mc is not a multiset but a set of 16 points of σ = pg(4, 3) corresponding to the columns of g, for γ0 = 1. considering the (7 − 3j)-hyperplanes as j-points in the dual space σ∗ of σ for j = 0, 1, 2, one can get a 9-divisible [69, 5, 45]3 code c∗. actually, [69, 5, 45]3 codes are unique up to equivalence, see [5] for the detail. lemma 2.4 ([27]). let c be an [n,k,d]q code and let ∪ γ0 i=0ci be the partition of σ = pg(k − 1,q) obtained from c. if ∪i≥1ci contains a t-flat ∆ and if d > qt, then there exists an [n−θt,k,d′]q code c′ with d′ ≥ d−qt. the punctured code c′ in lemma 2.4 can be constructed from c by removing the t-flat ∆ from the multiset mc. we denote the resulting multiset by mc − ∆. the method to construct new codes from a given [n,k,d]q code by deleting the coordinates corresponding to some geometric object in pg(k − 1,q) is called geometric puncturing, see [25]. lemma 2.5 ([3]). let c1 be an [n1,k,d1]q code containing a codeword of weight d1 + m with m > 0 and let c2 be an [n2,k− 1,d2]q code. then, adding mc2 to an (n1 −d1 −m)-hyperplane for c1 gives an [n1 + n2,k,d]q code with d = d1 + m if m < d2 and d = d1 + d2 if m ≥ d2. an [n,k,d]q code with generator matrix g is called extendable if there exists a vector h ∈ fkq such that the extended matrix [g ht] generates an [n + 1,k,d + 1]q code. the following theorems will be applied to prove the extendability of codes with certain parameters in sections 4 and 5. theorem 2.6 ([23],[32]). let c be an [n,k,d]q code with q ≥ 5, d ≡ −2 (mod q), k ≥ 3. then c is extendable if ai = 0 for all i 6≡ 0,−1,−2 (mod q). theorem 2.7 ([30]). let c be an [n,k,d]q code with gcd(d,q) = 1. then c is extendable if σi 6≡n,n−d (mod q) ai < q k−2. theorem 2.8 ([29]). let c be an [n,k,d]q code with q = 2h, h ≥ 3, d odd, k ≥ 3. then c is extendable if ai = 0 for all i 6≡ 0,d (mod q/2). a set of s lines in pg(2,q) is called an s-arc of lines if no three of which are concurrent. an fmultiset f in pg(2,q) is an (f,m)-minihyper if every line meets f in at least m points and if some line meets f in exactly m points with multiplicity. lemma 2.9 ([20]). for x = q 2 + 1 with q even, every (x(q + 1),x)-minihyper in pg(2,q) is either the sum of x lines or the union of the lines forming a (q + 2)-arc of lines. 3. construction results in this section, we prove theorems 1.3, 1.4(b) and a part of theorem 1.5. lemma 3.1. there exist [n = 2q3 − 2q2 + 1 − t(q + 1), 4, 2q3 − 4q2 + 2q − tq]q codes for 0 ≤ t ≤ q − 1 for q ≥ 7. proof. for q ≥ 7, let h be a hyperbolic quadric in pg(3,q), see [12] for hyperbolic quadric. let l1 and l2 be two skew lines contained in h. take two skew lines l3 and l4 contained in h meeting l1, l2 and four points p1, . . . ,p4 of h so that l1 ∩ l3 = p1, l1 ∩ l4 = p2, l2 ∩ l3 = p3, l2 ∩ l4 = p4. let l5 = 〈p1,p4〉, l6 = 〈p2,p3〉 and ∆ij = 〈li, lj〉, where 〈χ1,χ2, · · · 〉 denotes the smallest flat containing χ1,χ2, · · · . we set c0 = l1 ∪ l2 ∪·· ·∪ l6, c1 = (∆13 ∪ ∆14 ∪ ∆23 ∪ ∆24 ∪h) \c0 77 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 and c2 = pg(3,q) \ (c0 ∪ c1). then λ0 = 6q − 2, λ1 = 5q2 − 10q + 5, λ2 = q3 − 4q2 + 5q − 2, where λi = |ci|, and the multiset c1 + 2c2 gives a griesmer [2q3 −3q2 + 1, 4, 2q3 −5q2 + 3q]q code, say c. this construction is due to [16]. next, take a line l contained in h such that l is skew to l3 and l4. let l ∩ l1 = q1, l ∩ l2 = q2 and let δ1, . . . ,δq−1 be the planes through l other than 〈l, l1〉, 〈l, l2〉. then each δi meets l1 and l2 in the points q1 and q2, respectively, and meets l3, . . . , l6 in some points out of l. hence, we can take a line mi in δi with mi ∩ c0 = ∅ for 1 ≤ i ≤ q − 1 such that m1 ∩ l, · · · ,mq−1 ∩ l are distinct points. now, take an elliptic quadric e and let e′ be the projection of e from a point r ∈ e \ ∆13 on to ∆13. since mc(∆13) = q2 − 2q + 1, it follows from lemma 2.5 that the multiset m′ = mc + e′ gives a [2q3−2q2 + 1, 4, 2q3−4q2 + 2q]q code, say c′. applying lemma 2.4 by deleting t of the lines m1, . . . ,mq−1, we get an [n = 2q3 − 2q2 + 1 − tθ1, 4,d = 2q3 − 4q2 + 2q − tq]q code. the code constructed by lemma 3.1 is griesmer for t = 0, 1 and the length satisfies n = gq(4,d) + 1 for 2 ≤ t ≤ q−1. hence, theorem 1.3 follows from the existence of griesmer codes with d = 2q3 −4q2 + 2q, 2q3 − 4q2 + 3q by puncturing. we also have that nq(4,d) ≤ gq(4,d) + 1 for 2q3 − 5q2 + 2q + 1 ≤ d ≤ 2q3−4q2−2q. since theorem 1.4 (2) is already known for q ≥ 9 [19], it suffices to show the nonexistence of griesmer codes for d = 2q3 − 4q2 − 3q + 1 for q = 7, 8, which is given in section 4, see lemma 4.1. next, we give a method to construct good codes by some orbits of a given projectivity in pg(k−1,q). for a non-zero element α ∈ fq, let r = fq[x]/(xn − α) be the ring of polynomials over fq modulo xn −α. we associate the vector (a0,a1, ...,an−1) ∈ fnq with the polynomial a(x) = ∑n−1 i=0 aix i ∈ r. for g = (g1(x), ...,gm(x)) ∈ rm, cg = {(r(x)g1(x), ...,r(x)gm(x)) | r(x) ∈ r} is called the 1-generator quasi-twisted (qt) code with generator g. cg is usually called quasi-cyclic (qc) when α = 1. when m = 1, cg is called α-cyclic or pseudo-cyclic or constacyclic. all of these codes are generalizations of cyclic codes (α = 1, m = 1). take a monic polynomial g(x) = xk − ∑k−1 i=0 aix i in fq[x] dividing xn − α with non-zero α ∈ fq, and let t be the companion matrix of g(x). let τ be the projectivity of pg(k − 1,q) defined by t. we denote by [gn] or by [a0a1 · · ·ank−1] the k ×n matrix [p,tp,t2p,...,tn−1p ], where p is the column vector (1, 0, 0, ..., 0)t (ht stands for the transpose of a row vector h). then [gn ] generates an α−1-cyclic code. hence one can construct a cyclic or pseudo-cyclic code from an orbit of τ. for non-zero vectors pt2 , ...,p t m ∈ f k q, we denote the matrix [p,tp,t2p,...,tn1−1p ; p2,tp2, ...,t n2−1p2; · · · ; pm,tpm, ...,tnm−1pm] by [gn1 ] + pn22 + · · ·+ p nm m . then, the matrix [g n ] + pn2 + · · ·+ pnm defined from m orbits of τ of length n generates a qc or qt code, see [28]. it is shown in [28] that many good codes can be constructed from orbits of projectivities. example 3.2. take g(x) = 1 +x+x2 +x4 ∈ f2[x] and a point q(1, 0, 0, 1) ∈ pg(3, 2). then, the matrix [g7] generates a cyclic hamming [7, 4, 3]2 code and the matrix [g7] + q7 =   1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 1  . generates a qc [14, 4, 7]2 code with weight distribution 017887. let f8 = {0, 1,α,α2, . . . ,α6}, with α3 = α + 1. for simplicity, we denote α,. . . ,α6 by 2, 3, . . . , 7 so that f8 = {0, 1, 2, . . . , 7}. it sometimes happens that qc or qt codes are divisible or can be extended to divisible codes. lemma 3.3. there exists a [440, 4, 384]8 code. 78 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 proof. let c be the qc [40, 4, 32]8 code with generator matrix [11115] + 01215 + 01245 + 01415 + 01655 + 01715 + 10355 + 10535. then c is a 4-divisible code with weight distribution 0132115536280040140. applying lemma 2.2, as the projective dual of c, one can get a 16-divisible [440, 4, 384]8 code c∗ with weight distribution 013843815400280. lemma 3.4. there exist codes with parameters [156, 4, 134]8, [169, 4, 145]8, [208, 4, 179]8, [225, 4, 194]8 and [286, 4, 248]8. proof. the qc codes with generator matrices [146413] + 100413 + 150413 + 152413 + 162513 + 114513 + 127213 + 164313 + 112613 +106213 + 114413 + 101713, [146413] + 100413 + 150413 + 152413 + 152313 + 142713 + 147113 + 144513 + 164313 +112613 + 106213 + 151013 + 101713, [146413] + 100413 + 150413 + 152413 + 152313 + 162513 + 147113 + 144513 + 123213 +112613 + 106213 + 140113 + 175213 + 173113 + 151013 + 101713, [100115] + 100415 + 150415 + 152315 + 142315 + 113315 + 175715 + 127715 + 123215 +127315 + 103615 + 130715 + 170715 + 126515 + 114415, [146413] + 100413 + 150413 + 152413 + 152313 + 142313 + 162513 + 142713 + 146513 +113313 + 123213 + 116013 + 123113 + 133013 + 106213 + 126513 + 114413 + 174013 +105013 + 127413 + 173113 + 101713 give the desired codes with the following weight distributions 0113418201361183138364140364144182148182, 011451365146637147546148364149182150273152273154182156911579116091, 011791092180637181728182546184728193364, 0119417851961050198420200210202105204210206210208105, 012483003256100126491, respectively. since it is known that g8(4,d) + 1 ≤ n8(4,d) ≤ g8(4,d) + 2 for 381 ≤ d ≤ 384, theorem 1.5 (a) for 381 ≤ d ≤ 384, theorem 1.5 (b) and (c) for d = 178, 179, 247, 248 follow from lemmas 3.3 and 3.4. 4. nonexistence of some griesmer codes note that one can get an [n− 1,k,d− 1]q code from a given [n,k,d]q code by puncturing and that the nonexistence of an [n−1,k,d−1]q code implies the nonexistence of an [n,k,d]q code. hence, to prove (a) and (b) of theorem 1.4, it suffices to show the following. lemma 4.1. there exists no [gq(4,d), 4,d]q code for d = 2q3 −sq2 − 3q + 1 with s = 3, 4 for q ≥ 7. lemma 4.1 was proved for q ≥ 9 in [19]. it follows from theorem 1.5(a) that lemma 4.1 is valid for q = 8, see lemmas 4.14 and 4.17 in this section. we can also prove lemma 4.1 for q = 7, but we omit the proof here because it is quite similar to the proof for q = 8, see [6] for the detail. the existence of a [gq(4,d) + 1, 4,d]q code for d = 2q3 − 3q2 − 2q is obtained from the result in [15]. it is also known that nq(4,d) = gq(4,d) + 1 for 2q3 − 5q2 − q + 1 ≤ d ≤ 2q3 − 5q2 with q ≥ 7 except fpr q = 8 [18]. hence, theorem 1.4(c) follows from theorem 1.5(a), see lemma 4.12 below. in this section, we prove that there exists no [g8(4,d), 4,d]8 code for d = 173, 574, 633, 690, 697, 745, 809, giving theorem 1.5. n8(3,d) is already known for all d as follows, see [1, 7, 26]. 79 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 table 1. the spectra of some [n, 3, d]8 codes. parameters possible spectra reference [6, 3, 4]8 (a0, a1, a2) = (34, 24, 15) [14] [7, 3, 5]8 (a0, a1, a2) = (31, 21, 21) [14] [8, 3, 6]8 (a0, a1, a2) = (29, 16, 28) [14] [9, 3, 7]8 (a0, a1, a2) = (28, 9, 36) [14] [10, 3, 8]8 (a0, a2) = (28, 45) [14] [26, 3, 22]8 (a0, a2, a3, a4) = (10, 1, 16, 46) lemma 4.3 [33, 3, 28]8 (a0, a3, a5) = (9, 16, 48) [16] (a0, a1, a4, a5) = (4, 5, 28, 36) (a0, a3, a4, a5) = (6, 10, 18, 39) [42, 3, 36]8 (a0, a4, a5, a6) = (4, 6, 24, 39) [2] (a0, a3, a5, a6) = (3, 7, 21, 42) (a0, a4, a6) = (3, 21, 49) (a0, a2, a4, a6) = (2, 3, 18, 50) [60, 3, 52]8 (a4, a6, a8) = (3, 16, 54) [16] (a0, a4, a7, a8) = (1, 1, 32, 39) (a0, a5, a6, a7, a8) = (1, 1, 3, 27, 41) (a0, a6, a7, a8) = (1, 1, 6, 24, 42) [61, 3, 53]8 (a0, a5, a7, a8) = (1, 1, 24, 47) [16] (a0, a6, a7, a8) = (1, 3, 21, 48) [62, 3, 54]8 (a0, a6, a7, a8) = (1, 1, 16, 55) [16] [63, 3, 55]8 (a0, a7, a8) = (1, 9, 63) [9] [64, 3, 56]8 (a0, a8) = (1, 72) [9] [69, 3, 60]8 (a5, a8, a9) = (1, 32, 40) [9] (a6, a7, a8, a9) = (1, 3, 27, 42) (a7, a8, a9) = (6, 24, 43) [70, 3, 61]8 (a6, a8, a9) = (1, 24, 48) [9] (a7, a8, a9) = (3, 21, 49) [71, 3, 62]8 (a7, a8, a9) = (1, 16, 46) [9] [72, 3, 63]8 (a8, a9) = (9, 64) [9] [73, 3, 64]8 a9 = 73 [9] [92, 3, 80]8 (a0, a8, a12) = (1, 9, 63) [20] (a4, a12) = (6, 67) (a4, a8, a12) = (1, 10, 62) (a8, a12) = (12, 61) [101, 3, 88]8 (a5, a13) = (5, 68) [20] (a9, a13) = (10, 63) [108, 3, 94]8 (a4, a6, a13, a14) = (1, 3, 16, 53) [16] (a5, a6, a12, a13, a14) = (2, 2, 1, 14, 54) (a5, a6, a12, a13, a14) = (1, 3, 1, 15, 53) (a6, a12, a13, a14) = (4, 1, 16, 52) (a6, a12, a14) = (4, 9, 60) 80 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 theorem 4.2. n8(3,d) = g8(3,d) + 1 for d = 13-16, 29-32, 37-40, 43-48 and n8(3,d) = g8(3,d) for any other d. lemma 4.3. every [26, 3, 22]8 code has spectrum (a0,a2,a3,a4) = (10, 1, 16, 46). proof. let c be a [26, 3, 22]8 code. by (1), γ0 = 1 and γ1 = 4. since (γ1 − γ0)θ1 + γ0 − 2 = 26, any t-line though a fixed 1-point satisfies t ≥ γ1 − 2 = 2. hence, there is no 1-line. from (4)-(6), we obtain (a0,a2,a3,a4) = (s, 61 − 6s, 8s − 64, 76 − 3s) with 8 ≤ s ≤ 10. let l1, . . . , l8 be 0-lines. then, l = {l1, . . . , ls} forms an s-arc of lines, for (θ1 − 3)γ1 < 26. suppose s = 8. then, one can find a line l so that l∪{l} forms a 9-arc of lines since every 8-arc is contained in a 10-arc, see [13]. since l meets l1, . . . , l8 in different points, l must be a 1-line, a contradiction. similarly, we can rule out the case s = 9. hence, our assertion follows. lemma 4.4. there exists no [199, 4, 173]8 code. proof. let c be a putative griesmer [199, 4, 173]8 code. then, γ0 = 1, γ1 = 4, γ2 = 26 from (1). by lemma 4.3, the spectrum of a γ2-plane ∆1 is (τ0,τ2,τ3,τ4) = (10, 1, 16, 46). an i-plane with a t-line satisfies t ≤ i + 9 8 (10) by lemma 2.1. we have a1 = 0 from lemma 2.1(e) since ∆1 has no 1-line. if a 14-plane δ exists, it follows from (10) that m(δ) gives a [14, 3, 12]8 code, which does not exist. in this way, using theorem 4.2 and lemma 2.1, one can get ai = 0 for all i 6∈ {0, 7-10, 15, 23-26}. we refer to this procedure as the first sieve in the proofs of the nonexistence results. from (7), we get ∑ i≤24 ( 26 − i 2 ) ai = 4259. (11) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (26 − j)cj = w + 9 − 8t. (12) suppose a0 > 0. then, a0 = 1 and ai > 0 with i > 0 implies i ≥ 23. setting w = t = 0 in (12), the maximum possible contribution of cj’s to the lhs of (11) is (c23,c26) = (3, 5). hence we get 4259 = (lhs of (11)) ≤ 9 × 73 + 325 = 982, which contradicts (11). hence a0 = 0. now, setting w = 26 in (12), the maximum possible contribution of cj’s to the lhs of (11) are (c7,c10,c26) = (1, 1, 6) for t = 0; (c7,c26) = (1, 7) for t = 2; (c15,c26) = (1, 7) for t = 3; (c23,c26) = (1, 7) for t = 4. hence we get 4259 = (lhs of (11)) ≤ 291 × 10 + 171 × 1 + 55 × 16 + 3 × 46 = 4099, a contradiction. this completes the proof. the following lemma is needed to prove the nonexistence of a [657, 4, 574]8 code. lemma 4.5 ([24]). there exists no [658, 4, 575]8 code. lemma 4.6 ([24]). the spectrum of a [83, 3, 72]8 code satisfies ai = 0 for all i with i /∈{3, 5, 7, 9, 11}. lemma 4.7. there exists no [657, 4, 574]8 code. 81 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 proof. let c be a putative [657, 4, 574]8 code. using theorem 4.2 and lemmas 2.1 and 4.6, one can get ai = 0 for all i 6∈ {33, 49, 65-73, 81-83} by the first sieve. from (7), we get∑ i≤81 ( 83 − i 2 ) ai = 64λ2 − 2583. (13) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (83 − j)cj = w + 7 − 8t. (14) suppose a72 > 0. from table 1, the spectrum of a 72-plane is (τ8,τ9) = (9, 64). setting i = 72, the maximum possible contributions of cj’s in (14) to the lhs of (13) are (c68,c83) = (1, 7) for t = 8; (c81,c82,c83) = (3, 1, 4) for t = 9. hence we get 64λ2 − 2583 = (lhs of (13)) ≤ (105 × 1 + 0 × 7)9 + (1 × 3 + 0 × 1 + 0 × 4)64 + 55 = 1192, giving λ2 ≤ 58. on the other hand, we have λ2 = λ0 + 72 = 72 from (2), a contradiction. hence a72 = 0. similarly, we can prove a71 = a70 = a69 = a68 = 0. applying theorem 2.6, c is extendable, which contradicts lemma 4.5. this completes the proof. as in the above proof, we often obtain a contradiction to rule out the existence of some i-plane by eliminating the value of λ2 using (7), (8) and the possible spectra for a fixed w-plane. we refer to this proof technique as "(λ2,w)-ruling out method ((λ2,w)-rom)" in what follows. lemma 4.8. there exists no [725, 4, 633]8 code. proof. let c be a putative griesmer [725, 4, 633]8 code. then, γ0 = 2, γ1 = 12, γ2 = 92 by (1). from table 1, the spectrum of a γ2-plane ∆1 is one of the following: (a) (τ0,τ8,τ12) = (1, 9, 63) with λ′0 = 9, (b) (τ4,τ12) = (6, 67) with λ ′ 0 = 15, (c) (τ4,τ8,τ12) = (1, 10, 62) with λ′0 = 5, (d) (τ8,τ12) = (12, 61) with λ ′ 0 = 3, where λ′0 = λ0(∆1). using theorem 4.2 and lemma 2.1, one can get ai = 0 for all i 6∈ {0,21-28, 53-64, 69-73, 85-92} by the first sieve. from (7), we get ∑ i≤90 ( 92 − i 2 ) ai = 64λ2 − 5315. (15) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (92 − j)cj = w + 11 − 8t. (16) we first prove ai = 0 for 0 ≤ i ≤ 28. assume a t-plane δt with 0 ≤ t ≤ 28 exists. then, the multiset mc + δt gives an [n = 798, 4,d = 697]8 code c′ since mc′(δt) = t + θ2 ≤ 28 + 73 ≤ 101 and since n = n + θ2 = 725 + 73 = 798 and n −d = n−d + θ1 = 92 + 9 = 101. this contradicts that a [798, 4, 697]8 code does not exist by lemma 4.12. hence, ai = 0 for all 0 ≤ i ≤ 28. since ∆1 has no 9-line, we have a73 = 0. we can prove ai = 0 for i = 72, 71, 70, 69, 64, 63, 62, 61, 60 by (λ2, i)-rom using the possible spectra of an i-plane in table 1. next, we prove ai = 0 for 53 ≤ i ≤ 59. suppose a53 > 0 and let δ53 be a 53-plane with spectrum (τ0, . . . ,τ8). then, we have ∑ i≤7 ( 8 − i 2 ) τi = 83. (17) 82 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 setting w = 53 in (16), the maximum possible contribution of cj’s to the left hand side of (15) are (c53,c85,c88,c92) = (1, 3, 1, 3) for t = 0; (c53,c85,c87,c91) = (1, 1, 1, 5) for t = 1; (c53,c89,c91) = (1, 1, 6) for t = 2; (c59,c91) = (1, 7) for t = 3; (c85,c88,c92) = (4, 1, 3) for t = 4; (c85,c87,c91) = (2, 1, 5) for t = 5; (c85,c89,c91) = (1, 1, 6) for t = 6; c91 = 8 for t = 7; c92 = 8 for t = 8 since c92 = 0 for t = 1, 2, 3, 5, 6, 7. hence we get 64λ2 − 5315 = (lhs of (15)) ≤ 810τ0 + 772τ1 + 744τ2 + 528τ3 + 90τ4 + 52τ5 + 24τ6 < 53 × (17) = 4399 giving λ2 ≤ 151. on the other hand, we have λ2 = 140 + λ0 ≥ 140 + 73 − 53 ≥ 160, a contradiction. hence a53 = 0. we can prove a54 = a55 = a56 = a57 = a58 = a59 = 0 similarly, see [6] for the detail. now, we have ai = 0 for all i < 85. setting w = 92, (16) has no solution for t = 0, 4. hence every 92-plane has spectrum (d). then, we get a contradiction by (λ2, 92)-rom. this completes the proof. lemma 4.9. let c be a [101, 3, 88]8 code and let σ = pg(2, 8). then, (a) c has spectrum (a5,a13) = (5, 68) with λ0 = 10 and mc = 2σ − (l1 + · · · + l5), where {l1, . . . , l5} is a 5-arc of lines; or (b) c has spectrum (a9,a13) = (10, 63) with λ0 = 0 and mc = 2σ −l, where l is the union of a 10-arc of lines. proof. let c be a [101, 3, 88]8 code. then γ0 = 2 from (1) since c is griesmer. hence, our assertion follows from lemma 2.9 since the multiset 2σ −mc is a (45, 5)-minihyper. lemma 4.10. every [100, 3, 87]8 code c is extendable and its spectrum is one of the following: (a) (a5,a12,a13) = (5, 9, 59), (b) (a4,a5,a12,a13) = (1, 4, 8, 60), (c) (a8,a9,a12,a13) = (2, 8, 7, 56), (d) (a9,a12,a13) = (10, 9, 54). proof. let c be a [100, 3, 87]8 code. by lemma 1, γ0 = 2 and γ1 = 13. since (γ1 −γ0)θ1 + γ0 −1 = n, the lines though a fixed 2-point is one 12-line and eight 13-lines, and a10 = a11 = 0. let l be a t-line containing a 1-point p. considering the lines through p, we get n ≤ (γ1 − 1)8 + t, so 4 ≤ t. hence a1 = a2 = a3 = 0. suppose a 0-line l0 exists. since there is no 9-line, for a point p on l0, there are four 12-lines and four 13-lines through p. hence, the spectrum is (a0,a12,a13) = (1, 36, 36), then, from (6), we have λ2 = 648−4950/8, a contradiction. hence, there is no 0-line. next, assume a6 > 0 and let l6 be a 6-line. for a 1-point p on l6, there are exactly two 12-lines and six 13-lines through p . hence a9 = 0. for a 0-point q on l6, there are at most two lines whose multiplicities are less than 9. hence we have∑ i≡n,n−d ai ≤ (9 − 6)2 + 1 = 7, and c is extendable by theorem 2.7. one can prove this similarly when a7 > 0. finally, assume a6 = a7 = 0. then, we have ai = 0 for all i 6∈ {4, 5, 8, 9, 12, 13}, which implies that ai = 0 for all i 6≡ 0, 87 mod 4. hence, c is extendable by theorem 2.8. assume that adding a point p to the multiset mc gives a 101-plane δ corresponding to a [101, 3, 88]8 code. then, δ satisfies (a) or (b) in the previous lemma. so, one can get the spectra (a)-(d) according to the cases (a) p is a 2-point on δ with case (a); (b) p is a 1-point from a 5-line on δ with case (a); (c) p is a 1-point from a 9-line on δ with case (b); (d) p is a 2-point on δ with case (b), respectively. lemma 4.11. there exists no [790, 4, 690]8 code. proof. let c be a putative griesmer [790, 4, 690]8 code. then, we have γ0 = 2, γ1 = 13, γ2 = 100 from (1). since (γ1−γ0)θ2+γ0−15 = 790, an i-plane containing a 2-point satisfies i ≥ (γ1−γ0)θ1+γ0−15 = 86. from table 1, the spectrum of a γ2-plane ∆1 is one of the following: (a) (τ5,τ12,τ13) = (5, 9, 59), (b) (τ4,τ5,τ12,τ13) = (1, 4, 8, 60), (c) (τ8,τ9,τ12,τ13) = (2, 8, 7, 56), (d) (τ9,τ12,τ13) = (10, 9, 54). 83 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 by the first sieve, one can get ai = 0 for all i 6∈ {22-28, 30-33, 54-73, 86-92, 94-100}. from (7), we get∑ i≤98 ( 100 − i 2 ) ai = 64λ2 − 8685. (18) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (100 − j)cj = w + 10 − 8t. (19) we first prove ai = 0 for 22 ≤ i ≤ 33. assume a t-plane δt with 22 ≤ t ≤ 33 exists. then, the multiset mc + δt gives an [n = 863, 4,d = 754]8 code c′ since mc′(δt) = t + θ2 ≤ 33 + 73 ≤ 109 and since n = n + θ2 = 790 + 73 = 863 and n −d = n−d + θ1 = 790 − 690 + 9 = 109. this contradicts that a [863, 4, 754]8 code does not exist, see [26]. hence, ai = 0 for all i ≤ 33. we can prove ai = 0 for i = 73, 72, 71, 70, 64, 63, 62, 61, 60, 69 in this order by (λ2, i)-rom using the possible spectra of each i-plane from table 1. suppose a68 > 0 and let δ68 be a 68-plane. since δ68 corresponds to a griesmer [68, 3, 59]8 code, m(δ68) is obtained from δ68 by deleting five points, and the spectrum of δ68 is one of the following: (a) (τ4,τ8,τ9) = (1, 40, 32), (b) (τ5,τ7,τ8,τ9) = (1, 4, 33, 35), (c) (τ6,τ7,τ8,τ9) = (1, 7, 28, 37), (d) (τ6,τ7,τ8,τ9) = (2, 4, 31, 36), (e) (τ7,τ8,τ9) = (10, 25, 38). one can get a contradiction by the usual (λ2, 68)-rom for the possible spectra (b)-(e). hence δ68 has spectrum (a). from (19), there is at most one i-plane with i ≤ 68 other than δ68. we may assume that δ68 meets ∆1 in a 9-line. then ∆1 has spectrum (c) or (d). setting w = 100 in (19), the maximum possible contributions of cj’s to the lhs of (18) are (c54,c100) = (1, 7) for t = 8; (c86,c96,c100) = (3, 1, 4) for t = 8 when cj = 0 for j < 86; (c65,c97,c100) = (1, 1, 6) for t = 9; (c86,c90,c100) = (2, 1, 5) for t = 9 when cj = 0 for j < 86; (c86,c100) = (1, 7) for t = 12; (c94,c100) = (1, 7) for t = 13. hence, we get 64λ2 − 8685 = (lhs of (18)) ≤ 1035 + 279(τ8 − 1) + 227τ9 + 91τ12 + 15τ13 = 4607 for the spectrum (c), giving λ2 ≤ 207. on the other hand, we have λ2 = λ0 +205 ≥ 205+(73−69) = 209, a contradiction. similarly, we get a contradiction for spectrum (d). hence a68 = 0. one can also prove a67 = a66 = 0 as well. suppose a54 > 0. let δ54 be a 54-plane and l be 8-line in δ54. then, the other planes through l other than δ54 are 100-planes of spectrum (c), say ∆1, . . . , ∆8. suppose that there is no plane with no 2-point meeting l in a 1-point. then, one can get a contradiction by (λ2, 100)-rom using the spectrum (c) of a 100-plane. so, there is a plane δ with no 2-point meeting l in a 1-point p . since δ meets each of ∆1, . . . , ∆8 in a 9-line, we have mc(δ) ≥ (9 − 1)8 + 1 = 65, whence δ is a 65-plane with spectrum (τ1,τ8,τ9) = (1, 64, 8). then, we get a a contradiction by (λ2, 65)-rom. hence a54 = 0. similarly, we can prove a55 = a56 = a57 = a58 = a59 = 0. suppose a65 > 0 and let δ65 be a 65-plane. let l be a 9-line on δ65 and take a 100-plane ∆1 through l. since δ65 has no 2-point, there are eight 0-points in δ65, and there are at most two lines on δ65 whose multiplicities are at most 5. since any other 65-plane meets δ65 in some t-line with t ≤ 5 and since the spectrum of ∆1 is (c) or (d), we have a65 ≤ 3 from (19) with w = 100. setting w = 100 in (19), the maximum possible contributions of cj’s to the lhs of (18) are (c65,c89,c100) = (1, 1, 6) for t = 8; (c86,c96,c100) = (3, 1, 4) for t = 8 with c65 = 0; (c65,c97,c100) = (1, 1, 6) for t = 9; (c86,c90,c100) = (2, 1, 5) for t = 9 with c65 = 0; (c86,c100) = (1, 7) for t = 12; (c94,c100) = (1, 7) for t = 13. it follows from λ2 = λ0 + 205 ≥ 205 + (73 − 65) = 213 that one can get a contradiction by (λ2, 100)-rom as 64λ2 − 8685 = (lhs of (18)) ≤ 650τ8 + 227τ9 + 91τ12 + 15τ13 = 4593 when ∆1 has spectrum (c) and 64λ2 − 8685 = (lhs of (18)) ≤ 598 × 2 + 227(τ9 − 2) + 91τ12 + 15τ13 = 4641 84 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 when ∆1 has spectrum (d) since a65 ≤ 3, giving λ2 ≤ 208. hence, a65 = 0. now, we have ai = 0 for all i < 86. one can get a contradiction by (λ2, 100)-rom using the possible spectra (a)-(d) as usual. this completes the proof. lemma 4.12. there exists no [798, 4, 697]8 code. proof. let c be a putative griesmer [798, 4, 697]8 code. by lemma 4.9, the spectrum of a γ2-plane ∆1 is either (a) (τ5,τ13) = (5, 68) or (b) (τ9,τ13) = (10, 63). using theorem 4.2 and lemma 2.1, one can get ai = 0 for all i 6∈ {30-33, 62-73, 94-101} by the first sieve. it follows from (7) that ∑ i≤99 ( 101 − i 2 ) ai = 64λ2 − 9123. (20) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (101 − j)cj = w + 10 − 8t. (21) one can deduce that ai = 0 by (λ2, i)-rom for 70 ≤ i ≤ 73 using the possible spectra of the [73 − j, 3, 64 − j]8 codes for 0 ≤ j ≤ 3, see table 1. suppose a30 > 0 and let δ30 be a 30-plane. it follows from (21) that a30 > 0 implies a30 = 1 and aj = 0 for other j < 94. since γ1(δ30) = 5, one can find a 101-plane ∆ of spectrum (a) meeting δ30 in a 5-line. take another 5-line l5 on ∆. then, every plane through l5 has multiplicity at least 94, which is impossible from (21) with (w,t) = (101, 5). hence a30 = 0. one can get a31 = a32 = a33 = 0, similarly. then, using the possible spectra of the [70 − j, 3, 61 − j]8 codes, we can also prove that a70−j = 0 by (λ2, 70 − j)-rom for 1 ≤ j ≤ 3. now, we have ai = 0 for all i 6∈ {62-66, 94-101}. note that a (62 + e)-plane with 0 ≤ e ≤ 3 could have a 2-point because it corresponds to a [62 + e, 3, 53 + e]8 code which is not griesmer. suppose a (62 + e)-plane δ with 0 ≤ e ≤ 3 has a 2-point. then, one can find a 9-line l9 through the 2-point on δ and a 101-plane through l9 from (21) with (w,t) = (62 + e, 9). this contradicts that a 9-line in a 101-plane with spectrum (b) has no 2-point by lemma 4.9. thus, a (62 + e)-plane with 0 ≤ e ≤ 4 has no 2-point since a 66-plane corresponds to a griesmer code. suppose a62 > 0 and let δ62 be a 62-plane and let l be a 9-line on δ62. then, the other planes through l are 101-planes, say ∆1, . . . , ∆8. for a fixed 1-point p on l, one can take a 9-line lj( 6= l) on ∆j for 1 ≤ j ≤ 8 from the geometric structure described in lemma 4.9. suppose that the plane δ = 〈l1, l2〉 is a (62 + e)-plane with 0 ≤ e ≤ 3 and let δ∩δ62 be an α-line. since γ1(δ) = 9, δ contains all of l1, . . . , l8, and we have mc(δ) = 64 + α. one can rule out such cases by (λ2, 64 + α)-rom. hence, a62 > 0 implies that a62 = 1 and aj = 0 for other j < 94. setting w = 101, the maximum possible contributions of cj’s in (21) to the lhs of (20) are (c62,c101) = (1, 7) for t = 9 with c62 > 0; (c94,c97,c101) = (5, 1, 2) for t = 9 with c62 = 0; (c94,c101) = (1, 7) for t = 13. using the spectrum of a 101-plane of spectrum (b), one can get a contradiction by (λ2, 101)-rom. hence a62 = 0. one can similarly prove a63 = 0. to rule out a 101-plane of spectrum (a), let ∆1 be such a plane. from (21) with (w,t) = (101, 5), there exists a (64 + e)-plane with 0 ≤ e ≤ 2 through each of the 5-lines on ∆1. one can rule out such a 66-plane by (λ2, 66)-rom using all possible spectra of a 66-plane with a 5-line. hence a66 = 0. note that λ0 ≥ 8 − 4 + 10 = 14 since a 101-plane of spectrum (a) has ten 0-points. setting w = 101, the maximum possible contributions of cj’s in (21) to the lhs of (20) are (c64,c94,c95,c101) = (1, 4, 1, 2) for t = 5; (c94,c101) = (1, 7) for t = 13. using the spectrum of a 101-plane of spectrum (a), one can get a contradiction by (λ2, 101)-rom. hence every 101-plane has spectrum (b). suppose a66 > 0 and let δ66 be a 66-plane with spectrum (τ2, . . . ,τ9). then, from the three equalities (4)-(6), we obtain τ2 + τ3 + τ4 ≤ 2 and τ5 + τ6 + τ7 ≤ 21. setting w = 66, the maximum possible contributions of cj’s in (21) to the lhs of (20) are (c64,c94,c95,c99) = (1, 1, 1, 5) for t = 2 since a 100plane has no 2-line by lemma 4.10; (c94,c96,c100) = (4, 1, 3) for t = 5 since c101 = 0; (c96,c100) = (1, 7) 85 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 for t = 8; (c97,c101) = (1, 7) for t = 9. using (τ2,τ5,τ8,τ9) = (2, 21, 49, 1) instead of all possible spectra of a 66-plane, one can get a contradiction by (λ2, 66)-rom. hence a66 = 0. we can prove a65 = a64 = 0 similarly. hence, we have ruled out all possible i-planes with i < 94. finally, using the spectrum (b) of a 101-plane, one can get a contradiction by (λ2, 101)-rom. this completes the proof. lemma 4.13. a [107, 3, 93]8 code c satisfies λ0 > 0. proof. suppose λ0 = 0. it follows from lemma 2.4 that the multiset mc−pg(2, 8) gives a [34, 3, 29]8 code, which does not exist by theorem 4.2, a contradiction. lemma 4.14. there exists no [853, 4, 745]8 code. proof. let c be a putative griesmer [853, 4, 745]8 code. from table 1, the spectrum of a γ2-plane ∆1 is one of the following: (a) (τ4,τ6,τ13,τ14) = (1, 3, 16, 53), (b) (τ5,τ6,τ12,τ13,τ14) = (2, 2, 1, 14, 54), (c) (τ5,τ6,τ12,τ13,τ14) = (1, 3, 1, 15, 53), (d) (τ6,τ12,τ13,τ14) = (4, 1, 16, 52), (e) (τ6,τ12,τ14) = (4, 9, 60). one can get ai = 0 for all i 6∈ {21-33, 37-42, 61-64, 69-73, 85-108} by the first sieve. from (7), we get∑ i≤106 ( 108 − i 2 ) ai = 64λ2 − 12251. (22) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (108 − j)cj = w + 9 − 8t. (23) we first prove ai = 0 for 21 ≤ i ≤ 42. assume a t-plane δt with 21 ≤ t ≤ 42 exists. then, the multiset mc + δt gives an [n = 926, 4,d = 809]8 code c′ since mc′(δt) = t + θ2 ≤ 42 + 73 ≤ 115 and since n = n + θ2 = 853 + 73 = 926 and n −d = n−d + θ1 = 853 − 745 + 9 = 117. this contradicts that a [926, 4, 809]8 code does not exist by lemma 4.17. hence, ai = 0 for all i ≤ 60. if a73 > 0, then any line on a 73-plane is a 9-line from table 1, which contradicts that ∆1 has no 9-line. hence a73 = 0. similarly, a64 = a63 = a71 = a72 = 0. suppose a62 > 0. the spectrum of a 62-plane is (τ0,τ6,τ7,τ8) = (1, 1, 16, 55) and a 62-plane meets ∆1 in a 6-line since the possible multiplicities of lines in ∆1 are 4, 5, 6, 12, 13, 14. setting w = 62 in (23), the maximum possible contributions of cj’s to the lhs of (22) are (c106,c107) = (1, 7) for t = 8; (c98,c107) = (1, 7) for t = 7; (c85,c106,c108) = (1, 1, 6) for t = 6; (c42,c107) = (1, 7) for t = 0. using the spectrum of a 62-plane, one can get a contradiction by (λ2, 62)-rom since λ2 = λ0 + 268 ≥ 268. hence a62 = 0. similarly, we can prove a61 = a69 = a70 = 0 using the spectra from table 1. now, we have ai = 0 for all i < 85. using the possible spectra (a)-(e) of a 108-plane, one can get a contradiction as follows. take a 14-line l on a 108-plane so that l has no 0-point. setting (w,t) = (108, 14) in (23), the solutions of cj’s are (c101,c108) = (1, 7), (c102,c107,c108) = (1, 1, 6), (c107,c108) = (7, 1) and so on. counting the number of 0-points on the planes through l, we have λ0 ≥ 6 + 6 + 7 = 19 since a 108-plane has at least six 0-points and since a 107-plane has at least one 0-point by lemma 4.13. hence λ2 = λ0 + 268 ≥ 287. (24) using the spectra (a)-(d) of a 108-plane, we get a contradiction by (λ2, 108)-rom. hence every 108-plane has spectrum (e). then, we have 64λ2 − 12251 ≤ 6577, giving λ2 ≤ 294. (25) 86 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 next, we rule out a possible 85-plane. assume a 85-plane δ exists. then, δ has a 12-line ` and the other planes through ` are 108-planes. let s be the number of 0-points on `. since s ≤ 3 and since a 108-plane of spectrum (e) has seven 0-point, we obtain λ2 = λ0 + 268 ≥ 268 + (7−s)8 + s ≥ 304, which contradicts (25). hence, a85 = 0. counting the number of 0-points on the planes through a fixed 14-line, the lower bound (24) can be improved to λ2 ≥ 289 since a 108-plane of spectrum (e) has seven 0-point. on the other hand, since the maximum possible contributions of cj’s in (23) with w = 108 to the lhs of (22) are (c86,c103,c108) = (3, 1, 4) for t = 6 and (c86,c108) = (1, 1, 6) for t = 12, the upper bound (25) can be also improved to λ2 ≤ 287, a contradiction. this completes the proof. we recall that the multiset for a [2q2 − q − 1, 3, 2q2 − 3q]q code with q ≥ 5 consists of two copies of pg(2,q) with three non-concurrent lines deleted [16]. the following code is obtained from this code by deleting two (not necessarily distinct) points. lemma 4.15 ([16]). a [2q2 − q − 3, 3, 2q2 − 3q − 2]q code c′ with q ≥ 7 is extendable to a [2q2 − q − 1, 3, 2q2 − 3q]q code c and its spectrum is one of the following: (a) (aq−3,aq−1,a2q−2,a2q−1) = (1, 2, 2q,q2 −q − 2), (b) (aq−2,aq−1,a2q−3,a2q−2,a2q−1) = (2, 1, 1, 2q − 2,q2 −q − 1), (c) (aq−2,aq−1,a2q−3,a2q−2,a2q−1) = (1, 2, 1, 2q − 1,q2 −q − 2), (d) (aq−1,a2q−3,a2q−2,a2q−1) = (3, 1, 2q,q2 −q − 3), (e) (aq−1,a2q−3,a2q−1) = (3,q + 1,q2 − 3), according to the cases (a) p and q are 1-points on the same (q− 1)-line on δ; (b) p and q are 1-points from different (q − 1)-lines on δ; (c) p is a 1-point and q is a 2-point on δ; (d) p and q are distinct 2-points in on δ; (e) p and q are the same 2-points in on δ, respectively, where p and q are the points corresponding to the coordinates of c to be removed from the (2q2 −q− 1)-plane δ stated in the previous lemma. one can get the following similarly to lemma 4.13. lemma 4.16. a [116, 3, 101]8 code c satisfies λ0 > 0. lemma 4.17. there exists no [926, 4, 809]8 code. proof. let c be a putative [926, 4, 809]8 code. from lemma 4.15, the spectrum of a γ2-plane ∆ is one of the following: (a) (τ5,τ7,τ14,τ15) = (1, 2, 16, 54), (b) (τ6,τ7,τ13,τ14,τ15) = (2, 1, 1, 14, 55), (c) (τ6,τ7,τ13,τ14,τ15) = (1, 2, 1, 15, 54), (d) (τ7,τ13,τ14,τ15) = (3, 1, 16, 53), (e) (τ7,τ13,τ15) = (3, 9, 61). using theorem 4.2 and lemma 2.1, we obtain ai = 0 for all i 6∈ {30-33,38-42,46-49,62-64,70-73,94-117} by the first sieve. from (7), we get ∑ i≤115 ( 117 − i 2 ) ai = 64λ2 − 17083. (26) lemma 2.1(c) gives ∑ j cj = 8 and ∑ j (117 − j)cj = w + 10 −qt. (27) first, we prove ai = 0 for 30 ≤ i ≤ 33. suppose a30 > 0 and let δ30 be a 30-plane. then, it follows from (27) that a30 = 1 and any i-plane with i > 30 satisfies i ≥ 94. from lemma 2.1, δ30 meets ∆ in a 5-line, say l, and ∆ has the spectrum (a). recall from lemma 4.15 that there are two 7-lines in the 117-plane of spectrum (a) meeting the 5-line in 0-points. since the other planes ( 6= δ30, ∆) through l are 117-planes of spectrum (a), say ∆1, . . . , ∆7, and since there are four 0-points on l, one can take a 0-point q on l which 87 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 is on at least four 7-lines, say l1, l2, l3, l4. without loss of generality, we may assume that lj is on ∆j for 1 ≤ j ≤ 4. for the plane δ = 〈l1, l2〉, we have mc(δ) ≤ 7 + 7 + 5 + 15×6 = 109 since mc(δ30∩δ) ≤ 5. since a 109-plane has no 15-line, we have mc(δ ∩ ∆j) = 7 for 1 ≤ j ≤ 4, and mc(δ) ≤ 7 × 4 + 5 + 14 × 4 = 89, a contradiction. hence a30 = 0. one can similarly prove a31 = a32 = a33 = 0. if a73 > 0, then any line on a 73-plane is a 9-line from table 1, which contradicts that ∆ has no 9-line. hence a73 = 0. similarly, a64 = a72 = 0. using the spectrum of a w-plane from table 1, one can get a contradiction by (λ2,w)-rom for w = 62, 63, 70, 71. hence, a62 = a63 = a70 = a71 = 0. suppose a38 > 0. then, a38 = 1 and aj > 0 with j 6= 38 implies j ≥ 94. let δ0 be the 38-plane. then, δ0 contains a 6-line, say l, and the other planes through l other than δ0 are 117-planes of spectrum (b) or (c), say ∆1, . . . , ∆8. recall from lemma 4.15 that there are two 7-lines (resp. one 6-line and one 7-line) in the 117-plane of spectrum (c) (resp. (b)) meeting the 6-line l in 0-points. let lj and mj be the 6or 7-lines in ∆j other than l. since there are three 0-points on l, one can take a 0-point q on l which is on at least six 6or 7-lines. without loss of generality, we may assume that l2 and l3 meet l in q = l1 ∩l and that two of other lj,mj with j ≥ 2 meet l in q′ = m1 ∩l. note that there is no s-line with 7 < s < 14 in ∆1 through q or q′ by lemma 4.15. let δ be a t-plane through l1 other than ∆1. if t < 102, then δ meets ∆2 and ∆3 in l2 and l3, respectively, since a t-plane contains no 14nor 15-line, whence mc(δ) ≤ 7 + 7 + 7 + |δ∩δ0|+ 5×14 ≤ 97. then, from lemma 2.1, δ contains no 14-plane, and we have mc(δ) ≤ 7 + 7 + 7 + 6 + 5 × 13 = 92, a contradiction. hence any t-plane through l1 or m1 satisfies t ≥ 102. take ∆1 as π in lemma 2.1, the maximum possible contribution of cj’s in (27) with w = 117 to the lhs of (26) are (c38,c117) = (1, 7) for t = 6 with c38 > 0; (c102,c113,c117) = (5, 1, 2) for t = 6 with c38 = 0; (c102,c106,c117) = (4, 1, 3) for t = 7; (c94,c117) = (1, 7) for t = 13; (c102,c117) = (1, 7) for t = 14; (c110,c117) = (1, 7) for t = 15. note that λ2 = λ0 + 341 ≥ 341 + (73 − 38) + 8 = 384 since each of ∆1, . . . , ∆8 contains a 0-point out of l. using the spectrum of a 117-plane of spectrum (b) or (c), one can get a contradiction by (λ2, 117)-rom. hence a38 = 0. one can prove a39 = a40 = a41 = a42 = 0 similarly. (when δ0 is a 42-plane, there are four 117-planes of spectrum (b) or (c), say ∆1, . . . , ∆4, through a fixed 6-line l in δ0, which contain 6or 7-lines li,mj as above. it could happen that l2, l3, l4 meet l in q = l1 ∩l, m2 meets l in q′ = m1 ∩l and that m3 and m4 meet l in the remaining 0-point of l other than q,q′. in this case, any t-plane ( 6= 〈m1,m2〉) through m1 satisfies t ≥ 102. considering this situation, one can get a contradiction by (λ2, 117)-rom as above.) the above investigation for the case a38 > 0 is also valid to rule out possible i-planes for 46 ≤ i ≤ 49, see [6] for the detail. now, we have ai = 0 for all i without 94 ≤ i ≤ 117. using the spectrum (a)-(e) of a 117plane, one can get a contradiction as follows. take a 15-line with no 0-point on a 117-plane. since the possible contributions of cj’s with w = 117 in (27) to the lhs of (26) are (c110,c117) = (1, 7), (c116,c117) = (7, 1) and so on for t = 15 and since a 116-plane has at least one 0-point by lemma 4.16, we have λ2 = λ0 + 341 ≥ 341 + 3 + 3 × 1 + 1 × 7 ≥ 354. hence, we can get a contradiction by (λ2, 117)rom when the spectrum of the w-plane is one of (a)-(d). now, we may assume that any 117-plane has spectrum (e). we first rule out a possible 94-plane. assume a 94-plane δ exists. then, δ has a 13-line ` and the other planes through ` are 117-planes. since ` has at most two 0-points and since ` is on a 117-plane of spectrum (e) which has four 0-points, we get λ2 = λ0 + 341 ≥ 341 + (4−2)8 + 2 ≥ 359. on the other hand, we obtain λ2 ≤ 358 by (λ2, 117)-rom using the spectrum (e), a contradiction. hence there is no 94-plane. take a 15-line with no 0-point on a 117-plane. counting the number of 0-points on the planes through the 15-line, we get a lower bound on λ2 as λ2 = λ0 + 341 ≥ 341 + 4 + 4×1 + 1×7 ≥ 356 since a 117-plane of spectrum (e) has four 0-points. then, we get a contradiction by (λ2, 117)-rom. this completes the proof. now, theorem 1.5 follows from lemmas 4.4-4.17. acknowledgment: the authors wish to express their thanks to the anonymous reviewers for their careful reading and valuable comments that improved the presentation and the content of the paper. 88 n. bono et. al. / j. algebra comb. discrete appl. 8(2) (2021) 73–90 references [1] s. ball, table of bounds on three dimensional linear codes or (n,r)-arcs in pg(2,q), available at https://web.mat.upc.edu/people/simeon.michael.ball/codebounds.html. 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[32] y. yoshida, t. maruta, an extension theorem for [n,k,d]q codes with gcd(d,q) = 2, australas. j. combin. 48 (2010) 117–131. 90 https://doi.org/10.1016/j.disc.2007.07.045 https://doi.org/10.1016/j.disc.2007.07.045 https://mathscinet.ams.org/mathscinet-getitem?mr=3251934 https://mathscinet.ams.org/mathscinet-getitem?mr=3251934 https://doi.org/10.1007/s10623-011-9497-x https://doi.org/10.1007/s10623-011-9497-x https://doi.org/10.1016/j.disc.2007.07.044 https://doi.org/10.1016/j.disc.2007.07.044 https://mathscinet.ams.org/mathscinet-getitem?mr=2732106 https://mathscinet.ams.org/mathscinet-getitem?mr=2732106 introduction geometric methods construction results nonexistence of some griesmer codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729446 j. algebra comb. discrete appl. 7(2) • 161–181 received: 21 january 2019 accepted: 4 december 2019 journal of algebra combinatorics discrete structures and applications some bounds arising from a polynomial ideal associated to any t-design∗ research article william j. martin, douglas r. stinson abstract: we consider ordered pairs (x,b) where x is a finite set of size v and b is some collection of k-element subsets of x such that every t-element subset of x is contained in exactly λ “blocks” b ∈b for some fixed λ. we represent each block b by a zero-one vector cb of length v and explore the ideal i(b) of polynomials in v variables with complex coefficients which vanish on the set {cb | b ∈ b}. after setting up the basic theory, we investigate two parameters related to this ideal: γ1(b) is the smallest degree of a non-trivial polynomial in the ideal i(b) and γ2(b) is the smallest integer s such that i(b) is generated by a set of polynomials of degree at most s. we first prove the general bounds t/2 < γ1(b) ≤ γ2(b) ≤ k. examining important families of examples, we find that, for symmetric 2-designs and steiner systems, we have γ2(b) ≤ t. but we expect γ2(b) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying γ2(b) = k. 2010 msc: 05b05, 05e30, 13f20 keywords: design, steiner system, polynomial ideal, bounds 1. introduction let x be a finite set of size v and consider a k-uniform hypergraph (x,b) with vertex set x and block (hyperedge) set b. we aim to study polynomial functions on x which vanish on each element of b so that b may be viewed as the variety of some naturally defined ideal of polynomials in v variables. in order to do so, we identify each block b ∈ b with a 01-vector cb, with entries indexed by the elements ∗ the first author’s research is supported by a grant from the us national science foundation. the second author’s research is supported by nserc discovery grant rgpin-03882. william j. martin (corresponding author); department of mathematical sciences and department of computer science, worcester polytechnic institute, 100 institute road, worcester, ma 01609, usa (email: martin@wpi.edu). douglas r. stinson; david r. cheriton school of computer science, university of waterloo, waterloo, ontario, n2l 3g1, canada (email: dstinson@uwaterloo.ca). 161 https://orcid.org/0000-0002-2027-5859 https://orcid.org/0000-0001-5635-8122 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 of x, whose ith entry is equal to one if i ∈ b and equal to zero otherwise. in other words, cb is the bth column of the point-block incidence matrix, a, of the hypergraph. in this paper we work in characteristic zero and consider the polynomial ring r = c[x1, . . . ,xv]. the evaluation map ε : r→ cb given by ε(f) (b) = f(cb) is a ring homomorphism and its kernel is the ideal of all polynomials in v variables which evaluate to zero on each block of the hypergraph. denoting this kernel by i, we see that i is the ideal of the finite variety {cb | b ∈ b} and we write i = i(b). our goal in this paper is to explore connections between this ideal i and the hypergraph (x,b). our primary question involves the identification of good generating sets g for i based on the combinatorial structure of the design (x,b). we are not concerned here with the actual size of the generating set; in fact, we prefer a set of polynomials preserved by the automorphism group of the design. by “good” here, we mean principally that polynomials in the generating set are all of low degree. but we also seek polynomials that shed light on properties of the design. 1.1. background and related work algebraic geometry has a long history in the theory of error-correcting codes. we also have a theory of spherical codes and designs that involves the evaluation of multivariate polynomials at finite sets of points on the unit sphere. möller [20] constructs good quadrature rules for spherical integration by choosing zero sets of well-chosen families of polynomials. conder and godsil [5] studied the symmetric group as a polynomial space. the standard module of the johnson association scheme j(v,k) can be identified with polynomials of degree at most k in v variables in such a way that the polynomials of a given maximum degree j correspond to a sum of eigenspaces v0 + v1 + · · ·+ vj. this approach, which is implicit in [8] is worked out in more detail in a later paper of delsarte [9] (see also [3]). these phenomena motivated martin and steele [17] to consider the ideal of polynomials that vanish on the shortest vectors of certain lattices. in work in progress, martin [18] extends this to attach an ideal to any cometric association scheme by viewing the columns of the q-polynomial generator e1 in the bose-mesner algebra as a finite variety. several important examples are connected to combinatorial t-designs and – when disentangled from the language of association schemes – the problem of determining generating sets for the ideals of those association schemes reduces to the problem we address here. the “polynomial method” in combinatorics [24] is a powerful tool for deriving bounds on the size of combinatorial objects and for proving non-existence of extremal objects. in particular, a recent breakthrough by croot, lev and pach [7] (see also [11]) has stimulated interest in collections of multivariate polynomials that vanish on some configuration in a finite vector space. here, we work in characteristic zero and are interested in how the ideal we obtain is related to the structure of the design. by contrast, the authors of [7, 11] work over fields of positive characteristic. to see the connection, we note that, since our zero set lies in zv, the polynomial generators g ∈g may always be chosen from z[x]. so application of the reduction z → zp maps the ideal 〈g〉 ⊆ z[x] into the ideal of the same finite variety considered modulo p. it will be an interesting follow-up task to determine when the image under this map is the full ideal in positive characteristic. we have recently learned of previous gröbner basis approaches to algebraic aliasing in the statistical design of experiments. once a useful basis is chosen for the space of polynomial functions on the elements of the design, higher order effects can be mapped to more elementary interactions between factors. for a recent survey, see [19], with the caveat that the terminology used there differs from the terminology in this paper. 1.2. the trivial ideal we pause to introduce our notation for the standard operations of elementary algebraic geometry. (see, e.g., [10, chapter 15] for a basic introduction.) for a set s ⊆ cv, we let i(s) denote the ideal of all polynomials in c[x] := c[x1, . . . ,xv] that vanish at each point in s. and if g is any set of 162 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 polynomials in c[x1, . . . ,xv], we denote by z(g) the zero set of g, the collection of all points c in cv which satisfy f(c) = 0 for all f ∈g. note that, when s is finite, we have z(i(s)) = s. in the opposite direction, for any ideal j of polynomials, the nullstellensatz (see, e.g., [13, p21], [6, p179]) informs us that i(z(j)) = rad(j), where rad(j) denotes the radical of ideal j, the ideal of all polynomials g such that gn ∈ j for some positive integer n. a radical ideal is an ideal which is already closed under this process: rad(j) = j. our first example to consider is the complete uniform hypergraph (x,kvk). lemma 1.1. let x be a finite set of size v ≥ k and let kvk = ( x k ) consist of all k-subsets of x. let g0 = {x1 + · · ·+ xv −k}∪{x2i −xi | 1 ≤ i ≤ v}. (1) then i(kvk) = 〈g0〉 and z(〈g0〉) = {cb | b ∈k v k}. proof. one easily checks that each cb is a common zero of the polynomials in g0. conversely, any point in cv which is a zero of each polynomial in g0 must be a 01-vector with exactly k ones. in order to verify that g0 generates the full ideal, we check that the zariski tangent space at each point is fulldimensional. let b ⊂ x be a k-set; evaluating the gradient ∇f of f(x) = x2j −xj at cb we obtain ±ej where ej is the standard basis vector with a one in position j and all other entries zero. so the jacobian of the set g0 of v + 1 polynomials in v variables evaluated at cb takes the form jac(g0,cb) = [ ∂fi ∂xh ∣∣∣∣∣cb ] h,i =   1 ±1 0 · · · 0 1 0 ±1 · · · 0 ... ... 1 0 0 · · · ±1   which clearly has column rank equal to v. this guarantees that each cb is a simple zero of 〈g0〉 and so the ideal is indeed radical. it now follows from the nullstellensatz that i({cb | b ∈kvk}) = i(z(〈g0〉)) = rad(〈g0〉) = 〈g0〉. see section 3 for details of these last calculations. 2. two parameters in this paper, once we fix v and k, every ideal will contain the ideal we have just considered. we call this the trivial ideal and denote t = 〈g0〉 = 〈 x1 + · · ·+ xv −k, x21 −x1, . . . ,x 2 v −xv 〉 . (2) for any k-uniform hypergraph (x,b) on v points, the ideal i(b) := i({cb | b ∈b}) will contain t and a polynomial f ∈i(b) will be called non-trivial if f 6∈ t and trivial otherwise. to each c ⊆ {1,2, . . . ,v} we associate the monomial xc = ∏ j∈c xj and note that, for a block b ∈b, the value of xc at point cb is one if c ⊆ b and zero otherwise. a k-uniform hypergraph (x,b) is a t-(v,k,λ) design (or a block design of strength t) if, for every t-element subset t ⊆ x of points, there are exactly λ blocks b ∈b with t ⊆ b (so ∑ b∈b f(cb) = λ whenever f(x) = x t for some t-set t ⊆ x). every t-(v,k,λ) design is an s-(v,k,λs) design for each s ≤ t where λs ( k−s t−s ) = λ ( v−s t−s ) . the following characterization of t-designs is well-known (see, for example, godsil [14, cor. 14.6.3]). lemma 2.1 (cf. delsarte [9, theorem 7]). let x be a set of size v and let (x,b) be a k-uniform hypergraph defined on x with corresponding vectors cb (b ∈ b) as defined above. then (x,b) is a t-design on x if and only if the average over b of any polynomial f(x) in v variables of total degree at most t is equal to the average of f(x) over the complete uniform hypergraph kvk defined on x. 163 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 proof. let c ⊆ x with |c| = s ≤ t. exactly ( v−s k−s ) elements of kvk contain c so the average value of f(x) = xc over {cb | b ∈kvk} is( v −s k −s )/( v k ) = k(k −1) · · ·(k −s + 1) v(v −1) · · ·(v −s + 1) = λs λ0 which is exactly the average of f(x) over the block set b. so the result holds for monomials. but every polynomial in v variables of total degree at most t is a linear combination of such monomials, so the result holds for these as well by linearity. given (x,b), we seek combinatorially meaningful generating sets for i(b). two polynomials f(x) and g(x) have the same value at every point cb, b ∈b if and only if their difference belongs to the ideal i(b). for example, f(x) = x2j and g(x) = xj take the same value on every 01-vector so x 2 j −xj belongs to the ideal of any design. we say f(x) is a multilinear polynomial in v variables if f is linear in each xi: i.e., each monomial with non-zero coefficient in f is a product of distinct indeterminates. modulo the trivial ideal t , each polynomial in c[x1, . . . ,xv] is equivalent to some (not necessarily unique) multilinear polynomial with zero constant term. with a preference for polynomials of smallest possible degree, we define two fundamental parameters. definition 2.2. let (x,b) be a non-empty, non-complete k-uniform hypergraph on vertex set x = {1, . . . ,v} with corresponding ring of polynomials r = c[x1, . . . ,xv]. let i(b) and t be defined as above. define γ1(b) = min{deg f | f ∈i(b), f 6∈ t } and γ2(b) = min{max{deg f : f ∈g} | g ⊆r, 〈g〉 = i(b)} . so γ1(b) is the smallest possible degree of a non-trivial polynomial that vanishes on each block and γ2(b) is the smallest integer s such that i(b) admits a generating set all polynomials of which have degree at most s. obviously, γ1(b) ≤ γ2(b); designs satisfying equality here are particularly interesting. theorem 2.3. if (x,b) is a t-design (t ≥ 2) and f ∈i(b) is non-trivial, then deg f > t/2. so, for any non-trivial t-design (x,b), γ1(b) ≥ (t + 1)/2. proof. suppose f ∈ i(b) has degree at most t/2. write f(x) = f(x) + ig(x) where f,g ∈ r[x] each have degree at most t/2. since the entries of each cb are real, it is clear that f,g ∈ i(b). then f2 ∈ i(b) is a non-negative polynomial of degree at most t. by lemma 2.1, its average over b is zero hence its average over kvk is also zero. since f 2 is everywhere non-negative, it must evaluate to zero on the incidence vector cb of every k-set b. so it belongs to the ideal i(kvk). since this ideal is radical and contains f2, it also contains f. by lemma 1.1, f must be trivial. the same argument applies to g and, hence, to f. remark 2.4. the same sort of reasoning used in this proof shows that i(b) admits a vector space basis, even a generating set, of polynomials with integer coefficients. let f be a polynomial which vanishes on b and let {ζ1 . . . ,ζm} ⊆ c be a basis for the subspace of c, as a vector space over q, that contains all the coefficients of f. then there exist unique polynomials f1, . . . ,fm in q[x] with f = ∑ h ζhfh. since each cb is a vector with integer entries, the fact that f evaluates to zero at cb implies that each fh also vanishes at that point. so, scaling appropriately, we may assume each generator belongs to z[x]. a standard result in the theory of designs (see; cameron [4] and delsarte [8, theorem 5.21]) is the fact that a t-design with s distinct block intersection sizes satisfies t ≤ 2s. we now show that theorem 2.3 implies a stronger result, which we believe is new. 164 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 let c ⊆ x with characteristic vector c and suppose {|c ∩b| : b ∈ b} = {i1, . . . , is}. then every cb for b ∈b is a zero of the degree s zonal polynomial f(x) = (c ·x− i1) · · ·(c ·x− is). of course, if |c| is sufficiently small, this polynomial belongs to the trivial ideal. corollary 2.5. let (x,b) be a t-design and let c1, . . . ,c` ⊆ x and {i1,1, . . . , i1,s1, i2,1, . . . , i`,s`} be a multiset of integers such that, for every b ∈b there exist 1 ≤ h ≤ ` and 1 ≤ j ≤ sh with |b∩ch| = ih,j. if there exists some k-set s 6∈ b with |s∩ch| 6∈ {ih,1, . . . , ih,sh} for all h = 1, . . . ,`, then γ1(b) ≤ s1+· · ·+s` hence s1 + · · ·+ s` > t/2. proof. for y ⊆ x, define 01-vector χy by (χy )a = 1 if a ∈ y and (χs)a = 0 otherwise. e.g., χb = cb when b is a block. consider the product of ` zonal polynomials f(x) = ∏̀ h=1 sh∏ j=1 ( χch ·x− ih,j ) . by hypothesis, f(cb) = 0 for every b ∈ b. since f(χs) 6= 0, f is non-trivial. so, by theorem 2.3, deg f > t/2. lemma 2.6. let (x,b) be a t-(v,k,λ) design. let s denote the smallest integer such that ( v s ) > |b|. then γ1(b) ≤ s. proof. the vector space of functions on kvk representable by multilinear polynomials in r with zero constant term and total degree at most s has dimension ( v s ) . for the chosen value of s, there exists a non-zero multilinear polynomial f(x) ∈ r, of total degree at most s, which vanishes on each element of b. (we have |b| equations and ( v s ) unknowns.) being multilinear with zero constant term, f(x) is non-trivial with degree at most s. we finish this section with two instructive examples. example 2.7. let us construct the ideal of the fano plane. let x = z7 and take b = {{0,1,3},{1,2,4},{2,3,5},{3,4,6},{4,5,0},{5,6,1},{6,0,2}} . starting from g0, let us build up a meaningful generating set for i(b). the unique triple in b containing both 0 and 1 also contains 3; this combinatorial condition may be encoded as x0x1 − x0x1x3 ∈ i(b). alternatively, including the quadratic polynomial x0x1 − x0x3 in a generating set g for our ideal also guarantees that any vector c ∈z(〈g〉) with c0 = 1 and c1 = 1 must have c3 = 1 as well. up to sign, there are ( 7 2 ) quadratic generators of this form and these, together with those in g0, generate the full ideal. example 2.8. with x = {1, . . . ,9} and b = {{1,2,3},{4,5,6},{7,8,9}}, a 1-design, we find generating set g = g0 ∪{x1 −x2, x1 −x3, x4 −x5, x4 −x6, x7 −x8, x7 −x9} . while we will not see further examples where the ideal is generated by g0 and linear polynomials, we will see that the difference of two monomials appears again as a useful tool. 3. radical ideals in this section, we deal with a technicality which arises as we compute ideals of finite sets. we show here that every ideal containing the trivial ideal is radical, thereby eliminating any further need to check this property. 165 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 given a finite set s of points in cv, it is often easy to come up with polynomials that vanish at each of those points and, with a bit of work, we might find a generating set g for some ideal j = 〈g〉 whose zero set is exactly s: z(〈g〉) = s. hilbert’s nullstellensatz then tells us that i(s) = i(z(j)) = rad(j), (3) the radical of of ideal j given by rad(j) = {f ∈r | (∃n ∈ n) (fn ∈ j)} . the ideal j is a radical ideal if rad(j) = j. our goal then is achieved in three steps: given a finite set of points s, find a nice set g of small-degree polynomials that vanish on s; verify that z(g) = s and nothing more; verify also that 〈g〉 is a radical ideal. in this section, we discuss ways to achieve this last step. if j is an ideal in c[x1, . . . ,xv] with finite zero set z(j) = s, then c[x]/j is a finite-dimensional complex vector space and its dimension is equal to the sum of the multiplicities of all the zeros of i, dim c[x]/j = ∑ c∈s mult(c). the coordinate ring of a variety s ⊆ c v is defined as the quotient ring c[x]/i(s) and this is naturally identified with the ring of “polynomial functions” on the set s. if j is an ideal in c[x] with finite zero set z(j) = s, then it is well-known (e.g., section 5.3, proposition 7 in [6]) that ∑ c∈s mult(c) = dim c[x]/j ≥ dim c[x]/rad(j) = |s| (4) where mult(c) is the multiplicity of point c as a zero of j. this proves proposition 3.1. with notation as above: (i) if s ⊆z(j), then dim c[x]/j ≥ |s|; (ii) if j is a radical ideal and z(j) = s is finite, then j = i(s); (iii) if j is an ideal in c[x] with finite zero set z(j) ⊇s and the coordinate ring c[x]/j has dimension equal to |s|, then the ideal j is radical, z(j) = s, each point of s is a smooth point (multiplicity one), and i(s) = j. � let g = {f1, . . . ,f`} be a generating set for ideal j ⊆ c[x1, . . . ,xv]. the jacobian of the system {f1(x) = 0, . . . ,f`(x) = 0} of polynomial equations evaluated at point c ∈ cv is the v×` matrix jac(g,c) with (i,j)-entry equal to ∂fj ∂xi ∣∣∣ c , the partial derivative ∂fj/∂xi evaluated at point c. since we are dealing only with zero-dimensional varieties in this paper, we say the point c is smooth if jac(g,c) has column rank v, so that the zariski tangent space is zero-dimensional. a smooth point has multiplicity one. so another way to show that the ideal j is radical is to check that, at each point c of its zero set, the jacobian jac(g,c) has full row rank where g is some generating set for j. proposition 3.2. let g ⊆ c[x] be any set of polynomials such that g0 ⊆ g (cf. equation (1)). then 〈g〉 is a radical ideal. proof. in the proof of lemma 1.1, we proved that the jacobian jac(g0,cb) of g0 has column rank equal to v at any characteristic vector cb of any k-set b. since jac(g0,cb) is a submatrix of jac(g,cb), this latter jacobian also has full row rank and each point of the finite variety z(g) is smooth. we note here that this approach also provides another proof of the fundamental bound of raychaudhuri and wilson. lemma 3.3. let s ⊆ {1, . . . ,v} and t ≥ |s|. if i is an ideal containing t , then in the quotient ring c[x]/i, the coset xs + i is expressible as a linear combination of cosets xt + i where |t | = t. 166 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 proof. assume t = |s|+ 1. since 1 + i = 1 k ∑ j xj + i we have xs + i = ∏ s∈s xs + i = 1 k ∑ j xj ∏ s∈s xs + i = ( t−1 k xs + i ) +   ∑ s⊆t |t|=t xt   + i and we may solve for xs + i. theorem 3.4. if (x,b) is a 2s-(v,k,λ) design, then |b|≥ ( v s ) . proof. since the coordinate ring c[x]/i(b) has dimension |b| and the monomials {xc : |c| = s} represent linearly independent cosets in this quotient ring, we have |b|≥ ( v s ) . this language differs from that employed by ray-chaudhuri and wilson. consider the ( v s ) × |b| matrix a(s) with rows indexed by all c ⊆ x with |c| = s, with columns indexed by the set b of blocks of a t-(v,k,λ) design, and (c,b)-entry equal to one if c ⊆ b and equal to zero otherwise. the proof of theorem 1 in [21] establishes that the columns cb of a(s) span the space r( v s). the celebrated bound follows immediately for even t and, for odd t, one obtains |b| ≥ 2 ( v s ) whenever b is the block set of a t-(v,k,λ) design with t = 2s+ 1 by applying this idea to both the derived design and the residual design. as we will use the fact later, we record it here as a lemma. lemma 3.5. let (x,b) be a t-(v,k,λ) design with t ≥ 2s and consider the incidence matrix a(s) of s-subsets of x versus blocks as defined in the previous paragraph. then the column rank of a(s) is exactly( v s ) . � 4. steiner systems and partial designs for a subset c ⊆ x and f ∈ r, define f(c) in the obvious way, by setting xi = 1 if i ∈ c and xi = 0 if i 6∈ c (1 ≤ i ≤ v), and then evaluating f(χc) where χc = (x1, . . . ,xv). we want to construct small-degree generating sets g for the ideal i = i(b) where b is the block set of our design (x,b). we assume throughout that g contains g0 so that 〈g〉 contains the trivial ideal t . every zero of this latter ideal is a 01-vector with exactly k ones. as we choose the remaining generators, we need only search for multilinear polynomials: each monomial xe11 · · ·x ev v appearing in f(x) has all ei ∈ {0,1}. it is clear that the automorphism group of a design (x,b) acts on the ideal i(b) by permuting indeterminates; rather than minimizing |g|, we typically show a preference for sets of generators invariant under this action. recall, for a subset c ⊆ x, we have xc = ∏ i∈c xi. next define, for an integer j ≤ |c|, the polynomial xc,j = ∑ j⊆c,|j|=j xj. (this is a certain symmetric function — the elementary symmetric polynomial of degree j — in the variables {xi | i ∈ c}.) for example, xc,|c| = xc. since the trivial ideal contains ( xx,1 −k )j for all j it 167 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 also contains xx,j − ( k j ) for 1 ≤ j ≤ k. for example, modulo 〈g0〉, ( xx,1 −k )2 = v∑ i=1 x2i + 2  ∑ i<j xixj  −2k ( v∑ i=1 xi ) + k2 ≡ ( v∑ i=1 xi −k ) + 2  ∑ i<j xixj − ( k 2 )−2k ( v∑ i=1 xi −k ) ≡ ( xx,1 −k ) + 2 ( xx,2 − ( k 2 )) −2k ( xx,1 −k ) . lemma 4.1. suppose that b ⊆ x, |b| = k and j ⊆ b. denote j = |j|. define gb,j(x) = x b,j − ( k j ) xj . (5) then, for every c ⊆ x, we have that gb,j(c) 6= 0 if and only if j ≤ |c ∩b| < k. proof. if |c∩b| < j, then every monomial appearing in gb,j(x) evaluates to 0 on χc. if |c∩b| = k, then all the monomials in gb,j(x) take nonzero values on χc, and gb,j(c) = ( k j ) (1) − (1) ( k j ) = 0. if j ≤ |c∩b| < k, then some proper subset of the monomials occurring in gb,j(x) take nonzero value, and gb,j(c) cannot equal 0. theorem 4.2. for any k-uniform hypergraph (x,b), γ2(b) ≤ k. proof. in addition to our generators g0 for t given in (1), we will include one generator gy (x) for each set y of k −1 points. for y ⊆ x with |y | = k −1, define j = {j ∈ x : y ∪{j}∈b}. suppose j = {j1, . . . ,j`} (possibly the empty set). then consider the polynomial gy (x) = x y (xj,1 −1), (6) noting that gy (x) = −xy whenever y is contained in no block of the hypergraph. let i be the ideal generated by g = g0 ∪{gy (x) : |y | = k −1} . it follows from proposition 3.2 that i is a radical ideal. in order to prove that i = i(b) (and hence that γ2(b) ≤ k), we must verify • f(b) = 0 for every f ∈g and every b ∈b; • for any k-set c 6∈ b, there is some y with gy (c) 6= 0. suppose that b ∈ b; then it is easy to verify from (6) that gy (b) = 0 for any (k − 1)-subset y . now suppose that c ⊆ x, |c| = k, c 6∈ b. let y ⊆ c, |y | = k − 1. if y is contained in no blocks, then gy (c) = −1 since gy (x) = −xy . on the other hand, if y is contained in at least one block, then gy (c) = −1 from (6). we then have that i is a radical ideal with z(i) = b; by proposition 3.1(ii) we are done. a steiner system is a t-(v,k,λ) design with λ = 1. the question of existence of non-trivial steiner systems with t > 5 has been recently resolved in spectacular fashion by keevash [16]. 168 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 theorem 4.3. let (x,b) be any t-(v,k,1) design. for a block b ∈ b and any t-element subset t contained in b, define (as in (5)) gb,t (x) = x b,t − ( k t ) xt . then (i) i(b) is generated by g0 ∪{gb,t (x) : b ∈b,t ⊆ b, |t | = t}; (ii) γ2(b) ≤ t. proof. in view of theorem 4.2, we can assume that t < k. consider the generating set g = g0 ∪{gb,t (x) : b ∈b,t ⊆ b, |t | = t} . from lemma 4.1, if b ∈ b then gb,t (b) = 0 for each t-subset t of b, while if b′ ∈ b, b′ 6= b, then |b′ ∩b| ≤ t−1 and it follows that gb′,t (b) = 0 for any t-subset t of b′. now suppose that |c| = k, c 6∈ b, and choose t ⊆ c, |t | = t. there is a block b ∈b with t ⊆ b. then gb,t (c) 6= 0 from lemma 4.1 because t ≤ |b ∩c| ≤ k −1. so z(g) = {cb | b ∈b}. by proposition 3.2, 〈g〉 is a radical ideal. so equation (3) gives i(b) = 〈g〉 and the result follows. remark 4.4. let b be a block of the t-(v,k,1) design (x,b) and let t and t ′ be two t-element subsets of b. then it is easy to see that i(b) contains xt −xt ′ . since each of the generators gb,t in the above theorem is expressible as a sum of polynomials of this form, we have a perhaps simpler set of generators g0 ∪ { xt −xt ′ | t,t ′ ⊆ b ∈b, |t | = |t ′| = t } for the ideal. question: do the cosets { xb,t +i(b) | b ∈b } form a basis for the coordinate ring c[x]/i(b) in this case? next, we describe a couple of variations of theorem 4.3. a partial t-(v,k,1)-design is a k-uniform hypergraph in which any t-subset occurs in at most one block. a partial t-(v,k,1)-design, (x,b), is maximal if there does not exist a k-subset c ⊆ x, c 6∈ b such that (x,b∪{c}) is a partial t-(v,k,1)design. corollary 4.5. for any maximal partial t-(v,k,1)-design (x,b), γ2(b) ≤ t. proof. as we employ the same generating set as in the proof of theorem 4.3, we need only check that z(〈g〉) = b. as before, we have that gb,t (b′) = 0 for all blocks b,b′ ∈ b and all t-subsets t of b. now suppose that |c| = k, c 6∈ b. because (x,b) is a maximal partial t-(v,k,1)-design, it is possible to choose t ⊆ c, |t | = t such that there is a block b ∈b with t ⊆ b. then gb,t (c) 6= 0 as before. by a slight extension of our construction, we do not require the partial t-(v,k,1)-design to be maximal. theorem 4.6. for any partial t-(v,k,1)-design (x,b), γ2(b) ≤ t. proof. if (x,b) is maximal, then corollary 4.5 yields the desired result, so assume (x,b) is not maximal. let t = {t ⊆ x : |t | = t,(∀b ∈b)(t 6⊆ b)} . 169 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 now consider the generating set g = g0 ∪{gb,t (x) : b ∈b,t ⊆ b, |t | = t}∪{xt : t ∈ t}. since xt evaluates to zero on every block, we still have b ⊆ z(〈g〉). but if c is a k-set not belonging to b, either b∪{c} is again a partial t-design or some t-subset t of c is contained in some block b. in the latter case, gb,t (c) 6= 0 as above; in the former case, every t-subset of c belongs to t so we can take any t-subset t ⊆ c and, with f(x) = xt ∈g, we have f(c) 6= 0. so z(〈g〉) = b and the rest of the proof follows just as before. this result gives us another upper bound on γ2 for general t-designs. corollary 4.7. let (x,b) be a t-(v,k,λ) design with |b∩b′| < s for every pair b,b′ of distinct blocks. then γ2(b) ≤ s. � 5. symmetric balanced incomplete block designs a 2-(v,k,λ) design is traditionally called a balanced incomplete block design (bibd) [23, chapter 1]. fisher’s inequality states that, for any such 2-design, we have |b| ≥ |x| since, for t ≥ 2, the point-block incidence matrix a has rank v. a 2-design with equally many blocks and points is called a symmetric 2design. while theorem 2.3 implies here that γ1(b) > 1, we may use the invertibility of a to obtain more information in this case. if f(x) = w0 + ∑v i=1 wixi ∈i(b) then w = (w1, . . . ,wv) satisfies w >a = −w01 and w + w0 k 1 lies in the left nullspace of a. so w is a scalar multiple of 1 and f is trivial. the quadratic case is more interesting. let ri denote row i of matrix a. for any two distinct points i,j ∈ x, the entrywise product ri ◦rj is expressible as a linear combination of the rows of a. say ri ◦rj = v∑ h=1 whrh . then the polynomial f(x) given by f(x) = xixj − v∑ h=1 whxh is easily seen to belong to i(b): for a block b indexing column ` of matrix a, we have f(cb) = ai`aj` − ∑v h=1 whah` = 0. in fact, we may determine the coefficients wh explicitly to obtain a nice generating set for our ideal. theorem 5.1. let (x,b) be any non-trivial symmetric 2-(v,k,λ) design. for each pair i,j of distinct points from x define fi,j(x) = (k −λ)xixj − ∑ i,j∈b∈b xb,1 + λ2. then (i) i(b) is generated by g0 ∪{fi,j | i,j ∈ x}; (ii) γ1(b) = γ2(b) = 2; (iii) the coordinate ring c[x]/i(b) admits a basis consisting of cosets {xi +i(b) | 1 ≤ i ≤ v}. 170 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 proof. assume (x,b) is a 2-design with |b| = v and incidence matrix a. as aa> = (k −λ)i + λj, we see that the inverse of our incidence matrix is a−1 = 1 k −λ ( a> − λ k j ) . letting ri denote the ith row of a (i ∈ x), observe that ri ◦ rj is a 01-vector of length v with λ entries equal to one. so ri ◦rj = w>a gives w> = (ri ◦rj)a−1 = 1 k −λ ∑ i,j∈b c>b − λ2 k(k −λ) 1> with entries wh = −λ2 k(k −λ) + 1 k −λ |{b ∈b | h,i,j ∈ b}| ; that is, i(b) contains the quadratic polynomial xixj + 1 k −λ ∑ i,j∈b xb,1 − λ2 k(k −λ) xx,1. as xx,1 takes value k on each cb, this shows that ideal i contains fi,j(x) for every pair i,j of distinct points. set g = g0 ∪{fi,j | i,j ∈ x} and consider i = 〈g〉. the dimension of the quotient ring c[x]/i is v since every monomial of degree two is congruent, modulo 〈g〉, to some polynomial of degree one1. since we have dim c[x]/i = |b|, we use proposition 3.1(iii) to see that z(i) = b and each zero has multiplicity one. it then follows that i = i(b) and the cosets of the form xixj + i(b) (i 6= j) form a basis as claimed. example 5.2. consider the case where (x,b) is a symmetric 2-(v,k,2) design. let x = {i : 1 ≤ i ≤ v} be the set of points and let b = {br : 1 ≤ r ≤ v} be the set of blocks in the design. let i,j be any two distinct points. there are two blocks that contain i and j, say br and bs. note that br ∩bs = {i,j}. denote the symmetric difference by zi,j = br ∪bs \{i,j} and define fi,j(x) = (k −2)xixj + 4−2(xi + xj)− ∑ h∈zi,j xh. then i(b) is generated by xx,1 −k, xi(xi −1) (1 ≤ i ≤ v) and the polynomials fi,j(x). theorem 5.3. suppose that (x,b) consists of the points and e-dimensional subspaces of pg(d,q), where 1 ≤ e < d. let l denote the set of all lines (1-dimensional subspaces) of pg(d,q). for every line l in l and every 2-element subset j ⊆ l define gl,j(x) = xl,2 − ( q+1 2 ) xj. then (i) i(b) is generated by g := g0 ∪{gl,j | l ∈l,j ⊆ l, |j| = 2}; (ii) γ1(b) = γ2(b) = 2; 1 we choose some monomial ordering for the ring c[x] which refines the partial order by total degree. 171 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 (iii) the coordinate ring c[x]/i(b) admits a basis consisting of cosets {xi +i(b) | 1 ≤ i ≤ v}. proof. here we have v = (qd+1 − 1)/(q − 1) and k = (qe+1 − 1)/(q − 1). every line of pg(d,q) contains q + 1 points. suppose b is an e-dimensional subspace of pg(d,q) and l is a line. then |l∩b| ∈ {0,1,q + 1}. in each case, gl,j(b) = 0 for any 2-element j ⊆ l by lemma 4.1. a subset of points of pg(d,q) that intersects any line in 0,1 or q + 1 points is necessarily a subspace of pg(d,q). suppose that |c| = k but c is not an e-dimensional subspace of pg(d,q). then there exists a line l such that |l∩c| 6∈ 0,1,q + 1. in this case, gl,j(c) 6= 0 by lemma 4.1. we have shown that the ideal generated by g is a radical ideal with zero set b, so we are done by proposition 3.1(ii). remark 5.4. clearly we can build a much smaller generating set than our choice of g by selecting just one pair j of points in each line. we instead prefer here to choose a set g of polynomials which is invariant under the automorphism group of the design. 6. triple systems identifying each square-free monomial xc with the set c, the multilinear polynomials with real coefficients are in bijective correspondence with the real-valued functions on the boolean lattice. for multilinear f write c(f) = (fd : d ⊆{1, . . . ,v}) where f(x) = ∑ d fdx d. suppose that c ⊆ x and 0 ≤ s ≤ |c|. the s-incidence vector of c, denoted δ = δs(c), is the vector of length w = ∑s i=0 ( v i ) , whose coordinates correspond to the subsets of x of cardinality at most s, defined by δj = { 1 if j ⊆ c 0 otherwise, where |j| ≤ s. now suppose that f is a multilinear polynomial in x1, . . . ,xv of degree at most s. the vector of coefficients of f, denoted c = c(f), is also a w-dimensional vector whose coordinates correspond to the subsets of x of cardinality at most s. the following lemma is obvious. lemma 6.1. if f is a polynomial of degree at most s and c ⊆ x, then f(c) = δ · c, where δ = δs(c) and c = c(f). � example 6.2. suppose that x = {1,2,3,4,5}, c = {1,2,3} and s = 2. let f = 1 + x1 + 2x2 −3x3 + 4x4 −x1x2 + +3x1x5 + 2x2x3 −3x3x5. then c(f) = (1,1,2,−3,4,0,−1,0,0,3,2,0,0,0,−3,0) and δs(c) = (1,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0). it is easy to verify that f(c) = 1 + 1 + 2−3−1 + 2 = 2. 172 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 theorem 6.3. suppose that (x,b) is a 2-(v,3,2) design such that {{1,2,3},{1,4,5},{2,4,6},{3,5,6},{1,2,4},{1,3,5},{2,3,6}}⊆b, but {4,5,6} 6∈b. then γ2(b) = 3. proof. by theorems 2.3 and 4.2, we have 2 ≤ γ2(b) ≤ 3. we now show γ2(b) > 2. denote b1 = {1,2,3}, b2 = {1,4,5}, b3 = {2,4,6}, b4 = {3,5,6}, b5 = {1,2,4}, b6 = {1,3,5}, b7 = {2,3,6} and c = {4,5,6}. for s = 2, it is easy to verify that δs(b1) + δ s(b2) + δ s(b3) + δ s(b4) = δ s(b5) + δ s(b6) + δ s(b7) + δ s(c). (7) for any f ∈i(b), we have that f(b1) = f(b2) = · · · = f(b7) = 0. it then follows from (7) and lemma 6.1 that f(c) = 0. however, since c 6∈ b, it must be the case that f(c) 6= 0 for some f ∈ i(b). this contradiction establishes the desired result. note that this linearization technique may be applied more generally. if we can find c 6∈ b such that δs(c) is a linear combination of the w-dimensional vectors {δs(b) | b ∈b}, then γ2(b) > s. we now show one way to construct examples of 2-(v,3,2) designs that satisfy the hypotheses of theorem 6.3. our construction works for any v ≡ 1,3 (mod 6), v ≥ 15. suppose we have a 2-(v,3,2) design that satisfies the hypotheses of theorem 6.3. we first observe that {{1,2,3},{1,4,5},{2,4,6},{3,5,6}} (8) is a set of four blocks that forms a so-called quadrilateral (or pasch configuration). as well, {{1,2,4},{1,3,5},{2,3,6}} (9) is a set of three blocks that is not contained in a quadrilateral (because {4,5,6} is not a block). we require two ingredients: 1. the unique 2-(7,3,1) design is isomorphic to point-line structure of pg(2,2) and it contains a quadrilateral (in fact, it contains exactly seven distinct quadrilaterals). therefore the points of this design can be relabelled so it contains the four blocks in (8). now, from the doyen-wilson theorem, we can embed this 2-(7,3,1) design in a 2-(v,3,1) design for any v ≡ 1,3 (mod 6), v ≥ 15. 2. it is shown in [22, theorem 3.1] that the maximum number of quadrilaterals in a 2-(v,3,1) design is v(v − 1)(v − 3)/24, and this maximum is attained if an only if the design is isomorphic to the point-line structure of the projective geometry pg(n,2) for some integer n ≥ 2. take any 2-(v,3,1) design that is not isomorphic to the projective geometry pg(n,2) (this can be done provided v ≡ 1,3 (mod 6), v ≥ 9. it is easy to see that design must contain three non-collinear points that are not contained in a quadrilateral. by relabelling points in the design, we can assume that the three non-collinear points are denoted 1,2 and 3, and they are contained in the three blocks in (9). moreover, {4,5,6} is not a block in this design because the three points 1,2,3 are not contained in a quadrilateral. now we take the union of the blocks in the two 2-(v,3,1) designs constructed above. the result is a 2-(v,3,2) design that contains the seven blocks in (8) and (9). we have already noted that {4,5,6} is not a block in the second design. it is also not a block in the first design because the pairs {4,5}, {4,6} and {5,6} occur in three different blocks in this design. as a consequence of this discussion and theorem 6.3, the following result is immediate. 173 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 theorem 6.4. suppose v ≡ 1,3 (mod 6), v ≥ 15. then there exists a 2-(v,3,2) design (x,b) such that γ2(b) = 3. � we now give a more general version of this construction, starting from an arbitrary trade. we recall some definitions from [12]. a trade is a set t of two (finite) subsets of blocks of size three, say t = {t1,t2} that satisfies the following properties: 1. each t` (` = 1,2) is a partial steiner triple system (i.e., no pair of points occurs in more than one block) 2. t1 ∩t2 = ∅ 3. the set of pairs contained in the blocks in t1 is identical to the set of pairs contained in the blocks in t2. as an example, if t1 = {{1,2,3},{1,4,5},{2,4,6},{3,5,6}} and t2 = {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}, then t = {t1,t2} is a trade. the volume of a trade t = {t1,t2}, which is denoted vol(t), is the number of blocks in t1 (or, equivalently, the number of blocks in t2). the foundation of t, denoted found(t), is the set of points covered by the blocks in t1 (or, equivalently, the set of points covered by the blocks in t2). in the above example, vol(t) = 4 and found(t) = {1,2,3,4,5,6}. the following lemma is easy to prove. lemma 6.5. t = {t1,t2} is a trade. then∑ b∈t1 δ2(b) = ∑ b∈t2 δ2(b). proof. the definition of a trade t = {t1,t2} ensures that t1 and t2 cover the same set of pairs. so we just need to prove that t1 and t2 contain the same points with the same multiplicities. suppose i ∈ found(t). let n` = {j : {i,j} is contained in a block in t1}. then it is easy to see that i is contained in |n`|/2 blocks in each of t1 and t2. the following is a slight generalization of theorem 6.3. we omit the proof, which makes use of lemma 6.5, since it is essentially the same. theorem 6.6. let t = {t1,t2} be a trade. suppose b = {h,i,j}∈ t2. suppose that (x,b) is 2-(v,3,2) design such that t1 ∪ (t2 \{b}) ⊆b and b 6∈ b. then γ2(b) = 3. � theorem 6.7. suppose that t = {t1,t2} is a trade, where |found(t)| = n. let b ∈ t1. suppose v ≡ 1,3 (mod 6), v ≥ 2n + 3. then there exists a 2-(v,3,2) containing all the blocks in t1 ∪ (t2 \{b}), such that γ2(b) = 3. proof. t1 is a partial steiner triple system on n points. the famous result of bryant and horsley [2] shows that t1 can be embedded in a 2-(v,3,1) design for any v ≡ 1,3 (mod 6), v ≥ 2n + 1. 174 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 suppose b = {h,i,j} and let b∗ = {i,j,`}, where ` 6∈ found(t). define t∗2 = (t2\{b})∪{b∗}. t∗2 is a partial steiner triple system on n + 1 points, so (again, from [2]) it can be embedded in a 2-(v,3,1) design for any v ≡ 1,3 (mod 6), v ≥ 2(n + 1) + 1. so for v ≡ 1,3 (mod 6), v ≥ 2n + 3, we have two 2-(v,3,1) designs (which we can assume are defined on the same set of points), say (y,b1) and (y,b2), such that t1 ⊆ b1 and t∗2 ⊆ b2. then (y,b1 ∪b2) is a 2-(v,3,2) design that contains all the blocks in t1 ∪ (t2 \{b}). we claim that b is not a block in b1 ∪b2. first, there is a unique block b′ ∈b1 that contains the pair {i,j}, and this block b′ is one of the blocks in t1. because b′ ∈ t1 and b ∈ t2, it follows that b′ 6= b. therefore b 6∈ b1. to see that b 6∈ b2, we observe that the unique block in b2 that contains the pair {i,j} is b∗ 6= b. thus we have shown that (y,b1 ∪b2) satisfies the hypotheses of theorem 6.6, and the proof is complete. 7. strength greater than two we finish by addressing the ideals of non-trivial t-(v,k,λ) designs with t > 2. for steiner systems (where λ = 1), we have t + 1 2 ≤ γ1(b) ≤ γ2(b) ≤ t using theorem 2.3 and theorem 4.3. when λ > 1, the lower bound still holds and we may apply lemma 2.6: since the number of blocks is |b| = λ ( v t )( k t )−1 , we find γ1(b) ≤ s whenever ( v s )( k t ) > λ ( v t ) . we also have corollary 4.7 which tells us that γ2(b) ≤ m+ 1 when m is the maximum size of the intersection of two distinct blocks. let (x,b) be a t-(v,k,λ) design. for i ∈ x, the derived design of (x,b) with respect to i is the ordered pair (ẋ, ḃ) where ẋ = x \{i} and ḃ = {b \{i} | i ∈ b ∈b} . the residual design of (x,b) with respect to i has vertex set ẋ and block set {b ∈b | i 6∈ b}. if (x,b) is a t-design, then both its derived design and its residual design are (t−1)-designs. lemma 7.1. let (x,b) be a non-trivial t-design with i ∈ x. with notation as above γh(ḃ) ≤ γh(b) for h = 1,2. the same inequalities hold for the residual design. proof. we handle the case of the derived design; the computations for the residual design are similar. to simplify the notation, we take i = 1. let i = i(b) and define ideal j as the image of i under the ring homomorphism ϕ : c[x1, . . . ,xv] → c[x2, . . . ,xv] mapping x1 to 1 and mapping each xj to itself for j = 2, . . . ,v. for g ∈ i write ġ := ϕ(g) ∈ j. for any (k−1)-set c ⊆{2, . . . ,v}, we have ġ(c) = g(c∪{1}) and so, for c ∈ ḃ we have ġ(c) = 0 for all ġ ∈ j and, for c 6∈ ḃ, there exists ġ ∈ j for which ġ(c) 6= 0. it follows that, if g is a generating set for i, then ϕ(g) is a generating set for j. this shows γ2(ḃ) ≤ γ2(b). next, if g is a non-trivial polynomial in i of smallest degree, then ġ has degree no larger than the degree of g and is also non-trivial since ϕ maps trivial ideal to trivial ideal. we illustrate this and other results in this paper by recording, in the following table, the exact value of these parameters for the witt designs and the t-designs appearing as their derived designs. up to isomorphism, there are unique block designs with parameters 5-(24,8,1), 4-(23,7,1), 3-(22,6,1), 5(12,6,1), 4-(11,5,1) and 3-(10,4,1). one accessible source of information on the witt designs is the note [1] by andries brouwer. 175 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 t-(v,k,λ) γ1(b) γ2(b) notes 5-(24,8,1) 3 3 theorems 2.3, 7.2 4-(23,7,1) 3 3 theorem 7.3 3-(22,6,1) 2 2 theorem 7.4 2-(21,5,1) 2 2 theorem 5.3 5-(12,6,1) 3 3 discussion below 4-(11,5,1) 3 3 lemma 7.1 3-(10,4,1) 2 2 discussion below 2-(9,3,1) 2 2 theorems 2.3, 4.3 in the large witt design, with parameters 5-(24,8,1), blocks intersect in 0, 2 or 4 points. so we might start with the zonal polynomials (cb ·x)(cb ·x−2)(cb ·x−4)(cb ·x−8) where b ∈b. we know that the blocks of this design are the supports of the minimum weight codewords in the extended binary golay code. we may then use the fact that this is a self-dual code to show that these, together with the generators of t , generate our ideal. but we can do better. theorem 7.2. let (x,b) be the 5-(24,8,1) design. for a block b ∈b and points i,j ∈ b, define fb,i,j(x) = (xi −xj)(cb ·x−2)(cb ·x−4). then (i) i(b) is generated by g0 ∪{fb,i,j | i,j ∈ b ∈b}; (ii) γ1(b) = γ2(b) = 3. proof. we know γ1(b) ≥ 3 by theorem 2.3. for any block b ∈b, the number mi of blocks b′ ∈b with |b ∩b′| = i is given by i 8 7 6 5 4 3 2 1 0 mi 1 0 0 0 280 0 448 0 30 so the zonal polynomial (cf. corollary 2.5) ∏ i=8,4,2,0(cb · x − i) belongs to i(b) and the quadratic polynomial (cb ·x−2)(cb ·x−4) vanishes on every block except b itself and those blocks disjoint from b. but if i and j both belong to block b, the linear function xi −xj vanishes on cb and on cb′ for any block b′ disjoint from b. to show that the polynomials fb,i,j — as b ranges over the blocks and i, j range over the elements of b — together with the polynomials in the trivial ideal, generate our ideal, we employ two basic facts about the extended binary golay code g24. the blocks in b are precisely the supports of minimum weight codewords in this code. this is a self-dual code, so a binary tuple c ∈ f242 satisfies c ∈ g24 if and only if the mod 2 dot product c · c′ is zero for every c′ ∈ g24. since g24 is generated by its weight eight codewords, we may say c ∈ g24 if and only if its inner product with these 759 codewords is zero mod two. since we only want to recover the codewords of weight eight, we may omit integer inner product six. let i = 〈g0 ∪{fb,i,j | i,j ∈ b ∈b}〉 and observe that any element of z(i) has exactly eight entries equal to one and sixteen entries equal to zero. for c ∈ f242 , let c ∈ r24 be the corresponding 01-vector with real entries: cj = 1 if cj = 1 and cj = 0 if cj = 0. assuming c ∈z(i), we have fb,i,j(c) = 0 for each b ∈b and each i,j ∈ b which implies that either cb ·c ∈{2,4} or ci = 1 ⇔ cj = 1 for all i,j ∈ b. this latter alternative clearly means that either c = cb or c · cb = 0. for the corresponding binary vectors, this implies c ·c′ = 0 for each c′ ∈ g24 with hamming weight eight. as outlined above, this gives c ∈ g24 and, in turn, c = cb′ for some b′ ∈b. by propositions 3.2 and 3.1(ii), we have i = i(b). 176 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 theorem 7.3. let (x,b) be the 4-(23,7,1) design. for three distinct points i, j and k, let ci,j,k = {c = b \{i,j,k} | {i,j,k}⊆ b ∈b} and define hi,j,k(x) = 3 + 12xixjxk −3(xixj + xixk + xjxk)− ∑ c∈ci,j,k xc,2 . then (i) i(b) is generated by g0 ∪{hi,j,k | i,j,k}; (ii) γ1(b) = γ2(b) = 3; (iii) the coordinate ring c[x]/i(b) admits a basis consisting of those ( 23 2 ) cosets xixj +i(b) represented by multilinear monomials of degree two. proof. denote by b1,b2,b3,b4,b5 the five blocks containing t := {i,j,k} and set c` = b` \t . we know that two distinct blocks of our witt design intersect in either three points or one point. we now show that hi,j,k(b) = 0 for each b ∈b. to illustrate the simple arithmetic involved, we write hi,j,k(x) = 3 + 12(xixjxk)−3(xixj + xixk + xjxk)− (xc1,2)− (xc2,2)− (xc3,2)− (xc4,2)− (xc5,2) where b` = c` ∪t and we retain most of the parentheses here in our evaluation. case (1) t ⊆ b. here, we have b = b` for some ` ∈{1, . . . ,5} and hi,j,k(b) = 3 + 12(1)−3(1 + 1 + 1)− (1 + 1 + 1 + 1 + 1 + 1)− (0)− (0)− (0)− (0) = 0. case (2) |t ∩b| = 2. here we must have |b ∩b`| = 3 for all ` and, as b never contains two points from the same “sub-block” c` = b` \t , we have hi,j,k(b) = 3 + 12(0)−3(1)− (0)− (0)− (0)− (0)− (0) = 0. case (3) |t ∩b| = 1. in this case, as there are five blocks containing t and |b| = 7, we must have |b ∩b`| = 3 for exactly three values of ` and, as b contains two points from the same sub-block c` = b` \t in each of these three cases, we have hi,j,k(b) = 3 + 12(0)−3(0)− (1)− (1)− (1)− (0)− (0) = 0. case (3) t ∩ b = ∅. in this case, as there are five blocks containing t and |b| = 7, we must have |b ∩c`| = 3 for some unique sub-block c` = b` \t and b must contain a unique point from each of the other four. in this case, we have hi,j,k(b) = 3 + 12(0)−3(0)− (1 + 1 + 1)− (0)− (0)− (0)− (0) = 0. on the other hand, if s is a 7-set of points which is not a block, then hi,j,k(s) 6= 0 for any {i,j,k}⊆ s. for if t := {i,j,k} is contained in s and hi,j,k(s) = 0, then we have 0 = hi,j,k(s) = 3 + 12(1)−3(1 + 1 + 1)− ( m1 2 ) − ( m2 2 ) − ( m3 2 ) − ( m4 2 ) − ( m5 2 ) where m` = |s ∩ c`|. but m1 + · · · + m5 = 4 and we see that the only arrangement that achieves the stated equality is where some m` = 4. but then s = b` and we are done. so, if i = 〈g0 ∪{hi,j,k | i,j,k}〉 we have shown z(i) = {cb | b ∈b}. by proposition 3.2, i is a radical ideal. so proposition 3.1(ii) gives us i = i(b). this proves that γ2(b) = 3 and, by theorem 2.3, γ1(b) = 3 as well. by lemma 3.3, each coset xi + i can be expressed as a linear combination of cosets xixj + i. since the number of blocks of the design is ( 23 2 ) , we see that the cosets {xixj +i | i,j} form a vector space basis for c[x]/i. 177 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 next, if (x,b) denotes the witt design on 22 points, lemma 7.1 tells us 2 ≤ γ1(b) ≤ γ2(b) ≤ 3. we now show that both values are equal to two. theorem 7.4. let (x,b) be the 3-(22,6,1) design. for two distinct points i, j and a block b containing them, say b = {i,j,r,s,t,u}, define hi,j,b(x) = (xi −xj)(xr + xs + xt + xu −1). then (i) i(b) is generated by g0 ∪{hi,j,b | i 6= j, i,j ∈ b, b ∈b}; (ii) γ1(b) = γ2(b) = 2; (iii) the coordinate ring c[x]/i(b) admits a basis consisting of the 77 cosets xb,2 +i(b) obtained as b ranges over the blocks of the design. proof. by theorem 2.3, we have γ1(b) ≥ 2. consider b ∈ b and two distinct points i,j ∈ b. write b = {i,j,r,s,t,u}. since any two blocks of this design intersect in zero or two points, any block b′ that contains exactly one of i,j contains exactly one element from {r,s,t,u}. so hi,j,b(b′) = 0. the same holds if |b′∩{i,j}| is even. on the other hand, let c be a 6-element subset of x and choose three distinct points i,t,u ∈ c. there is a unique block b containing these three points, say b = {i,j,r,s,t,u}. if all three polynomials hi,j,b(x), hi,r,b(x), hi,s,b(x) vanish on c, then we must have c = b. this finishes the proof that z(g) = {cb | b ∈b} and, as g0 ⊆g, the ideal 〈g〉 is radical, proving (i) and (ii). to show that the functions on b represented by the polynomials {xb,2 | b ∈ b} are linearly independent, consider the 77 × 77 matrix m with (b,b′)-entry equal to the value the polynomial xb ′,2 takes at the point cb. then m −i is the adjacency matrix of a well-known2 strongly regular graph with eigenvalues 60, 5 and −3. it follows that m is invertible and the 77 cosets {xb,2 + i(b) | b ∈ b} are linearly independent in the coordinate ring. for the small witt designs, we do not have a computer-free proof of our claims. let us instead describe some generators for the ideals. first consider the unique witt design on twelve points. let (x,b) be the 5-(12,6,1) design. for three distinct points i, j and k, let c = x \ {i,j,k}. the twelve blocks containing i, j and k yield a 2-(9,3,1) design (c,b′), b′ = {b \{i,j,k} | i,j,k ∈ b ∈b} on the point set c and the four parallel classes of this affine plane may be oriented in a total of sixteen ways (each resulting in a 4-set of directed triples of blocks). we find that certain orientations yield polynomials of degree three which, together with those polynomials in g0, generate the ideal i(b). this shows γ1(b) = γ2(b) = 3. to be precise, let m12 be the subgroup of s12 generated by {(1 4)(3 10)(5 11)(6 12), (1 8 9)(2 3 4)(5 12 11)(6 10 7)} and consider the 132 6-sets in the orbit containing {1,2,3,4,5,9}. since m12 is 5-transitive, this is a 5-(12,6,1) design. two parallel lines in the derived design consisting of all blocks containing points 1, 2 and 3 are {4,5,9} and {8,10,11}. with a computer, one easily verifies that the polynomial f(x) = x1x4(x10 −x11) + x1x5(x11 −x8) + x1x9(x8 −x10) + x2x9(x10 −x11) + x2x4(x11 −x8) + x2x5(x8 −x10) + x3x5(x10 −x11) + x3x9(x11 −x8) + x3x4(x8 −x10) 2 see, for example, https: // www. win. tue. nl/ ~aeb/ graphs/ srg/ srgtab51-100. html 178 https://www.win.tue.nl/~aeb/graphs/srg/srgtab51-100.html w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 vanishes on each block of the design. it follows that any image [1π,2π,3π], [4π,5π,9π], [10π,11π,8π] of the three triples of indices under any π ∈ m12 yields another polynomial in the ideal. it requires a bit more computation, using the singular computer algebra system [15], to check that the ideal i(b) is generated by these polynomials together with those in g0. the relative orderings within the three triples above is important and we do not have an intrinsic description of the permissible orderings that yield vanishing polynomials. if we take the above computation as correct, we may determine γ1 and γ2 for the witt design on eleven points. if (x,b) now denotes a 4-(11,5,1) design, we may use theorem 2.3 to see that γ1(b) ≥ 3. by lemma 7.1, we have equality, and γ2(b) = 3 as well. without proof, we note that if (x,b) is the unique 3-(10,4,1) design [23, fig. 9.1], we find γ1(b) = γ2(b) = 2. in addition to the generators of the trivial ideal, we build certain quadratic generators from any pair b1,b2 of disjoint blocks. in order to explain these generators, we first describe the design. for the construction in [23], we take x = f23 ∪{∞} with the numbering 0 1 2 3 4 5 6 7 8 9 ∞ (0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2) and blocks {0} ∪ ` where ` is a line of ag(2,3) and the following eighteen symmetric differences of orthogonal lines: 1245, 1278, 1269, 1346, 1379, 1358, 2356, 2389, 2347, 4578, 4679, 5689, 1567, 2468, 3459, 1489, 2579, 3678. for any pair b1,b2 ∈ b with b1 ∩ b2 = ∅, the number mi,j of blocks b ∈ b with |b ∩ b1| = i and |b ∩b2| = j is given in the following table: i\j 0 1 2 3 4 0 0 0 2 0 1 1 0 0 8 0 0 2 2 8 8 0 0 3 0 0 0 0 0 4 1 0 0 0 0 for each i ∈ b1, the two blocks meeting b1 only in this point partition b2 into two sets of size two by their intersections. we select one of these two to determine two neighbours of i along an octagon as in figure 1. once b1 and b2 have been selected and this choice of a pair of neighbours has been made, this determines a quadratic generator for our ideal. we illustrate this with b1 = {0,1,2,3} and b2 = {4,5,7,8}. the resulting polynomial is g(x) = x0x5 −x5x1 + x1x7 −x7x3 + x3x8 −x8x2 + x2x4 −x4x0. we leave it to the reader to check that the way in which any block intersects this configuration guarantees that g(cb) = 0 for any b ∈ b. in fact, just five of these polynomials are needed — along with g0 — to generate the ideal. 8. conclusion we have introduced an algebraic approach to the study of t-designs which builds on existing machinery tied to the space of polynomial functions on blocks. for a design (x,b), we introduced the ideal 179 w. j. martin, d. r. stinson / j. algebra comb. discrete appl. 7(2) (2020) 161–181 0 13 2 4 5 7 8 − + − +− + − + figure 1. two disjoint blocks {0,1,2,3} and {4,5,7,8} and the quadratic polynomial obtained from the pair i(b) and proposed two parameters γ1(b) and γ2(b) which we claim capture essential information in the case of designs where the number of blocks achieves, or is close to, the bound of ray-chaudhuri and wilson. we prove, among other things, that t + 1 2 ≤ γ1(b) ≤ γ2(b) ≤ k with the upper bound of k replaced by t in the case of steiner systems or partial steiner systems. we determine the exact value of these parameters for symmetric 2-designs and the witt designs. by constructing many triple systems with γ2(b) = k, we indicate that γ2(b) can be larger than t. while we expect the value to be more typically close to k, we leave this as an open problem. one may also investigate the ideal vanishing on the codewords of an error-correcting code. in order to compute γ1 and γ2 in such a situation, we have some degree of freedom as these parameters are invariant under affine transformations (provided one is careful with the definition of the trivial ideal). representing the codewords of a binary linear [n,k,d] code c by ±1 vectors in rn, we see that each dual codeword c = [c1, . . . ,cn] corresponds to an element fc(x) = −1 + ∏ j x cj j in the ideal of our code. in the linear case, γ1(c) is the minimum distance of c⊥ and γ2(c) seems tied to the smallest g such that c⊥ is generated by its codewords of weight g or less. so it seems interesting to classify those codes c for which γ1(c) = γ2(c) as these seem related to tight designs. acknowledgment: the authors thank padraig ó catháin, bill kantor and brian kodalen for useful comments on the work presented here. we are grateful to the referee for several improvements to the manuscript. 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https://doi.org/10.4171/emss/1 https://doi.org/10.4171/emss/1 introduction two parameters radical ideals steiner systems and partial designs symmetric balanced incomplete block designs triple systems strength greater than two conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.864902 j. algebra comb. discrete appl. 8(1) • 1–8 received: 28 december 2019 accepted: 24 august 2020 journal of algebra combinatorics discrete structures and applications reversible dna codes from skew cyclic codes over a ring of order 256 research article yasemin cengellenmis, nuh aydin, abdullah dertli abstract: we introduce skew cyclic codes over the finite ring f2+uf2+vf2+wf2+uvf2+uwf2+vwf2+uvwf2, where u2 = 0, v2 = v, w2 = w, uv = vu, uw = wu, vw = wv and use them to construct reversible dna codes. the 4-mers are matched with the elements of this ring. the reversibility problem for dna 4-bases is solved and some examples are provided. 2010 msc: 94b05, 94b60 keywords: dna codes, skew codes, reversibility 1. introduction it is well known that dna contains genetic program for the biological development of life and has two strands which are linked by watson-crick pairing so that every a is linked with a t and every c with a g, and vice versa, where a,t,c,g are the four bases of a dna sequence. dna computing started in 1994 when adleman showed how to solve a computationally difficult problem (traveling salesman problem, a well-known np-complete problem) by manipulations of dna molecules in [2]. later, more applications of dna codes were discovered such as using dna codes to break a cryptosystem known as des [3, 5], and using dna codewords as high density storage media [16]. devising methods to design dna codes for dna computing has been a major topic of research since the beginning of the century. a block code is called a dna code if it satisfies the following constraints [8, 22]. 1. the hamming constraint for minimum distance, yasemin cengellenmis; department of mathematics, trakya university, edirne, turkey (email: ycengellenmis@gmail.com). nuh aydin; department of mathematics and statistic, kenyon college, gambier,oh, united states (email: aydinn@kenyon.edu). abdullah dertli (corresponding author); department of mathematics, ondokuz mayıs university, samsun, turkey (email: abdullah.dertli@gmail.com). 1 https://orcid.org/0000-0002-8133-9836 https://orcid.org/0000-0002-5618-2427 https://orcid.org/0000-0001-8687-032x y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 2. the reverse-complement constraint, 3. the reverse constraint, and 4. the fixed gc content. in [8], all of these constraints are translated in terms of coding theory. this enabled researchers to use the results from classical coding theory to design codes for dna computation. the last decade witnessed an increased activity in this direction. since dna uses a four-letter alphabet, {a,g,t,c}, it is most natural to use a ring of size 4 in employing classical coding theory techniques for the design of dna codes. later, alphabets of size 4k for k ≥ 1 have been considered (e.g. [1, 4, 6–9, 13, 21? , 22]). it was also observed that having a cyclic structure in dna codes has certain advantages in terms of complexities of algorithms [17, 21]. one of the challenging problems in the field is the reversibility problem [18]. this problem arises from the fact that the pairing of nucleotides in two different strands of a dna sequence is done in opposite direction and reverse order. for example, let us consider the codeword (dna string) gttaggca which corresponds to a codeword (a1,a2). the reverse of (a1,a2) is (a2,a1). however, the vector (a2,a1) corresponds to ggcagtta which is not the reverse of gttaggca. the reverse of gttaggca is acggattg. some authors solved this problem by considering skew cyclic codes. the reversibility problem for dna 8-bases and dna 2s+1k-bases is solved in [10] and [11] respectively by using skew cyclic codes over the finite rings f16 + uf16 + vf16 + uvf16, where u2 = u,v2 = v,uv = vu and f42k [u1, ...,us]/〈u12 − u1, ...,us 2 −us〉 where k,s > 1,uiuj = ujui. motivated by the previous work [10, 11], we study the reversibility problem for dna 4-bases using skew cyclic codes over the finite ring r := f2 + uf2 + vf2 + wf2 + uvf2 + uwf2 + vwf2 + uvwf2 of order 256 where u2 = 0,v2 = v,w2 = w,uv = vu,uw = wu,vw = wv. in [6], cyclic dna codes over r are studied. a map from r to r21 is given where r1 = f2 + uf2 + vf2 + uvf2 and u2 = 0,v2 = v,uv = vu. moreover, cyclic codes of arbitrary length over r satisfying the reverse constraint and reverse complement constraint are studied and a one to one correspondence between the elements of the ring r and sd256 is established where sd256 = {aaaa,...,gggg}. the binary image of a cyclic code over r is also determined. in this paper, by defining a non-trivial automorphism on r, skew cyclic codes over r are introduced. thanks to these type of codes, reversible dna codes are obtained and some examples are provided. 2. preliminaries in [6], the finite ring r = f2 + uf2 + vf2 + wf2 + uvf2 + uwf2 + vwf2 + uvwf2 = {a1 + ua2 + va3 + wa4 + uva5 + uwa6 + vwa7 + uvwa8 : ai ∈ f2, i = 1, 2, ..., 8} with u2 = 0,v2 = v,w2 = w,uv = vu,uw = wu,vw = wv is introduced. the ring r is commutative with characteristic 2 and 256 elements. it can be viewed as r = (f2 + uf2 + vf2 + uvf2) + w(f2 + uf2 + vf2 + uvf2) = r1 + wr1, w2 = w where r1 is the ring f2 + uf2 + vf2 + uvf2, u2 = 0,v2 = v,uv = vu, introduced in [23]. by using the dna alphabet sd4 = {a,t,c,g}, the authors define a correspondence τ between the elements of the finite ring r1 and dna double pairs as in the following table, by means of a gray map from r1 to (f2 + uf2) 2 with u2 = 0. 2 y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 ring elements α dna double pairs τ(α) 0 aa 1 gg u tt v ag uv at 1 + u cc 1 + v ga u + v tc u + uv ta v + uv ac 1 + uv gc 1 + u + uv cg 1 + u + v ct 1 + v + uv gt u + v + uv tg 1 + u + v + uv ca we define a gray map as follows φ : r −→ r21 x + yw 7−→ (x,x + y) where x,y ∈ r1. by using the matching and the gray map above, we get a matching ψ between the elements of r and a set of dna 4-bases sd256 = {aaaa,tttt,....} as follows. ψ : r −→ sd256 x + wy 7−→ ψ(x + wy) = γ(x,x + y) = (τ(x),τ(x + y)) where γ : r21 −→ sd256 (s,t) 7−→ (τ(s),τ(t)) for s,t ∈ r1. that is, ψ = γ ◦φ. 3. skew cyclic codes over r definition 3.1. let b be a finite ring and θ be a non-trivial automorphism on b. a subset c of bn is called a skew cyclic code of length n if c satisfies the following conditions, 1. c is a submodule of bn 2. if c = (c0,c1, ...,cn−1) ∈ c, then σθ(c) = (θ(cn−1),θ(c0), ...,θ(cn−2)) ∈ c, where σθ is the skew cyclic shift operator. by defining a non-trivial automorphism θ on r as follows, we can define skew cyclic codes over r. let θ : r −→ r x + yw 7−→ θ(x + yw) = θ ′ (x + y) + wθ ′ (y) 3 y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 where x,y ∈ r1 and θ ′ is a non-trivial authomorphism on r1 defined by θ ′ : r1 −→ r1 a + bu + v(c + du) 7−→ (a + c + (b + d)u) + v(c + du) the order of each θ and θ ′ is 2. the set of polynomials r[x,θ] = {a0 + a1x + ... + an−1xn−1 : ai ∈ r,n ∈ n} is the skew polynomial ring over r with the usual addition of polynomials and the non-commutative multiplication given by (axi)(bxj) = aθi(b)xi+j. in polynomial representation, a skew cyclic code of length n over r is defined as a left ideal of the quotient ring rθ,n = r[x,θ]/〈xn − 1〉, if the order of θ divides n, that is, if n is even. if the order of θ does not divide n, a skew cyclic code of length n over r is defined as a left r[x,θ]-submodule of rθ,n, since the set rθ,n = r[x,θ]/〈xn − 1〉 = {f(x) + 〈xn − 1〉 : f(x) ∈ r[x,θ]} is a left r[x,θ]-module with the multiplication from left defined by r(x)(f(x) + 〈xn − 1〉) = r(x)f(x) + 〈xn − 1〉 for any r(x) ∈ r[x,θ]. in either case, the following holds. theorem 3.2. let c be a skew cyclic code over r and let f(x) be a polynomial in c of minimal degree. if the leading coefficient of f(x) is a unit in r, then c = 〈f(x)〉, where f(x) is a right divisor of xn − 1. proof. it can be proven similarly to the proof of theorem 4 in [20]. 4. reversible dna codes from skew cyclic codes over r definition 4.1. for x = (x0,x1, ...,xn−1) ∈ rn, the vector (xn−1,xn−2, ...,x1,x0) is called the reverse of x and is denoted by xr. a linear code c of length n over r is said to be reversible if xr ∈ c for every x ∈ c. each element α of r1 and θ ′ (α) are mapped to dna pairs, which are reverses of each other. for example, τ(v) = ag, while τ ( θ ′ (v) ) = ga. this map can be extended to a map γ from r21 to 4-mers as follows, γ(a,b) = (τ(a),τ(b)) where a,b ∈ r1. by means of the map ψ = γ ◦φ, we can find a relationship between skew cyclic codes over r and dna codes. we note that ψ(r) and ψ (θ(r)) are dna reverses of each other. indeed, for r = x+yw ∈ r, we have ψ(r) = γ (φ(x + yw)) = γ (x,x + y) = (τ(x),τ(x + y)) on the other hand, ψ (θ(r)) = ψ ( θ ′ (x + y) + wθ ′ (y) ) = γ ( φ ( θ ′ (x + y) + wθ ′ (y) )) = γ ( θ ′ (x + y),θ ′ (x) ) = ( τ ( θ ′ (x + y) ) ,τ ( θ ′ (x) )) 4 y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 this map can be extended as follows. for any r = (r0, ...,rn−1) ∈ rn, (ψ(r0), ψ(r1), . . . , ψ(rn−1)) r = (ψ(θ(rn−1)), . . . , ψ(θ(r1)), ψ(θ(r0))) example 4.2. if r = u + uv + w(1 + uv) ∈ r, then we get ψ(r) = γ (φ(r)) = γ (u + uv, 1 + u) = (τ (u + uv) ,τ (1 + u)) = (ta,cc) on the other hand, ψ (θ(r)) = ψ ( θ ′ (1 + u) + wθ ′ (1 + uv) ) = γ ◦φ ( θ ′ (1 + u) + wθ ′ (1 + uv) ) = γ ( θ ′ (1 + u),θ ′ (u + uv) ) = ( τ ( θ ′ (1 + u) ) ,τ ( θ ′ (u + uv) )) = (cc,at) definition 4.3. let c be a code of length n over r. if ψ(c)r ∈ ψ(c) for all c ∈ c, then c or equivalently ψ(c) is called a reversible dna code. definition 4.4. let g(x) = a0 + a1x + a2x2 + ... + asxs be a polynomial of degree s over r. g(x) is called a palindromic polynomial if ai = as−i for all i ∈{0, 1, ...,s}. g(x) is called a θ-palindromic polynomial if ai = θ(as−i) for all i ∈{0, 1, ...,s}. as the order of θ is 2, a skew cyclic code of odd length n over r with respect to θ is an ordinary cyclic code. so we will take the length n to be even. the next two theorems show that palindromic and θ-palindromic polynomials generate reversible dna codes. they are analogous to theorem 1 and theorem 2 in [12] stated for codes over a field. theorem 4.5. let c = 〈f(x)〉 be a skew cyclic code of length n over r, where f(x) is a right divisor of xn−1 and deg(f(x)) is odd. if f(x) is a θ-palindromic polynomial, then ψ(c) is a reversible dna code. proof. let f(x) be a θ-palindromic polynomial and f(x) = a0 + a1x + ... + a2s−1x2s−1. so ai = θ(a2s−1−i), for all i = 0, 1, ...,s− 1. let h(x) = h0 + h1x + · · · + h2k−1x2k−1. let bl be the coefficient of xl in h(x)f(x), where l = 0, 1, . . . ,n− 1. for any t < n/2, the coefficient of xt in h(x)f(x) is bt = t∑ j=0 hjθ j(at−j) and the coefficient of xn−t is bn−t = ∑t j=0 h2k−1−jθ 2k−1−j(a2s−1−(t−j)). the polynomial h(x)f(x) = ∑2k−1 d=0 hdx df(x) corresponds to a vector b = (b0,b1, ..., bn−1) ∈ c. the vector ψ(b)r = ((ψ(b0), ..., ψ(bn−1)))r is equal to the vector ψ(z), where the vector z corresponds to the polynomial ∑2k−1 d=0 θ(hd)x 2k−1−df(x). so, ψ(c) is a reversible dna code. theorem 4.6. let c = 〈f(x)〉 be a skew cyclic code of length n over r, where f(x) is a right divisor of xn − 1 and deg(f(x)) is even. if f(x) is a palindromic polynomial, then ψ(c) is a reversible dna code. 5 y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 proof. let f(x) be a palindromic polynomial with even degree so that f(x) = a0 + a1x + ... + a2sx2s and ai = a2s−i, for all i = 0, 1, ...,s. let h(x) = h0 + h1x + ... + h2kx2k. let bl be the coefficient of xl in h(x)f(x), where l = 0, 1, ..,n− 1. for any t < n/2, the coefficient of xt in h(x)f(x) is bt = t∑ j=0 hjθ j(at−j) and the coefficient of xn−t is bn−t = ∑t j=0 h(2k)−jθ (2k)−j(a2s−(t−j)). the polynomial h(x)f(x) = ∑2k d=0 hdx df(x) corresponds to a vector b = (b0,b1, . . . ,bn−1) ∈ c. the vector ψ(b)r = ((ψ(b0), ..., ψ(bn−1)))r is equal to the vector ψ(z), where the vector z corresponds to the polynomial ∑2k d=0 θ(hd)x 2k−df(x). so, ψ(c) is a reversible dna code. the next two theorems show that palindromic and θ-palindromic polynomials come in pairs. they are analogous to theorem 4 and theorem 3 in [12] stated for skew polynomials over a field. theorem 4.7. let xn−1 = h(x)f(x) ∈ r[x,θ], where the degree of f(x) is odd. if f(x) is a θ-palindromic polynomial, then h(x) is a palindromic polynomial. proof. let f(x) = a0 + a1x + ... + a2s−1x2s−1. as the length n is even, then h(x) = h0 + h1x + ... + h2k−1x 2k−1. since f(x) is a θ-palindromic polynomial, then ai = θ(a2s−1−i) for all i = 0, 1, ...,s− 1. let bl be the coefficient of xl in h(x)f(x), where l = 0, 1, ..,n− 1. for any t < n/2, the coefficient of xt in h(x)f(x) is bt = t∑ j=0 hjθ j(at−j) and the coefficient of xn−t is bn−t = ∑t j=0 h2k−1−jθ 2k−1−j(a2s−1−(t−j)). by using the fact that b0 = bn = 0 and bi = 0 for all i = 1, 2, ...,n− 1, it can be shown that hi = h2k−1−i for all i = 0, 1, ..,k − 1 as in the proof of theorem 4 in [12], by induction. a polynomial that is in the center z(r[x,θ]) of r[x,θ] is called a central polynomial. a central polynomial commutes with every element of r[x,θ]. as in the field case, we can prove that xn − 1 is central when n is even. proposition 4.8. the polynomial xn − 1 is central if and only if n is even. proof. let n be even. let s(x) = a0 + a1x + ... + amxm ∈ r[x,θ]. since n is even, θn(a) = a for any a ∈ r. so (xn−1)s(x) = xna0+xna1x+...+xnamxm−s(x) = xna0+θn(a1)xnx+...+θn(am)xnxm−s(x) = (a0 + a1x + ... + amx m)xn − s(x) = s(x)(xn − 1). hence (xn − 1) ∈ z(r[x,θ]). conversely, assume (xn − 1) ∈ z(r[x,θ]). then (xn − 1) commutes with every element in r[x,θ]. in particular, we have (xn − 1)amxm = amxm(xn − 1) for any m and any am ∈ r. as (xn − 1)amxm = θn(am)xn+m −amxm and amxm(xn − 1) = (am)xn+m −amxm, we have θn(am) = am. this implies that n is even. lemma 4.9. let xn − 1 = hf ∈ z(r[x,θ]). then hf = fh in r[x,θ]. proof. since hf is a central element, we have (hf)h = h(hf). so h(fh−hf) = 0. since the leading coefficient of h is a unit, h is not a zero divisor. hence, fh = hf in r[x,θ]. theorem 4.10. let xn−1 = h(x)f(x) ∈ r[x,θ], where the degree of f(x) is even. if h(x) is a palindromic polynomial then so is f(x). 6 y. cengellenmis, n. aydin, a. dertli / j. algebra comb. discrete appl. 8(1) (2021) 1–8 proof. this can be proven similarly to theorem 3 in [12]. since n is even, xn − 1 = hf ∈ z(r[x,θ]) . so we get that any right divisor of xn − 1 is also a left divisor of xn − 1. corollary 4.11. let xn − 1 = h(x)f(x) ∈ r[x,θ], where the degree of f(x) is even. if f(x) is a palindromic polynomial, then h(x) is a palindromic polynomial as well. finally, we give a few examples of palindromic and theta-palindromic polynomials over r. hence, these polynomials generate reversible dna codes. we found these polynomials using magma software [15]. example 4.12. there are at least 576 different factorizations of x4 − 1 in the form x4 − 1 = f(x)h(x) where deg(f(x)) = deg(h(x)) = 2, in the skew polynomial ring over r. of these, 64 of the factorizations involve palindromic polynomials. one of these 64 pairs is f(x) = h(x) = x2 + ux + 1. example 4.13. there are at least 990 different factorizations of x8 − 1 in the form x8 − 1 = h(x)f(x) over r where deg(h(x)) = 3 and deg(f(x)) = 5. of these, 1 factorization involves theta-palindromic polynomials, namely h(x) = x3 + (uw + (uv + (u + 1)))x2 + (uw + (uv + (u + 1)))x + 1 and f(x) = x5 + (uw + (uv + (u + 1)))x4 + (uw + (uv + u))x3 + (uw + (uv + u))x2 + (uw + (uv + (u + 1)))x + 1. example 4.14. the polynomial f(x) = x4 + (v + 1)x3 + (uvw + u)x2 + (v + 1)x + 1 divides x16 − 1 in the skew polynomial ring over r and it is palindromic. hence it generates a reversible dna code. additionally, h(x) = x 16−1 f(x) (division in the skew polynomial ring over r) is also palindromic. example 4.15. there are at least 1600 different factorizations of x12−1 in the form x12−1 = f(x)h(x) in the skew polynomial ring over r where deg(f(x)) = 2 and deg(h(x)) = 10. of these, 144 of factorizations involve palindromic polynomials, one of which is f(x) = x2 + (uv + u)x + 1. 5. conclusion we have shown that skew cyclic codes over the ring r = f2 + uf2 + vf2 + wf2 + uvf2 + uwf2 + vwf2 + uvwf2, where u2 = 0,v2 = v,w2 = w,uv = vu,uw = wu,vw = wv, can be used to construct the reversible dna codes. we have also provided several specific examples of such codes. references [1] t. abualrub, a. ghrayeb, x. n. zeng, construction of cyclic codes over gf(4) for dna computing, j. frankl. inst. 343(4-5) (2006) 448–457. 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(this lower bound of 5 was later improved by currie and eggleton to 6.) the de bruijn erdős theorem (which relies on the axiom of choice) then guarantees the existence, for each �, of a finite subgraph h� of g� such that χ(h�) ≥ 5. in this paper, we explicitly construct such finite graphs h�. we find that the number of vertices needed to create such a graph is no more than 2π(15 + 14�−1)2. our proof can be done by hand without the aid of a computer. 2010 msc: 05c15, 05c10 keywords: chromatic number of the plane, �-unit distance graph 1. introduction the much-studied “chromatic number of the plane” problem asks for the smallest number of colors needed to color every point in the plane in such a way that no two points of unit distance apart from each other have the same color. one can easily think of this as a graph coloring problem by forming a graph g0 whose vertex set is r2, where two vertices p and q are adjacent if and only if d(p,q) = 1. what is χ(g0)? here d(p,q) denotes the euclidean distance from p to q, and χ(g) denotes the chromatic number of a graph g. the best known bounds are 5 ≤ χ(g0) ≤ 7. (1) the book [7] thoroughly details the history of this problem up to 2009. for many decades the bestknown lower and upper bounds were 4 and 7, respectively. the lower bound of 5 was first proved in mike krebs; department of mathematics, california state university los angeles, 5151 state university drive, los angeles, california 90032, usa (email: mkrebs@calstatela.edu) 161 https://orcid.org/0000-0002-5097-2647 m. krebs / j. algebra comb. discrete appl. 8(3) (2021) 161–166 [4], in which de grey explicitly constructs a finite unit-distance graph that cannot be properly 4-colored. unfortunately, the proof requires extensive machine computation. in [2], exoo formulates the following variation on this problem. let � be a real number with 0 < � < 1, and let g� be the graph whose vertex set is r2, where two vertices p and q are adjacent if and only if 1− � ≤ d(p,q) ≤ 1 + �. we now seek χ(g�). one immediately has the inequality χ(g0) ≤ χ(g�), and so by (1) we have that χ(g�) ≥ 5 for all � > 0. exoo conjectures that in fact χ(g�) = 7 for all �. in [2], exoo shows that χ(g�) is finite whenever 0 < � < 1. in [5], grytczuk et al. prove that whenever 0 < � < 1, we have that χ(g�) ≥ 5. (we note that the stronger result χ(g�) ≥ 6 was proved in [1].) their proof is indirect and nonconstructive; in section 2, we present a simplified version of their (rather elegant) proof. the classical de bruijn erdős theorem states that any graph k with finite chromatic number ` must possess a finite subgraph with chromatic number `. this theorem depends on the axiom of choice. following [2], we define an �-unit distance graph to be a subgraph of g�. (this mirrors the use of the term unit distance graph to refer to a subgraph of g.) together, the three theorems mentioned in this paragraph imply the existence, for each �, of a finite �-unit distance graph h� with χ(h�) ≥ 5. in section 3, we present our main theorem (theorem 3.2), which gives an explicit construction of finite �-unit distance graphs h� with χ(h�) ≥ 5. we thereby obtain an upper bound of 2π(15+14�−1)2 for the number of vertices in h�. this upper bound is o(n2), where � = n−1 and n ranges over the positive integers. the results of this paper and of [5] are, of course, inferior to de grey’s landmark theorem, but to their credit the proofs do not require machine computation. it would be tempting to use some sort of limiting argument, letting � → 0 in the inequality χ(g�) ≥ 5, to try to obtain a proof (one that does not require computer assistance) that χ(g0) ≥ 5. the following example illustrates the difficulties in such a line of attack. let j be the graph whose vertex set is r, where two vertices p and q are adjacent if and only if |p − q| = 1. note that χ(j) = 2, because the mapping x 7→ bxc modulo 2 is a proper 2-coloring of j. here bxc denotes the largest integer less than or equal to x. whenever 0 < � < 1, let j� be the graph whose vertex set is r, where two vertices p and q are adjacent if and only if 1−� ≤ |p−q| ≤ 1 +�. for any �, let n be a positive integer such that n−1 ≤ �. then j� has a (2n + 1)-cycle with vertices 0,1 + n−1,2(1 + n−1), . . . ,(n−1)(1 + n−1),n + 1,n,n−1,n−2, . . . ,2,1. it follows that χ(j�) ≥ 3. 2. a lower bound of 5 for the chromatic number of g� we begin by establishing a basic lemma that will be used twice later on. given a coloring of a graph, we say that a set of three distinct vertices is a monochromatic triangle if any two of them are adjacent to each other (that is, they induce a complete graph k3), and they all have the same color. lemma 2.1. let g be a graph. suppose that for any 2-coloring of g, there is a monochromatic triangle in g. then χ(g) ≥ 5. proof. assume that there is a proper 4-coloring c: v (g) →{1,2,3,4} of g. define c′ : v (g) →{1,2} by c′(v) = 1 if c(v) ∈{1,2} and c′(v) = 2 if c(v) ∈{3,4}. then g has a monochromatic triangle {p,q,r} with respect to c′. but then {p,q,r} uses only two colors from the coloring c, which is impossible. the fact that χ(g�) ≥ 5 for all � > 0 appears as theorem 2 in [5]. the proof given in [5] invokes theorem 2 from [6], from which the desired bound follows quickly. in fact, a less powerful tool—namely, a special case of theorem 1 from [6]—suffices. we present here a slightly simplified proof. we do so not only to keep this paper more self-contained but also to motivate the idea of the proof of our main theorem in section 3. 162 m. krebs / j. algebra comb. discrete appl. 8(3) (2021) 161–166 theorem 2.2 ([5], theorem 2). let � be a real number such that 0 < � < 1. then χ(g�) ≥ 5. proof. partition r2 into two sets w and b. vertices in w we call white; those in b we call black. by lemma 2.1, it suffices to show that either w or b contains a monochromatic triangle. this clearly holds if either w = r2 or b = r2, so assume that w and b are both proper and nonempty. because r2 is connected in the euclidean topology, w has a boundary point p. then there exists a point q with d(p,q) ≤ �/2 such that p and q have opposite colors. let c be the circle of radius 1 with center p. if c ⊂ w or c ⊂ b, then the points p + (1,0) and p + (1/2, √ 3/2), together with either p or q, form a monochromatic triangle. otherwise, because c is connected as a topological subspace of r2, it follows that w has a boundary point r with r ∈ c. then there exists a point s with d(r,s) ≤ �/2 such that r and s have opposite colors. choose a point t of distance 1 from both p and r. then t shares a color with one of p and q and one of r and s; these three points form a monochromatic triangle. 3. finite �-unit distance graphs with chromatic number ≥ 5 carefully reading the proof of theorem 2.2, we see that we do not require the full power of the point p being in the boundary of w . for we do not need to find points of color opposite to that of p arbitrarily close to p; we need only find a single such point of distance ≤ �/2 from p. the same goes for r. this observation allows us to discretize the argument, replacing topological connectedness with graphical connectedness. we now make this precise. lemma 3.1. let h and l be graphs with a common vertex set v such that |v | ≥ 2. suppose that l is connected. moreover, suppose that for any two l-adjacent vertices x and y, there exists a connected induced subgraph j of l with order ≥ 2 such that: 1. every vertex in j is h-adjacent to both x and y, and 2. whenever a and b are l-adjacent vertices in j, there exists a vertex z of j such that z is h-adjacent to both a and b. then χ(h) ≥ 5. proof. partition v into two sets w and b, as in the proof of theorem 2.2. by lemma 2.1, it suffices to show that either w or b contains a monochromatic triangle with respect to h. if v ⊂ w or v ⊂ b, then we are done, because h contains k3 as a subgraph. so assume that w and b are both proper and nonempty. because l is connected, there must be two l-adjacent vertices x and y of opposite colors. take j as in the statement of the lemma. first suppose that j ⊂ w . because j is a connected induced subgraph of l with order ≥ 2, we know that there exist l-adjacent vertices a and b in j. let z be as in (2). then z, a, and one of x and y form a monochromatic triangle with respect to h. a similar argument works if j ⊂ b. so we now assume that j contains both black and white points. because j is a connected induced subgraph of l with order ≥ 2, it follows that there exist l-adjacent vertices a and b in j of opposite color. let z be as in (2). figure 1 illustrates the situation. then z, one of a and b, and one of x and y form a monochromatic triangle with respect to h. theorem 3.2. let � be a real number such that 0 < � < 1. let s = � 14 √ 2 . let v = {(as,bs) | a,b ∈ z and (as)2 + (bs)2 ≤ (1 + �)2}. define h� to be the graph with vertex set v where two vertices p and q are adjacent if and only if 1− � ≤ d(p,q) ≤ 1 + �. then χ(h�) ≥ 5. moreover, h� has no more than 2π(15 + 14�−1)2 vertices. 163 m. krebs / j. algebra comb. discrete appl. 8(3) (2021) 161–166 figure 1. the points x,y,a,b, and z. edges are solid in l, dashed in h. proof. let l be the graph with vertex set v where two vertices p and q are adjacent if and only if 0 < d(p,q) ≤ 4�/7. we will show that h� and l satisfy the conditions of lemma 3.1. the set v contains the points (0,0) and (s,0), so |v | ≥ 2. note that l is connected, for the following reason: if b > 0, then (as,bs) is l-adjacent to (as,(b−1)s). if b < 0, then (as,bs) is l-adjacent to (as,(b + 1)s). if a > 0, then (as,0) is l-adjacent to ((a− 1)s,0). and if a < 0, then (as,0) is l-adjacent to ((a + 1)s,0). so we can find a path in l from any vertex to (0,0) by first traveling vertically until we reach the x-axis, then moving horizontally until we reach the origin. given two l-adjacent vertices x and y, we now construct an induced subgraph j with the properties required in lemma 3.1. take x = (as,bs). without loss of generality, we may assume that x 6= (0,0). let p1 be the unique point of distance 1 from x such that p1 lies on the ray from x to (0,0). for any point q in the plane other than x, let α(q) be the signed measure of the angle ∠qxp1. let p2 and p3 be the unique points in the plane of distance 1 from x such that α(p2) = −π/3 and α(p3) = π/3. let a be the arc, running from p2 to p3, of the circle of radius 1 centered at x. note that a subtends an angle of 2π/3. to construct j, the essential idea is to find a set of vertices of v that discretely approximates a. we now make this precise. let o = (0,0). let b(o,1 + �) be the open disk of radius 1 + � centered at (0,0). observe that if an open disk d of radius �/7 is contained in b(o,1 + �), then d contains an element of v . (the value of s was chosen for this reason. indeed, we could have taken s to be any positive real number less than � 7 √ 2 .) let q be any point on the arc a. applying the law of cosines to the triangle qxo, we find that (1 + �) − d(q,o) > �/7. thus the open disk of radius �/7 centered at q is contained in b(o,1 + �). cover a with finitely many open disks of radius �/7, each centered at a point of a. for each disk in this covering, choose an element of v . let j be the subgraph of l induced by this collection of elements, as shown in figure 2. the union of two intersecting disks of radius �/7 has diameter no greater than 4�/7. this fact, together with the fact that the disks used to define j cover a, imply that j is connected. we have that d(p1,p2) = 1. let v1,v2 be vertices of j such that d(v1,p1) < 2�/7 and d(v2,p2) < 2�/7. hence v1 6= v2, so j contains at least two distinct vertices. let v be any vertex of j. then there is some point q ∈ a such that d(v,q) < �/7. hence 1− � < 1− �/7 < d(x,v) < 1 + �/7 < 1 + �, so x and v are h�-adjacent. then using that d(x,y) < 4�/7, together with triangle inequality, we see that y and v are also h�-adjacent. finally, let a and b be l-adjacent vertices in j. we know that there is some point q ∈ a such that d(a,q) < �/7. because a subtends an angle of 2π/3, there exists a point r ∈ a such that d(q,r) = 1. (to find r, move either clockwise or counterclockwise along a by an angle of π/3.) let z be a vertex of 164 m. krebs / j. algebra comb. discrete appl. 8(3) (2021) 161–166 figure 2. construction of j j with d(r,z) < 2�/7. then 1 − 3�/7 ≤ d(a,z) ≤ 1 + 3�/7, so a and z are h�-adjacent. the fact that d(a,b) ≤ 4�/7 then implies that b and z are h�-adjacent, too. therefore, by lemma 3.1, we have that χ(h�) ≥ 5. estimating |h�| (the number of vertices in h�) is essentially the gauss circle problem. we obtain the simple upper bound in the statement of the theorem via a standard method. for each vertex (as,bs) in h�, consider the square with vertices (as,bs),((a+1)s,bs),(as,(b+1)s),((a+1)s,(b+1)s). the union u of all such squares has area |h�|s2. but u is contained in the disk of radius 1 + � + s √ 2 centered at the origin, so |h�|s2 ≤ π(1 + � + s √ 2)2. from this we get that |h�| ≤ 2π(15 + 14�−1)2. we remark that the graph h� cannot be realized as a unit distance graph, because it contains a complete bipartite graph k2,3 as a subgraph. indeed, no graph h satisfying the conditions of lemma 3.1 is a unit distance graph, for the same reason: the vertices a,b in the statement of the lemma have x,y, and z as common neighbors, but in a unit distance graph two distinct vertices can have no more than 2 common neighbors. 4. future directions many possible approches to the chromatic number of the plane problem remain to be investigated. denote by χc(r2) the “closure chromatic number of the plane,” that is, the smallest number of sets into which r2 can be partitioned such that the closure of none of these sets contains two points of distance 1 from each other. nielsen’s argument (the proof of theorem 2.2) shows that χc(r2) ≥ 5. does the argument in [1] show that χc(r2) ≥ 6? more generally, if χ(g�) ≥ k for all � > 0, does this imply that χc(r2) ≥ k? we remark nielsen’s argument holds for any topology in which every unit circle (and hence the entire plane) is connected. the definition of χc(r2) extends to arbitrary topologies. if χc(r2) ≤ k for every 165 m. krebs / j. algebra comb. discrete appl. 8(3) (2021) 161–166 topology in some suitable collection of topologies on r2, does this imply that χ(r2) ≤ k? the closure chromatic number of the plane is related to the “measurable chromatic number of the plane,” denoted χm(r2), which equals the smallest number of lebesgue measurable sets into which r2 can be partitioned such that none of these sets contains two points of distance 1 from each other. in [3], falconer showed that χm(r2) ≥ 5 — yet another result superseded by [4]. the fact that χm(r2) ≤ χc(r2) follows immediately from the facts that closed sets are measurable and that the set difference of measurable sets is measurable. it may be worth looking into whether the arguments of nielsen or of currie and eggleton shed any light on further connections between χc(r2) and χm(r2). in section 1, we discussed the difficulties in simply taking the limit of the chromatic numbers as � → 0. we can view this set-up as starting with too many edges (those that correspond to distances 6= 1), then gradually removing edges until we are left only with those edges we wish to keep (those that correspond to distance 1). one way to rectify the situation is to instead start with too few edges, then gradually add edges until all desired edges appear. the corresponding limit-of-the-chromatic numbers argument will go through in this case. for example, rather than expand the set of allowable distances, we can contract the set of allowable angles, perhaps as follows. let θ ∈ [0,π/2). given two distinct points p and q in r2, let ` be the line that contains both. we declare p and q to be adjacent if and only if the distance between them is 1 and 0 ≤ α ≤ θ, where α is the unique angle ≤ π/2 formed by ` and the x-axis. then the chromatic number of the plane equals the limit as θ → π/2 of the chromatic number of the “angle-restricted graph” corresponding to θ. acknowledgment: the author would like to thank the referee for many helpful suggestions. references [1] j. d. currie and r. b. eggleton, chromatic properties of the euclidean plane, arxiv:1509.03667 11 sep 2015. [2] g. exoo, �-unit distance graphs, discrete comput. geom. 33(1) (2005) 117-123. [3] k. j. falconer, the realization of distances in measurable subsets covering rn, j. combin. theory ser. a 31(2) (1981) 184-189. [4] a. d. de grey, the chromatic number of the plane is at least 5, geombinatorics 28(1) (2018) 18–31. [5] j. grytczuk, k. junosza-szaniawski, j. sokół, k. węsek, fractional and j-fold coloring of the plane, discrete comput. geom. 55(3) (2016) 594-609. [6] m. j. nielsen, approximating monochromatic triangles in a two-colored plane, acta math. hungar. 74(4) (1997) 279-286. [7] a. soifer, the mathematical coloring book, springer, new york (2009). 166 https://arxiv.org/abs/1509.03667 https://arxiv.org/abs/1509.03667 https://doi.org/10.1007/s00454-004-1092-8 https://doi.org/10.1016/0097-3165(81)90014-5 https://doi.org/10.1016/0097-3165(81)90014-5 https://doi.org/10.1007/s00454-016-9769-3 https://doi.org/10.1007/s00454-016-9769-3 https://doi.org/10.1023/a:1006520219668 https://doi.org/10.1023/a:1006520219668 introduction a lower bound of 5 for the chromatic number of g finite -unit distance graphs with chromatic number 5 future directions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.790748 j. algebra comb. discrete appl. 7(3) • 229–236 received: 18 april 2020 accepted: 28 may 2020 journal of algebra combinatorics discrete structures and applications classification of optimal quaternary hermitian lcd codes of dimension 2 research article keita ishizuka abstract: hermitian linear complementary dual codes are linear codes whose intersections with their hermitian dual codes are trivial. the largest minimum weight among quaternary hermitian linear complementary dual codes of dimension 2 is known for each length. we give the complete classification of optimal quaternary hermitian linear complementary dual codes of dimension 2. 2010 msc: 94b05 keywords: linear complementary dual code, hermitian linear complementary dual code, optimal codes 1. introduction let f4 := {0, 1, ω, ω2} be the finite field of order four, where ω satisfies ω2 +ω + 1 = 0. the conjugate of x ∈ f4 is defined as x := x2. a quaternary [n, k, d] code is a linear subspace of fn4 with dimension k and minimum weight d. throughout this paper, we consider only linear quaternary codes and omit the term “linear quaternary”. given a code c, a vector c ∈ c is said to be a codeword of c. the weight of a codeword c is denoted by wt(c). let u := (u1, u2, . . . , un), v := (v1, v2, . . . , vn) be vectors of fn4 . the hermitian inner product is defined as (u, v)h := u1v1 + u2v2 + · · · + unvn. given a code c, the hermitian dual code of c is c⊥h := {x ∈ fn4 | (x, y)h = 0 for all y ∈ c}. a generator matrix of the code c is any matrix whose rows form a basis of c. moreover, a generator matrix of the hermitian dual code c⊥h is said to be a parity check matrix of c. given a matrix g, we denote the transpose of g by gt and the conjugate of g by g. hermitian linear complementary dual codes, hermitian lcd codes for short, are codes whose intersections with their hermitian dual codes are trivial. the concept of lcd codes was invented by massey [7] in 1992. lcd codes have been applied in data storage, communication systems and cryptography. for example, it is known that lcd codes can be used against side-channel attacks and keita ishizuka; research center for pure and applied mathematics graduate school of information sciences, tohoku university, sendai 980–8579, japan (email: keita.ishizuka.p5@dc.tohoku.ac.jp). 229 https://orcid.org/0000-0001-5943-6245 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 fault injection attacks [3]. we note that a code c is a hermitian lcd code if and only if its generator matrix g satisfies det gg t 6= 0 [5]. two codes c, c′ are equivalent if one can be obtained from the other by a permutation of the coordinates and a multiplication of any coordinate by a nonzero scalar. we denote the equivalence of two codes c, c′ by c ' c′. let g, g′ be generator matrices of two codes c, c′ respectively. it is known that c ' c′ if and only if g can be obtained from g′ by an elementary row operation, a permutation of the columns and multiplication of any column by a nonzero scalar. it was shown in [6] that the upper bound of the minimum weight of the hermitian lcd [n, 2, d] codes is given as follows: d ≤ {⌊ 4n 5 ⌋ if n ≡ 1, 2, 3 (mod 5),⌊ 4n 5 ⌋ − 1 otherwise. (1) also, it was proved that for all n 6= 1, there exists a hermitian lcd [n, 2, d] code which meets this upper bound. we say that a hermitian lcd [n, 2, d] code is optimal if it meets this upper bound. it was shown in [4] that any code over fq2 is equivalent to some hermitian lcd code for q ≥ 3. furthermore, it was proved in [6] that a hermitian lcd code leads to a construction of a maximal-entanglement entanglement-assisted quantum error correcting code. motivated by the results, we are concerned with the complete classification of optimal hermitian lcd codes of dimension 2. this paper is organized as follows. in section 2, we present a method to construct optimal hermitian lcd codes of dimension 2, including all inequivalent codes. also, a method to classify optimal hermitian lcd codes of dimension 2 is given. in section 3, we classify optimal hermitian lcd codes of dimension 2. up to equivalence, the complete classification of optimal hermitian lcd codes of dimension 2 is given. it is shown that all inequivalent codes have distinct weight enumerators, which is used for the classification. 2. classification method let 0n be the zero vector of length n and 1n be the all-ones vector of length n. let (a0, a1, a2, a3, a4, a5) be a tuple of nonnegative integers. we introduce the following notation: g(a0, a1, a2, a3, a4, a5) := ( 1 0 0a0 0a1 1a2 1a3 1a4 1a5 0 1 0a0 1a1 0a2 1a3 ω1a4 ω 21a5 ) . we denote by c(a0, a1, a2, a3, a4, a5) the code whose generator matrix is g(a0, a1, a2, a3, a4, a5). by the same argument as in [1], we obtain the following lemma. lemma 2.1. given a code c, define c∗ := {(x, 0) | x ∈ c}. let c∗n,k denote the set of all inequivalent hermitian lcd [n, k] codes c such that the minimum weight of c⊥h is 1. then there exists a set cn−1,k of all inequivalent hermitian lcd [n− 1, k] codes such that c∗n,k = {c ∗ | c ∈cn−1,k}. we assume a0 = 0 by lemma 2.1 and omit a0. furthermore, throughout this paper, we use the following notations: g(a) := g(a1, a2, a3, a4, a5), c(a) := c(a1, a2, a3, a4, a5), (2) respectively, to save space. proposition 2.2. let c be an [n, 2, d] code. then there exist nonnegative integers a1, a2, a3, a4, a5 such that c ' c(a) and 1 + a2 + a3 + a4 + a5 = d. proof. let g be a generator matrix of the code c. by multiplying rows by some non-zero scalars, g is changed to a generator matrix which consists only of the columns of g(a). permuting the columns, 230 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 g(a) is obtained from g. hence it holds that c ' c(a). since the minimum weight of c is d, we may assume that the first row of g is a codeword with weight d, which yields 1 + a2 + a3 + a4 + a5 = d. given a code c(a), we may assume without loss of generality that g(a) satisfies 1 + a2 + a3 + a4 + a5 = d, (3) by proposition 2.2. this assumption on a generator matrix reduces computations later. lemma 2.3. let c be an [n, 2, d] code c(a). then c is a hermitian lcd code if and only if c satisfies the following conditions: a1 = n−d− 1, (4) a2 ≤ n−d− 1, (5) a3, a4, a5 ≤ n−d, (6){ a3 + a4 + a5 + a3a4 + a4a5 + a5a3 6≡ 0 (mod 2) if d is even, a3a4 + a4a5 + a5a3 6≡ n−d (mod 2) otherwise. (7) proof. suppose c is a hermitian lcd [n, 2, d] code. let g be a generator matrix of the code c. let r1, r2 be the first and second rows of g respectively. the number of columns of g equals to the length n. thus, it holds that 2 + a1 + a2 + a3 + a4 + a5 = n. (8) since the minimum weight of c is d, the following holds: wt(r2) ≥ d, wt(r1 + r2) ≥ d, wt(r1 + ωr2) ≥ d, wt(r1 + ω 2r2) ≥ d. by (3), we have wt(r1) = d. substituting (8) in each equation, we obtain (4) through (6). the code c is a hermitian lcd code if and only if det gg t = (r1, r1)h(r2, r2)h − (r2, r1)h(r1, r2)h 6= 0, where (r1, r1)h = 1 + a2 + a3 + a4 + a5 = d, (r1, r2)h = a3 + ωa5 + ω 2a4 = a3 + a4 + ω(a4 + a5), (r2, r1)h = a3 + ωa4 + ω 2a5 = a3 + a5 + ω(a4 + a5), (r2, r2)h = 1 + a1 + a3 + a4 + a5. here we regard n, d, a3, a4, a5 as elements of f4. therefore, (4) through (7) hold if c is a hermitian lcd [n, 2, d] code and vice versa. given an [n, 2, d] code c(a), we define the following: bi := (n−d) −ai for 1 ≤ i ≤ 5. (9) lemma 2.4. let c be an [n, 2, d] code c(a). then c is a hermitian lcd code if and only if c satisfies the following conditions: b1 = 1, (10) b2 ≥ 1, (11) b3, b4, b5 ≥ 0,{ b3 + b4 + b5 + b3b4 + b4b5 + b5b3 6≡ 0 (mod 2) if d is even, b3b4 + b4b5 + b5b3 6≡ 0 (mod 2) otherwise. 231 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 proof. the result follows from lemma 2.3. given an [n, 2, d] code c(a), we define the following: ∆ := 4n− 5d. (12) lemma 2.5. let c be a hermitian lcd [n, 2, d] code. then c is optimal if and only if the value of ∆ with respect to n is given as follows: ∆ =   5 if n ≡ 0 (mod 5), 4 if n ≡ 1 (mod 5), 3 if n ≡ 2 (mod 5), 2 if n ≡ 3 (mod 5), 6 if n ≡ 4 (mod 5). proof. the result follows from (12). lemma 2.6. let c be a code c(a). if c is a hermitian lcd code, then it holds that 0 ≤ b3, b4, b5 ≤ ∆. proof. substituting b2, b3, b4, b5, ∆ in (3), we obtain b2 = ∆ + 1 − (b3 + b4 + b5). (13) combining with (11), we obtain b3 +b4 +b5 ≤ ∆. since 0 ≤ b3, b4, b5, it follows that 0 ≤ b3, b4, b5 ≤ ∆. lemma 2.7. c(a) ' c(a1, a2, a5, a3, a4). (14) proof. multiply the second row of g(a) by ω. permuting the columns, the result follows. note that we may assume that the nonzero entry of a column is 1, provided that the entry of the other column is 0. by lemma 2.7, we may assume a3 ≥ a4, a5. notice that a3 ≥ a4, a5 if and only if b3 ≤ b4, b5 by (9). 3. optimal hermitian lcd codes of dimension 2 by lemmas 2.4 through 2.7, it suffices to calculate all b3, b4, b5 satisfying 0 ≤ b3 ≤ b4, b5 ≤ ∆, (15){ b3 + b4 + b5 + b3b4 + b4b5 + b5b3 6= 0 if d is even, b3b4 + b4b5 + b5b3 6= 0 otherwise, (16) in order to obtain optimal hermitian lcd codes of dimension 2, including all inequivalent codes. notice that b1, b2 are obtained by (10), (13) respectively. our computer search found all integers b3, b4, b5 satisfying (15) and (16). this calculation was done by magma [2]. recall that a1, a2, a3, a4, a5 are obtained from b1, b2, b3, b4, b5 by (9). for optimal hermitian lcd codes of dimension 2, the integers a1, a2, a3, a4, a5 are listed in table 1, where the rows are in lexicographical order with respect to a1, a2, a3, a4, a5, and m is a nonnegative integer. 232 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 table 1: optimal hermitian lcd codes of dimension 2 n code (a1, a2, a3, a4, a5) m n = 5m c5m,1 (m, m, m, m, m−2) m ≥ 2 c5m,2 (m, m, m, m−2, m) m ≥ 2 c5m,3 (m, m−1, m + 1, m, m−2) m ≥ 2 c5m,4 (m, m−1, m + 1, m−2, m) m ≥ 2 c5m,5 (m, m−1, m, m, m−1) m ≥ 1 c5m,6 (m, m−1, m, m−1, m) m ≥ 1 c5m,7 (m, m−2, m, m, m) m ≥ 2 c5m,8 (m, m−3, m + 1, m, m) m ≥ 3 n = 5m + 1 c5m+1,1 (m, m, m + 1, m, m−2) m ≥ 2 c5m+1,2 (m, m, m + 1, m−2, m) m ≥ 2 c5m+1,3 (m, m, m, m, m−1) m ≥ 1 c5m+1,4 (m, m, m, m−1, m) m ≥ 1 c5m+1,5 (m, m−1, m + 1, m + 1, m−2) m ≥ 2 c5m+1,6 (m, m−1, m + 1, m, m−1) m ≥ 1 c5m+1,7 (m, m−1, m + 1, m−1, m) m ≥ 1 c5m+1,8 (m, m−1, m + 1, m−2, m + 1) m ≥ 2 c5m+1,9 (m, m−2, m + 1, m, m) m ≥ 2 c5m+1,10 (m, m−3, m + 1, m + 1, m) m ≥ 3 c5m+1,11 (m, m−3, m + 1, m, m + 1) m ≥ 3 n = 5m + 2 c5m+2,1 (m, m, m, m, m) m ≥ 0 c5m+2,2 (m, m−1, m + 1, m, m) m ≥ 1 n = 5m + 3 c5m+3,1 (m, m−1, m + 1, m + 1, m) m ≥ 1 c5m+3,2 (m, m, m + 1, m, m) m ≥ 0 c5m+3,3 (m, m−1, m + 1, m, m + 1) m ≥ 1 n = 5m + 4 c5m+4,1 (m + 1, m + 1, m + 1, m + 1, m−2) m ≥ 2 c5m+4,2 (m + 1, m + 1, m + 1, m, m−1) m ≥ 1 c5m+4,3 (m + 1, m + 1, m + 1, m−1, m) m ≥ 1 c5m+4,4 (m + 1, m + 1, m + 1, m−2, m + 1) m ≥ 2 c5m+4,5 (m + 1, m + 1, m + 2, m + 1, m−3) m ≥ 3 c5m+4,6 (m + 1, m + 1, m + 2, m−1, m−1) m ≥ 1 c5m+4,7 (m + 1, m + 1, m + 2, m−3, m + 1) m ≥ 3 c5m+4,8 (m + 1, m, m + 1, m, m) m ≥ 0 c5m+4,9 (m + 1, m, m + 2, m + 1, m−2) m ≥ 2 c5m+4,10 (m + 1, m, m + 2, m + 2, m−3) m ≥ 3 c5m+4,11 (m + 1, m, m + 2, m, m−1) m ≥ 1 c5m+4,12 (m + 1, m, m + 2, m−1, m) m ≥ 1 c5m+4,13 (m + 1, m, m + 2, m−2, m + 1) m ≥ 2 c5m+4,14 (m + 1, m, m + 2, m−3, m + 2) m ≥ 3 c5m+4,15 (m + 1, m−1, m + 1, m + 1, m) m ≥ 1 c5m+4,16 (m + 1, m−1, m + 1, m, m + 1) m ≥ 1 c5m+4,17 (m + 1, m−1, m + 2, m + 1, m−1) m ≥ 1 c5m+4,18 (m + 1, m−1, m + 2, m−1, m + 1) m ≥ 1 c5m+4,19 (m + 1, m−2, m + 2, m + 1, m) m ≥ 2 c5m+4,20 (m + 1, m−2, m + 2, m + 2, m−1) m ≥ 2 c5m+4,21 (m + 1, m−2, m + 2, m, m + 1) m ≥ 2 c5m+4,22 (m + 1, m−2, m + 2, m−1, m + 2) m ≥ 2 c5m+4,23 (m + 1, m−3, m + 2, m + 1, m + 1) m ≥ 3 c5m+4,24 (m + 1, m−4, m + 2, m + 1, m + 2) m ≥ 4 c5m+4,25 (m + 1, m−4, m + 2, m + 2, m + 1) m ≥ 4 lemma 3.1. suppose a3 is a positive integer. then c(a) ' c(a1, a3 −1, a2 + 1, a5, a4). (17) proof. add the second row of g(a) to the first row. permuting the columns, the result follows. recall that c(a) is defined in (2). 233 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 table 2. equivalent optimal hermitian lcd codes of dimension 2 n code n = 5m c5m,7 '17,14 c5m,6 '14 c5m,5 c5m,8 '17,14,17 c5m,3 '17 c5m,2 '14 c5m,1 '17 c5m,4 n = 5m + 1 c5m+1,9 '17,14,17 c5m+1,6 '17 c5m+1,4 '14 c5m+1,3 '17 c5m+1,7 c5m+1,11 '17,14,17 c5m+1,5 '17,14 c5m+1,1 '18 c5m+1,2 '14,17 c5m+1,8 '17,14,17 c5m+1,10 n = 5m + 2 c5m+2,1 '17 c5m+2,2 n = 5m + 3 c5m+3,1 '17,14 c5m+3,2 '14,17 c5m+3,3 n = 5m + 4 c5m+4,8 '14,17 c5m+4,16 '14 c5m+4,15 c5m+4,6 '14,17 c5m+4,22 '14 c5m+4,20 c5m+4,21 '17,14,17 c5m+4,17 '17,14,17 c5m+4,12 '17 c5m+4,2 '18 c5m+4,3 '17 c5m+4,11 '17,14,14,17 c5m+4,19 '17,14,14,17 c5m+4,18 c5m+4,23 '17,14,17 c5m+4,9 '17 c5m+4,4 '18 c5m+4,1 '17 c5m+4,13 c5m+4,24 '17,14,17 c5m+4,10 '17,14 c5m+4,5 '18 c5m+4,7 '14,17 c5m+4,14 '17,14,17 c5m+4,25 lemma 3.2. suppose 1 + a1 + a3 + a4 + a5 = d. then c(a) ' c(a2, a1, a3, a5, a4). (18) proof. interchange the first row and the second row of g(a). permuting the columns, the result follows. by lemmas 2.7 through 3.2, we have the equivalences among some codes listed in table 1, which are displayed in table 2. note that c 'i c′ denotes the two codes c, c′ are equivalent by (i). also, c 'i,j c′ denotes that, given two codes c, c′, there exists a code c′′ such that c 'i c′′ 'j c′. table 3 gives the weight enumerators of representatives in table 2. the weight enumerator is given by 1 + 3ywt(r1) + 3ywt(r2) + 3ywt(r1+r2) + 3ywt(r1+ωr2) + 3ywt(r1+ω 2r2), where r1, r2 be the first and second rows of g(a) respectively. since the weight enumerators are distinct, the codes in table 3 are inequivalent. table 4 gives the classification of optimal hermitian lcd codes of dimension 2, with the case where n is so small that some codes in table 1 do not exist. recall that we have assumed a0 = 0 by lemma 2.1. it follows from (1) that there exists an optimal hermitian lcd [n, 2] code c such that the minimum weight of c⊥h equals to 1 if and only if n ≡ 4 (mod 5). consequently, we obtain the following theorem. theorem 3.3. (i) up to equivalence, there exist two optimal hermitian lcd [5m, 2, 4m − 1] codes for every integer m with m ≥ 2. (ii) up to equivalence, there exist two optimal hermitian lcd [5m + 1, 2, 4m] codes for every integer m with m ≥ 2. (iii) up to equivalence, there exists a unique optimal hermitian lcd [5m + 2, 2, 4m + 1] code for every integer m with m ≥ 0. (iv) up to equivalence, there exists a unique optimal hermitian lcd [5m + 3, 2, 4m + 2] code for every integer m with m ≥ 0. (v) up to equivalence, there exist six optimal hermitian lcd [5m + 4, 2, 4m + 2] codes for every integer m with m ≥ 3. one of them is the code such that the minimum weight of the hermitian dual code is 1. 234 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 table 3. weight enumerators of representatives n code weight enumerator n = 5m c5m,7 1 + 3y 4m−1 + 9y4m + 3y4m+1 c5m,8 1 + 6y 4m−1 + 6y4m + 3y4m+2 n = 5m + 1 c5m+1,9 1 + 6y 4m + 6y4m+1 + 3y4m+2 c5m+1,1 1 + 9y 4m + 3y4m+1 + 3y4m+3 n = 5m + 2 c5m+2,1 1 + 6y 4m+1 + 6y4m+2 + 3y4m+3 n = 5m + 3 c5m+3,1 1 + 9y 4m+2 + 6y4m+3 n = 5m + 4 c5m+4,8 1 + 3y 4m+2 + 6y4m+3 + 6y4m+4 c5m+4,6 1 + 9y 4m+2 + 6y4m+5 c5m+4,21 1 + 6y 4m+2 + 3y4m+3 + 3y4m+4 + 3y4m+5 c5m+4,23 1 + 6y 4m+2 + 6y4m+3 + 3y4m+6 c5m+4,24 1 + 9y 4m+2 + 3y4m+3 + 3y4m+7 table 4. classification of optimal hermitian lcd codes of dimension 2 n m code n = 5m m = 1 c5m,7 m ≥ 2 c5m,7, c5m,8 n = 5m + 1 m = 1 c5m+1,9 m ≥ 2 c5m+1,9, c5m,1 n = 5m + 2 m ≥ 0 c5m+2,1 n = 5m + 3 m ≥ 0 c5m+3,1 n = 5m + 4 m = 0 c5m+4,8 m = 1 c5m+4,8, c5m+4,6, c5m+4,21 m = 2 c5m+4,8, c5m+4,6, c5m+4,21, c5m+4,23 m ≥ 3 c5m+4,8, c5m+4,6, c5m+4,21, c5m+4,23, c5m+4,24 4. concluding remarks a natural extension of this work is to classify the quaternary hermitian lcd codes of larger dimensions. in the case where the dimension is 3, there are 64 different codewords. by a method similar to that in proposition 2.2, the number of column vectors we need consider is reduced to 22. however, this is still a large number. therefore, it is difficult to extend our method to classify hermitian lcd codes of larger dimensions. acknowledgment: the author would like to thank supervisor professor masaaki harada for introducing the problem, useful discussions and his encouragement. also, the author would like to thank the reviewers for their thoughtful comments. 235 k. ishizuka / j. algebra comb. discrete appl. 7(3) (2020) 229–236 references [1] m. araya, m. harada, on the classification of linear complementary dual codes, discrete math. 342 (2019) 270–278. 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[7] j. l. massey, linear codes with complementary duals, discrete math. 106/107 (1992) 337–342. 236 https://doi.org/10.1016/j.disc.2018.09.034 https://doi.org/10.1016/j.disc.2018.09.034 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1007/978-3-319-17296-5_9 https://doi.org/10.1007/978-3-319-17296-5_9 https://doi.org/10.1109/tit.2018.2789347 https://doi.org/10.1109/tit.2018.2789347 https://doi.org/10.1016/j.ffa.2016.07.005 https://doi.org/10.1016/j.ffa.2016.07.005 https://doi.org/10.1007/s11128-014-0830-y https://doi.org/10.1007/s11128-014-0830-y https://doi.org/10.1016/0012-365x(92)90563-u introduction classification method optimal hermitian lcd codes of dimension 2 concluding remarks references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.867532 j. algebra comb. discrete appl. 8(1) • 9–22 received: 30 june 2020 accepted: 9 september 2020 journal of algebra combinatorics discrete structures and applications hyper-zagreb indices of graphs and its applications research article girish v. rajasekharaiah, usha p. murthy abstract: the first and second hyper-zagreb index of a connected graph g is defined by hm1(g) =∑ uv∈e(g)[d(u) + d(v)] 2 and hm2(g) = ∑ uv∈e(g)[d(u).d(v)] 2. in this paper, the first and second hyper-zagreb indices of certain graphs are computed. also the bounds for the first and second hyper-zagreb indices are determined. further linear regression analysis of the degree based indices with the boiling points of benzenoid hydrocarbons is carried out. the linear model, based on the hyper-zagreb index, is better than the models corresponding to the other distance based indices. 2010 msc: 05c12, 92e10 keywords: degree of a vertex, hyper-zagreb index, molecular graph 1. introduction in theoretical chemistry, a molecular graph represents the topology of a molecule, by considering how the atoms are connected. this can be modeled by a graph taking vertices as atoms and edges as covalent bonds. the properties of these graph-theoretic models can be used in the study of quantitative structure– property relationship (qspr) and quantitative structure–activity relationship (qsar) of molecules by obtaining numerical graph invariants. many such graph invariant indices have been studied. the oldest well known parameter is the wiener index introduced by harold wiener in 1947, to study the chemical properties of paraffins [32]. for a graph theoretic terminology, we refer the books [3, 16]. let g be a connected graph of order n and size m. let v (g) be the vertex set and e(g) be the edge set of g. the edge joining the vertices u and v is denoted by uv. the degree of a vertex u is the number of edges incident to it and is denoted by d(u). as usual pn,k1,n−1,cn, kn, and wn denote path, star, cycle, complete graph and wheel graph on n vertices and fn be the friendship graph with n blocks, respectively. girish v. rajasekharaiah (corresponding author); department of science and humanities, pes university(ec campus), electronic city, bengaluru, karnataka, india (email: girishvr1@pes.edu, giridsi63@gmail.com). usha p. murthy; department of mathematics, siddaganga institute of technology, b.h. road, tumakuru, karnataka, india (email: ushapmurhty@yahoo.com). 9 https://orcid.org/0000-0002-0036-6542 https://orcid.org/0000-0001-9855-1887 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 the cartesian product g�h of two graphs g and h is the graph with vertex set v (g)×v (h) and edge set contains the edge (u,v)(u ′ ,v ′ ) if and only if u = u ′ and v and v ′ are adjacent in h or v = v ′ and u and u ′ are adjacent in g. the wiener index w(g) of a connected graph g is defined as the sum of the distances between all pairs of vertices of g [32]. that is, w = ∑ u,v∈v (g) d(u,v),d(u,v) is the shortest distance between u and v. the wiener index is also called as gross status [15] and total status[3]. for more about the wiener index one can refer [4, 7, 14, 24–26, 31]. the first and second zagreb indices of a graph g are defined as [13], m1(g) = ∑ uv∈e(g)[d(u) + d(v)] and m2(g) = ∑ uv∈e(g)[d(u).d(v)] the zagreb indices were used in the structure property model [12, 29]. recent results on the zagreb indices can be found in [5, 10, 11, 19, 22, 33]. the eccentric connectivity indices of a connected graph g are defined as [1, 30] ξ1(g) = ∑ uv∈e(g)[e(u) + e(v)] and ξ2(g) = ∑ uv∈e(g)[e(u).e(v)]. details on mathematical properties and chemical applications of eccentric connectivity indices can be found in [2, 6, 8, 9, 17, 18, 20, 21, 28, 34]. the first status connectivity index s1(g) and second status connectivity index s2(g) [27] of a connected graph g is defined as: s1(g) = ∑ uv∈e(g)[σ(u) + σ(v)] and s2(g) = ∑ uv∈e(g)[σ(u).σ(v)], where σ(u) = ∑ uv∈e(g) d(u,v). harishchandra s. ramane and ashwini s. yalnaik [27] had applied linear regression analysis of the distance based indices with the boiling points of benzenoid hydrocarbons and they have shown that it is better than any other distance based indices. motivated by this, we applied linear regression analysis of the degree based indices with the boiling points of benzenoid hydrocarbons. the first and second hyper-zagreb index of a connected graph g [19] is defined by hm1(g) = ∑ uv∈e(g)[d(u) + d(v)] 2 and hm2(g) = ∑ uv∈e(g)[d(u).d(v)] 2. in this paper, the first and second hyper-zagreb indices of certain graphs are computed. also the bounds for the first and second hyper-zagreb indices are determined. further linear regression analysis of the degree based indices with the boiling points of benzenoid hydrocarbons is carried out. the linear model, based on the hyper-zagreb index, is better than the models corresponding to the other distance based indices. 2. computation of first and second hyper-zagreb indices of standard graphs (i) for any path pn with n vertices, hm1(pn) = { 4 n = 2 16p + 2 n = p + 2, p ≥ 1 hm2(pn) = { 1 n = 2 16p−8 n = p + 2, p ≥ 1 10 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 (ii) for any cycle cn, hm1(cn) = 16n=hm2(cn). (iii) for any star graph k1,n−1, hm1(k1,n−1) = n 2(n−1) hm2(k1,n−1) = (n−1)3. (iv) for any complete graph kn, hm1(kn) = 2n(n−1)3 hm2(kn) = n(n−1)5 2 . (v) for any wheel graph wn, hm1(wn) = (n−1)[(n + 2)2 + 62] hm2(wn) = 9(n−1)[(n−1)2 + 32]. (vi) for any friendship graph fn with n blocks, hm1(fn) = 8n 3 + 16n2 + 24n. hm2(fn) = 32n 3 + 16n. 3. bounds for first and second hyper-zagreb indices theorem 3.1. let g be the connected graph with n vertices and m edges, then 4m ≤ hm1(g) ≤ 4m(n−1)2. equality holds for k2. proof. for the lower bound, since for the connected graph g, the degree of each vertex is greater than or equal 1. hence hm1(g) ≥ ∑ uv∈e(g) [d(u) + d(v)]2 = ∑ uv∈e(g) [1 + 1]2 = ∑ uv∈e(g) 4 = 4m. for the upper bound, since for the connected graph g, the degree of each of vertex is less than or 11 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 equal to n−1. hence hm1(g) ≤ ∑ uv∈e(g) [d(u) + d(v)]2 = ∑ uv∈e(g) [n−1 + n−1]2 = ∑ uv∈e(g) 4(n−1)2 = 4m(n−1)2. equality: for k2, m = 1,n = 2, the result follows from 2(i) and 2(iv). theorem 3.2. let g be the connected graph with n vertices and m edges, then m ≤ hm2(g) ≤ m(n−1)4. equality holds for k2. proof. for the lower bound, since for the connected graph g, the degree of each vertex is greater than or equal 1. hence hm1(g) ≥ ∑ uv∈e(g) [d(u).d(v)]2 = ∑ uv∈e(g) [1.1]2 = ∑ uv∈e(g) 1 = m. for the upper bound, since for the connected graph g, the degree of each of vertex is less than or equal to n−1. hence hm1(g) ≤ ∑ uv∈e(g) [d(u) + d(v)]2 = ∑ uv∈e(g) [(n−1)(n−1)]2 = ∑ uv∈e(g) (n−1)4 = m(n−1)4. equality: for k2, m = 1,n = 2, the result follows from 2(i) and 2(iv). corollary 3.3. let g be a connected graph with n vertices, the 2n ≤ hm1(g) ≤ 2n(n−1)3 n−1 ≤ hm2(g) ≤ n(n−1)5 2 . theorem 3.4. for the connected graph g = pm�pn,m,n ≥ 3, hm1(g) = 128mn−150(m + n) + 144 hm2(g) = 512mn−830(m + n) + 1236. 12 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 proof. let v (g) = {(ui,vj),j = 1,2,3, . . . ,n}i=mi=1 and e(g) = a ∪ b ∪ c ∪ d, where a = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 2,d((ur,vs)) = 3} with |a| = 8, b = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 3,d((ur,vs)) = 3} with |b| = 2[(m−3) + (n−3)], c = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 3,d((ur,vs)) = 4} with |c| = 2[(m−2)+(n−2)], d = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 4,d((ur,vs)) = 4} with |d| = (m−3)(n−2)+(n−3)(m−2) such that |a|+|b|+|c|+|d| = |e(g)| = n(m−1)+(n−1)m. case 1: the first hyper-zagreb index is hm1(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)) + d((ur,vs))] 2 . = ∑ (ui,vj)(ur,vs)∈a[d((ui,vj)) + d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈b[d((ui,vj)) + d((ur,vs))] 2+ ∑ (ui,vj)(ur,vs)∈c[d((ui,vj)) + d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈d[d((ui,vj)) + d((ur,vs))] 2. = ∑ (ui,vj)(ur,vs)∈a[2 + 3] 2 + ∑ (ui,vj)(ur,vs)∈b[3 + 3] 2 + ∑ (ui,vj)(ur,vs)∈c[3 + 4] 2+ ∑ (ui,vj)(ur,vs)∈d[4 + 4] 2. = 8.52 + 2[(m−3) + (n−3)].62 + 2[(m−2) + (n−2)].72 +[(m−3)(n−2) + (n−3)(m−2)].82. = 128mn−150(m + n) + 144. case 2: the second hyper-zagreb index is hm2(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)).d((ur,vs))] 2 . = ∑ (ui,vj)(ur,vs)∈a[d((ui,vj)).d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈b[d((ui,vj)).d((ur,vs))] 2+ ∑ (ui,vj)(ur,vs)∈c[d((ui,vj)).d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈d[d((ui,vj)).d((ur,vs))] 2. = ∑ (ui,vj)(ur,vs)∈a[2.3] 2 + ∑ (ui,vj)(ur,vs)∈b[3.3] 2 + ∑ (ui,vj)(ur,vs)∈c[3.4] 2+ ∑ (ui,vj)(ur,vs)∈d[4.4] 2. = 8.62 + 2[(m−3) + (n−3)].92 + 2[(m−2) + (n−2)].122 +[(m−3)(n−2) + (n−3)(m−2)].162. = 512mn−830(m + n) + 1236. theorem 3.5. for the connected graph g = cm�pn,m,n ≥ 3, hm1(g) = 128mn−150m 13 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 hm2(g) = 512mn−830m. proof. let v (g) = {(ui,vj),j = 1,2,3, . . . ,n}i=mi=1 and e(g) = a∪b ∪c, where a = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 3,d((ur,vs)) = 3} with |a| = 2(m− 1) + 2, b = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 3,d((ur,vs)) = 4} with |b| = 2m, c = {((ui,vj)(ur,vs))/(ui,vj),(ur,vs) ∈ v (g),d((ui,vj)) = 4 with |c| = [(m−1)(n−2)+m(n−3)+(n−2)] such that |a|+ |b|+ |c| = |e(g)| = n(m−1) + (n−1)m + n. case 1: the first hyper-zagreb index is hm1(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)) + d((ur,vs))] 2 . = ∑ (ui,vj)(ur,vs)∈a[d((ui,vj)) + d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈b[d((ui,vj))+ d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈c[d((ui,vj)) + d((ur,vs))] 2. = ∑ (ui,vj)(ur,vs)∈a[3 + 3] 2 + ∑ (ui,vj)(ur,vs)∈b[3 + 4] 2+ ∑ (ui,vj)(ur,vs)∈c[4 + 4] 2 = [2(m−1) + 2].62 + 2m.72 + [(m−1)(n−2) + m(n−3) + (n−2)].82 = 128mn−150m. case 2: the second hyper-zagreb index is hm2(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)).d((ur,vs))] 2 . = ∑ (ui,vj)(ur,vs)∈a[d((ui,vj)).d((ur,vs))] 2 + ∑ (ui,vj)(ur,vs)∈b[d((ui,vj)).d((ur,vs))] 2+ ∑ (ui,vj)(ur,vs)∈c[d((ui,vj)).d((ur,vs))] 2. = ∑ (ui,vj)(ur,vs)∈a[3.3] 2 + ∑ (ui,vj)(ur,vs)∈b[3.4] 2 + ∑ (ui,vj)(ur,vs)∈c[4.4] 2+ = [2(m−1) + 2].92 + 2m.122 + [(m−1)(n−2) + m(n−3) + (n−2)].162 = 512mn−830m. theorem 3.6. for the connected graph g = cm�cn, hm1(g) = 192mn−64(m + n) hm2(g) = 768mn−256(m + n). proof. let v (g) = {(ui,vj),j = 1,2,3, . . . ,n}i=mi=1 . for the graph g = cm�cn, the degree of each vertex is 4. 14 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 case 1: the first hyper-zagreb index is hm1(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)) + d((ur,vs))] 2 = ∑ (ui,vj)(ur,vs)∈e(g)[4 + 4] 2 = [(m−1)n + m(n−1) + mn]82 = 192mn−64(m + n). case 2: the second hyper-zagreb index is hm2(g) = ∑ (ui,vj)(ur,vs)∈e(g)[d((ui,vj)) + d((ur,vs))] 2 = ∑ (ui,vj)(ur,vs)∈e(g)[4 + 4] 2 = [(m−1)n + m(n−1) + mn]162 = 768mn−256(m + n). 4. regression model for boiling point here we investigate the correlation between the boiling point (bp) of benzenoid hydrocarbons and the distance based indices of the corresponding molecular graphs. experimental values of boiling points of benzenoid hydrocarbons represented in fig.1 are taken from [23]. the scatter plot between bp and indices hm1(g),hm2(g),s1,s2,ξ1,ξ2 and w are shown in figs. 2, 3, 4, 5, 6, 7 and 8. figure 1. molecular graphs of benzenoid hydrocarbons under consideration 15 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 figure 2. scatter plot between the boiling point (bp) and the first hyper-zagreb index (hm1) figure 3. scatter plot between the boiling point (bp) and the second hyper-zagreb index (hm2) 16 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 figure 4. scatter plot between the boiling point (bp) and the first status connectivity index (s1) figure 5. scatter plot between the boiling point (bp) and the second status connectivity index (s2) 17 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 figure 6. scatter plot between the boiling point (bp) and the first eccentricity index (s2) figure 7. scatter plot between the boiling point (bp) and the second eccentricity index (s2) 18 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 figure 8. scatter plot between the boiling point (bp) and index (w) the linear regression models for the boiling point (bp) using the data of table 1 are obtained using the least square fitting procedure as implemented in ncss statistics programme. in table 2, the model (1), shows that the correlation of the experimental boiling point of benzenoid hydrocarbons with first hyper zagreb index is better (r = 0.974) than the correlation with other distance based indices considered in this paper. the linear model (2) is also good (r = 0.945) compared to the models (4), (5), (6) and (7). 5. conclusion for the degree based topological indices namely first and second hyper-zagreb index of graphs, we computed these indices for some specific graphs. also the bounds for these indices are reported. further a regression analysis of the boiling points of benzenoid hydrocarbons with the degree based indices have been carried out and compared the linear models. the linear model obtained, based on the status index, is better than the corresponding model based on the other distance indices. among the distance based topological indices considered in this paper, the first hyper-zagreb index has good correlation with the boiling point of benzenoid hydrocarbons. 19 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 table 1. the values of experimental boiling points, degree and distance based indices of 21 benzenoid hydrocarbons table 2. correlation coefficient and standard error of the estimation 20 g. v. rajasekharaiah, u. p. murthy / j. algebra comb. discrete appl. 8(1) (2021) 9–22 references [1] a. r. ashrafi, m. ghorbani, eccentric connectivity index of fullerenes. in: gutman, i., furtula, b. 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[34] b. zhou, z. du, on eccentric connectivity index, match commun. math. comput. chem. 63 (2010) 181–198. 22 http://match.pmf.kg.ac.rs/electronic_versions/match76/n1/match76n1_19-22.pdf http://match.pmf.kg.ac.rs/electronic_versions/match76/n1/match76n1_19-22.pdf http://dx.doi.org/10.22342/jims.18.1.110.57-66 http://dx.doi.org/10.22342/jims.18.1.110.57-66 https://doi.org/10.1007/s12190-016-1052-5 https://doi.org/10.1007/s12190-016-1052-5 https://doi.org/10.1007/s12190-016-1052-5 http://match.pmf.kg.ac.rs/electronic_versions/match43/match43_85-98.pdf http://match.pmf.kg.ac.rs/electronic_versions/match43/match43_85-98.pdf http://match.pmf.kg.ac.rs/electronic_versions/match43/match43_85-98.pdf https://onlinelibrary.wiley.com/doi/book/10.1002/9783527613106 http://europepmc.org/article/med/24061796 http://europepmc.org/article/med/24061796 http://match.pmf.kg.ac.rs/electronic_versions/match50/match50_117-132.pdf http://match.pmf.kg.ac.rs/electronic_versions/match50/match50_117-132.pdf https://doi.org/10.1021/ja01193a005 http://match.pmf.kg.ac.rs/electronic_versions/match54/n1/match54n1_233-239.pdf http://match.pmf.kg.ac.rs/electronic_versions/match54/n1/match54n1_233-239.pdf http://match.pmf.kg.ac.rs/electronic_versions/match63/n1/match63n1_181-198.pdf http://match.pmf.kg.ac.rs/electronic_versions/match63/n1/match63n1_181-198.pdf introduction computation of first and second hyper-zagreb indices of standard graphs bounds for first and second hyper-zagreb indices regression model for boiling point conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.858732 j. algebra comb. discrete appl. 8(1) • 31–39 received: 16 february 2020 accepted: 13 september 2020 journal of algebra combinatorics discrete structures and applications a new construction of anticode-optimal grassmannian codes research article ben paul dela cruz, john mark lampos, herbert palines, virgilio sison abstract: in this paper, we consider the well-known unital embedding from fqk into mk(fq) seen as a map of vector spaces over fq and apply this map in a linear block code of rate ρ/` over fqk . this natural extension gives rise to a rank-metric code with k rows, k` columns, dimension ρ and minimum distance k that satisfies the singleton bound. given a specific skeleton code, this rank-metric code can be seen as a ferrers diagram rank-metric code by appending zeros on the left side so that it has length n−k. the generalized lift of this ferrers diagram rank-metric code is a grassmannian code. by taking the union of a family of the generalized lift of ferrers diagram rank-metric codes, a grassmannian code with length n, cardinality q n−1 qk−1 , minimum injection distance k and dimension k that satisfies the anticode upper bound can be constructed. 2010 msc: 94b05, 94b60, 94b65 keywords: ferrers diagram, rank-metric code, grassmannian, constant dimension, anticode bound 1. introduction let fq be the finite field of order q. the projective space of order n over fq, denoted by pq(n), is the set of all subspaces of fnq . given an integer k such that 0 ≤ k ≤ n, the set of all k-dimensional subspaces of fnq is known as a grassmannian, denoted by gq(n,k). a subspace code is a nonempty subset of pq(n), while a grassmannian code is a nonempty subset of gq(n,k). subspace codes are used in network coding, a method that is far more efficient than classical coding. this paper aims to generalize the results of [4], i.e. to construct maximum rank distance (mrd) codes whose generalized lifts form an anticode-optimal grassmannian code. the paper is organized as follows. the next section gives some preliminaries and the construction of subspace codes in [2]. section 3 shows how to construct mrd codes from linear block codes. given ben paul dela cruz (corresponding author), john mark lampos, herbert palines, virgilio sison; institute of mathematical sciences and physics, university of the philippines, los baños, college, laguna 4031, philippines (email: bbdelacruz2@up.edu.ph, jtlampos@up.edu.ph, hspalines@up.edu.ph, vpsison@up.edu.ph). 31 https://orcid.org/0000-0002-0351-1231 https://orcid.org/0000-0002-8354-5772 https://orcid.org/0000-0002-7407-2539 https://orcid.org/0000-0003-2955-2311 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 a specific skeleton code, these mrd codes turn out to be ferrers diagram maximum rank distance (fdmrd) codes. in section 4, anticode-optimal grassmannian code will be constructed using these fdmrd codes and the multi-level construction in [2]. instead of using pending dots, we will use the multi-level construction as presented in [6]. lastly, we give our conclusion in section 5. 2. preliminaries a [k × `] matrix code over fq is a nonempty subset of mk×`(fq). the rank distance between two k × ` matrices over fq, say a and b, is given by dr(a,b) = rank(a−b). the minimum rank distance of a matrix code c, denoted by δ, is defined by δ = min{dr(a,b)|a,b ∈ c,a 6= b}. a [k × `,δ] rank-metric code is a [k × `] matrix code with minimum rank distance δ. it is worth noting that a linear code in mk×`(fq) is a subspace of the vector space mk×`(fq). a [k × `,ρ,δ] rank-metric code c is a linear code in mk×`(fq) with dimension ρ and minimum distance δ. the following theorem gives the relationship of the minimum distance of a rank-metric code with its minimum nonzero rank. theorem 2.1. [4] let c be a [k×`,ρ,δ] rank-metric code with minimum nonzero rank ω. then δ = ω. theorem 2.2. [1] for a [k × `,ρ,δ] rank−metric code c, ρ ≤ min{k(`−δ + 1),`(k −δ + 1)}. a code that attains the bound in theorem 2.2 is called a maximum rank distance code or an mrd code. before we go to grassmannian codes, we first give two definitions which are vital to our construction. let a ∈ mk×`(fq). the lift of a, denoted by l(a), is the standard matrix (ik a). we adapted this definition from [4]. for a given matrix a, we denote the row space of a by 〈a〉. example 2.3. let a = ( 1 0 1 0 1 1 ) ∈ m2×3(f2). then l(a) = ( 1 0 1 0 1 0 1 0 1 1 ) . moreover, the rowspace generated by l(a) is the linear block code of length 5, rate 2/5 over f2. 〈l(a)〉 = {(0,0,0,0,0),(1,0,1,0,1),(0,1,0,1,1),(1,1,1,1,0)}. definition 2.4. [4] let c be a [k × `] rank-metric code. the set λ(c) = {〈l(a)〉|a ∈ c} is called the lift of c. it is well known that the cardinality of gq(n,k) is given by the q-ary gaussian coefficient |gq(n,k)| = [ n k ] q = k−1∏ i=0 qn−i − 1 qk−i − 1 . (1) consequently, |pq(n)| = |∪nk=0 gq(n,k)|. furthermore, the subspace distance and the injection distance, defined as ds(a,b) = dim(a + b) − dim(a∩b) 32 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 and di(a,b) = max{dim a, dim b}− dim(a∩b) respectively, for any a,b in pq(n) are metrics on pq(n), and so in gq(n,k). it is clear that if a and b have the same dimension, then di(a,b) = 1 2 ds(a,b). we say that c ⊆ gq(n,k) is an (n,m,d,k)q code in the grassmannian, or a constant-dimension code, if |c| = m and the minimum injection distance of c is d = min{di(a,b)|a,b ∈ c,a 6= b}. since di(a,b) = 1 2 ds(a,b), then the minimum subspace distance of a grassmannian code is twice of its minimum injection distance. equivalently, one may opt to use the subspace distance instead of injection distance. the next theorem gives the parameters of the resulting grassmannian code from a lifted rank-metric code. theorem 2.5. [6] let c be a [k×`,ρ,δ] rank-metric code. then λ(c) is a (k+`,qρ,δ,k)q grassmannian code. the maximum number of codewords in an (n,m,d,k)q code is denoted by aq(n,d,k). bounds for aq(n,d,k) were given in [7], [3] and [11]. the lift of maximum rank distance (mrd) codes in [10] asymptotically attains the bounds given in [3] and [7]. the next theorem gives the bound that were used to check the optimality of our constructed code. theorem 2.6. [11] anticode bound. aq(n,d,k) ≤ [ n k ] q[ n−k + d− 1 d− 1 ] q = [ n k −d + 1 ] q[ k k −d + 1 ] q (2) etzion and silberstein provided a multi-level construction of grassmannian codes in [2]. the codes constructed using multi-level construction are called lifted ferrers diagram (fd) codes in [6]. furthermore, an alternative form of the construction of lifted fd codes which uses matrices instead of the usual pending dots were presented in [6]. in this paper, we opt to use the alternative form of the said construction. as defined in [6], for a nonzero x ∈ mk×`(fq), there corresponds a vector prof(x) ∈ {0, 1}n, called the profile vector of x, in which supp(prof(x)) is the set of column positions of the leading ones in the rows of the row reduced echelon form of x. if x = 0, then we set prof(x) to be the zero vector. associated with a vector space u ∈gq(n,k),k > 0, is a unique k ×n matrix xu in row reduced echelon form such that u = 〈xu〉. now define the profile vector of u, denoted by prof(u), given by prof(u) = prof(xu ). in addition, prof(u) = 0 ∈ fn2 if dim(u) = 0. let b ∈ fn2 , the schubert cell in pq(n) corresponding to b is the set sq(b) = {u ∈pq(n)|prof(u) = b}. some papers denote this set as prof−1(b). given any binary vector b of length n and hamming weight k, the permutation matrix with respect to b, denoted by p(b), is the n×n permutation matrix whose rows indexed by supp(b) form p(b)supp(b) =( ik 0k×(n−k) ) and the remaining rows of p(b) form p(b)supp(b̄) = ( 0(n−k)×k i(n−k) ) , where b + b̄ is the all one vector of length n. 33 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 example 2.7. let b = (1, 0, 0, 1, 0). so b̄ = (0, 1, 1, 0, 1). p(b) =   1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1   we changed some notations in the next definition to be consistent with our earlier notations but the essence is exactly the same. definition 2.8. [6] let b be a binary vector of length n and hamming weight k. for x ∈ mk×(n−k)(fq), define the generalized lifting of x with respect to b, denoted by λb(x), as λb(x) = 〈l(x)〉p(b)−1 = 〈l(x)p(b)−1〉. definition 2.9. [6] let b be a binary vector of length n, hamming weight k and let cb be a [k×(n−k)] rank-metric code. the set λb(cb) = {λb(c)|c ∈cb} is called the generalized lift of cb. we now present some necessary conditions for the definition of lifted fd codes which were established in [6]. let q = [aij] be the n×n upper triangular matrix with aij = 1 if j ≥ i and aij = 0 otherwise. given a binary profile vector b of length n and hamming weight k, regarded as an element of z1×n, define the vector c(b) ∈ z1×n via c(b) = bqp(b). let x = [xij] ∈ mk×(n−k)(fq). according to [6], λb(x) is guaranteed to be in the schubert cell corresponding to b provided that for 1 ≤ i ≤ k and 1 ≤ j ≤ n−k, i > c(b)j+k implies that xij = 0. (3) an fd(b) code is a rank-metric code cb ⊆ mk×(n−k)(fq) in which each codeword satisfies (3) while the code λb(cb) is referred to as lifted fd(b) code. we now consider the minimum distance between elements in distinct schubert cells. let u,v ∈ fn2 and u 6= v. now define the logical and of u and v, denoted by u∧v, as (u∧v)i = uivi. also the asymmetric distance between u and v, denoted by da(u,v), is given by da(u,v) = max{wth(u),wth(v)}−wth(u∧v). theorem 2.10. [5] let u,v ∈ fn2 , u 6= v, u ∈ sq(u) and v ∈ sq(v) and d(u,v ) be the injection distance of u and v . then d(u,v ) ≥ da(u,v). the following steps to construct an (n,m,d,k)q code c are from [5]. 1. choose a binary constant weight code b of length n, hamming weight k, and minimum asymmetric distance d. 2. for each b ∈b, consider an fd(b) code with minimum rank distance d. 3. construct the lifted fd(b) code λb(cb) for each b ∈b. 4. set c = ⋃ b∈b λb(cb) the cardinality m of c greatly depends on the choice of b. theorem 2.11. [2] c is an (n,m,d,k)q constant-dimension code, where m = ∑ b∈b |λb(cb)|. 34 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 3. rank-metric codes and grassmannian codes the following well-known theorem will be the cornerstone of our construction. theorem 3.1. [9] let f(x) = k∑ i=0 aix i ∈ fq[x] be a monic irreducible polynomial and x be its companion matrix. then the mapping π : fq[x] → mk(fq),g(x) 7→ g(x) (4) induces a unital embedding τ : fq[x]/(f) = fqk → mk(fq). (5) let n be a positive integer. note that τ can be extended to the following monomorphism φ defined by φ : fn qk → mk×kn (fq) where φ (α1,α2, ...,αn) = ( τ(α1) τ(α2) ... τ(αn) ) . lemma 3.2. if c is a linear block code of length n over fqk then c ∼= φ (c) as fq−vector spaces. the following theorem is a generalization of theorem 3.7 of [4]. theorem 3.3. let c be a linear block code of length n over fqk and ρ its dimension as an fq-vector space. then i. φ (c) is a [k ×kn,ρ,k] rank-metric code, ii. λ (φ (c)) is a (k + kn,qρ,k,k)q code, iii. the pairwise intersection of codewords of λ (φ (c)) is trivial. proof. let c be a linear block code of length n over fqk and ρ its dimension as an fq-vector space. by lemma 3.2, c and φ (c) are isomorphic as fq−vector spaces. hence, the dimension of φ (c) is ρ. consider the case n = 1. since fqk is a field, then ∀α ∈ fqk −{0},φ (α) = τ(α) ∈ mk(fq) is a unit and thus has rank k, where τ is as defined in (5). thus, ( τ (α1) τ (α2) ... τ (αn) ) has rank k for any positive integer n where αi ∈ fqk −{0} for some i, 1 ≤ i ≤ n. therefore, by theorem 2.1, the minimum distance of φ (c) is k. clearly, ii. follows directly from theorem 2.5 if λ (φ (c)) is a (k + kn,qρ,k,k)q code, the minimum injection distance of λ (φ (c)) is k. let a,b ∈ λ (φ (c)). we have k ≤ d (a,b) = max{dim a, dim b}−dim (a∩b) . hence, k ≤ k−dim (a∩b) which makes dim (a∩b) to be equal to 0. therefore, the pairwise intersection of codewords of λ (φ (c)) is trivial. note that for any positive integer n, the rank-metric code φ(fn qk ) meets the singleton bound as given in theorem 9 of [7]. example 3.4. consider the irreducible polynomial x2 + x + 1 over f2 and its companion matrix x =( 1 1 1 0 ) , and the linear block code f24. then φ(f 2 4) is a [2 × 4, 4, 2] rank-metric code which satisfies the singleton bound. moreover, λ(φ(f24)) is a (6, 16, 2, 2)2−code. 35 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 4. anticode-optimal grassmannian code now we construct grassmannian codes using the multi-level construction in [2] and the result in theorem 3.3. let n be a multiple of k and n > k. for the skeleton code b, choose b = {b` = (0, · · · , 0︸ ︷︷ ︸ n−k−k` , 1, · · · , 1︸ ︷︷ ︸ k , 0, · · · , 0︸ ︷︷ ︸ k` )|0 ≤ ` ≤ n−k k }. clearly, b is a binary constant weight code of length n, weight k, and minimum asymmetric distance k. then for 0 ≤ ` ≤ n−k k p(b`) =   in−k−k`ik ik`   . let q = [aij] be the n×n upper triangular matrix with aij = 1 if j ≥ i and aij = 0 otherwise. then, c(b`) = b`qp(b`) = (0, · · · , 0︸ ︷︷ ︸ n−k−k` , 1, · · · , 1︸ ︷︷ ︸ k , 0, · · · , 0︸ ︷︷ ︸ k` )qp(b`) = (0, · · · , 0︸ ︷︷ ︸ n−k−k` , 1, 2, · · · ,k,k, · · · ,k︸ ︷︷ ︸ k` )p(b`) = (1, 2, · · · ,k, 0, · · · , 0︸ ︷︷ ︸ n−k−k` ,k, · · · ,k︸ ︷︷ ︸ k` ) . lemma 4.1. let n be a multiple of k, n > k and cb` = {( 0k×(n−k−k`) φ(v) ) v ∈ f`qk } where 1 ≤ ` ≤ n−k k . then, for 1 ≤ ` ≤ n−k k , cb` is an fd(b`) code and has minimum rank distance k. proof. recall that for a rank-metric code c to be an fd(b) code, c must be a subset of mk×(n−k)(fq) and all of its codewords must satisfy (3). let 1 ≤ ` ≤ n−k k . clearly, cb` ⊆ mk×(n−k)(fq). now let x = [xij] ∈ cb`. by (3), for 1 ≤ i ≤ k, 1 ≤ j ≤ n−k, i > c(b`)j+k implies that xij = 0. notice that in (3), we only check the last n−k components of c(b`). now c(b`)j+k = { 0 if 1 ≤ j ≤ n−k −k` k if n−k −k` < j ≤ n−k. hence the entries in the first n − k − k` columns of the elements of cb` must be all zero. clearly, cb` satisfies this. lastly, by theorem 3.3, φ(v) has minimum rank distance k. clearly, cb` has minimum rank distance k. theorem 4.2. the rank-metric code cb0 = {(0k×(n−k))} is an fd(b0) code. although cb0 does not have a minimum distance k as required in [2] and [6], it will not pose any problem since the resulting grassmannian will still have a minimum distance k. ironically, b0 is included in example 10 of [2]. note also that cb0 is the only rank-metric code that will satisfy b0. now we are ready to construct the lifted fd(b) code λb(cb). by definition 2.8 and definition 2.9, 36 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 λb` (cb` ) =   {〈 0k×(n−k−k`) ik φ(v) 〉 v ∈ f` qk } , 1 ≤ ` ≤ n−k k{〈 0k×(n−k) ik 〉} , ` = 0. the lifted fd code, denoted by ω(b), is given by ω(b) = ⋃ b∈b λb(cb). as an immediate consequence of lemma 4.1 and theorem 4.2, we have the following theorem. theorem 4.3. ω(b) is an ( n, q n−1 qk−1 ,k,k ) q code. since cb0 does not have a minimum distance k and b0 appeared only as an example in [2], we will separate the case of b0 in our proof. proof. for 1 ≤ ` ≤ n−k k , cb` is an fd(b`) code with minimum rank distance k by lemma 4.1. now, ω(b) = ⋃ b∈b λb(cb) =   ⋃ b∈b,b6=b0 λb(cb)  ∪ λb(cb0 ). by theorem 2.11, ⋃ b∈b,b6=b0 λb(cb) is an (n,m,k,k)q code where m = ∑ b∈b,b 6=b0 |λb(cb)|. clearly, λb0 (cb0 ) ⊆ gq(n,k). now we compute for the distance of codewords u ∈ ⋃ b∈b,b6=b0 λb(cb) and v ∈ λb0 (cb0 ). since u ∈ ⋃ b∈b,b 6=b0 λb(cb), then u ∈ λb` (cb` ) for some b` ∈ b −{b0}. by theorem 2.10, d(u,v ) ≥ da(b`,b0) ≥ k − 0 = k. note that λb0 (cb0 ) has only one codeword. therefore, ω(b) has minimum distance k. observe that λb` (cb` ) ∩ λbj (cbj ) = ∅, where ` 6= j. hence, |ω(b)| = ∑ b∈b |λb(cb)| = n−k k∑ i=0 qki = qn − 1 qk − 1 . note that ω(b) attains the anticode bound. the following illustrates the construction of an anticode-optimal grassmannian code. example 4.4. consider the irreducible polynomial x3 + x + 1 over f2, its companion matrix x =  0 1 11 0 0 0 1 0  , φ1 : f23 → m3×3(f2),α1 7→ τ(α1), φ2 : f223 → m3×6(f2), (α1,α2) 7→ ( τ(α1) τ(α2) ) , and the skeleton code b = {b` = (0, · · · , 0︸ ︷︷ ︸ 6−3` , 1, · · · , 1︸ ︷︷ ︸ 3 , 0, · · · , 0︸ ︷︷ ︸ 3` )|0 ≤ ` ≤ 2}. 37 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 now, cb0 = {03×6} cb1 = {(03×3|φ1(v)) : v ∈ f23} cb2 = {φ2(v) : v ∈ f 2 23}. so λb0 (cb0 ) = {〈03×6|i3〉} λb1 (cb1 ) = {〈03×3|i3|φ1(v)〉 : v ∈ f23} λb2 (cb2 ) = {〈i3|φ2(v)〉 : v ∈ f 2 23}. finally, ω(b) = λb0 (cb0 ) ∪ λb1 (cb1 ) ∪ λb2 (cb2 ) with cardinality 1 + 8 + 64 = 73 = a2(9, 3, 3). theorem 4.3 is a generalization of theorem 3.16 in [4] and is similar with the construction of spread codes in [8]. note that by choosing v to be in a different linear block code c of length ` over fqk instead of f` qk in lemma 4.1, one may construct a grassmannian code that is not a spread code. 5. summary and conclusion we presented two constructions of grassmannian codes for any length, over any finite field and whose dimension is equal to its minimum injection distance. these two constructions are generalizations of some constructions in [4]. the first construction uses a linear block code to construct a rank-metric code. the resulting rank-metric code was then used to create a grassmannian code that meets the singleton bound. the second construction uses the results in the first construction together with the concept of ferrers diagram to get an anticode-optimal grassmannian code. the resulting code from the second construction is similar to the construction of spread codes found in [8]. acknowledgment: the authors would like to thank the reviewer for his/her valuable comments. references [1] t. etzion, subspace codes − bounds and constructions, 1st european training school on network coding, barcelona, spain, (2013). 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[6] a. khaleghi, d. silva, f. r. kschischang, subspace codes, ima int. conf. 49(4) (2009) 1–21. 38 https://doi.org/10.1109/tit.2009.2021376 https://doi.org/10.1109/tit.2009.2021376 https://doi.org/10.1109/tit.2010.2095232 https://doi.org/10.1109/tit.2010.2095232 http://www.ripublication.com/gjpam16/gjpamv12n2_52.pdf http://www.ripublication.com/gjpam16/gjpamv12n2_52.pdf https://doi.org/10.1109/cwit.2009.5069509 https://doi.org/10.1109/cwit.2009.5069509 https://doi.org/10.1007/978-3-642-10868-6_1 b. p. dela cruz et al. / j. algebra comb. discrete appl. 8(1) (2021) 31–39 [7] r. koetter, f. r. kschischang, coding for errors and erasures in random network coding, ieee trans. inform. theory 54(8) (2008) 3579–3591. [8] f. manganiello, e. gorla, j. rosenthal, spread codes and spread decoding in network coding, in: proc. 2008 ieee isit, toronto, canada, (2008) 851–855. [9] a. j. menezes, i. f. blake, x. gao, r. c. mullen, s. a. vanstine, t. yaghoobian, applications of finite fields, boston, ma: kluwer academic publishers 1993. [10] d. silva, f. r. kschischang, r. koetter, a rank-metric approach to error control in random network coding, ieee trans. inform. theory 54(9) (2008) 3951–3967. [11] h. wang, c. xing, r. safavi-naini, linear authentication codes: bounds and constructions, ieee trans. inform. theory 49(4) (2003) 866–872. 39 https://doi.org/10.1109/tit.2008.926449 https://doi.org/10.1109/tit.2008.926449 https://doi.org/10.1109/isit.2008.4595113 https://doi.org/10.1109/isit.2008.4595113 https://doi.org/10.1109/tit.2008.928291 https://doi.org/10.1109/tit.2008.928291 https://doi.org/10.1109/tit.2003.809567 https://doi.org/10.1109/tit.2003.809567 introduction preliminaries rank-metric codes and grassmannian codes anticode-optimal grassmannian code summary and conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056485 j. algebra comb. discrete appl. 9(1) • 1–7 received: 10 may 2020 accepted: 4 november 2021 journal of algebra combinatorics discrete structures and applications note on the permutation group associated to e-polynomials∗ research article hirotaka imamura, masashi kosuda, manabu oura abstract: this is a continuation of our project which focuses on e-polynomials and the related combinatorics. a pair of groups appearing in the definition of e-polynomials yields the permutation group. in this paper, we determine the multi-matrix structures of the centralizer algebras of the tensor representations of this permutation group. 2010 msc: 05e10, 05e30 keywords: centralizer algebra, permutation group 1. introduction the purpose of this note is to investigate the centralizer algebras of the tensor representation of the finite group g that arises from e-polynomials in genus 1. we also discuss the association scheme which appears as the first case. our results can be regarded as combinatorial properties of e-polynomials. the study of eisenstein series is of interest because of its importance in number theory. we defined and studied finite analogue of eisenstein series from combinatorial point of view. we call it an epolynomial in some of our previous papers [8, 10, 11]. let g be a positive integer. then, an action of γg = sp(2g,z) on the theta series induces a finite subgroup hg of gl(2g,c). it is known that the invariant ring of hg is closely related to the ring of modular forms for γg. as to this direction in higher genus, see [9, 14]. there exist some results based on the fact that the weight enumerator of a self-dual and doubly even binary code is invariant under the action of hg [3, 4, 13, 15]. the correspondence existing among the invariant theory of the finite groups, ∗ this work was supported by jsps kakenhi grant numbers 17k05164, 19k03398. hirotaka imamura; pfu limited, japan (email: h.i.ishikawa01@gmail.com). masashi kosuda; department of engineering, yamanashi university, japan (email: mkosuda@yamanashi.ac.jp). manabu oura (corresponding author); institute of science and engineering, kanazawa university, japan (email: oura@se.kanazawa-u.ac.jp). 1 https://orcid.org/0000-0003-0028-6785 https://orcid.org/0000-0002-0498-9667 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 the theory of modular forms, and combinatorics such as coding theory motivates the notion and study of e-polynomials. let kg be a subgroup of hg fixing x0. an e-polynomial of degree ` in degree g is defined by ϕ`(xa : a ∈ f g 2) = 1 |hg| ∑ kg\hg3σ (σx0) `. the rings generated by e-polynomials have been studied in [10, 11] (cf. [8]). moreover, the representation theory of h1 is investigated and the centralizer rings of the tensor representation of h1 are determined in [6]. since the character table of h1 is used in this study, it is reproduced at the end of this paper from [6]. as usual, let c denote the complex number field. we denote by md the matrix algebra of degree d over c. for simpliticy, let nmd denote md ⊕···⊕md︸ ︷︷ ︸ n . 2. preliminaries we follow the notations in [6]. let h1 be the group of order 96 generated by 1+i2 ( 1 −1 1 1 ) , ( 1 0 0 i ) and k1 = 〈 ( 1 0 0 i ) 〉 be its subgroup of orer 4. a set of representatives of k1\h1 is given by 1,t,t 2,t 3,t 4,t 5,t 6,t 7,td,td2,td3,t 3d,t 3d2,t 3d3, t 5d,t 5d2,t 5d3,t 7d,t 7d2,t 7d3,td2t,t 3d2t,t 5d2t,t 7d2t and we name each of the classes 1, 2, . . . , 24, for example, 24 = k1t 7d2t . now, we set ω = {1, 2, . . . , 24}, then the action of h1 on ω gives a transitive permutation representation of h1. we observe that t and d correspond to t = (1, 2, 3, 4, 5, 6, 7, 8)(9, 14, 12, 17, 15, 20, 18, 11)(10, 21, 13, 22, 16, 23, 19, 24), and d = (2, 9, 10, 11)(4, 12, 13, 14)(6, 15, 16, 17)(8, 18, 19, 20)(21, 22, 23, 24), respectively. consequently, we obtain a faithful permutation representation g = 〈t,d〉 of h1. the group g is an imprimitive permutation group of order 96 on the 24 points. under the correspondence g 7→ (δαg,β)α,β∈ω , we may regard g as a matrix group. 3. results we follow the argument presented in the papers [6, 7]. let us denote by χ the permutation character of g. it is known that χ(g) is the number of α ∈ ω which is fixed by g. proposition 3.1. χ = χ1 + χ5 + χ9 + χ10 + χ12 + χ13 + χ14 + χ15 + χ16. 2 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 proof. let χ = m1χ1 + m2χ2 + · · · + m16χ16. here we remind that χ1 is the identity character of g. since the characters depend on the conjugacy classes, χ(c) denotes the value of χ at a conjugacy class c. then, we have that( χ(c1) χ(c2) . . . χ(c16) ) = ( m1 m2 . . . m16 ) x, where x denotes the character table of g presented at the end of this paper. if we know the explicit values χ(ci)’s on the left-hand side, we multiply x−1 on both sides from the right to get the result. suppose g ∈ c for a conjugacy class c, then χ(c) is the number of α ∈ ω which is fixed by g. therefore the values of χ(ci)’s can be given by( χ(c1) χ(c2) . . . χ(c16) ) = (24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 4, 0). this completes the proof of proposition 3.1. before proceeding further, we shall interpret this proposition from the perspective of wielandt [17]. let a be the centralizer ring of the matrix group g. then proposition 3.1 shows that a is commutative and of dimension 9 and a basis can be obtained as follows. let g1 be the stabilizer of a point 1 ∈ ω in g. then g1 has 9 orbits {1},{3},{5},{7},{2, 9, 10, 11},{4, 12, 13, 14},{6, 15, 16, 17},{8, 18, 19, 20},{21, 22, 23, 24}. with each orbit ∆ we associate a matrix v(∆) = (v∆α,β) by v∆α,β = { 1 ∃g ∈ g such that 1g = β and αg −1 ∈ ∆, 0 otherwise. in particular, for ∆ = {1}, we have that v({1}) = i where i is the 24 × 24 identity matrix. then the matrices v(∆)’s form a basis for a. next we shall connect a with the theory of association schemes ([1]. cf. [16]). under the ordering of the orbits given above, we denote the matrices v(∆) by a0 = i,a1, . . . ,a8. we observe that the matrices a0,a2,a8 are symmetric and ta1 = a3, ta4 = a7, ta5 = a6. then the set of matrices x = {a0,a1, . . . ,a8} forms a non-symmetric commutative association scheme of class 8. the algebra a is also called the bose-mesner algebra of the association scheme x . with regards to the commutativity and the normality of ai’s, x has another basis e0 = 124j,e1, . . . ,e8 of the primitive idempotents, where j is a 24 × 24 matrix with every entry 1. the transformation matrix p is defined by ( a0 a1 . . . a8 ) = ( e0 e1 . . . e8 ) p which can also be called the first eigenmatrix of x . in our case, the first eigenmatrix is  1 1 1 1 4 4 4 4 4 1 1 1 1 −2 −2 −2 −2 4 1 1 1 1 0 0 0 0 −4 1 i −1 −i 2 + 2i −2 + 2i −2 − 2i 2 − 2i 0 1 i −1 −i −1 − i 1 − i 1 + i −1 + i 0 1 −i −1 i 2 − 2i −2 − 2i −2 + 2i 2 + 2i 0 1 −i −1 i −1 + i 1 + i 1 − i −1 − i 0 1 −1 1 −1 2i −2i 2i −2i 0 1 −1 1 −1 −2i 2i −2i 2i 0   . 3 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 it turns out that x is isomorphic to no.349 in the list [5]. summing up, we have obtained the following theorem. theorem 3.2. the centralizer ring of the transitive permutation group g1 on the 24 points is a commutative association scheme of degree 8 which is isomorphic to no.349 in [5]. we proceed to the centralizer algebra of the tensor representation of g. let a(k) be the centralizer algebra of the k-th tensor representation of g. notice a = a(1). we set χ⊗k = 16∑ i=1 d (k) i χi and −−→ d(k) = (d (k) 1 ,d (k) 2 , . . . ,d (k) 16 ). we are going to determine the coefficients d(k)i explicitly. by proposition 3.1, we already knew −−→ d(1) = (1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1). we shall consider the matrix a such that  χχ1 χχ2 ... χχ16   = a   χ1 χ2 ... χ16   . the matrix a can be calculated explicitly as a = x.diagonal([χ(c1),χ(c2), . . . ,χ(c16)]).x−1 = x.diagonal([24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 4, 0]).x−1 =   1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 0 2 0 1 1 1 1 1 1 2 2 2 2 0 0 1 1 0 2 1 1 1 1 2 2 1 1 2 2 0 1 1 0 1 1 2 1 0 1 2 1 2 1 2 2 0 1 0 1 1 1 1 2 1 0 2 1 1 2 2 2 1 0 0 1 1 1 0 1 2 1 1 2 1 2 2 2 1 0 1 0 1 1 1 0 1 2 1 2 2 1 2 2 0 1 1 1 1 2 2 2 1 1 3 2 2 2 3 3 1 0 1 1 1 2 1 1 2 2 2 3 2 2 3 3 1 1 1 0 2 1 2 1 1 2 2 2 3 2 3 3 1 1 0 1 2 1 1 2 2 1 2 2 2 3 3 3 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4   . we have thus gotten −−→ d(k) = −−−→ d(k−1)a (k ≥ 2). then we have only to apply linear algebra. the matrix a is diagonalizable and we get1 p−1ap = diagonal([4, 4, 4, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) 1 the matrices x, a, and p can be found at [12]. 4 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 for some non-singular matrix p . we apply this calculation to −−→ d(k) = −−→ d(1)pak−1p−1 = (ak,bk,bk,bk,ck,dk,dk,dk,ek,ek,fk,gk,gk,gk,hk,hk), where ak = 6 · 24k−2 + 3 · 4k−2, bk = 6 · 24k−2 − 4k−2, ck = 12 · 24k−2 + 2 · 4k−2, dk = 12 · 24k−2 − 2 · 4k−2, ek = 12 · 24k−2 + 2 · 4k−2, fk = 18 · 24k−2 − 3 · 4k−2, gk = 18 · 24k−2 + 4k−2, hk = 24 k−1. we have thus obtained the following multi-matrix structures of a(k). theorem 3.3. we have that a(k) ∼= { 9m1 k = 1, mak ⊕ 3mbk ⊕mck ⊕ 3mdk ⊕ 2mek ⊕mfk ⊕ 3mgk ⊕ 2mhk k ≥ 2, where ak,bk, . . . ,hk are given above. corollary 3.4. (1) a(k) is commutative if and only if k = 1. (2) dim a(k) = 3456 · 242(k−2) + 48 · 42(k−2) (k ≥ 1). the second assertion of corollary 3.4 is obtained by taking the sum of the dimensions of the simple components. it would be interesting if we interpret this number from combinatorial point of view, see [6]. this paper is concluded with a small table of dim a(k). k 1 2 3 4 5 6 dim a(k) 9 3504 1991424 1146630144 660452081664 380420288937984 acknowledgment: this work was supported by jsps kakenhi grant numbers 17k05164, 19k03398. the computations were done with magma [2] and maple. references [1] e. bannai, t. ito, algebraic combinatorics i: association schemes, the benjamin/cummings publishing co, california (1984). [2] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symbolic comput. 24(3-4) (1997) 235–265. [3] m. broué, m. enguehard, polynômes des poids de certains codes et fonctions thêta de certains réseaux, ann. sci. école norm. sup. 5(1) (1972) 157–181. [4] w. duke, on codes and siegel modular forms, internat. math. res. notices 5 (1993) 125–136. 5 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.24033/asens.1223 https://doi.org/10.24033/asens.1223 https://doi.org/10.1155/s1073792893000121 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 [5] a. hanaki, http://math.shinshu-u.ac.jp/ hanaki/as/. [6] m. kosuda, m. oura, on the centralizer algebras of the primitive unitary reflection group of order 96, tokyo j. math. 39(2) (2016) 469–482. [7] m. kosuda, m. oura, centralizer algebras of the group associated to z4-codes, discrete math. 340(10) (2017) 2437–2446. [8] t. motomura, m. oura, e-polynomials associated to z4-codes, hokkaido math. j. 47(2) (2018) 339–350. [9] m. oura, the dimension formula for the ring of code polynomials in genus 4, osaka j. math. 34(1) (1997) 53–72. [10] m. oura, eisenstein polynomials associated to binary codes, int. j. number theory 5(4) (2009) 635–640. [11] m. oura, eisenstein polynomials associated to binary codes (ii), kochi j. math. 11 (2016) 35–41. [12] m. oura, http://sphere.w3.kanazawa-u.ac.jp [13] b. runge, on siegel modular forms i, j. reine angew. math. 436 (1993) 57–86. [14] b. runge, on siegel modular forms ii, nagoya math. j. 138 (1995) 179–197. [15] b. runge, codes and siegel modular forms, discrete math. 148(1-3) (1996) 175–204. [16] c. c. sims, graphs and finite permutation groups, math. z. 95 (1967) 76–86. [17] h. wielandt, finite permutation groups, academic press, new york-london (1964). 6 http://math.shinshu-u.ac.jp/~hanaki/as/ https://doi.org/10.3836/tjm/1484903133 https://doi.org/10.3836/tjm/1484903133 https://doi.org/10.1016/j.disc.2017.06.001 https://doi.org/10.1016/j.disc.2017.06.001 https://doi.org/10.14492/hokmj/1529308822 https://doi.org/10.14492/hokmj/1529308822 https://mathscinet.ams.org/mathscinet-getitem?mr=1438997 https://mathscinet.ams.org/mathscinet-getitem?mr=1438997 https://doi.org/10.1142/s1793042109002298 https://doi.org/10.1142/s1793042109002298 https://mathscinet.ams.org/mathscinet-getitem?mr=3559475 http://sphere.w3.kanazawa-u.ac.jp https://doi.org/10.1515/crll.1993.436.57 https://doi.org/10.1017/s0027763000005237 https://doi.org/10.1016/0012-365x(94)00271-j https://doi.org/10.1007/bf01117534 https://doi.org/10.1016%2fc2013-0-11702-3 h. imamura et.al. / j. algebra comb. discrete appl. 9(1) (2022) 1–7 h 1 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 1 t t 2 t 3 t 4 t 6 d d t d t 2 d t 3 d t 4 d t 5 d t 6 d t 7 d 2 d 2 t 2 or d er 1 8 4 8 2 4 4 6 4 12 4 3 4 12 2 4 χ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ 2 1 − 1 1 − 1 1 1 − 1 1 − 1 1 − 1 1 − 1 1 1 1 χ 3 1 − i − 1 i 1 − 1 i 1 − i − 1 i 1 − i − 1 − 1 1 χ 4 1 i − 1 − i 1 − 1 − i 1 i − 1 − i 1 i − 1 − 1 1 χ 5 2 0 2 0 2 2 0 − 1 0 − 1 0 − 1 0 − 1 2 2 χ 6 2 0 − 2 0 2 − 2 0 − 1 0 1 0 − 1 0 1 − 2 2 χ 7 2 0 − 2 i 0 − 2 2 i − 1 + i 1 1 + i − i 1 − i − 1 − 1 − i i 0 0 χ 8 2 0 2 i 0 − 2 − 2 i − 1 − i 1 1 − i i 1 + i − 1 − 1 + i − i 0 0 χ 9 2 0 − 2 i 0 − 2 2 i 1 − i 1 − 1 − i − i − 1 + i − 1 1 + i i 0 0 χ 1 0 2 0 2 i 0 − 2 − 2 i 1 + i 1 − 1 + i i − 1 − i − 1 1 − i − i 0 0 χ 1 1 3 1 3 1 3 3 − 1 0 − 1 0 − 1 0 − 1 0 − 1 − 1 χ 1 2 3 − 1 3 − 1 3 3 1 0 1 0 1 0 1 0 − 1 − 1 χ 1 3 3 i − 3 − i 3 − 3 i 0 − i 0 i 0 − i 0 1 − 1 χ 1 4 3 − i − 3 i 3 − 3 − i 0 i 0 − i 0 i 0 1 − 1 χ 1 5 4 0 − 4 i 0 − 4 4 i 0 − 1 0 i 0 1 0 − i 0 0 χ 1 6 4 0 4 i 0 − 4 − 4 i 0 − 1 0 − i 0 1 0 i 0 0 7 introduction preliminaries results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.645015 j. algebra comb. discrete appl. 7(1) • 3–20 received: 27 june 2019 accepted: 20 september 2019 journal of algebra combinatorics discrete structures and applications construction of quasi-twisted codes and enumeration of defining polynomials research article t. aaron gulliver, vadlamudi ch. venkaiah abstract: let dq(n,k) be the maximum possible minimum hamming distance of a linear [n,k] code over fq. tables of best known linear codes exist for small fields and some results are known for larger fields. quasi-twisted codes are constructed using m×m twistulant matrices and many of these are the best known codes. in this paper, the number of m × m twistulant matrices over fq is enumerated and linear codes over f17 and f19 are constructed for k up to 5. 2010 msc: 94b05, 94b65, 94a55 keywords: finite fields, twistulant matrices, quasi-twisted codes, optimal codes, griesmer bound 1. introduction let fq denote the finite field of q elements, and v (n,q) the vector space of n-tuples over fq. a linear [n,k] code c of length n and dimension k over fq is a k-dimensional subspace of v (n,q). the elements of c are called codewords. the (hamming) weight of a codeword is the number of non-zero coordinates, and the minimum distance of c is the smallest weight among all non-zero codewords of c. an [n,k,d] code is an [n,k] code with minimum distance d. let ai be the number of codewords of weight i in c. then the numbers a0,a1, . . . ,an are called the weight distribution of c. a central problem in coding theory is that of optimizing one of the parameters n,k and d for given values of the other two. one can find dq(n,k), the largest value of d for which there exists an [n,k,d] code over fq, or nq(k,d), the smallest value of n for which there exists an [n,k,d] code over fq. a code which achieves either of these values is called optimal. tables of best known linear codes exist for q = 2 to 9 [6], 11 [3] and 13 [4]. t. aaron gulliver (corresponding author); department of electrical and computer engineering, university of victoria, po box 1700, stn csc, victoria, bc v8w 2y2, canada (email: agullive@ece.uvic.ca). vadlamudi ch. venkaiah; school of computer and information sciences, university of hyderabad, gachibowli, hyderabad 500 046, india (email: venkaiah@hotmail.com, vvcs@uohyd.ernet.in) 3 https://orcid.org/0000-0001-9919-0323 https://orcid.org/0000-0001-5440-0200 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 the griesmer bound is a well-known lower bound on nq(k,d) nq(k,d) ≥ gq(k,d) = k−1∑ j=0 ⌈ d qj ⌉ , (1) where dxe denotes the smallest integer ≥ x. for k ≤ 2, there exist codes that attain equality in the griesmer bound for all q and d. the singleton bound [12] is a lower bound on nq(k,d) and is given by nq(k,d) ≥ d + k −1. (2) codes that meet this bound are called maximum distance separable (mds). mds codes exist for all values of n ≤ q + 1. thus, for q = 17 mds codes exist for all lengths 18 or less, and for q = 19 all lengths 20 or less. note that all mds codes are optimal. for larger lengths and dimensions, far less is known about codes over f17 and f19. mds self-dual codes (k = n/2), of lengths 2, 4, 6, 8, 10 and 18 over f17 are known [1], as well as self-dual [12,6,6], [14,7,7], [16,8,8], [20,10,10], [22,11,10] and [24,12,10] codes. the [14,7,8] and [20,10,10] extended quadratic residue (qr) codes are given in [13]. using magma [2], it was determined that the next extended qr code has parameters [44,22,18]. mds self-dual codes of lengths 4, 8, 12 and 20 over f19 are known [1], as well as self-dual [16,8,8] and [24,12,9] codes. the [18,9,9] and [32,16,14] extended qr codes are given in [13]. in this paper, codes over f17 and f19 for k up to 5 are presented. these codes establish lower bounds on the minimum distance. many of these meet the singleton and/or griesmer bounds, and so are optimal. 2. quasi-twisted codes a constacyclic shift of an m-tuple (x0,x1, . . . ,xm−1), is the m-tuple (λxm−1,x0, . . . ,xm−2), where λ ∈ fq\{0}, and a constacyclic shift by p positions is the m-tuple (λxm−p, . . . ,λxm−1,x0, . . . ,xm−p−1). a linear code c is said to be quasi-twisted (qt) if a constacyclic shift of any codeword by p positions is also a codeword in c [8]. note that quasi-twisted codes generalize the classes of constacyclic codes (p = 1), quasi-cyclic codes (λ = 1), cyclic codes (λ = 1, p = 1), and negacyclic codes (λ = −1, p = 1). the length of a qt code considered here is n = mp. with a suitable permutation of coordinates, many qt codes can be characterized in terms of m×m twistulant matrices. in this case, a qt code can be transformed into an equivalent code with generator matrix g = [b0 b1 b2 . . . bp−1] , (3) where bi, i = 0,1, . . . ,p− 1, is an m×m twistulant matrix (also known as a constacyclic matrix), over fq of the form [9] b =   b0 b1 b2 · · · bm−1 λbm−1 b0 b1 · · · bm−2 λbm−2 λbm−1 b0 · · · bm−3 ... ... ... ... λb1 λb2 λb3 · · · b0   , 4 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 where λ ∈ fq\{0} and bi,0 ≤ i ≤ m − 1, are elements of fq. when λ = 1, a twistulant matrix is a circulant matrix, and when λ = −1, a twistulant matrix is known as a negacirculant matrix [8]. the algebra of m×m twistulant matrices over fq is isomorphic to the algebra of polynomials in the ring fq[x]/(xm −λ) if bi is mapped onto the polynomial bi(x) = b0,i + b1,ix + b2,ix2 + . . . + bm−1,ixm−1 formed from the entries in the first row of bi. the bi(x) associated with a qt code are called the defining polynomials [7]. the set {b0(x),b1(x), . . . ,bp−1(x)} defines an [mp,m] qt code with k ≤ m. 3. defining polynomials the construction of qt codes requires a representative set of defining polynomials. these are the equivalence class representatives of a partition of the set of polynomials of degree less than m. for defining polynomials, multiplication by a non-zero element of fq does not change the weight and hence does not change the equivalence class. thus, two polynomials rj(x) and ri(x) are said to be equivalent if rj(x) = γx lri(x) mod (x m −λ), for some integer l ≥ 0 and scalar γ ∈ fq\{0}. a closed-form expression for the number of defining polynomials is now given. let g be the permutation (1,2, . . . ,m) so that g maps i to i + 1, for 1 ≤ i ≤ m− 1, and m to 1. therefore, gi, 1 ≤ i ≤ m, is also a permutation and has order m gcd(m,i) in the symmetric group of degree m. thus, the action of g on the m-tuple x = (x1,x2, . . . ,xm) changes x to (xm,x1,x2, . . . ,xm−1) where xi ∈ fq. now let λg be such that the action of λg on the m-tuple changes x to (λxm,x1,x2, . . . ,xm−1), the action of (λg)2 on x results in (λxm−1,λxn,x1,x2, . . . ,xm−2), and similarly for other powers. then the order of λg is ord(λ)m, where ord(λ) is the order of λ in fq. further, let t(λg), t ∈ fq \{0}, be such that it changes x to (tλxm, tx1, tx2, . . . , txm−1). the action of t(λg)2 changes x to (tλxm−1, tλxm, tx1, tx2, . . . , txm−2), and similarly for other powers. the equivalence relation is induced by the action of the group consisting of the elements t(λg)i, 1 ≤ i ≤ ord(λ)m, t ∈ fq \{0}. distinct equivalence classes correspond to distinct orbits under the action of this group and so can be enumerated using burnside’s lemma [5, 9]. definition 3.1. an ordered m-tuple (or word of length m), x = (x1,x2, . . . ,xm), is said to be fixed by t(λg)i, t,λ ∈ fq \{0}, 1 ≤ i ≤ ord(λ)m, if the m-tuple x remains unchanged by the action of t(λg)i. theorem 3.2. the number of words of length m over the alphabet fq fixed by t(λg)i for some fixed λ ∈ fq \{0}, t ∈ fq \{0}, 1 ≤ i ≤ ord(λ)m, is qgcd(m,i) if t ( m gcd(m,i) ) λ ( i gcd(m,i) ) = 1. otherwise, it is 1. proof. let x = (x1,x2, . . . ,xm) be a word of length m over fq. then the relation between the components of x before and after the action of t(λg)i is x1 = tλxm−i+1 x2 = tλxm−i+2 x3 = tλxm−i+3 ... xi−1 = tλxm−1 xi = tλxm xi+1 = tx1 xi+2 = tx2 ... xm−1 = txm−i−1 xm = txm−i 5 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 if 1 ≤ i ≤ m. if m + 1 ≤ i ≤ 2m, the relation between the components of x is x1 = tλ 2xm−i+1 x2 = tλ 2xm−i+2 x3 = tλ 2xm−i+3 ... xi−1 = tλ 2xm−1 xi = tλ 2xm xi+1 = tλx1 xi+2 = tλx2 ... xm−1 = tλxm−i−1 xm = tλxm−i. thus in general, the relation between the components of x is x1 = tλ jxm−i+1 x2 = tλ jxm−i+2 x3 = tλ jxm−i+3 ... xi−1 = tλ jxm−1 xi = tλ jxm xi+1 = tλ j−1x1 xi+2 = tλ j−1x2 ... xm−1 = tλ j−1xm−i−1 x = tλj−1xm−i if (j −1)m + 1 ≤ i ≤ jm, 1 ≤ j ≤ ord(λ). let gcd(m,i) = h. then, from the expressions above, the orbit of xm is xm = txm−i = t 2xm−2i = · · · = tb m i cxm−bm i ci = t(b m i c+1)λx2m−(bm i c+1)i = t (bm i c+2)λx2m−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−(b2m i c−1)i = t(b 2m i c)λx2m−(b2m i c)i = t (b2m i c+1)λ2x3m−(b2m i c+1)i = t(b 2m i c+2)λ2x3m−(b2m i c+2)i = · · · = t (b3m i c−1)λ2x3m−(b3m i c−1)i = t(b 3m i c)λ2x3m−(b3m i c)i = t (b3m i c+1)λ3x4m−(b3m i c+1)i = t(b 3m i c+2)λ3x4m−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−(b4m i c−1)i = t(b 4m i c)λ3x4m−(b4m i c)i = · · · = t (b( i h −1) m i c+1)λ( i h −1)x im h −(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)x im h −(b im h i c−1)i = t(b( i h m i )c)λ( i h )xm, 6 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 and so we also have xm−1 = txm−1−i = t 2xm−1−2i = · · · = tb m i cxm−1−bm i ci = t(b m i c+1)λx2m−1−(bm i c+1)i = t (bm i c+2)λx2m−1−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−1−(b2m i c−1)i = t (b2m i c)λx2m−1−(b2m i c)i = t(b 2m i c+1)λ2x3m−1−(b2m i c+1)i = t (b2m i c+2)λ2x3m−1−(b2m i c+2)i = · · · = t(b 3m i c−1)λ2x3m−1−(b3m i c−1)i = t (b3m i c)λ2x3m−1−(b3m i c)i = t(b 3m i c+1)λ3x4m−1−(b3m i c+1)i = t (b3m i c+2)λ3x4m−1−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−1−(b4m i c−1)i = t (b4m i c)λ3x4m−1−(b4m i c)i = · · · = t(b( i h −1) m i c+1)λ( i h −1)x im h −1−(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −1−(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)x im h −1−(b im h i c−1)i = t(b( i h m i )c)λ( i h )xm−1. similar expressions exist for xm−2,xm−3, . . . ,xm−h+2, and in general xm−h+1 = txm−h+1−i = t 2xm−h+1−2i = · · · = tb m i cxm−h+1−bm i ci = t(b m i c+1)λx2m−+1−(bm i c+1)i = t (bm i c+2)λx2m−h+1−(bm i c+2)i = · · · = t(b 2m i c−1)λx2m−h+1−(b2m i c−1)i = t (b2m i c)λx2m−h+1−(b2m i c)i = t(b 2m i c+1)λ2x3m−h+1−(b2m i c+1)i = t (b2m i c+2)λ2x3m−h+1−(b2m i c+2)i = · · · = t(b 3m i c−1)λ2x3m−h+1−(b3m i c−1)i = t (b3m i c)λ2x3m−h+1−(b3m i c)i = t(b 3m i c+1)λ3x4m−h+1−(b3m i c+1)i = t (b3m i c+2)λ3x4m−h+1−(b3m i c+2)i = · · · = t(b 4m i c−1)λ3x4m−h+1−(b4m i c−1)i = t (b4m i c)λ3x4m−h+1−(b4m i c)i = · · · = t(b( i h −1) m i c+1)λ( i h −1)x im h −h+1−(b ( i h −1)m i c+1)i = t(b( i h −1) m i c+2)λ( i h −1)x im h −h+1−(b ( i h −1)m i c+2)i = · · · = t(b( i h m i )c−1)λ( i h −1)xim h −h+1−( im ih −1)i = t( i h m i )λ( i h )xm−h+1. thus, the orbits of xm, xm−1, . . ., xm−h+1 are fixed by the action of t(λg)i if and only if t m h λ i h = t m gcd(m,i) λ i gcd(m,i) = 1. since there are h = gcd(m,i) independent orbits and each orbit can take on q values, qh = qgcd(m,i) words are fixed by t(λg)i. if t m gcd(m,i) λ i gcd(m,i) 6= 1, then there is only one orbit consisting of all m elements of the word. in this case, since λ 6= 1, the only word that is fixed is the zero word. theorem 3.3. the number of defining polynomials of length m over fq is mq,λ(m) = 1 (q −1)ord(λ)m ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + 1 (4) 7 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 proof. there are (q − 1)ord(λ)m permutations given by t(λg)i. thus, by burnside’s lemma [5], the number of orbits of words of length m over an alphabet of size q is equal to the average number of words fixed by each t(λg)i, 1 ≤ i ≤ ord(λ)m, t ∈ fq \{0}. therefore, we have mq,λ(m) = 1 (q −1)ord(λ)m ord(λ)m∑ i=1 t∈fq\{0} |fix t(λg)i|, where |fix t(λg)i| denotes the number of words fixed by t(λg)i. from theorem 1, the number of words fixed by t(λg)i is either qgcd(m,i) or 1 depending on whether t m gcd(m,i) λ i gcd(m,i) = 1 or not. therefore mq,λ(m) = 1 (q −1)ord(λ)m   ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 qgcd(m,i) +   ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) 6=1 1     = 1 (q −1)ord(λ)m   ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 qgcd(m,i) +  (q −1)ord(λ)m− ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 1     = 1 (q −1)ord(λ)m   ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + (q −1)ord(λ)m   = 1 (q −1)ord(λ)m ord(λ)m∑ i=1 t m gcd(m,i) λ i gcd(m,i) =1 (qgcd(m,i) −1) + 1 example 3.4. let q = 13, m = 2, and λ = 4. since 6 is the least integer such that λ6 = 46 = 1, we 8 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 have ord(λ) = ord(4) = 6. then m13,4(2) = 1 144 12∑ i=1 t∈f13\{0},t 2 gcd(2,i) 4 i gcd(2,i) =1 (13gcd(2,i) −1) + 1 = 1 144 {[ (13gcd(2,12) −1) ] + [ (13gcd(2,5) −1) + (13gcd(2,11) −1) ]} + 1 144 {[ (13gcd(2,8) −1) ] + [ (13gcd(2,10) −1) ]} + 1 144 {[ (13gcd(2,3) −1) + (13gcd(2,9) −1) ] + [ (13gcd(2,1) −1) + (13gcd(2,7) −1) ]} + 1 144 {[ (13gcd(2,1) −1) + (13gcd(2,7) −1) ]} + 1 144 {[ (13gcd(2,3) −1) + (13gcd(2,9) −1) ]} + 1 144 {[ (13gcd(2,4) −1) ] + [ (13gcd(2,2) −1) ]} + 1 144 {[ (13gcd(2,5) −1) + (13gcd(2,11) −1) ] + [ (13gcd(2,6) −1), ]} + 1. because 1 2 gcd(2,12) 4 12 gcd(2,12) = 2 2 gcd(2,5) 4 5 gcd(2,5) = 2 2 gcd(2,11) 4 11 gcd(2,11) = 3 2 gcd(2,8) 4 8 gcd(2,8) = 4 2 gcd(2,10) 4 10 gcd(2,10) = 5 2 gcd(2,3) 4 3 gcd(2,3) = 5 2 gcd(2,9) 4 9 gcd(2,9) = 6 2 gcd(2,1) 4 1 gcd(2,1) = 6 2 gcd(2,7) 4 7 gcd(2,7) = 7 2 gcd(2,1) 4 1 gcd(2,1) = 7 2 gcd(2,7) 4 7 gcd(2,7) = 8 2 gcd(2,3) 4 3 gcd(2,3) = 8 2 gcd(2,9) 4 9 gcd(2,9) = 9 2 gcd(2,4) 4 4 gcd(2,4) = 10 2 gcd(2,2) 4 2 gcd(2,2) = 11 2 gcd(2,5) 4 5 gcd(2,5) = 11 2 gcd(2,11) 4 11 gcd(2,11) = 12 2 gcd(2,6) 4 6 gcd(2,6) = 1, and therefore m13,4(2) = 1 144 [(132 −1) + [(13−1) + (13−1)] + (132 −1) + (132 −1) + [(13−1) + (13−1)]] + 1 144 [[(13−1) + (13−1)] + [(13−1) + (13−1)] + [(13−1) + (13−1)]] + 1 144 [(132 −1) + (132 −1) + [(13−1) + (13−1)] + (132 −1)] + 1 = 9. note that setting λ = 1 in (4) gives the number of defining polynomials for quasi-cyclic codes [14] mq,1(m) = 1 (q −1)m ∑ i|m φ(i) gcd(i,q −1)(qm/i −1) + 1. (5) table 1 gives the number of defining polynomials over f2, f3, f5, f7, and f11 with λ = 1, table 2 gives the number of defining polynomials over f4, f8, f9, and f16 with λ = 1, and table 3 gives the number of defining over f13, f17, and f19 with λ = 1. to illustrate the effect of λ, tables 4 and 5 give the number of defining polynomials over f3 and f4. 9 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 1. the number of defining polynomials over f2, f3, f5, f7, and f11 with λ = 1 q m 2 3 5 7 11 1 2 2 2 2 2 2 3 4 5 6 8 3 4 6 12 22 46 4 6 14 45 106 374 5 8 26 158 562 3226 6 14 68 665 3298 29576 7 20 158 2792 19610 278390 8 36 424 12255 120206 2679860 9 60 1098 54262 747330 26199450 10 108 2980 244301 4708486 259377496 11 188 8054 1109732 29959498 2593742462 12 352 22218 5086965 192243598 26153599626 table 2. the number of defining polynomials over f4, f8, f9, and f16 with λ = 1 q m 4 8 9 16 1 2 2 2 2 2 4 6 7 10 3 10 26 32 94 4 24 150 213 1098 5 70 938 1478 13986 6 238 6258 11107 186478 7 782 42806 85412 2556530 8 2744 299670 672825 35791946 9 9726 2130458 5380862 509033346 10 34990 15339642 43586287 7330084546 11 127102 111557594 356602952 106619309362 12 466198 818092242 2941985613 1563749966062 4. quasi-twisted codes over f17 and f19 in this section, the defining polynomials given above are used to construct quasi-twisted codes over f17 and f19. the number of defining polynomials over f17 for m = 1 to 5 is given in table 6 and over f19 for m = 1 to 5 in table 7. note that the zero polynomial is not considered in constructing codes. considering a code structure (i.e. qt), results in a search space that is smaller than for the general code design problem. the more restrictions on the structure, the smaller the search, but this creates a tradeoff since good codes may be missed if too much structure is imposed on the code. the qt codes presented here were constructed using a stochastic optimization algorithm, namely tabu search, which is similar to that in [10, 11, 14]. by restricting the search for good codes to the class of qt codes, and using a stochastic heuristic, codes with high minimum distance can be found with a reasonable amount of computational effort. based on the results obtained here, this approach provides a good tradeoff. the search for a (pm,m) qt code begins with a random set of p defining polynomials. a polynomial 10 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 3. the number of defining polynomials over f13, f17, and f19 with λ = 1 q m 13 17 19 1 2 2 2 2 9 11 12 3 64 104 130 4 605 1317 1822 5 6190 17750 27514 6 67117 251543 435760 7 747008 3663740 7094222 8 8497807 54499433 117943232 9 98189934 823526990 1991899630 10 1148826961 12599979635 34061506732 11 13576972684 194726683568 588334640902 12 161792326165 3034491071421 10246828768390 table 4. the number of defining polynomials over f3 with λ ∈{1,2} m λ number 1 1,2 2 2 1 4 2 2 3 3 1,2 6 4 1 14 4 2 11 5 1,2 26 6 1 68 6 2 63 7 1,2 158 8 1 424 8 2 411 9 1,2 1098 10 1 2980 10 2 2955 11 1,2 8054 12 1 22218 12 2 22151 is replaced with a new polynomial if this results in an increase in the minimum distance. this process is repeated until a code with the desired minimum distance is found or an iteration threshold is reached. the search is restarted periodically to ensure good coverage of the search space. it is not necessary to check the weight of every codeword in a qt code in order to determine the minimum distance d. only a subset of the codewords need be considered since the hamming weights of the polynomials rj(x) and ri(x) are the same if rj(x) = γx lri(x) mod (x m −λ), for integer l ≥ 0 and scalar γ ∈ fq\{0}. 11 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 5. the number of defining polynomials over f4 with λ ∈{1,α,α2} m λ number 1 1,α,α2 2 2 1,α,α2 4 3 1 10 3 α,α2 8 4 1,α,α2 24 5 1,α,α2 70 6 1 238 6 α,α2 232 7 1,α,α2 782 8 1,α,α2 2744 9 1 9726 9 α,α2 9710 10 1,α,α2 34990 11 1,α,α2 127102 12 1 466198 12 α,α2 466152 table 6. the number of defining polynomials over f17 m λ number 1 all 2 2 1,2,4,8,9,13,15,16 11 2 3,5,6,7,10,11,12,14 10 3 all 104 4 1,4,13,16 1317 4 2,8,9,15 1315 4 3,5,6,7,10,11,12,14 1306 5 all 17750 6 1,2,4,8,9,13,15,16 251543 6 3,5,6,7,10,11,12,14 251440 the best qt codes found over f17 are given in tables 8 to 10, and over f19 in tables 11 to 13. the defining polynomials are listed with the lowest degree coefficient on the left, i.e. 7321 corresponds to the polynomial x3 + 2x2 + 3x+ 7, with leading zeroes left out for brevity. the digits 10, 11,. . . , 18 are denoted by (10), (11), . . . , (18), respectively. as an example, consider the [24,4] code in table 12 with m = 4, λ = 1 and p = 6 defining polynomials. these polynomials give the following generator matrix g =   1129 1596 1632 01(14)(11) 0016 169(12) 9112 6159 2163 (11)01(14) 6001 (12)169 2911 9615 3216 (14)(11)01 1600 9(12)16 1291 5961 6321 1(14)(11)0 0160 69(12)1   with weight distribution 12 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 7. the number of defining polynomials over f19 m λ number 1 all 2 2 1,4,5,6,7,9,11,16,17 12 2 2,3,8,10,12,13,14,15,18 11 3 1,7,8,11,12,18 130 3 2,3,4,5,6,9,10,13,14,15,16,17 128 4 1,4,5,6,7,9,11,16,17 1822 4 2,3,8,10,12,13,14,15,18 1811 5 all 27514 i wi 0 1 20 6372 21 10944 22 28296 23 49680 24 35028 this code is optimal since it meets the griesmer bound (1), and so establishes that d19(24,4) = 20. the codes that meet the griesmer bound are indicated by ∗ in the tables. the codes given in the tables are qc codes (λ = 1) when a qc code has the highest minimum distance among all qt codes with the same length and dimension. in two cases for q = 19 and m = 4, a code with λ = −1 was found with a minimum distance higher than the corresponding qc code. 5. conclusion closed-form expressions for the number of twistulant matrices and 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[14] v. ch. venkaiah, t. a. gulliver, quasi-cyclic codes over f13 and enumeration of defining polynomials, j. discrete algorithms 16 (2012) 249–257. 14 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 http://www.codetables.de http://www.codetables.de http://www.codetables.de https://doi.org/10.1007/bf00124211 https://doi.org/10.1007/bf00124211 https://doi.org/10.1109/18.391267 https://doi.org/10.1109/18.391267 https://doi.org/10.1109/18.144719 https://doi.org/10.1109/18.144719 https://doi.org/10.1023/a:1018090707115 https://doi.org/10.1023/a:1018090707115 https://doi.org/10.1016/0097-3165(88)90078-7 https://doi.org/10.1016/0097-3165(88)90078-7 https://doi.org/10.1016/j.jda.2012.04.006 https://doi.org/10.1016/j.jda.2012.04.006 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 appendix a: table 8. qt codes over f17 with m = 3 code λ d bi(x) [6,3] 1 4∗ 11, 1(13)8 [9,3] 1 7∗ 18(11), 1, 117 [12,3] 1 10∗ 14(15), 1(12), 16, 157 [15,3] 1 13∗ 1, 118, 125, 152, 164 [18,3] 1 16∗ 118, 125, 1, 152, 13(12), 164 [21,3] 1 18∗ 152, 14(10), 13(16), 17(12), 127, 172, 115 [24,3] 1 21∗ 11, 137, 11(10), 145, 17(16), 18(11), 15(16), 13 [27,3] 1 24∗ 116, 12(11), 12(16), 18(11), 118, 11(13), 12, 147, 16(10) [30,3] 1 26 147, 11(14), 12(10), 16, 11, 164, 164, 1(13)8, 11(12), 182 [33,3] 1 29 184, 17(12), 14, 114, 17(10), 11(10), 19(16), 18, 12(13), 11(12), 1(13)4 [36,3] 1 32 18, 1(13)4, 12, 1(14), 1(12)8, 11(13), 14(13), 157, 18(12), 198, 1(10), 16(11) [39,3] 1 35∗ 116, 124, 117, 1(13)8, 11(11), 11(10), 12, 145, 19, 12(14), 1(12), 1(10)4, 15(14) [42,3] 1 38∗ 1(13), 15(12), 1, 152, 11, 125, 1(13)4, 12(11), 116, 175, 118, 198, 132, 137 [45,3] 1 40 1, 1(15), 118, 113, 1(12)8, 14(11), 124, 114, 19(16), 17, 126, 125, 154, 14, 11(16) [48,3] 1 43 15, 126, 14(15), 1(10)4, 15(16), 17(16), 11(14), 125, 12(12), 13(14), 143, 11(10), 1(10)(11), 11(16), 16(15), 157 [51,3] 1 46 18, 126, 198, 11(13), 157, 13(14), 15(14), 13(15), 12(16), 12(14), 12, 152, 13(10), 11, 16(11), 16(14), 14(11) [54,3] 1 49 11, 11(13), 154, 1, 135, 12, 152, 124, 143, 137, 14(11), 116, 16(14), 16(15), 13(12), 11(12), 1(10)8, 1(10) [57,3] 1 52∗ 11, 13(16), 1(13)4, 11(15), 1(13)8, 12, 15(14), 16(16), 12(11), 11(12), 18(12), 14(14), 1, 15(16), 125, 115, 147, 145, 162 [60,3] 1 54 12, 1, 113, 152, 12(10), 11(14), 114, 16(11), 15(12), 135, 1(10)(11), 17, 16, 126, 11, 134, 16(14), 157, 125, 165 [63,3] 1 57 11(13), 11, 12(15), 154, 19(16), 12, 12(10), 114, 125, 12(12), 15(14), 15(12), 117, 113, 14(10), 11(15), 16(15), 162, 1(13), 13(16), 123 [66,3] 1 60 13, 145, 15(12), 13(13), 117, 1(15), 12(10), 13(10), 143, 116, 14(13), 168, 16(11), 11, 11(12), 13(15), 1(14), 13(14), 115, 182, 17, 142 [69,3] 1 63 11(13), 1(12), 154, 1, 19, 128, 12(10), 182, 12(16), 12(15), 15(14), 18(12), 112, 11(12), 117, 198, 15(11), 1(11), 13(16), 11(14), 138, 1(13), 145 [72,3] 1 65 15, 175, 11, 13(12), 1(10), 11(12), 118, 19(14), 17(12), 16(11), 182, 154, 137, 138, 132, 16(15), 145, 18, 11(15), 125, 142, 147, 116, 17(10) [75,3] 1 68 1(10), 12, 12(13), 13(15), 126, 1(11), 12(16), 112, 11(13), 117, 12(12), 16(14), 14(13), 175, 1(10)4, 11, 14(11), 168, 11(16), 17(10), 17(16), 1(12), 1(12)8, 114, 127 15 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 9. qt codes over f17 with m = 4 code λ d bi(x) [8,4] 1 5∗ 1, 1128 [12,4] 1 9∗ 16(11)(14), 123, 1685 [16,4] 1 13∗ 12(10)(16), 1275, 113, 11(16)7 [20,4] 1 16 18, 13(11)7, 113(12), 1179, 12(15)2 [24,4] 1 20∗ 134, 1798, 18(11), 1(11)1(14), 126(10), 1(10)(11)(12) [28,4] 1 23 1(11), 1(13)1, 11(10)6, 137(15), 1(15)(11), 1(11)1(16), 1(10)5 [32,4] 1 27 1, 124, 13(14)7, 1(10)(11)(12), 126(16), 1(15)(16), 1437, 13(13)2 [36,4] 1 30 13, 15(16)(11), 13(15)(14), 1(10)2, 16(16), 11, 112(14), 12(13)(16), 1734 [40,4] 1 34 14(10)7, 1453, 1317, 1, 1(13)1, 11(15)4, 1598, 12(15)4, 1(12)2, 118(10) [44,4] 1 38 114(15), 12(10)(16), 11, 118(14), 113, 17(13)(16), 1(16)7, 14(15)(10), 1(12)7, 19(14), 147(16) [48,4] 1 41 12(12), 12, 11(16)(14), 181, 14, 111(14), 1254, 12(10)(13), 1114, 1(10)5, 156(10), 147(12) [52,4] 1 45 124(16), 14(13)3, 1114, 16, 1(15)(16), 12, 183(10), 118(16), 1192, 13(10), 12(14), 173(11), 14(15)(10) 116(15), 14(10)2, 1(11)64, 14(14)7 [60,4] 1 52 11, 11(13), 13(16), 146(13), 153, 13, 192, 164(10), 1386, 137(13), 1372, 125(16), 11(10)5, 1272, 12(13)3 [64,4] 1 56 13, 158(14), 11(15)(14), 1982, 1(13)5, 1(10)1(16), 12(14)5, 1517, 13(13)(12), 103, 11(12)9, 1(12)(15)8, 116(16), 11(15)7, 1239, 1(12)(10) [68,4] 1 59 12, 1(10)1(12), 12(13)(12), 1(11)2, 151(11), 17(12)(15), 16(11)(12), 1(15)(16), 1187, 1472, 124(15), 13(10)(13), 11(14), 1462, 1(11)3, 172(14), 11 [72,4] 1 63 1(15), 12(10)2, 11(16)(11), 11(16)7, 1(13), 15(16)5, 1838, 12(15), 14(15), 13(13)(11), 1682, 1(12)38, 1468, 135(15), 1795, 137(10), 14(10), 175 [76,4] 1 67 1(10), 12(16)7, 15(12)8, 1127, 13(15)5, 12(15)6, 1358, 126(16), 1123, 121(13), 14, 1387, 187, 1654, 11(12)(14), 1193, 163(11), 1(12)(13), 1468 [80,4] 1 70 11(13), 1, 13, 1(10)1(11), 15(12)4, 1268, 167, 11(12), 141(10), 184(15), 1517, 17(12)(11), 16(15)(14), 12(14)(15), 18(14), 1188, 17(10)(14), 138(12), 111(16), 176(16) [84,4] 1 74 1418, 17, 1356, 135(15), 13(16)2, 12(11)(13), 136, 1162, 16(10)(15), 149(16), 147(13), 12(16)(13), 17(16), 11(14)5, 1164, 1213, 121(15), 1838, 11(11)5, 1547, 17(10)4 [88,4] 1 78 1(10), 12(11)(16), 1(11)(13), 15(14), 13(16)8, 138(11), 133, 172(15), 1568, 13(16)(10), 132(14), 1(13)(15), 1169, 1(11)1, 17(15), 1146, 14(13)2, 14(10)(11), 11(10)(11), 14(11)(15), 14(13)(14), 15(11)(12) [92,4] 1 81 13, 11(16)2, 17(12)5, 16(15)(12), 11(10)(11), 1(13)(16), 1(10)(11), 119(11), 16(14), 163(11), 139, 1(12)2, 13(13)8, 15(16), 162, 128(13), 1652, 14(12), 16(16)5, 162(14), 118(15), 11(10)7, 1(10)1 [96,4] 1 85 1, 16(11)(14), 1145, 16(10)(14), 102, 14(11)(12), 1(10)8(11), 111, 11(11)2, 13(13)8, 176(10), 16(11)8, 1376, 12(11)5, 1(10)8(12), 187, 15(12)3, 1(12)3(11), 11(12)4, 17(12)(10), 111(12), 1(10)(12), 125(14), 1832 [100,4] 1 89 105, 1(12)3, 1235, 198(12), 13(15)3, 151(15), 134(10), 11, 146(10), 1(13)(15), 1(11)4, 11(12)(11), 184(15), 143(15), 113(14), 1158, 17(12)(10), 1173, 116(10), 183(10), 119(11), 19(15)4, 11(15)(13), 1584, 1425 16 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 10. qt codes over f17 with m = 5 code λ d bi(x) [10,5] 1 6∗ 18, 16(15)(16)2 [15,5] 1 10 11, 162(15)(14), 11(11)(15)(12) [20,5] 1 14 131, 11(13)96, 1041, 11(11) [25,5] 1 19 15, 11(11)(16)(12), 135(12)4, 1126(10), 1278(14) [30,5] 1 23 11(13)(10)(15), 1(11)(10)(16), 115, 115(11)4, 112(11)8, 17(12)8(11) [35,5] 1 28 111(16)6, 143(13)(16), 121, 106(13), 15(14)68, 119(16)6, 13(13)(16)7 [40,5] 1 32 19, 13415, 1576(16), 1547, 17(13)(15)(14), 12(12)(13)(12), 13(12)85, 12(13)(11)4 [45,5] 1 36 1(16)58, 131, 13(12)9, 1(15)7(16), 11(13)(15)(14), 12(15)3(15), 1(10)1(10), 13(14)3(12), 141(15)(12) [50,5] 1 41 16, 121(15)(13), 15217, 172(13)(12), 11132, 17(10)1(16), 13(11)75, 14(14)2(15), 12(11)26, 11(15)7(14) [55,5] 1 45 1, 12, 11(16)(14)(15), 1(15)(11)4, 1416(10), 11636, 19(10)7, 111(15)8, 11(13)7(16), 12495, 1748 [60,5] 1 50 19, 10(11)4, 1(10)83(11), 1(14)74, 11(13)(16)3, 11452, 13(10)(15)(14), 13(12)(14)(16), 12(12)34, 12(13)8(15), 17(10), 154(15)3 [65,5] 1 54 13, 11, 1058, 14(14)4, 126(14)8, 13(11)(12)(10), 121(11)8, 1389(12), 19(15)3, 129(12), 13(14)94, 12(15)27, 1212(13) [70,5] 1 59 11, 13475, 1(12)(10)(14), 11(13)4, 18(12)(15)2, 14(12)4, 14(13)(16)3, 173(11)(12), 11235, 1586(11), 1(16)6(12), 18(14)(10), 15(12)1(12), 1(14)9(10) [75,5] 1 63 1(10)(11), 1956, 19, 16(15)(12)(16), 19(16)4, 11(14)59, 112(10), 15(11)(12)8, 17(14)3, 17(16)42, 11(10)4(15), 11(11)3, 17(16)1(14), 12(15)1, 19(14)8 [80,5] 1 68 1, 1(16)(14)6, 14, 13(14)6(15), 163(11)(12), 111, 17595, 179(15)(14), 14382, 1(12)5(12)(10), 1(13)(14)9, 1(12)(15)6(16), 14(10)2, 12(14)15, 19(13)82, 1825(11) [85,5] 1 72 1(10), 1327(16), 11, 1(11)11, 177, 15(10)4, 11513, 13263, 155(14), 12(16)8(14), 1342(11), 123(15)9, 125(13)(14), 12(14)38, 1(12)5(12), 17(10)4(10), 1641(14) [90,5] 1 77 1(11), 119(13), 1933, 18198, 13714, 115(15)(15), 195(13), 17(11)3, 11475, 13(15)7, 156(10)2, 1(11)(16)(10), 15(16)2(15), 15(11)34, 113, 11(10)(12)3, 1(10)5(13), 11(14)(10)3 [95,5] 1 81 1, 12432, 1438(10), 13(11)1(12), 1(11)42, 131(10)(11), 14(11)1(15), 1425(11), 10(10), 1185(13), 1(12)18, 15982, 118(15)3, 1794(14), 12927, 1321(14), 13(14)(16)(14), 12(14)(12)(15), 1976(16) [100,5] 1 86 16(14)(13)(11), 15(14)72, 11, 1, 1431(11), 12(10)(13)3, 113(12)(14), 1156(10), 115(11), 1752(16), 12(12)(10)(13), 1247, 173(13)4, 172(15)(16), 1(13)(11)(11), 116(11)(14), 146(12), 16(15)(16)8, 11(11)(15)3, 157(16)4 [105,5] 1 90 13(13)(10)5, 1(11)37, 11, 1(14)(11)3, 117(14)3, 13(11)(16)(12), 13, 123(16)7, 1195, 19(12)9, 15(16)(12), 11(14)39, 11(14)6(14), 1(12)8, 15932, 1517(10), 1(12)(11)9, 156(10)(12), 13(10)6(13), 1183(13), 11(10)(16) [110,5] 1 95 12, 14, 13(12)9, 1(15)7(16), 11(13)(15)(14), 12(15)3(15), 1(10)1(10), 13(14)3(12), 141(15)(12), 15168, 11534, 1568(14), 118(15)(12), 1(10)44, 135(11)3, 163(11)(12), 11(15)(10)(16), 13(11)(15)(14), 169(16)(10), 117(11), 12(11)(14)(10), 14234 [115,5] 1 99 18, 1, 11(10)3(16), 1(11)(12)(15)8, 13(16)9(11), 13(16)95, 13(16)54, 1012, 1127(14), 15942, 131(16)2, 1187, 12(12)9(12), 1326, 15(14)1(14), 11112, 18(11)(12)8, 15(11)32, 164(10)8, 159(12)5, 1(14)(16)(13), 103(13), 17(11)4 [120,5] 1 104 14, 117(15)8, 1071, 15(13)(11)(15), 15(12)1, 13(12)(16)(13), 14(15)62, 134(10)(13), 15(16)85, 15(14)(16), 12, 19(11)(12), 1815, 112(14)(13), 14(11)2, 15(14)(10)8, 12(12)(11)(15), 1256(13), 1(12)3(12), 1452(16), 1243(10), 1166(14), 13(11)(10)4, 11(10)5(15) [125,5] 1 109 1(10), 13184, 13(13)(16), 12(13)53, 1081, 1(11)(10)(10), 13(11)2, 15(14)(15)2, 11(14)(11)9, 127(16)(13), 1415(12), 12972, 15175, 11(11)(12)6, 1(13)(16)(13), 19832, 14295, 13(16)(11), 12(16)4(15), 132(14)(15), 156(14)2, 11(12)87, 131(11)4, 15(16)64, 1656 17 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 11. qt codes over f19 with m = 3 code λ d bi(x) [6,3] 1 4∗ 1, 112 [9,3] 1 7∗ 135, 11, 16(16) [12,3] 1 10∗ 1, 15(12), 17(15), 138 [15,3] 1 13∗ 1(11), 154, 11(16), 149, 14(17 [18,3] 1 16∗ 17, 159, 11(16), 157, 145, 176 [21,3] 1 18 1, 114, 142, 12, 137, 159, 16(15) [24,3] 1 21∗ 11, 15(12), 1(11)(18), 138, 15(17), 139, 1(10)6, 159 [27,3] 1 24∗ 12, 18(14), 116, 162, 189, 154, 14(14), 13(12), 11(14) [30,3] 1 27∗ 1(11), 1, 15(11), 1(10)7, 1(13)6, 135, 11(13), 1(10)(18), 15(16), 128 [33,3] 1 29 112, 115, 1(12)(14), 12, 11, 15(12), 138, 1(17), 1(16), 128, 192 [36,3] 1 32 14(15), 15, 125, 117, 12(13), 158, 18(17), 11(13), 13(12), 1, 17(15), 11(17) [39,3] 1 35 167, 16, 11(11), 1(14), 17(12), 14, 1(12)(14), 173, 17, 13(16), 146, 127, 152 [42,3] 1 38∗ 1(11), 11(17), 17(18), 114, 12, 12(12), 15(17), 1(13)6, 18(17), 162, 11(12), 19, 116 1(10)(18) [45,3] 1 41∗ 14, 1(14)6, 16(14), 17, 11(11), 162, 17(12), 187, 16, 1(16), 1(10)(15), 1(10)6, 11(12) 178, 169 [48,3] 1 43 15(16), 1, 168, 12(12), 19(14), 129, 1(16), 12, 1(12)9, 154, 135, 189, 112, 1(10)9 1(10)8, 117 [51,3] 1 46 12, 18(17), 162, 15(11), 11(11), 137, 13(14), 15, 17(17), 11, 163, 1, 11(13), 12(15) 19(14), 135, 18(15) [54,3] 1 49 1, 18, 176, 13(16), 162, 12(12), 1(10), 125, 154, 192, 147, 1(10)6, 135, 159, 134, 169 13(12), 17 [57,3] 1 52 14, 13, 1, 1(13)6, 129, 143, 1(10), 168, 118, 158, 124, 1(10)(18), 18(14), 19(14), 14(15) 1(11)(18), 159, 12(13), 14(14) [60,3] 1 55 1(14), 1(10), 14, 1(17), 115, 126, 17(18), 154, 17, 1(13)6, 129, 12(15), 137, 11(16), 128 1(11)(18), 1(13)(18), 127, 113, 149 [63,3] 1 57 17, 1(12)9, 1(16), 1(14), 193, 135, 1(10)6, 12(16), 19, 1(18), 162, 116, 145, 126, 129 169, 14(17), 15(12), 11(16), 1(12), 1(14)6 [66,3] 1 60 1(12), 11(15), 15(18), 176, 137, 193, 138, 12(11), 1(10)9, 1(11)(18), 13(16), 135, 162 11(13), 16(15), 1(16)6, 17(18), 132, 17, 1(12)(14), 147, 125 [69,3] 1 63 1, 18, 1(16), 134, 1(13)(18), 14(15), 114, 1(14), 115, 11(11), 139, 186, 15(16), 12(15) 1(13), 15(17), 129, 1(10)8, 17(18), 14(11), 132, 14(14), 126 [72,3] 1 66 1(18), 12(15), 173, 117, 139, 1(15), 15(18), 1(12)9, 167, 1(13), 11, 127, 1(13)6, 11(16) 11(14), 13(16), 19, 168, 129, 13(17), 1(11)(18), 138, 14(11), 16(16) [75,3] 1 69 15, 12(14), 13(17), 149, 1, 1(14), 193, 11(13), 134, 16(16), 1(13)(18), 15(16), 115, 1(13) 14(15), 12(17), 1(11)(18), 128, 116, 16(14), 17(17), 15(12), 186, 137, 11 18 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 12. qt codes over f19 with m = 4 code λ d bi(x) [8,4] 1 5∗ 12, 114 [12,4] 1 9∗ 1(10), 1193, 1(10)6(15) [16,4] -1 13∗ 12(12), 16, 134(12) , 111(14) [20,4] -1 17∗ 14(17), 115(17), 1281, 11(11)9, 1146 [24,4] 1 20∗ 1129, 1596, 1632, 1(14)(11), 16, 169(12) [28,4] 1 23 18, 135(14), 19, 16(14)(17), 11(15), 145(11), 14(11)(15) [32,4] 1 27 104, 112, 1534, 1629, 1737, 17(15)6, 12(16)6, 1356 [36,4] 1 30 18, 119(12), 12(16), 112, 15(11)(15), 13(16)3, 121(18), 15(18)(11), 1719 [40,4] 1 34 106, 14(14)3, 1627, 11(12)(18), 16(10)(14), 14(11)(12), 157(15), 152(16), 1(16)9 1923 [44,4] 1 38 16, 127(14), 12(14)(16), 15, 1(10)(18)7, 18(17)6, 149(15), 1398, 158, 1279, 141(18) [48,4] 1 41 159(14), 11, 15(17), 121(13), 15(13)(16), 14(10)(17), 12(14)6, 124(18), 146(11), 13(11) 1168, 118(15) [52,4] 1 45 1(12), 1(12)5, 14(11), 123(18), 14(10)(18), 13(15)(12), 12(14)(11), 15, 187(17) 163(17), 159(11), 17(12), 1174 [56,4] 1 49 13, 11(15)(13), 1(10)5(14), 1(11)59, 161(14), 1619, 13(10)(15), 1(12)(16), 11(12)(11) 1(10)(12), 12(15)(18), 13(16)6, 126(11), 1272 [60,4] 1 52 1843, 169, 17(17), 1153, 11(11)2, 1(11)8(14), 183(17), 16(16)2, 141(12), 12(14)(15) 11(15), 1416, 1576, 148, 162(17) [64,4] 1 56 1(14), 153(15), 169(16), 11(12)(16), 1157, 19, 1(12)4, 1427, 124(11), 1144, 12(16)2 1(16)7, 14(17)9, 12(11)7, 169(15), 1188 [68,4] 1 60 14, 1283, 1642, 15(17)(16), 1(16)(10), 1(11)7, 1(12)29, 12, 1923, 1287, 17(15)8, 113 12(14)3, 131(18), 17(10), 1582, 149(12) [72,4] 1 63 14, 15(14)6, 1(11)9, 1215, 19(15), 15(17)(18), 1, 168(18), 1278, 11(17)4, 13(12)8 1(11)7(15), 16(14)(15), 1192, 164, 1(11)(16)6, 111(12), 132 [76,4] 1 67 1(10), 152(17), 1(12)(13), 113(17), 1(11)4, 12(13)(14), 1(11)78, 191(14), 191, 13(12)3 14(12), 1358, 1696, 11(18)(15), 1(12)2(15), 1(12)8, 1(18)7, 1194, 1(13)(13) [80,4] 1 71 15, 14(11)4, 12(18)2, 1(15)9, 1(13)6(14), 129(14), 11(13)(16), 1153, 12(14)(13), 128 1(16)(17), 167(12), 12(10)(16), 13, 11(11)(15), 14(14)(15), 12(17), 117(10), 1(12)2(15) 14(18)(12) [88,4] 1 78 1(11), 1629, 11(11)(15), 178(12), 14(17)9, 113(11), 12(18)(16), 12(13)(15), 1923 113(12), 11(17), 14(15)4, 1287, 125(13), 14(10)5, 102, 1(12)78, 14(14)(17), 1(11)53 19(17), 154(17), 12(10)(12) [92,4] 1 82 1(12), 13(18)(14), 13(13)6, 1319, 115(15), 106, 164, 12(17)7, 19(12)7, 14(14)3, 13(13)3 14(17)9, 19(12)(18), 112(17), 138(17), 1(11)(17), 1175, 15(13)(12), 168, 1(10)58, 181 16(13)3, 178(12) [96,4] 1 86 17, 1(11)(10)8, 18(17)(14), 1(18)4, 145(17), 13(17)(18), 1(11)(15), 12(14)(15), 1293 11(16)9, 164(14), 1532, 1(10)78, 12(10)(11), 14(13)4, 14(14)2, 186(14), 111(16) 15(13)9, 1569, 132(17), 13, 11(16)(14), 12(11) [100,4] 1 89 15, 1, 1(13)9, 131(15), 14(10)7, 1(10)5(18), 11(16)7, 159(16), 16(14), 138(12), 11(18)6 11(11)(13), 118(10), 12(12)7, 115, 19(11)2, 11(14)(15), 13, 12(17)4, 1(10)(18), 17(18)3 17(10)(17), 15(12)8, 1(11)78, 141(12) 19 t. a. gulliver, v. ch. venkaiah / j. algebra comb. discrete appl. 7(1) (2020) 3–18 table 13. qt codes over f19 with m = 5 code λ d bi(x) [10,5] 1 6∗ 104, 1(10)(18)59 [15,5] 1 11∗ 114, 11(15)(13)(10), 14(14)(10)3 [20,5] 1 15 105, 11(15)(18)(12), 12(18)(15)(18), 131(12)6 [25,5] 1 19 12, 117(10)(15), 198, 12(12)(11)(13), 1(15)(14)4 [30,5] 1 23 11, 11(15)8(15), 11(16)(13)9, 1(11)12, 11634, 1(16)(18)8 [35,5] 1 28 132(14)(15), 11, 10(13)(18), 11(13)(11)5, 135(12)(15), 159(12), 1(14)(18)1 [40,5] 1 32 1(13)1, 1185(15), 17637, 16(13)(11)8, 1(15)(13)2, 131(14)(11), 17(12)1(14) 1(18)(13)(14) [45,5] 1 37 1(12), 1(16)(10)7, 13(11)(15)4, 17(10)53, 1567(18), 12(17)62, 1(11)71(15) 16(10)8(17), 119(16)3 [50,5] 1 41 1(15)(15)6, 1298(11), 13(18)4(14), 13, 143(18)9, 1(14)(13)(11), 11766, 1(13)9(12) 124(10)(16), 19(13)(11)2 [55,5] 1 46 1(16), 14(13)(17)(12), 143(18)(11), 11(12)7(10), 13(17)(16)(11), 12347, 1(10)(15)93 138(12)7, 138(14)(17), 11(13)(11)6, 1(15)(11) [60,5] 1 50 1, 15(12)4(18), 11345, 1562(15), 12145, 12(10)5(17), 108, 18(16)5, 13(18)8(12) 12(16)(12)(16), 15(18)(13)(15), 12634 [65,5] 1 55 11(10), 11(12)8(10), 16843, 137(13)4, 18(14)(16)6, 1(10)2, 13(11)(10)5, 1237(15) 1287, 17(18)57, 13(17)95, 13(15)(18)8, 13235 [70,5] 1 59 13, 13(11)(18)8, 1146(13), 11524, 13739, 12(10)(13)(16), 108(10), 16953, 17(11)(17) 13(17)74, 1535, 18387, 16(13)(12)(11), 141(15)(12) [75,5] 1 64 13, 1(13)(11)(13)(18), 12, 13(16)(18)4, 189(13)(17), 12(18)(15), 1177(17), 145(13) 12(13)(12)2, 15(17)(16)6, 1013, 11(17)(15)(11), 117(15), 11112, 14(14)57 [80,5] 1 68 128(17)3, 1934(18), 18, 132(10)(18), 184(14)6, 1(16)52, 133(16), 13(13)(15)2 12(15)43, 13(16)17, 14(14)15, 17586, 12(15)15, 11962, 17(14), 117(12)9 [85,5] 1 73 1(15), 13(16)2, 143(11)9, 12(16)(17)4, 1136(11), 13(18)46, 15(17)59, 1163(17), 1 126(15)6, 13(16)56, 161(12)8, 10(16), 15(18)5(18), 12(15)(16)(18), 135(14)(11) 17(17)(15) [90,5] 1 77 101, 102, 11(11)(11)(18), 1418(17), 1274, 12(16)7, 13(17)(12)(18), 11(13)(16)(11) 1693, 13(18)(14)(12), 15(17)1(11), 11778, 1(15)1, 13854, 1457, 11(14)(17)4, 1135 12(13)(12)(14) [95,5] 1 82 1, 12(14)3(17), 101, 11(17)7(12), 14(17)(14)(15), 137(17)(16), 12(10)49 1(13)(14)(16), 12(14)68, 114, 12(16)1(12), 134(16)5, 18(12)4, 18(13)2, 161(15)2 1(10)(18)(11), 11(14)4(14), 112(15)(14), 12(12)43 [100,5] 1 87 111, 14, 183(16), 192(13)(17), 17(17)(14)6, 11(15)38, 13(13)(18)(15), 13(18)7(12) 1(12)(16)1, 13(13)(15)5, 1354(14), 164(18)(11), 1585, 145(11)5, 1(11)(17)4 1(10)(18)(17), 11(15)(17)(11), 17(18)7(18), 117(18)(12), 13(14)46 [105,5] 1 91 10(14), 11, 14(12)(15), 1118(17), 12, 1(13)(14)3, 18(14)6(15), 14(13)46, 137(13)9 11(16)(17)6, 1247(12), 1354(16), 11(10)27, 115(14)8, 131(12)3, 12(18)78, 114(14)2 11(11)75, 1538, 1(16)(16)2, 16(11)4 [110,5] 1 96 1025, 14, 18, 193(10)(17), 12(10)4(11), 1(10)8(17)(14), 14(13)2(18), 1(18)(18)(16) 137(10)(14), 11(18)78, 158(11)(15), 111(17)6, 1(13)(11)(13)6, 12(18)42 13(12)(16)(12), 11484, 1383(15), 142(12)6, 12(15)2(17), 11279, 14(11)(16)(17) 1693(17) [115,5] 1 100 12, 1(14)5, 1(11), 145(16)5, 131(14)(18), 11(16)(17)(10), 177(12), 198(13), 161(14)6 14(13)(10)(14), 181(13), 1239, 11(13)9(14), 18(11)7(15), 12(10)1(16), 13(12)3(15) 167(12)(18), 11277, 19(10)1, 1442, 115(13)(17), 1(17)(10)4, 15(11)(12)6 [120,5] 1 105 118, 1769(12), 1, 142(10)3, 1(14)(17)(14), 12(12)(17)(18), 14(10)8(17), 1616 1529(16), 1(10)(17)4(14), 1144(16), 135(12), 15(13)4(14), 11853, 1(12)(10)(17) 11(10), 153(13)8, 13(14)34, 18(14)(10)(14), 13(11)1(18), 12(17)5(14), 11(15)(17)(16) 1(17)42, 123(17)(16) [125,5] 1 109 102, 17(16)7, 13894, 13(15)(18)(14), 13(18)(10)(15), 12(14)76, 11, 15696, 142(13)3 11(11)(12)9, 15376, 11(13)76, 115(10), 1(10)52, 112(17)(10), 1145(10), 18957 132(11)2, 132(10)6, 118(11)5, 12(16), 13(14)2(17), 13(13)(10)3, 15(18)32 134(11)(15) 20 introduction quasi-twisted codes defining polynomials quasi-twisted codes over f17 and f19 conclusion references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(1) • 65-73 received: 3 august 2014; accepted: 30 december 2014 doi 10.13069/jacodesmath.66836 journal of algebra combinatorics discrete structures and applications graphical properties of the bipartite graph of spec(z[x])\{0} research article christina eubanks-turner1∗, aihua li2∗∗ 1. department of mathematics, loyola marymount university, los angeles, california 2. department of mathematical sciences, montclair state university, montclair, new jersey abstract: consider spec(z[x]), the set of prime ideals of z[x] as a partially ordered set under inclusion. by removing the zero ideal, we denote gz = spec(z[x])\{0} and view it as an infinite bipartite graph with the prime ideals as the vertices and the inclusion relations as the edges. in this paper, we investigate fundamental graph theoretic properties of gz. in particular, we describe the diameter, circumference, girth, radius, eccentricity, vertex and edge connectivity, and cliques of gz. the complement of gz is investigated as well. 2010 msc: 05c25, 05c63, 06a11 keywords: bipartite graph, prime spectrum, poset, ring theory 1. introduction throughout, r is a commutative noetherian ring with identity 1. the prime spectrum of r, spec(r), is the partially ordered set (poset) of prime ideals of r ordered by inclusion. considerable research has been done in order to determine which partially ordered sets (posets) arise as spectra of two-dimensional integral domains, such as, spectra of polynomial and power series rings [5–8, 10, 11]. in the past, diagrams were used to represent the prime spectra of certain rings. we refer to these diagrams as spec graphs. for example, in 1989, william heinzer and sylvia wiegand used spec graphs to help illustrate the relationships of the elements of spec(r[y]), where r is a semi-local countable onedimensional noetherian domain and y is an indetereminate. similarly, spec graphs have been used to describe certain polynomial rings and power series rings (cf. [2–4, 7, 8]). the primary aim of those papers is to find properties that the spectra satisfy as partially ordered sets, in particular those properties that determine the posets. ∗ e-mail: ceturner@lmu.edu, supported by the association for women in mathematics. ∗∗ e-mail: lia@mail.montclair.edu 65 graphical properties of gz recently, researchers have begun to give more attention to graphs derived from the ring structure. they have also studied connections between the graph properties and the original rings. for example, in 1999 anderson and livingston introduced zero-divisor graphs [1]. in this direction we expand the idea of applying graph theory to spec graphs. for a ring r, we consider spec(r) as an undirected graph whose vertices are the prime ideals of r and two distinct vertices p and q are adjacent if and only if either p ⊂ q or q ⊂ p. here the inclusion “⊂” is strict, since spec(r) is a poset. this implies there are no loops in the graph. we investigate spec graphs of commutative rings in a graph theoretical way, which may lead to information about the ring itself. in this paper we focus on the bipartite graph derived from spec(z[x]) \ {0}, which we call gz. we give background on the poset spec(z[x]) in section 2. in section 3, we give results describing fundamental graph theoretic properties of gz. particularly, we show that the radius of gz is 3, the diameter of gz is 4, gz has infinite circumference and infinite vertex and edge connectivity. in section 4 we look at certain interesting subgraphs of gz and describe properties related to those subgraphs, in particular, completeness. we also show that the complement, gz, retains certain properties similar to those of gz, although gz is not bipartite. 2. overview of spec(z[x]) definition 2.1. let u be a partially ordered set with a unique minimal element u0. the height of an element u in u, ht(u), is the length of the longest chain to u from u0. we denote the dimension of the set u by dim(u) = max{ht(u) ∣∣ u ∈ u}. the set of height-i elements of u is denoted hi(u) = {u ∈ u ∣∣ ht(u) = i}. notation 2.2. for u a partially ordered set and u ∈ u, we let u↑ = {v ∈ u|u < v} and u↓ = {v ∈ u|u > v}. the spectrum of the polynomial ring z[x] has been well described by roger wiegand in 1986 [11]. in [11] roger wiegand characterized spec(z[x]) as a partially ordered set satisfying five specific axioms, called czp axioms. definition 2.3. [the czp axioms] for any poset u the following axioms are called the czp axioms: (p1) u is countable with a unique minimal element. (p2) u has dimension two. (p3) every height-one element of u is below countably infinitely many height-two elements. (p4) for each pair of distinct elements of height one, there are only finitely many height-two elements above both of them. (p5) given finite subsets s and t, with ∅ 6= s ⊂ h1(u) and t ⊂ h2(u), there is a height-one element w in u such that w < m, ∀m ∈ t, and whenever m′ ∈ u is greater than both w and s for some s in s, then m′ ∈ t. as in [8], we call the element w from (p5) the radical element of the pair (s, t) and it can be generalized to higher dimension posets. definition 2.4. let u be a partially ordered set of dimension n. let ∅ 6= s ⊆ hi(u) and t ⊆ hi+1(u) with 0 < i < n. an element w ∈ u is called a radical element if w < m and ∀m ∈ t, then w↑ ∩ s↑ ⊆ t, for all s ∈ s. thus the fifth axiom (p5) can be restated: (p5′) given a pair of finite subsets s, t of u such that ∅ 6= s ⊂ h1(u) and t ⊂ h2(u), there exists a height-one radical element for the pair (s, t). we call such a pair (s, t) in (p5′) a (1, 2)-pair. 66 c. eubanks-turner, a. li figure 1. illustration of axiom (p5) theorem 2.5. if u is a poset satisfying (p1)-(p5) of definition 2.3, then u ∼= spec(z[x]) [11]. the following properties from [8] are useful for our investigation. theorem 2.6. let u be a poset that satisfies the czp axioms in definition 2.3. then 1. if w is a radical element for a (1, 2)-pair (s, t), then w /∈ s. 2. if s ⊆ s′ ⊆ h1(u) and t ⊆ t ′ ⊆ h2(u), where t ≯ s,∀t ∈ t ′ − t,∀s ∈ s 6= ∅, then a radical element w for (s′, t ′) is also a radical element for (s, t). 3. in spec(z[x]), any (1, 2)-pair (s, t) has infinitely many radical elements. next we give a proposition useful for our investigation, which is related to theorem 2.6. proposition 2.7. let u be a poset that satisfies the czp axioms. 1. ∀u ∈ h1(u), there exists infinitely many u′ ∈ h1(u) such that u↑ ∩u′↑ = ∅. 2. every height-two element of u has infinitely many height-one elements below it. 3. every height-two element of u has infinitely many height-one elements not below it. proof. 1. follows from theorem 2.6: any radical element u′ for the (1, 2)-pair ({u},∅) will work. for 2. and 3., we use the fact that u ∼= spec(z[x]) as posets. consider v = 〈f, p〉 ∈ spec(z[x]), where p is a prime number and f is an irreducible polynomial in z[x]. then by theorem 2.6 (3), there are infinitely many radical elements for the (1, 2)-pair (〈p〉, v) and so we have statement 2. for all prime integers q 6= p, 〈q〉 ∈ h1(u) and 〈q〉 ≮ v. since there are infinitely many such q, 3 holds. theorem 2.8. the following noetherian integral domains have spectra isomorphic to spec(z[x]): 1. the polynomial ring d[x], where d is an order in an algebraic number field [10]; 2. two-dimensional integral domains which are finitely generated algebras over an algebraic extension of a finite field [11]; 3. finitely generated birational extensions of the polynomial ring z[x] [8]. after characterizing spec(z[x]), roger wiegand gave the following conjecture: for every twodimensional finitely generated z-algebra d, spec(d) ∼= spec(z[x]), c.f. [10]. this conjecture remains open. 67 graphical properties of gz 3. properties of the graph gz in this section, we treat gz := spec(z[x]) \ {0} as an undirected bipartite graph and investigate basic graph theory properties related to gz. first, we define the graph as follows: definition 3.1. let u = spec(z[x]) be the poset of the prime ideals of z[x] ordered by inclusion. the graph gz = (v (gz), e(gz)) derived from u, with partites h1(u), h2(u), is given by v (gz) = u \{0} = h1(u)∪h2(u) and e(gz) = {st | s ∈ h1(u), t ∈ h2(u), and s < t in u}. all edges of gz have the form: st, where s ∈ h1(u), t ∈ h2(u). remark 3.2. note that since u is a partially ordered set this ensures the graph gz is bipartite. that is, every edge of gz has one vertex in h1(u) and one vertex in h2(u). we now denote hi(u) as hi(gz) for i = 1, 2. many basic graph theory properties of gz can be obtained from the ring theory properties of spec(z[x]), for example, proposition 2.7 (2) and (3) shown above. we investigate the connectivity, existence of cycles of different sizes, cut-edges, cut-vertices, values of girth, diameter, circumference and so on. existence of radical elements plays an important role in the process. here, we define the latter mentioned terms. definition 3.3. let g be a graph. 1. the girth of g is the length of the shortest cycle contained in g. 2. the circumference of a graph g is the longest cycle contained in g. 3. a cut-edge in a graph g is an edge that increases the disconnectivity when removing it. 4. a cut-vertex in a graph g is a vertex that increases the disconnectivity when removing it. theorem 3.4. gz has no cut-edges. furthermore, every edge of gz is contained in a cycle of length 4. furthermore, girth(gz) = 4 and circumference(gz) = ∞. proof. it is known that an edge is not a cut-edge if and only if it is included in a cycle. assume e = uv ∈ e(gz), where u ∈ h1(gz), v ∈ h2(gz). by theorem 2.5 (p3), there are infinitely many elements in h2(gz) adjacent to u. choose v′ ∈ h2(gz), where v 6= v′, such that uv′ ∈ e(gz). by theorem 2.7 (3), there are infinitely many radical elements w ∈ h1(gz) of the (1, 2)-pair ({u},{v, v′}) which are adjacent to each of v and v′, that is, wv, wv′ ∈ e(gz). thus we obtain infinitely many cycles of length four of the form uvwv′u including the edge e = uv. so e is not a cut-edge. furthermore, a cycle of length 4 must be the shortest cycle since gz is bipartite and there is no cycle of length 3. thus, girth(gz) = 4. to see that the circumference(gz) = ∞, consider a vertex u1 ∈ h2(gz). by proposition 2.7 (2) |u↓1| = ∞ and so there exist u2 ∈ h1(gz) such that u1u2 ∈ e(gz). we can choose u3 ∈ u ↑ 2 \ {u1}, since |u↑2| = ∞ by theorem 2.5 (p3). note that in this setting u2, u4, . . . , u2i, . . . are height-one and u1, u3, . . . , u2i+1, . . . are height-two. continuing in this manner for any r ∈ n, choose a radical element w := u2r for ({u2, . . . , u2r−2}, {u1, . . . , u2r−1}). note u1, . . . , u2r−1 are all distinct. thus we have a cycle u1u2u3 · · ·u2r−1u2ru1 of length n = 2r. since this holds for any n, where n is even, circumference(gz) = ∞. the next theorem states that gz has no finite vertex cut-set. theorem 3.5. consider the graph gz. then 68 c. eubanks-turner, a. li 1. every pair of vertices are connected by infinitely many paths of length ≤ 4. 2. gz is connected. 3. there is no finite vertex cut-set of gz, that is, the vertex-connectivity of gz is infinite. 4. the edge-connectivity of gz is infinite. proof. for 1, if uv ∈ e(gz), u, v are connected by an edge, a path of length one. moreover, by the proof of theorem 3.4, uv is contained in infinitely many cycles of length 4. thus u and v are connected by infinitely many paths of length 3. next we assume u, v are not adjacent. we consider the following cases. case 1: u, v ∈ h1(gz). by theorem 2.5 (p3) we can choose t, t′ ∈ h2(gz) such that ut, vt′ ∈ e(gz) and t 6= t′. thus we take a radical element w for the pair ({u, v},{t, t′}). obviously w /∈ {u, v}. so we have a path of length 4 connecting u and v: utwt′v. note, there are infinitely many choices for such t and t′. case 2: u, v ∈ h2(gz). by proposition 2.7 (2), there exist s ∈ h1(gz) with su ∈ e(gz). similarly, there are infinitely many radical elements w for the pair ({s},{u, v}) and we have the path uwv, of length 2. case 3: u ∈ h1(gz) and v ∈ h2(gz). by theorem 2.5 (p3), ∃t ∈ u↑\{v}. there are infinitely many radical elements w for the pair ({u},{t, v}), which builds infinitely many paths of the form utwv of length 3. thus, 1 holds. the above also implies that every two vertices of gz is connected. thus gz is connected and statement 2 is true. to see 3, let u′ be any finite subset of v (gz) and consider u, v ∈ v (gz) \ u′. by using the above proof for 1, we can obtain infinitely many paths from u to v, not using any vertex from u′. thus u and v are connected in gz \u′ and so u′ is not a vertex-cut set of gz. item 3 implies 4 by the well-known fact that for any graph g, vertex connectivity of g ≤ edge connectivity of g, see [9]. next we give some basic concepts of graph theory and explore these parameters for gz. definition 3.6. let g be a graph. 1. the distance between two vertices u, v of g is the number of edges in a shortest path connecting them. we denote the distance of u and v as d(u, v). 2. the eccentricity of a vertex v, ecc(v), is the greatest distance from v to any other vertex. 3. the radius of a graph is the minimum eccentricity over all vertices in the graph. we denote the radius of g as rad(g). a central vertex in a graph is a vertex that achieves the radius. 4. the diameter of a graph is the maximum eccentricity over all the vertices in the graph. that is, it is the greatest distance over all pairs of vertices. we denote the diameter of a graph g as diam(g). a peripheral vertex in a graph is a vertex that achieves the diameter. theorem 3.7. gz satisfies the following properties: 1. for u, v ∈ v (gz), d(u, v) ≤ 4. 2. ecc(v) = { 3, if v ∈ h2(gz) 4, if v ∈ h1(gz) 3. rad(gz) = 3. 69 graphical properties of gz 4. diam(gz) = 4. 5. h1(gz) = {all peripheral vertices of gz} and h2(gz) = {all central vertices of gz}. 6. gz has no finite maximal path. proof. 1. and 2. follow from the proof of theorem 3.5. it is straightforward that 2. implies 3,4 and 5. to see 6., let pn = u1 . . . un+1 be a path of length n ≥ 1. if un+1 ∈ h1(gz), we choose un+2 ∈ u ↑ n+1\{u1, . . . , un} to extend pn into a path u1 . . . un+1un+2 of length n+1. the existence of un+2 is based on the fact that |u↑n+1| = ∞ by theorem 2.5 (p3). if un+1 ∈ h2(gz), ∃un+2 ∈ u ↓ n+1\{u1, . . . , un} by proposition 2.7 (2). a longer path is produced again: u1 . . . un+1un+2. another topic that generates a lot of interest in graph theory is the concept of graph matchings. below we define what a matching is and show that gz has a perfect matching. definition 3.8. a matching in a graph g is a set of pair-wise vertex disjoint edges. a perfect matching is a matching which covers all vertices of the graph. that is, every vertex of the graph is incident to exactly one edge of the matching. theorem 3.9. gz has a perfect matching. proof. we build a matching step by step that can cover all the vertices of gz. start with any edge u1v1 of gz, which is a matching of size 1, where u1 ∈ h1(gz) and v1 ∈ h2(gz). call it m1 = {u1v1}. we then list all the rest of the vertices of h1(gz) in a countable way: u2, u3, . . .. similarly, we build the countable list for the rest of the vertices of h2(gz): v2, v3, . . . .. next we pick u2 ∈ h1(gz) and choose v′2 from u ↑ 2 \{v1}. this is doable because ∣∣∣u↑2∣∣∣ = ∞. now we have a matching m2 = {u1v1, u2v′2} of size 2. in the next step, we will pick v2 and find u′3 ∈ v ↓ 2 \ { u1, u2 } to extend the mating m2 into m3 = { u1v1, u2v ′ 2, v2u ′ 3, . . . } . continue picking vertices alternatively from the countable lists of the two partites to enlarge the matching that eventually can cover all the vertices of gz. 4. completeness in gz and the complement of gz a complete bipartite graph is a bipartite graph where every vertex of the first partite is adjacent to every vertex of the second set. we adopt the commonly used notation km,n for the complete bipartite graph with vertex partitions of size m and n. note that for any n ∈ n, k1,n ⊆ gz, by (p3) theorem 2.5. also we consider the complement graph of gz and give some interesting properties of it. 4.1. completeness here, we give a theorem showing that there are infinitely many finite complete subgraphs of gz. theorem 4.1. for every m, n ∈ n, there are countably infinitely many subgraphs of gz which are isomorphic to km,n. proof. without loss of generality, assume m ≤ n. we first consider the corresponding situation in spec(z[x]). let p1, p2, . . . , pn be distinct prime numbers. then xi + p1p2 . . . pn is irreducible in z[x],∀i, 1 ≤ i ≤ n. so 〈xi + p1p2 . . . pn〉 ∈ h1(spec(z[x])), for each i. let s = {〈xi + p1p2 . . . pn〉}ni=1 and t = {〈x, pj〉}mj=1 ⊆ h2(spec(z[x])). then 〈xi + p1p2 . . . pn〉 ⊂ 〈x, pj〉, for all i, j, 1 ≤ i ≤ n, 1 ≤ j ≤ m. thus the corresponding subgraph of gz induced by s ∪ t is complete and isomorphic to km,n. since there are infinitely many choices for such p1, p2, . . . , pn, we have infinitely many of these subgraphs. 70 c. eubanks-turner, a. li an immediate corollary of theorem 4.1 is that gz is not planar. first, recall that a graph g is planar if g can be embedded into a plane. that is, g can be drawn on a plane in such a way that edges only intersect at their common vertices. also recall that a graph is planar if and only if it does not contain k3,3 or k5 [9]. corollary 4.2. gz is not planar and it has infinitely many non-planar subgraphs. proof. by theorem 4.1, there are infinitely subgraphs of gz isomorphic to k3,3. thus we have the claim by kuratowski’s theorem, see [9]. next, we give a proposition that shows we can extend any finite complete bipartite subgraph of gz into a larger complete bipartite subgraph of gz. recall that for a graph g, a clique of a graph g is a complete subgraph of g. proposition 4.3. let kn,m be a complete bipartite subgraph of gz with partites s ⊆ h1(gz) and t ⊆ h2(gz), n, m ≥ 1. 1. let r1 be a positive integer. then kn,m can be extended to a complete subgraph of gz that is isomorphic to kn+r1,m. 2. let s↑ = ⋂ s∈s s↑. assume |s↑| = r0 > m. then for every integer r2, 0 < r2 ≤ r0 −m, kn,m can be extended to a subgraph that is isomorphic to kn,m+r2. 3. gz has no maximum clique. however, for a fixed finite clique in gz, all the extended cliques have a maximum partite in h2(gz). proof. for 1., choose a radical element w1 for the (1, 2)-pair (s, t). let t↓ = ⋂ t∈t t↓. since w1 ∈ t↓, the subgraph of gz induced by s∪{w1}∪t is a complete bipartite subgraph. continuing in this process, we can choose distinct w1, w2, . . . , wr1 not in s∪t such that s∪{w1, w2, . . . , wr1}∪t generates a clique of gz isomorphic to kn+r1,m. to see 2., we can choose t0 ⊆ s↑\t with |t0| = r2. then s ∪t0 ∪t will induce a clique of gz which is isomorphic to kn,m+r2. for 3, without loss of generality, we assume s has at least two vertices. from the proof of part 2, any added vertex from h2(gz) in an enlarged clique must be in s↑ \ t , which is finite by the czp axiom (p4) since |s| ≥ 2. thus only finitely many vertices from h2(gz) can be added to any extended clique. 4.2. the complement of gz recall that the complement g of a graph g is a graph with the same vertex set as g such that two vertices u, v are adjacent in g if and only if u, v are not adjacent in g. since gz is bipartite, the two partites of gz induce two complete subgraphs of gz. proposition 4.5 asserts that gz is connected. in gz, we denote ngz(u) = {v ∈ v (gz)|uv ∈ e(gz)} = {v ∈ v (gz)|uv 6∈ e(gz)}. proposition 4.4. for every u ∈ v (gz), |ngz(u)| = ∞. proof. case 1. u ∈ h1(gz). note that ngz(u) = h2(gz) \ u ↑. assume ngz(u) is finite. then t = h2(gz) \ u↑ is finite. let s = {u} and consider the (1, 2)-pair (s, t) in gz. since both s, t are finite and s 6= ∅, there exists a radical element w for the pair (s, t). also since w ∈ h1(gz), |w↑| = ∞, and h2(gz) = u↑ ∪t , ∃m ∈ w↑ ∩u↑ which implies m /∈ t , contradiction. case 2. u ∈ h2(gz). by proposition 1(3), there are infinitely many vertices in h1(gz) not adjacent to u in gz thus adjacent to u in gz. thus in both cases, |ngz(u)| = ∞. 71 graphical properties of gz proposition 4.5. the complement gz is connected and both the vertex and edge connectivity of gz are infinite. proof. it is obvious that gz is made of two complete subgraphs induced from the vertex sets h1(gz) and h2(gz) and there are infinitely many edges connecting vertices between the two cliques. thus gz is connected. to see that gz has infinite vertex-connectivity, let u′ be a finite vertex cut set and consider any two vertices u, v ∈ v (gz) \ u′. we show that u and v are connected in gz. since each of h1(gz), h2(gz) induces the empty graph in gz , we have that each induces a complete subgraph of gz. it is sufficient to assume u ∈ h1(gz) and v ∈ h2(gz). by proposition 4.4 since ∣∣∣ngz(v)∣∣∣ = ∞, and ∣∣∣u′∣∣∣ < ∞, we can choose w ∈ ngz(v)\u ′ to make a cycle uwvu in gz. since there are infinitely many such w, v and u are connected in v (gz)\u′, we have u′ is not a cut-set of gz. thus, gz has infinite vertex-connectivity. the latter implies infinite edge-connectivity, by the well-known fact that for any graph g, vertex connectivity of g ≤ edge connectivity of g, see [9]. 5. what does the graph gz reveal about the ring z[x]? much research has been done in describing the structures of rings whose spectra are isomorphic to spec(z[x]) as posets. the work in this paper gives a set of graph theoretic properties of the derived graph gz by spec(z[x]) \ {0} as an infinite bipartite graph. is it shown that gz is a graph with many good properties. although it is infinite, it has fairly small graph measurements such as girth, diameter, radius, etc. it contains many different types of finite bipartite subgraphs such as perfect matching, cliques of any size, etc. it is natural to ask: “what does the graph gz reveal about the ground rings represented by z[x]?" in this regard, we give some descriptions from the ring theoretic point of view about any ring whose spectrum satisfies the czp axioms, based on the graph theory results obtained in earlier sections of this paper. proposition 5.1. let d be a ring such that spec(d) satisfies the czp axioms given in definition 2, as a poset. then the following are true: 1. assume p1, p2 are two co-maximal height-1 prime ideals (so p1 + p2 = d). then there exist a height-1 prime p and two maximal ideals m1, m2 such that pi + p ⊆ mi for i = 1, 2. 2. for any pair (p, m) of prime ideals, where p is of height one, m is maximal, and p 6⊆ m, there exisit a height-1 prime p′ and a maximal ideal m′ such that p ⊆ m′ and p′ ⊆ m∩m′. 3. we can order all the height-1 ideals and all the maximal ideals as two corresponding countable lists: h2(spec(d)) : m1, m2, . . . , mn, . . . h1(spec(d)) : p1, p2, . . . , pn, . . . such that for each i, pi ⊆ mi. 4. for any positive integers n and r, there are n height-1 prime ideals p1, p2, . . . , pn and r maximal ideals m1, m2, . . . , mr such that pi ⊆ mj for all i, j where 1 ≤ i ≤ n and 1 ≤ j ≤ r. proof. statement 1 is immediately from theorem 3.5 (1) (refer to case 1 in the proof). statement 2 is also by theorem 3.5 (1) (refer to case 3 in the proof). statement 3 is from theorem 3.9: gz has a perfect matching. the last statement follows directly from the existence of cliques of any size in gz (theorem 4.1). 72 c. eubanks-turner, a. li 6. conclusion this research shows that graphs derived from ring spectra may provide rich structural properties and may contain various types of graphs of interest, thus worthy of study. the graphic properties may tell something about the ring structure itself and vice versa. in the future, we will examine more types of spectra and apply more advanced graph theory techniques, such as graph polynomials, in the study. references [1] d. f. anderson and p. livingston, the zero-divisor graph of a commutative ring, j. algebra, 217, 434-447, 1999. [2] e. celikabs and c. eubanks-turner, the projective line over the integers, progress in commutative algebra ii: ring theory, homology, and decompositions, 221-240, de gruyter, 2012. [3] c. eubanks-turner, m. luckas, s. saydam, prime ideals in birational extensions of two-dimensional power series rings, communications in algebra, 41(2), 703-735, 2013. [4] w. heinzer, c. rotthaus, s. wiegand, mixed polynomial/power series rings and relations among their spectra, multiplicative ideal theory in commutative algebra, springer, new york, 227-242, 2006. [5] w. heinzer, s. wiegand, prime ideals in two-dimensional polynomial rings, proc. amer. math. soc., 577-586, 1989. [6] w. heinzer, s. wiegand, prime ideals in polynomial rings over one-dimensional domains, trans. amer. math. soc., 347(2), 639-650, 1995. 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[11] r. wiegand, the prime spectrum of a two-dimensional affine domain, j. pure appl. algebra, 40(2), 209-214, 1986. 73 introduction overview of spec(z[x]) properties of the graph gz completeness in gz and the complement of gz what does the graph gz reveal about the ring z[x]? conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.617244 j. algebra comb. discrete appl. 6(3) • 163–172 received: 20 october 2018 accepted: 21 august 2019 journal of algebra combinatorics discrete structures and applications a note on constacyclic and skew constacyclic codes over the ring zp[u, v]/〈u2 −u, v2 −v, uv −vu〉∗ research article tushar bag, habibul islam, om prakash, ashish k. upadhyay abstract: for odd prime p, this paper studies (1 + (p−2)u)-constacyclic codes over the ring r = zp[u,v]/〈u2 − u,v2 − v,uv − vu〉. we show that the gray images of (1 + (p − 2)u)-constacyclic codes over r are cyclic and permutation equivalent to a quasi cyclic code over zp. we derive the generators for (1 + (p − 2)u)-constacyclic and principally generated (1 + (p − 2)u)-constacyclic codes over r. among others, we extend our results for skew (1 + (p − 2)u)-constacyclic codes over r and exhibit the relation between skew (1 + (p− 2)u)-constacyclic codes with the other linear codes. finally, as an application of our study, we compute several non trivial linear codes by using the gray images of (1 + (p− 2)u)-constacyclic codes over this ring r. 2010 msc: 94b05, 94b15 , 94b60 keywords: constacyclic codes, skew constacyclic codes, gray map, quasi-cyclic codes 1. introduction in algebraic coding theory, one of the main goals is to produce good error-correcting linear codes by means of larger minimum distance and code rate. towards this, the cyclic code is one of important class of linear codes and researchers have been studying it for last six decades due to their successful applications in the theory of error-correcting codes. the constacyclic code is one of the prominent generalization of cyclic codes by which many good error-correcting codes can be developed over finite fields and rings. in 2006, qian et al. [12] introduced constacyclic codes over f2 + uf2. later, abualrub and siap [1] also studied structural properties of constacyclic codes over f2 + uf2. in 2011, karadeniz and yildiz [9] studied (1 + v)-constacyclic codes over f2 + uf2 + vf2 + uvf2 and constructed some new optimal binary codes as the gray images of (1 + v)-constacyclic codes over f2 + uf2 + vf2 + uvf2. in 2015, ashraf and ∗ the research was supported by the university grant commission (ugc), govt. of india. tushar bag, habibul islam, om prakash (corresponding author), ashish k. upadhyay; department of mathematics, indian institute of technology patna, patna–801 103, bihar, india (email:tushar.pma16@iitp.ac.in, habibul.pma17@iitp.ac.in, om@iitp.ac.in, upadhyay@iitp.ac.in). 163 https://orcid.org/0000-0002-7613-8351 https://orcid.org/0000-0002-2196-1586 https://orcid.org/0000-0002-6512-4229 https://orcid.org/0000-0001-6307-6799 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 mohammed [2] studied (1 + 2u)-constacyclic codes over z4 + uz4. recently, researchers [11, 13, 14] have studied constacyclic codes over the extension ring z4 + uz4 of z4 and obtained several new linear codes over z4 from gray images of these codes. for more works on the topic, we refer [3, 4, 6–8]. motivated by the above works, we study λ = (1 + (p− 2)u)-constacyclic codes over the finite nonchain ring r = zp[u,v]/〈u2 −u,v2 −v,uv−vu〉 and find some good codes over zp, where zp denotes the finite field with p elements for odd prime p. this ring can also be seen as zp + uzp + vzp + uvzp, where u2 = u,v2 = v,uv = vu. the paper is organized as follows: in section 2, we define two gray maps over the ring r. section 3 contains some results on gray images of λ = (1+(p−2)u)-constacyclic codes over the ring r. in section 4, we derive the generators for λ = (1+(p−2)u)-constacyclic and one-generator λ = (1+(p−2)u)-constacyclic codes, respectively over r. in section 5, we extend our results for skew λ = (1 + (p− 2)u)-constacyclic codes over r and finally, in section 6 some non trivial examples are included by using our gray maps. 2. preliminaries for an odd prime p, let r = zp + uzp + vzp + uvzp, where u2 = u,v2 = v,uv = vu. then r is a commutative non-chain semi-local ring with maximal ideals 〈u,v〉 and 〈u, 1 − v〉. recall that a non empty subset c of rn is called a linear code of length n over r if it forms an r-submodule of rn, and elements of c are referred as codewords. a linear code c of length n over r is said to be a λ-constacyclic code if c is closed under the constacyclic shift operator υ : rn −→ rn, defined by υ(c0,c1, . . . ,cn−1) = (λcn−1,c0, . . . ,cn−2), where λ is a unit in r. note that a constacyclic code is a cyclic code for λ = 1 and a negacyclic code for λ = −1. by identifying a codeword c = (c0,c1, . . . ,cn−1) ∈ rn to a polynomial c(x) = c0 + c1x + · · · + cn−1xn−1 in r[x] 〈xn−λ〉, a linear code c is a λ-constacyclic code of length n over r if and only if it is an ideal of the ring r[x]〈xn−λ〉. for the rest of this article, we denote λ = (1 + (p− 2)u). here, we define two new gray maps over r. the first gray map is φ1 : r → z2p define by φ1(a + ub + vc + uvd) = (a + b + c + d, (p− 1)(a + b + c + d)), (1) where a,b,c,d ∈ zp. it is easy to see that φ1 is a zp-linear map and can be extended component-wise as follows. φ1 : r n → z2np (r0,r1, . . . ,rn−1) 7−→ (a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 +c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)), where ri = ai + ubi + vci + uvdi ∈ r and ai,bi,ci,di ∈ zp for i = 0, 1, . . . ,n− 1. the next gray map is φ2 : r → z3p define by φ2(a + ub + vc + uvd) = (b + c + d, (p− 1)(2a + b + c + d),a), (2) where a,b,c,d ∈ zp. this φ2 is also a zp-linear map and can be extended component-wise like the φ1-gray map. note that these two gray maps are zp-linear but not bijection, similar to the gray map in [13]. the lee weight of any element a+ub+vc+uvd ∈ r is defined as wl(r) = wh(φi(a+ub+vc+uvd)), i = 1, 2, where wh denotes the hamming weight over zp. lee weight for r = (r0,r1, . . . ,rn−1) ∈ rn is defined by wl(r) = ∑n−1 i=0 wl(ri). the lee distance between any two elements r ′,r′′ ∈ rn is dl(r′,r′′) = wl(r′−r′′) and the minimum lee distance of c is defined as dl(c) = min{dl(r′,r′′) | r′ 6= r′′; r′,r′′ ∈ c}. by this discussion, one can check that φ1,φ2 are distance preserving zp-linear maps from (rn,dl) to (z2np ,dh) and (z3np ,dh), respectively, where dh denotes the minimum hamming distance of codes over zp. 164 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 3. gray images of (1 + (p−2)u)-constacyclic codes in this section, we explore the connection between cyclic, quasi cyclic and (1 + (p−2)u)-constacyclic codes via the gray maps φ1 and φ2, defined in the previous section. proposition 3.1. let υ be the (1 + (p− 2)u)-constacyclic shift on rn and ρ be the cyclic shift on z2np . if φ1 is the gray map from rn to z2np as defined in equation (1), then φ1υ=ρφ1. proof. let r = (r0,r1, . . . ,rn−1) ∈ rn, where ri = ai + ubi + vci + uvdi and ai,bi,ci,di ∈ zp for i = 0, 1, . . . ,n− 1. then φ1(r) =(a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). applying ρ on both sides, we get ρφ1(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). on the other hand, φ1υ(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−2 + bn−2 + cn−2 + dn−2,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). therefore, φ1υ=ρφ1. the derived relation in proposition 3.1, will help us to find the gray images of (1 + (p − 2)u)constacyclic codes of r. in that way, we have the following result. theorem 3.2. the φ1-gray image of (1 + (p−2)u)-constacyclic code of length n over r is a cyclic code of length 2n over zp. proof. let c be a (1 + (p−2)u)-constacyclic code of length n over r. then υ(c) = c, applying φ1 on both sides, we get φ1υ(c) = φ1(c). by proposition 3.1, ρφ1(c) = φ1υ(c) = φ1(c). therefore, φ1(c) is a cyclic code of length 2n over zp. definition 3.3. let a ∈ zmnp , where a = (a1|a2| . . . |am−1|am) and each ai ∈ znp for i = 1, 2, . . . ,m. let ηm be a map from zmnp to z mn p defined by ηm(a) = (ρ(a1)|ρ(a2)| . . . |ρ(am)), where ρ is the cyclic shift from znp to z n p and ′ |′ is the usual vector concatenation. a linear code c of length mn over zp is called a qc or quasi cyclic code of index m if ηm(c) = c. similar to proposition 3.1, here we derive a relation based on the gray map φ2, which will help us to find the φ2-gray images of (1 + (p− 2)u)-constacyclic codes of r. proposition 3.4. let υ be the (1 + (p−2)u)-constacyclic shift on rn, φ2, the gray map from rn to z3np defined in equation (2), and η3, the map defined in the preliminary section. then φ2υ = δη3φ2, where the permutation δ on z3np is defined as δ(x1,x2, . . . ,x3n) = (xβ(1),xβ(2), . . . ,xβ(3n)) with the permutation β = (1,n + 1) on the set {1, 2, . . . , 3n}. proof. let r = (r0,r1, . . . ,rn−1) ∈ rn, where ri = ai + ubi + vci + uvdi and ai,bi,ci,di ∈ zp for i = 0, 1, . . . ,n− 1. then φ2(r) =(b0 + c0 + d0, . . . ,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−1 + bn−1 + cn−1 + dn−1),a0, . . . ,an−1). 165 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 now, applying η3 on both sides, we get η3φ2(r) =(bn−1 + cn−1 + dn−1,b0 + c0 + d0, . . . ,bn−2 + cn−2 + dn−2, (p− 1)(2an−1 + bn−1 + cn−1 + dn−1), (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). on the other hand, φ2υ(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2 + cn−2 + dn−2, bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). now, applying δ on η3φ2(r), we get δη3φ2(r) = φ2υ(r). therefore, φ2υ = δη3φ2. in theorem 3.2, we presented the φ1-gray images of (1 + (p−2)u)-constacyclic codes over r. similar to that, using proposition 3.4, we present the φ2-gray image of (1 + (p− 2)u)-constacyclic code over r as below. theorem 3.5. the φ2-gray image of (1 + (p−2)u)-constacyclic code of length n over r is permutation equivalent to a qc code of index 3 over zp. proof. let c be a (1 + (p− 2)u)-constacyclic code of length n over r. then υ(c) = c. by applying φ2, we have φ2(υ(c)) = φ2(c). now, by proposition 3.4, φ2(υ(c)) = δη3(φ2(c)) = φ2(c), therefore, φ2(c) is permutation equivalent to a qc code of length 3n and index 3 over zp. now, we present the permutation version φπ of φ1 defined as φπ(r0,r1, . . . ,rn−1) =(a0 + b0 + c0 + d0, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, (p− 1)(a1 + b1 + c1 + d1), . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(an−1 + bn−1 + cn−1 + dn−1)), where ri = ai + ubi + vci + uvdi ∈ r and ai,bi,ci,di ∈ zp for i = 0, 1, . . . ,n− 1. using similar arguments used in the proof of [11, theorem 4.3], one can easily show the following result. proposition 3.6. for any r ∈ rn, we have φπρ(r) = ρ2φπ(r). the following corollary is a direct consequence of proposition 3.6. corollary 3.7. let c be a cyclic code of length n over r. then φπ(c) is equivalent to a quasi cyclic code of length 2n and index 2 over zp. 4. generators of (1 + (p−2)u)-constacyclic codes in this section, we derive the generators of λ = (1 + (p− 2)u)-constacyclic codes of length n over r, when gcd(n,p) = 1. we start with the approach shown in [2, 11]. let n be an odd integer. then ψ : rn = r[x] 〈xn − 1〉 −→ rn,λ = r[x] 〈xn −λ〉 166 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 defined by ψ(c(x)) = c(λx), is a ring isomorphism. by this isomorphism it is evident that i is an ideal of the ring rn if and only if ψ(i) is an ideal of ring rn,λ. note that λn = 1 when n is an even integer and λn = λ when n is an odd integer. then the map µ : rn → rn defined by µ(c0,c1, . . . ,cn−1) = (c0,λc1,λ 2c2, . . . ,λ n−1cn−1). (3) corresponds to the map ψ in the polynomial form. then it is easy to see that c is a cyclic code of odd length n over r if and only if µ(c) is a λ-constacyclic code of length n over r. theorem 4.1. [10, theorem 3.4] let c be a cyclic code of length n over r. if gcd(n,p) = 1, then c = 〈g(x) + ua1(x) + uvr1(x),va2(x) + uva3(x)〉, where a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1). theorem 4.1 gives the generators of cyclic codes of length n over r, see [10]. now, extending this result for λ-constacyclic codes of length n over r, we present the generators for λ-constacyclic codes of length n over r, where gcd(n,p) = 1. theorem 4.2. let c be a λ-constacyclic code of length n over r. if gcd(n,p) = 1, then c is an ideal of rn,λ which is generated by c = 〈g(x̄) + ua1(x̄) + uvr1(x̄),va2(x̄) + uva3(x̄)〉, where x̄ = λx and g(x),ai(x),r1(x) are polynomials in zp[x]/〈xn − 1〉, satisfying the conditions a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1). using the φ1-gray map, we give the one-generated λ-constacyclic code of length n over r as follows: theorem 4.3. let c = 〈a(x) + ub(x) + vc(x) + uvd(x)〉 be a λ-constacyclic code of length n over r, where a(x),b(x),c(x),d(x) ∈ zp[x]. then φ1(c) is a cyclic code of length 2n over zp generated by the polynomial (a(x) + b(x) + c(x) + d(x)) + xn[(p− 1)(a(x) + b(x) + c(x) + d(x))]. proof. the polynomial version of the gray map φ1 is φ1 : r[x]/〈xn −λ〉−→ zp[x]/〈xn − 1〉×zp[x]/〈xn − 1〉 defined by φ1(a(x) + ub(x) + vc(x) + uvd(x)) =(a(x) + b(x) + c(x) + d(x), (p− 1)a(x) + b(x) + c(x) + d(x)). rest of this proof is straightforward. note that for ri(x) ∈ zp[x], we have φ1((r1 + ur2 + vr3 + uvr4)(a1 + ub2 + vc3 + uvd4)) = r1[(a + b + c + d), (p− 1)(a + b + c + d)] + r2[(a + b + c + d), (p− 1)(a + b + c + d)] + r3[(a + b + c + d), (p− 1)(a + b + c + d)] + r4[(a + b + c + d), (p− 1)(a + b + c + d)], where [(a(x) + b(x) + c(x) + d(x)), (p−1)(a(x) + b(x) + c(x) + d(x))] represents the element (a(x) + b(x) + c(x) + d(x)) + xn[(p− 1)(a(x) + b(x) + c(x) + d(x))] in zp[x]/〈x2n − 1〉. example 4.4. let n = 4, p = 5 and the one generated (1 + 3u)-constacyclic code be c = 〈(1 + u + v + uv) + (1 + uv)x + (u + uv)x2 + (v + uv)x3〉. by theorem 4.3, φ1(c) is a cyclic code of length 8 over z5 generated by the polynomial 3x7 + 3x6 + 3x5 + x4 + 2x3 + 2x2 + 2x + 4 with minimum lee distance 8. 167 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 now, we study about a permutation over zp, referred as nechaev’s permutation, which is defined as π(c0,c1, . . . ,c2n−1) = (cτ(0),cτ(1), . . . ,cτ(2n−1)), where n is odd and τ = (1,n + 1)(3,n + 3) · · ·(2i + 1,n + 2i + 1) · · ·(n− 2, 2n− 2) is a permutation on the set {0, 1, . . . , 2n− 1}. proposition 4.5. let µ be the map defined in the equation (3). if π is the nechaev permutation and n is odd, then φ1µ = πφ1. proof. let r = (r0,r1, . . . ,rn−1) ∈ rn, where ri = ai + ubi + vci + uvdi ∈ r and ai,bi,ci,di ∈ zp for i = 0, 1, . . . ,n− 1. since n is odd, we have µ(r) = (r0, (1 + (p− 2)u)r1,r2, (1 + (p− 2)u)r3,r4, . . . , (1 + (p− 2)u)rn−2,rn−1). also, φ1(r) = (a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p − 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). therefore, φ1µ(r) =(a0 + b0 + c0 + d0, (p− 1)(a1 + b1 + c1 + d1),a2 + b2 + c2 + d2, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). on the other hand, πφ1(r) =(a0 + b0 + c0 + d0, (p− 1)(a1 + b1 + c1 + d1),a2 + b2 + c2 + d2, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0),a1 + b1 + c1 + d1, . . . , (p− 1)(an−1 + bn−1 + cn−1 + dn−1)). hence, φ1µ = πφ1. corollary 4.6. let π be the nechaev permutation and n be an odd. if λ is the gray image of a cyclic code of length n over r, then π(λ) is a cyclic code. proof. let c be a cyclic code of length n over r and λ = φ1(c). then by proposition 4.5, φ1µ(c) = πφ1(c) = π(λ). also, from our discussion in the beginning of this section, µ(c) is a (1 + (p − 2)u)constacyclic code. hence, by theorem 3.2, φ1(µ(c)) = π(λ) is a cyclic code. analogous to nechaev permutation, we present another permutation over zp, defined as χ(c1, . . . ,c3n) = (c%(1), . . . ,c%(3n)), where % = (2,n + 2)(4,n + 4) · · ·(n− 1, 2n− 1) is a permutation on the set {1, 2, . . . , 3n}. proposition 4.7. let µ and χ be the maps defined above. then φ2µ = χφ2. moreover, if n is odd and α is the φ2-gray image of a cyclic code of length n over r, then α is permutation equivalent to a qc code of length 3n and index 3 over zp. proof. it can be easily shown using similar procedure adopted in the proof of [3, proposition 4]. 5. skew (1 + (p−2)u)-constacyclic codes over r in section 3, we have derived some relations to study the (1 + (p−2)u)-constacyclic codes over r in terms of cyclic and quasi cyclic codes over zp. extending these discussion, here we obtain some relations to study the skew (1 + (p− 2)u)-constacyclic codes over r as an extension in noncommutative set up. 168 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 definition 5.1. let aut(r) be the set of all automorphisms defined over the ring r and θ ∈ aut(r). the set r[x; θ] = {r0 +r1x+· · ·+rn−1xn−1 | ri ∈ rn} forms a ring under the usual addition of polynomial and the multiplication defined as (rxi)(sxj) = rθi(s)xi+j. this ring is called skew polynomial ring over r. also, it is a non-commutative ring unless θ is the identity. we define a non-trivial automorphism θ : r −→ r θ(0) = 0, θ(1) = 1, θ(u) = v, θ(v) = u; i.e., θ(a + ub + vc + uvd) = a + vb + uc + uvd for a,b,c,d ∈ zp. note that the order of the automorphism is 2. for the rest of this section, we consider this automorphism θ over r. definition 5.2. a subset c of rn is called a skew λ-constacyclic code of length n over r if c satisfies the following conditions: i) c is a r-submodule of rn; ii) if c = (c0,c1, . . . ,cn−1) ∈ c, then σθ,λ(c) = (θ(λcn−1),θ(c0), . . . ,θ(cn−2)) ∈ c. in polynomial representation of a skew λ-constacyclic code of length n over r, identifying a codeword c = (c0,c1, . . . ,cn−1) ∈ rn by a polynomial c(x) = c0 + c1x + · · ·+ cn−1xn−1 in r[x; θ]/〈xn−λ〉, we have the following result. theorem 5.3. let c be a linear code of length n over r. then c is a skew λ-constacyclic code of length n over r if and only if c is a left r[x; θ]-submodule of r[x; θ]/〈xn −λ〉. in proposition 3.1 and proposition 3.4, we derived relations using λ-constacyclic shift over r to study the φ1 and φ2-gray images of λ-constacyclic codes over r. similarly, here also we derive some relations using skew λ-constacyclic shift over r which will help us to study φ1 and φ2-gray images of skew λ-constacyclic codes over r. these results can be seen as extension of proposition 3.1 and proposition 3.4. proposition 5.4. if σθ,λ is the skew constacyclic shift on rn and φ1,φ2 are the gray maps defined in equation (1) and equation (2), respectively, then 1. ρφ1 = φ1σθ,λ. 2. δη3φ2 = φ2σθ,λ, where ρ,δ and η3 are as in proposition 3.1 and proposition 3.4. proof. 1. from proposition 3.1, we have ρφ1(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). also, σθ,λ(r) = (θ(λrn−1),θ(r0),θ(r1), . . . ,θ(rn−2)). applying φ1 on both sides, we get φ1σθ,λ(r) =((p− 1)(an−1 + bn−1 + cn−1 + dn−1),a0 + b0 + c0 + d0, . . . ,an−1 + bn−1 + cn−1 + dn−1, (p− 1)(a0 + b0 + c0 + d0), . . . , (p− 1)(an−2 + bn−2 + cn−2 + dn−2)). hence, ρφ1 = φ1σθ,λ. 169 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 2. from proposition 3.4, we have δηφ2(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2+ cn−2 + dn−2,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2),an−1,a0, . . . ,an−2). also, σθ,λ(r) = (θ(λrn−1),θ(r0),θ(r1), . . . ,θ(rn−2)). applying φ2 on both sides, we get φ2σθ,λ(r) =((p− 1)(2an−1 + bn−1 + cn−1 + dn−1),b0 + c0 + d0, . . . ,bn−2+ cn−2 + dn−2,bn−1 + cn−1 + dn−1, (p− 1)(2a0 + b0 + c0 + d0), . . . , (p− 1)(2an−2 + bn−2 + cn−2 + dn−2), an−1,a0, . . . ,an−2). hence, δηφ2 = φ2σθ,λ. in theorem 3.2 and theorem 3.5, we discussed the φ1 and φ2-gray images of λ-constacyclic codes over r. extending these results for skew λ-constacyclic codes over r, we present the φ1 and φ2-gray image of a skew λ-constacyclic code over r as below: theorem 5.5. let φ1,φ2 be the gray maps defined in equation (1) and equation (2), respectively. then 1. φ1-gray image of a skew λ-constacyclic code of length n over r is a cyclic code of length 2n over zp. 2. φ2-gray image of a skew λ-constacyclic code of length n over r is permutation equivalent to a quasi-cyclic code of length 3n and index 3 over zp. proof. 1. let c be a skew (1 + (p − 2)u)-constacyclic code of length n over r. then σθ,λ(c) = c. now, applying φ1 on both sides, we get φ1σθ,λ(c) = φ1(c). by proposition 5.4, ρφ1(c) = φ1σθ,λ(c) = φ1(c). therefore, φ1(c) is a cyclic code of length 2n over zp. 2. let c be a skew (1 + (p − 2)u)-constacyclic code of length n over r. then σθ,λ(c) = c. now, applying φ2 on both sides, we get φ2σθ,λ(c) = φ2(c). also, by proposition 5.4, φ2(σθ,λ(c)) = δη(φ2(c)) = φ2(c). thus, φ2(c) is permutation equivalent to a qc code of length 3n and index 3 over zp. 6. examples for better understanding of our study, we present some non trivial examples which are computed by using the φ1 and φ2-gray images of the λ-constacyclic codes over r. all computations of this section are carried out by magma software [5]. example 6.1. let p = 3,λ = 1 + u and n = 8. following theorem 4.2, assume that c = 〈(1 + u)x5 + (2 + 2u)x3 + (1 + u)x2 + 2 + u + uv, (v + uv)x3 + 2vx2 + vx + v + 2uv〉 be the (1 + u)-constacyclic code of length 8 over r = z3[u,v]/〈u2 −u,v2 −v,uv −vu〉. therefore, the φ2-gray image of c is a [24, 8, 4] linear code over z3. example 6.2. let p = 17,λ = 1 + 15u and n = 9. further, assume c = 〈(1 + 15u)x5 + 3x4 + (1 + 15u)x3 + 16x2 + (1 + 15u)x + 16 + 16u + uv, (v + 5uv)x3 + 16uvx + 16v + 16uv〉 is a (1 + 15u)-constacyclic code of length 9 over r = z17[u,v]/〈u2 − u,v2 − v,uv − vu〉. then φ1(c) and φ2(c) have parameters [18, 10, 5] and [27, 10, 6] over z17, respectively. 170 t. bag et al. / j. algebra comb. discrete appl. 6(3) (2019) 163–172 table 1. linear codes as gray images of (1 + (p− 2)u)-constacyclic codes over zp λ n h(x̄) k(x̄) φ1(c) φ2(c) 1 + 3u 6 [1, 1 + 3u,u, 4 + 2u, 1 + 4u + uv] [v + 3uv, 2v, 2v + 4uv,v + uv] [12, 5, 2]5 [18, 5, 6]5 1 + 3u 8 [1 + 3u, 3, 2, 4 + uv] [4v, 3 + uv] [16, 8, 2]5 [24, 12, 5]5 1 + 3u 9 [1 + 3u, 4, 0, 1, 4 + 2u, 0, 1 + 2u, 3 + uv] [v, 0, 0,v + 3uv, 0, 0,v + uv] [18, 5, 2]5 [27, 5, 6]5 1 + 5u 4 [1 + 5u, 1, 1 + 4u, 1 + u + uv] [v, 0,v + uv] [8, 4, 2]7 [12, 4, 4]7 1 + 5u 10 [1 + 5u, 0, 0, 0, 6u, 1 + u + uv] [v, 6v + 2uv,v, 6v + 2uv,v + uv] [20, 10, 2]7 [30, 11, 5]7 1 + 9u 5 [1 + 9u, 8, 10u, 8 + 10u + uv] [v,v + 8uv, 3v + 2uv] [10, 5, 2]11 [15, 5, 6]11 1 + 9u 8 [1 + 9u, 2, 8 + 6u, 3 + u, 7 + 8u, 1 + u + uv] [v + 9uv, 10v,v + 8uv, 10v + 10uv] [16, 8, 2]11 [24, 8, 6]11 1 + 11u 4 [1, 9 + 7u, 8 + u] [v + 11uv, 8v + uv] [8, 3, 4]13 [12, 5, 4]13 1 + 11u 7 [1 + 11u, 6, 1 + 11u, 0, 11 + 3u, 11 + 12u] [v, 8v + 10uv, 4v + uv, 8v + 5uv,v + uv] [14, 5, 4]13 [21, 5, 8]13 1 + 15u 6 [1, 1 + 15u, 0, 16 + u, 16 + u] [v + 15uv, 16v + uv] [12, 5, 2]17 [18, 7, 2]17 1 + 15u 4 [1 + 15u, 1, 1 + 14u, 1 + u] [v, 5v + 6uv, 4v + 4uv] [8, 3, 4]17 [12, 3, 6]17 we recall from theorem 4.2 that generator of a λ-constacyclic code of length n with gcd(n,p) = 1 is given by c = 〈h(x̄),k(x̄)〉, where h(x̄) = g(x̄) + ua1(x̄) + uvr1(x̄),k(x̄) = va2(x̄) + uva3(x̄) with a1(x) | g(x) | (xn − 1) and a3(x) | a2(x) | g(x) | (xn − 1), x̄ = λx. in table 1, we obtain some linear codes from the λ-constacyclic codes over zp. first column includes the value of λ, second column is the length of the constacyclic code while third and fourth column gives the coefficients of generator polynomials h(x̄),k(x̄). lastly, fifth and sixth column shows the parameters of φ1 and φ2-gray images, respectively. we write coefficients of generator polynomials in decreasing order, for example, we write [u, 0, 1 + uv,v, 1 + u + uv] to represent the polynomial ux4 + (1 + uv)x2 + vx + 1 + u + uv. 7. conclusion and future works in this paper, we studied λ = (1 + (p − 2)u)-constacyclic codes over r = zp + uzp + vzp + uvzp for odd prime p. we have constructed two new gray maps over r and have shown some results based on their definitions. we have derived generators for λ = (1 + (p − 2)u)-constacyclic and one-generated λ = (1 + (p − 2)u)-constacyclic codes over r. using some permutation maps over zp, we have shown some results to understand cyclic and quasi cyclic codes by λ = (1 + (p − 2)u)-constacyclic codes over this ring in simpler way. at last, we have discussed skew (1 + (p − 2)u)-constacyclic codes over this ring and extended our results from section 3. as a future work, finding the generators of these skew λ = (1 + (p− 2)u)-constacyclic code over r would be interesting. we hope our results would be useful to find some good codes over zp via these gray maps over r. acknowledgment: the authors would like to thank 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[14] h. yu, y. wang, m. shi, (1 +u)–constacyclic codes over z4 +uz4, springer plus 5 (2016) 1325(1–8). 172 https://doi.org/10.1142/s1793830918500568 https://doi.org/10.1142/s1793830918500568 https://www.math.uzh.ch/sepp/magma-2.19.8-cr/handbook.pdf https://doi.org/10.1504/ijicot.2018.095017 https://doi.org/10.1504/ijicot.2018.095017 https://doi.org/10.1007/s12190-018-1211-y https://doi.org/10.1007/s12190-018-1211-y https://doi.org/10.1142/s1793830919500307 https://doi.org/10.1142/s1793830919500307 https://doi.org/10.1016/j.jfranklin.2011.08.005 https://doi.org/10.1016/j.jfranklin.2011.08.005 https://doi.org/10.1016/j.ffa.2015.01.005 https://doi.org/10.1016/j.ffa.2015.01.005 https://doi.org/10.1016/j.ffa.2015.12.003 https://doi.org/10.1016/j.ffa.2015.12.003 https://doi.org/10.1016/j.aml.2005.10.011 https://doi.org/10.1016/j.aml.2005.10.011 https://doi.org/10.1016/j.ffa.2016.11.016 https://doi.org/10.1016/j.ffa.2016.11.016 https://doi.org/10.1186/s40064-016-2717-0 introduction preliminaries gray images of (1+(p-2)u)-constacyclic codes generators of (1+(p-2)u)-constacyclic codes skew (1+(p-2)u)-constacyclic codes over r examples conclusion and future works references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(3) • 163-168 received: 25 march 2015; accepted: 11 july 2015 doi 10.13069/jacodesmath.90080 journal of algebra combinatorics discrete structures and applications skew cyclic codes over fq + ufq + vfq + uvfq∗ research article ting yao1∗∗, minjia shi1∗∗∗, patrick solé2§ 1. school of mathematical sciences of anhui university, china 2. telecom paris tech, france abstract: in this paper, we study skew cyclic codes over the ring r = fq +ufq +vfq +uvfq, where u2 = u, v2 = v, uv = vu, q = pm and p is an odd prime. we investigate the structural properties of skew cyclic codes over r through a decomposition theorem. furthermore, we give a formula for the number of skew cyclic codes of length n over r. 2010 msc: 94b15, 11a15 keywords: linear codes, skew cyclic codes, gray map, generator polynomial 1. introduction cyclic codes form an important subclass of linear block codes, studied from the fifties onward. their clear algebraic structures as ideals of a quotient ring of a polynomial ring makes for an easy encoding. a landmark paper [11] has shown that some important binary nonlinear codes with excellent error-correcting capabilities can be identified as images of linear codes over z4 under the gray map. recently, in [3], d. boucher et al. gave skew cyclic codes defined by using the skew polynomial ring with an automorphism θ over the finite field with q elements. the definition generalizes the concept of cyclic codes over non-commutative polynomial rings. soon afterwards, d. boucher et al. studied skew constacyclic codes in [5]. later, in [4], some important results on the duals of the skew cyclic codes over fq[x; θ] are given. in [12], i. siap et al. presented the structure of skew cyclic codes of arbitrary length. further, s. jitman et al. in [10] defined skew constacyclic codes over the skew polynomial ring with coefficients from finite rings. in [1], t. abualrub and p. seneviratne studied skew cyclic codes over ring ∗ supported by nnsf of china (61202068), technology foundation for selected overseas chinese scholar, ministry of personnel of china (05015133) and the project of graduate academic innovation of anhui university (no. yfc100008). ∗∗ e-mail: yaoting_1649@163.com ∗∗∗ e-mail: smjwcl.good@163.com (corresponding author) § e-mail: sole@enst.fr 163 skew cyclic codes over fq + ufq + vfq + uvfq f2 + vf2 with v2 = v. moreover, j. gao [6] and f. gursoy et al. [8] presented skew cyclic codes over fp + vfp and fq + vfq with different automorphisms, respectively. in [7], j. gao et al. also studied skew generalized quasi-cyclic codes over finite fields. in this article, we mainly study skew cyclic codes over ring r = fq + ufq + vfq + uvfq, where u2 = u,v2 = v,uv = vu and q = pm. in our work, the automorphism θ on the ring r is defined to be θ(b0 + b1u + b2v + b3uv) = b p 0 + b p 2u + b p 1v + b p 3uv, for all b0 +b1u+b2v+b3uv ∈ r, where bi ∈ fq, and i = 0, 1, 2, 3. in fact, for any a1η1 +a2η2 +a3η3 +a4η4 ∈ r, we have θ(a1η1 + a2η2 + a3η3 + a4η4) = θ(a1)η1 + θ(a2)η2 + θ(a4)η3 + θ(a3)η4. note that if m is even, the order of the ring automorphism |〈θ〉| is m, otherwise, 2m. the material is organized as follows. in section 2, we show the basics of codes over ring r that we need for further reference. section 3 derives the structure of linear codes over r. in section 4, we introduce skew cyclic codes over ring r and give the structural properties of skew cyclic codes over r through a decomposition theorem. section 5, we give a example to illustrate the discussed results. 2. preliminary let fq be a finite field with q elements, where q = pm, p is an odd prime. throughout, we let r denote the commutative ring fq +ufq +vfq +uvfq, where u2 = u,v2 = v, and uv = vu. let η1 = 1−u−v +uv, η2 = uv, η3 = u−uv, η4 = v −uv. it is easy to verify that η2i = ηi,ηiηj = 0, and ∑4 k=1 ηk = 1, where i,j = 1, 2, 3, 4, and i 6= j. according to [2], we have r = η1r ⊕ η2r ⊕ η3r ⊕ η4r. by calculating, we can easily obtain that ηir ∼= fq, i = 1, 2, 3, 4. therefore, for any r ∈ r, r can be expressed uniquely as r = ∑4 i=1 ηiai, where ai ∈ fq for i = 1, 2, 3, 4. we recall the definition of the gray map over r in [13] φ : r = fq + ufq + vfq + uvfq → f4q η1a + η2b + η3c + η4d → (a,a + b,a + c,a + b + c + d). equivalently, if r = a′ + b′u + c′v + d′uv ∈ r, then φ(r) = (a′, 2a′ + b′ + c′ + d′, 2a′ + b′, 4a′ + 2b′ + 2c′ + d′). this map can be naturally extended to the case over rn. for any element r = a + bu + cv + duv ∈ r, we define the lee weight of r as wl(r) = wh (a,a + b,a + c,a + b + c + d), where wh denotes the ordinary hamming weight for q-ary codes. the lee distance of r ∈ r can be similarly defined. from the definition of the gray map φ, we can easily check that φ is fq-linear and it is also a distance-reserving isometry from (rn,dl) to (f4nq ,dh ), where dl and dh denote the lee and hamming distance in rn and f4nq , respectively. 3. linear codes over r in this section, we mainly show some familiar structural properties of r. the proofs of the following theorems can be found in [13], so we omit them here. 164 t. yao, m-j. shi, p. solé if ai (i = 1, 2, 3, 4) are codes over r, we denote their direct sum by a1 ⊕a2 ⊕a3 ⊕a4 = {a1 + a2 + a3 + a4|ai ∈ ai, i = 1, 2, 3, 4}. definition 3.1. let c be a linear code of length n over r, we define that c1 = {a ∈ fnq |∃b,c,d ∈ f n q |η1a + η2b + η3c + η4d ∈ c}, c2 = {b ∈ fnq |∃a,c,d ∈ f n q |η1a + η2b + η3c + η4d ∈ c}, c3 = {c ∈ fnq |∃a,b,d ∈ f n q |η1a + η2b + η3c + η4d ∈ c}, c4 = {d ∈ fnq |∃a,b,c ∈ f n q |η1a + η2b + η3c + η4d ∈ c}. it is clear that ci (i = 1, 2, 3, 4) are linear codes over fnq . furthermore, c = η1c1⊕η2c2⊕η3c3⊕η4c4, and |c| = |c1|·|c2|·|c3|·|c4|. throughout the paper ci (i = 1, 2, 3, 4) will be reserved symbols referring to these special subcodes. according to definition 3.1 and [13], we have the following theorem. theorem 3.2. let c = η1c1 ⊕ η2c2 ⊕ η3c3 ⊕ η4c4 be a linear code of length n over r. then c⊥ = η1c ⊥ 1 ⊕η2c⊥2 ⊕η3c⊥3 ⊕η4c⊥4 . according to the definition of the gray map φ, we can easily obtain the following theorem. theorem 3.3. let c be a linear code of length n over r, |c| = qk and dl(c) = d. then φ(c) is a q-ary linear code with parameter [4n,k,d]. let c = η1c1 ⊕η2c2 ⊕η3c3 ⊕η4c4 be a linear code of length n over r. since c is a fq-module, then we have the following lemma. lemma 3.4. if gi are generator matrices of q-ary linear codes ci (i = 1, 2, 3, 4), respectively, then the generator matrix of c is g =   η1g1 η2g2 η3g3 η4g4   . moreover, if g1 = g2 = g3 = g4, then g = g1. in the light of the definition of gray map φ, we can easily obtain the following proposition. proposition 3.5. if c is a linear code of length n over r with generator matrix g, then we have φ(g) =   φ(η1g1) φ(η2g2) φ(η3g3) φ(η4g4)   =   g1 g1 g1 g1 0 g2 0 g2 0 0 g3 g3 0 0 0 g4   . 4. skew cyclic codes over r in this section, we assume c3 and c4 are equal. before studying skew cyclic codes over r, we define a skew polynomial ring r[x; θ] and skew cyclic codes over r. next, we determine the structural properties of skew cyclic codes over r through a decomposition theorem. 165 skew cyclic codes over fq + ufq + vfq + uvfq definition 4.1. we define the skew polynomial ring as r[x; θ] = {a0 + a1x + · · · + anxn|ai ∈ r,i = 0, 1, · · · ,n}, where the coefficients are written on the left of the variable x. the multiplication is defined by the basic rule (axi)(bxj) = aθi(b)xi+j, and the addition is defined to be the usual addition rule of polynomials. it is easily checked that the ring r[x; θ] is not commutative unless θ is the identity automorphism on r. definition 4.2. a nonempty subset c of rn is called a skew cyclic code of length n if c satisfies the following conditions: (1) c is a submodule of rn; (2) if r = (r0,r1, · · · ,rn−1) ∈ c, then skew cyclic shift ρ(r) = (θ(rn−1),θ(r0), · · · ,θ(rn−2)) ∈ c. theorem 4.3. let c = η1c1 ⊕η2c2 ⊕η3c3 ⊕η4c4 be a linear code of length n over r, where ci (i = 1, 2, 3, 4) are codes over fq of length n. then c is a skew cyclic code with respect to the automorphism θ if and only if ci are skew cyclic codes over fq with respect to the automorphism θ. proof. for any r = (r0,r1, · · · ,rn−1) ∈ c, let ri = η1ai + η2bi + η3ci + η4di for 0 ≤ i ≤ n − 1, where a = (a0,a1, · · · ,an−1) ∈ c1, b = (b0,b1, · · · ,bn−1) ∈ c2, c = (c0,c1, · · · ,cn−1) ∈ c3 and d = (d0,d1, · · · ,dn−1) ∈ c4. if ci are skew cyclic codes, then ρ(r) = ρ(η1a + η2b + η3c + η4d) = η1ρ(a) + η2ρ(b) + η3ρ(d) + η4ρ(c) = η1ρ(a) + η2ρ(b) + η3ρ(c) + η4ρ(d) ∈ c. this implies that c is a skew cyclic code over r. on the other hand, if c is a skew cyclic code over r, we have ρ(r) = (θ(rn−1),θ(r0), · · · , θ(rn−2)) = η1ρ(a) + η2ρ(b) + η3ρ(c) + η4ρ(d) ∈ c, which implies ρ(a) ∈ c1, ρ(b) ∈ c2, ρ(c) ∈ c3, ρ(d) ∈ c4. thus ci are skew cyclic codes over fq. according to ([4], corollary 18), we know that the dual code of every skew cyclic code over fq is also skew cyclic. by using this connection and theorem 4.3, we get the following corollary. corollary 4.4. if c is a skew cyclic code over r, then the dual code c⊥ is also skew cyclic. the following theorem determines the generator polynomials of a skew cyclic code of length n over r. theorem 4.5. let c = η1c1 ⊕ η2c2 ⊕ η3c3 ⊕ η4c4 be a skew cyclic code of length n over r and suppose that gi(x) are generator polynomials of ci (i=1, 2, 3, 4) respectively. then c = 〈η1g1(x),η2g2(x),η3g3(x),η4g4(x)〉 and |c| = q4n− ∑4 i=1 deg(gi(x)). proof. since ci = 〈gi(x)〉, for i = 1, 2, 3, 4, and c = η1c1 ⊕η2c2 ⊕η3c3 ⊕η4c4, then c = { c(x) = 4∑ i=1 ηiri(x)gi(x)|ri(x) ∈ fq[x; θ] } . hence c ⊆〈η1g1(x),η2g2(x),η3g3(x),η4g4(x)〉. conversely, for any ∑4 i=1 ηiki(x)gi(x) ∈ 〈η1g1(x),η2 · g2(x),η3g3(x),η4g4(x)〉, where ki(x) ∈ r[x; θ]/(xn − 1), then there exist ri ∈ fq[x; θ] such that ηiki(x) = ηiri(x), i = 1, 2, 3, 4. thus 〈η1g1(x),η2g2(x),η3g3(x),η4g4(x)〉 ⊆ c, which implies c = 〈η1g1(x),η2g2(x),η3g3(x),η4g4(x)〉. since |c| = |c1| · |c2| · |c3| · |c4|, we obtain that |c| = q4n− ∑4 i=1 deg(gi(x)). theorem 4.6. let ci (i = 1, 2, 3, 4) be skew cyclic codes over fq and gi(x) be the monic generator polynomials of these codes respectively, then there is a unique polynomial g(x) ∈ r[x; θ] such that c = 〈g(x)〉 and g(x) is a right divisor of xn − 1, where g(x) = ∑4 i=1 ηigi(x). 166 t. yao, m-j. shi, p. solé proof. by theorem 4.5, we know c = 〈η1g1(x),η2g2(x),η3g3(x),η4g4(x)〉. we take g(x) = η1g1(x) + η2g2(x) + η3g3(x) + η4g4(x), obviously, we have 〈g(x)〉 ⊆ c. on the other hand, one can check that ηigi(x) = ηig(x)(i = 1, 2, 3, 4), which implies c ⊆〈g(x)〉. hence c = 〈g(x)〉. since gi(x) are monic right divisors of xn − 1 ∈ fq[x; θ], then there exist ri(x) ∈ fq[x; θ] such that xn − 1 = ri(x)gi(x). thus [η1r1(x) + η2r2(x) + η3r3(x) + η4r4(x)]g(x) = 4∑ i=1 ηiri(x) · 4∑ i=1 ηigi(x) = 4∑ i=1 ηiri(x)gi(x) = 4∑ i=1 ηi(x n − 1) = xn − 1. this implies g(x) is a right divisor of xn − 1. corollary 4.7. every left submodule of r[x; θ]/(xn − 1) is principally generated. let g(x) = g0 + g1x + · · · + gtxt and h(x) = h0 + h1x + · · · + hn−txn−t be polynomials in fq[x; θ] such that xn − 1 = h(x)g(x) and c be the skew cyclic code generated by g(x) in fq[x; θ]/(xn − 1), according to corollary 18 in [4], then the dual code of c is a skew cyclic code generated by h̃(x) = hn−t + θ(hn−t−1)x + · · · + θn−t(h0)xn−t. therefore we have the following corollary. corollary 4.8. let ci be skew cyclic codes over fq and gi(x) be their generator polynomial such that xn − 1 = hi(x)gi(x) in fq[x; θ]. if c is a skew cyclic code over r, then c⊥ = 〈 ∑4 i=1 ηih̃i(x)〉 and |c⊥| = q ∑4 i=1 deg(gi(x)). let t be the order of θ. the following theorem can be obtain by applying similar steps of the theorem 3.7 in [6]. theorem 4.9. let (n,t) = 1 and c be a skew cyclic code of length n, then c is a cyclic code of length n over r. in [8], the factorization of xn−1 in fq[x; θi] is unique if (n,ti) = 1. let c = η1c1⊕η2c2⊕η3c3⊕η4c4 be a skew cyclic code of length n over r and suppose that gi(x) are generator polynomials of ci(i = 1, 2, 3, 4) respectively. then each gi(x) is a right divisor of xn − 1 in fq[x; θ]. θ acts on fq as follows, θ(a) = ap for all a ∈ fq. thus the order of θ on fq is m. hence if (n,m) = 1 then the factorization of xn − 1 in fq[x; θ] is unique. now we can determine the number of distinct skew cyclic codes of length n over r, where (n,m) = 1. corollary 4.10. let (n,m) = 1 and xn−1 = ∏r i=1 p si i (x), where pi(x) ∈ fq[x; θi] is irreducible, then the number of distinct skew cyclic codes of length n over r is equal to the number of ideals in r[x]/(xn −1), i.e. ∏r i=1(si + 1) 3. 5. application example in this section, we will exhibit a example of skew cyclic codes and their gray images over gf(9). before giving a example, we first give the definition of plotkin sum. let c⊕p d denote the plotkin sum of two linear codes c and d, also called (u|u + v) construction, where u ∈ c,v ∈ d. for more information on the plotkin sum, one can see a good survey [9]. 167 skew cyclic codes over fq + ufq + vfq + uvfq in the following, we assume gi are generator matrices of 9-ary linear codes ci for i = 1, 2, 3, 4, respectively. let c = η1c1 ⊕ η2c2 ⊕ η3c3 ⊕ η4c4 be a linear code of length n over r, then its gray image φ(c) is none other than (c1 ⊕p c2) ⊕p (c3 ⊕p c4). we construct skew cyclic codes over gf(9) with some conditions. if c1 is a [20, 1, 20] code, c2 is a [20, 9, 4] code, c3 is a [20, 10, 2] code and c4 is a [20, 10, 2] code, then the gray image of c has parameters [80, 30, 4] over gf(9). 6. conclusion this paper is devoted to studying skew cyclic codes over r = fq + ufq + vfq + uvfq, where u2 = u,v2 = v,uv = vu,q = pm and p is an odd prime. first, we introduce the structure of linear codes over r and show the structural properties of skew cyclic codes over r. next, we give the enumeration of distinct skew cyclic codes over r when n is odd. references [1] t. abualrub and p. seneviratne, skew codes over rings, in proc. imecs, hong kong, ii, 2010. 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[10] s. jitman, s. ling and p. udomkavanich, skew constacyclic over finite chain rings, adv. math. commun., 6(1), 29-63, 2012. [11] a. r. hammons jr., p. v. kumar, a. r. calderbank, n. j. a. sloane and p. solé, the z4-linearity of kerdock, preparata, goethals, and related codes, ieee trans. inform. theory, 40(2), 301-319, 1994. [12] i. siap, t. abualrub, n. aydin and p. seneviratne, skew cyclic codes of arbitrary length, int. nat. sci., 2(1), 10-20, 2011. [13] y. t. zhang, research on constacyclic codes over some classes of finite non-chain rings, master’s thesis, hefei university of technology, 2013. 168 introduction preliminary linear codes over r skew cyclic codes over r application example conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.617235 j. algebra comb. discrete appl. 6(3) • 135–145 received: 15 january 2018 accepted: 23 july 2019 0 journal of algebra combinatorics discrete structures and applications asymptotically good homological error correcting codes research article jason mccullough∗, heather newman abstract: let ∆ be an abstract simplicial complex. we study classical homological error correcting codes associated to ∆, which generalize the cycle codes of simple graphs. it is well-known that cycle codes of graphs do not yield asymptotically good families of codes. we show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least 2. we also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields. 2010 msc: 94b25, 05e45 keywords: error correcting codes, simplicial complexes, simplicial homology 1. introduction fix a field (usually finite) k. given an abstract simplicial complex ∆, one can define the reduced chain complex of ∆ with coefficients in k, which naturally defines the cycles, boundaries and homology of ∆. these are the classical homological codes, seemingly first studied by salzer [10]. in the case when ∆ is a one-dimensional simplicial complex, ∆ may be viewed as a simple graph. since the moore bound gives a logarithmic bound on the girth of a graph in terms of the number of edges and vertices, it follows that one cannot define asymptotically good families of codes from cycle codes of graphs. by applying recent work in [4] and [2], we show in theorem 4.2 that one can define asymptotically good families of codes from cycle codes of simplicial complexes of dimension two or higher. the paper is organized as follows: section 2 gathers preliminary results, definitions and related work. section 3 contains our results about arbitrary homological codes over arbitrary fields. in it we recover the parameters for the ith cycle and boundary codes (and their duals) of a simplex of arbitrary dimension. this was previously done in [10] and [13] over f2. section 4 contains our main results on the existence of asymptotically good homological codes. ∗ this author was supported by a grant from the simons foundation (576107, jgm). jason mccullough(corresponding author); department of mathematics, iowa state university,ames, iowa 50011, united states (email: jmccullo@iastate.edu). heather newman; department of mathematics, drexel university, philadelphia, pa 19104, united states (email: hn385@drexel.edu). 135 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 2. preliminaries 2.1. simplicial complexes let ∆ be an abstract simplicial complex on vertex set [m] := {1, . . . ,m}; that is, ∆ is a collection of subsets of [m], closed under taking subsets. elements σ ∈ ∆ are called faces or simplices. maximal faces are called facets. we note that ∆ is determined by its facets. for σ ∈ ∆, we define the dimension of σ by setting dim(σ) = |σ|−1. thus dim(∅) = −1. then the dimension of ∆ is dim(∆) = max{dim(σ) |σ ∈ ∆}. we denote by ∆i the set of i-dimensional faces of ∆, called the i-skeleton of ∆, and call an element of ∆i an i-face. an m-simplex is then a simplicial complex on [m + 1] consisting of all possible facets with one m-dimensional facet. set fi(∆) to be the number of i-faces, that is, fi(∆) = |∆i|. the vector (f−1(∆),f0(∆),f1(∆), . . . ,fdim(∆)(∆)) is called the f-vector of ∆. we sometimes refer to 0-faces as vertices and 1-faces as edges. we say that an i-face σ ∈ ∆i has degree r if σ is contained in exactly r many (i+ 1)-faces of ∆; that is, deg(σ) = ∣∣{τ ∈ ∆i+1 : σ ⊂ τ}∣∣ . finally we define the i-degree of ∆ to be deg(i, ∆) = min{deg(σ) |σ ∈ ∆i}. thus deg(1, ∆) is just the minimal degree of a vertex of the 1-skeleton of ∆ viewed as a simple graph. over any field k, we may define a chain complex (c•(∆,k),∂•) by setting ci(∆,k) to be the kvector space whose basis is identified with ∆i and such that given σ ∈ ∆i, write σ = {j0,j1, . . . ,ji}⊆ [m], where j0 < · · · < ji, then ∂i(σ) = i∑ k=0 (−1)k{j0, . . . , ĵk, . . . ,ji}, where ĵk denotes that jk has been removed. thus c−1(∆,k) ∼= k as ∅ is the only −1-dimensional face. we extend this linearly to define a map ∂i : ci(∆,k) → ci−1(∆,k). since ∂i−1 ◦∂i = 0 ∀i, the resulting sequence is a chain complex of k-vector spaces. let zi(∆,k) = ker(∂i) and bi(∆,k) = im(∂i+1). note that bi(∆,k) ⊆ zi(∆,k) ⊆ ci(∆,k) ∀i. then we define the ith reduced simplicial homology of ∆ with coefficients in k as hi(c•(∆,k)) = h̃i(∆,k) = zi(∆,k) bi(∆,k) . we define the co-chain complex of ∆ over k to be the vector space dual c•(∆,k) = (c•(∆,k)) ∗ of the chain complex, with coboundary maps ∂i = ∂∗i . we set z i(∆,k) = ker(∂i+1) and bi(∆,k) = im(∂i). then the ith reduced simplicial cohomology of ∆ over k is hi(c•(∆,k)) = h̃i(∆,k) = zi(∆,k) bi(∆,k) . since homk( ,k) is exact, there is a canonical isomorphism h̃i(∆,k) ∼= h̃i(∆,k). when the simplicial complex ∆ and the field k are clear from context, we omit them from the notation, e.g. writing h̃i for h̃i(∆,k). let s(∆) denote the suspension of the simplicial complex ∆; that is, s(∆) is the simplicial complex on vertex set ∆0 ∪{a,b}, where a,b are two new vertices. for any facet σ ∈ ∆, we assert that σ ∪{a} and σ ∪{b} are facets of s(∆). this uniquely determines s(∆). the suspension can be viewed as the union of two cones of ∆ glued along their bases. 2.2. linear codes a linear code c of length n over a (usually finite) field k is a k-vector subspace of kn. elements x = (x1, . . . ,xn) ∈ c are called codewords. we say that c is an [n,k,d]-linear code if c has length n, the dimension dimk(c) is k, and the minimum hamming distance of c is d. the minimum hamming 136 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 distance is the minimum over all pairs x,y ∈ c of distinct codewords of the number of positions i such that xi 6= yi. the weight w(x) of a codeword x ∈ c is the number of positions i such that xi 6= 0. the minimum distance of a linear code c is equal to the minimum weight of a nonzero codeword of c. a generator matrix g of a linear code c over k is a matrix whose rowspace is c. a parity check matrix h of c is a matrix whose null space is c. in other words x ∈ c if and only if hxt = 0. the dual code c⊥ of c is the code whose generator matrix is h. thus g is a parity check matrix for c⊥. we do not require the rows of g or h to be linearly independent. we make use of the following well-known fact relating minimum distance of a code to its parity check matrix. proposition 2.1 ([8, lemma 1.2.3]). let h be a parity-check matrix for a linear code c over a field k. then every set of d−1 columns of h are linearly independent if and only if c has minimum distance at least d. one of the main challenges in classical error correcting code theory is to find codes with large minimum distance and dimension relative to their length. more precisely, we seek a family of asymptotically good codes. a family of codes ci with parameters [ni,ki,di] with limi→∞ni →∞ is asymptotically good if there exists α,β > 0 such that lim i→∞ ki ni > α and lim i→∞ di ni > β, where ki ni and di ni define the information rate and relative minimum distance of ci, respectively. of particular interest are low density parity check (ldpc) codes. a linear code c (usually defined over f2) is an ldpc code if c has a parity check matrix h with relatively few nonzero entries. if the hamming weight (number of nonzero entries) in each row and column of h is constant, c is called a regular ldpc code. if we relax one of these conditions, c is called an irregular ldpc code. the cycle codes in this paper have parity check matrices with fixed column weight (since the image of the boundary of any i-face has weight i + 1), but only certain ones will have fixed row weight as well. 2.3. homological error correcting codes let ∆ be an m-dimensional simplicial complex and let k be a field. for each 0 ≤ i ≤ m, we define four potentially distinct linear codes. these codes will be the main object of study in this paper. the definitions parallel those in [10] and [13]. we define the ith cycle code (respectively boundary code) as zi(∆,k) (resp. bi(∆,k)). similarly, we define the ith cocycle code (respectively coboundary code) as zi(∆,k) (resp. bi(∆,k)). note that all four codes have length fi as they are subvector spaces of ci or c∗i . moreover, by their definitions it is clear that z i(∆,k) = bi(∆,k)⊥ and bi(∆,k) = zi(∆,k)⊥. in particular for all i we have, dimk z i(∆,k) + dimk bi(∆,k) = dimk zi(∆,k) + dimk bi(∆,k) = fi(∆). 2.4. relation to previous work the definition of ldpc codes originates in the thesis of gallager [6], [5]. such codes have recently attracted more attention because of the good iterative decoding algorithms associated to them which approach the shannon limit. asympotically good codes can be shown to exist by pseudorandom methods and constructed algorithmically. (see e.g. [7]). our theorem 4.2 shows that they can be constructed as cycle codes of two-dimensional simplicial complexes. the earliest references we can find to the definition of homological error-correcting codes of a simplicial complex are the papers by salzer [10] and thomeier [13], where the authors use the terms “topological codes” and “polyhedral codes,” respectively. both describe the general construction of cycle and boundary 137 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 codes of a simplicial complex. both authors then compute the parameters of the ith cycle codes for an m-dimensional simplex. we recover this as a special case of our more general formulas for the dimension and minimum distance. see theorem 3.7. in [9], rytíř showed that any linear code over q or fp can be realized as a truncated homological code of some two dimensional simplicial complex. without truncation, there are linear codes which cannot be realized. our results give both restrictions on the parameters of homological codes and existence results for asymptotically good homological codes. when ∆ is a one-dimensional simplicial complex, we may view ∆ as a simple graph g = (v,e) with vertex set v = ∆0 and edge set e = ∆1. in this case z1(∆,f2) is called the cycle code of the graph g and has been well-studied. see [14] for a concise survey. it is easy to see that if g = (v,e) is a graph, then its cycle code has parameters [n,k,d], where n = |e|, k = |e|−|v |+ 1 and d is the girth of g, that is, the minimum length of a cycle in g. the well-known moore bound states that for an r-regular graph g, that is a graph which has r edges incident to each vertex, the girth of g is bounded above by a term logarithmic in r and |v |. a similar bound for non-regular graphs was given in [1]. thus if we have a family of graphs gi = (vi,ei) with linear growth for |vi| and |ei|, the girth of gi is at most logarithmic in |vi| and |ei|. therefore, it is impossible to construct a family of asymptotically good error correcting codes from graphs or one-dimensional simplicial complexes. our theorem 4.2 shows that it is possible to do so with two-dimensional simplicial complexes. finally we comment that a there is great interest in finding quantum error correcting codes, introduced in [11], with similar properties. a well-studied family are the css codes [3], [12]. some constructions of css codes, also called homological codes, are defined from graphs, simplicial or polyhedral complexes. in the quantum setting, the existence of asymptotically good codes is still an open question. we do not discuss quantum codes further in this paper and refer the interested reader to the survey [14] and the references therein. 3. results on arbitrary homological codes in this section we collect general results which hold for each of the four types of homological codes for an arbitrary simplicial complex ∆ and an arbitrary field k. when clear from context, we suppress ∆ and k from the notation. theorem 3.1. let ∆ be an m-dimensional simplicial complex, k a field and 0 ≤ i ≤ m. then zi = zi(∆,k) is an [n,k,d]-code with n = fi, k = i∑ j=−1 (−1)i+jfj + i−1∑ j=0 (−1)i+j+1 dimk h̃j, d ≥ i + 2. proof. zi is a sub-vector space of ci and dimk ci = fi. so n = fi. by definition, the matrix h associated to ∂i is a parity check matrix for zi. since two i-faces of ∆ can only share at most one (i− 1)-face, any two columns of h can share at most one index where that entry is nonzero. thus each set of i + 1 columns of ∂i are linearly independent. by proposition 2.1, d ≥ i + 2. for the computation of k, we proceed by induction on i. for i = 0, we have the short exact sequence 0 → z0 → c0 → c−1 → 0. thus, dimk z0 = dimk c0 − dimk c−1 = f0 −f−1 138 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 as desired. now suppose dimk zp = p∑ j=−1 (−1)j+pfj + p−1∑ j=0 (−1)j+p+1 dimk h̃j for some p ∈ n. for i ≥ 0, we have the short exact sequences 0 → bi → zi → h̃i → 0 and 0 → zi → ci → bi−1 → 0. therefore, dimk bi = dimk zi − dimk h̃i and dimk zi = dimk ci − dimk bi−1. then by induction we have dimk zp+1 = dimk cp+1 − dimk bp = fp+1 − (dimk zp − dimk h̃p) = fp+1 −     p∑ j=−1 (−1)j+pfj + p−1∑ j=0 (−1)j+p+1 dimk h̃j  − dimk h̃p   = p+1∑ j=−1 (−1)j+p+1fj + p∑ j=0 (−1)j+p dimk h̃j. corollary 3.2. let ∆ be a contractible simplicial complex and let k be a field. then, dimk zi(∆,k) = i∑ j=−1 (−1)i+jfj. proof. if ∆ is contractible then h̃i(∆,k) = 0 ∀i. the following corollary provides an explicit formula for computing the dimension of cycle codes of graphs discussed in section 2.3. corollary 3.3. let ∆ be a connected simplicial complex. then, dimk z1(∆,k) = f1(∆) −f0(∆) + 1. proof. by theorem 3.1, we have dimk z1 = f1 −f0 + f−1 + dimk h̃0. since ∆ is connected, h̃0 = 0. moreover, f−1 = 1 for any simplicial complex. the result follows. 139 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 in general, as the following example shows, the minimum distance of zi(∆,k) can be strictly bigger than i + 2. when i = 1, d represents the girth of the 1-skeleton of ∆. example 3.4. consider the petersen graph p viewed as a simplicial complex of dimension 1. it has 10 vertices (0-faces) and 15 edges (1-faces). the graph p is 3-regular with girth 5. it follows from theorem 3.1 and the discussion in section 2.4 that z1(p,k) is a [15, 6, 5]-code for any field k. note that the minimum distance here is 5 > 3 = 1 + 2. 1 2 34 5 6 7 89 10 if k = f2, then the parity check matrix of z1(p,f2) is the incidence matrix of p, which is also the matrix associated to ∂1(p,f2). we index the rows by the vertices and the columns by the edges of p.   1,3 1,4 1,6 2,4 2,5 2,7 3,5 3,8 4,9 5,10 6,7 6,10 7,8 8,9 9,10 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 4 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 7 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 10 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1   since the dimensions of the homologies of a simplicial complex can depend on the field k, the parameters for a given cycle code can also depend on k. example 3.5. let ∆ be the standard triangulation of the real projective plane on vertex set {1, 2, 3, 4, 5, 6} with the following 2-faces: ∆2 = {123, 124, 126, 134, 135, 156, 235, 236, 346, 456}. if k = f2, then h̃1(∆,k) = k, whereas if k = fp with p 6= 2, then h̃1(∆,k) = 0. so by theorem 3.1, when k = f2, z2(∆,k) has dimension k = f2 −f1 + f0 −f−1 + h̃1(∆,f2) − h̃0(∆,f2) = 10 − 15 + 6 − 1 + 1 − 0 = 1, whereas when k = fp for p 6= 2, z2(∆,k) has dimension k = 0. 140 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 theorem 3.6. let ∆ be an m-dimensional simplicial complex, k a field and 0 ≤ i < m. then bi(∆,k) is an [n,k,d]-code with n = fi, k = i∑ j=−1 (−1)i+jfj + i∑ j=0 (−1)i+j+1 dimk h̃j, d = i + 2. proof. bi is a sub-vector space of ci and dimk ci = fi. so n = fi. since dimk bi = dimk zi−dimk h̃i, the formula for k follows from theorem 3.1. since dim(∆) ≥ i + 1, the image of ∂i+1 is nonempty. each (i + 1)-face has boundary with i + 2 distinct faces. hence there is at least one vector in bi of weight exactly i + 2. so d ≤ i + 2. but bi ⊆ zi and the minimum distance of zi was at least i + 2. thus d = i + 2. corollary 3.7. let ∆ be an m-dimensional simplex, k a field and 0 ≤ i < m. then zi(∆,k) = bi(∆,k) is an [n,k,d]-code with n = ( m + 1 i + 1 ) , k = ( m i + 1 ) , d = i + 2. proof. by theorem 3.1, n = fi = ( m+1 i+1 ) . since ∆ is contractible, h̃i(∆) = 0 ∀i. hence, zi = bi ∀i and d = i + 2 by theorem 3.6. finally, we have k = i∑ j=−1 (−1)i+jfj = i∑ j=−1 (−1)i+j ( m + 1 j + 1 ) = ( m i + 1 ) , by pascal’s identity. note that the parameters do not depend on the field k. remark 3.8. the family of m-simplex codes described here are not asymptotically good according to the definition given in section 2.2. if i = m− 1, then k = ( m m ) = 1; that is, a one-dimensional code. in the most interesting scenario, we set i = m− 2 and obtain the following: n = (m2 + m) 2 , k = m, d = m. note that there is quadratic growth in n, but linear growth in k and d. thus, lim i→∞ ki ni = 0 and lim i→∞ di ni = 0. other choices of i < m− 2 produce codes with similar asymptotics. a similar analysis can be carried out on the dual codes zi(∆,k) and bi(∆,k). in general, the minimum distance depends on the minimum degree of an i-face in ∆. 141 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 theorem 3.9. let ∆ be an m-dimensional simplicial complex, k a field and 0 ≤ i < m. then zi = zi(∆,k) is an [n,k,d]-code with n = fi, k = i−1∑ j=−1 (−1)i+j+1fj + i∑ j=0 (−1)i+j dimk h̃j, d ≥ deg(i, ∆) + 1. proof. since zi ⊆ ci, n = fi. since zi = b⊥i , dimk z i + dimk bi = fi and the value of k follows from theorem 3.6. each column of the matrix associated to ∂∗i+1 has at least deg(i, ∆) nonzero entries. since any two distinct i-faces are contained in at most one (i + 1)-face, every set of deg(i, ∆) columns of ∂∗i+1 is linearly independent. indeed suppose r columns of ∂∗i+1 are linearly dependent. the first such column has at least deg(i, ∆) nonzero entries and each other column can cancel at most one of these entries, meaning r > deg(i, ∆). applying proposition 2.1 yields that d ≥ deg(i, ∆) + 1. a similar argument gives the parameters for the coboundary codes. theorem 3.10. let ∆ be an m-dimensional simplicial complex, k a field and 0 ≤ i ≤ m. then bi = bi(∆,k) is an [n,k,d]-code with n = fi, k = i−1∑ j=−1 (−1)i+j+1fj + i−1∑ j=0 (−1)i+j dimk h̃j, deg(i, ∆) + 1 ≤ d ≤ min{deg(σ) |σ ∈ ∆i−1 and deg(σ) > 0}. proof. the formulas for n and k follow from theorem 3.1. that deg(i, ∆) + 1 ≤ d follows from theorem 3.9 since bi ⊆ zi. on the other hand, by definition there is a nonzero element in bi of weight min{deg(σ) |σ ∈ ∆i−1 and deg(σ) > 0}. when ∆ is the m-dimensional simplex, each i-face is contained in exactly m−i many (i + 1)-dimensional faces. applying the previous two theorems yields the following calculation, which together with theorem 3.7, recovers calculations in [10] and [13]. corollary 3.11. let ∆ be an m-simplex, k a field and 0 ≤ i ≤ m. then zi(∆,k) = bi(∆,k) is an [n,k,d]-code with n = ( m + 1 i + 1 ) , k = ( m i ) , d = m− i + 1. it is easy to check that none of these codes form a regular ldpc family of codes that are asymptotically good. 4. asymptotically good homological codes the goal of this section is to prove the existence of asymptotically good cycle codes for specially chosen simplicial complexes ∆ with dim(∆) ≥ 2. we rely on the following result of dotterer, guth, and kahle derived from work of aronshtam, linial, łuczak and meshulam [2]. since a proof is omitted from [4] we sketch a proof here. for notation and details, we refer the reader to [2]. 142 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 theorem 4.1 ([4, theorem 6.2(1)]). for any α > 0, and for all n sufficiently large, there exist 2dimensional simplicial complexes ∆ with n vertices and with at least αn2 faces, such that every cycle in in h2(∆,f2) is supported on at least cαn2 faces (where cα > 0 is a constant depending only on α). proof. let y2(n,p) denote the probability space of simplicial complexes of that contain every (d− 1)dimensional face and also each d-dimensional face with independent probability p. fix α > 0 and let c > 6α. by [2, theorem 4.1], there exists a constant δ (depending only on c) such that asymptotically almost surely, every every minimal core subcomplex (and thus every minimal cycle) k of y ∈ y2(n, cm) with f2(k) ≤ δn2 is the boundary of a 3-dimensional simplex. let xm denote the number of 2-cycles in y with exactly m faces so that xm = 0 for m ≤ 3 and x4 counts the number of boundaries of 3-dimensional simplices in y . the expected number of such subcomplexes is ex4 = ( n 4 ) p4 = n(n− 1)(n− 2)(n− 3) 3! c4 n4 ≤ c4 6 . thus for sufficiently large n, we can find a d-dimensional simplicial complex y with n vertices and no cycles of size ≤ δn2 except for the at most c 4 6 boundaries of 3-simplices. by removing one face from each cycle, we get a simplicial complex y ′ with no 2-cycles of size ≤ δn2. the expected number of 2-faces in y ′ is ef2(y ′) ≥ ( n 3 ) p− c4 6 = n(n− 1)(n− 2) 6 c n − c4 6 = c(n− 1)(n− 2) − c4 6 , and for n sufficiently large c(n−1)(n−2)−c 4 6 > αn2. thus we may take cα = δ. with this result and theorem 3.1 in hand, we can now prove our main result about the existence of asymptotically good cycle codes. theorem 4.2. there is a positive constant c ∈ r such that, for all m ∈ n sufficiently large, there exists a 2-dimensional simplicial complex ∆m on m vertices such that z2(∆m,f2) is an [n,k,d]-code with n = m2, k ≥ m2 2 , d ≥ cm2. in particular, z2(∆m,f2) give a family of asymptotically good codes over f2. proof. let α = 1 and apply theorem 4.1. we get an infinite family of 2-dimensional simplicial complexes ∆m for m � 0 such that ∆m has m vertices and at least m2 2-faces and such that every cycle of h2(∆m,f2) = z2(∆m,f2) is supported on at least cm2 faces where c = c1 above. first note that we may assume that f2(∆m) = m2 since deleting faces can only increase the size of the support of a cycle in h2(∆m,f2). moreover, we may assume that all ( m 2 ) edges are present since adding edges does not affect z2(∆m,f2). thus by theorem 3.1, z2(∆m,f2) is an [n,k,d]-code with n = f2(∆m) = m2. since every cycle is supported on at least cm2 faces, d ≥ cm2. finally we have k = f2 −f1 + f0 −f−1 + h̃1 − h̃0 = m2 − ( m 2 ) + m− 1 + h̃1 − h̃0 ≥ m2 2 where the last inequality uses that h̃0 = 0, since ∆m is connected, and that m ≥ 2. 143 j. mccullough, h. newman / j. algebra comb. discrete appl. 6(3) (2019) 135–145 the results in [2] and [4] rely on probabilistic methods to prove the existence of such simplicial complexes. it would be interesting to have explicit constructions of such simplicial complexes. finally we remark that the previous theorem can be extended to higher dimension by taking suspensions. proposition 4.3. let ∆ be an m-dimensional simplicial complex and let k be a field. if zm(∆,k) had parameters [n,k,d], then zm+1(s(∆),k) has parameters [2n,k, 2d]. proof. for all i ≥ −1, there is a canonical isomorphism φ : h̃i(∆,k) ∼= h̃i+1(s(∆),k), coming from the mayer-vietoris sequence induced by sending an i-face σ to the sum σ∪{a} + σ∪{b}, where a,b are the two additional vertices. in particular, zm(∆,k) = h̃m(∆,k) ∼= h̃m+1(s(∆),k) = zm+1(s(∆),k) and the dimensions of the 2 codes agree. if n is the length of zm(∆,k), then by construction the length of s(∆) is fm+1(s(∆)) = 2fm(∆) = 2n. finally, it is clear that φ doubles the support of any cycle in zm(∆,k) and hence doubles the minimum weight of the associated code. note that since the dimension is preserved, we do not get asymptotically good families by taking repeated suspensions of a given simplicial complex. however, combining with theorem 4.2 yields the following consequence. corollary 4.4. for any integer r ≥ 2, there is a family of r-dimensional simplicial complexes {∆m} whose cycle codes zr(∆m,f2) are asymptotically good. proof. fix r ≥ 2. let sr(∆) denote the r-fold suspension of ∆. let {∆m} be the family of 2dimensional simplicial complexes from theorem 4.2. then by proposition 4.3, zr(sr−2(∆m),f2) has parameters [2r−2m2,m2/2,c2r−2m2], and hence forms an asymptotically good family. finally we remark that since the parity check matrix of zr(sr−2(∆m),f2) is the matrix associated to ∂r(sr−2(∆m),f2), it has exactly r + 1 ones per column. so the corresponding family of codes is a (non-regular) ldpc family of codes. acknowledgment: we are very grateful to tony bahri and lance miller for feedback on an earlier version of this paper and matthew kahle for useful 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[14] g. zémor, on cayley graphs, surface codes, and the limits of homological coding for quantum error correction, coding and cryptology, lecture notes in comput. sci., vol. 5557, springer, berlin (2009) 259–273. 145 https://doi.org/10.1016/j.aim.2015.06.011 https://mathscinet.ams.org/mathscinet-getitem?mr=237222 https://mathscinet.ams.org/mathscinet-getitem?mr=237222 https://doi.org/10.1103/physreva.52.r2493 https://doi.org/10.1103/physreva.52.r2493 https://www.jstor.org/stable/52827 https://www.jstor.org/stable/52827 https://mathscinet.ams.org/mathscinet-getitem?mr=1121382 https://doi.org/10.1007/978-3-642-01877-0_21 https://doi.org/10.1007/978-3-642-01877-0_21 https://doi.org/10.1007/978-3-642-01877-0_21 introduction preliminaries results on arbitrary homological codes asymptotically good homological codes references issn 2148-838x j. algebra comb. discrete appl. 10(2) • 73–86 received: 26 november 2020 accepted: 31 december 2022 journal of algebra combinatorics discrete structures and applications some upper and lower bounds for dα-energy of graphs research article abdollah alhevaz, maryam baghipur, ebrahim hashemi, yilun shang abstract: the generalized distance matrix of a connected graph g, denoted by dα(g), is defined as dα(g) = αtr(g) + (1 − α)d(g), 0 ≤ α ≤ 1. here, d(g) is the distance matrix and tr(g) represents the vertex transmissions. let ∂1 ≥ ∂2 ≥ · · · ≥ ∂n be the eigenvalues of dα(g) and let w(g) be the wiener index. the generalized distance energy of g can be defined as edα(g) = n∑ i=1 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣. in this paper, we develop some new theory regarding the generalized distance energy edα(g) for a connected graph g. we obtain some sharp upper and lower bounds for edα(g) connecting a wide range of parameters in graph theory including the maximum degree ∆, the wiener index w(g), the diameter d, the transmission degrees, and the generalized distance spectral spread dαs(g). we characterized the special graph classes that attain the bounds. 2010 msc: 05c50, 05c12, 15a18 keywords: generalized distance matrix, generalized distance energy, distance (signless laplacian) matrix, transmission regular graph, generalized distance spectral spread 1. introduction we consider simple connected graphs in this paper. let g = (v (g),e(g)) be a graph with v (g) = {v1,v2, . . . ,vn} being its vertex set and e(g) its edge set. n = |v (g)| is called the order and m = |e(g)| the size. the neighborhood of a vertex v is the collection of vertices adjacent to it and is denoted by n(v). dg(v) or simply dv is the degree of v, meaning the cardinality of its neighborhood. a regular graph has all degrees the same. the adjacency matrix a(g) = a = (aij) of g has (i,j)-element one if vi is abdollah alhevaz; faculty of mathematical sciences, shahrood university of technology, p.o. box: 3163619995161, shahrood, iran(email: a.alhevaz@shahroodut.ac.ir). maryam baghipur; faculty of mathematical sciences, shahrood university of technology, shahrood, iran (email: maryamb8989@gmail.com). ebrahim hashemi; faculty of mathematical sciences, shahrood university of technology, shahrood, iran (email: eb_hashemi@shahroodut.ac.ir). yilun shang (corresponding author); department of computer and information sciences, northumbria university, newcastle ne1 8st, uk (email: yilun.shang@northumbria.ac.uk). 73 https://orcid.org/0000-0001-6167-607x https://orcid.org/0000-0002-9069-9243 https://orcid.org/0000-0002-8673-9556 https://orcid.org/0000-0002-2817-3400 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 adjacent to vj and zero if not. hence, a is an n by n symmetric matrix. the degree diagonal matrix is deg(g) = diag(d1,d2, . . . ,dn). two well known matrices associated with a are the laplacian l(g) = deg(g) − a(g) and the signless laplacian q(g) = deg(g) + a(g). they are symmetric and positive semi-definite. their spectra are organized as 0 = µn ≤ µn−1 ≤ ··· ≤ µ1 and 0 ≤ qn ≤ qn−1 ≤ ··· ≤ q1, respectively. the length of a shortest path between two vertices u and v is commonly known as the distance and is denoted by duv. if we take maximum among all such distances in a graph, we have the diameter. the matrix d(g) = (duv)u,v∈v (g) is called the distance matrix of g. for a comprehensive survey of recent results on distance matrix and its spectrum, we refer the reader to [6]. the sum of the distances from v to all other vertices in g is called the transmission and it is denoted by trg(v) = ∑ u∈v (g) duv. if trg(v) = k for all v, then g is called k-transmission regular. the well known wiener index is defined by w(g) = 1 2 ∑ v∈v (g) trg(v) and is also called transmission of a graph. the transmission trg(vi) or shortly tri for a vertex vi is also called the transmission degree. the transmission degree sequence of g is {tr1,tr2, . . . ,trn}. the distance laplacian and the distance signless laplacian matrices of connected graphs have been introduced in m. aouchiche and p. hansen [7]. two matrices are of utmost relevance: the distance laplacian matrix dl(g) = tr(g)−d(g) and the distance signless laplacian matrix dq(g) = tr(g)+d(g). here, tr(g) = diag(tr1,tr2, . . . ,trn) characterizes the vertex transmission of g. many important spectral properties of these matrices have been intensively explored in the recent years; see e.g. [4, 5, 7, 8]. an effort has been made in [9] to merge the different spectra theory of distance matrix, distance laplacian etc. by introducing the so-called generalized distance matrix dα(g), where dα(g) = αtr(g)+ (1 − α)d(g), for 0 ≤ α ≤ 1. note that d0(g) = d(g), 2d1 2 (g) = dq(g), d1(g) = tr(g) and dα(g)−dβ(g) = (α−β)dl(g). therefore, the spectral properties of individual graph matrices can be reproduced from the spectral theory of generalized distance matrix. we will re-arrange the eigenvalues of dα(g) as ∂1 ≥ ∂2 ≥ ···≥ ∂n. the largest eigenvalue ∂1 is referred to as the generalized distance spectral radius. when no confusion will be caused, we simply write ∂(g). the spectral properties of dα(g) have attracted much more attention of the researchers. for some recent works we refer to [1–3, 9, 11, 22, 23] and the references therein. the topic of graph energy [13] was put forward by ivan gutman. it is rooted in the theory of mathematical chemistry. assume the adjacency spectrum of a graph g is represented by λ1,λ2, . . . ,λn. the graph energy is defined as e(g) = n∑ i=1 |λi| [14]. graph energy has been intensively studied in mathematical chemistry and some early bounds for e(g) have been reported in e.g. [17]. energy-like graph invariants with respect to other matrices (in addition to the adjacency matrix) have been discussed in [5, 10, 12, 15, 16, 21, 24]. this paper aims to study a new energy-like quantity on the basis of the eigenvalues of generalized distance matrix dα(g). we define auxiliary eigenvalues θi corresponding to the generalized distance eigenvalues as θi = ∂i − 2αw(g) n . the following definition of generalized distance energy is given in [3], which is inspired by the distance laplacian energy el(g) and the distance signless laplacian eq(g). namely, edα(g) = n∑ i=1 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ = n∑ i=1 |θi|, which is the average deviation of the generalized distance eigenvalues. let t be the largest positive integer satisfying ∂t ≥ 2αw(g) n . let mk(g) = ∑k i=1 ∂i be the sum of k 74 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 largest generalized distance eigenvalues. it is shown in [3] that edα(g) = 2 ( mt − 2αtw(g) n ) = 2 max 1≤j≤n ( j∑ i=1 ∂i − 2αjw(g) n ) . it can be seen that ∑n i=1 θi = 0. noting ∑n i=1 ∂i = 2αw(g) and n∑ i=1 ∂2i = trace[dα(g)] 2 = 2(1 − α)2 n∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i , in view of θi we obtain ∑n i=1 θ 2 i = 2ζ, where ζ = (1 − α)2 ∑n 1≤i<j≤n(dij) 2 + α 2 2 n∑ i=1 tr2i − 2α2w2(g) n . moreover, ed0 (g) = ed(g) and 2e d 1 2 (g) = eq(g). the generalized distance graph energy can be viewed as merging the theories of distance graph energy and distance signless laplacian graph energy. it is therefore appealing to investigate edα(g) and explore the influence of the graph structure and parameter α on edα(g). in [22], the authors defined the generalized distance spectral spread as dαs(g) = ∂1 −∂n. some lower and upper bounds for the parameter dαs(g) can be found in [22]. in this paper, we continue on this line and study the generalized distance energy edα(g) of a connected graph g. we establish some sharp upper and lower bounds for edα(g) using graph-theoretic parameters including wiener index w(g), maximum degree ∆, minimum degree δ, diameter d, as well as generalized distance spectral spread dαs(g) and transmission degrees. we characterize the graphs that attain our bounds. 2. new bounds on generalized distance energy of graphs we start this section with some essential lemmas which will be used along the paper. the following lemma characterizes the graphs with two generalized distance eigenvalues. lemma 2.1. a connected graph g has two different dα(g) eigenvalues if and only if g is complete. proof. the proof is similar to the proof given in [15, lemma 2], and is excluded. lemma 2.2. [9] suppose that g is an n-vertex connected graph. then ∂(g) ≥ 2w(g) n , with equality holding if and only if g is transmission regular. lemma 2.3. [18] suppose that a is an n×n non-negative matrix with the spectral radius λ(a) and row sums r1,r2, . . . ,rn. we have min1≤i≤n ri ≤ λ(a) ≤ max1≤i≤n ri. if a is irreducible, then one of the equalities holds if and only if all the row sums of a are equal. since the i-th row sum of dα(g) is ri = αtri + (1 −α) ∑n j=1 dij = tri, by lemma 2.3 we have the following statement. corollary 2.4. suppose g is an n-vertex simple connected graph. let trmax and trmin be the largest and the smallest transmissions of g, respectively. we have trmin ≤ ∂(g) ≤ trmax. furthermore, any equality above holds if and only if g is transmission regular. our first aim is to giving some upper bounds for edα(g). the following theorem presents an upper bound for edα(g). some parameters like wiener index w(g) and transmission degrees are used. 75 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 theorem 2.5. suppose g is a graph of order n. we have edα(g) ≤ √√√√√n  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  . (1) the above equality holds if and only if g is a complete graph. proof. let ∂1 ≥ ∂2 ≥ ··· ≥ ∂n be the eigenvalues of dα(g). by applying the cauchy-schwarz inequality to these n−1 vectors (1, 1, . . . , 1) and (∣∣∣∂2 − 2αw(g)n ∣∣∣ ,∣∣∣∂3 − 2αw(g)n ∣∣∣ , . . . ,∣∣∣∂n − 2αw(g)n ∣∣∣) we obtain, ( edα(g) − ( ∂1 − 2αw(g) n ))2 ≤ (n− 1) ( 2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i − 4α2w2(g) n − ( ∂1 − 2αw(g) n )2 ) . (2) then edα(g) ≤ ∂1 − 2αw(g) n + √√√√√(n− 1)  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n − ( ∂1 − 2αw(g) n )2. the function f(x) = x + √√√√√(n− 1)  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n −x2   decreases if and only if x ≥ √ 2(1−α)2 ∑ 1≤i<j≤n(dij) 2+α2 ∑ n i=1 tr2 i −4α 2w2(g) n n . therefore edα(g) ≤ f   √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n n   . hence the result follows. now, suppose equality holds in (1). then ∂1 = √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n n + 2αw(g) n . from equality in (2), we get ∣∣∣∂2 − 2αw(g)n ∣∣∣ = ∣∣∣∂3 − 2αw(g)n ∣∣∣ = · · · = ∣∣∣∂n − 2αw(g)n ∣∣∣ and hence n∑ i=2 ( ∂i − 2αw(g) n )2 = ( 2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i − 4α2w2(g) n − ( ∂1 − 2αw(g) n )2 ) , 76 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 that is, (n− 1) ( ∂i − 2αw(g) n )2 = ( 2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i − 4α2w2(g) n − ( ∂1 − 2αw(g) n )2 ) . therefore for i = 2, . . . ,n, we get∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ = √ 2(1−α)2 ∑ 1≤i<j≤n(dij) 2+α2 ∑ n i=1 tr2 i −4α 2w2(g) n −(∂1− 2αw(g) n ) 2 n−1 = √ 2(1−α)2 ∑ 1≤i<j≤n(dij) 2+α2 ∑ n i=1 tr2 i −4α 2w2(g) n n . thus, we have ∂1 = p + 2αw(g) n and ∂i = 2αw(g) n + p or ∂i = 2αw(g) n − p, where p =√ 2(1−α)2 ∑ 1≤i<j≤n(dij) 2+α2 ∑ n i=1 tr2 i −4α 2w2(g) n n . in the first case g has one distinct dα-eigenvalue, which is so if g = k1. in the second case g has two distinct eigenvalues namely, p + 2αw(g) n and 2αw(g) n −p with multiplicity 1 and n−1, respectively and so by lemma 2.1, we have g = kn. conversely, it is easy to see that if g = kn, the equality occurs. this completes the proof. corollary 2.6. suppose g is a connected graph over n vertices. assume the diameter is d. we have edα(g) ≤ n √ (n− 1) ( (1 −α)2d2 + α2n2(n− 1) 4 −α2(n− 1) ) . (3) the above equality holds if and only if g ∼= k2. proof. since dij ≤ d for i 6= j, and there are n(n−1) 2 pairs of vertices in g, from the upper bound part in theorem 2.5, and also using the inequality tri ≤ n(n−1) 2 , we conclude edα(g) ≤ √√√√√n  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n   ≤ √ n ( 2(1 −α)2d2 n(n− 1) 2 + α2n3(n− 1)2 4 −α2n(n− 1)2 ) = n √ (n− 1) ( (1 −α)2d2 + α2n2(n− 1) 4 −α2(n− 1) ) . equality occurs in the inequality tri ≤ n(n−1) 2 if and only if g ∼= pn. this together with theorem 2.5, gives that equality occurs in the inequality (3) if and only if g ∼= k2. the following result provides an upper bound for edα(g) via transmission degrees together with θ1 and θn. theorem 2.7. let g be a connected graph of order n. then edα(g) ≤ √√√√√n  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  − n 2 ( |θ1|− |θn| )2 . (4) 77 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 proof. according to n ∑n i=1 |θi| 2 − (∑n i=1 |θi| )2 = ∑ 1≤i<j≤n(|θi|− |θj|) 2, we derive that n n∑ i=1 |θi|2 − ( n∑ i=1 |θi| )2 ≥ n−1∑ i=2 (( |θ1|− |θi| )2 + ( |θi|− |θn| )2) + ( |θ1|− |θn| )2 . (5) thanks to jensen’s inequality (see [19]), we obtain n−1∑ i=2 (( |θ1|− |θi| )2 + ( |θi|− |θn| )2) ≥ n− 2 2 ( |θ1|− |θn| )2 . (6) by inequalities (5) and (6), we arrive at the inequality (4). remark 2.8. since for every connected graph with the property |∂1| 6= |∂n|, we have√√√√√n  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  − n 2 (|θ1|− |θn|)2 ≤ √√√√√n  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  , it follows that the given upper bound in (4) is better than the given bound in (1). corollary 2.9. let g be a connected graph satisfying |∂1| 6= |∂n|. then edα(g) ≤ √ n 2  n ( 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n ) |θ1|− |θn|   . (7) proof. inequality (4) can be rewritten as (edα(g))2 + n 2 (|θ1|− |θn|)2 ≤ n  2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i − 4α2w2(g) n   . from the inequality between arithmetic and geometric means [19] we obtain 2 √ n 2 (edα(g))2(|θ1|− |θn|)2 ≤ n  2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i − 4α2w2(g) n   , where from the inequality (7) follows. our next result presents a sharp upper bound on the generalized distance energy. invariants like transmission degrees and α are involved. theorem 2.10. let g be an n-vertex connected graph with n ≥ 3. we have edα(g) ≤ (1 −α) √∑n i=1 tr 2 i n + (1 −α) √∑n i=1 tr 2 i n + √ (n− 2) ( 2ζ − (1 −α)2 ∑n i=1 tr 2 i n − (1 −α)2 ∑n i=1 tr 2 i n2 ) . (8) if equality holds in (8) then g must have exactly three or exactly four distinct dα-eigenvalues. 78 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 proof. using the cauchy-schwarz inequality on two (n− 2)-vectors (1, 1, . . . , 1︸ ︷︷ ︸ (n-2) times ) and (∣∣∣∣∂2 − 2αw(g)n ∣∣∣∣ , ∣∣∣∣∂3 − 2αw(g)n ∣∣∣∣ , . . . , ∣∣∣∣∂n−1 − 2αw(g)n ∣∣∣∣ ) , we have ( n−1∑ i=2 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ )2 ≤ ( n−1∑ i=2 1 )( n−1∑ i=2 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣2 ) , (9) and then edα(g) ≤ ∣∣∣∣∂1 − 2αw(g)n ∣∣∣∣ + ∣∣∣∣∂n − 2αw(g)n ∣∣∣∣ + √√√√(n− 2) ( 2ζ − ( ∂1 − 2αw(g) n )2 − ( ∂n − 2αw(g) n )2) , where 2ζ = 2(1 − α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n . let x = ∣∣∣∂1 − 2αw(g)n ∣∣∣ and y =∣∣∣∂n − 2αw(g)n ∣∣∣. define a function f(x,y) = x + y + √ (n− 2)(2ζ −x2 −y2). differentiating f(x,y) with respect to x and y, we have fx = 1 − x(n− 2)√ (n− 2) (2ζ −x2 −y2) , fy = 1 − y(n− 2)√ (n− 2) (2ζ −x2 −y2) , fxx = − ( 2ζ −y2 )√ n− 2√ (2ζ −x2 −y2)3 , fyy = − ( 2ζ −x2 )√ n− 2√ (2ζ −x2 −y2)3 and fxy = − xy √ n− 2√ (2ζ −x2 −y2)3 . for maxima or minima, fx = 0 and fy = 0 which implies x = y = √ 2ζ n . at this point the values of fxx,fyy,fxy and ∆ = fxxfyy −f2xy are fxx = − (n− 1) √ n− 2√ 2(n−2)3ζ n ≤ 0, fyy = − (n− 1) √ n− 2√ 2(n−2)3ζ n ≤ 0 fxy = − √ n− 2√ 2(n−2)3ζ n ≤ 0, and ∆ = n(n2 − 3n + 3) 2(n− 2)2ζ ≥ 0. therefore f(x,y) attains its maximum value at this point, hence f (√ 2ζ n , √ 2ζ n ) = √ 2nζ. however f(x,y) decreases in the intervals √ 2ζ n ≤ x ≤ √ ζ and 0 ≤ y ≤ √ 2ζ n . since 2 ∑ 1≤i<j≤n(dij) 2 < ( ∑n i=1 tri) 2 n (see 79 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 [10]), and by the cauchy-schwartz inequality we have (∑n i=1 tri )2 ≤ n ∑n i=1 tr 2 i , hence for 0 ≤ α ≤ 1 2 we get √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n n < √ (1−α)2( ∑ n i=1 tri) 2 n + α2 ∑n i=1 tr 2 i − α2( ∑ n i=1 tri) 2 n n = √ (1 − 2α) ( ∑ n i=1 tri) 2 n + α2 ∑n i=1 tr 2 i n ≤ √ (1 − 2α) ∑n i=1 tr 2 i + α 2 ∑n i=1 tr 2 i n = (1 −α) √∑n i=1 tr 2 i n , then √ 2ζ n ≤ (1 −α) √∑ n i=1 tr2 i n ≤ x ≤ √ ζ. now, by cauchy-schwarz inequality, we have tr2i = (∑n j=1 dij )2 ≤ n ∑n j=1 d 2 ij. hence ∑n i=1 tr 2 i ≤ n ∑n i=1 ∑n j=1 d 2 ij, and then we get ∑ 1≤i<j≤n d 2 ij ≥ 1 2n ∑n i=1 tr 2 i . then we have√ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n n ≥ √√√√2(1 −α)2 ∑1≤i<j≤n(dij)2 + α2(∑ni=1 tri)2n − α2(∑ni=1 tri)2n n = √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 n ≥ (1 −α) √∑n i=1 tr 2 i n . hence 0 ≤ y ≤ (1−α) √∑ n i=1 tr 2 i n ≤ √ 2ζ n . therefore f(x,y) ≤ f ( (1 −α) √∑n i=1 tr 2 i n , (1 −α) √∑n i=1 tr 2 i n ) ≤ f (√ 2ζ n , √ 2ζ n ) . consequently, edα(g) ≤ (1 −α) √∑n i=1 tr 2 i n + (1 −α) √∑n i=1 tr 2 i n + √ (n− 2) ( 2ζ − (1 −α)2 ∑n i=1 tr 2 i n − (1 −α)2 ∑n i=1 tr 2 i n2 ) . the first part is complete. suppose the equality holds in (8). from equality in (9), we get√∣∣∣∣∂2 − 2αw(g)n ∣∣∣∣ = √∣∣∣∣∂3 − 2αw(g)n ∣∣∣∣ = · · · = √∣∣∣∣∂n−1 − 2αw(g)n ∣∣∣∣, and hence n−1∑ i=2 ( ∂i − 2αw(g) n )2 = ( 2ζ − ( ∂1 − 2αw(g) n )2 − ( ∂n − 2αw(g) n )2) . 80 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 that is, (n− 2) ( ∂i − 2αw(g) n )2 = 2ζ − ( ∂1 − 2αw(g) n )2 − ( ∂n − 2αw(g) n )2 n− 2 , i = 2, . . . ,n− 1. therefore, ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ = √√√√2ζ −(∂1 − 2αw(g)n )2 −(∂n − 2αw(g)n )2 n− 2 , i = 2, . . . ,n− 1. hence ∣∣∣∂i − 2αw(g)n ∣∣∣ can have at most two distinct values. thus, if equality holds in (8), then g must be a connected graph with exactly three or exactly four distinct dα-eigenvalues. next, we consider some lower bounds for the generalized distance energy of graphs. the following theorem describes a lower bound based on the wiener index w(g) and the parameter α. theorem 2.11. suppose g is an n-vertex connected graph. its wiener index is w(g). we have edα(g) > 4αw(g) n − 2 √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i −∂ 2 1 n− 1 . (10) proof. let ∂1 ≥ ∂2 ≥ . . . ≥ ∂n be the generalized distance eigenvalues and t is the number of generalized distance eigenvalues of g which are greater than or equal to 2αw(g) n . note that 2(1 −α)2 ∑ 1≤i<j≤n (dij) 2 + α2 n∑ i=1 tr2i = n∑ i=1 ∂2i ≥ ∂ 2 1 + (n− 1)∂ 2 n, (11) hence ∂n ≤ √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i −∂ 2 1 n− 1 . in view of generalized distance energy, we see that edα(g) = 2 max 1≤j≤n ( j∑ i=1 ∂i − 2αjw(g) n ) ≥ 2 ( n−1∑ i=1 ∂i − 2α(n− 1)w(g) n ) = 2 ( 2αw(g) −∂n − 2α(n− 1)w(g) n ) = 4αw(g) n − 2∂n ≥ 4αw(g) n − 2 √ 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i −∂ 2 1 n− 1 . by contradiction, we will prove that the inequality in (10) is strict. for this we assume that the equality in (10) holds true. therefore, all inequalities above are essentially equalities. thus we arrive at max 1≤j≤n ( j∑ i=1 ∂i − 2αjw(g) n ) = n−1∑ i=1 ∂i − 2α(n− 1)w(g) n . (12) the equality in (12) holds if and only if t = n−1 and equality occurs in (11) if and only if ∂2 = · · · = ∂n. in other words, this happens if and only if g has exactly two distinct eigenvalues. since t = n− 1, we see that ∂n < 2αw(g) n ≤ ∂n−1 = ∂n, which is clearly a contradiction. the proof is complete. 81 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 the following observation is immediate from lemma 2.2 and theorem 2.11. corollary 2.12. suppose g is an n-vertex connected graph with minimum transmission degree trmin and wiener index w(g). we have edα(g) > 4αw(g) n − 2 √ n2(2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i ) − 4w2(g) n− 1 . next, we establish the lower bound for edα(g) involving wiener index w(g), transmission degrees as well as generalized distance spectral spread dαs(g). theorem 2.13. suppose that g is an n-vertex connected graph. we have edα(g) ≥ 2 ( 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n ) dαs(g) . (13) proof. let ∂1 ≥ ∂2 ≥ . . . ≥ ∂n be the generalized distance eigenvalues of g. assume x = (xi) and a = (ai), i = 1, 2, . . . ,n, be real number sequences satisfying n∑ i=1 |xi| = 1 and n∑ i=1 xi = 0. it is proved in [20] that ∣∣∣∣∣ n∑ i=1 aixi ∣∣∣∣∣ ≤ 12 ( max 1≤i≤n ai − min 1≤i≤n ai ) . (14) let ai = ∂i − 2αw(g) n and xi = ∂i− 2αw(g) n∑ n i=1|∂i−2αw(g)n | , i = 1, 2, . . . ,n. since n∑ i=1 xi = ∑n i=1 ( ∂i − 2αw(g) n ) ∑n i=1 ∣∣∣∂i − 2αw(g)n ∣∣∣ = 0 and n∑ i=1 |xi| = ∑n i=1 ∣∣∣∂i − 2αw(g)n ∣∣∣∑n i=1 ∣∣∣∂i − 2αw(g)n ∣∣∣ = 1, the conditions for inequality (14) are satisfied. hence, we have∣∣∣∣∣∣∣ ∑n i=1 ( ∂i − 2αw(g) n )2 ∑n i=1 ∣∣∣∂i − 2αw(g)n ∣∣∣ ∣∣∣∣∣∣∣ ≤ 1 2 ( max 1≤i≤n ( ∂i − 2αw(g) n ) − min 1≤i≤n ( ∂i − 2αw(g) n )) , that is 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n edα(g) ≤ 1 2 ( ∂1 − 2αw(g) n −∂n + 2αw(g) n ) , then, we get edα(g) ≥ 2 ( 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n ) dαs(g) . 82 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 the following bound for dαs(g) was established in [22]: dαs(g) ≤ √√√√√2  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  . (15) hence, using (15) and (13), we obtain the following lower bound for the generalized distance energy. corollary 2.14. suppose that g is an n-vertex connected graph. we have edα(g) ≥ √√√√√2  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i − 4α2w2(g) n  . corollary 2.15. assume g is connected and has n vertices. we obtain edα(g) ≥ (1 −α) √ 2n(n− 1). proof. since dij ≥ 1 for i 6= j and there are n(n−1) 2 pairs of vertices in g, from the lower bound of corollary 2.14, we get edα(g) ≥ √√√√√2  2(1 −α)2 ∑ 1≤i<j≤n (dij)2 + α2 n∑ i=1 tr2i −α2 n∑ i=1 tr2i   ≥ √4(1 −α)2 n(n− 1) 2 = (1 −α) √ 2n(n− 1). we conclude by giving the following lower bound for the generalized distance energy edα(g) involving some graph-theoretic invariants such as w(g), trmax and trmin. theorem 2.16. suppose that g is an n-vertex graph. let trmax and trmin be respectively the largest and the least transmissions of g. let γ = ∣∣∣det(dα(g) − 2αw(g)n i)∣∣∣. we have edα(g) ≥ trmin − 2αw(g) n + √ 2ζ −ω2 + (n− 1)(n− 2) ( γ ω ) 2 n−1 , (16) where 2ζ = 2(1 − α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n and ω = trmax − 2αw(g) n . equality holds if and only if either g is a complete graph or a k-transmission regular graph with three distinct dα-eigenvalues ( k, √ m −k2(1 −α)2 n− 1 + αk,− √ m −k2(1 −α)2 n− 1 + αk ) , where m = 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2. proof. it is not difficult to see that ( edα(g) )2 = n∑ i=1 ( ∂i − 2αw(g) n )2 + 2 ∑ 1≤i<j≤n ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ ∣∣∣∣∂j − 2αw(g)n ∣∣∣∣ = 2ζ + 2 ( ∂1 − 2αw(g) n ) n∑ i=2 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ + 2 ∑ 2≤i<j≤n ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ ∣∣∣∣∂j − 2αw(g)n ∣∣∣∣ , (17) 83 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 where 2ζ = 2(1−α)2 ∑ 1≤i<j≤n(dij) 2 +α2 ∑n i=1 tr 2 i − 4α2w2(g) n . it follows from the arithmetic-geometric mean inequality that ∑ 2≤i<j≤n ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ ∣∣∣∣∂j − 2αw(g)n ∣∣∣∣ ≥ (n− 1)(n− 2)2 ( n∏ i=2 ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ ) 2 n−1 (18) = (n− 1)(n− 2) 2 ( γ ∂1 − 2αw(g) n ) 2 n−1 . (19) applying the inequality (19), we get from (17) that ( edα(g) )2 ≥ 2ζ + 2 ( ∂1 − 2αw(g) n )( edα(g) − ( ∂1 − 2αw(g) n )) + (n− 1)(n− 2) ( γ ∂1 − 2αw(g) n ) 2 n−1 , which implies that ( edα(g) )2 − 2 ( ∂1 − 2αw(g) n ) edα(g) − [ 2ζ − 2 ( ∂1 − 2αw(g) n )2 +(n− 1)(n− 2) ( γ ∂1 − 2αw(g) n ) 2 n−1 ] ≥ 0. by solving the above inequality, we get edα(g) ≥ ∂1 − 2αw(g) n + √√√√ 2ζ − ( ∂1 − 2αw(g) n )2 + (n− 1)(n− 2) ( γ ∂1 − 2αw(g) n ) 2 n−1 . applying corollary 2.4, we have edα(g) ≥ trmin − 2αw(g) n (20) + √√√√ 2ζ − ( trmax − 2αw(g) n )2 + (n− 1)(n− 2) ( γ trmax − 2αw(g) n ) 2 n−1 . hence we get the required result in (16). the first part is complete. assume that the equality in (16) holds. the rest inequalities above are essentially equalities. hence, ∂1 = trmax = trmin, and g is a transmission regular graph. it follows from the equality in (18), we see that ∣∣∣∂2 − 2αw(g)n ∣∣∣ = ∣∣∣∂3 − 2αw(g)n ∣∣∣ = · · · = ∣∣∣∂n − 2αw(g)n ∣∣∣ , and hence n∑ i=2 ( ∂i − 2αw(g) n )2 = ( 2ζ − ( ∂1 − 2αw(g) n )2) , that is, (n− 1) ( ∂i − 2αw(g) n )2 = ( 2ζ − ( ∂1 − 2αw(g) n )2) , 84 a. alhevaz et.al. / j. algebra comb. discrete appl. 10(2) (2023) 73–86 then for each i = 2, . . . ,n, we have ∣∣∣∣∂i − 2αw(g)n ∣∣∣∣ = √√√√2ζ −(∂1 − 2αw(g)n )2 n− 1 . consequently, ∣∣∣∂i − 2αw(g)n ∣∣∣ can have at most two distinct values and we arrive at the following: (i) g has exactly one distinct dα-eigenvalue. thus, g = k1. (ii) g has exactly two distinct dα-eigenvalues. using lemma 2.1, g = kn. (iii) g has exactly three distinct dα-eigenvalues. we have ∂1 = trmax, and ∣∣∣∂i − 2αw(g)n ∣∣∣ =√ 2ζ−(trmax− 2αw(g) n ) 2 n−1 , i = 2, . . . ,n. moreover, g is k-transmission regular graph. we have g as a graph with three distinct dα-eigenvalues( k, √ m −k2(1 −α)2 n− 1 + αk,− √ m −k2(1 −α)2 n− 1 + αk ) , in which m = 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2. hence the result follows. since for any i, we have n− 1 ≤ tri ≤ n(n−1) 2 , hence from theorem 2.16, the following observation is immediate. corollary 2.17. suppose that g is an n-vertex graph and γ = ∣∣∣det(dα(g) − 2αw(g)n i)∣∣∣. we have edα(g) ≥ n− 1 − 2αw(g) n + √ 2ζ −h2 + (n− 1)(n− 2) ( γ h ) 2 n−1 , where 2ζ = 2(1 −α)2 ∑ 1≤i<j≤n(dij) 2 + α2 ∑n i=1 tr 2 i − 4α2w2(g) n and h = n(n−1) 2 − 2αw(g) n . acknowledgements: the authors would like to thank the anonymous referee(s) and the editor for their valuable comments to improve the presentation of the manuscript. y. shang was supported in part by the uoa flexible fund no. 201920a1001 from northumbria university. references [1] a. alhevaz, m. baghipur, k. c. das and y. shang, sharp bounds on (generalized) distance energy of graphs, mathematics 8(3) (2020) article id: 426. 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[24] j. yang, l. you and i. gutman, bounds on the distance laplacian energy of graphs, kragujevac j. math. 37 (2013) 245–255. 86 https://doi.org/10.1016/j.laa.2013.02.030 https://doi.org/10.1016/j.laa.2013.02.030 https://doi.org/10.1016/j.laa.2013.01.023 https://doi.org/10.1016/j.laa.2013.01.023 https://doi.org/10.1016/j.laa.2018.10.01 https://doi.org/10.1016/j.laa.2018.10.01 https://doi.org/10.1016/j.laa.2018.01.032 https://doi.org/10.1016/j.laa.2018.01.032 https://doi.org/10.1142/s1793830920501001 https://doi.org/10.1142/s1793830920501001 https://doi.org/10.1016/j.dam.2016.09.030 https://doi.org/10.1016/j.dam.2016.09.030 https://www.scirp.org/(s(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=2058507 https://doi.org/10.1007/978-3-642-59448-9_13 https://doi.org/10.1007/978-3-642-59448-9_13 https://doi.org/10.1007/978-3-642-59448-9_13 https://doi.org/10.1016/j.laa.2008.07.005 https://doi.org/10.1016/j.laa.2008.07.005 https://match.pmf.kg.ac.rs/electronic_versions/match60/n2/match60n2_461-472.pdf https://match.pmf.kg.ac.rs/electronic_versions/match60/n2/match60n2_461-472.pdf https://doi.org/10.1016/j.laa.2015.08.032 https://doi.org/10.1016/j.laa.2015.08.032 https://doi.org/10.1080/03081087.2020.1814194 https://doi.org/10.1080/03081087.2020.1814194 https://doi.org/10.1016/j.laa.2020.05.022 https://doi.org/10.1016/j.laa.2020.05.022 http://elib.mi.sanu.ac.rs/files/journals/kjm/41/kjom3702-04.pdf http://elib.mi.sanu.ac.rs/files/journals/kjm/41/kjom3702-04.pdf introduction new bounds on generalized distance energy of graphs references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1000837 j. algebra comb. discrete appl. 8(3) • 167–195 received: 21 september 2020 accepted: 08 march 2021 journal of algebra combinatorics discrete structures and applications minimum distance and idempotent generators of minimal cyclic codes of length p1 α1p2 α2p3 α3 research article pankaj kumar, pinki devi abstract: let p1, p2, p3, q be distinct primes and m = p1α1p2α2p3α3 . in this paper, it is shown that the explicit expressions of primitive idempotents in the semi-simple ring rm = fq[x]/(xm − 1) are the trace function of explicit expressions of primitive idempotents from r p αi i . the minimal polynomials, generating polynomials and minimum distances of minimal cyclic codes of length m over fq are also discussed. all the results obtained in [1], [3], [4], [5], [11] and [14] are simple corollaries to the results obtained in the paper. 2010 msc: 94b05, 94b15 keywords: cyclotomic cosets, primitive idempotents, cyclic codes, trace function 1. introduction let p1,p2,p3 and q be distinct primes and m = p1α1p2α2p3α3.then rm = fq[x]/(xm −1) is a semisimple ring. every cyclic code c of length m over fq is a direct sum of minimal cyclic codes in rm. therefore, we need to compute the minimal cyclic codes of length m over fq. a minimal cyclic code always has a primitive idempotent generator [12, theorem 1, chapter 8], therefore, the study of minimal cyclic codes need the computation of explicit expressions of primitive idempotents in rm. for m = pn, pruthi and arora [13] obtained all the minimal cyclic codes of length m in rpn = fq[x]/(xp n −1); q is a primitive root modulo pn. in [7], chen et al. studied the minimal cyclic codes of length lm over fq, where l is a prime divisor of q − 1 and m is a positive integer. many authors have worked to compute the primitive central idempotents in semi-simple group algebra. in [2], bakshi et al. have computed primitive central idempotents in semi-simple group algebra fg; f is finite fields and g is arbitrary meta-cyclic group. in another paper [9], they developed an algorithm to compute the primitive central idempotents in a semi-simple group algebra. broche and rio [6] developed a method to compute the primitive central idempotents and the wedderburn decomposition of a semi simple finite group algebra of pankaj kumar, pinki devi (corresponding author); department of mathematics, guru jambheshwar university of science and technology, hisar 125001, india (email: joshi78023@yahoo.com, pinkinarwal123@gmail.com). 167 https://orcid.org/0000-0002-3371-1875 https://orcid.org/0000-0002-3245-8863 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 an abelian-by-supersolvable group g. ferraz and milies [8] discussed a simple method of computing the idempotent generator of minimal abelain codes and discussed all the results as obtained in [1]. sharma et al. [15] gave an algorithm to compute all the primitive idempotents over fq in rpn by choosing fpn−1 as the order of q modulo pn. since m = p1α1p2α2p3α3, therefore it is natural to ask that “can one use the explicit expressions of primitive idempotents from rpαi i to compute the explicit expressions of primitive idempotents in rm?” in this regard bakshi et al. [4] used trace function to compute the explicit expressions of primitive idempotents in rm; m = m1m2 with gcd ( om1(q),om2(q) ) = 1 or d, where omi(q) denotes the order of q modulo mi. kumar and arora [10] defined a λ-mapping to compute the explicit expressions of primitive idempotents in rm; m = p α1 1 p α2 2 ...p αr r with the help of primitive idempotents from rpαi i in a single step under the following conditions: (i) for each i, q is a primitive root modulo pαii , (ii) at most one prime factor of m is of the form 4k + 1 and the other prime factors of m are of the form 4k + 3, (iii) gcd ( φ(pαii ),φ(p αj j ) ) = 2, 1 ≤ i 6= j ≤ r. in this paper, we choose m = p1α1p2α2p3α3, where pi’s are distinct primes and, for each i,1 ≤ i ≤ 3, opαi i (q) = φ(p αi i ) ri and obtain the primitive idempotents and minimum distances of minimal cyclic codes in rm over the finite field fq. this paper is organized as follows: in section 2, we use λmapping to obtain the q-cyclotomic cosets and in theorem 2.5, we count the number of distinct q-cyclotomic cosets modulo m. in section 3, the primitive idempotents are obtained. in theorem 3.5, we use λ-mapping and in theorem 3.6 we use trace function to compute the primitive idempotents in rm with the help of primitive idempotents from rpαi i . the minimal polynomials and the generating polynomials of some cyclic codes of length m are obtained in section 4. in section 5, we give a lower bound on minimum distances of cyclic codes of length m. we include examples in the last section. in example 6.1, we choose m = 715 and count the total number of 3-cyclotomic cosets modulo 715. then we obtain the explicit expression of θ7151 (x) in r715 by λ product of polynomials and by trace function (the explicit expressions of all primitive idempotents are shown in appendix 1). in example 6.2, the minimal cyclic codes of length 30 are discussed in r30 = f7[x]/(x30 −1). all the results obtained in [1], [3], [4], [5], [11] and [14] are simple corollaries to the results obtained in the paper. 2. cyclotomic cosets modulo m in this section we use λ-mapping [10, definition(2.2)] to compute the q-cyclotomic cosets modulo m with the help of qvi-cyclotomic cosets modulo pαii . then we count the number of distinct q-cyclotomic cosets modulo m. throughout this paper p1,p2,p3 and q are distinct primes, m = p1α1p2α2p3α3, for 1 ≤ i ≤ 3, opαi i (q) = φ(p αi i ) ri , mi = mpαi i , a = {0,1,2, ...,m − 1} and ai = {0,1, ...,pαii − 1}, δ is a fixed primitive mth root of unity in some extension of fqt and δi = δmi is a fixed primitive piαith root of unity in the same extension of fqt. we denote the q-cyclotomic coset modulo m containing s; 0 ≤ s ≤ m−1 by c m(q) s = {s,sq,sq2, ...,sqr−1}, where r is the smallest positive integer such that sqr ≡ s(mod m) and, for a divisor i of r, cm(q i) s = {s,sqi,sq2i, ...,sqi( r i −1)} denote the qi-cyclotomic coset modulo m containing s. definition 2.1. [λ-mapping] the mapping λ from a1 ×a2 ×a3 −→ a defined by λ(b1,b2,b3) = b1m1 + b2m2 + b3m3(mod m), where bi ∈ ai is called the λ-mapping. it is a one-one and onto mapping. lemma 2.2. let v1,v2 and v3 be positive integers. if l = lcm(v1,v2,v3), l1 = lcm(v2,v3), l2 = lcm(v1,v3) and l3 = lcm(v1,v2) then, for 1 ≤ i ≤ 3 (i) li divides l. (ii) l divides vili and, if vili = uil, where ui is some positive integer, then l1l2l3 = l2lcm(u1,u2,u3). 168 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 theorem 2.3. let q be an odd prime with gcd(q,m) = 1. further, let om(q) = d, omi(q) = di, vi = o p αi i (q)di d . then c m(q) 1 = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) , where t = lcm(v1,v2,v3). proof. by definition 2.1, for 1 ∈ a , there exists (a1,a2,a3) ∈ a1 ×a2 ×a3 such that λ(a1,a2,a3) = 1. (1) as om(q) = d, therefore, multiplying (1) by qk; 0 ≤ k ≤ d−1, we get λ(a1,a2,a3) = 1 λ(a1q,a2q,a3q) = q λ(a1q 2,a2q 2,a3q 2) = q2 ... λ(a1q d−1,a2q d−1,a3q d−1) = qd−1. the left hand side of above equations has three columns namely b1 = {a1,a1q,a1q2, . . . ,a1qd−1}, b2 = {a2,a2q,a2q2, . . . ,a2qd−1}, b3 = {a3,a3q,a3q2, . . . ,a3qd−1}. as om1(q) = d1 and v1 = o p α1 1 (q)d1 d , therefore, the subset cp α1 1 (q v1 ) a1 of b1 will repeat with the set {(a2,a3),(a2q,a3q),(a2q2,a3q2), . . . ,(a2qd1−1,a3qd1−1)}. similarly, the subset cp α2 2 (q v2 ) a2 of b2 will repeat with the set {(a1,a3),(a1q,a3q),(a1q2,a3q2), . . . ,(a1qd2−1,a3qd2−1)} and cp α3 3 (q v3 ) a3 of b3 will repeat with the set {(a1,a2),(a1q,a2q),(a1q2,a2q2), . . . ,(a1qd3−1,a2qd3−1)}. hence c p α1 1 (q v1 ) a1 ×c p α2 2 (q v2 ) a2 ×c p α3 3 (q v3 ) a3 ⊂{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. 169 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 similarly, we get c p α1 1 (q v1 ) a1q ×c p α2 2 (q v2 ) a2q ×c p α3 3 (q v3 ) a3q ⊂{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. as the set {(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)} is finite, therefore, there exists a positive integer t; t = lcm(v1,v2,v3) such that t−1⋃ j=0 ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) ⊂{(a1,a2,a3),(a1q,a2q,a3q), ...,(a1qd−1,a2qd−1,a3qd−1)}. since c p αi i (qvi) aiqj contains d di elements, therefore, t−1⋃ j=0 ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) contains td 3 d1d2d3 elements. (by lemma 2.2) td 3 d1d2d3 = d, therefore, t−1⋃ j=0 ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) = {(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. equivalently, t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) = λ{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)} = c m(q) 1 . corollary 2.4. let s ∈ a. if ais ≡ bip βi i (mod p αi i ) then cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) . proof. by theorem 2.3, c m(q) 1 = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1qj ×cp α2 2 (q v2 ) a2qj ×cp α3 3 (q v3 ) a3qj ) , therefore, for s ∈ a, we have cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1sqj ×cp α2 2 (q v2 ) a2sqj ×cp α3 3 (q v3 ) a3sqj ) . 170 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 if ais ≡ bip βi i (mod p αi i ) then clearly cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) . theorem 2.5. let q be an odd prime with gcd(q,m) = 1. further, let opαi i (q) = φ(p αi i ) ri , om(q) = d, omi(q) = di, vi = o p αi i (q)di d and lcm(v1,v2,v3) = t. then, the number of distinct q-cyclotomic cosets modulo m is ∏3 i=1 riαivi t + ∑ 1≤i<j≤3 gcd ( opαi i (q),o p αj j (q) ) αiαjrirj + 3∑ i=1 αiri + 1. proof. since om(q) = d, omi(q) = di, vi = o p αi i (q)di d , therefore, by corollary 2.4, cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) . consequently, the number of distinct q-cyclotomic cosets modulo m is equal to the number of distinct combinations of the form t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) . we now count all such combinations by considering the following cases: case (1): 0 ≤ βi ≤ αi − 1; 1 ≤ i ≤ 3. in this case the number of distinct choices for c p αi i (qvi) bip βi i qj is αiviri. therefore, the number of distinct combinations of the form t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) is ∏r i=1 riαivi t . these q-cyclotomic cosets are of the form cm(q) sp β1 1 p β2 2 p β3 3 ; gcd(s,p1p2p3) = 1. case (2): β1 = α1,0 ≤ β2 ≤ α2 − 1,0 ≤ β3 ≤ α3 − 1. by choosing β1 = α1, we have fixed c p α1 1 (q v1 ) b1p β1 1 q j = {0}. in other words we have a single choice of qv1-cyclotomic coset modulo pα11 . consequently, t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) = t23−1⋃ j=0 λ ( c p α1 1 (q v1 ) 0 ×c p α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) , where t23 = v2 = v3 = gcd ( opα22 (q),opα33 (q) ) . therefore, the number of q-cyclotomic cosets mod m of the form cm(q) sp α1 1 p β2 2 p β3 3 , where gcd(s,p1p2p3) = 1, is gcd ( opα22 (q),opα33 (q) ) α2α3r2r3. case (3): 0 ≤ β1 ≤ α1 − 1,β2 = α2,0 ≤ β3 ≤ α3 − 1. in this case the number of distinct q cyclotomic cosets of the form cm(q) sp β1 1 p α2 2 p β3 3 is gcd ( opα11 (q),opα33 (q) ) α1α3r1r3. case (4): 0 ≤ β1 ≤ α1 − 1,0 ≤ β2 ≤ α2 − 1,β3 = α3. in this case the number of distinct q cyclotomic cosets of the form cm(q) sp β1 1 p β2 2 p α3 3 is gcd ( opα11 (q),opα22 (q) ) α1α2r1r2. 171 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 case (5): β1 = α1,β2 = α2,0 ≤ β3 ≤ α3 −1. for this case t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) = λ ( c p α1 1 (q v1 ) 0 ×c p α2 2 (q v2 ) 0 ×c p α3 3 (q v3 ) b3p β3 3 q j ) . therefore, the number of cosets of the form cm(q) sp α1 1 p α2 2 p β3 3 is α3r3. case (6): β1 = α1,0 ≤ β2 ≤ α2 −1,β3 = α3. in this case c m(q) sp α1 1 p β2 2 p α3 3 is α2r2. case (7): 0 ≤ β1 ≤ α1 −1,β2 = α2,β3 = α3. in this case c m(q) sp β1 1 p α2 2 p α3 3 is α1r1. case (8): β1 = α1,β2 = α2,β3 = α3. for this case t−1⋃ j=0 λ ( c p α1 1 (q v1 ) b1p β1 1 q j ×cp α2 2 (q v2 ) b2p β2 2 q j ×cp α3 3 (q v3 ) b3p β3 3 q j ) = λ ( c p α1 1 (q v1 ) 0 ×c p α2 2 (q v2 ) 0 ×c p α3 3 (q v3 ) 0 ) . therefore, cm(q)0 is 1. as a result of above discussion, the number of distinct q-cyclotomic cosets modulo m is ∏3 i=1 riαivi t + ∑ 1≤i<j≤3 gcd ( opαi i (q),o p αj j (q) ) αiαjrirj + 3∑ i=1 αiri + 1. corollary 2.6. let q be an odd prime with gcd(q,m) = 1. further, let opαi i (q) = φ(p αi i ) ri , om(q) = d, omi(q) = di, vi = o p αi i (q)di d and lcm(v1,v2,v3) = t. if v1 = v2 = v3, then, the number of distinct q-cyclotomic cosets modulo m is ∏3 i=1(riαivi + 1) + t−1 t . proof. by theorem 2.5,∏3 i=1 riαivi t + ∑ 1≤i<j≤3 gcd ( opαi i (q),o p αj j (q))αiαjrirj + 3∑ i=1 αiri + 1. therefore, for lcm(v1,v2,v3) = t and v1 = v2 = v3. as gcd(opαi i (q),o p αj j (q) ) ×lcm ( opαi i (q),o p αj j (q) ) = opαi i (q)×o p αj j (q). use the above result in theorem 2.5, we have (α1v1r1)(α2v2r2)(α3v3r3) t + (α1v1r1)(α2v2r2) t + (α1v1r1)(α3v3r3) t + (α2v2r2)(α3v3r3) t + (α1v1r1) t + (α2v2r2) t + (α3v3r3) t + 1 further solve it. we get ∏3 i=1(riαivi + 1) + t−1 t . 172 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 3. primitive idempotents in rm for 1 ≤ i ≤ 3, let θp αi i (qvi) s (xi) and θ m(q) s (x) denote the primitive idempotents in rpαi i = fqvi [x]/(x p αi i −1) and rm = fq[x]/(xm −1) corresponding to c p αi i (qvi) s and c m(q) s respectively. in this section, we give two different proofs to show that θm(q)s (x) can be computed with the help of suitably chosen primitive idempotents θ p αi i (qvi) s (xi) from rpαi i . the composition of cm(q)s helps in choosing these θ p αi i (qvi) s (xi). by theorem 2.5, cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1sqj ×cp α2 2 (q v2 ) a2sqj ×cp α3 3 (q v3 ) a3sqj ) , therefore, in theorem 3.5, we use λ product of polynomials [definition 3.1] and show that θm(q)s (x) = t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) . in theorem 3.6, we use trace function trfqt|fq from fqt[x] to fq[x] to show that θms (x) = trfqt|fq ( θ p α1 1 (q v1 ) s (x m1)θ p α2 2 (q v2 ) s (x m2)θ p α3 3 (q v3 ) s (x m3) ) , where θ p αi i (qvi) s (x mi) is the polynomial obtained from θ p αi i (qvi) s (xi) by replacing xi by xmi. definition 3.1. [λ-product of xbii ’s] let x1,x2,x3 be three variables. define λ ( (α1x b1 1 ),(α2x b2 2 ),(α3x b3 3 ) ) = α1α2α3x λ(b1, b2, b3), where αi ∈ fqt, bi ∈ ai and call it the λ-product [10, definition 3.1] of xbii ’s. definition 3.2. let f(x) = ∑ aix i ∈ fqt[x]. then, the trace function trfqt|fq : fqt[x] → fq[x] is defined as trfqt|fq ( f(x) ) = ∑ trfqt|fq(ai)x i. further, for f(x),g(x),h(x) ∈ fqt[x], if f(x)g(x)h(x) = ∑ cix i, then trfqt|fq ( f(x)g(x)h(x) ) = ∑ trfqt|fq(ci)x i. definition 3.3. [3, theorem 1] the θm(q)s (x) = ∑m−1 i=0 �ix i, where �i = � cm(q)s i = 1 m (∑ s∈cm(q)s δ−is ) and δ is a fixed primitive mth root of unity in some extension of fq, is a primitive idempotent in rm corresponding to cm(q)s . note 3.4. in the following theorems, δ is a fixed primitive mth root of unity in some extension of fqt and therefore, δi = δmi is a primitive p αi i th root of unity in the same extension of fqt. theorem 3.5. if λ(a1,a2,a3) = 1 and cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1sqj ×cp α2 2 (q v2 ) a2sqj ×cp α3 3 (q v3 ) a3sqj ) , then θm(q)s (x) = t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) , where vi and t are same as defined in theorem 2.3. 173 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 proof. first we compute the coefficient of xk in t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) . by definition 3.3 θ p αi i (qvi) sqj (xi) = p αi i −1∑ si=0 �six si i ; 1 ≤ i ≤ 3, where �si = � c p αi i (qvi) sqj si = 1 pαii ∑ sqj∈c p αi i (qvi) sqj δi −sisqj. since 1 = λ(a1,a2,a3), therefore, k = λ(a1k,a2k,a3k). then, by definition 3.1 the coefficient of xk in t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) is t−1∑ j=0 � c p α1 1 (qv1 ) sqj a1k � c p α2 2 (qv2 ) sqj a2k � c p α3 3 (qv3 ) sqj a3k = t−1∑ j=0 ( 1 pα11 ∑ sqj∈c p α1 1 (qv1 ) sqj δ1 −a1ksqj )( 1 pα22 ∑ sqj∈c p α2 2 (qv2 ) sqj δ2 −a2ksqj ) ( 1 pα33 ∑ sqj∈c p α3 3 (qv3 ) sqj δ3 −a3ksqj ) δi = δ mi, therefore, we get t−1∑ j=0 � c p α1 1 (qv1 ) sqj a1k � c p α2 2 (qv2 ) sqj a2k � c p α3 3 (qv3 ) sqj a3k = t−1∑ j=0 1 pα11 p α2 2 p α3 3 ∑ (sqj,sqj,sqj)∈ ( c p α1 1 (qv1 ) sqj ×c p α2 2 (qv2 ) sqj ×c p α3 3 (qv3 ) sqj )δ−(m1a1ksqj+m2a2ksqj+m3a3ksqj) = t−1∑ j=0 1 m ∑ (sqj,sqj,sqj)∈ ( c p α1 1 (qv1 ) sqj ×c p α2 2 (qv2 ) sqj ×c p α3 3 (qv3 ) sqj )δ−k(m1a1+m2a2+m3a3)sqj = t−1∑ j=0 1 m ∑ (sqj,sqj,sqj)∈ ( c p α1 1 (qv1 ) sqj ×c p α2 2 (qv2 ) sqj ×c p α3 3 (qv3 ) sqj )δ−kλ(a1,a2,a3)sqj = t−1∑ j=0 1 m ∑ (sqj,sqj,sqj)∈ ( c p α1 1 (qv1 ) sqj ×c p α2 2 (qv2 ) sqj ×c p α3 3 (qv3 ) sqj )δ−ksqj. (2) 174 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 now we obtain the coefficient of xk in θm(q)s (x). by definition 3.3, the coefficient of xk in θ m(q) s (x) is �k = � cm(q)s k = 1 m ∑ s∈cm(q)s δ−ks = 1 m ∑ s∈ ⋃t−1 j=0 λ ( c p α1 1 (qv1 ) a1sq j ×c p α2 2 (qv2 ) a2sq j ×c p α3 3 (qv3 ) a3sq j )δ−ks = 1 m t−1∑ j=0 ∑ sqj∈λ ( c p α1 1 (qv1 ) a1sq j ×c p α2 2 (qv2 ) a2sq j ×c p α3 3 (qv3 ) a3sq j )δ−ksqj. since sqj = λ(a1sqj,a2sqj,a3sqj), therefore, �k = 1 m t−1∑ j=0 ∑ λ(a1sqj,a2sqj,a3sqj)∈λ ( c p α1 1 (qv1 ) a1sq j ×c p α2 2 (qv2 ) a2sq j ×c p α3 3 (qv3 ) a3sq j )δ−ksqj = 1 m t−1∑ j=0 ∑ (a1sqj,a2sqj,a3sqj)∈ ( c p α1 1 (qv1 ) a1sq j ×c p α2 2 (qv2 ) a2sq j ×c p α3 3 (qv3 ) a3sq j )δ−ksqj = 1 m t−1∑ j=0 ∑ (sqj,sqj,sqj)∈ ( c p α1 1 (qv1 ) sqj ×c p α2 2 (qv2 ) sqj ×c p α3 3 (qv3 ) sqj )δ−ksqj. (3) by (2) and (3) we get that the coefficient of xk is same in θm(q)s (x) and in t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) . hence θm(q)s (x) = t−1∑ j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) . theorem 3.6. if λ(a1,a2,a3) = 1 and cm(q)s = t−1⋃ j=0 λ ( c p α1 1 (q v1 ) a1sqj ×cp α2 2 (q v2 ) a2sqj ×cp α3 3 (q v3 ) a3sqj ) , then θm(q)s (x) = trfqt|fq ( θ p α1 1 (q v1 ) s (x m1)θ p α2 2 (q v2 ) s (x m2)θ p α3 3 (q v3 ) s (x m3) ) , where vi and t are same as defined in theorem 2.3. 175 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 proof. since t = lcm(v1,v2,v3), the fqvi ⊂ fqt and therefore, we can multiply θ p α1 1 (q v1 ) s (x m1) ∈ fqv1 [x],θ p α2 2 (q v2 ) s (x m2) ∈ fqv2 [x] and θ p α3 3 (q v3 ) s (x m3) ∈ fqv3 [x]. let θp α1 1 (q v1 ) s (x m1)θ p α2 2 (q v2 ) s (x m2)θ p α3 3 (q v3 ) s (x m3) = ∑ ckx k. we compute the coefficient of xk in trfqt|fq ( θ p α1 1 (q v1 ) s (x m1)θ p α2 2 (q v2 ) s (x m2)θ p α3 3 (q v3 ) s (x m3) ) . by definition 3.3 θ p αi i (qvi) s (x mi) = p αi i −1∑ si=0 �six misi; 1 ≤ i ≤ 3, where �si = � c p αi i (qvi) s si = 1 pαii ∑ s∈c p αi i (qvi) s δi −sis. (4) as 1 = λ(a1,a2,a3), therefore, k = λ(a1k,a2k,a3k) = ( m1a1k + m2a2k + m3a3k ) (mod m). therefore, ck = � c p α1 1 (qv1 ) s a1k � c p α2 2 (qv2 ) s a2k � c p α3 3 (qv3 ) s a3k = ( 1 pα11 ∑ s∈c p α1 1 (qv1 ) s δ1 −a1ks )( 1 pα22 ∑ s∈c p α2 2 (qv2 ) s δ2 −a2ks )( 1 pα33 ∑ s∈c p α3 3 (qv3 ) s δ3 −a3ks ) . δi = δ mi, therefore, ck = 1 pα11 p α2 2 p α3 3 ∑ (s,s,s)∈ ( c p α1 1 (qv1 ) s ×c p α2 2 (qv2 ) s ×c p α3 3 (qv3 ) s )δ−(m1a1ks+m2a2ks+m3a3ks) = 1 m ∑ (s,s,s)∈ ( c p α1 1 (qv1 ) s ×c p α2 2 (qv2 ) s ×c p α3 3 (qv3 ) s )δ−(m1a1+m2a2+m3a3)ks = 1 m ∑ (s,s,s)∈ ( c p α1 1 (qv1 ) s ×c p α2 2 (qv2 ) s ×c p α3 3 (qv3 ) s )δ−ks. since (s,s,s) ∈ ( c p α1 1 (q v1 ) s ×c p α2 2 (q v2 ) s ×c p α3 3 (q v3 ) s ) ⇐⇒ (a1s,a2s,a3s) ∈ ( c p α1 1 (q v1 ) a1s ×c p α2 2 (q v2 ) a2s ×c p α3 3 (q v3 ) a3s ) ⇐⇒ λ(a1s,a2s,a3s) ∈ λ ( c p α1 1 (q v1 ) a1s ×c p α2 2 (q v2 ) a2s ×c p α3 3 (q v3 ) a3s ) 176 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 ⇐⇒ s ∈ λ ( c p α1 1 (q v1 ) a1s ×c p α2 2 (q v2 ) a2s ×c p α3 3 (q v3 ) a3s ) therefore, ck = 1 m ∑ s∈λ ( c p α1 1 (qv1 ) a1s ×c p α2 2 (qv2 ) a2s ×c p α3 3 (qv3 ) a3s )δ−ks. then, trfqt|fq ( θ p α1 1 (q v1 ) s (x m1)θ p α2 2 (q v2 ) s (x m2)θ p α3 3 (q v3 ) s (x m3) ) = ∑ trfqt|fq(ck)x k = ∑ trfqt|fq ( 1 m ∑ s∈λ ( c p α1 1 (qv1 ) a1s ×c p α2 2 (qv2 ) a2s ×c p α3 3 (qv3 ) a3s )δ−ks ) xk = ∑( 1 m ∑ s∈ ⋃t−1 j=0 λ ( c p α1 1 (qv1 ) a1sq j ×c p α2 2 (qv2 ) a2sq j ×c p α3 3 (qv3 ) a3sq j )δ−ks ) xk = ∑( 1 m ∑ s∈cs δ−ks ) xk = ∑ �kx k = θm(q)s (x). it completes the proof. as a result of theorem 3.5 and theorem 3.6, we have following results. corollary 3.7. if v1 = v2 = v3 = 1 and if 1 = λ(a1,a2,a3) and cm(q)s = λ ( c p α1 1 (q v1 ) a1s ×c p α2 2 (q v2 ) a2s ×c p α3 3 (q v3 ) a3s ) , then θm(q)s (x) = λ ( θ p α1 1 (q v1 ) s (x1)θ p α2 2 (q v2 ) s (x2)θ p α3 3 (q v3 ) s (x3) ) . corollary 3.8. if v1 = v2 = v3 = 1 and if 1 = λ(a1,a2,a3) and cm(q)s = λ ( c p α1 1 (q) a1s ×c p α2 2 (q) a2s ×c p α3 3 (q) a3s ) , then θm(q)s (x) = trfq|fq ( θ p α1 1 (q) s (x m1)θ p α2 2 (q) s (x m2)θ p α3 3 (q) s (x m3) ) = θ p α1 1 (q) s (x m1)θ p α2 2 (q) s (x m2)θ p α3 3 (q) s (x m3). note 3.9. if θm(q)s (x) = ∑t−1 j=0 λ ( θ p α1 1 (q v1 ) sqj (x1)θ p α2 2 (q v2 ) sqj (x2)θ p α3 3 (q v3 ) sqj (x3) ) , then practically θm(q)s (x) =∑t−1 j=0 λ ( θ p α1 1 (q v1 ) s1qj (x1)θ p α2 2 (q v2 ) s2qj (x2)θ p α3 3 (q v3 ) s3qj (x3) ) , where s ≡ si(mod pαii ). we have shown it by calculating expressions of all the primitive idempotents in r715 over f3 in appendix 1. 4. minimal and generating polynomials of minimal cyclic codes of length m let mms denote the minimal cyclic code of length m corresponding to c m(q) s . further, let ηms (x) and gms (x) denote the minimal polynomial and generating polynomial of m m s respectively. then η m s (x) =∏ s∈cm(q)s (x−δs) and gms (x) = (xm−1) ηms (x) , where δ is a primitive mth root of unity. in this section we obtain minimal polynomials and generating polynomials of minimal cyclic codes of length m. 177 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 note 4.1. since m = pα11 p α2 2 p α3 3 and gcd(m,q) = 1, therefore, the polynomials x p β1 1 p β2 2 p β3 3 −1; 0 ≤ βi ≤ αi and 1 ≤ i ≤ 3 are separable over fq. lemma 4.2. the expression f(x) h(x) , where f(x) = ( xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1 ) ( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 )( xp α1−β1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) , and h(x) = ( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1 1 p α2−β2−1 2 p α3−β3 3 −1 ) ( xp α1−β1 1 p α2−β2 2 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1), is a polynomial over fq. proof. to prove that f(x) h(x) is a polynomial over fq, it is sufficient to show that every root of h(x) is a root of f(x) also. let s = ap β1 1 p β2 2 p β3 3 ; 0 ≤ a ≤ p α1−β1 1 p α2−β2 2 p α3−β3 3 −1, be an integer. we consider the following cases: case (1): gcd(a,p1p2p3) = p1(or p2 or p3). in this case the δs is a multiple root of h(x) with multiplicity 1. moreover, it is a root of f(x) also having multiplicity 1. hence in this case δs has same multiplicity. case (2): gcd(a,p1p2p3) = p1p2(or p1p3 or p2p3). in this case the δs is a multiple root of h(x) with multiplicity 2. moreover, it is a root of f(x) also having multiplicity 2. hence in this case δs has same multiplicity. case (3): gcd(a,p1p2p3) = p1p2p3. in this case the δs is a multiple root of h(x) and f(x) with multiplicity 4. by above discussion we conclude that h(x)|f(x). further, for every a with gcd(a,p1p2p3) = 1, the δs is a root of f(x) only. therefore, f(x) h(x) is a polynomial of degree φ(pα1−β11 p α2−β2 2 p α3−β3 3 ) over fq, where φ is euler’s phi function. theorem 4.3. let gcd(s,m) = pβ11 p β2 2 p β3 3 . then, ∏ s η m s (x) = f(x) h(x) , where f(x) and h(x) are same as defined in lemma 4.2. the following results are easy to prove. theorem 4.4. let gcd(s,m) = pα11 p β2 2 p β3 3 .then, ∏ s ηms (x) = ( xp α2−β2 2 p α3−β3 3 −1 )( xp α2−β2−1 2 p α3−β3−1 3 −1 ) ( xp α2−β2 2 p α3−β3−1 3 −1 )( xp α2−β2−1 2 p α3−β3 3 −1 ). theorem 4.5. if gcd(s,m) = pβ11 p α2 2 p β3 3 , then, ∏ s ηms (x) = ( xp α1−β1 1 p α3−β3 3 −1 )( xp α1−β1−1 1 p α3−β3−1 3 −1 ) ( xp α1−β1 1 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α3−β3 3 −1 ). 178 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 theorem 4.6. if gcd(s,m) = pβ11 p β2 2 p α3 3 , then, ∏ s ηms (x) = ( xp α1−β1 1 p α2−β2 2 −1 )( xp α1−β1−1 1 p α2−β2−1 2 −1 ) ( xp α1−β1 1 p α2−β2−1 2 −1 )( xp α1−β1−1 1 p α2−β2 2 −1 ). theorem 4.7. if gcd(s,m) = pα11 p α2 2 p β3 3 , then ∏ s ηms (x) = ( xp α3−β3 3 −1 ) ( xp α3−β3−1 3 −1 ). theorem 4.8. if gcd(s,m) = pα11 p β2 2 p α3 3 , then ∏ s ηms (x) = ( xp α2−β2 2 −1 ) ( xp α2−β2−1 2 −1 ). theorem 4.9. if gcd(s,m) = pβ11 p α2 2 p α3 3 , then, ∏ s ηms (x) = ( xp α1−β1 1 −1 ) ( xp α1−β1−1 1 −1 ). theorem 4.10. if gcd(s,m) = pα11 p α2 2 p α3 3 , then, η m s (x) = ( x−1 ) . theorem 4.11. let gcd(s,m) = pβ11 p β2 2 p β3 3 . then ∏ s g m s (x) = p(x) q(x) , where p(x) = ( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1 1 p α2−β2−1 2 p α3−β3 3 −1 ) ( xp α1−β1 1 p α2−β2 2 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) (pβ11 pβ22 pβ33 −1∑ i=0 xp α1−β1 1 p α2−β2 2 p α3−β3 3 i ) and q(x) = ( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 ) ( xp α1−β1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) . proof. since ∏ s gms (x) = ( xm −1 )∏ s η m s (x) = ( xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 )(∑pβ11 pβ22 pβ33 −1 i=0 x p α1−β1 1 p α2−β2 2 p α3−β3 3 i ) ∏ s η m s (x) , the result follows from theorem 4.3, we have ∏ s η m s (x) = f(x) h(x) , where f(x) and h(x) are same as defined in lemma 4.2 thus, we obtain ∏ s g m s (x) = p(x) q(x) , where p(x) = ( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1 1 p α2−β2−1 2 p α3−β3 3 −1 ) ( xp α1−β1 1 p α2−β2 2 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) 179 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 (pβ11 pβ22 pβ33 −1∑ i=0 xp α1−β1 1 p α2−β2 2 p α3−β3 3 i ) and q(x) = ( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 ) ( xp α1−β1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) . theorem 4.12. let gcd(s,m) = pα11 p β2 2 p β3 3 . then∏ s gms (x) = ( xp α2−β2 2 p α3−β3−1 3 −1) ( xp α2−β2−1 2 p α3−β3 3 −1 )(∑pα11 pβ22 pβ33 −1 i=0 x p α2−β2 2 p α3−β3 3 i ) ( xp α2−β2−1 2 p α3−β3−1 3 −1 ) . proof. ∏ s gms (x) = xm −1∏ s η m s (x) = ( xp α2−β2 2 p α3−β3 3 −1 )(∑pα11 pβ22 pβ33 −1 i=0 x p α2−β2 2 p α3−β3 3 i ) ∏ s η m s (x) . the result follows from theorem 4.4. by using the same argument which we have used in theorem 4.12, we have following results: theorem 4.13. let gcd(s,m) = pβ11 p α2 2 p β3 3 . then∏ s gms (x) = ( xp α1−β1 1 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α3−β3 3 −1 )(∑pβ11 pα22 pβ33 −1 i=0 x p α1−β1 1 p α3−β3 3 i ) ( xp α1−β1−1 1 p α3−β3−1 3 −1 ) . theorem 4.14. let gcd(s,m) = pβ11 p β2 2 p α3 3 . then∏ s gms (x) = ( xp α1−β1 1 p α2−β2−1 2 −1 )( xp α1−β1−1 1 p α2−β2 2 −1 )(∑pβ11 pβ22 pα33 −1 i=0 x p α1−β1 1 p α2−β2 2 i ) ( xp α1−β1−1 1 p α2−β2−1 2 −1 ) . theorem 4.15. let gcd(s,m) = pα11 p α2 2 p β3 3 . then ∏ s gms (x) = ( xp α3−β3−1 3 −1 )(pα11 pα22 pβ33 −1∑ i=0 xp α3−β3 3 i ) . 180 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 proof. ∏ s gms (x) = xm −1∏ s η m s (x) = ( xp α3−β3 3 −1 )(∑pα11 pβ22 pβ33 −1 i=0 x p α3−β3 3 i ) ∏ s η m s (x) , the result follows from theorem 4.7. theorem 4.16. let gcd(s,m) = pα11 p β2 2 p α3 3 . then ∏ s gms (x) = ( xp α2−β2−1 2 −1 )(pα11 pβ22 pα33 −1∑ i=0 xp α2−β2 2 i ) . theorem 4.17. let gcd(s,m) = pβ11 p α2 2 p α3 3 . then ∏ s gms (x) = ( xp α1−β1−1 1 −1 )(pβ11 pα22 pα33 −1∑ i=0 xp α1−β1 1 i ) . theorem 4.18. let gcd(s,m) = pα11 p α2 2 p α3 3 . then ∏ s gms (x) = p α1 1 p α2 2 p α3 3 −1∑ i=0 xi. 5. minimum distance of minimal cyclic codes of length m in this section, we compute a lower bound on minimum distance of minimal cyclic codes of length m = p1 α1p2 α2p3 α3. first we prove some results. lemma 5.1. let f(x) = a1xt1 + a2xt2 + ... + akxtk be a polynomial over fq and {1,2,3, ...k} = {i1, i2, i3, ...ik}. further, let m be a positive integer such that ti ≡ ri(mod m). then xm − 1 divides f(x) if and only if ri1 = ri2 = ... = rin1 , rin1+1 = rin1+2 = ... = rin2 , rin2+1 = rin2+2 = ... = rin3 , ..., rint+1 = rint+2 = ... = rik, ai1 + ai2 + ... + ain1 = 0, ain1+1 + ain1+2 + ... + ain2 = 0, ain2+1 + ain2+2 + ... + ain3 = 0, ..., aint+1 + aint+2 + ... + aik = 0. proof. since xm ≡ 1(mod xm − 1) and ti ≡ ri(mod m), therefore, ∑k j=1 ajx tj ≡∑k j=1 ajx rj(mod xm −1). consequently, xm −1 divides f(x) if and only if ∑k j=1 ajx rj = 0. therefore, xm −1 divides f(x) if and only if ri1 = ri2 = ... = rin1 , rin1+1 = rin1+2 = ... = rin2 , rin2+1 = rin2+2 = ... = rin3 , ..., rint+1 = rint+2 = ... = rik, ai1 + ai2 + ... + ain1 = 0, ain1+1 + ain1+2 + ... + ain2 = 0, ain2+1 + ain2+2 + ... + ain3 = 0, ..., aint+1 + aint+2 + ... + aik = 0. lemma 5.2. let t1 ≡ t2 ≡ t3 ≡ t4(mod pα11 p α2 2 ), t1 ≡ t3 and t2 ≡ t4(mod pα22 p α3 3 ), t1 ≡ t2 and t3 ≡ t4(mod pα11 p α3 3 ). then t1 ≡ t2 ≡ t3 ≡ t4(mod pα11 p α2 2 p α3 3 ). 181 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 lemma 5.3. the expression( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) ( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1) ( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 ) , where all the above polynomials are separable over fq, is a polynomial over fq. proof. we know that if the polynomials xab − 1, xa − 1 and xb − 1 are separable over fq, then (xab−1)(x−1) (xa−1)(xb−1) is a polynomial over fq. therefore, if we choose x p α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 = y, then ( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) (xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 ) = ( yp2p3 −1 )( y −1 )( yp3 −1 )( yp2 −1 ). since all the polynomials are separable over fq, the result follows from above discussion. theorem 5.4. let c be the code of length pα1−β11 p α2−β2 2 p α3−β3 3 generated by g(x) = p(x)(xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1) q(x) , where p(x) = ( xp α1−β1−1 1 p α2−β2 2 p α3−β3 3 −1 )( xp α1−β1 1 p α2−β2−1 2 p α3−β3 3 −1 ) ( xp α1−β1 1 p α2−β2 2 p α3−β3−1 3 −1 ) and q(x) = ( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3 3 −1 )( xp α1−β1−1 1 p α2−β2 2 p α3−β3−1 3 −1 ) ( xp α1−β1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ) . then, the minimum distance of c is 8. proof. as p(x) = g(x) q(x)( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 ), therefore, p(x) is a codeword in c. the weight of p(x) is 8, therefore, c has a codeword of weight 8. we will now show that c does not has a nonzero codeword of weight less than 8. by lemma 5.3, xp αi−βi i p αj−βj j −1; 1 ≤ i < j ≤ 3 divides g(x), therefore, xp αi−βi i p αj−βj j −1 divides c(x) also. we now show that c has no codeword of weight 2, 3, 4, 5, 6 and 7. case (1): c has no codeword of weight 2. let c(x) = a1xt1 + a2xt2 be a codeword in c. as x p αi−βi i p αj−βj j −1; 1 ≤ i < j ≤ 3 divides c(x), therefore, by lemma 5.1, t1 ≡ t2(mod p αi−βi i p αj−βj j ) and a1 + a2 = 0. by lemma 5.2), t1 ≡ t2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and a1 + a2 = 0. 182 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 then, by lemma 5.1, xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 divides c(x). therefore, c(x) is a zero codeword. hence c has no codeword of weight 2. case (2): c has no codeword of weight 3. let c(x) = a1xt1 + a2xt2 + a3xt3. the x p αi−βi i p αj−βj j −1; 1 ≤ i < j ≤ 3 divides c(x). by lemma 5.1, it is possible only when t1 ≡ t2 ≡ t3(mod p αi−βi i p αj−βj j ) and a1 + a2 + a3 = 0. lemma 5.2, t1 ≡ t2 ≡ t3(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and a1 + a2 + a3 = 0. then, by lemma 5.1, xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 divides c(x). therefore, c(x) is a zero codeword. hence c has no codeword of weight 3. case (3): c has no codeword of weight 4. let c(x) = a1xt1 + a2xt2 + a3xt3 + a4xt4. then, proceeding on similar steps as discussed in case (1) and in case (2), we get that either ti1 ≡ ti2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 = 0 and ti3 ≡ ti4(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai3 + ai4 = 0, or ti1 ≡ ti2 ≡ ti3 ≡ ti4(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and ai1 + ai2 + ai3 + ai4 = 0, where {1,2,3,4} = {i1, i2, i3, i4}. then, by lemma 5.1, xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 divides c(x). therefore, c(x) is a zero codeword. hence c has no codeword of weight 4. case (4): c has no codeword of weight 5. let c(x) = a1xt1 + a2xt2 + a3xt3 + a4xt 4 + a5x t5. then we get following congruence relations: ti1 ≡ ti2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 = 0 and ti3 ≡ ti4 ≡ ti5(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai3 + ai4 + ai5 = 0, or ti1 ≡ ti2 ≡ ti3 ≡ ti4 ≡ ti5(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and ai1 + ai2 + ai3 + ai4 + ai5 = 0, where {1,2,3,4,5} = {i1, i2, i3, i4, i5}. again, by lemma 5.1 xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 divides c(x). therefore, c(x) is a zero codeword. hence c has no codeword of weight 5. case (5): c has no codeword of weight 6. let c(x) = a1xt1 + a2xt2 + a3xt3 + a4xt4 + a5xt5 + a6xt6. then we obtain the following congruence relations: ti1 ≡ ti2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 = 0 and 183 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 ti3 ≡ ti4(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai3 + ai4 = 0 and ti5 ≡ ti6(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai5 + ai6 = 0 or ti1 ≡ ti2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 = 0 and ti3 ≡ ti4 ≡ ti5 ≡ ti6(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai3 + ai4 + ai5 + ai6 = 0, or ti1 ≡ ti2 ≡ ti3(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ), ai1 + ai2 + ai3 = 0 and ti4 ≡ ti5 ≡ ti6(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai4 + ai5 + ai6 = 0, or ti1 ≡ ti2 ≡ ti3 ≡ ti4 ≡ ti5 ≡ ti6(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and ai1 + ai2 + ai3 + ai4 + ai5 + ai6 = 0, where {1,2,3,4,5,6} = {i1, i2, i3, i4, i5, i6}. again, by lemma 5.1, xp α1−β1 1 p α2−β2 2 p α3−β3 3 − 1 divides c(x). therefore, c(x) is a zero codeword. hence c has no codeword of weight 6. case (6): c has no codeword of weight 7. let c(x) = a1xt1 + a2xt2 + a3xt3 + a4xt4 + a5xt5 + a6xt6 + a7xt7. then we obtain the following congruence relations: ti1 ≡ ti2(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 = 0 and ti3 ≡ ti4 ≡ ti5 ≡ ti6 ≡ ti7(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ), ai3 + ai4 + ai5 + ai6 + ai7 = 0, or ti1 ≡ ti2 ≡ ti3(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 + ai3 = 0 and ti4 ≡ ti5(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai4 + ai5 = 0 and ti6 ≡ ti7(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai6 + ai7 = 0 or ti1 ≡ ti2 ≡ ti3 ≡ ti4(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai1 + ai2 + ai3 + ai4 = 0 and ti5 ≡ ti6 ≡ ti7(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ),ai5 + ai6 + ai7 = 0 or ti1 ≡ ti2 ≡ ti3 ≡ ti4 ≡ ti5 ≡ ti6 ≡ ti7(mod p α1−β1 1 p α2−β2 2 p α3−β3 3 ) and ai1 + ai2 + ai3 + ai4 + ai5 + ai6 + ai7 = 0. by lemma 5.1 xp α1−β1 1 p α2−β2 2 p α3−β3 3 −1 divides c(x). consequently, c(x) is a zero codeword. hence c has no codeword of weight 7. it completes the proof. 184 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 corollary 5.5. let c∗ be the code of length pα11 p α2 2 p α3 3 generated by g∗(x) = p(x) ( xp α1−β1−1 1 p α2−β2−1 2 p α3−β3−1 3 −1 )(∑pβ11 pβ22 pβ33 −1 i=0 x p α1−β1 1 p α2−β2 2 p α3−β3 3 i ) q(x) , where p(x) and q(x) are same as defined in theorem 5.4, then the minimum distance of c∗ is 8pβ11 p β2 2 p β3 3 . proof. since c∗ is a repetition code of c, repeated pβ11 p β2 2 p β3 3 times, where c is same as defined in theorem 5.4, the result follows by theorem 5.4. similarly, we have the following easy results. corollary 5.6. let c∗ be the code of length pα11 p α2 2 p α3 3 generated by( xp α2−β2 2 p α3−β3−1 3 −1 )( xp α2−β2−1 2 p α3−β3 3 −1 )(∑pα11 pβ22 pβ33 −1 i=0 x p α2−β2 2 p α3−β3 3 i ) ( xp α2−β2−1 2 p α3−β3−1 3 −1 ) or generated by ( xp α1−β1 1 p α3−β3−1 3 −1 )( xp α1−β1−1 1 p α3−β3 3 −1 )(∑pβ11 pα22 pβ33 −1 i=0 x p α1−β1 1 p α3−β3 3 i ) ( xp α1−β1−1 1 p α3−β3−1 3 −1 ) , or generated by ( xp α1−β1 1 p α2−β2−1 2 −1 )( xp α1−β1−1 1 p α1−β1 1 −1 )(∑pβ11 pβ22 pα33 −1 i=0 x p α1−β1 1 p α2−β2 2 i ) ( xp α1−β1−1 1 p α2−β2−1 2 −1 ) . then the minimum distance of c∗ is 4pα11 p β2 2 p β3 3 or 4p β1 1 p α2 2 p β3 3 or 4p β1 1 p β2 2 p α3 3 . corollary 5.7. let c∗ be the code of length pα11 p α2 2 p α3 3 generated by ( xp α3−β3−1 3 −1 )(pα11 pα22 pβ33 −1∑ i=0 xp α3−β3 3 i ) or generated by ( xp α2−β2−1 2 −1 )(pα11 pβ22 pα33 −1∑ i=0 xp α2−β2 2 i ) , or generated by ( xp α1−β1−1 1 −1 )(pβ11 pα22 pα33 −1∑ i=0 xp α1−β1 1 i ) . then, the minimum distance of c∗ is 2pα11 p α2 2 p β3 3 or 2p α1 1 p β2 2 p α3 3 , or 2p β1 1 p α2 2 p α3 3 . 185 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 by theorem 4.11, theorem 4.12, theorem 4.13, theorem 4.14, theorem 4.15, theorem 4.16, theorem 4.17 and theorem 4.18 the codes mms are sub-codes of the codes c ∗ defined in corollary 5.5, corollary 5.6 and corollary 5.7 therefore, we have the following results minimal cyclic code :: minimum distance d m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = p β1 1 p β2 2 p β3 3 :: d ≥ 8p β1 1 p β2 2 p β3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = pα11 p β2 2 p β3 3 :: d ≥ 4p α1 1 p β2 2 p β3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = p β1 1 p α2 2 p β3 3 :: d ≥ 4p β1 1 p α2 2 p β3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = p β1 1 p β2 2 p α3 3 :: d ≥ 4p β1 1 p β2 2 p α3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = pα11 p α2 2 p β3 3 :: d ≥ 2p α1 1 p α2 2 p β3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = pα11 p β2 2 p α3 3 :: d ≥ 2p α1 1 p β2 2 p α3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = p β1 1 p α2 2 p α3 3 :: d ≥ 2p β1 1 p α2 2 p α3 3 m p α1 1 p α2 2 p α3 3 s ; gcd(s,m) = pα11 p α2 2 p α3 3 :: d = p α1 1 p α2 2 p α3 3 remark 5.1 (i) one can observe that the results obtained in this paper are sufficient to discuss all the parameters of all the minimal cyclic codes of length m = pα11 p α2 2 ...p αk k . (ii) one can easily observe that the minimum distance of the cyclic codes of length m = pα11 p α2 2 ...p αk k is at least 2k. 6. examples example 6.1. in this example we choose α1 = α2 = α3 = 1, q = 3,p1 = 5,p2 = 11 and p3 = 13. first we count the number of 3-cyclotomic cosets modulo 715. as o5(3) = 4, o11(3) = 5 and o13(3) = 3, therefore, r1 = 1,r2 = 2,r3 = 4, d = 60, d1 = 15, d2 = 12, d3 = 20. then v1 = v2 = v3 = 1 and t = 1. hence by corollary 2.6, the number of 3-cyclotomic cosets modulo 715 is (1.1.1 + 1)(2.1.1 + 1)(4.1.1 + 1) + (1−1) 1 = 30. since λ(2,10,9) = 1, therefore, by corollary 2.4, the distinct 3-cyclotomic cosets modulo 715 are as: c 715(3) 1 = λ(c 5(3) 1 ×c 11(3) 2 ×c 13(3) 1 ),c 715(3) 383 = λ(c 5(3) 1 ×c 11(3) 2 ×c 13(3) 2 ), c 715(3) 493 = λ(c 5(3) 1 ×c 11(3) 2 ×c 13(3) 4 ),c 715(3) 713 = λ(c 5(3) 1 ×c 11(3) 2 ×c 13(3) 8 ), c 715(3) 263 = λ(c 5(3) 1 ×c 11(3) 1 ×c 13(3) 1 ),c 715(3) 318 = λ(c 5(3) 1 ×c 11(3) 1 ×c 13(3) 2 ), 186 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 c 715(3) 428 = λ(c 5(3) 1 ×c 11(3) 1 ×c 13(3) 4 ),c 715(3) 648 = λ(c 5(3) 1 ×c 11(3) 1 ×c 13(3) 8 ), c 715(3) 120 = λ(c 5(3) 0 ×c 11(3) 1 ×c 13(3) 1 ),c 715(3) 175 = λ(c 5(3) 0 ×c 11(3) 1 ×c 13(3) 2 ), c 715(3) 285 = λ(c 5(3) 0 ×c 11(3) 1 ×c 13(3) 4 ),c 715(3) 505 = λ(c 5(3) 0 ×c 11(3) 1 ×c 13(3) 8 ), c 715(3) 185 = λ(c 5(3) 0 ×c 11(3) 2 ×c 13(3) 1 ),c 715(3) 240 = λ(c 5(3) 0 ×c 11(3) 2 ×c 13(3) 2 ), c 715(3) 350 = λ(c 5(3) 0 ×c 11(3) 2 ×c 13(3) 4 ),c 715(3) 570 = λ(c 5(3) 0 ×c 11(3) 2 ×c 13(3) 8 ), c 715(3) 198 = λ(c 5(3) 1 ×c 11(3) 0 ×c 13(3) 1 ),c 715(3) 253 = λ(c 5(3) 1 ×c 11(3) 0 ×c 13(3) 2 ), c 715(3) 363 = λ(c 5(3) 1 ×c 11(3) 0 ×c 13(3) 4 ),c 715(3) 583 = λ(c 5(3) 1 ×c 11(3) 0 ×c 13(3) 8 ), c 715(3) 208 = λ(c 5(3) 1 ×c 11(3) 1 ×c 13(3) 0 ),c 715(3) 273 = λ(c 5(3) 1 ×c 11(3) 2 ×c 13(3) 0 ), c 715(3) 55 = λ(c 5(3) 0 ×c 11(3) 0 ×c 13(3) 1 ),c 715(3) 110 = λ(c 5(3) 0 ×c 11(3) 0 ×c 13(3) 2 ), c 715(3) 220 = λ(c 5(3) 0 ×c 11(3) 0 ×c 13(3) 4 ),c 715(3) 440 = λ(c 5(3) 0 ×c 11(3) 0 ×c 13(3) 8 ), c 715(3) 65 = λ(c 5(3) 0 ×c 11(3) 1 ×c 13(3) 0 ),c 715(3) 130 = λ(c 5(3) 0 ×c 11(3) 2 ×c 13(3) 0 ), c 715(3) 143 = λ(c 5(3) 1 ×c 11(3) 0 ×c 13(3) 0 ),c 715(3) 0 = λ(c 5(3) 0 ×c 11(3) 0 ×c 13(3) 0 ), where c 5(3) 0 = {0},c 5(3) 1 = {1,2,3,4},c 11(3) 0 = {0},c 11(3) 1 = {1,3,4,5,9}, c 11(3) 2 = {2,6,7,8,10},c 13(3) 0 = {0},c 13(3) 1 = {1,3,9}, c 13(3) 2 = {2,6,5},c 13(3) 4 = {4,12,10} and c 13(3) 8 = {8,11,7}. we now compute the explicit expression of primitive idempotent θ7151 (x) in r715 = f3[x]/(x 715 −1). first we use λproduct of polynomials to compute θ7151 (x). the c 715(3) 1 = λ ( c 5(3) 1 ×c 11(3) 2 ×c 13(3) 1 ) therefore, by theorem 3.5, θ7151 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 1 (x3) ) . as the expression of θ5(3)1 (x1) in r5 = f3[x1]/(x 5 1 −1) is θ 5(3) 1 (x1) = 2σ0(x1) + σ1(x1), 187 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 where σs(x1) = ∑ s∈c5(3)s xs1, the expression of θ11(3)1 (x2) in r11 = f3[x2]/(x 11 2 −1) is θ 11(3) 1 (x2) = σ0(x2) + σ1(x2), where σs(x2) = ∑ s∈c11(3)s xs2, and, the expression of θ13(3)1 (x3) in r13 = f3[x3]/(x 13 3 −1) is θ 13(3) 1 (x3) = 2σ2(x3) + 2σ4(x3) + σ8(x3), where σs(x3) = ∑ s∈c13(3)s xs3, therefore, by definition 3.1, the λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 1 (x3) ) λ (( 2σ0(x1) + σ1(x1) )( σ0(x2) + σ1(x2) )( 2σ2(x3) + 2σ4(x3) + σ8(x3) )) = σλ(0,0,2)(x) + σλ(0,0,4)(x) + 2σλ(0,0,8)(x) + σλ(0,1,2)(x) + σλ(0,1,4)(x) + 2σλ(0,1,8)(x) + 2σλ(1,0,2)(x) + 2σλ(1,0,4)(x) + σλ(1,0,8)(x) + 2σλ(1,1,2)(x) +2σλ(1,1,4)(x) + σλ(1,1,8)(x). hence θ 715(3) 1 (x) = σ110(x) + σ220(x) + 2σ440(x) + σ175(x) + σ285(x) + 2σ505(x) +2σ253(x) + 2σ363(x) + σ583(x) + 2σ318(x) + 2σ428(x) + σ648(x). we now compute the θ715(3)1 (x) by trace function of the product of explicit expressions of( θ 5(3) 1 (x 143)θ 11(3) 1 (x 65)θ 13(3) 1 (x 55) ) . as θ5(3)1 (x) = 2 + x + x 2 + x3 + x4, therefore, θ 5(3) 1 (x 143) = 2 + x143 + x283 + x429 + x572. similarly, θ11(3)1 (x) = 1 + x + x 3 + x4 + x5 + x9, therefore, θ 11(3) 1 (x 65) = 1 + x65 + x195 + x260 + x325 + x585 and θ 13(3) 1 (x) = 2x 2 + 2x6 + 2x5 + 2x4 + 2x10 + 2x12 + x8 + x11 + x7 implies that θ 13(3) 1 (x 55) = 2x110 + 2x330 + 2x275 + 2x220 + 2x550 + 2x660 + x440 + x605 + x385. on writing the the product ( θ 5(3) 1 (x 143)θ 11(3) 1 (x 65)θ 13(3) 1 (x 55) ) 188 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 in terms of σs(x) = ∑ s∈c715(3)s xs in r715, we get( θ 5(3) 1 (x 143)θ 11(3) 1 (x 65)θ 13(3) 1 (x 55) ) = σ110(x) + σ220(x) + 2σ440(x) + σ175(x) + σ285(x) + 2σ505(x) +2σ253(x) + 2σ363(x) + σ583(x) + 2σ318(x) + 2σ428(x) + σ648(x). as t = 1, therefore, by theorem 3.6, θ7151 (x) = trf3|f3 ( θ 5(3) 1 (x 143)θ 11(3) 1 (x 65)θ 13(3) 1 (x 55) ) = σ110(x) + σ220(x) + 2σ440(x) + σ175(x) + σ285(x) + 2σ505(x) + 2σ253(x) + 2σ363(x) + σ583(x) + 2σ318(x) + 2σ428(x) + σ648(x). example 6.2. in this example we obtain 7-cyclotomic cosets modulo 30, all the explicit expressions of primitive idempotents in r30 = f7[x]/(x30 −1), minimal polynomials, generating polynomials and minimum distances of all the minimal cyclic codes of length 30 over f7. here 7 is primitive root modulo 2, quadratic residue modulo 3 and primitive root modulo 5. therefore, o2(7) = 1,o3(7) = 1 and o5(7) = 4. thus d = 4,d1 = 4,d2 = 4,d3 = 1, v1 = 1,v2 = 1,v3 = 1, t = 1 and λ(1,1,1) = 1. hence λ(c 2(7) 1 ×c 3(7) 1 ×c 5(7) 1 ) = c 30(7) 1 ,λ(c 2(7) 1 ×c 3(7) 2 ×c 5(7) 1 ) = c 30(7) 11 , λ(c 2(7) 0 ×c 3(7) 0 ×c 5(7) 1 ) = c 30(7) 6 ,λ(c 2(7) 0 ×c 3(7) 1 ×c 5(7) 0 ) = c 30(7) 10 , λ(c 2(7) 0 ×c 3(7) 1 ×c 5(7) 1 ) = c 30(7) 4 ,λ(c 2(7) 0 ×c 3(7) 2 ×c 5(7) 0 ) = c 30(7) 20 , λ(c 2(7) 0 ×c 3(7) 2 ×c 5(7) 1 ) = c 30(7) 2 ,λ(c 2(7) 1 ×c 3(7) 0 ×c 5(7) 0 ) = c 30(7) 15 , λ(c 2(7) 1 ×c 3(7) 0 ×c 5(7) 1 ) = c 30(7) 3 ,λ(c 2(7) 1 ×c 3(7) 1 ×c 5(7) 0 ) = c 30(7) 25 , λ(c 2(7) 1 ×c 3(7) 2 ×c 5(7) 0 ) = c 30(7) 5 ,λ(c 2(7) 0 ×c 3(7) 0 ×c 5(7) 0 ) = c 30(7) 0 , where c2(7)0 = {0},c 2(7) 1 = {1}, c 3(7) 0 = {0},c 3(7) 1 = {1},c 3(7) 2 = {2}, c 5(7) 0 = {0} and c 5(7) 1 = {1,2,3,4}. 1. the twelve primitive idempotents are: θ (30) 0 (x) = 1 30 [σ0(x) + σ6(x) + σ10(x) + σ4(x) + σ20(x) + σ2(x) +σ15(x) + σ3(x) + σ25(x) + σ1(x) + σ5(x) + σ11(x)], 189 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 θ (30) 1 (x) = 1 30 [4σ0(x)−σ6(x) + 2σ10(x)−4σ4(x) + σ20(x)−2σ2(x) −4σ15(x) + σ3(x)−2σ25(x) + 4σ1(x)−σ5(x) + 2σ11(x)], θ (30) 11 (x) = 1 30 [4σ0(x)−σ6(x) + σ10(x)−2σ4(x) + 2σ20(x)−4σ2(x) −4σ15(x) + σ3(x)−σ25(x) + 2σ1(x)−2σ5(x) + 4σ11(x)], θ (30) 4 (x) = 1 30 [4σ0(x)−σ6(x) + 2σ10(x)−4σ4(x) + σ20(x)−2σ2(x) +4σ15(x)−σ3(x) + 2σ25(x)−4σ1(x) + σ5(x)−2σ11(x)], θ (30) 2 (x) = 1 30 [4σ0(x)−σ6(x) + σ10(x)−2σ4(x) + 2σ20(x)−4σ2(x) +4σ15(x)−σ3(x) + σ25(x)−2σ1(x) + 2σ5(x)−4σ11(x)], θ (30) 3 (x) = 1 30 [4σ0(x)−σ6(x) + 4σ10(x)−σ4(x) + 4σ20(x)−σ2(x) −4σ15(x) + σ3(x)−4σ25(x) + σ1(x)−4σ5(x) + σ11(x)], θ (30) 25 (x) = 1 30 [σ0(x) + σ6(x) + 4σ10(x) + 4σ4(x) + 2σ20(x) + 2σ2(x) −σ15(x)−σ3(x)−4σ25(x)−4σ1(x)−2σ5(x)−2σ11(x)], θ (30) 5 (x) = 1 30 [σ0(x) + σ6(x) + 2σ10(x) + 2σ4(x) + 4σ20(x) + 4σ2(x) −σ15(x)−σ3(x)−2σ25(x)−2σ1(x)−4σ5(x)−4σ11(x)], θ (30) 6 (x) = 1 30 [4σ0(x)−σ6(x) + 4σ10(x)−σ4(x) + 4σ20(x)−σ2(x) +4σ15(x)−σ3(x) + 4σ25(x)−σ1(x) + 4σ5(x)−σ11(x)], θ (30) 10 (x) = 1 30 [σ0(x) + σ6(x) + 4σ10(x) + 4σ4(x) + 2σ20(x) + 2σ2(x) +σ15(x) + σ3(x) + 4σ25(x) + 4σ1(x) + 2σ5(x) + 2σ11(x)], θ (30) 20 (x) = 1 30 [σ0(x) + σ6(x) + 2σ10(x) + 2σ4(x) + 4σ20(x) + 4σ2(x) +σ15(x) + σ3(x) + 2σ25(x) + 2σ1(x) + 4σ5(x) + 4σ11(x)], θ (30) 15 (x) = 1 30 [σ0(x) + σ6(x) + σ10(x) + σ4(x) + σ20(x) + σ2(x) −σ15(x)−σ3(x)−σ25(x)−σ1(x)−σ5(x)−σ11(x)]. 190 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 2. the minimal polynomials of m30s are: η300 (x) = (x−1),η 30 1 (x) = (x 4 + 5x3 + 4x2 + 6x + 2), η3011(x) = (x 4 + 3x3 + 2x2 + 6x + 4),η302 (x) = (x 4 + 2x3 + 4x2 + x + 2), η304 (x) = (x 4 + 4x3 + 2x2 + x + 4),η303 (x) = (x 4 −x3 + x2 −x + 1), η306 (x) = (x 4 + x3 + x2 + x + 1),η3010(x) = (x−2), η3020(x) = (x−4),η 30 5 (x) = (x−3),η 30 25(x) = (x−5),η 30 15(x) = (x + 1). 3. the generating polynomials of m30s are: g300 (x) = 29∑ i=0 xi, g3015(x) = 29∑ i=0 (−1)ixi, g305 (x) = (x 24 + x18 + 2x12 + 6x6 + 1)(x5 + 3x4 + 2x3 + 6x2 + 4x + 5), g3025(x) = (x 24 + x18 + x12 + x6 + 1)(x5 + 5x4 + 4x3 + 6x2 + 2x + 3), g3010(x) = (x 27 + x24 + 2x21 + x18 + x15 + x12 + 2x9 + x6 + x3 + 1)(x2 + 2x + 4), g3020(x) = (x 27 + x24 + 2x21 + x18 + x15 + x12 + 2x9 + x6 + x3 + 1)(x2 + 4x + 2), g301 (x) = (x 25 −4x20 + 2x15 −x10 + 4x5 −2)(x + 2), g3011(x) = (x 25 −2x20 + 4x15 −x10 + 2x5 −4)(x + 4), g302 (x) = (x 25 + 4x20 + 2x15 + x10 + 4x5 + 2)(x−2), g304 (x) = (x 25 + 2x20 + 4x15 + x10 + 2x5 + 4)(x−4), g303 (x) = (x 25 −x20 + x15 −x10 + x5 −1)(x + 1), g306 (x) = (x 25 + x20 + x15 + x10 + x5 + 1)(x−1). 4. the dimension and the minimum distance of m30s are: code m300 m 30 1 m 30 11 m 30 2 m 30 4 m 30 3 dimension 1 4 4 4 4 4 minimum distance 30 12 12 12 12 12 code m306 m 30 5 m 30 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https://doi.org/10.1080/00927872.2012.674590 https://doi.org/10.1080/00927872.2012.674590 https://doi.org/10.1504/ijicot.2014.066087 https://doi.org/10.1504/ijicot.2014.066087 https://doi.org/10.1504/ijicot.2014.066087 https://doi.org/10.1006/ffta.1996.0156 https://doi.org/10.1006/ffta.1996.0156 https://doi.org/10.1016/j.ffa.2012.07.003 https://doi.org/10.1016/j.ffa.2012.07.003 https://doi.org/10.1016/j.ffa.2004.01.005 https://doi.org/10.1016/j.ffa.2004.01.005 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 appendix 1 the explicit expressions of 30 primitive idempotents in r715 are as follows: since c 715(3) 1 = ( c 5(3) 1 ×c 11(3) 2 ×c 13(3) 1 ) , by theorem 3.5 θ 715(3) 1 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 1 (x3) ) . hence θ 715(3) 1 (x) = σ110(x) + σ220(x) + 2σ440(x) + σ175(x) + σ285(x) + 2σ505(x) +2σ253(x) + 2σ363(x) + σ583(x) + 2σ318(x) + 2σ428(x) + σ648(x); if we denote aσs(x) by a(s), then under the above notation, θ 715(3) 1 (x)= (110)+ (220)+ 2(440)+(175)+(285)+ 2(505)+ 2(253)+ 2(363)+ (583)+ 2(318) +2(428) +(648). since c 715(3) 1 = ( c 5(3) 1 ×c 11(3) 2 ×c 13(3) 1 ) and c715(3)383 = ( c 5(3) 1 ×c 11(3) 2 ×c 13(3) 2 ) , by theorem 3.6, θ 715(3) 383 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 2 (x3) ) = (55) + (110) + 2(220) + (120) + (175) + 2(285) + 2(198) + 2(253) +(363) + 2(263) + 2(318) + (428). similarly, θ715(3)493 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 4 (x3) ) = (55) + 2(110) + (440) + (120) + 2(175) + (505) + 2(198) + (253) + 2(583) + 2(263) + (318) + 2(648), θ 715(3) 713 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 8 (x3) ) = 2(55) + (220) + (440) + 2(120) + (285) + (505) + (198) + 2(363) + 2(583) + (263) + 2(428) + 2(648), θ 715(3) 263 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 2 (x2)θ 13(3) 1 (x3) ) = (110) + (220) + 2(440) + (240) + (350) + 2(570) + 2(253) + 2(363) + (583) + 2(383) + 2(493) + (713), θ 715(3) 318 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 2 (x2)θ 13(3) 2 (x3) ) = (55) + (110) + 2(220) + (185) + (240) + 2(350) + 2(198) + 2(253) + (363) + 2(1) + 2(383) + (493), θ 715(3) 428 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 2 (x2)θ 13(3) 4 (x3) ) = (55) + 2(110) + (440) + (185) + 2(240) + (570) + 2(198) + (253) + 2(583) + 2(1) + (383) + 2(713), θ 715(3) 648 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 2 (x2)θ 13(3) 8 (x3) ) = 2(55) + (220) + (440) + 2(185) + (350) + (570) + (198) + 2(363) + 2(583) + (1) + 2(493) + 2(713), θ 715(3) 120 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 2 (x2)θ 13(3) 1 (x3) ) = (110) + (220) + 2(440) + (240) + (350) + 2(570) + 193 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 (253) + (363) + 2(583) + (383) + (493) + 2(713), θ 715(3) 175 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 2 (x2)θ 13(3) 2 (x3) ) = (55) + (110) + 2(220) + (185) + (240) + 2(350) + (198) + (253) + 2(363) + (1) + (383) + 2(493) , θ 715(3) 285 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 2 (x2)θ 13(3) 4 (x3) ) = (55) + 2(110) + (440) + (185) + 2(240) + (570) + (198) + 2(253) + (583) + (1) + 2(383) + (713), θ 715(3) 505 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 2 (x2)θ 13(3) 8 (x3) ) = 2(55) + (220) + (440) + 2(185) + (350) + (570) + 2(198) + (363) + (583) + 2(1) + (493) + (713), θ 715(3) 185 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 1 (x2)θ 13(3) 1 (x3) ) = (110) + (220) + 2(440) + (175) + (285) + 2(505) + (253) + (363) + 2(583) + (318) + (428) + 2(648), θ 715(3) 240 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 1 (x2)θ 13(3) 2 (x3) ) = (55) + (110) + 2(220) + (120) + (175) + 2(285) + (198) + (253) + 2(363) + (263) + (318) + 2(428), θ 715(3) 350 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 1 (x2)θ 13(3) 4 (x3) ) = (55) + 2(110) + (440) + (120) + 2(175) + (505) + (198) + 2(253) + (583) + (263) + 2(318) + (648), θ 715(3) 570 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 1 (x2)θ 13(3) 8 (x3) ) = 2(55) + (220) + (440) + 2(120) + (285) + (505) + 2(198) + (363) + (583) + 2(263) + (428) + (648), θ 715(3) 198 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 0 (x2)θ 13(3) 1 (x3) ) = 2(110) + 2(220) + (440) + 2(175) + 2(285) + (505) + 2(240) + 2(350) + (570) + (253) + (363) + 2(583) + (318) + (428) + 2(648) + (383) + (493) + 2(713), θ 715(3) 253 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 0 (x2)θ 13(3) 2 (x3) ) = 2(55)+2(110)+(220)+2(120)+2(175)+(285)+2(185)+ 2(240) + (350) + (198) + (253) + 2(363) + (263) + (318) + 2(428) + (1) + (383) + 2(493), θ 715(3) 363 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 0 (x2)θ 13(3) 4 (x3) ) = 2(55)+(110)+2(440)+2(120)+(175)+2(505)+2(185)+ (240) + 2(570) + (198) + 2(253) + (583) + (263) + 2(318) + (648) + (1) + 2(383) + (713), θ 715(3) 583 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 0 (x2)θ 13(3) 8 (x3) ) = (55) +2(220) + 2(440) + (120) + 2(285) +2(505) + (185) + 2(350) + 2(570) + 2(198) + (363) + (583) + 2(263) + (428) + (648) + 2(1) + (493) + (713), θ 715(3) 208 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 2 (x2)θ 13(3) 0 (x3) ) = 2(0) + 2(55) + 2(110) + 2(220) + 2(440) + 2(130) + 2(185)+2(240)+2(350)+2(570)+(143)+(198)+(253)+(363)+(583)+(273)+(1)+(383)+(493)+(713), θ 715(3) 273 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 1 (x2)θ 13(3) 0 (x3) ) = 2(0) + 2(55) + 2(110) + 2(220) + 2(440) + 2(65) + 2(120)+2(175)+2(285)+2(505)+(143)+(198)+(253)+(363)+(583)+(208)+(263)+(318)+(428)+(648), θ 715(3) 55 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 0 (x2)θ 13(3) 1 (x3) ) = 2(110) + 2(220) + (440) + 2(175) + 2(285) + (505) + 2(240) + 2(350) + (570) + 2(253) + 2(363) + (583) + 2(318) + 2(428) + (648) + 2(383) + 2(493) + (713), θ 715(3) 110 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 0 (x2)θ 13(3) 2 (x3) ) = 2(55)+2(110)+(220)+2(120)+2(175)+(285)+2(185)+ 2(240) + (350) + 2(198) + 2(253) + (363) + 2(263) + 2(318) + (428) + 2(1) + 2(383) + (493), θ 715(3) 220 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 0 (x2)θ 13(3) 4 (x3) ) = 2(55)+(110)+2(440)+2(120)+(175)+2(505)+2(185)+ 194 p. kumar, p. devi / j. algebra comb. discrete appl. 8(3) (2021) 167–195 (240) + 2(570) + 2(198) + (253) + 2(583) + 2(263) + (318) + 2(648) + 2(1) + (383) + 2(713), θ 715(3) 440 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 0 (x2)θ 13(3) 8 (x3) ) = (55) +2(220) + 2(440) + (120) + 2(285) +2(505) + (185) + 2(350) + 2(570) + (198) + 2(363) + 2(583) + (263) + 2(428) + 2(648) + (1) + 2(493) + 2(713), θ 715(3) 65 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 2 (x2)θ 13(3) 0 (x3) ) = 2((0) + (55) + (110) + (220) + (440) + (130) + (185)+(240)+(350)+(570)+(143)+(198)+(253)+(363)+(583)+(273)+(1)+(383)+(493)+(713)), θ 715(3) 130 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 1 (x2)θ 13(3) 0 (x3) ) = 2((0) + (55) + (110) + (220) + (440) + (65) + (120) + (175) + (285) + (505) + (143) + (198) + (253) + (363) + (583) + (208) + (263) + (318) + (428) + (648)), θ 715(3) 143 (x) = λ ( θ 5(3) 1 (x1)θ 11(3) 0 (x2)θ 13(3) 0 (x3) ) = (0) + (55) + (110) + (220) + (440) + (65) + (120) + (175) + (285) + (505) + (130) + (185) + (240) + (350) + (570) + 2((143) + (198) + (253) + (363) + (583) + (208) + (263) + (318) + (428) + (648) + (273) + (1) + (383) + (493) + (713)), θ 715(3) 0 (x) = λ ( θ 5(3) 0 (x1)θ 11(3) 0 (x2)θ 13(3) 0 (x3) ) = (0) + (55) + (110) + (220) + (440) + (65) + (120) + (175) + (285) + (505) + (130) + (185) + (240) + (350) + (570) + (143) + (198) + (253) + (363) + (583) + (208) + (263) + (318) + (428) + (648) + (273) + (1) + (383) + (493) + (713). 195 introduction cyclotomic cosets modulo m primitive idempotents in rm minimal and generating polynomials of minimal cyclic codes of length m minimum distance of minimal cyclic codes of length m examples references appendix 1 issn 2148-838x j. algebra comb. discrete appl. 10(2) • 87–96 received: 16 march 2021 accepted: 31 october 2022 journal of algebra combinatorics discrete structures and applications lee weight distributions of trace codes over fq + ufq from irreducible cyclic codes research article gerardo vega abstract: the construction of two or few-weight codes from trace codes over the ring fq + ufq, where u2 = 0, was recently presented in [4]. for such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the lee weight distributions of the corresponding trace codes. motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field fp, the lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring fp + ufp, was recentely found in [11]. in this work, we prove that the lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. with this equivalence in mind, and by using the already known weight distributions of an infinite family of irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the conclusion of [11] to determine the lee weight distribution of an infinite family of trace codes over the ring fq + ufq, that includes the infinite family found in [11]. 2010 msc: 94b15 · 11t71 keywords: codes over a ring, trace codes, semiprimitive codes and cyclotomic classes, irreducible cyclic codes 1. introduction consider the ring r = fq + ufq, with u2 = 0, and, for any positive integer m, the ring extension r = fqm + ufqm. a trace code, cl, with defining set l = {d1,d2, · · · ,dn}⊆ r∗ is defined by cl = {(tr(xd1),tr(xd2), · · · ,tr(xdn)) |x ∈ r} , (1) where r∗ is the group of units of r, and tr is the trace function from r to r defined as gerardo vega; dirección general de cómputo y de tecnologías de información y comunicación, universidad nacional autónoma de méxico, 04510 ciudad de méxico, mexico (gerardov@unam.mx). 87 https://orcid.org/0000-0002-4957-6575 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 tr(a + ub) = m−1∑ j=0 (aq j + ubq j ) . for any a,b ∈ fqm. it is clear that tr(a + ub) = trfqm/fq (a) + utrfqm/fq (b) , where “trfqm/fq " denotes the standard trace mapping from fqm to fq. note that cl is a linear code over r of length n, that is cl is a particular kind of r-submodule of rn. in analogy to the ring z4, the gray map φ from r to f2q is defined by φ(a + ub) = (b,a + b) , for all a,b ∈ fq . this map is extended naturally into a map from rn to f2nq . remark 1.1. the lee weight, wl, over rn is defined in terms of the standard hamming weight, wh, of the gray image as follows wl(a + ub) = wh(b) + wh(a + b) , for all a,b ∈ fnq . similar to the hamming distance, dh, the lee distance, dl, over rn is defined as dl(x,y) = wl(x−y) for all x,y ∈ rn. moreover if x = a1 + ub1 and y = a2 + ub2, then dl(x,y) = wh(b1 − b2) + wh(a1 − a2 + b1 −b2) = wh(b1 −b2,a1 −a2 + b1 −b2) = wh((b1,a1 + b1)− (b2,a2 + b2)) = dh(φ(x),φ(y)). thus φ is a distance-preserving map or isometry from (rn,dl) to (f2nq ,dh). we recall that the hamming weight enumerator, hamc(z), of a linear code c of length n over a finite field is defined as the polynomial hamc(z) = ∑n j=0 ajz j, where aj (0 ≤ j ≤ n) denote the number of codewords with hamming weight j in the code c. the sequence (a0 = 1,a1, · · · ,an) is called the weight distribution of the code. in a similar way for a trace code cl, defined through (1), let aj (0 ≤ j ≤ 2n) be the number of codewords with lee weight j in the linear code cl of length n over r, then the lee weight enumerator of cl, leecl(z), is defined by leecl(z) = ∑2n j=0 ajz j. through different choices of the defining set l, and particularly when fq is either the binary field (f2) or a prime field (fp), several optimal or nearly optimal codes from trace codes of the form cl where recently found ([4, 5, 8, 9, 11–16]). particularly, the construction of two or few-weight codes from trace codes over the ring fq + ufq, was recently presented in [4]. for such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the lee weight distributions of the corresponding trace codes. motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field fp, the lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring fp + ufp, was recently constructed in [11]. one of the purposes of this work is to show that there exists a direct identity between lee weight enumerators of the trace codes constructed by [4], [8], [9] and [11], and the hamming weight enumerators of the irreducible cyclic codes. in other words, we are going to prove that the lee weight distribution problem for the trace codes constructed following [4], [8], [9] or [11], is equivalent to the standard hamming weight distribution problem for the irreducible cyclic codes. with this equivalence in mind, we then follow the open problem suggested in the conclusion of [11], and determine the lee weight distribution of trace codes of the form cl in terms of other class of irreducible cyclic codes, with known or well-understood 88 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 weight distribution. for this purpose, we use the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive, [17]), to determine the lee weight distribution of a new infinite family of trace codes of the form cl, that includes the infinite family constructed in [11]. this work is organized as follows: in section 2, we set up some new notations and recall some definitions, particularly those related with the irreducible cyclic codes. section 3 will show the relationship between the lee weight distribution problem for some trace codes and the hamming weight distribution problem for the irreducible cyclic codes. in section 4, it is shown that all the lee weight distributions reported in [4], [8], [9], and [11] can be obtained easily as particular instances of such relationship, which gives us a simplified view for all these lee weight distributions. in section 5 we use the already known weight distributions of an infinite family irreducible cyclic codes to determine the lee weight distribution of a new infinite family of trace codes. finally, section 6 is devoted to conclusions. 2. background material let fq be a finite field of order q, where q = pt for some prime p and for some positive integer t. a subset c of vectors or codewords in fnq is called an [n,l] linear code over fq if c is an l-dimensional subspace of fnq : n is called the length and l is called the dimension of c. an m-weight code (either over a finite field or over a ring) is a code such that the cardinality of the set of nonzero weights is m. a linear code c over a finite field fq of length n, is cyclic if (c0,c1, . . . , cn−1) ∈ c implies (cn−1,c0,c1, . . . ,cn−2) ∈ c. cyclic codes have wide applications in storage and communication systems because, unlike encoding and decoding algorithms for linear codes, encoding/decoding algorithms for cyclic codes can be implemented easily and efficiently by employing shift registers with feedback connections ([6, p. 209]). by identifying the vector (c0,c1, . . . ,cn−1) ∈ fnq with the polynomial c0+c1x+. . .+cn−1xn−1 ∈ fq[x], it follows that any linear code c of length n over fq corresponds to a subset of the residue class ring fq[x]/〈xn−1〉. moreover, it is well known that the linear code c is cyclic if and only if the corresponding subset is an ideal of fq[x]/〈xn −1〉 (see for example [3, theorem 9.36]). now, note that every ideal of fq[x]/〈xn − 1〉 is principal. in consequence, if c is a cyclic code of length n over fq, then c = 〈g(x)〉, where g(x) is a monic polynomial, such that g(x) | (xn − 1). this polynomial is unique, and it is called the generator polynomial of c ([6, theorem 1, p. 190]). on the other hand, the polynomial h(x) = (xn −1)/g(x) is referred to as the parity check polynomial of c. as usual in cyclic codes, we always assume that the length n of any cyclic code is relatively prime to q. thus, xn −1 has no repeated factors ([6, p. 196]). a cyclic code over fq is called irreducible (reducible) if its parity check polynomial is irreducible (reducible) over fq. let v and w be integers, such that gcd(v,w) = 1. then, the smallest positive integer i, such that wi ≡ 1 (mod v), is called the multiplicative order of w modulo v, and is denoted by ordv(w). from now on, m will denote a positive integer, and by using γ we will denote a fixed primitive element of fqm. for any integer a, the polynomial ha(x) ∈ fq[x] will denote the minimal polynomial of γ−a ([3, definition 1.81]). in addition, c(a) will denote the irreducible cyclic code whose parity check polynomial is ha(x). note that c(a) is an [n,l] linear code, where n = qm−1 gcd(qm−1,a), and l = deg(ha(x)) = ordn(q) is a divisor of m. for any positive divisor e of qm − 1 and for any 0 ≤ i ≤ e− 1, we define d(e,q m) i := γ i〈γe〉, where 〈γe〉 denotes the subgroup of f∗qm generated by γe. the cosets d (e,qm) i are called the cyclotomic classes of order e in fqm. an alternative definition for an irreducible cyclic code is as follows: definition 2.1. [7, definition 2.2] let n be a positive divisor of qm−1, write e = (qm−1)/n, and let ω 89 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 be a primitive n-th root of unity in fqm. then, an irreducible cyclic code of length n over fq, is the set {(trfqm/fq (yω i))n−1i=0 |y ∈ fqm} . remark 2.2. note that, thanks to delsarte’s theorem (see for example [1]), the parity-check polynomial of the irreducible cyclic code under the previous definition is h−e(x), if ω = γe. let v and w be integers, such that gcd(v,w) = 1. we say that w is semiprimitive modulo v, if there exists a positive integer j, such that wj ≡−1 (mod v). according to definition 3 in [17], an irreducible cyclic code c(a) of dimension m over fq is called semiprimitive if µ ≥ 2 and p is semiprimitive modulo µ, where µ = gcd(q m−1 q−1 ,a) (recall q = p t). 3. a relationship between the hamming and the lee weight enumerators let n be a positive divisor of qm − 1, and write e = (qm − 1)/n. for each β ∈ fqm define c(n,e,β) as the vector of length n over fq, given by c(n,e,β) = (trfqm/fq (βγ ei))n−1i=0 . (2) now, since the elements γei (0 ≤ i < n) are the n roots of the unity in fqm, we have, in accordance with definition 2.1, that the irreducible cyclic code c(e) can be described as c(e) = {c(n,e,β) |β ∈ fqm} . remark 3.1. c(e) is an irreducible cyclic code of length n, whose dimension will be m iff ordn(q) = m. we will now focus our attention on the trace codes of the form cl, and for this purpose we are going to fix the defining set in terms of the cyclotomic class d(e,q m) 0 , where e is any divisor of q m − 1. therefore, from now on cle will denote the trace code defined through (1), where the defining set le = d (e,qm) 0 + ufqm. since, |d (e,qm) 0 | = qm−1 e , the length of trace code cle is q2m−qm e . for a ∈ r (recall r = fqm + ufqm) define the evaluation map ev(a) as ev(a) = (tr(ax))x∈le . thus cle = {ev(a) |a ∈ r}. next, we will show that there exists a direct relationship between the lee weight of some codewords in cle and the hamming weight of the codewords in the irreducible cyclic code c(e). theorem 3.2. let n be a positive divisor of qm − 1, and write e = (qm − 1)/n. consider cle and c(e) as before. let a ∈ r, then (i) if a = 0, then wl(ev(a)) = 0. (ii) if β ∈ f∗qm and a = uβ, then wl(ev(a)) = 2qmwh(c(n,e,β)), where c(n,e,β) is the codeword in c(e) given by (2). 90 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 (iii) if α ∈ f∗qm, β ∈ fqm, and a = α + uβ, then wl(ev(a)) = 2 q−1 eq (q2m −qm). proof. part (i) is direct. suppose that a = uβ, for some β ∈ f∗qm. let x = w + uw′ ∈ le, where w ∈ d(e,q m) 0 and w ′ ∈ fqm. therefore ax = uβw and tr(ax) = utrfqm/fq (βw). taking gray map yields φ(ev(a)) = φ((utrfqm/fq (βw))w,w′) = (trfqm/fq (βw),trfqm/fq (βw))w,w′ , where (utrfqm/fq (βw))w,w′ is the vector of length q2m−qm e whose elements are all possible evaluations of utrfqm/fq (βw) by taking w ′ ∈ fqm and w ∈ d (e,qm) 0 = 〈γ e〉 (in a similar way for (trfqm/fq (βw),trfqm/fq (βw))w,w′) . now, by considering remark 1.1, we have wl(ev(a)) = 2wh((trfqm/fq (βw))w,w′) = 2qmwh((trfqm/fq (βw))w) = 2qmwh((trfqm/fq (βγ ei))n−1i=0 ) = 2qmwh(c(n,e,β)) , where the last equality comes from (2). lastly, suppose a = α + uβ ∈ r∗, with α ∈ f∗qm and β ∈ fqm. let x = w + uw′ ∈ le, where w ∈ d (e,qm) 0 and w ′ ∈ fqm. therefore, ax = αw + u(αw′ + βw) and tr(ax) = trfqm/fq (αw) + utrfqm/fq (αw ′ + βw). taking gray map yields φ(ev(a)) = φ((trfqm/fq (αw) + utrfqm/fq (αw ′ + βw))w,w′) = (trfqm/fq (αw ′ + βw),trfqm/fq (αw ′ + βw + αw))w,w′ . by considering remark 1.1 again, wl(ev(a)) = wh((trfqm/fq (αw ′ + βw),trfqm/fq (αw ′ + βw + αw))w,w′) . thus the lee weight of the codeword ev(a) ∈ cle is equal to 2 q2m−qm e −z(a), where z(a) = ]{(w,w′) ∈ d(e,q m) 0 ×fqm |trfqm/fq (αw ′ + βw) = 0}+ ]{(w,w′) ∈ d(e,q m) 0 ×fqm |trfqm/fq (αw ′ + βw + αw) = 0} . if χ′ and χ are, respectively, the canonical additive characters of fq and fqm (see for example [3, chapter 5]), then χ′ and χ are related by χ′(trfqm/fq (ε)) = χ(ε) for all ε ∈ fqm. therefore 91 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 z(a) = 1 q ∑ w∈d(e,q m) 0 ∑ w′∈fqm ∑ s∈fq χ′(trfqm/fq (s(αw ′ + βw))) + 1 q ∑ w∈d(e,q m) 0 ∑ w′∈fqm ∑ s∈fq χ′(trfqm/fq (s(αw ′ + βw + αw))) = 2 q2m −qm eq + 1 q ∑ w∈d(e,q m) 0 ∑ s∈f∗q χ(sβw) ∑ w′∈fqm χ(sαw′) + 1 q ∑ w∈d(e,q m) 0 ∑ s∈f∗q χ(s(βw + αw)) ∑ w′∈fqm χ(sαw′) = 2 q2m −qm eq , because ∑ w′∈fqm χ(sαw ′) = 0. thus wl(ev(a)) = 2 q−1 eq (q2m −qm). the previous theorem strongly suggests the existence of a relationship between hamming and the lee weight enumerators. we establish this result by means of the following: corollary 3.3. with our current notation, let n be a positive divisor of qm −1 such that ordn(q) = m. fix e = (qm − 1)/n. let hamc(e)(z) and leecle (z) be, respectively, the hamming and the lee weight enumerators of c(e) and cle. then the gray image of cle is a code over fq of length 2 q2m−qm e and size q2m. furthermore, the lee weight enumerator of cle (and therefore the hamming weight enumerator of its gray image) is leecle (z) = hamc(e)(z 2qm) + (q2m −qm)z2 q−1 eq (q2m−qm) . (3) proof. this is a direct consequence of theorem 3.2 and remark 3.1. what (3) is saying is that if c(e) is an m-weight code, then the trace code cle (or alternatively its gray image) is an (m + 1)-weight code. as elaborated below, this property and particularly (3), is in complete accordance with the lee weight enumerators recently reported in [4], [8], [9] and [11]. 4. in perspective with some already reported lee weight distributions let q and n be as before, and suppose that gcd(q,n) = 1. just by looking at the length n it is possible to determine the existence or nonexistence of either a one-weight or a semiprimitive two-weight irreducible cyclic code of dimension ordn(q) over any finite field fq. we recall such characterization by means of the following: theorem 4.1. [17, theorem 4] let n and m be positive integers, such that gcd(n,q) = 1 and m = ordn(q). fix e = (qm − 1)/n and µ = gcd(q m−1 q−1 ,e). then, either there exists a one-weight, or a semiprimitive two-weight irreducible cyclic code c(e) of length n, and dimension m iff µ = 1, or p is semiprimitive modulo µ. if such code exists then its hamming weight enumerator is 1 + (qm −1) µ ((µ−1)z (q−1)qm/2 eq (qm/2−(−1)s) + z (q−1)qm/2 eq (qm/2+(−1)s(µ−1))) , 92 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 where s = (mt)/ordµ(p) (recall that q = pt). from previous theorem note that if µ = 1, then e | (q−1), and c(e) is a one-weight irreducible cyclic code of length n and dimension m, whose hamming weight enumerator is hamc(e)(z) = 1+(q m−1)z q−1 eq qm. but if e | (q − 1) then it is easy to see that gcd(m,e) = gcd(q m−1 q−1 ,e) = µ = 1, and the condition in [4, theorem 3.3] is fulfilled. this, of course, is right because corollary 3.3 tells us that cle is a two-weight trace code of length q 2m−qm e over r, whose lee weight enumerator is leecle (z) = hamc(e)(z 2qm) + (q2m −qm)z2 q−1 eq (q2m−qm) = 1 + (qm −1)z2 q−1 eq q2m + (q2m −qm)z2 q−1 eq (q2m−qm) , which is the same lee weight enumerator reported in [4, theorem 3.3]. if µ = 2 in theorem 4.1, then q and p are an odd integers. therefore, clearly, p is semiprimitive modulo µ. thus, c(e) is a two-weight irreducible cyclic code of length n and dimension m, whose hamming weight enumerator is hamc(e)(z) = 1 + (qm −1) 2 (z (q−1)(qm−qm/2) eq + z (q−1)(qm+qm/2) eq ) , and by corollary 3.3, cle is a three-weight trace code of length q2m−qm e over r, whose lee weight enumerator is leecle (z) = 1 + (qm −1) 2 (z 2(q−1)(q2m−q3m/2) eq + z 2(q−1)(q2m+q3m/2) eq ) + (q2m −qm)z2 q−1 eq (q2m−qm) , which is the same lee weight enumerator reported in [4, theorem 3.7]. in fact, in the particular case when e = 2, µ = gcd(q m−1 q−1 ,e) = 2. that is, in this case, we have (µ = e = 2)|(p− 1) and d (2,qm) 0 = q, where q is the set of all square elements of f∗qm. therefore, as is correctly pointed out in [4, remark 3.8], [9, theorem 1] is a special case of [4, theorem 3.7]. now, suppose that q = p and that e > 2 is an integer such that e | pr + 1 | pm −1, for some integer r, and let l be the smallest integer such that e | pl + 1. but, under these conditions, it is easy to see that 2l | m, and therefore µ = gcd(p m−1 p−1 ,e) = e, and ordµ(p) = 2l. thus, theorem 4.1 tell us that c(e) is a two-weight irreducible cyclic code over the prime field fp, of length p m−1 e and dimension m, whose hamming weight enumerator, hamc(e)(z), is 1 + (pm −1) e ((e−1)z (p−1)pm/2 ep (pm/2−(−1) m 2l ) + z (p−1)pm/2 ep (pm/2+(−1) m 2l (e−1))) , and by corollary 3.3, cle is a three-weight trace code of length p2m−pm e over the prime ring r = fp+ufp, whose lee weight enumerator is leecle (z) = 1 + (pm −1) e [(e−1)z 2(p−1)p3m/2 ep (pm/2−(−1) m 2l ) + z 2(p−1)p3m/2 ep (pm/2+(−1) m 2l (e−1))] + (p2m −pm)z2 p−1 ep (p2m−pm) , 93 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 which is the same lee weight enumerator reported in [11, table 1] (in such table n = e). note that the conditions e | (q −1) or µ = e are quite restrictive, but fortunately theorem 4.1 tells us that we can get rid of it, and we show this by means of the following: example 4.2. let q = p = 5 and n = 3. then m = ord3(5) = 2, e = (p2 − 1)/3 = 8, and µ = gcd(p 2−1 p−1 ,e) = 2. by theorem 4.1, c(8) is a two-weight irreducible cyclic code over f5, of length 3 and dimension 2, whose hamming weight enumerator is hamc(8)(z) = 1 + 12z 2 + 12z3. now, by corollary 3.3, the gray image of cl8 is a three-weight code over the prime field f5, of length 2 p2m−pm e = 150, and size p2m = 625, whose hamming weight enumerator is leecl8 (z) = 1 + 12z 100 + 12z150 + 600z120. in this example, in addition that e (q−1) and e 6= µ, also note that (e = 8) 5r +1, for any positive integer r. therefore the three-weight trace code in example 4.2 is new in the context of [4], [9] and [11], but not the binary case of [8]. up to now we used theorem 4.1 to construct two or three-weight trace codes of the form cle. in fact, since theorem 4.1 is a characterization, there are no others two or three-weight trace codes of the form cle. however, as is outlined below, it possible to construct m-weight trace codes of the form cle, with m > 3. let e be a divisor of qm−1. suppose that µ = gcd(m,e) = gcd(q m−1 q−1 ,e) = 3, and that p ≡ 1 (mod 3) (that is p is not semiprimitive modulo µ). under these conditions, and with the help of [2, theorem 18], c(e) is a three-weight irreducible cyclic code over fq, of length qm−1 e and dimension m, whose hamming weight enumerator is 1 + qm −1 3 (z (q−1)(qm−c1q m 3 ) eq + z (q−1)(qm−1 2 (c1−9d1)q m 3 ) eq + z (q−1)(qm−1 2 (c1+9d1)q m 3 ) eq ) , where c1 and d1 are uniquely given by 4qm/3 = c21 + 27d 2 1, c1 ≡ 1 (mod 3) and gcd(c1,p) = 1. through a direct application of corollary 3.3, over the earlier hamming weight enumerator, it is easy to see that cle is a four-weight trace code, whose lee weight enumerator is precisely the lee weight distribution reported in the first part of [4, table ii]. in a quite similar way, it is easy to see that the lee weight distribution reported in the first part of [4, table iii], is just the result of direct application of corollary 3.3 over the hamming weight enumerator reported in [2, theorem 20]. lastly, note that the families of codes in the second parts of [4, table ii and table iii] are threeweight trace codes that come, as we already explained above, from two-weight semiprimitive irreducible cyclic codes. 5. the lee weight distribution of an extended family of trace codes over fq + ufq we are now going to follow the open problem suggested in the conclusion of [11], and determine the lee weight distribution of trace codes of the form cle in terms of other class of irreducible cyclic codes, with known or well-understood weight distribution. for this purpose, we recall the following result. theorem 5.1. [17, theorem 10] let n, m and r be three positive integers, such that gcd(n,q) = 1, m = ordn(q), and r ≥ 1. if r ≥ 2, suppose that the prime factors of r divide n but not (qm − 1)/n, and that qm ≡ 1 (mod 4), if 4 | r. fix µ = gcd(q m−1 q−1 , qm−1 n ). assume also that µ = 1 or p is semiprimitive modulo µ. then, the weight enumerator polynomial of any [nr,mr] irreducible cyclic code is (1 + (qm −1) µ ((µ−1)z (q−1)qm/2 eq (qm/2−(−1)s) + z (q−1)qm/2 eq (qm/2+(−1)s(µ−1))))r , 94 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 where s = (mt)/ordµ(p). combining previous theorem with corollary 3.3 we get: theorem 5.2. let n, m and r be three positive integers, such that gcd(n,q) = 1, m = ordn(q), and r ≥ 1. if r ≥ 2, suppose that the prime factors of r divide n but not (qm − 1)/n, and that qm ≡ 1 (mod 4), if 4 | r. fix µ = gcd(q m−1 q−1 , qm−1 n ), e = q m−1 n , and e′ = q mr−1 nr . assume also that µ = 1 or p is semiprimitive modulo µ. then the gray image of cle′ is a code over fq of length 2 q2mr−qmr e′ and size q2mr. furthermore, the lee weight enumerator of cle′ is (1 + (qm −1) µ ((µ−1)zf(q)(q m/2−(−1)s) + zf(q)(q m/2+(−1)s(µ−1))))r + (q2mr −qmr)z2 q−1 e′q (q 2mr−qmr) , where s = (mt)/ordµ(p), and f(q) = 2qmr(q−1)qm/2 eq . proof. this is a direct consequence of theorem 5.1 and corollary 3.3. example 5.3. let q = p = 5, n = 8, and r = 2. then m = ord8(5) = 2, µ = gcd( qm−1 q−1 , qm−1 n ) = 3, e = q m−1 n = 3, e′ = q mr−1 nr = 39, s = 1, and clearly p is semiprimitive modulo µ, and r divide n but not (qm − 1)/n. thus, the gray image of cl39 is a five-weight code over the prime field f5, of length 2q 2mr−qmr e′ = 20000, and size q2mr = 58, whose hamming weight enumerator is leecl39 (z) = (1 + 8z5000 + 16z10000)2 + 390000z16000 = 1 + 16z5000 + 96z10000 + 256z15000 + 390000z16000 + 256z20000. finally, note that theorem 5.1 includes all the semiprimitive irreducible cyclic codes, when r = 1. therefore, the new infinite family of trace codes of the form cle, described in theorem 5.2, includes the infinite family found in [11]. 6. conclusion in this work, we showed that there exists an identity between lee weight enumerators of the trace codes of the form cle, and the hamming weight enumerators of the irreducible cyclic codes of the form c(e). in other words, we proved that the lee weight distribution problem for the trace codes constructed following [4], [8], [9] or [11], is equivalent to the standard hamming weight distribution problem for the irreducible cyclic codes. in fact, this identity allowed us to present a simplified view for all the lee weight distributions reported in these works. finally, we used the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive), to determine the lee weight distribution of a new infinite family of trace codes of the form cle, that includes the infinite family found in [11]. as a future work, it could be interesting to explore the possible existence of an identity, like that in corollary 3.3, for trace codes of the form cle when they are defined over a different ring (for example, a semi-local ring similar to that in [10]). acknowledgment: the author want to express his gratitude to the anonymous referee for his valuable suggestions. 95 g. vega / j. algebra comb. discrete appl. 10(2) (2023) 87–96 references [1] p. delsarte, on subfield subcodes of reed-solomon codes, ieee trans. inf. theory 21(5) (1975) 575–576. 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[17] g. vega, a characterization of all semiprimitive irreducible cyclic codes in terms of their lengths, applicable algebra eng. commun. computin. 30(5) (2019) 441–452. 96 https://doi.org/10.1109/tit.1975.1055435 https://doi.org/10.1109/tit.1975.1055435 https://doi.org/10.1016/j.disc.2012.11.009 https://www.amazon.com/finite-fields-encyclopedia-mathematics-applications/dp/0521065674 https://doi.org/10.1109/tit.2019.2891562 https://doi.org/10.1109/tit.2019.2891562 https://doi.org/10.1016/j.dam.2017.11.020 https://doi.org/10.1016/j.dam.2017.11.020 https://www.elsevier.com/books/the-theory-of-error-correcting-codes/macwilliams/978-0-444-85193-2 https://www.elsevier.com/books/the-theory-of-error-correcting-codes/macwilliams/978-0-444-85193-2 https://doi.org/10.1006/ffta.2000.0293 https://doi.org/10.1006/ffta.2000.0293 https://doi.org/10.1109/lcomm.2016.2614934 https://doi.org/10.1109/lcomm.2016.2614934 https://doi.org/10.1007/s12095-016-0206-5 https://doi.org/10.1007/s12095-016-0206-5 https://doi.org/10.1109/tit.2017.2742499 https://doi.org/10.1109/tit.2017.2742499 https://doi.org/10.1017/s0004972718000291 https://doi.org/10.1017/s0004972718000291 https://doi.org/10.1016/j.dam.2016.09.050 https://doi.org/10.1016/j.dam.2016.09.050 https://doi.org/10.1007/s00200-017-0345-8 https://doi.org/10.1007/s00200-017-0345-8 https://mathscinet.ams.org/mathscinet-getitem?mr=mr3792337 https://mathscinet.ams.org/mathscinet-getitem?mr=mr3792337 https://doi.org/10.1007/s40840-017-0553-1 https://doi.org/10.1007/s40840-017-0553-1 https://doi.org/10.1007/s00200-019-00385-z https://doi.org/10.1007/s00200-019-00385-z introduction background material a relationship between the hamming and the lee weight enumerators in perspective with some already reported lee weight distributions the lee weight distribution of an extended family of trace codes over fq+ufq conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.560410 j. algebra comb. discrete appl. 6(2) ● 105–122 received: 9 march 2019 accepted: 16 april 2019 journal of algebra combinatorics discrete structures and applications complexity of neural networks on fibonacci-cayley tree∗ research article jung-chao ban, chih-hung chang abstract: this paper investigates the coloring problem on fibonacci-cayley tree, which is a cayley graph whose vertex set is the fibonacci sequence. more precisely, we elucidate the complexity of shifts of finite type defined on fibonacci-cayley tree via an invariant called entropy. we demonstrate that computing the entropy of a fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. what is more, the entropy of a fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. we apply the result to neural networks defined on fibonacci-cayley tree, which reflect those neural systems with neuronal dysfunction. aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on fibonacci-cayley tree, we address the formula of the boundary in the parameter space. 2010 msc: 37a35, 37b10, 92b20 keywords: neural networks, learning problem, cayley tree, separation property, entropy 1. introduction for the past few decades, neural networks have been developed to mimic brain behavior so that they are capable of exhibiting complex and various phenomena; they are widely applied in many disciplines ∗ this work is partially supported by the ministry of science and technology, roc (contract no most 1072115-m-259-004-my2 and 107-2115-m-390-002-my2). jung-chao ban; department of mathematical sciences, national chengchi university, taipei 11605, taiwan, roc and math. division, national center for theoretical science, national taiwan university, taipei 10617, taiwan, roc (email: jcban@nccu.edu.tw). chih-hung chang (corresponding author); department of applied mathematics, national university of kaohsiung, kaohsiung 81148, taiwan, roc (email:chchang@nuk.edu.tw). 105 https://orcid.org/0000-0002-4920-6945 https://orcid.org/0000-0001-7352-5148 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 such as signal propagation between neurons, deep learning, image processing, pattern recognition, and information technology [1]. while the overwhelming majority of neural network models adopt n-dimensional lattice as network topology, gollo et al. [15–17] propose a neural network with tree structure and show that the dynamic range attains large values. (see [2, 18, 19] for more references.) recent literature also provides shreds of evidence that excitable media with a tree structure performed better than other network topologies [15, 16]. mathematically speaking, a tree is a group/semigroup with finitely free generators, and group is one of the most ubiquitous structures found both in mathematics and in nature. a cayley graph is a graph which represents the structure of a group via the notions of generators of the group. remarkably, pomi proposes a neural representation of mathematical group structures which store finite group through their cayley graphs [20]. since neural networks with tree structure come to researcher’s attention, it is natural to ask how the complexity of a tree structure neural network can be measured. alternatively, it is of interest to know how much information a neural network can store. this motivates the investigation of the notion of tree-shifts introduced by aubrun and béal [3, 4]. a tree-shift is a shift space defined on a cayley tree, and a shift space is a dynamical system which consists of patterns avoiding a set of local patterns (such a set is called a forbidden set). roughly speaking, a tree-shift consists of colored cayley trees. in [3], aubrun and béal indicate that tree-shifts constitute an intermediate class in between one-sided and multidimensional shifts. ban and chang propose an algorithm for the computation of the entropy of a class of tree-shifts called tree-shifts of finite type [8]. they also demonstrate that the entropy of a tree-shift of finite type is the logarithm of a perron number and vice versa [6]. in other words, tree-shifts of finite type achieves rich phenomena. nowadays, we have known that the entropy of a tree structure neural network reflects its complexity. it is natural to consider the complex nature of a neural network representing disordered brain such as alzheimer’s disease, which is one of the most prevalent neurodegenerative disorders causing dementia and related severe public health concerns. it is well-known that alzheimer’s disease is an irreversible, progressive brain disorder that slowly destroys the patient’s memory and thinking skills; although the greatest known risk factor is aging, alzheimer’s disease is not just a disease for elders. recent experimental investigations support the hypothesis that the concept of illness progression via neuron-to-neuron transmission and transsynaptic transport of pathogens from affected neurons to anatomically interconnected nerve cells is compatible with the inordinately drawn out prodromal phase and the uniform progression of the pathological process in alzheimer’s disease. also, a neuron-to-neuron transfer and propagation via seeding offer the most straightforward explanation for both the predictable distribution pattern of the intraneuronal tau lesions in the brain as well as the prolonged rate of disease progression that characterize alzheimer’s disease neuropathologically. readers who are interested in recent development about alzheimer’s disease are referred to [10, 11, 14] and the references therein for more details. to investigate the complexity of a system with neuronal dysfunction, we propose neural networks whose topologies are cayley graphs. the basic idea is that an infected neuron cannot transfer signal correctly. hence we treat those inherit neurons as dead ones. in other words, if x is a node representing some infected neuron, then no node with prefix x is a node in the graph. to clarify the investigation, we study neural networks whose topology is fibonacci-cayley tree, which is a semigroup with generators {1,2} and a relation 2∗2 = 2. namely, the nodes of a fibonacci-cayley tree come from the fibonacci sequence. in section 2, we recall some definitions and notions about tree-shifts and extend the concept to define a fibonacci tree-shift, which is a shift space whose topology is fibonacci-cayley tree. it follows from the definition that a tree-shift is a dynamical viewpoint of coloring problem on a cayley tree under a given rule. section 3 provides the idea of the entropy of a fibonacci tree-shift. we demonstrate therein that the computation of the entropy of a fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and propose an algorithm for computing entropy. it is remarkable that, except for 106 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 extending the methodology developed in [6] to fibonacci tree-shifts of finite type, we can generalize such an algorithm to tree-shifts of finite type with more complex topology. our result might help in the investigation of the multidimensional coloring problem. section 4 applies the results to elucidate the complexity of neural networks on fibonacci-cayley tree. the fibonacci tree-shifts came from neural networks are constrained by the so-called separation property; after demonstrating there are only two possibilities of entropy for neural networks on fibonacci-cayley tree, we reveal the formula of the boundary in the parameter space. 2. fibonacci-cayley tree we introduce below some basic notions of symbolic dynamics on fibonacci-cayley tree. the main difference between fibonacci-cayley tree and infinite cayley trees is their topologies. we refer to [3, 7] for an introduction to symbolic dynamics on infinite cayley trees. 2.1. cayley tree let σ ={1,2, . . . ,d}, d ∈ n, and let σ∗ =⋃n≥0 σn be the set of words over σ, where σ0 ={ϵ} consists of the empty word ϵ. an infinite (cayley) tree t over a finite alphabet a is a function from σ∗ to a; a node of an infinite tree is a word of σ∗, and the empty word relates to the root of the tree. suppose x is a node of a tree. each node xi, i ∈ σ, is known as a child of x while x is the parent of xi. a sequence of words (wk)1≤k≤n is called a path if, for all k ≤ n−1, wk+1 = wkik for some ik ∈ σ and w1 ∈ σ∗. let t be a tree and let x be a node, we refer tx to t(x) for simplicity. a subtree of a tree t rooted at a node x is the tree t′ satisfying t′y = txy for all y ∈ σ ∗, where xy = x1⋯xmy1⋯yn means the concatenation of x = x1⋯xm and y1⋯yn. given two words x = x1x2 . . .xi and y = y1y2 . . .yj, we say that x is a prefix of y if and only if i ≤ j and xk = yk for 1 ≤ k ≤ i. a subset of words l ⊂ σ∗ is called prefix-closed if each prefix of l belongs to l. a function u defined on a finite prefix-closed subset l with codomain a is called a pattern, and l is called the support of the pattern; a pattern is called an n-block if its support l = x∆n−1 for some x ∈ σ∗, where ∆n = ⋃ 0≤i≤n σi. let aς ∗ be the set of all infinite trees over a. for i ∈ σ, the shift transformations σi ∶aς ∗ →aς ∗ is defined as (σit)x = tix for all x ∈ σ∗. the set aς ∗ equipped with the shift transformations σi is called the full tree-shift over a. suppose w = w1⋯wn ∈ σ∗. define σw = σwn ○σwn−1 ○⋯○σw1 ; it follows immediately that (σwt)x = twx for all x ∈ σ∗. suppose that u is a pattern and t is a tree. let supp(u) denote the support of u. we say that u is accepted by t if there exists x ∈ σ∗ such that uy = txy for every node y ∈ supp(u). in this case, we say that u is a pattern of t rooted at the node x. a tree t is said to avoid u if u is not accepted by t; otherwise, u is called an allowed pattern of t. given a collection of finite patterns f, let xf denote the set of trees avoiding any element of f. a subset x ⊆aς ∗ is called a tree-shift if x = xf for some f, and f is a forbidden set of x. a tree-shift xf is called a tree-shift of finite type (tsft) if the forbidden set f is finite; we say that xf is a markov tree-shift if f consists of two-blocks. suppose a1,a2, . . . ,ad are binary matrices indexed by a, the vertex tree-shift xa1,a2,...,ad is defined as xa1,a2,...,ad ={t ∈a σ ∗ ∶ ai(tx,txi)= 1 for all x ∈ σ∗,1 ≤ i ≤ d}. (1) it follows immediately that each vertex tree-shift is a markov tree-shift and each markov tree-shift is a tsft. in [7], the authors indicate that every tsft is conjugated to a vertex tree-shift. 107 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 1 2 figure 1. each node of the fibonacci-cayley tree is a finite walk in the fibonacci graph. 2.2. fibonacci-cayley tree let σ ={1,2} and let a =(1 1 1 0 ). denote by σa the collection of finite walks of graph representation g (cf. figure 1) of a together with the empty word ϵ. a fibonacci-cayley tree (fibonacci tree) t over a finite alphabet a is a function from σa to a; in other words, a fibonacci tree is a pattern whose support is the fibonacci lattice σa. a pattern u is called an n-block if supp(u) = x∆n−1⋂σa for some x ∈ σa. notably, there are two different supports for n-blocks, say l1 = ∆n−1⋂σa and l2 = 1∆n−1⋂σa, up to a shift of node. it is seen that the shift transformation σ2 is not well-defined in this case. we consider the shift transformation σ ∶ aσa × σa → aσa given by σ(t,x)y = txy if and only if xy ∈ σa. the set aσa equipped with the new defined shift transformation σ is called the full fibonacci tree-shift over a. a subset x ⊆aσa is called a fibonacci tree-shift if x = xf for some forbidden set f. similar to the definitions above, a fibonacci tree-shift xf is called a fibonacci tree-shift of finite type (ftsft) if f is a finite set. if f consists of two-blocks, then xf is a markov-fibonacci tree-shift. suppose a1 and a2 are binary matrices indexed by a, the vertex-fibonacci tree-shift xa1,a2 is defined as xa1,a2 ={t ∈a σa ∶ ai(tx,txi)= 1 for all x,xi ∈ σa}. (2) see figure 2 for the support of a fibonacci tree. example 2.1. suppose that the alphabet a = {r,g} consists of red and green two colors. let a1 and a2, the coloring rules on the left and right directions, respectively, be given as a1 = a2 =( 1 1 1 0 ) . that is, we can not color green on any two consecutive nodes. 3. complexity of colored fibonacci-cayley tree this section investigates the complexity of fibonacci tree-shifts of finite type. we use an invariant known as entropy introduced in [6, 8]. 3.1. entropy of fibonacci tree-shifts let x be a fibonacci tree-shift. for n ∈ n and w ∈ σa, let γ [w] n (x) denote the set of n-blocks of x rooted at w. more explicitly, γ[w]n (x)={u ∶ u is allowed and supp(u)= w∆n−1⋂σa}. fix c ∈a, set γ[w]c;n(x)={u ∈ γ [w] n (x) ∶ uw = c} and γ [w] c;n = ∣γ [w] c;n(x)∣, where ∣ ⋅ ∣ means the cardinality. for simplicity, we refer to γ[ϵ]c;n(x) and γ [ϵ] c;n as γc;n(x) and γc;n, respectively. the entropy of x is defined 108 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 ϵ 1 2 11 12 21 111 112 121 211 212 figure 2. the support of fibonacci-cayley tree. it is seen that nodes 22, 122, 221, 222,⋯ are absent in this case. as follows. definition 3.1. suppose x is a fibonacci tree-shift. the entropy of x, denoted by h(x), is defined as h(x)= limsup n→∞ ln 2 γn n , (3) where ln2 = ln○ ln. notably, the growth rate of the number of nodes of ∆n⋂σa is exponential; this makes the speed of increase of γn doubly exponential. hence, the entropy h(x) measures the growth rate of feasible patterns concerning their height. let a = {c1,c2, . . . ,ck} for some k ≥ 2 ∈ n. given two binary matrices a1 = (ai,j)ki,j=1 and a2 = (bi,j)ki,j=1, it comes immediately that γn = k ∑ ℓ=1 γℓ;n and γi;n = k ∑ j1,j2=1 ai,j1bi,j2γ [1] j1;n−1γ [2] j2;n−1 (4) for 1 ≤ i ≤ k and n ≥ 3 ∈ n, herein x = xa1,a2 is a vertex-fibonacci tree-shift, γ [w] ℓ;n refers to γ[w]cℓ;n, and γi;2 = ( k ∑ j=1 ai,j) ⋅( k ∑ j=1 bi,j) for 1 ≤ i ≤ k. for w ∈ σa, the follower set of w is defined as fw ={w′ ∈ σa ∶ ww′ ∈ σa}. since each element w = w1w2⋯wn ∈ σa is a finite walk in the fibonacci graph g (cf. figure 1), it is easily seen that fw ={ σa, wn = 1; f2, otherwise. 109 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 this makes γ [1] i;n = γi;n and γ [2] i;n = k ∑ j=1 ai,jγ [21] j;n−1 = k ∑ j=1 ai,jγj;n−1 (5) for 1 ≤ i ≤ k and n ≥ 3 ∈ n. substituting (4) with (5) derives γi;n = k ∑ j1,j2,j3=1 ai,j1bi,j2aj2,j3γj1;n−1γj3;n−2 for 1 ≤ i ≤ k and n ≥ 4 ∈ n. alternatively, computing the entropy of x is equivalent to the investigation of the following nonlinear recursive system. ⎧⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ γi;n = k ∑ j1,j2,j3=1 ai,j1bi,j2aj2,j3γj1;n−1γj3;n−2, γi;2 = ( k ∑ j=1 ai,j) ⋅( k ∑ j=1 bi,j), (6) for 1 ≤ i ≤ k and n ≥ 4 ∈ n. we call (6) the recurrence representation of xa1,a2 . example 3.2. suppose that a={c1,c2} and a1 = a2 =( 1 1 1 0 ) . observe that ⎧⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎩ γ1;n = (γ [1] 1;n−1 +γ [1] 2;n−1) ⋅(γ [2] 1;n−1 +γ [2] 2;n−1), γ2;n = γ [1] 1;n−1 ⋅γ [2] 1;n−1, γ1;2 = 4,γ2;2 = 1. since γ[1]i;n = γi;n, γ [2] 1;n = γ [21] 1;n−1 +γ [21] 2;n−1 = γ1;n−1 +γ2;n−1, and γ [2] 2;n = γ [21] 1;n−1 = γ1;n−1, examining the entropy of xa1,a2 is equivalent to studying ⎧⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎩ γ1;n = (γ1;n−1 +γ2;n−1) ⋅(2γ1;n−2 +γ2;n−2), γ2;n = γ1;n−1 ⋅(γ1;n−2 +γ2;n−2), γ1;2 = 4,γ2;2 = 1. in the next subsection, we introduce an algorithm for solving the growth rate of the nonlinear recursive system (6). 3.2. computation of entropy the previous subsection reveals that computing the entropy of a fibonacci tree-shift of finite type is equivalent to studying a corresponding nonlinear recursive system. to address an algorithm for the computation of the entropy, we start with the following proposition. 110 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 proposition 3.3. suppose that x = xa1,a2 is a vertex-fibonacci tree-shift over a with respect to a1,a2. then the limit (3) exists and h(x)= limsup n→∞ ln∑ki=1 lnγi;n n . (7) proof. from the cauchy inequality γ1;n ⋅γ2;n⋯γk;n ≤( ∑ki=1 γi;n k ) k we derive that k ∑ i=1 lnγi;n ≤ k(ln k ∑ i=1 γi;n − lnk). it can be verified without difficulty that limsup n→∞ ln∑ki=1 lnγi;n n ≤ limsup n→∞ ln 2∑ki=1 γi;n n = h(x). it is noteworthy that if there exist 1 ≤ i ≤ k and m ≥ 2 such that γi;m ≥ γj;m for all j ≠ i, then γi;n ≥ γj;n for all j ≠ i and n ≥ m ∈ n. without loss of generality, we may assume that γ1;n ≥ γi;n for all n ≥ 2 ∈ n and 1 ≤ i ≤ k. the inequality γ1;n ≤ k ∑ i=1 γi;n ≤ kγ1;n reveals that limsup n→∞ ln 2∑ki=1 γi;n n = limsup n→∞ ln 2 γ1;n n . meanwhile, k ∑ i=1 lnγi;n = ln k ∏ i=1 γi;n ≥ lnγ1;n. therefore, we conclude that limsup n→∞ ln∑ki=1 lnγi;n n ≥ limsup n→∞ ln 2 γ1;n n = limsup n→∞ ln 2∑ki=1 γi;n n = h(x). this completes the proof. a symbol c ∈a is called essential (resp. inessential) if γc;n ≥ 2 for some n ∈ n (resp. γc;n = 1 for all n ∈ n). proposition 3.3 infers the alphabet a can be expressed as the disjoint union of two subsets ae and ai consisting of essential and inessential symbols, respectively. in addition, it is easily seen from (7) that h(x)= limsup n→∞ ln 2∑c∈a γc;n n = limsup n→∞ ln 2∑c∈ae γc;n n . for the simplicity, we assume that a=ae; that is, each symbol is essential. 111 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 now we turn back to the elaboration of the nonlinear recursive system (6). first, we consider a subsystem γi;n = γij1 ;n−1γij2 ;n−1, 1 ≤ i ≤ k, (8) arose from (6). notably, for each i, the above subsystem contains only one term in the original system and the coefficient is 1. we call such a system simple. generally speaking, a fibonacci tree-shift has many simple recurrence representations. the following theorem reveals the entropy of a vertex-fibonacci tree-shift with simple recurrence representation relates to the spectral radius of some integral matrix. theorem 3.4. suppose that x = xa1,a2 is a fibonacci tree-shift over a such that each symbol is essential. if the recurrence representation of x is simple, then there exists an integral matrix m such that h(x)= log ρm, where ρm is the spectral radius of m. proof. since ci is essential for 1 ≤ i ≤ k, we may assume that γi;2 ≥ 2 for simplicity. proposition 3.3 demonstrates that h(x)= limsup n→∞ ln∑ki=1 lnγi;n n . let θn = (lnγ1;n, lnγ1;n−1, lnγ2;n, lnγ2;n−1, . . . , lnγk;n, lnγk;n−1)t (9) be a 2k × 1 vector. the recursive system (5) suggests that there exists a 2k × 2k nonnegative integral matrix m such that θn = mθn−1 for n ≥ 3. (10) note that γi;2 = ( k ∑ j=1 ai,j) ⋅( k ∑ j=1 bi,j)≤ k2 for 1 ≤ i ≤ k. therefore, h(x)= limsup n→∞ ln∑ki=1 lnγi;n n ≤ lim n→∞ ln∑2ki,j=1 m n(i,j) n ≤ lnρm. on the other hand, h(x)= limsup n→∞ ln∑ki=1 lnγi;n n = limsup n→∞ ln∑ki=1 2lnγi;n n ≥ limsup n→∞ ln∑ki=1(lnγi;n + lnγi;n−1) n = lim n→∞ ln∑2ki,j=1 m n(i,j) n = lnρm. this completes the proof. we call the matrix m sketched in theorem 3.4 the adjacency matrix of the simple recursive system (5). this makes theorem 3.4 an extension of a classical result in symbolic dynamical systems. 112 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 example 3.5. suppose that the alphabet a and a1,a2 are given, and the recurrence representation of xa1,a2 is { γ1;n = γ1;n−1 ⋅γ1;n−2, γ2;n = γ1;n−1 ⋅γ2;n−2. (11) furthermore, each symbol in a is essential. let θn = ⎛ ⎜⎜⎜ ⎝ lnγ1;n lnγ1;n−1 lnγ2;n lnγ2;n−1 ⎞ ⎟⎟⎟ ⎠ . it comes immediately that m = ⎛ ⎜⎜⎜ ⎝ 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 ⎞ ⎟⎟⎟ ⎠ since θn = mθn−1. in addition, the characteristic polynomial of m is f(λ) = (λ − 1)(λ + 1)(λ2 − λ − 1) and the spectral radius of m is ρm = g, where g = 1+ √ 5 2 is the golden mean. theorem 3.4 asserts that the entropy of xa1,a2 is h(xa1,a2)= lng. theorem 3.6. suppose that x = xa1,a2 is a fibonacci tree-shift over a such that each symbol is essential. then h(x)= max{lnρm ∶ m is the adjacency matrix of a simple subsystem of x}. (12) the main idea of the proof of theorem 3.6 can be examined via the following example. example 3.7. suppose that a={c1,c2} and a1 = a2 =( 1 1 1 0 ) are the same as considered in example 3.2. recall that the recurrence representation of xa1,a2 is ⎧⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎩ γ1;n = (γ1;n−1 +γ2;n−1) ⋅(2γ1;n−2 +γ2;n−2), γ2;n = γ1;n−1 ⋅(γ1;n−2 +γ2;n−2), γ1;3 = 15,γ2;3 = 8,γ1;2 = 4,γ2;2 = 1. it can be verified without difficulty that γ1;n ≥ γ2;n ≥ γ1;n−1 ≥ γ2;n−1 for n ≥ 3. therefore, we have two inequalities γ1;n = (γ1;n−1 +γ2;n−1) ⋅(2γ1;n−2 +γ2;n−2) = 2γ1;n−1γ1;n−2 +γ1;n−1γ2;n−2 +2γ2;n−1γ1;n−2 +γ2;n−1γ2;n−2 = γ1;n−1γ1;n−2(2+ γ2;n−2 γ1;n−2 +2 γ2;n−1 γ1;n−1 + γ2;n−1γ2;n−2 γ1;n−1γ1;n−2 )≤ 6γ1;n−1γ1;n−2, and γ2;n = γ1;n−1 ⋅(γ1;n−2 +γ2;n−2) = γ1;n−1γ1;n−2 +γ1;n−1γ2;n−2 = γ1;n−1γ1;n−2(1+ γ2;n−2 γ1;n−2 )≤ 2γ1;n−1γ1;n−2 < 6γ1;n−1γ1;n−2. 113 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 consider the nonlinear recursive system αi;n = 6α1;n−1α1;n−2, i = 1,2,n ≥ 3, let θn = (lnα1;n, lnα1;n−1, lnα2;n, lnα2;n−1)t and m = ⎛ ⎜⎜⎜ ⎝ 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 ⎞ ⎟⎟⎟ ⎠ . it is noteworthy that m is also the adjacency matrix of the simple recurrence representation γi;n = γ1;n−1γ1;n−2, i = 1,2,n ≥ 3, of x. meanwhile, θn = mθn−1 +(ln6)14 = mn−2θ2 +(ln6)(mn−3 +⋯+m +i4)14, where 14 = (1,1,1,1)t . therefore, we derive from proposition 3.3 that h(xa1,a2)= limsup n→∞ ln∑2i=1 lnγi;n n ≤ limsup n→∞ ln∑2i=1 lnαi;n n ≤ lim n→∞ ln∑n−2i=1 ∥m i∥ n = lng, where g = 1+ √ 5 2 is the golden mean. obviously, h(xa1,a2)≥ lng. this concludes that h(xa1,a2)= lng. proof of theorem 3.6. since every symbol is essential, we may assume that, for 1 ≤ i ≤ k, γi;2 ≥ 2 for simplicity. set h̵ = max{lnρm ∶ m is the adjacency matrix of a simple subsystem of x}. it suffices to show that h(x)≤ h̵. we may assume without losing the generality that γ1;n ≥ γ2;n ≥⋯≥ γk;n ≥ γ1;n−1 ≥ γ2;n−1 ≥⋯≥ γk;n−1 (13) for n ≥ 3. recall that, for 1 ≤ i ≤ k and n ≥ 4, γi;n = k ∑ j1,j2,j3=1 ai,j1bi,j2aj2,j3γj1;n−1γj3;n−2, where a1 = (ai,j)1≤i,j≤k,a2 = (bi,j)1≤i,j≤k. let γī1;n−1γī2;n−2 be the maximum in the above equation. then γi;n = k ∑ j1,j2,j3=1 ai,j1bi,j2aj2,j3γj1;n−1γj3;n−2 = γī1;n−1γī2;n−2 k ∑ j1,j2,j3=1 ai,j1bi,j2aj2,j3 γj1;n−1γj3;n−2 γī1;n−1γī2;n−2 ≤ κγī1;n−1γī2;n−2 114 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 for some constant κ ∈ n. notably, κ is independent of i and n. consider the nonlinear recursive system αi;n = καī1;n−1αī2;n−2 for 1 ≤ i ≤ k. (14) let θn = (lnα1;n, lnα1;n−1, lnα2;n, lnα2;n−1, . . . , lnαk;n, lnαk;n−1)t . similar to the discussion in the proof of theorem 3.4, there exists a 2k×2k integral matrix m such that θn = mθn−1 +(lnκ)12k = mn−2θ2 +(lnκ)(mn−3 +⋯+m +i2k)12k, where 12k = (1, . . . ,1)t is a 2k × 1 vector. combining the computation above with the fact γi;n ≥ 2 indicates that h(x)= limsup n→∞ ln 2∑ki=1 γi;n n = limsup n→∞ ln∑ki=1 lnγi;n n ≤ limsup n→∞ ln∑ki=1 lnαi;n n ≤ lim n→∞ ln∑n−2i=1 ∥m i∥ n = lnρm. observe that (14) is a simple recurrence representation of x if we replace κ in (14) by 1, and m is the adjacency matrix of such a new system. therefore, we conclude that h(x)≤ h̵. the proof is thus complete. remark 3.8. a key presumption in theorem 3.6 is that every symbol is essential. if there are inessential symbols, the proof of theorem 3.6 infers that we only need to replace m by m′, where m′ is obtained by deleting all the rows and columns indexed by those inessential symbols. 4. neural networks on fibonacci-cayley tree in this section, we apply the algorithm developed in the previous section to neural networks with the fibonacci-cayley tree as their underlying space. for some reasoning, the overwhelming majority of considered underlying spaces of neural networks are zn lattice for some n ∈ n; nevertheless, it is known that the characteristic shape of neurons is tree. some medical experiments suggest that the signal transmission might be blocked by those diseased neurons, which motivate the investigation of neural networks on “incomplete" trees such as fibonacci-cayley tree. 4.1. neural networks on cayley tree to clarify the discussion, we focus on the neural networks proposed by chua and yang [13], which is known as cellular neural network and is widely applied to many disciplines such as signal propagation between neurons, pattern recognition, and self-organization; the elaboration of other models such as hopfield neural networks can be carried out analogously. let σ∗ be the set of nodes of cayleys as described in the previous section. a cellular neural network on cayley tree (ctcnn) is represented as d dt xw(t)=−xw(t)+z + ∑ v∈n avf(xwv(t)), (15) for some finite set n ⊂ σ∗ known as the neighborhood, v ∈n , and t ≥ 0. the transformation f(s)= 1 2 (∣s+1∣− ∣s−1∣) (16) 115 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 xw xw1 xw2 a1 a2 a figure 3. a cellular neural network with the nearest neighborhood defined on binary trees. in this case, the neighborhood n ={ϵ, 0, 1} and a = aϵ. is called the output function or activation function, and z is called the threshold. the weighted parameters a = (av)v∈n is called the feedback template, and figure 3 shows the connection of a binary ctcnn with the nearest neighborhood. a mosaic solution x = (xw)w∈σ∗ of (15) is an equilibrium solution which satisfies ∣xw∣ > 1 for all w ∈ σ∗; its corresponding pattern y = (yw)w∈σ∗ = (f(xw))w∈σ∗ is called a mosaic output pattern. since the output function (16) is piecewise linear with f(s)= 1 (resp. −1) if s ≥ 1 (resp. s ≤−1), the output of a mosaic solution x = (xw)w∈σ∗ must be an element in {−1,+1} σ ∗ , which is why we call it a pattern. given a ctcnn, we refer to y as the output solution space; namely, y ={(yw)w∈σ∗ ∶ yw = f(xw) and (xw)w∈σ∗ is a mosaic solution of (15)} . 4.2. learning problem learning problem (also called the inverse problem) is one of the most investigated topics in a variety of disciplines. from the mathematical point of view, determining whether a given collection of output patterns can be exhibited by a ctcnn is essential for the study of learning problem. this subsection reveals the necessary and sufficient condition for the capability of exhibiting the output patterns of ctcnns. for the compactness and consistency of this paper, we focus on cellular neural networks on the binary tree with the nearest neighborhood. the discussion of general cases is similar to the investigation in [5, 9, 12], thus it is omitted. a ctcnn with the nearest neighborhood is realized as d dt xw(t)=−xw(t)+z +af(xw(t))+a1f(xw1(t))+a2f(xw2(t)), (17) where a,a1,a2 ∈ r and w ∈ σ∗. considering the mosaic solution x = (xw)w∈σ∗, the necessary and sufficient condition for yw = f(xw)= 1 is a−1+z >−(a1yw1 +a2yw2). (18) 116 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 similarly, the necessary and sufficient condition for yw = f(xw)=−1 is a−1−z > a1yw1 +a2yw2. (19) let v 2 ={v ∈ r2 ∶ v = (v1,v2), and ∣vi∣= 1,1 ≤ i ≤ 2}, and let α = (a1,a2) represent the feedback template without the self-feedback parameter a. the admissible local patterns with the +1 state in the parent neuron is denoted by b̃+(a,z)={v ∈ v 2 ∶ a−1+z >−α ⋅v}, (20) where “⋅" is the inner product in euclidean space. similarly, the admissible local patterns with the −1 state in the parent neuron is denoted by b̃−(a,z)={v ∈ v 2 ∶ a−1−z > α ⋅v}. (21) furthermore, the admissible local patterns induced by (a,z) can be denoted by b(a,z)=b+(a,z)⋃b−(a,z), (22) where b+(a,z)={v ∶ vϵ = 1 and (v1,v2) ∈ b̃+(a,z)}, b−(a,z)={v ∶ vϵ =−1 and (v1,v2) ∈ b̃−(a,z)}. namely, b(a,z) consists of two-blocks over a = {1,−1}. for simplicity, we omit the parameters (a,z) and refer to b(a,z) as b. meanwhile, we denote the output space y by xb to specify the set of local patterns b. suppose u is a subset of v n, where n ≥ 2 ∈ n. let uc = v n ∖u. we say that u satisfies the linear separation property if there exists a hyperplane h which separates u and uc. more precisely, u satisfies the separation property if and only if there exists a linear functional g(z1,z2, . . . ,zn)= c1z1+c2z2+⋯+cnzn such that g(v)> 0 for v ∈ u and g(v)< 0 for v ∈ uc. figure 4 interprets those u ⊂ v 2 satisfying the linear separation property. proposition 4.1 elucidates the necessary and sufficient condition for the learning problem of ctcnns, which follows from straightforward examination. thus the proof is omitted. such a property holds for arbitrary n provided n is prefix-closed. readers are referred to [5] for more details. proposition 4.1. a collection of patterns b =b+⋃b− can be realized in (17) if and only if either of the following conditions is satisfied: (inv1) −b̃+ ⊆ b̃− and b̃− satisfies linear separation property; (inv2) −b̃− ⊆ b̃+ and b̃+ satisfies linear separation property. herein, b̃+ and b̃− are defined in (20) and (21), respectively. we emphasize that proposition 4.1 demonstrate the parameter space can be partitioned into finitely many equivalent regions. indeed, whenever the parameters a1 and a2 are determined, (18) and (19) partition the a-z plane into 25 regions; the “order" (i.e., the relative position) of lines a − 1 + (−1)ℓz = (−1)ℓ(a1yw1 +a2yw2), ℓ = 1,2, can be uniquely determined by the following procedures: 1) the signs of a1,a2 (i.e., the parameters are positive or negative). 117 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 (-1,-1) (1,-1) (-1,1) (1,1) (-1,-1) (1,-1) (-1,1) (1,1) (-1,-1) (1,-1) (-1,1) (1,1) (-1,-1) (1,-1) (-1,1) (1,1) (-1,-1) (1,-1) (-1,1) (1,1) (-1,-1) (1,-1) (-1,1) (1,1) b b bb b b bb b b bb b b bb b b b b b b b b b b bb bb b b bb b b figure 4. suppose u is a proper subset of v 2 = {−1, 1}2. there are only 12 choices of u that satisfies the linear separation property 2) the magnitude of a1,a2 (i.e., ∣a1∣> ∣a2∣ or ∣a1∣< ∣a2∣). this partitions a-z plane into 8×25 = 200 regions. as a brief conclusion, the parameter space is partitioned into at most 200 equivalent regions. example 4.2. suppose that 0 < −a1 < a2. it follows from a1 − a2 < −a1 − a2 < a1 + a2 < −a1 + a2 that, whenever a and z are fixed, the “ordered” basic set of admissible local patterns b =b+⋃b− must obey b+ ⊆{(+;−,+),(+;+,+),(+;−,−),(+;+,−)} and b− ⊆{(−;+,−),(−;−,−),(−;+,+),(−;−,+)}, herein, we denote a two-block u by (uϵ;u1,u2) and use symbols “+" and “−" to represent +1 and −1, respectively, for clarity. if the parameters a and z locate in the region [3,2](cf. figure 5), then the set of feasible local patterns is b[3,2] ={(+;−,+),(+;+,+),(+;−,−),(−;+,−),(−;−,−)}. a careful but straightforward examination indicates that two symbols are both inessential if and only if the pair of parameters (a,z) is in the plane below the red w -shape line. 4.3. neural networks on fibonacci-cayley tree the definition of cellular neural networks on fibonacci-cayley tree is similar to the definition of fibonacci-cayley tree-shifts. let a and σa be the same as defined in the previous section. a cellular neural network on fibonacci-cayley tree (ftcnn) with the nearest neighborhood is realized as d dt xw(t)={ −xw(t)+z +af(xw(t))+a1f(xw1(t))+a2f(xw2(t)), wn = 1; −xw(t)+z +af(xw(t))+a1f(xw1(t)), wn = 2; (23) 118 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 a−1 z [0, 0] [1, 0] [2, 0] [3, 0] [4, 0] [0, 1] [0, 2] [0, 3] [0, 4] [1, 1] [2, 1] [3, 1] [4, 1] [1, 2] [1, 3] [1, 4] [2, 4] [3, 4] [4, 4] [4, 3] [4, 2] [3, 2] [3, 3] [2, 3] [2, 2] figure 5. for each chosen pair of parameters (a1, a2), the a-z plane is partitioned into 25 equivalent regions. where a,a1,a2 ∈ r and w = w1⋯wn ∈ σa. suppose the parameters a = (a,a1,a2) and z are given. let b be the set of feasible local patterns corresponding to (a,z) which is studied in the previous subsection. it can be verified without difficulty that the output space of (23) is yf ={t ∈aσa ∶ t = t′∣σa for some t ′ ∈ y = xb}. in other words, the output space yf of (23) is a markov-fibonacci tree-shift. 4.4. entropy of neural networks on fibonacci tree it is of interest that how much information a diseased neural network can store. based on the algorithm developed in the previous section, we study the entropy of ftcnns with the nearest neighborhood. 119 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 suppose that the parameters a1,a2 are given. a pair of parameters (a,z) is called critical if, for each r > 0, there exists (a′,z′),(a′′,z′′) such that h(xb′) ≠ h(xb′′) = 0, where b′ =b(a′,a1,a2,z′),b′′ = b(a′′,a1,a2,z′′). notably, the a-z plane is partitioned into 25 equivalent regions whenever a1 and a2 are given and is indexed as [p,q] for 0 ≤ p,q ≤ 4; we use y[p,q] instead of xb for simplicity. theorem 4.3. suppose that the parameters a1,a2 are given. let m = min{∣a1∣, ∣a2∣},m = max{∣a1∣, ∣a2∣}. then h(y[p,q])={ 0, if min{p,q}= 0 or max{p,q}= 1; lng, otherwise. furthermore, (a,z) is critical if and only if a−1 = ∣∣z∣−m∣−m. (24) proof. obviously, h(y[p,q])= 0 if min{p,q}= 0 or max{p,q}= 1. we only need to show that h(y[p,q])= lng provided p,q > 0 and max{p,q}≥ 2. actually, it suffices to consider the cases where [p,q]= [1,2] or [p,q] = [2,1] since the entropy function is increasing. we demonstrate that h(y[1,2]) = lng; the other case can be derived analogously, thus it is omitted. suppose that both symbols in a are essential. let θn = (lnγ1;n, lnγ2;n, lnγ1;n−1, lnγ2;n−1)t , where γ1;n (resp. γ2;n) denotes the number of n-blocks whose rooted symbol is + (resp. −). notably, for each simple recurrence representation of y[1,2], there exist 2 × 2 binary matrices b1 and b2 satisfying 2 ∑ k=1 bi(j,k) = 1 for 1 ≤ i,j ≤ 2 and m = ( b1 b2 i2 02 ) such that θn = mθn−1 for n ≥ 2. set v = (g,g,1,1)t . then mv = (g +1,g +1,g,g)t = (g2,g2,g,g)t = gv. perron-frobenius theorem and theorem 3.6 imply that h(y[1,2])≥ lng. it is easily seen that h(y[1,2])≤ lng, thus we can conclude that h(y[1,2])= lng. suppose that there is exactly one essential symbol. (in this case, the essential symbol is −.) it can be verified from (18), (19), and proposition 4.1 that b[1,2] ={(+;+,+),(−;−,−),(−;−,+)} or b[1,2] ={(+;+,+),(−;−,−),(−;+,−)}. for either case, γi;n = γi;n−1γi;n−2 for i = 1,2, is a simple recurrence representation of y[1,2], and its corresponding adjacency matrix is m = ⎛ ⎜⎜⎜ ⎝ 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 ⎞ ⎟⎟⎟ ⎠ . deleting the first and third rows and columns of m delivers m′ =(1 1 1 0 ). this shows that h(y[1,2])≥ lnρm′ = lng, 120 j.-c. ban, c.-h. chang / j. algebra comb. discrete appl. 6(2) (2019) 105–122 which is followed by h(y[1,2])= lng. next, we show that (a,z) is critical if and only if (a,z) satisfies (24). let c = {ℓ1a1 +ℓ2a2 ∶ ℓ1,ℓ2 ∈ {−1,1}}, and let k1 = maxc and k2 = maxc ∖{k1}. be the largest and the second largest elements in c, respectively. a careful but straightforward verification asserts that (a,z) is critical if and only if a−1 = ∣∣z∣− k1 −k2 2 ∣− k1 +k2 2 . (see figure 5 for more information.) the desired result follows from the fact that k1 = ∣a1∣+ ∣a2∣ and k2 = ∣a1∣+ ∣a2∣−2m. this completes the proof. 5. conclusion this paper studies the entropy of fibonacci-cayley tree-shifts, which are shift spaces defined on fibonacci-cayley trees. entropy is one of the most frequently used invariant that reveals the growth rate of patterns stored in a system. followed by demonstrating that the computation of the entropy of a fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system, we propose an algorithm for computing entropy. it is seen that the so-called simple recursive system plays an important role. as an application, we elucidate the complexity of neural networks whose topology is fibonaccicayley tree. such a model reflects a brain going with neuronal dysfunction such as alzheimer’s disease. a fibonacci tree-shift came from a neural network is constrained by the so-called separation property. aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on fibonacci-cayley tree, we reveal the formula of the boundary in the parameter space. a related interesting problem is “does the black-or-white entropy spectrum of neural networks still hold for other network topologies?” further discussion is in preparation. references [1] m. anthony, p. l. bartlett, neural network learning: theoretical foundations, cambridge university press, cambridge, 1999. 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https://doi.org/10.1103/physrevlett.106.058101 https://doi.org/10.1103/physrevlett.106.058101 https://doi.org/10.1007/s11538-016-0202-0 https://doi.org/10.1007/s11538-016-0202-0 introduction fibonacci-cayley tree complexity of colored fibonacci-cayley tree neural networks on fibonacci-cayley tree conclusion references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 1(1) • 1-12 received: 21 april 2014; accepted: 26 august 2014 doi 10.13069/jacodesmath.31486 journal of algebra combinatorics discrete structures and applications cyclic and constacyclic codes over a non-chain ring research article aysegul bayram1∗, irfan siap1∗∗ 1. yildiz technical university, faculty of arts and science, department of mathematics, istanbul, turkey abstract: in this study, we consider linear and especially cyclic codes over the non-chain ring zp[v]/〈vp − v〉 where p is a prime. this is a generalization of the case p = 3. further, in this work the structure of constacyclic codes are studied as well. this study takes advantage mainly from a gray map which preserves the distance between codes over this ring and p-ary codes and moreover this map enlightens the structure of these codes. furthermore, a macwilliams type identity is presented together with some illustrative examples. 2010 msc: 94b05, 94b15, 11t71 keywords: non-chain rings, linear codes, cyclic codes, constacycllic codes, macwilliams type identity 1. introduction recently, codes over some special finite rings especially chain rings have been studied. more recently, codes over finite non-chain rings have been also considered. however, the study on non-chain rings has proved to be challenging due to the algebraic structure of these rings which does not allow to give a nice and compact presentation of linear codes over these rings. study on codes over such rings or rings in general is motivated by the existence of some special maps called gray maps whose images give codes over fields. the existence of such maps is not guaranteed in general. first substantial paper which relates codes over the quaternary ring z4 to binary codes is studied initially in [9] where a gray map is presented. some important results of this study are the generation of some optimal non-binary codes such as kerdock, preparata codes via a gray map. this particular work motivated the researchers and since then codes over rings have been of great importance to the study. we can list some related studies on this subject that study codes over chain rings such as the ring of four elements f2 + uf2, the ring of 8 elements f2 + uf2 + u2f2, and a more general chain ring f2[u]/〈us〉 are presented in [2–4, 6, 11, 13]. some euclidean and hermitian self-dual codes over the ring f2[v]/〈v2 −v〉 are related to binary self-dual and formally self-dual codes and optimal self-dual binary codes obtained in [5] which inspired the original work of the authors [4]. gao studied a new generalization of [4] over fp under the restriction v3 −v in ∗ e-mail: abayram@yildiz.edu.tr, aaysegulbayram@gmail.com ∗∗ e-mail: isiap@yildiz.edu.tr 1 cyclic and constacyclic codes over a non-chain ring [8]. here, the authors mainly further generalize the results in [4] to codes over the ring zp[v]/〈vp − v〉 and study algebraic structure, that is, its ideals, units, etc. furthermore, the authors also determine the algebraic structure of linear, cyclic and constacyclic codes over this generalized ring by means of a gray map. in this work, we consider codes over the non-chain ring zp[v]/〈vp − v〉 = {a0 + a1v + ... + ap−1v p−1|a0,a1, ...,ap−1 ∈ zp and vp = v}. in the first section we analyze the structure of the ring and investigate its algebraic properties. furthermore, linear codes over zp[v]/〈vp − v〉 are taken into account and the generator matrices of their gray images are examined. then, the dual of a linear code is defined by defining an inner product and relation between linear code and further its dual is presented. the relation between cyclic codes and their duals over zp[v]/〈vp −v〉 are also studied. finally, a class of constacyclic codes have been introduced and dual codes of them are studied. 2. preliminaries the ring rp = zp[v]/〈vp − v〉 has pp elements where p is a prime number. in order to study the structure of this ring, we introduce a linear map φ which we refer as a gray map in the following way: φ :rp = zp[v]/〈vp −v〉→ zpp α = a0 + a1v + ... + ap−1v p−1 → φ(α) = φ(a0 + a1v + ... + ap−1vp−1) = (α(0),α(1), ...,α(p−1)) (1) where α(i) = a0 + a1i + ... + ap−1ip−1 (mod p) for all i ∈ {0,1, ...,p− 1}.. indeed, this map is basically the natural one that gives the chinese remainder theorem and hence this map relates the rings rp and zpp. due to the fact that the map φ is a ring isomorphism, we have rp ∼= zp[v]/〈v〉⊕zp[v]/〈v −1〉⊕ ·· ·⊕zp[v]/〈v − (p−1)〉∼= zpp. it is not easy to find the structure of lattices of ideals of non-chain rings in general. here by using the gray map introduced above, we are able to give the structure of ideals of rp and further count the number of ideals as follows: lemma 2.1. rp has exactly 2p ideals. proof. since zp is a field then its ideals are exactly the zero ideal and zp itself, then the number of ideals of zpp is the product of the number of the trivial ideals. therefore the number of ideals of rp is 2p. example 2.2. consider the ring r5. prior to listing the ideals of r5 we introduce a short notation such as 11010 which means that the ideal in z55 which is composed by the zero ideals in its third and fifth coordinates and the all ring in the rest. also, we note that a1a20a40 where ai 6= 0 for i ∈ {1,2,4} gives the same ideals since the nonzero elements in the field generate the all field. therefore, the ideals that generate the all ring have 55 = 3125 elements and naturally the elements that generate these ideals are units of r5: • 11111 → 〈1〉 = 〈2〉 = 〈3〉 = 〈4〉 = 〈2 + v2〉 = ... = 〈1 + v + 2v2〉. the maximal ideals with 625 elements: • 11110 → 〈1 +v〉 = 〈2 + 2v〉 = 〈3 + 3v〉 = 〈4 + 4v〉 = 〈1 + 2v +v2〉 = ... = 〈4 + 3v + 4v2 + 4v3 + 4v4〉. • 11101 → 〈2 +v〉 = 〈4 + 2v〉 = 〈1 + 3v〉 = 〈3 + 4v〉 = 〈4 + 4v +v2〉 = ... = 〈3 + 3v + 4v2 + 4v3 + 4v4〉. • 11011 → 〈3 + v〉 = 〈1 + 2v〉 = 〈4 + 3v〉 = 〈2 + 4v〉 = 〈4 + v + v2〉 = ... = 〈2 + 3v + 4v2 + 4v3 + 4v4〉. • 10111 → 〈4 +v〉 = 〈3 + 2v〉 = 〈2 + 3v〉 = 〈1 + 4v〉 = 〈1 + 3v +v2〉 = ... = 〈4 + 4v + 4v2 + 4v3 + 4v4〉. • ... 2 a. bayram, i. siap • 11100→〈2+3v+v2〉 = 〈4+v+2v2〉 = 〈1+4v+3v2〉 = 〈3+2v+4v2〉 = ... = 〈1+3v+2v2+4v3+4v4〉. the ideals with 25 elements: • 00011 → 〈2v + 2v2 + (v3)〉 = 〈4v + 4v2 + 2v3〉 = 〈v + v2 + 3v3〉 = ... = 〈1 + 3v + 2v2 + 4v3 + 4v4〉. • 00101 →〈3v+v2+v3〉 = 〈v+2v2+2v3〉 = 〈4v+3v2+3v3〉 = 〈2v+4v2+4v3〉 = ... = 〈2v2+4v3+4v4〉. • ... • 11000 →〈1+v+v2+v3〉 = 〈1+2v+2v2+2v3〉 = 〈3+3v+3v2+3v3〉 = ... = 〈4+3v+3v2+3v3+4v4〉. the ideals with 5 elements: • 10000 → 〈4 + v4〉 = 〈3 + 2v4〉 = 〈2 + 3v4〉 = 〈1 + 4v4〉. • 01000 → 〈v+v2 +v3 +v4〉 = 〈2v+2v2 +2v3 +2v4〉 = 〈3v+3v2 +3v3 +3v4〉 = 〈4v+4v2 +4v3 +4v4〉. • 00100 → 〈3v+4v2 +2v3 +v4〉 = 〈v+3v2 +4v3 +2v4〉 = 〈4v+2v2 +v3 +3v4〉 = 〈2v+v2 +3v3 +4v4〉. • 00010 → 〈2v+4v2 +3v3 +v4〉 = 〈4v+3v2 +v3 +2v4〉 = 〈v+2v2 +4v3 +3v4〉 = 〈3v+v2 +2v3 +4v4〉. • 00001 → 〈4v+v2 +4v3 +v4〉 = 〈3v+2v2 +3v3 +2v4〉 = 〈2v+3v2 +2v3 +3v4〉 = 〈v+4v2 +v3 +4v4〉. and the zero ideal: • 00000 → 〈0〉. let r be a ring and a ∈ r. if a is nonzero then its hamming weight denoted by w(a) equals to 1 otherwise it is equal to 0. this is generalized to an n-tuple such that if a = (a1,a2, . . . ,an) ∈ rn, then the hamming weight of a is defined by w(a) = ∑n i=1 w(ai). the hamming distance between two n-tuples is d(x,y) = w(x−y) where x,y ∈ rn. it is well known that the hamming distance is a metric on rn. it is possible to characterize the unit elements of rp and further give the number of elements in an ideal by considering the definition of φ together with its properties. lemma 2.3. suppose that i = 〈α〉 where α = a0 + a1v + . . . + ap−1vp−1 ∈ rp. |i| = p ∑p−1 i=0 w(α(i)). especially, if α(i) 6= 0 for all i, then α is a unit in rp and vice versa. since the map φ is a ring isomorphism, the inverse map of φ denoted by φ−1 : zpp → rp exists. in the following example we present the inverse map explicitly: example 2.4. the inverse map is defined by φ−1 : z55 → r5 (k,l,m,n,t) → k+(4l+2m+3n+t)v+(4l+m+n+4t)v2 +(4l+3m+2n+t)v3 +4(k+l+m+n+t)v4. definition 2.5. (gray weight) let α = a0 + a1v + ... + ap−1vp−1 ∈ rp. then wg(α) = w(φ(α)) (2) is called the gray weight of α. the gray distance between two elements α and β of rp is described by dg(α,β) = w(φ(α)−φ(β)) which also happens to be a linear distance preserving map from (rnp ,dg) to (z pn p ,d). example 2.6. let p = 7. if α = 1 + v + 5v2 + 5v3 and β = 6v + 4v2 + 5v3, then wg(α) = w(φ(1 + v + 5v2 + 5v3)) = w(1,5,0,2,6,0,0) = 4 and wg(β) = w(φ(6v + 4v2 + 5v3)) = w(0,1,5,0,2,6,0) = 4 hence dg(α,β) = w(φ(α) − φ(β)) = w((1,5,0,2,6,0,0) − (0,1,5,0,2,6,0)) = w((1,4,2,2,4,1,0)) = w(1 + 2v + v2) = 6. definition 2.7. let a = (a1,a2, ..,ap) ∈ zpp. then, supp(a) = {i|ai 6= 0}⊆{1,2, ...,p}. 3 cyclic and constacyclic codes over a non-chain ring we can easily check that: • if supp(φ(α)) = supp(φ(β)) then wg(α) = wg(β) where α,β ∈ rp. • assume that 〈α〉 and 〈β〉 are two ideals in rp. then, supp(φ(α)) = supp(φ(β)) if and only if 〈α〉 = 〈β〉. therefore, rp is a principal ideal ring, that is, all ideals in rp are generated by a single element of rp similar to the special case p = 3 [4] . theorem 2.8. if i = 〈α1,α2, . . . ,αs〉 is a finitely generated ideal of rp, then i = 〈β〉 for some β ∈ rp where supp(φ(β)) = ⋃s i=1 supp(φ(αi)). example 2.9. let i = 〈α1,α2〉 where α1 = 3v + v2 + v3 and α2 = 1 + 3v + 3v2 + 3v3 + 2v4 ∈ r5. since supp(φ(α1)) = supp(φ(3v + v 2 + v3)) = supp((0,0,3,0,2)) = {3,5} and supp(φ(α2)) = supp(φ(1 + 3v + 3v 2 + 3v3 + 2v4)) = supp((1,2,0,0,0)) = {1,2} supp(φ(β)) = 2⋃ i=1 supp(φ(αi)) = {1,2,3,5}, then β can be selected as 4 + 2v + 3v2 + 3v3 + 2v4 which generates a maximal ideal in r5. the units and the elements which generate the maximal ideals in rp can be classified by means of their gray images: lemma 2.10. let α ∈ rp. the follows hold: i) supp(φ(α)) = {1, ...,p} if and only if α is a unit. hence, rp has exactly (p−1)p units . ii) suppose i = 〈α〉. then, | supp(φ(α))| = p−1, if and only if i is maximal. 3. linear codes over rp a minimal generating set is comprised for all linear codes by a set of linearly independent and spanning vectors called basis for codes over fields. however, in the case for codes over rings, this is a challenging problem and in most cases impossible since we do not have basis in general for modules. in [2] and in [3], authors gave a basis or a minimal spanning set for the codes of even length over z2 + uz2 and z2 + uz2 + u2z2 + · · ·+ uk−1z2, respectively. these are all chain rings, that is, the set of all ideals is a chain under set-theoretic inclusion. since rp is not a chain ring, we can not get a generating matrix, easily. to overcome this problem in linear code case some special definitions (modular dependence) and cases of codes over rings are presented in [12] and in [15]. here, based on the gray image of the code, the generator matrix of the image code is presented and some results are obtained: theorem 3.1. assume that the set {g1,g2, . . . ,gk}⊂ rnp is a generating set of a linear code c over rp of length n where gi = (gi1,gi2, . . . ,gin). then, the matrix 4 a. bayram, i. siap φ(g) =   φ(g11) φ(g12) · · · φ(g1n) φ(vg11) φ(vg12) · · · φ(vg1n) φ(v2g11) φ(v 2g12) · · · φ(v2g1n) ... ... ... φ(vp−1g11) φ(v p−1g12) · · · φ(vp−1g1n) ... ... ... φ(gk1) φ(gk2) · · · φ(gkn) φ(vgk1) φ(vgk2) · · · φ(vgkn) φ(v2gk1) φ(v 2gk2) · · · φ(v2gkn) ... ... ... φ(vp−1gk1) φ(v p−1gk2) · · · φ(vp−1gkn   generates φ(c). example 3.2. let p = 5 and suppose that g = [ 2v + 4v2 + 3v3 + v4 0 0 4v + v2 + 4v3 + v4 ] is a generator matrix of c of length 2 over r5 = z5[v]/〈v5 −v〉. then φ(g) =   00000 00000 00000 00000 00000 00000 00000 00000 00000 00004 00000 00000 00000 00000 00000 00000 00040 00000 00000 00000   . hence φ(g) is a generator matrix of φ(c), with length 10, dimension 2 and size 52 = 25. another simple and compact way to represent the structure of a generator matrix of φ(g) is given below. let α = g0 + g1v + ... + gp−1vp−1 ∈ rp and α(i) = g0 + g1i + ... + gp−1ip−1 (mod p). φ(α) = φ(g0 + g1v + ... + gp−1v p−1) = (α(0),α(1), ...,α(p−1)). alternatively, after some row operations the generator matrix is then equivalent to a block matrix with blocks (gij)p×p = diag(αij(0),αij(1), . . . ,αij(p−1)). as mentioned above, we again emphasize that it is a difficult problem to determine the minimal independent sets that generate a linear code over rp in general due to the fact that rp is not a chain ring. however, one can adopt a similar approach as presented in both [12] and [15] to capture the size of linear codes over r for some special cases. definition 3.3. a set {g1,g2, . . . ,gk} ⊂ rnp is called a minimal independent generating set for a code c, if {φ(g1),φ(vg1), . . . ,φ(vp−1g1),φ(g2),φ(vg2), . . . ,φ(vp−1g2), . . . ,φ(gk),φ(vgk), . . . ,φ(vp−1gk)}⊂ zpnp is a zp-linearly independent set. 5 cyclic and constacyclic codes over a non-chain ring now, having this definition at hand one can determine the size of a code with a generating set which is zp-linearly independent: lemma 3.4. if c = 〈{g1,g2, . . . ,gk}〉 where the set {g1,g2, . . . ,gk} ⊂ rnp is a minimal independent generating set, then |c| = ppk. 3.1. the dual code in this subsection, an inner product which is introduced as below helps us to construct the dual code of a linear code where the inner product is obtained with the gray image. we also show the proof of a lemma which relates dual of the code and its gray image: let g = [g1,g2, . . . ,gn],h = [h1,h2, . . . ,hn] ∈ rnp , gi = gi1 + gi2v + . . . + gipvp−1,hi = hi1 + hi2v + . . .+hipv p−1 , gi(j) = gi1 +gi2j + . . .+gipjp−1 (mod p) and hi(j) = hi1 +hi2j + . . .+hipjp−1 (mod p). 〈g, h〉φ = n∑ i=1 p−1∑ j=1 (gi(j)hi(j)) . if c is a linear code of length n over the ring rp, then the dual code is defined by c⊥ = {h ∈ rnp |〈g,h〉φ = 0 for all g ∈ c}. (3) lemma 3.5. φ(c)⊥ = φ(c⊥). proof. the proof follows from the definitions: if h ∈ c⊥, then, 〈g,h〉φ = 0 for all g ∈ c. this implies that 〈φ(g),φ(h)〉 = 0 for all φ(g). hence, φ(h) ∈ (φ(c))⊥ . thus, φ(c⊥) ⊂ φ(c)⊥. the reverse follows directly by reversing the steps. example 3.6. let g = [ 2 + 4v4 3 + 3v + v2 + 2v3 + 3v4 2v3 + 2v4 3 + 4v + 3v4 ] be a generator matrix of a linear code c over r5 = z5[v]/〈v5−v〉, then the image of this code is generated by φ(g) =   φ(2 + 4v4) φ(3 + 3v + v2 + 2v3 + 3v4) φ(v(2 + 4v4)) φ(v(3 + 3v + v2 + 2v3 + 3v4)) φ(v2(2 + 4v4)) φ(v2(3 + 3v + v2 + 2v3 + 3v4)) φ(v3(2 + 4v4)) φ(v3(3 + 3v + v2 + 2v3 + 3v4)) φ(v4(2 + 4v4)) φ(v4(3 + 3v + v2 + 2v3 + 3v4)) φ(2v3 + 2v4) φ(3 + 4v + 3v4) φ(v(2v3 + 2v4)) φ(v(3 + 4v + 3v4)) φ(v2(2v3 + 2v4)) φ(v2(3 + 4v + 3v4)) φ(v3(2v3 + 2v4)) φ(v3(3 + 4v + 3v4)) φ(v4(2v3 + 2v4)) φ(v4(3 + 4v + 3v4))   =   2 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2   6 a. bayram, i. siap ∼   1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1   over z5, then |c| = 59. let h = [ 0 0 0 1 0 0 0 0 3 0 ] . is a parity check matrix of φ(c). hence ∣∣φ(c)⊥∣∣ = 5. conversely, for all h = [h] ∈ c⊥ such that h = [ h1 + h2v + h3v 2 + h4v 3 + h5v 4 h ′ 1 + h ′ 2v + h ′ 3v 2 + h ′ 4v 3 + h ′ 5v 4 ] then c⊥ = { h = [ (2v + 4v2 + 3v3 + v4)h5 (v + 2v 2 + 4v3 + 3v4)h5 ] ,h5 ∈ z5} hence,∣∣∣c⊥∣∣∣ = 5. therefore, φ(c⊥) = φ(c)⊥. 3.2. macwilliams identity for codes over rp the macwilliams identity is one of the prominent results in coding theory, which supplies the relationship between the weight enumerator of a linear code and that of its dual code [10]. the distribution of weights for a linear code is crucial to its performance analysis such as, linear programming bound, error correcting capabilities, the extremal weight enumerators related to the dual codes, etc. in this section, we state several lemmas and the main theorem. we also illustrate the theorem with a moderate example. in this work, we assume that the character χ is described by χ(a) = ξa(0)+a(1)+a(2)+...+a(p−1) where ξ = e2πi/p. the gray weight enumerator of a linear code c over rp is defined by w (x,y) = ∑ c∈c xpn−wg(c)ywg(c). this section is a generalization of section 3.2 in [4], so we will not give all proofs in detail here. therefore we present the statement of lemmas and the main theorem and state an example to show the result. lemma 3.7. 1. assume that i 6= {0} be an ideal of the ring rp. then,∑ a∈i χ(a) = 0. 2. for a ∈ rp, we have ∑ r∈rp χ(ar) = { pp, a = 0 0, a 6= 0. 7 cyclic and constacyclic codes over a non-chain ring 3. if β ∈ rp, then ∑ α∈rp χ(〈β,α〉)xp−wg(α)ywg(α) = (x + (p−1)y)p−wg(β) (x−y)wg(β) . the following well known result plays an important role in finalizing the proof of the main theorem: lemma 3.8. [10] if c and its dual c⊥ are linear codes over the ring rp with f̂ (u) = ∑ v∈rnp χ(〈u,v〉)f (v) , then ∑ v∈c⊥ f (v) = 1 |c| ∑ u∈c f̂ (u). by combining the lemmas above we get the main theorem that relates the gray weight enumerators of the code and its dual: theorem 3.9. suppose that c is a linear code over rp, then wc⊥ (x,y) = 1 |c| wc (x + (p−1)y,x−y) . example 3.10. assume that g = [ 2 + 3v4 0 0 3v + 3v2 + 3v3 + 3v4 ] generates a linear code c over r5. then, its gray weight enumerator is wc(x,y) = x 10 + 8x9y + 16x8y2. therefore, by applying the necessary change of variables in the main theorem, we obtain wc⊥(x,y) = x 10 + 32x9y + 448x8y2 + 3584x7y3 + 17920x6y4 + 57344x5y5 + 114688x4y6 + 131072x3y7 +65536x2y8. 4. cyclic codes over rp a very significant and well know class of linear codes is the class of cyclic codes which plays a crucial role in coding theory due to their easy implementation. since cyclic codes can be described as ideals in some polynomial rings, they have considerable inherent algebraic structure. in this part we consider the algebraic structure of cyclic codes over the ring rp. we also study the structure of their duals. definition 4.1. let σ be a cyclic right shift on the entries of an n-tuple in rn such that σ(c0,c1, . . . , cn−1) = (cn−1,c0, . . . ,cn−2). for a linear code c, if σ(c) = c, then c is called a cyclic code of length n. 8 a. bayram, i. siap after associating a polynomial c(x) = c0 + c1x + · · ·+ cn−1xn−1 to a codeword c = (c0,c1, . . . ,cn−1) ∈ c, if c is a cyclic code then, c becomes an ideal of the quotient ring r[x]/〈xn −1〉. let r(n,p) = rp[x]/〈xn −1〉. since rp ∼= zpp, then rp[x]/〈xn −1〉∼= zp[x]/〈xn −1〉×zp[x]/〈xn −1〉× . . .×zp[x]/〈xn −1〉. let l(n,p) = zp[x]/〈xn −1〉×zp[x]/〈xn −1〉× . . .×zp[x]/〈xn −1〉. now as a natural extension of φ ,we can get an isomorphism between the rings r(n,p) and l(n,p). we define a projection map πi : z p p → zp, such that πi((a1,a2, . . . ap)) = ai for 1 ≤ i ≤ p. then, we identify φ :r(n,p) → l(n,p) φ( n∑ i=0 aix i) = ( n∑ i=0 π1(φ(ai))x i, n∑ i=0 π2(φ(ai))x i, . . . , n∑ i=0 πp(φ(ai))x i). example 4.2. let f(x) = (1 + v2)x3 + (1 + 3v + v3)x2 + (3v2 + v4)x + 1 in r(4,5). then, φ(f(x)) = (x3 + x2 + 1,2x3 + 4x + 2,3x + 1,2x2 + 3x + 1,2x3 + 2x2 + 4x + 1). it is easy to get the structure of r[x]/〈xn −1〉 since this map is an isomorphism. r(n,p) is a principal ideal ring. we can determine the generator of ideals as follows: suppose that i = 〈f1(x),f2(x), . . . ,fs(x)〉 is a finitely generated ideal of r(n,p) where fi(x) = ∑n j=0 fijx j. then, for i = 1,2, . . . , p let gi = gcd1≤j≤s(πi(φ(fj))),xn −1). hence, i = 〈g(x)〉 where g(x) = φ−1((g1(x),g2(x), . . . ,gp(x))). example 4.3. let i = 〈f1(x) = (1+v+2v2+x+(1+2v+v2)x2,f2(x) = 2+2v2+(1+v+v2)x+(v+2v2)x2〉 be an ideal of r(4,3). then, φ(f1(x)) = ((2+x)2,(2+x)2,2+x) and φ(f2(x)) = (2+x,1,(2+x)2). next, g1 = gcd((2+x) 2,2+x,x3−1) = 2+x, g2 = gcd((2+x)2,1,x3−1) = 1, g3 = gcd(2+x,(2+x)2,x3−1) = 2 + x. so we have φ(i) = 〈(2 + x,1,2 + x)〉. therefore, i = φ−1(φ(i)) = 〈φ−1(2 + x,1,2 + x)〉 = 〈2 + v + v2 + (1 + v + v2)x〉. the following lemma can be observed as a straightforward result of the above statements and the example: lemma 4.4. if c = 〈g(x)〉 is a cyclic code of length n over rp and φ(g(x)) = (g1,g2, . . . , gp) with deg(gcd(gi,x n −1)) = n−ki for 1 ≤ i ≤ p, then |c| = p ∑p i=1 ki. 4.1. the dual of cyclic codes in this subsection, we study the algebraic structure of the dual of a cyclic code over rp. let c = 〈g(x)〉 be a cyclic code of length n over rp. assume that, φ(c) = j = 〈(g1(x),g2(x), . . . gp(x))〉 where gi = πi(φ(g(x))). the dual of j is the cyclic code j⊥ = 〈(h1r(x),h2r(x), . . . , hpr (x))〉, where hi(x) = (xn − 1)/(gcd(xn − 1,gi) and hi r (x) is the reciprocal polynomial of hi(x). hence, c⊥ = 〈φ−1(h1 r (x),h2 r (x), . . . , hp r (x))〉. 9 cyclic and constacyclic codes over a non-chain ring example 4.5. let i = 〈f(x) = (3v4+3v3+4v)x3+(4v4+4v3+2v2+1)x2+(2v4+4v2+3)x+(v3+4v+2) > be an ideal of r(5,4). then, φ(f(x)) = (x2+3x+2,x2+4x+2,x+3,x3+x2+x+1,x3+3x2+4x+2). next, g1 = gcd(x 2+3x+2,x4−1) = x2+3x+2, g2 = gcd(x2+4x+2,x4−1) = x2+4x+2, g3 = gcd(x+3,x4−1) = x+3, g4 = gcd(x3+x2+x+1,x4−1) = x3+x2+x+1, g5 = gcd(x3+3x2+4x+2,x4−1) = x3+3x2+4x+2. thus, |〈i〉| = 59. c⊥ = 〈φ−1(h1r(x),h2r(x),h3r(x),h4r(x),h5r(x))〉 = 〈φ−1((2x2 + 2x + 1,3x2 + 4x + 1,x3+x2+x+1,4x+1,2x+1))〉 = 〈(4v4+3v3+v2+2v)x3+(4v4+3v2+4v+2)x2+(4v3+4v2+2v+2)x+1〉. therefore, ∣∣〈c⊥〉∣∣ = 511. 5. constacyclic codes over rp in this section, we study constacyclic codes over rp. definition 5.1. let α = a0 +a1v+...+ap−1vp−1 be a unit element of rp and c be a linear code of length n over rp. if for all c = (c0,c1, . . . ,cn−1) ∈ c and a unit in rp we have (αcn−1,c0,c1, . . . ,cn−2) ∈ c, then c is called an α-constacyclic code or shortly constacyclic code. similar to the cyclic codes case if we associate each codeword c = (c0,c1, . . . ,cn−1) ∈ c with a polynomial c = c0 + c1x + · · · + cn−1xn−1 ∈ r[x], then c can be viewed as an ideal in s(n,p) = rp[x]/〈xn −α〉. by applying the chinese remainder theorem, we have the following result: let s(n,p) = rp[x]/〈xn −α〉. since rp ∼= zpp, then s ∼= zp[x]/〈xn −α(0)〉×zp[x]/〈xn −α(1)〉× . . .×zp[x]/〈xn −α(p−1)〉. for example, let i = 〈f(x) = (4+3v+v3 +v4 +(1+3v+v2 +4v3 +4v4)x+(3v+4v2 +2v3 +2v4)x2 +(2v+ v2+3v3+4v4)x3)〉, be an ideal of r(4,5) then φ(f(x)) = (4+x,4+3x+x2,4+2x+x3,1+x+x2,1+4x+x2). since α is a unit element of rp, then α(i) 6= 0 for all 0 ≤ i ≤ p−1, the number of constacyclic codes over rp can be obtain as follows: theorem 5.2. the number of α-constacyclic codes of length n is equal to ∏p−1 i=0 δ(i) where δ(i) = { σn, i = 1, η(n,i), i = 2, ...,(p−1). σn and η(n,i) are equal to the number of cyclic and α(i)−constacyclic codes of length n over zp, respectively. proof. since α is a unit element of rp, the gray image consists of non zero elements of zp. if the gray image contains 1 as a component then the projection code corresponding to that particular component is cyclic which has a generator polynomial as a divisor of xn − 1 over zp. in addition, if gray image contains non zero elements different from 1, call it α(i), then the projection is a α(i)−constacyclic code of length n over zp. therefore, if σn and η(n,i) are equal to the number of cyclic and α(i)−constacyclic codes of length n over zp, respectively, then the number of α-constacyclic codes of length n is equal to∏p−1 i=0 δ(i). example 5.3. let φ(4 + v + 2v3) = (4,2,2,1,1) ∈ z5. since the number of 2−constacyclic , negacyclic and cyclic codes over z5 are δ(2) = η(4,2) = 2, δ(4) = η(4,4) = 22 and δ(1) = σ4 = 8, respectively then the number of all (4 +v + 2v3)−constacyclic code length 4 over r5 is equal to 4.2.2.8.8 = 22+1+1+3+3 = 210. the algebraic structure of a dual constacyclic code can be obtained as follows: suppose c = 〈g(x)〉 is an α-constacyclic code of length n over rp. let gi = πi(φ(g(x))), then φ(c) = 〈(g1(x),g2(x), . . . gp(x))〉. the dual of c is an α-constacyclic code which is equal to 〈φ−1(h1(x),h2(x), . . . hp(x))〉, where hi(x) = (xn −α(i))/(gi(x)). 10 a. bayram, i. siap example 5.4. let c be a (1+v+4v2 +3v3)−constacyclic code over r7 generated by f(x) = (4v6 +4v5 + 6v3 +v)x3 +(5v6 +v5 +6v4 +v3 +6v2 +v+1)x+(v6 +5v5 +4v3 +v2 +5v+3) which is an ideal of r(3,7). since φ(1 + v + 4v2 + 3v3) = (1,2,1,2,2,5,1) and φ(f(x)) = (x + 3,x3 + 5,0,0,x3 + 5,x3 + 2,x + 5) then g1 = x + 3 and h1(x) = (x3 −1)/(x + 3) = (x2 + 4x + 2), g2 = x3 + 5 and h2(x) = (x3 −2)/(x3 + 5) = 1, g3 = 0 and h3(x) = 0, g4 = 0 and h4(x) = 0, g5 = x3 + 5 and h5(x) = (x3 −2)/(x3 + 5) = 1, g6 = x3 + 5 and h6(x) = (x3 − 5)/(x3 + 2) = 1, g7 = x + 5 and h7(x) = (x3 − 1)/(x + 5) = x2 + 2x + 4. so, φ−1(h1(x),h2(x),h3(x),h4(x),h5(x),h6(x),h7(x)) = φ −1(x2 + 4x + 2,1,0,0,1,1,x2 + 2x + 4) = (5v6 + v5 +6v4 +v3 +6v2 +v+1)x2 +(v6 +2v5 +5v4 +2v3 +5v2 +2v+4)x+(5v6 +v5 +3v4 +3v3 +3v2 +5v+2) therefore, c⊥ = 〈(5v6 + v5 + 6v4 + v3 + 6v2 + v + 1)x2 + (v6 + 2v5 + 5v4 + 2v3 + 5v2 + 2v + 4)x + (5v6 + v5 + 3v4 + 3v3 + 3v2 + 5v + 2)〉. 6. conclusions we have explored further a new family of codes over a special non-chain ring by generalizing some results in [4]. in general, non-chain rings are very complicated to be studied. here, by introducing a gray map the problem has been resolved. linear, cyclic and constacyclic codes have been introduced. a macwilliams type identity is also proven. this results can be easily generalized to codes over the ring fq[v]/〈vq −v〉 where fq is a field with q elements. acknowledgment: the preliminary results of this paper are presented in proceedings of the 2013 international conference on computational and mathematical methods in science and engineeringcmmse 2013, june 24-27 2013, almeria, spain. references [1] t. abualrub, i. siap, on the construction of cyclic codes over the ring z2 + uz2, wseas trans. on math., 5(6), 750-756, 2006. 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[15] b. yildiz, s. karadeniz, cyclic codes over f2 +uf2 +vf2 +uvf2, des. codes crypt., 58(3), 221-234, 2011. [16] s.-x. zhu, y. wang, m.-j. shi, cyclic codes over f2 + vf2, isit’09 proceedings of the 2009 ieee international conference on symposium on information theory, 3, 1719-1722, 2009. 12 introduction preliminaries linear codes over rp cyclic codes over rp constacyclic codes over rp conclusions references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.940192 j. algebra comb. discrete appl. 8(2) • 139–149 received: 19 july 2020 accepted: 4 march 2021 journal of algebra combinatorics discrete structures and applications algebraic methods in difference sets and bent functions research article pradipkumar h. keskar, priyanka kumari abstract: we provide some applications of a polynomial criterion for difference sets. these include counting the difference sets with specified parameters in terms of hilbert functions, in particular a count of bent functions. we also consider the question about the bentness of certain boolean functions introduced by carlet when the c-condition introduced by him doesn’t hold. 2010 msc: 05b10, 6e30,13p25, 94c10, 11t71 keywords: hilbert functions, c-condition, flat, difference set, bent functions 1. introduction in [5], a criterion was developed for subsets of a finite abelian group to be difference sets with specified parameters. the criterion was in terms of polynomial conditions on their characteristic vectors. this opens up the possibilities of algebro-geometric methods in studying difference sets. recall that for a finite group g of order v and integers k,λ, a subset d of g is called a difference set of g with parameters (v,k,λ), or (v,k,λ)-difference set of g, if |d| = k and |{(g1,g2) ∈ d × d : g1g−12 = g}| = λ for any non-identity element g ∈ g. this paper discusses its applications in the study of difference sets and bent functions, which are cryptographically significant. recall that for an even positive integer t > 2, a boolean function of t variables is a bent function if and only if its support is a difference set in (z/2z)t with parameters (v,k,λ) where v = 2t,k = 2(t−1)±2(t−2)/2 and λ = 2(t−2)±2(t−2)/2(where signs are chosen consistently). in section 2, we show that the number of difference sets in an abelian group is given by the affine hilbert function of a certain ideal in a polynomial ring. as a special case, the same holds for the number of bent functions of an even dimension. it may be pointed out that the count of bent functions is an important unresolved issue and even the known bounds for it are quite weak, see [9] for the details. it is hoped that computer algebra software like macaulay2, singular, which facilitate computation of hilbert functions, will play a role in enhancing our knowledge in this direction. pradipkumar h. keskar, priyanka kumari (corresponding author); department of mathematics, birla intitute of technology and science, pilani (pilani campus), pilani 333031, india (email: keskar@pilani.bits-pilani.ac.in, priyanka.kumari@pilani.bits-pilani.ac.in). 139 https://orcid.org/0000-0001-5463-4189 https://orcid.org/0000-0003-4453-8518 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 in section 3, the criterion of [5] is reformulated to characterize those exchanges of elements of a difference set which again lead to a difference set. the formulation is in terms of certain values of a polynomial function. in subsequent sections, this criterion is applied to establish non-bentness of an infinite familiy of certain functions introduced by carlet in [2], which we now introduce. let f2 = {0, 1} be the field with two elements and let m be a positive integer and t = 2m. for any x = (x1, . . . ,xm),y = (y1, . . . ,ym) ∈ fm2 , let x · y = ∑m i=1 xiyi ∈ f2. let l be an f2-subspace of f m 2 , l⊥ = {y ∈ fm2 : x ·y = 0 for all x ∈ l} be the orthogonal complement of l and let 1l⊥ : fm2 → f2 be defined by 1l⊥(x) = 1 if x ∈ l⊥ and 1l⊥(x) = 0 otherwise. for a permutation π of fm2 , consider the function f(π,l) : ft2 = f m 2 ×fm2 → f2 be defined by f(π,l)(x,y) = x ·π(y) + 1l⊥(x). carlet found a class of bent functions called c-class of bent functions. for this purpose, he introduced c-condition on (π,l) thus : (π,l) satisfies the c-condition if and only if for any a ∈ fm2 , π−1(a + l) is a flat (i.e. an affine subspace) in fm2 . he then showed that c-condition is sufficient for bentness of f(π,l), that is, if (π,l) satisfies c-condition then f(π,l) is a bent function. the class of bent functions obtained in this manner is called the c-class of bent functions. the c-condition was further explored in [7]. but it is not known that failure of c-condition by (π,l) implies non-bentness of f(π,l). thus we need another machinary to prove non-bentness of f(π,l). as supports of bent functions are difference sets, the results of [5] become relevant. in this paper, we consider π defined by π(x1, . . . ,xm) = ((x1 + p(x2, . . . ,xm)),x2, . . . ,xm) for all (x1, . . . ,xm) ∈ fm2 , where p(x2, . . . ,xm) ∈ f2[x2, . . . ,xm]. thus π is induced by an important type of polynomial automorphisms of fm2 , called elementary automorphisms, a generating set of the so called tame automorphism group, see [10]. for several classes of (π,l), we decide when the c-condition is satisfied and in some examples where it is not satisfied, we conclude the non-bentness of f(π,l). 2. counting the difference sets let g = ∏t l=1 ( z nlz ) be an abelian group and let v = |g|. for any il ∈ znlz ; 1 ≤ l ≤ t, let i∗l ∈ {0, 1, . . . ,nl − 1} be such that il = i ∗ l + nlz. for any subset t of g, α = (αg : g ∈ g) ∈ c v is called the point representation or characteristic vector of t if αg = 1 for g ∈ t and αg = 0 otherwise. let x1, . . . ,xt be independent variables over c and let ag,g ∈ g be v independent variables over c[x1, . . . ,xt]. letting x = (x1, . . . ,xt) and a = (ag : g ∈ g), we define ψ = ψ(x,a) ∈ c[x,a] by ψ =   ∑ (i1,...,it)∈g ai1···itx i∗1 1 · · ·x i∗t t     ∑ (i1,...,it)∈g ai1···itx n1−i∗1 1 · · ·x nt−i∗t t   −λ   ∑ (i1,...,it)∈g x i∗1 1 · · ·x i∗t t  − (k −λ). further let u = {ξ = (ξ1, . . . ,ξt) ∈ ct : ξnll = 1 for all 1 ≤ l ≤ t} and pg(a) = a 2 g −ag ∈ c[a] for all g ∈ g. in theorem 3.2 of [5], we have given a polynomial criterion for (v,k,λ) difference set in g as follows : theorem 2.1. for α = (αg : g ∈ g) ∈ cv, α is a point representation of a (v,k,λ) difference set in g if and only if α satisfies the equations pg(a) = 0 for all g ∈ g, and ψ(ξ,a) = 0 for all ξ = (ξ1, . . . ,ξt) ∈ u. 140 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 as a consequence, (v,k,λ) difference sets in g are in one-to one correspondence with the points of the zero-dimensional affine algebraic set v (ψ(ξ,a),pg(a) : ξ ∈ u,g ∈ g) = {α ∈ cv : ψ(ξ,α) = 0 for all ξ ∈ u,pg(α) = 0 for all g ∈ g}. this brings us to the concept of affine hilbert function of an ideal in a polynomial ring. to define this, let r = k[z1, . . . ,zr] be the polynomial ring in r variables over a field k and for a non-negative integer s, let r≤s = {f(z1, . . . ,zr) ∈ r : deg f ≤ s}. for an ideal i of r, let i≤s = i ∩r≤s. note that r≤s is a finite dimensional k-vector space and i≤s is its subspace. we define the affine hilbert function of i as the integer valued function ahfi of non-negative integers such that ahfi(s) = dimk ( r≤s i≤s ) . it turns out that there is a polynomial, called affine hibert polynomial of i and denoted by ahpi, whose values coincide with the values of the affine hilbert function of i for large integer values of s. letting i(v ) = {f(z1, . . . ,zr) ∈ r : f(z1, . . . ,zr) = 0 for all (z1, . . . ,zr) ∈ v} for v ⊂ kr, exercise 11 on p. 475 of [3] shows that if |v | is finite then ahpi(v ) is the constant polynomial |v |. we now apply this to v = v (ψ(ξ,a),pg(a) : ξ ∈ u,g ∈ g) to get the following corollary 2.2. the number of (v,k,λ)-difference sets in the group g = ∏t l=1 ( z nlz ) is given by ahpj(s) for any s ∈ c where j is the ideal of c[a] generated by {ψ(ξ,a),pg(a) : ξ ∈ u,g ∈ g}. proof. in the light of the above discussion, the proof will be complete if we show that i(v (j)) = j. by strong hilbert nullstellensatz (theorem 6 on p. 176 of [3]), i(v (j)) = √ j. this reduces our work to showing that √ j = j. this follows from theorem 8.14 on p. 343 of [1], since pg(a) is square-free and c is perfect. remark 2.3. in stead of the affine hilbert polynomial of j, we could use the hilbert polynomial of the homogenization jh of j as well. in some computational software, hilbert polynomial of a homogeneous ideal in a graded ring is easier to deal with, hence we also give an alternate formulation of the above corollary. let b be an indeterminate over c[a] and let c[a]h = c[a][b]. the homogenization jh is the ideal of c[a]h generated by {fh : f ∈ j}, where for any f ∈ c[a], fh ∈ c[a]h is defined by fh(a,b) = bdeg(f)f ( ag b : g ∈ g ) . to obtain a finite generating set of jh, note that by theorem 4 on p. 388 of [3], if s is a grob̈ner basis of j then {fh : f ∈ s} is the grob̈ner basis of jh. to define hilbert function hfjh of j h, for any non-negative integer s, consider the k-vector spaces c[a]hs = {f ∈ r h : f = 0 or f is homogeneous of degree s} and jhs = j h∩c[a]hs. we define hfjh(s) = dimk ( c[a]hs jhs ) . by theorem 12 on p. 464 of [3], ahfj(s) = hfjh(s) for all non-negative integers s. this allows us to replace the affine hilbert function ahfj by the hilbert function hfjh(s). remark 2.4. corollary 2.2 also gives a count of all bent functions of t variables. since the set of all bent functions with supports of size 2(t−1) + 2(t−2)/2 and the set of those with supports of size 2(t−1) −2(t−2)/2 are disjoint of same cardinality, the count of the bent functions in t variables for an even t is given by 2hfjh(s) where (v,k,λ) = ( 2t, 2(t−1) + 2(t−2)/2, 2(t−2) + 2(t−2)/2 ) . 3. a difference set criterion theorem 2.1 imposes some restrictions on exchanges of elements of a (v,k1,λ1) difference set to get another (v,k2,λ2) difference set. by introducing a complex valued polynomial ∆(d1,d2)(x1, . . . ,xt), we make these restrictions explicit, in terms of its certain values. alternately, theorem 2.1 can also be phrased in the language of group characters, following theorem 11.18 on p. 224 of [8]. let g = ∏t l=1 ( z nlz ) be an abelian group. for any il ∈ znlz ; 0 ≤ l ≤ t, let i ∗ l ∈ {0, 1, . . . ,nl − 1} be such that il = i∗l + nlz. 141 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 for any t ⊂ g, let ρg(t)(x1, . . . ,xt) = ∑ (i1,...,it)∈t x i∗1 1 · · ·x i∗t t ∈ c[x1, . . . ,xt]. let u = {(ξ1, . . . ,ξt) ∈ ct : ξnll = 1 for all 1 ≤ l ≤ t}. for any (ξ1, . . . ,ξt) ∈ u and (i1, . . . , it) ∈ g, we define ξi11 · · ·ξ it t = ξ i∗1 1 · · ·ξ i∗t t . now let v = |g| and k,λ be non-negative integers. for any d ⊂ g, let d(−1) = {−d : d ∈ g}. in (3.2∗) of [5] it was proved that d is a (v,k,λ) difference set in g if and only if ρg(d)(ξ1, . . . ,ξt)ρg(d (−1))(ξ1, . . . ,ξt) −λρg(g)(ξ1, . . . ,ξt) − (k −λ) = 0 for all (ξ1, . . . ,ξt) ∈ u. note that for any ξ ∈ c with |ξ| = 1, we have ξ−1 = ξ̄, the complex conjugate of ξ. it follows that ρg ( d(−1) ) (ξ1, . . . ,ξt) is the complex conjugate of ρg(d)(ξ1, . . . ,ξt) and hence we get d is a (v,k,λ) difference set in g if and only if |ρg(d)(ξ1, . . . ,ξt)|2 −λρg(g)(ξ1, . . . ,ξt) − (k −λ) = 0 for all (ξ1, . . . ,ξt) ∈ u. (1) now let us assume ni = 2 for all i = 1, . . . , t then ρg(d)(ξ1, . . . ,ξt) ∈ r. further ρg(g)(ξ1, . . . ,ξt) = 0 if ξi 6= 1 for some i ∈{1, . . . , t}, while ρg(d)(ξ1, . . . ,ξt) = k and ρg(g)(ξ1, . . . ,ξt) = v if ξi = 1 for all i ∈{1, . . . , t}. this has the following consequence : if ni = 2 for all i = 1, . . . , t then d is a (v,k,λ) difference set in g if and only if for any (ξ1, . . . ,ξt) ∈ u (ρg(d)(ξ1, . . . ,ξt)) = { k if ξi = 1 for all i ∈{1, . . . , t} ± √ k −λ otherwise. (2) now suppose d1 is a (v,k1,λ1) difference set in g and d2 ⊂ g. let ∆(d1,d2)(x1, . . . ,xt) =ρg(d1 \d2)(x1, . . . ,xt) −ρg(d2 \d1)(x1, . . . ,xt). then we have ∆(d1,d2)(x1, . . . ,xt) = ρg(d1)(x1, . . . ,xt) −ρg(d2)(x1, . . . ,xt) and hence: if ni = 2 for all i = 1, . . . , t and d1 is a(v,k1,λ1) difference set in g then d2 is a (v,k2,λ2) difference set in g if and only if for any ξ = (ξ1, . . . ,ξt) ∈ u ∆(d1,d2)(ξ) ∈   {k1 −k2} if ξi = 1 for all i ∈{1, . . . , t}; { √ k1 −λ1 − √ k2 −λ2, √ k1 −λ1 + √ k2 −λ2} if ρg(d1)(ξ) = √ k1 −λ1; {− √ k1 −λ1 − √ k2 −λ2,− √ k1 −λ1 + √ k2 −λ2} if ρg(d1)(ξ) = − √ k1 −λ1. (3) 142 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 moreover, if (v,k1,λ1) = (v,k2,λ2) then d2 is a (v,k1,λ1) difference set in g if and only if for any ξ = (ξ1, . . . ,ξt) ∈ u ∆(d1,d2)(ξ) ∈   {0} if ξi = 1 for all i ∈{1, . . . , t}; {0, 2 √ k1 −λ1} if ρg(d1)(ξ) = √ k1 −λ1; {−2 √ k1 −λ1, 0} if ρg(d1)(ξ) = − √ k1 −λ1. (4) 4. the analysis of c-condition in the rest of the paper, we continue with the notation and terminology introduced in section 1. moreover we identify fr12 ×···× f ru 2 with f r1+···+ru 2 in a natural way. also for any integer u ≥ 0, we denote by 0u the element of fu2 whose all components are 0 and by 1u the element of f u 2 whose all components are 1. to search for examples when c-condition is not satisfied, we study the c-condition for (π,l). since for any x ∈ f2, x2 = x, by reducing all the exponents of variables mod 2, without loss of generality we can assume p(x2, . . . ,xm) = m−1∑ `=0 ∑ 2≤i1<i2<···<i`≤m αi1···i`xi1 · · ·xi` where αi1···i` ∈ f2 for all 2 ≤ i1 < i2 < · · · < i` ≤ m. to search for examples when c-condition is not satisfied, the search space is provided by the following: theorem 4.1. let m ≥ 2 and s ∈ [1,m] be integers. (a) let l = {(x1, . . . ,xs,0m−s) : xi ∈ f2, 1 ≤ i ≤ s} be a linear subspace of fm2 . then c-condition is satisfied by (π,l). (b) let l = {(0s,xs+1, . . . ,xm) : xi ∈ f2,s + 1 ≤ i ≤ m} be a linear subspace of fm2 . if αi1···i` = 0 for all (i1, . . . , i`) such that |{ij : ij > s}|≥ 2 then c-condition is satisfied by (π,l). (c) let l = {(0s,xs+1, . . . ,xm) : xi ∈ f2,s + 1 ≤ i ≤ m} be linear subspace of fm2 . if αi1···i` = 1 for some (i1, . . . , i`) such that |{ij : ij > s}|≥ 2 then c-condition is not satisfied by (π,l). moreover in (a) and (b), f(π,l) is a c-class bent function. proof. the proof is based on the following observation : a nonempty subset f ⊂ fm2 f is a flat ⇔ f − b is a subspace of fm2 for some b ∈ f ⇔ f − b is a subspace of fm2 for all b ∈ f. (5) we will apply this when f = π−1(a + l) and b = π−1(a) where a ∈ fm2 . (a) for any a = (a1, . . . ,am) ∈ fm2 we see that a + l = a∗ + l where a∗j = aj if j > s and a∗j = 0 otherwise. thus we can assume, without loss of generality, that aj = 0 for j ≤ s. therefore a + l = {(x1, . . . ,xs,as+1, . . . ,am) : x1, . . . ,xs ∈ f2}. it is enough to show that π−1(a + l) −π−1(a) is a subspace of fm2 for all a = (0s,as+1, . . . ,am) ∈ fm2 . 143 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 clearly, π−1(a + l)−π−1(a) ⊂ l and they have same cardinality. since |l| is finite, we get π−1(a + l)− π−1(a) = l. thus π−1(a + l) −π−1(a) is a subspace for all a = (0s,as+1, . . . ,am) ∈ fm2 . hence (π,l) satisfies the c-condition. (b) for any a = (a1, . . . ,am) ∈ fm2 we see that a + l = a∗ + l where a∗j = aj if j ≤ s and a∗j = 0 otherwise. thus we can assume, without loss of generality, that aj = 0 for j > s. therefore a + l = {(a1, . . . ,as,xs+1, . . . ,xm) : xs+1, . . . ,xm ∈ f2}. consequently it is enough to show that π−1(a + l) −π−1(a) is a subspace of fm2 for all a = (a1, . . . ,as,0m−s) ∈ fm2 . for any (i1, . . . , i`), if αi1···i` = 1 then ∏ i∈{ij:ij≤s}ai is a constant in f2, and |{ij : ij > s}|≤ 1. so for any a = (a1, . . . ,as,0m−s) ∈ fm2 , π−1(a + l) −π−1(a) = {(l(xs+1, . . . ,xm),0s−1,xs+1, . . . ,xm) : xs+1, . . . ,xm ∈ f2} where l(xs+1, . . . ,xm) ∈ f2[xs+1, . . . ,xm] is a polynomial of degree ≤ 1. now for any u,v ∈ π−1(a + l) −π−1(a) and α,β ∈ f2, where u = (l(xs+1, . . . ,xm),0s−1,xs+1, . . . ,xm) and v = (l(x∗s+1, . . . ,x ∗ m),0s−1,x ∗ s+1, . . . ,x ∗ m), αu + βv = α(l(xs+1, . . . ,xm),0s−1,xs+1, . . . ,xm) + β(l(x ∗ s+1, . . . ,x ∗ m),0s−1,x ∗ s+1, . . . ,x ∗ m) = (αl(xs+1, . . . ,xm) + βl(x ∗ s+1, . . . ,x ∗ m),0s−1,αxs+1 + βx ∗ s+1, . . . ,αxm + βx ∗ m) now π−1(a+0m)−π−1(a) = 0m. therefore l(xs+1, . . . ,xm) is a polynomial with no constant term. hence l(xs+1, . . . ,xm) is a linear transformation. then αu + βv = (l(αxs+1 + βx ∗ s+1, . . . ,αxm + βx ∗ m),0s−1,αxs+1 + βx ∗ s+1, . . . ,αxm + βx ∗ m), therefore αu + βv ∈ π−1(a + l) −π−1(a). as a consequence, π−1(a + l) −π−1(a) is a subspace of fm2 for all a = (a1, . . . ,as,0m−s) ∈ fm2 and hence (π,l) satisfy c-condition. (c) we classify nonzero terms of p(x2, . . . ,xm) in two types. type 1 : corresponding to (i1, . . . , i`) such that |{ij : ij > s}|≥ 2, type 2 : corresponding to (i1, . . . , i`) such that |{ij : ij > s}| < 2. let t1 = xi∗1 · · ·xi∗`∗ be minimal among all nonzero terms of p(x2, . . . ,xm) of type 1 with the divisibility partial order. hence for every nonzero term t 6= t1 of type 1 of p(x2, . . . ,xm) corresponding to (i1, . . . , i`) , there exists 1 ≤ j ≤ ` such that ij 6∈ {i∗1, . . . , i∗`∗}, and therefore t is divisible by xij. for any a = (a1, . . . ,am) ∈ fm2 we see that a + l = a∗ + l where a∗j = aj if j ≤ s and a ∗ j = 0 otherwise. thus we can assume, without loss, that aj = 0 for j > s. therefore a + l = {(a1, . . . ,as,xs+1, . . . ,xm) : xs+1, . . . ,xm ∈ f2}. in view of (5), we want to show that π−1(a + l) −π−1(a) is not a subspace of fm2 for some a = (a1, . . . ,as,0m−s) ∈ fm2 . let aj = 1 for all j = i∗u ≤ s and aj = 0 for any j ∈ {1, 2, . . . ,s}\{i∗1, . . . , i∗`∗}. since any term of p(x2, . . . ,xm) of type 1 except t1 is divisible by xij for some ij 6∈ {i∗1, . . . , i∗`∗}, in addition if we let xj = 0 for all j ∈{s + 1, . . . ,m}\{i∗1, . . . , i∗`∗} then π−1(a1, . . . ,as,xs+1, . . . ,xm) = (( a1 + l(xs+1, . . . ,xm) + `∗∏ r=s0 xi∗r ) ,a2, . . . ,as,xs+1, . . . ,xm ) 144 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 where s0 = min{j ∈{1, . . . ,`∗} : i∗j > s} and l(xs+1, . . . ,xm) is a polynomial of degree ≤ 1 coming from terms of type 2. therefore, in this case, π−1(a + l) −π−1(a) = {(( l(xs+1, . . . ,xm) − l(0m−s) + `∗∏ r=s0 xi∗r ) ,0s−1,xs+1, . . . ,xm ) : xs+1, . . . ,xm ∈ f2 } . for s0 ≤ j ≤ `∗, let ei∗ j = (xs+1, . . . ,xm) be such that xi∗ j = 1 and xi = 0 for i 6= i∗j and let fi∗ j = π−1(a + (0s,ei∗ j )) − π−1(a). then fi∗ j = ( l(ei∗ j ) − l(0m−s),0s−1,ei∗ j ) , as xi∗u = 0 for u 6= j. now fi∗ j ∈ π−1(a + l) − π−1(a) for all s0 ≤ j ≤ `∗. on the other hand, ∑`∗ j=s0 fi∗ j =( l( ∑`∗ j=s0 ei∗ j ) − l(0m−s),0s−1, ∑`∗ j=s0 ei∗ j ) , as l(xs+1, . . . ,xm) − l(0m−s) is a homogeneous linear polynomial. denoting ∑`∗ j=s0 ei∗ j by (ys+1, . . . ,ym), we have ∑`∗ j=s0 fi∗ j = (y1,0s−1,ys+1, . . . ,ym) where y1 = l(ys+1, . . . ,ym) − l(0m−s). since ∏`∗ r=s0 yi∗r = 1, we see that ∑`∗ j=s0 fi∗ j 6∈ π−1(a + l) −π−1(a). as a consequence, π−1(a + l) − π−1(a) is not a subspace of fm2 and hence (π,l) does not satisfy c-condition. 5. non-bentness of an infinite family the violation of c-condition by (π,l) is not sufficient to show f(π,l) is not bent. using the adaptation of difference set criterion from section 3, we will show the non-bentness of f(π,l) for several (π,l) in every even dimension. we require the following lemma 5.1. let m ≥ 3 and 1 ≤ s ≤ m− 2 be integers. then∑ (xs+1,...,xm)∈fm−s2 (−1)( ∑m i=s+1 xi+ ∏m i=s+1(xi+1)) = −2. proof. more generally, we will prove : for any j = s,s + 1, · · · ,m− 1,∑ (xj+1,...,xm)∈f m−j 2 (−1)( ∑m i=j+1 xi+ ∏m i=j+1(xi+1)) = −2. (†) we prove (†) by induction on u = m− j. if u = 1, we have j = m− 1. since∑ xm∈f2 (−1)(1+(xm+1))(−1)xm = 2, (†) holds for u = 1. assume (†) for u = ν where ν ≤ m− 2 and let u = ν + 1, that is, j = m−ν − 1. now ∑ (xj+1,...,xm)∈f m−j 2 (−1)( ∑m i=j+1 xi+ ∏m i=j+1 (xi+1)) = ∑ (xm−ν,...,xm)∈fν+12 (−1)( ∑m i=m−ν xi+ ∏m i=m−ν(xi+1)) = − ∑ (xm−ν,...,xm)∈fν+12 ( (−1)(1+ ∏m i=m−(ν−1)(xi+1)) )(xm−ν+1) (−1)( ∑m i=m−(ν−1) xi) 145 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 = − ∑ (x(m−ν)+1,...,xm)∈fν2 (−1)(1+ ∏m i=(m−ν)+1(xi+1))(−1)( ∑m i=(m−ν)+1 xi) − ∑ (x(m−ν)+1,...,xm)∈fν2 (−1)( ∑m i=(m−ν)+1 xi) since ∑ xm∈f2 (−1) xm = 0, ∑ (x(m−ν)+1,...,xm)∈fν2 (−1)( ∑m i=(m−ν)+1) xi) = m∏ i=(m−ν)+1 ( ∑ xi∈f2 (−1)xi ) = 0. moreover as (†) holds for u = ν, that is j = m−ν, we have∑ (x(m−ν)+1,...,xm)∈fν2 (−1)( ∏m i=(m−ν)(xi+1))(−1)( ∑m i=(m−ν)+1 xi) = −2. as a consequence, (†) holds for u = ν + 1. this completes the proof. now we come to the main result. theorem 5.2. let m ≥ 3 and 1 ≤ r ≤ s ≤ m − 2 be integers. further let l = {(0s,xs+1, . . . ,xm) : xi ∈ f2,s + 1 ≤ i ≤ m} be an m − s dimensional linear subspace of fm2 and π(x) =( (x1 + ∏m i=r+1 xi),x2, . . . ,xm ) be a permutation of fm2 . then f(π,l) : f 2m 2 → f2 is not a bent function. proof. since l = {(0s,xs+1, . . . ,xm) : xi ∈ f2,s + 1 ≤ i ≤ m} we have l⊥ = {(x1, . . . ,xs,0m−s) : xi ∈ f2, 1 ≤ i ≤ s}. also for any y = (y1, . . . ,ym) ∈ fm2 , π−1(y) = (( y1 + ∏m i=r+1 yi ) ,y2, . . . ,ym ) . therefore f(π,l)(x,y) = m∑ i=1 xiyi + x1 m∏ i=r+1 yi + m∏ i=s+1 (xi + 1) = f(x,y) + x1 m∏ i=r+1 yi where f(x,y) = ∑m i=1 xiyi + ∏m i=s+1(xi + 1) is a m-class bent function in 2m variables, see p. 90 of [4]. let d(π,l) and d denote the supports of f(π,l) and f respectively.then d \d(π,l) = {(x,y) ∈ d : x1 m∏ i=r+1 yi = 1}. we know x1 m∏ i=r+1 yi = 1 ⇐⇒ x1 = yr+1 = · · · = ym = 1. therefore (x,y) ∈ d and x1 m∏ i=r+1 yi = 1 ⇐⇒ y1 + r∑ i=2 xiyi + m∑ i=r+1 xi + m∏ i=s+1 (xi + 1) = 1 and 146 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 x1 = yr+1 = · · · = ym = 1 ⇐⇒ y1 = 1 + r∑ i=2 xiyi + m∑ i=r+1 xi + m∏ i=s+1 (xi + 1) and x1 = yr+1 = · · · = ym = 1. consequently d \d(π,l) = {(1,x2, . . . ,xm, 1 + r∑ i=2 xiyi + m∑ i=r+1 xi + m∏ i=s+1 (xi + 1),y2, . . . ,yr,1m−r) : x2, . . . ,xm,y2, . . . ,yr ∈ f2} and hence |d \d(π,l)| = 2m+r−2. now, if d̄ denotes the complement of d in f2m2 , d(π,l) \d = {(x,y) ∈ d̄ : x1 m∏ i=r+1 yi = 1} = {(1,x2, . . . ,xm, r∑ i=2 xiyi + m∑ i=r+1 xi + m∏ i=s+1 (xi + 1),y2, . . . ,yr,1m−r) : x2, . . . ,xm,y2, . . . ,yr ∈ f2} and hence |d(π,l) \d| = 2m+r−2. let u = {1,−1}2m. then for any (ξ,η) ∈ u we have ∆(d,d(π,l))(ξ,η) = ξ1ηr+1 · · ·ηm(η1 − 1)×∑ (x2,...,xm,y2,...,yr)∈fm+r−22 ξx22 · · ·ξ xm m η ( ∑r i=2 xiyi+ ∑m i=r+1 xi+ ∏m i=s+1 (xi+1)) 1 η y2 2 · · ·η yr r . henceforth let ξ1 = · · · = ξr = 1,ξr+1 = · · · = ξs = −1,ξs+1 = · · · = ξm = 1,η1 = −1,η2 = · · · = ηm = 1. further let λ1 = ∑ (x2,...,xr,y2,...,yr)∈f2r−22 (−1)( ∑r i=2 xiyi) and λ2 = ∑ (xs+1,...,xm)∈fm−s2 (−1)( ∑m i=s+1 xi+ ∏m i=s+1 (xi+1)). then ∆(d,d(π,l))(ξ,η) = −2 ∑ (xr+1,...,xs)∈fs−r2 (λ1λ2) = −2s−r+1 (λ1λ2) . 147 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 now λ1 = (∑ (x2,y2)∈f22 (−1)x2y2 ) · · · (∑ (xr,yr)∈f22 (−1)xryr ) and∑ (xi,yi)∈f22 (−1)xiyi = 2 for any i = 2, . . . ,r. hence by lemma 5.1, ∆(d,d(π,l))(ξ,η) = −2sλ2 = 2s+1. since |d \d(π,l)| = |d(π,l) \d| = 2m−1, we have |d| = |d(π,l)|. if f(π,l) is a bent function, then d(π,l) is a difference set. thus by ryser’s condition (see section 3 of [5]), it follows that parameters (v,k,λ) of d(π,l) are same as those of d, hence k−λ = 2t−2 = 22(m−1). consequently, by (4) of section 3, for any (ξ,η) ∈ u, ∆(d,d(π,l)) ∈{0,±2 √ (k −λ)} = {0,±2m}. but we have found (ξ,η) ∈ u such that ∆(d,d(π,l)) = 2s+1 6∈ {0,±2m} for any 1 ≤ s ≤ m − 2. therefore f(π,l) is not bent function. remark 5.3. the alternate tools for proving the non-bentness of f(π,l) could have been theorem on p. 94 of [2] or proposition 1 on p. 398 of [6]. as far as [2] is concerned, checking the required conditions is tedious. on the other hand, to prove non-bentness of f(π,l) using [6],it is sufficient to have either (a) the hamming distance d(f,f(π,l)) < 2m or (b) d(f,f(π,l)) = 2m and either support a of f + f(π,l) is not a flat or the restriction of f to a is not an affine function. now d ( f,f(π,l) ) = |d\d(π,l)|+|d(π,l)\d| = 2m+r−1. so if r > 1 then proposition 1 of [6] doesn’t help, though it suffices for r = 1. 6. computational results in this section, we report some more (π,l) such that f(π,l) is not bent. this was established through implementations of methods of previous sections as well as [5] using programs in c++ language. in section 5, the polynomial p(x2, · · · ,xm) contained only one term of type 1 (as described in case (c) of theorem 4.1). examples 6.1 and 6.2 contain more than one term of type 1. example 6.1. let l = {(0, 0,x3,x4) : x3,x4 ∈ f2} be a linear subspace of f42 and π(x) = (x1 + αx2x3 + βx2x4 + γx3x4 + δx2x3x4,x2,x3,x4) : α,β,γ,δ ∈ f2 be a permutation of f42. then f(π,l) : f42 ×f42 → f2 is a c-class bent function when γ = δ = 0 and f(π,l) is not a bent function otherwise. let us provide some justification for this. clearly when γ = δ = 0, then by (b) of theorem 4.1, (π,l) satisfies c-condition. let d(π,l) denote the support of f(π,l). when γ = 1, for ξ = (1, 1, 1, 1,−1, 1, 1, 1) ∈ u, we see that ( ρg(d(π,l))(ξ) )2 −λρg(g)(ξ) − (k −λ) = −64 6= 0. when (γ,δ) = (0, 1), (α,β) 6= (1, 1), for ξ = (1,−1, 1, 1,−1, 1, 1, 1) we have ( ρg(d(π,l))(ξ) )2 −λρg(g)(ξ)−(k−λ) = −64 6= 0. further when α,β,γ,δ) = (1, 1, 0, 1), for ξ = (1,−1, 1, 1,−1, 1, 1, 1) we have ( ρg(d(π,l))(ξ) )2 −λρg(g)(ξ) − (k−λ) = 192 6= 0. when α = β = γ = 0 and δ = 1, we get a special case of the family in section 5. using matlab, we have also determined that its walsh-hadamard spectrum (see [7]) contains -16 with multiplicity 104, 16 with multiplicity 88, each of -32 and 32 with multiplicity 8 and 0 with multiplicity 48. it has also been verified that in the other non-bentness cases of example 6.1, walsh-hadamard spectrum is not contained in {±2m}. this provides another verification of non-bentness, following the definition in [7]. example 6.2. let l = {(0, 0,x3,x4,x5,x6) : x3,x4,x5,x6 ∈ f2} be a linear subspace of f62 and π(x) = (x1 + x3x4 + x5x6,x2,x3,x4,x5,x6) be a permutation of f62. then f(π,l) : f 6 2 × f62 → f2 is not a bent function. this can be justified by observing that ( ρg(d(π,l))(ξ) )2 −λρg(g)(ξ) − (k −λ) = −768 6= 0 where d(π,l) is the support of f(π,l) and ξ = (1, 1, 1, 1, 1, 1,−1, 1, 1, 1, 1, 1) ∈ u. 148 p. h. keskar, p. kumari / j. algebra comb. discrete appl. 8(2) (2021) 139–149 7. concluding remarks in this paper, we have connected the count of abelian difference sets with given parameters to computation of hilbert functions of an ideal. there are algorithms for computation of hilbert function. while diffculties in implementation for large values of parameters need to be addressed, the theoretical consequences of this connection can also be explored. we also undertook to explore bentness of f(π,l) when (π,l) does not satisfy c-condition. theorem 4.1 (c) helped us determine the choice of (π,l) for exploration and sections 5 and 6 provided the results of exploration.this work was prompted by the following questions which still await the answers. question 1. is c-condition necessary for bentness of f(π,l)? if yes, then provide the proof or else provide the counter-example. question 2. as a consequence of theorem 4.1 (b) and (c), it follows that if l = {(0s,xs+1, . . . ,xm) : xi ∈ f2,s + 1 ≤ i ≤ m} and πi(x1, . . . ,xm) = (x1 + pi(x2, . . . ,mm),x2, . . . ,xm) for i = 1, 2 are such that (πi,l) satisfies c-condition for i = 1, 2 then (π1 ◦π2,l) also satisfies c-condition. what can we say about bentness of f(π1◦π2,l) if we know bentness of f(πi,l) for i = 1, 2? in general, for a given subspace l of fm2 , is there a semigroup structure on the set of all permutations π of f m 2 such that f(π,l) is bent? if not, what are the counterexamples? we hope to continue our exploration further guided by these questions. acknowledgment: both the authors thank the support from fist programme vide sr/fst/msi090/2013 of dst, govt. of india. the second author thanks ugc, govt of india for the support under srf programme (sr. no. 2061540979, ref. no. 21/06/2015(1)eu-v r. no. 426800). references [1] t. becker, v. weispfennig, groebner bases a computational approach to commutative algebra, springer (1993). [2] c. carlet, two new classes of bent functions, advances in cryptology-eurocrypt’93, lncs vol 765 (ed. t. hellseth) springer-verlag (1994) 77–101. [3] d. cox, j. little, d. o’shea, ideals, varieties and algorithms, springer verlag, new york inc (2007). [4] j. f. dillon, elementary hadamard difference sets, ph. d. thesis, university of maryland (1974). [5] p. h. keskar, p. kumari, polynomial criterion for abelian difference sets, indian journal of pure and applied mathematics 51(1) (2020) 233–249. [6] n. kolomeec, the graph of minimal distances of bent functions and its properties, designs, codes and cryptography 85 (2017) 395–410. [7] b. mandal, p. stănică, s. gangopadhyay, e. pasalic, an analysis of the c class of bent functions, fundamenta informaticae 146(3) (2016) 271–292. [8] e. h. moore, h. s. pollatsek, difference sets, connecting algebra, combinatorics, and geometry, american mathematical society (2013). [9] n.tokareva, bent functions: results and applications to cryptography, elsevier (2015). [10] a. van den essen, polynomial automorphisms and the jacobian conjecture, birkhauser (2000). 149 https://doi.org/10.1007/3-540-48285-7_8 https://doi.org/10.1007/3-540-48285-7_8 https://doi.org/10.1007/s13226-020-0397-5 https://doi.org/10.1007/s13226-020-0397-5 https://doi.org/10.1007/s10623-016-0306-4 https://doi.org/10.1007/s10623-016-0306-4 https://doi.org/10.3233/fi-2016-1386 https://doi.org/10.3233/fi-2016-1386 introduction counting the difference sets a difference set criterion the analysis of c-condition non-bentness of an infinite family computational results concluding remarks references issn 2148-838x j. algebra comb. discrete appl. 9(3) • 161–174 received: 11 january 2021 accepted: 31 march 2022 journal of algebra combinatorics discrete structures and applications a note on two-dimensional cyclic and constacyclic codes research article om prakash, shikha patel abstract: during the study of the two-dimensional cyclic (tdc) codes of length n = ls over a finite field fq where s = 2k, sepasdar and khashyarmanesh (2016, [11]) arose a problem that the technique used by them to characterize tdc codes of length n = ls does not work for tdc codes of length 3l. it naturally motivates us to study the tdc codes of other lengths together with 3l. further, (λ1,λ2)constacyclic codes are the generalization of constacyclic codes. thus, we study two-dimensional cyclic codes of length 3l and (λ1,λ2)-constacyclic codes of length 2l, respectively over finite fields. here, the generating set of polynomials for these two-dimensional codes and their duals are obtained. finally, with the help of our derived results, we have constructed many mds codes corresponding to the two-dimensional codes. 2010 msc: 12e20, 94b05, 94b15, 94b60 keywords: cyclic codes, two-dimensional cyclic codes, constacyclic codes, dual codes, generator matrix 1. introduction cyclic codes were introduced by prange [9] in 1957 and have been studied extensively till now. these codes have been studied over several finite rings and produce a huge amount of new and optimal codes, refer [2, 3, 8, 14, 15]. one of the powerful generalizations of cyclic codes over a finite field is constacyclic codes. this class of linear codes has a rich algebraic structure and is easy to recognize and implement. note that one can specify cyclic codes and λ-constacyclic codes of length n over the finite field fq by ideals of the polynomial ring r := fq[u]/〈un −1〉 and r′ := fq[u]/〈un −λ〉, respectively. since the above rings are principal ideal rings, the cyclic and constacyclic codes are generated by a unique polynomial. these generator polynomials help us to find the important parameters of the codes. in 1975, ikai et al. [5] observed the two-dimensional cyclic (tdc) codes as a generalized class of cyclic codes. moreover, just after 2 years, imai [4] introduced the concept of binary two-dimensional om prakash (corresponding author), shikha patel; department of mathematics, indian institute of technology patna, patna–801106, bihar, india (email: om@iitp.ac.in, shikha_1821ma05@iitp.ac.in). 161 https://orcid.org/0000-0002-6512-4229 https://orcid.org/0000-0001-8443-2596 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 cyclic codes. the two-dimensional theory has many applications in the analysis and generation of twodimensional periodic arrays that help us to construct the two-dimensional feedback shift register with a minimum number of storage devices. therefore, the study of these codes over finite rings has got the attention of many researchers and hence many new techniques have been discovered to produce cyclic codes over the finite commutative rings with better parameters, we refer [7, 10, 11, 14, 15]. in 2014, xiuli and hongyan [13] generalized the concept as two-dimensional skew cyclic codes over a finite field. in 2016, sepasdar and khashyarmanesh [11] further studied two-dimensional cyclic codes corresponding to the ideals of the ring r′ := f[x,y]/〈xl − 1,y2 k − 1〉. in 2019, sharma and bhaintwal [12] studied the structural behaviour of two-dimensional skew cyclic codes over the ring fq + ufq with u2 = 1. in [11], sepasdar and khashyarmanesh studied the algebraic structure of tdc codes of length n = l2k over the finite field. moreover, they claimed that the method used by them is not applicable for tdc codes of length 3l. these works motivate us to attempt over the problem raised by them in [11] and extend the study for (λ1,λ2)-constacyclic codes as well. further, we also study the algebraic properties of two-dimensional cyclic and constacyclic codes and their duals. the article is structured as follows: section 2 contains some basic definitions and notations. section 3 presents the study of two-dimensional cyclic codes and their duals of length 3l. here, we obtain the generating polynomials and the generator matrices of these codes. similarly, in section 4, we characterize the structure of two-dimensional (λ1,λ2)-constacyclic codes and their duals and calculate the generating polynomials. as an application of our results, some examples are presented in section 5 while section 6 concludes the paper. 2. notation and background for a prime p and an integer m ≥ 1, let fq be a finite field where q = pm and λ1,λ2 ∈ f∗q. throughout the paper, r and r′ represent the quotient ring fq[u,v]/〈ul − 1,v3 − 1〉 and fq[u,v]/〈ul −λ1,v2 −λ2〉, respectively. now, we recall some definitions for two-dimensional codes of flsq . definition 2.1. a two-dimensional code c ⊆ flsq is a set of l×s arrays over fq. these arrays are known as codewords (or code arrays). a two-dimensional code c is said to be linear if it is a subspace of the lsdimensional linear space flsq . later, in 1977, imai [4] introduced the notion of the binary two-dimensional cyclic codes as follows: definition 2.2. let c ⊆ flsq be a linear code of length n = ls over fq whose codewords are viewed as l×s arrays, i.e., c ∈c is written as c =   c0,0 c0,1 . . . c0,s−1 c1,0 c1,1 . . . c1,s−1 ... ... ... ... cl−1,0 cl−1,1 . . . cl−1,s−1   . each codeword c ∈c written in above matrix form is also defined by its polynomial representation c(u,v) =∑l−1 i=0 ∑s−1 j=0 ci,ju ivj = f0(u) + f1(u)v + · · · + fs−1(u)vs−1 ∈ fq[u,v]/〈ul − 1,vs − 1〉, where fj(u) = c0,j+c1,ju+· · ·+cl−1,jul−1 ∈ fq[u]/〈ul−1〉. these representations provide an explicit algebraic description for two-dimensional linear codes. • if c is closed under row-shift and column-shift of codewords, then c is called a two-dimensional cyclic code of length ls over fq, i.e., if for every l×s array c = (cij) ∈c, we have  c0,s−1 c0,0 . . . c0,s−2 c1,s−1 c1,0 . . . c1,s−2 ... ... ... ... cl−1,s−1 cl−1,0 . . . cl−1,s−2   ∈c 162 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 and   cl−1,0 cl−1,1 . . . cl−1,s−1 c0,0 c0,1 . . . c0,s−1 ... ... ... ... cl−2,0 cl−2,1 . . . cl−2,s−1   ∈c. • let λ1 ∈ f∗q = fq \{0}. then c is said to be a column λ1-constacyclic code of length ls if for every l×s array c = (cij) ∈c, we have  λ1cl−1,0 λ1cl−1,1 . . . λ1cl−1,s−1 c0,0 c0,1 . . . c0,s−1 ... ... ... ... cl−2,0 cl−2,1 . . . cl−2,s−1   ∈c. • let λ2 ∈ f∗q. then c is said to be a row λ2-constacyclic code of length ls if for every l × s array c = (cij) ∈c, we have   λ2c0,s−1 c0,0 . . . c0,s−2 λ2c1,s−1 c1,0 . . . c1,s−2 ... ... ... ... λ2cl−1,s−1 cl−1,0 . . . cl−1,s−2   ∈c. • let λ1,λ2 ∈ f∗q. if c is both column λ1-constacyclic and row λ2-constacyclic, then c is said to be a two-dimensional (λ1,λ2)-constacyclic code of length ls. clearly, when λ1 = λ2 = 1, then two-dimensional (λ1,λ2)-constacyclic code coincides with two-dimensional cyclic (tdc) code. it is noted that there is a one-one correspondence between cyclic codes of length l over fq and ideals of the polynomial ring fq[u]/〈ul−1〉. similarly, in case of tdc codes, there is also a one-one correspondence between tdc codes of length ls over fq and ideals of the polynomial ring fq[u,v]/〈ul−1,vs−1〉. hence, a tdc code c ⊆ flsq of length n = ls over the finite field fq can be viewed as an ideal of the quotient ring fq[u,v]/〈ul − 1,vs − 1〉. further, for λ1,λ2 ∈ f∗q, a two-dimensional (λ1,λ2)-constacyclic code c ⊆ flsq of length n = ls over the finite field fq is an ideal of the quotient ring fq[u,v]/〈ul −λ1,vs −λ2〉. recall that for c = (c0,c1, ...,cn−1) ∈ c ⊆ fnq , the hamming weight wh(c) is equal to the number of non-zero components of c and for any two codewords c and c′ of c, the hamming distance is defined as dh(c,c ′) = wh(c− c′). also, the hamming distance for the code c is dc = min {d(c,c′) | c, c′ ∈c,c 6= c′}. let c = (c0,c1, ...,cn−1) and c′ = (c′0,c ′ 1, . . . ,c ′ n−1) be two elements of c. then the inner product of c and c′ in fnq is defined as c · c′ = ∑n−1 j=0 cjc ′ j. the dual code of c is c ⊥ = {c ∈ fnq | c · c′ = 0, for all c′ ∈c}. it is well known that a code c is self-orthogonal if c ⊆c⊥ and self dual if c = c⊥. now, we recall the construction of tdc codes of length 2l over fq from [11], which is as follows: let m be a non-zero ideal of r = fq[u,v]/〈ul −1,v2 −1〉 and char(fq) 6= 2. it is known that fq[u,v]/〈ul −1,v2 −1〉∼= (fq[u]/〈ul −1〉)[v]/〈v2 −1〉. any arbitrary element of m can be uniquely written as g(u,v) = g0(u)+g1(u)v, where gi(u) ∈ fq[u]/〈ul− 1〉 for i = 0,1. here, we find a set of generator polynomials for m. to get the generator polynomials, we use the following identity in r. g(u,v) = 2−1 (g(u,v)(1 + v) + g(u,v)(1−v)) (1) where 163 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 (x) g(u,v)(1 + v) = (g0(u) + g1(u))(1 + v), (y) g(u,v)(1−v) = (g0(u)−g1(u))(1−v). next, we consider the following ideals m1 and m2 of fq[u]/〈ul −1〉, m1 = {f(u) ∈ fq[u]/〈ul −1〉 | f(u)(1 + v) ∈ m}, m2 = {f(u) ∈ fq[u]/〈ul −1〉 | f(u)(1−v) ∈ m}. since fq[u]/〈ul − 1〉 is a principal ideal ring, m1 and m2 are principal ideals. thus, there exist unique monic polynomials p1(u) and p2(u) in fq[u]/〈ul −1〉 such that mi = 〈pi(u)〉 for i = 1,2. also, p1(u) and p2(u) are divisors of ul −1. therefore, there exists p′i(u) such that pi(u)p ′ i(u) = u l −1 for i = 1,2. now, from (x), we have g0(u) + g1(u) ∈ m1, and hence g0(u) + g1(u) = p ′′ 1(u)p1(u), for some polynomial p′′1(u) ∈ fq[u]/〈ul −1〉. also, from (y ), we have g0(u) −g1(u) ∈ m2, i.e., there exists a polynomial p′′2(u) in fq[u]/〈ul − 1〉 such that g0(u)−g1(u) = p′′2(u)p2(u). therefore, from equation (1), we have g(u,v) = 2−1 ( g(u,v)(1 + v) + g(u,v)(1−v) ) = 2−1 ( (g0(u) + g1(u))(1 + v) + (g0(u)−g1(u))(1−v) ) = 2−1 ( p′′1(u)p1(u)(1 + v) + p ′′ 2(u)p2(u)(1−v) ) . since g(u,v) is an arbitrary element of m and hence m is an ideal of r generated by the polynomials p1(u)(1 + v) and p2(u)(1−v), i.e., m = 〈p1(u)(1 + v), p2(u)(1−v)〉. (2) these polynomials p1(u)(1 + v) and p2(u)(1 − v) are generators of m or of the corresponding twodimensional cyclic code of length 2l. the above discussed generating set of polynomials for tdc codes of length 2l will be used in the next section 3 for the construction of tdc codes of length 3l. 3. two-dimensional cyclic codes of length 3l in this section, we study two-dimensional cyclic (tdc) codes of length 3l over fq with char(fq) 6= 2,3. here, our main target is to find the generator polynomials for these codes to explore the structural properties and their duals. 3.1. generator matrix let i be a non-zero ideal of r = fq[u,v]/〈ul −1,v3 −1〉. it is known that fq[u,v]/〈ul −1,v3 −1〉∼= (fq[u]/〈ul −1〉)[v]/〈v3 −1〉. 164 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 therefore, each element g(u,v) ∈ i can be uniquely written as g(u,v) = g0(u) + g1(u)v + g2(u)v2, where gi(u) ∈ fq[u]/〈ul − 1〉 for i = 0,1,2. in order to obtain the set of generator polynomials of i, we have the following identity in r: g(u,v) = 3−1 ( g(u,v)(1 + v + v2) + g(u,v)(1−v) + g(u,v)(1−v2) ) (3) where (a) g(u,v)(1 + v + v2) = (g0(u) + g1(u) + g2(u))(1 + v + v2); (b) g(u,v)(1−v) = (g0(u)−g2(u) + (g1(u)−g2(u))v)(1−v); (c) g(u,v)(1−v2) = (g0(u)−g2(u) + (g1(u)−g2(u))v)(1−v2). next, we consider the ideals i1 in fq[u]/〈ul −1〉 and i2, i3 in fq[u,v]/〈ul −1,v2 −1〉, as follows: i1 = {f(u) ∈ fq[u]/〈ul −1〉 | f(u)(1 + v + v2) ∈ i}; i2 = {f(u,v) ∈ fq[u,v]/〈ul −1,v2 −1〉 | f(u,v)(1−v) ∈ i}; i3 = {f(u,v) ∈ fq[u,v]/〈ul −1,v2 −1〉 | f(u,v)(1−v2) ∈ i}. since fq[u]/〈ul−1〉 is a principal ideal ring, there exists a unique monic polynomial p0(u) ∈ fq[u]/〈ul−1〉 such that i1 = 〈p0(u)〉. also, p0(u) is a divisor of ul −1, there exists p′0(u) such that p0(u)p′0(u) = ul −1. further, in i2 and i3, f(u,v) ∈ fq[u,v]/〈ul − 1,v2 − 1〉. hence, from equation (2), we can find the generators of i2 and i3, i.e., i2 is generated by two polynomials p1(u)(1 + v), p2(u)(1 − v) and i3 is generated by the polynomials p3(u)(1 + v) and p4(u)(1 − v), respectively for some monic polynomials p1(u), p2(u), p3(u) and p4(u) in fq[u]/〈ul −1〉. therefore, (a) gives us g0(u) + g1(u) + g2(u) ∈ i1. hence, g0(u) + g1(u) + g2(u) = p0(u)p ′′ 0(u) for some polynomial p′′0(u) ∈ fq[u]/〈ul −1〉. thus, g(u,v)(1 + v + v2) = p0(u)p ′′ 0(u)(1 + v + v 2). (4) further, from (b), g0(u)−g2(u) + (g1(u)−g2(u))v ∈ i2. also, g0(u)−g2(u) + (g1(u)−g2(u))v = p1(u)p′′1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v) for some polynomials p′′1(u) and p ′′ 2(u) in fq[u]/〈ul −1〉. hence, g(u,v)(1−v) = (p1(u)p′′1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v))(1−v). (5) again, by (c), g0(u)−g2(u) + (g1(u)−g2(u))v ∈ i3, i.e., g0(u)−g2(u) + (g1(u)−g2(u))v = p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v) for some polynomials p′′3(u) and p ′′ 4(u) in fq[u]/〈ul −1〉. hence, g(u,v)(1−v2) = (p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v))(1−v 2). (6) from equations (4), (5) and (6), equation (3) can be viewed as g(u,v) = 3−1 ( p0(u)p ′′ 0(u)(1 + v + v 2) + (p1(u)p ′′ 1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v)) (1−v) + (p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v))(1−v 2) ) = 3−1 ( p0(u)p ′′ 0(u)(1 + v + v 2) + p1(u)p ′′ 1(u)(1−v 2) + p2(u)p ′′ 2(u) (1 + v2 −2v) + p3(u)p′′3(u)(1 + v)(1−v 2) + p4(u)p ′′ 4(u)(1−v) (1−v2) ) . 165 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 as g(u,v) was arbitrary in i and g(u,v) is written as a linear combination of the elements p0(u)(1 + v + v2), p1(u)(1 − v2), p2(u)(1 + v2 − 2v), p3(u)(1 + v)(1 − v2), p4(u)(1 − v)(1 − v2) over fq[u]/〈ul − 1〉. therefore, we have i = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v),p3(u) (1 + v)(1−v2),p4(u)(1−v)(1−v2)〉. since p4(u)(1−v)(1−v2) ∈ i, then p4(u)(1−v)(1−v2)(1−v2) ∈ i and this implies p4(u)(1−v)(1−v2)2 = 3p4(u)(1−v2) ∈ i and hence p4(u)(1−v2) ∈ i. also, from the definition of i3, p4(u) ∈ i3. hence, there exist polynomials m(u) and m′(u) in fq[u]/〈ul −1〉 such that p4(u) = m(u)p3(u)(1 + v) + m ′(u)p4(u)(1−v). (7) comparing the degree of v in equation (7), we get p4(u) = m(u)p3(u) + m ′(u)p4(u) and m(u)p3(u) = m′(u)p4(u), i.e., p4(u) = 2m(u)p3(u), implies p3(u)|p4(u). as p3(u)(1+v)(1−v2) ∈ i, so p3(u)(1+v)(1−v2)(1+v2) ∈ i and this gives p3(u)(1 − v2) ∈ i. in the same manner, we have p4(u)|p3(u). therefore, p3(u) = p4(u). hence, i3 = 〈p3(u)(1 + v),p4(u)(1−v)〉 = 〈p3(u)〉. thus, i = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v),p3(u)(1−v2)〉 further, we note that p1(u)(1−v2) ∈ i and hence p1(u) ∈ i3. therefore, there exist polynomials m1(u) and m′1(u) in fq[u]/〈ul −1〉 such that p1(u) = m1(u)p3(u)(1 + v) + m ′ 1(u)p3(u)(1−v). (8) comparing the degree of v in equation (8), we get p1(u) = m1(u)p3(u) + m ′ 1(u)p3(u) and m1(u)p3(u) = m ′ 1(u)p3(u), i.e., p1(u) = 2m1(u)p3(u), and hence p3(u)|p1(u). next, we have p3(u)(1 − v2) ∈ i. this implies that p3(u)(1−v2)(1 +v2) = p3(u)(1−v) ∈ i, i.e., p3(u) ∈ i2. thus, there exist polynomials m2(u) and m′2(u) in fq[u]/〈ul −1〉 such that p3(u) = m2(u)p1(u)(1 + v) + m ′ 2(u)p2(u)(1−v). (9) comparing the degree of v in equation (9), we have p3(u) = m2(u)p1(u) + m ′ 2(u)p2(u) and m2(u)p1(u) = m ′ 2(u)p2(u), i.e., p3(u) = 2m2(u)p1(u), implies p1(u)|p3(u) and hence p1(u) = p3(u). therefore, i = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v)〉. moreover, p2(u)v2(v −1)(1 + v2 −2v) ∈ i as p2(u)(1 + v2 −2v) ∈ i. this implies p2(u)3(1−v) ∈ i and hence p2(u)(1−v) ∈ i. finally, the set of generator polynomials for i (the associated tdc code) is{ p0(u)(1 + v + v 2),p1(u)(1−v2),p2(u)(1−v) } . also, any element of i can be written as an fq[u]/〈ul −1〉-combination of {p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v)}. in view of the above demonstration, the following theorem determines the generator matrix for the tdc code of length 3l. 166 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 theorem 3.1. suppose c is a tdc code of length n = 3l and its generating set of polynomials is{ p0(u)(1 + v + v 2),p1(u)(1−v2),p2(u)(1−v) } with deg pi(u) = ai for i = 0,1,2. then {p0(u)(1 + v + v2),up0(u)(1 + v + v2), . . . ,ul−a0−1p0(u)(1 + v + v2),p1(u)(1−v2), up1(u)(1−v2), . . . ,ul−a1−1p1(u)(1−v2),p2(u)(1 + v2 −2v),up2(u)(1 + v2 −2v), . . . ,ul−a2−1 p2(u)(1 + v 2 −2v)}, is an independent set, and hence elements of this set form the rows of the generator matrix of the code c. proof. here, it is enough to show that the above set is independent. for this, we consider m0(u)p0(u)(1 + v + v 2) + m1(u)p1(u)(1−v2) + m2(u)p2(u)(1 + v2 −2v) = 0, (10) for some polynomials mi(u) ∈ fq[u]/〈ul −1〉 with deg mi(u) ≤ l−ai −1 for i = 0,1,2. claim: mi(u) = 0 for i = 0,1,2. from equation (10), there exist polynomials a(u,v) and b(u,v) in fq[u,v] such that m0(u)p0(u)(1 + v + v 2) + m1(u)p1(u)(1−v2) + m2(u)p2(u)(1 + v2 −2v) = a(u,v)(ul −1) + b(u,v)(v3 −1) (11) in fq[u,v]. let a(u,v) = ∑t i=0 ai(u)v i and b(u,v) = ∑t′ i=0 bi(u)v i in fq[u,v] and m′ be the maximum degree of v in the right hand side of equation (11). then we have the following table for coefficients of each vi in the right hand side of equation (11). index coefficient i=0 a0(u)(ul −1)− b0(u) i=1 a1(u)(ul −1)− b1(u) i=2 a2(u)(ul −1)− b2(u) i=3 a3(u)(ul −1)− b3(u) + b0(u) i=4 a4(u)(ul −1)− b4(u) + b1(u) i=5 a5(u)(ul −1)− b5(u) + b2(u) ... ... i=m′ −1 am′−1(u)(ul −1) + bm′−4(u) i=m′ am′(u)(ul −1) + bm′−3(u) since the maximum degree of v in the left hand side of equation (11) is 2. therefore, for i ≥ 3, the coefficients of vi in right hand side of equation (11) must be zero. finally, we have the following equalities: bm′−3(u) = −am′(u)(ul −1), bm′−4(u) = −am′−1(u)(ul −1), bm′−5(u) = −am′−2(u)(ul −1), bm′−6(u) = −am′−3(u)(ul −1), ... b0(u) = −a3(u)(ul −1) + b3(u). 167 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 it can be easily seen that the coefficients of each vi in right hand side of equation (11) is zero or has the factor ul − 1. hence, comparing the coefficients of vi’s in both side of equation (11), we get the following equations: m0(u)p0(u) + m1(u)p1(u) + m2(u)p2(u) = 0, m0(u)p0(u)−2m2(u)p2(u) = 0, m0(u)p0(u)−m1(u)p1(u) + m2(u)p2(u) = 0. since char(fq) 6= 2,3, the above equations give mi(u) = 0 for i = 0,1,2. thus, we get the required result. 3.2. generator matrix of c⊥ our next aim is to find a generator matrix for the dual of a tdc code. towards this, we recall the following proposition. proposition 3.2. [4] the dual of a tdc code is also a tdc code. since dim(c) + dim(c⊥) = 3l, by theorem 3.1, dim(c) = 3l −a0 −a1 −a2. therefore, dim(c⊥) = a0 + a1 + a2. recall that the reciprocal polynomial f∗(u) of f(u) with deg(f(u)) = k is defined as f∗(u) := ukf(1/u). further, if g(u) is a generator polynomial of c, then g(u) is a factor of ul −1 and so ul −1 = g(u)h(u) for some polynomial h(u). this polynomial h(u) is called the check polynomial of c and also h∗(u) is a codeword in c⊥. the following theorem gives the generator matrix for the dual c⊥. theorem 3.3. let c be a tdc code of length n = 3l with generator polynomials p0(u)(1 + v + v2), p1(u)(1−v2) and p2(u)(1 +v2 −2v) such that deg pi(u) = ai and also, pi(u)p′i(u) = u l −1 for i = 0,1,2. then h =   (p′0) ∗(u)(1 + v + v2) u(p′0) ∗(u)(1 + v + v2) ... ua0−1(p′0) ∗(u)(1 + v + v2) (p′1) ∗(u)(1−v2) u(p′1) ∗(u)(1−v2) ... ua1−1(p′1) ∗(u)(1−v2) (p′2) ∗(u)(1 + v2 −2v) u(p′2) ∗(u)(1 + v2 −2v) ... ua2−1(p′2) ∗(u)(1 + v2 −2v)   is the generator matrix of c⊥. proof. with the help of a similar argument used to prove theorem 3.1, one can easily show that the rows of h are independent. also, (p′i) ∗(u) ∈ c⊥ and (1 + v + v2)∗ = 1 + v + v2, (1 −v2)∗ = v2 − 1 and (1 + v2 −2v)∗ = 1 + v2 −2v. hence, (p′0)∗(u)(1 + v + v2), (p′1)∗(u)(1−v2) and (p′2)∗(u)(1 + v2 −2v) are codewords in c⊥. therefore, the result follows from [6]. 168 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 4. two-dimensional (λ1,λ2)-constacyclic codes of length 2l in this section, we study (λ1,λ2)-constacyclic code c of length 2l over fq with char(fq) 6= 2. 4.1. generator matrix of c suppose j is a non-zero ideal of the ring r′ = fq[u,v]/〈ul − λ1,v2 − λ2〉 and α ∈ f∗q such that λ2 = α 2 in f∗q. as fq[u,v]/〈ul −λ1,v2 −λ2〉∼= (fq[u]/〈ul −λ1〉)[v]/〈v2 −α2〉, an arbitrary element of j can be uniquely written as g(u,v) = g0(u)+g1(u)v where gi(u) ∈ fq[u]/〈ul−λ1〉 for i = 0,1. first we find a set of generator polynomials for j. in order to get the generator polynomials, we use the following identity in r′. g(u,v) = 2−1α−1 (g(u,v)(α + v) + g(u,v)(α−v)) (12) where (a) g(u,v)(α + v) = (g0(u) + αg1(u))(α + v); (b) g(u,v)(α−v) = (g0(u)−αg1(u))(α−v). next, we consider the two ideals j1 and j2 of fq[u]/〈ul −λ1〉 where j1 = {f(u) ∈ fq[u]/〈ul −λ1〉 | f(u)(α + v) ∈ j}; j2 = {f(u) ∈ fq[u]/〈ul −λ1〉 | f(u)(α−v) ∈ j}. since fq[u]/〈ul −λ1〉 is a principal ideal ring, j1 and j2 are principal ideals. hence, there exist unique monic polynomials p1(u) and p2(u) in fq[u]/〈ul −λ1〉 such that ji = 〈pi(u)〉 for i = 1,2. also, p1(u) and p2(u) are divisors of ul −λ1. therefore, there exists p′i(u) such that pi(u)p ′ i(u) = u l −λ1 for i = 1,2. now, from (a), we have g0(u) + αg1(u) ∈ j1, and hence g0(u) + αg1(u) = p ′′ 1(u)p1(u) for some polynomial p′′1(u) ∈ fq[u]/〈ul −λ1〉. also from (b), we have g0(u)−αg1(u) ∈ j2, i.e., there exists a polynomial p′′2(u) in fq[u]/〈ul −λ1〉 such that g0(u)−αg1(u) = p′′2(u)p2(u). therefore, we have the following equality in r′ : g(u,v) = 2−1α−1 ( g(u,v)(α + v) + g(u,v)(α−v) ) = 2−1α−1 ( (g0(u) + αg1(u))(α + v) + (g0(u) + αg1(u))(α−v) ) = 2−1α−1 ( p′′1(u)p1(u)(α + v) + p ′′ 2(u)p2(u)(α−v) ) . since g(u,v) is an arbitrary element of j, hence 169 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 j = 〈p1(u)(α + v), p2(u)(α−v)〉. the polynomials p1(u)(α+v) and p2(u)(α−v) are generators of j or of the corresponding two-dimensional (λ1,λ2)-constacyclic code c. further, based on the above construction, we have the following theorem. theorem 4.1. let j be an ideal of fq[u]/〈ul−λ1,v2−λ2〉 with generating set {p1(u)(α+v),p2(u)(α−v)} where p1(u) and p2(u) are monic polynomials in fq[u]/〈ul −λ1〉 with deg(p1(u)) = c and deg(p2(u)) = d. then every element of j has the form c1(u)p1(u)(α + v) + c2(u)p2(u)(α−v), where c1(u) and c2(u) are polynomials in the ring fq[u]/〈ul−λ1〉 with deg(c1(u)) < l−c and deg(c2(u)) < l−d. lemma 4.2. [6, theorem 7.2.12] let l = tpa and λp a 1 = λ1 where p is the characteristic of fq and does not divide t. assume ut −λ1 = ∏r i=1 pi(u) where pi(u) are distinct monic irreducible polynomials for i = 1,2, . . . ,r. then ul − λ1 = (ut − λ1)p a = ∏r i=1 pi(u) pa. thus, the number of monic divisors of ul −λ1 ∈ fq[u] is (pa + 1)r. theorem 4.3. the number of two-dimensional (λ1,λ2)-constacyclic codes of length n = 2l with exactly two generators is ((pa + 1)r) ((pa + 1)r −1) , where ul − λ1 = ∏r i=1 pi(u) pa and λp a 1 = λ1 for some distinct monic irreducible polynomials pi(u) for i = 1,2, . . . ,r. proof. initially, we show that for distinct divisors m1(u) and m2(u) of ul − λ1, the polynomials m1(u)(α + v) and m2(u)(α − v) are the generator polynomials of some two-dimensional (λ1,λ2)constacyclic code. for this, we suppose j is an ideal of fq[u,v]/〈ul − λ1,v2 − λ2〉 generated by {m1(u)(α+v),m2(u)(α−v)} and consider the corresponding two-dimensional (λ1,λ2)-constacyclic code c of fq[u,v]/〈ul −λ1,v2 −λ2〉. we can assume p1(u)(α + v) and p2(u)(α − v) are the generator polynomials of c as we discussed in the starting of this section. also, p1(u)(α + v) ∈ j, there exist polynomials g1(u,v) and g2(u,v) in fq[u,v]/〈ul −λ1,v2 −λ2〉 such that p1(u)(α + v) = g1(u,v)m1(u)(α + v) + g2(u,v)m2(u)(α−v) + a(u,v)(ul −λ1) + b(u,v)(v2 −λ2). next, consider the evaluation map ψ1 : fq[u,v] → fq[u] defined by ψ1(g(u,v)) = g(u,α). we have 2αp1(u) = 2αg1(u,α)m1(u) + a(u,α)(u l −λ1). since char(fq) 6= 2 and m1(u) is a divisor of ul −λ1, i.e., m1(u)|p1(u). as j1 = {f(u) ∈ fq[u]/〈ul −λ1〉 | f(u)(α + v) ∈ j} =< p1(u) >, and m1(u) ∈ j1, so p1(u)|m1(u), and hence p1(u) = m1(u). similarly, we can prove that p2(u) = m2(u). moreover, if m1(u) = m2(u), then p1(u) = p2(u) and j =< p1(u) >. thus, the number of two-dimensional (λ1,λ2)-constacyclic codes of length n = 2l with exactly two generators is ((pa + 1)r) ((pa + 1)r −1) . 170 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 now, we provide the generator matrix for a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l. theorem 4.4. let c be a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l with generator polynomials p1(u)(α + v) and p2(u)(α−v) where deg(p1(u)) = c and deg(p2(u)) = d. then g =   p1(u)(α + v) up1(u)(α + v) ... ul−c−1p1(u)(α + v) p2(u)(α−v) up2(u)(α−v) ... ul−d−1p2(u)(α−v)   is the generator matrix of c. proof. under the same notations given in theorem 4.1, suppose c1(u)p1(u)(α+v) +c2(u)p2(u)(α−v) and d1(u)p1(u)(α + v) + d2(u)p2(u)(α−v) are two elements of j such that c1(u)p1(u)(α + v) + c2(u)p2(u)(α−v) = d1(u)p1(u)(α + v) + d2(u)p2(u)(α−v). then we find the following in fq[u,v]: (c1(u)−d1(u))p1(u)(α + v) + (c2(u)−d2(u))p2(u)(α−v) = e(u,v)(ul −λ1) + e′(u,v)(v2 −λ2) for some e(u,v), e′(u,v) in fq[u,v]. now, we define the evaluation map ψm : fq[u,v] → fq[u] by ψm(g(u,v)) = g(u,m) for m = −α,α. then 2α(c1(u)−d1(u))p1(u) = e(u,α)(ul −λ1), and 2α(c2(u)−d2(u))p2(u) = e(u,−α)(ul −λ1). since degree of u in the right hand side of the above equalities is at least l whereas the corresponding degree in the left hand side is at most l − 1. therefore, ci(u) = di(u) for i = 1,2. hence, j has ql−cql−d elements. thus, the dimension of the corresponding two-dimensional (λ1,λ2)-constacyclic code is 2l− c−d. 4.2. generator matrix of c⊥ it is known that dim(c) + dim(c⊥) = 2l, then from theorem 4.4, we have dim(c) = 2l − c − d. therefore, dim(c⊥) = c + d. further, if g(u) is a generator polynomial of a λ1-constacyclic code c, then g(u) is a factor of ul −λ1 and so ul −λ1 = g(u)h(u) for some polynomial h(u) and h∗(u) is a codeword in c⊥. the following theorem gives the generator matrix for the dual c⊥. 171 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 theorem 4.5. let c be a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l with generator polynomials p1(u)(α + v) and p2(u)(α−v) such that deg p1(u) = c, p1(u)p′1(u) = ul −λ1, deg p2(u) = d and p2(u)p′2(u) = u l −λ1. then h =   (p′1) ∗(u)(1 + vα) u(p′1) ∗(u)(1 + vα) ... uc−1(p′1) ∗(u)(1 + vα) (p′2) ∗(u)(1−vα) u(p′2) ∗(u)(1−vα) ... ud−1(p′1) ∗(u)(1−vα)   is the generator matrix of c⊥. proof. one can easily check that the rows of h are independent by using the method given in theorem 4.4. now, (p′i) ∗(u) ∈c⊥ for i = 1,2; (α+v)∗ = 1+vα and (α−v)∗ = vα−1. therefore, (p′1)∗(u)(1+vα) and (p′2) ∗(u)(1−vα) are codewords in c⊥. hence, the result follows from [6]. 5. examples a linear code c of length n over fq with minimum distance d and dimension k is represented by [n,k,d]. if c is an [n,k,d] code, then from the singleton bound, its minimum distance is bounded above by d ≤ n−k + 1. a code meeting the above bound is known as maximum-distance-separable (mds). we call a code almost mds if its minimum distance is one unit less than the mds. a code is called optimal if it has the highest possible minimum distance for its length and dimension. we provide here a few examples in which all computations are carried out by using magma software [1]. example 5.1. we consider a two-dimensional cyclic code of length 6 over f7. in f7, we have u2 −1 = (u + 1)(u + 6). suppose p0(u) = p1(u) = u + 1 and p2(u) = u + 6 = u−1, then from theorem 3.1, the generator matrix of the tdc code is given by g =   (u + 1)(1 + v + v 2) (u + 1)(1−v2) (u−1)(1 + v2 −2v)   and codewords corresponding to rows of that matrix are c1 = ( 1 1 1 1 1 1 ) , c2 = ( 1 0 −1 1 0 −1 ) , c3 = ( −1 2 −1 1 −2 1 ) . hence, g =   1 1 1 1 1 11 0 −1 1 0 −1 −1 2 −1 1 −2 1   . here, the obtained tdc code [6,3,4] is an mds code. 172 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 table 1. two-dimensional cyclic and (λ1,λ2)-constacyclic codes q (λ1,λ2) (l,n) p0(u) p1(u) p2(u) obtained tdc/ remark constacyclic code 7 (1,1) (2,6) u + 1 u + 1 u − 1 [6,3,4] mds 5 (1,1) (3,9) u − 1 u − 1 1 + u + u2 [9,5,4] optimal 5 (1,1) (14,42) u + 1 u + 1 u + 4 [42,39,2] optimal 11 (1,1) (4,12) u + 1 u − 1 u2 + 1 [12,8,4] almost mds 9 (1,1) (5,10) u + 2 u2 + w3u + 1 [10,7,4] mds 5 (3,4) (3,6) u + 3 u2 + 2u + 4 [6,3,4] mds example 5.2. we consider a two-dimensional (3,4)-constacyclic code of length 6 over f5. in f5, we have u3 −3 = (u + 3)(u2 + 2u + 4) and also α = 2. suppose p1(u) = u + 3 and p2(u) = u2 + 2u + 4, then from theorem 4.4, the generator matrix of the two-dimensional (3,4)-constacyclic code is given by g =   (u + 3)(2 + v)u(u + 3)(2 + v) (u2 + 2u + 4)(2−v)   and codewords corresponding to rows of that matrix are c1 =  1 32 1 0 0  , c2 =  0 01 3 2 1  , c3 =  3 −44 −2 2 −1  . hence, g =  1 3 2 1 0 00 0 1 3 2 1 3 −4 4 −2 2 −1   . here, the obtained two-dimensional (3,4)-constacyclic code [6,3,4] is an mds code. 6. conclusion in this paper, we have obtained the generating set of polynomials for the ideals corresponding to the two-dimensional cyclic codes of length 3l and (λ1,λ2)-constacyclic codes of length n = 2l, respectively. in future, we would like to develop a technique by which we can find the generating set of polynomials for two-dimensional codes of arbitrary length. 173 o. prakash, s. patel / j. algebra comb. discrete appl. 9(3) (2022) 161–174 acknowledgment: the authors are thankful to the department of science and technology, govt. of india for financial support under ref no. dst/inspire/03/2016/001445 and indian institute of technology patna for providing the research facilities. references [1] w. bosma, j. cannon, handbook of magma functions, univ. of sydney, sydney (1995). 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[15] x. zheng, b. kong, cyclic codes and λ1 +λ2u+λ3v+λ4uv-constacyclic codes over fp +ufp +vfp + uvfp, appl. math. comput. 306 (2017) 86–91. 174 https://books.google.sk/books?id=xjfhyaaacaaj https://doi.org/10.1016/s0019-9958(72)90223-9 https://doi.org/10.1016/s0019-9958(75)80001-5 https://doi.org/10.1016/s0019-9958(77)90232-7 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://doi.org/10.1017/cbo9780511755279 https://doi.org/10.1504/ijicot.2020.10032824 https://doi.org/10.1504/ijicot.2020.10032824 https://doi.org/10.1007/s12095-022-00572-9 https://openlibrary.org/books/ol20255744m/cyclic_error-correcting_codes_in_two_symbols https://openlibrary.org/books/ol20255744m/cyclic_error-correcting_codes_in_two_symbols https://doi.org/10.1016/j.ffa.2017.11.008 https://doi.org/10.1016/j.ffa.2017.11.008 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1007/s00200-019-00388-w https://doi.org/10.1007/s00200-019-00388-w https://doi.org/10.1016/j.ffa.2013.08.001 https://doi.org/10.1016/j.ffa.2013.08.001 https://www.inderscienceonline.com/doi/abs/10.1504/ijicot.2014.066107 https://www.inderscienceonline.com/doi/abs/10.1504/ijicot.2014.066107 https://doi.org/10.1016/j.amc.2017.02.017 https://doi.org/10.1016/j.amc.2017.02.017 introduction notation and background two-dimensional cyclic codes of length 3l two-dimensional (1,2)-constacyclic codes of length 2l examples conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.938105 j. algebra comb. discrete appl. 8(2) • 119–138 received: 12 october 2020 accepted: 8 february 2021 journal of algebra combinatorics discrete structures and applications the exact annihilating-ideal graph of a commutative ring research article subramanian visweswaran, premkumar t. lalchandani abstract: the rings considered in this article are commutative with identity. for an ideal i of a ring r, we denote the annihilator of i in r by ann(i). an ideal i of a ring r is said to be an exact annihilating ideal if there exists a non-zero ideal j of r such that ann(i) = j and ann(j) = i. for a ring r, we denote the set of all exact annihilating ideals of r by ea(r) and ea(r)\{(0)} by ea(r)∗. let r be a ring such that ea(r)∗ 6= ∅. with r, in [exact annihilating-ideal graph of commutative rings, j. algebra and related topics 5(1) (2017) 27-33] p.t. lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of r, denoted by eag(r) whose vertex set is ea(r)∗ and distinct vertices i and j are adjacent if and only if ann(i) = j and ann(j) = i. in this article, we continue the study of the exact annihilating-ideal graph of a ring. in section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. in section 3, we determine the structure of eag(r), where either r is a special principal ideal ring or r is a reduced ring which admits only a finite number of minimal prime ideals. 2010 msc: 13a15, 05c25 keywords: exact annihilating ideal, exact annihilating-ideal graph, connectedness, reduced ring, special principal ideal ring 1. introduction the rings considered in this article are commutative with identity which are not integral domains. let r be a ring. for an element a ∈ r, the annihilator of a in r, denoted by annr(a) or simply by ann(a) is defined as ann(a) = {r ∈ r | ra = 0}. recall from [12] that an element x ∈ r is said to be an exact zero-divisor if there exists y ∈ r\{0} such that ann(x) = ry and ann(y) = rx. it is clear that any exact zero-divisor of r is a zero-divisor of r. we denote the set of all zero-divisors of a ring r by subramanian visweswaran (corresponding author); department of mathematics, saurashtra university, rajkot, india (retired) (email: s_visweswaran2006@yahoo.co.in). premkumar t. lalchandani; department of mathematics, dr. subhash science college, junagadh, india (email: finiteuniverse@live.com). 119 https://orcid.org/0000-0002-4905-809x https://orcid.org/0000-0001-8938-7552 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 z(r) and z(r)\{0} by z(r)∗. as in [15], we denote the set of all exact zero-divisors of r by ez(r) and ez(r)\{0} by ez(r)∗. let r be a ring such that ez(r)∗ 6= ∅. recall from [15] that the exact zerodivisor graph of r, denoted by eγ(r) is an undirected graph whose vertex set is ez(r)∗ and distinct vertices x and y are adjacent in eγ(r) if and only if ann(x) = ry and ann(y) = rx. several properties of the exact zero-divisor graph of a commutative ring were investigated in [15, 16]. let r be a ring. recall from [7] that an ideal i of r is said to be an annihilating ideal if there exists r ∈ r\{0} such that ir = (0). as in [7], we denote the set of all annihilating ideals of r by a(r) and a(r)\{(0)} by a(r)∗. the concept of annihilating-ideal graph of a commutative ring was introduced and investigated by m. behboodi and z. rakeei in [7]. let r be a ring. recall from [7] that the annihilating-ideal graph of r, denoted by ag(r) is an undirected graph whose vertex set is a(r)∗ and distinct vertices i and j are adjacent in this graph if and only if ij = (0). motivated by the interesting results proved on the annihilating-ideal graph of a ring in [7, 8], several researchers contributed to the study of annihilating-ideal graphs of commutative rings (for example, refer [1], [2], [11]). inspired by the above mentioned work on annihilating-ideal graphs of rings and by the work on exact zero-divisor graphs of rings in [15, 16], in [17], p.t. lalchandani introduced and studied the concept of the exact annihilating-ideal graph of a commutative ring. let r be a ring. recall from [17] that an ideal i of r is said to be an exact annihilating ideal if there exists a non-zero ideal j of r such that ann(i) = j and ann(j) = i, where for an ideal a of r, the annihilator of a in r, denoted by annr(a) or simply by ann(a) is defined as ann(a) = {r ∈ r | ra = (0)} [4, page 19]. as in [17], we denote the set of all exact annihilating ideals of a ring r by ea(r) and we denote ea(r)\{(0)} by ea(r)∗. it is clear that for any ring r, ea(r)∗ ⊆ a(r)∗. let r be a ring such that ea(r)∗ 6= ∅. recall from [17] that the exact annihilating-ideal graph of r, denoted by eag(r) is an undirected graph whose vertex set is ea(r)∗ and distinct vertices i and j are adjacent in eag(r) if and only if ann(i) = j and ann(j) = i. the graphs considered in this article are undirected and simple. for a graph g, we denote the vertex set of g by v (g) and the edge set of g by e(g). for a ring r with ea(r)∗ 6= ∅, it is clear that v (eag(r)) = ea(r)∗ ⊆ a(r)∗ = v (ag(r)). observe that if i,j ∈ ea(r)∗ are such that i and j are adjacent in eag(r), then ann(i) = j and ann(j) = i. hence, ij = (0) and so, i and j are adjacent in ag(r). therefore, eag(r) is a subgraph of ag(r). the aim of this article is to continue the study of the exact annihilating-ideal graph of a commutative ring which was carried out in [17]. throughout this article, we consider rings r such that ea(r)∗ 6= ∅ (it is noted in a remark which appears just preceding the statement of corollary 2.2 that for a ring r, ea(r)∗ 6= ∅ if and only if r is not an integral domain) and study the interplay between the graph-theoretic properties of eag(r) and the ring-theoretic properties of r. this article consists of three sections including the introduction. in section 2 of this article, we discuss some results on the exact annihilating ideals of r, where r is a commutative ring which is not an integral domain. let i ∈ a(r)∗. it is proved in lemma 2.1 that the statements (1) i ∈ ea(r)∗; (2)i = ann(j) for some non-zero ideal j of r; and (3) ann(ann(i)) = i are equivalent. for a ring r, we denote the set of all proper ideals of r by i(r) and i(r)\{(0)} by i(r)∗. many examples of rings r are provided in section 2 such that i(r)∗ = a(r)∗ = ea(r)∗ (see examples 2.3, 2.8, lemmas 2.4 and 2.6). we denote the cardinality of a set a by |a|. whenever a set a is a subset of a set b and a 6= b, then we denote it by a ⊂ b. it is well-known that for a ring t , |a(t)∗| = 1 if and only if (t,z(t))) is a special principal ideal ring (spir) with (z(t))2 = (0)[7, corollary 2.9(a)]. for a ring r, we denote the set of all prime ideals of r by spec(r) and the set of all maximal ideals of r by max(r). let i be a non-zero proper ideal of a ring r. motivated by [7, corollary 2.9(a)], it is shown in theorem 2.9 that the statements (1) ea(r)∗ = {i} and (2) i ∈ spec(r),i2 = (0), and z(r) = i are equivalent. in example 2.14, a ring r is provided to illustrate that (2) ⇒ (1) of theorem 2.9 can fail to hold if the assumption that i = z(r) is omitted. it is verified that the ring r given in example 2.14 is such that |ea(r)∗| = 2. inspired by this example, it is natural to try to determine necessary and sufficient conditions on the ideals i,j of a ring r such that ea(r)∗ = {i,j}. it is well-known that the set of all nilpotent elements of a ring r is an ideal of r [4, proposition 1.7] and is called the nilradical of r. we denote the nilradical of a ring r by nil(r). a ring r is said to be reduced if nil(r) = (0). we denote the set of all units of r by u(r). we denote the set of all minimal primes ideals of a ring r by min(r). for non-zero proper ideals i,j of a reduced ring r which is not an integral domain, it is proved in theorem 2.16 that the statements (1) ea(r)∗ = {i,j}; (2) j = ann(i), i,j ∈ spec(r); and (3) min(r) = {i,j} are equivalent. let r be a ring. let p ∈ spec(r) be such that p = rp is principal, 120 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 n ≥ 2 is least with the property that pn = (0), and p = z(r). then it is shown in proposition 2.10 that ea(r)∗ = {pi | i ∈{1, . . . ,n− 1}} and moreover, it is verified in proposition 2.10 that a(r)∗ = ea(r)∗ if and only if p ∈ max(r). let r be a reduced ring. let n ≥ 2 and let min(r) = {pi | i ∈{1, 2, . . . ,n}}. let c denote the collection of all non-empty proper subsets of {1, 2, . . . ,n}. it is proved in proposition 2.15 that ea(r)∗ = { ∏ i∈a pi | a ∈ c}. moreover, it is verified in proposition 2.15 that a(r)∗ = ea(r)∗ if and only if pi ∈ max(r) for each i ∈{1, 2, . . . ,n}. let t be a unique factorization domain (ufd). it is shown in theorem 2.17 that the statements (1) for each prime element p of t , a( t t p2 )∗ = ea( t t p2 )∗; (2) t is a principal ideal domain (pid); and (3) for each i ∈ i(t)∗ with i /∈ max(t), a( t i )∗ = ea( t i )∗ are equivalent. let t be a ufd with at least two non-associate prime elements. it is proved in theorem 2.18 that the statements (1) for all non-associate prime elements p1,p2 of t , a( tt p1p2 ) ∗ = ea( t t p1p2 )∗; (2) t is a pid; and (3) for any i ∈ i(t)∗ with i /∈ max(t), a( t i )∗ = ea( t i )∗ are equivalent. let r be a von neumann regular ring which is not a field. it is shown in corollary 2.19 that |ea(r)∗| < ∞ if and only if there exist n ≥ 2 and fields f1,f2, . . . ,fn such that r ∼= f1 ×f2 ×···×fn as rings. let r be a ring such that ea(r)∗ 6= ∅. the aim of section 3 of this article is to discuss some results regarding the properties of eag(r). let i,j ∈ ea(r)∗ be such that i 6= j. it is proved in proposition 3.1 that there is a path in eag(r) between i and j if and only if i and j are adjacent in eag(r). if i−j is an edge of eag(r), then for any a ∈ ea(r)∗\{i,j}, it is shown in lemma 3.2 that i and a are not adjacent in eag(r) and j and a are not adjacent in eag(r). as a consequence of lemma 3.2, it is deduced in corollary 3.3 that if g is any component of eag(r), then g is a complete graph with at most two vertices. let r be a ring. let p ∈ spec(r)\{(0)} be such that p2 = (0), and z(r) = p. it is noted in proposition 3.4 that ea(r)∗ = {p} and moreover, it is verified in proposition 3.4 that its conclusion holds for a spir (r,m) with m 6= (0) but m2 = (0). for a real number x, we denote the integer part of x by [x]. let r be a ring. let p ∈ spec(r) be such that p = rp is principal. let n ≥ 3 be least with the property that pn = (0) and z(r) = p. then it is proved in proposition 3.5 that the following statements hold: (1) if n is odd, then eag(r) has exactly [ n 2 ] components and each component is a complete graph with two vertices. (2) if n ≥ 4 is even, then eag(r) has exactly n 2 components g1, . . . ,gn 2 −1,gn 2 such that gj is a complete graph with two vertices for each j ∈ {1, . . . , n2 − 1} and gn2 is a complete graph on a single vertex. moreover, it is noted in proposition 3.5 that the statements (1) and (2) hold for a spir (r,m) with the property that mn = (0) but mn−1 6= (0). let r,p = rp = z(r) be as in the statement of proposition 3.5. let n ≥ 2 be least with the property that pn = (0). then it is shown in theorem 3.7 that the statements (1) eag(r) = ag(r) and (2) (r,p) is a spir and n ∈ {2, 3} are equivalent. it is verified in example 3.8 that the ring r provided by d.d. anderson and m. naseer in [3, page 501] is such that eag(r) 6= ag(r) which illustrates that (2) ⇒ (1) of theorem 3.7 can fail to hold if the hypothesis that p is principal is omitted. let r be a reduced ring which is not an integral domain. it is shown in lemma 3.9 that each component of eag(r) is a complete graph with two vertices. it is proved in corollary 3.10 that eag(r) is connected if and only if |min(r)| = 2 and it is shown in corollary 3.11 that eag(r) = ag(r) if and only if r ∼= f1 × f2 as rings, where fi is a field for each i ∈ {1, 2}. if |min(r)| = n ≥ 2, then it is proved in corollary 3.12 that eag(r) has exactly 2n−1 − 1 components. let r be a ring such that ea(r)∗ 6= ∅. it is shown in theorem 3.14 that the statements (1) eag(r) = ag(r) and (2) either r ∼= f1 ×f2 as rings, where fi is a field for each i ∈ {1, 2} or (r,m) is a spir and if n ≥ 2 is least with the property that mn = (0), then n ∈{2, 3} are equivalent. let r be a ring. the krull dimension of r is simply referred to as the dimension of r and is denoted by dimr. let r be a ring such that dimr = 0. if eag(r) is connected, then it is proved in proposition 3.16 that |max(r)| ≤ 2 and if |max(r)| = 2 then it is shown in corollary 3.17 that eag(r) is connected if and only if r ∼= f1×f2 as rings, where fi is a field for each i ∈{1, 2}. let r be a ring such that ea(r)∗ 6= ∅. it is noted in corollary 3.20 that girth(eag(r)) = ∞ and eag(r) is perfect. 2. some basic properties of ea(r)∗ as mentioned in the introduction, the rings considered in this article are commutative with identity. let r be a ring such that ea(r)∗ 6= ∅. the aim of this section is to discuss some basic properties of the 121 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 exact annihilating ideals of r. let r be a ring which is not an integral domain. let i ∈ a(r)∗. in lemma 2.1, we provide a necessary and sufficient condition for i to be in ea(r)∗. lemma 2.1. let r be a ring and let i ∈ a(r)∗. the following statements are equivalent: (1) i ∈ ea(r)∗. (2) i = ann(j) for some non-zero ideal j of r. (3) ann(ann(i)) = i. proof. (1) ⇒ (2) we are assuming that i ∈ ea(r)∗. hence, by definition, there exists a non-zero ideal j of r such that ann(i) = j and ann(j) = i. (2) ⇒ (3) we are assuming that i = ann(j) for some non-zero ideal j of r. note that ann(ann(i)) = ann(ann(ann(j)) = ann(j) = i. (3) ⇒ (1) we are assuming that ann(ann(i)) = i. let us denote ann(i) by j. since i ∈ a(r)∗ by hypothesis, ann(i) 6= (0). thus j 6= (0) and is such that ann(i) = j and ann(j) = i. this proves that i ∈ ea(r)∗. let r be a ring which is not an integral domain. then a(r)∗ 6= ∅. let a ∈ a(r)∗. then ann(a) 6= (0) and as a(ann(a)) = (0), it follows that ann(a) ∈ a(r)∗. it follows from (2) ⇒ (1) of lemma 2.1 that ann(a) ∈ ea(r)∗. the above arguments imply that for a ring r, ea(r)∗ 6= ∅ if and only if r is not an integral domain. corollary 2.2. let r be a ring such that a(r)∗ 6= ∅. the following statements are equivalent: (1) a(r)∗ = ea(r)∗. (2) if i ∈ a(r)∗, then i = ann(j) for some non-zero ideal j of r. (3) for any i ∈ a(r)∗, ann(ann(i)) = i. proof. the statements (1) ⇒ (2) and (2) ⇒ (3) follow respectively from (1) ⇒ (2) and (2) ⇒ (3) of lemma 2.1. for any ring t , as ea(t)∗ ⊆ a(t)∗, the proof of (3) ⇒ (1) follows immediately from (3) ⇒ (1) of lemma 2.1. we illustrate corollary 2.2 with the help of the example provided by d.d. anderson and m. naseer in [3, page 501]. we verify that i(r)∗ = a(r)∗ = ea(r)∗ for the ring r provided in [3, page 501] in example 2.3. for any n ≥ 2, we denote the ring of integers modulo n by zn. example 2.3. let t = z4[x,y,z] be the polynomial ring in three variables x,y,z over z4. let i be the ideal of t generated by {x2 − 2,y 2 − 2,z2,xy,y z − 2,xz, 2x, 2y, 2z}. let r = t i . then i(r)∗ = a(r)∗ = ea(r)∗. proof. it is convenient to denote x + i,y + i,z + i by x,y,z, respectively. it was already noted in [3, page 501] that r is local with m = rx + ry + rz as its unique maximal ideal, m2 = {0 + i, 2 + i}, m3 = (0 + i), and |r| = 32. observe that z(r) = m and from m3 = (0), we get that each proper ideal of r is an annihilating ideal of r. therefore, i(r)∗ = a(r)∗. the ring r was also considered in [8, proposition 2.1] and it was noted there that i(r)∗ = {(2 +i), (x), (y), (z), (x+y), (y +z), (z +x), (x+y + z), (x,y), (y,z), (z,x), (x,y +z), (y,z +x), (z,x+y), (x+y,y +z), (x,y,z)}. from the multiplication table provided in [3, page 503], it follows that ann(2+i) = m,ann(x) = {0+i, 2+i,y,y+2,z,z+2,y+z,y+z+ 2},ann(y) = {0+1, 2+i,x,x+2,y+z,y+z+2,x+y+z,x+y+z+2},ann(z) = {0+i, 2+i,x,x+2,z,z+ 2,x+z,x+z+2},ann(x+y) = {0+i, 2+i,y+z,y+z+2,z+x,z+x+2,x+y,x+y+2},ann(y+z) = {0+ i, 2+i,x,x+2,y,y+2,x+y,x+y+2},ann(z+x) = {0+i, 2+i,z,z+2,x+y,x+y+2,x+y+z,x+y+z+2}, and ann(x + y + z) = {0 + i, 2 + i,x + z,x + z + 2,y,y + 2,x + y + z,x + y + z + 2}. note that 122 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 x(x + z) = y(x + z) = z(y + z) = x(x + y) = x(x + y + z) = 2 + i. from the above given arguments, it is clear that ann(m2) = m,ann(m) = m2,ann(rx) = ry + rz,ann(ry + rz) = rx,ann(ry) = rx + r(y + z),ann(rx + r(y + z)) = ry,ann(rz) = rx + rz,ann(rx + rz) = rz,ann(r(x + y)) = r(y + z) + r(z + x),ann(r(y + z) + r(z + x)) = r(x + y),ann(r(y + z)) = rx + ry,ann(rx + ry) = r(y+z),ann(r(z+x)) = rz+r(x+y),ann(rz+r(x+y)) = r(z+x),ann(r(x+y+z)) = ry+r(x+z), and ann(ry + r(x + z)) = r(x + y + z). from the above discussion, we obtain that i(r)∗ = a(r)∗ and each proper a of r is such that ann(ann(a)) = a. hence, we obtain from (3) ⇒ (1) of corollary 2.2 that a(r)∗ = ea(r)∗. therefore, i(r)∗ = a(r)∗ = ea(r)∗. recall that a principal ideal ring r is said to be a special principal ideal ring (spir) if r has a unique prime ideal. if m is the unique prime ideal of r, then it follows from [4, proposition 1.8] that nil(r) = m. since m is principal, we get that m is nilpotent. suppose that r is not a field. then m 6= (0). let n ≥ 2 be least with the property that mn = (0). then it follows from the proof of (iii) ⇒ (i) of [4, proposition 8.8] that {mi | i ∈{1, . . . ,n−1}} is the set of all non-zero proper ideals of r. if r is a spir with m as its only prime ideal, then we denote it by the notation (r,m) is a spir. let (r,m) be a spir which is not a field. we verify in lemma 2.4 that i(r)∗ = a(r)∗ = ea(r)∗. lemma 2.4. let (r,m) be a spir which is not a field. then i(r)∗ = a(r)∗ = ea(r)∗. proof. let n ≥ 2 be least with the property that mn = (0). note that i(r)∗ = {mi | i ∈{1, . . . ,n−1}}. from mn = (0), it follows that i(r)∗ = a(r)∗. let i ∈ {1, . . . ,n − 1}. observe that ann(mi) = mn−i and so, ann(ann(mi)) = ann(mn−i) = mi. therefore, we obtain from (3) ⇒ (1) of corollary 2.2 that a(r)∗ = ea(r)∗. this proves that i(r)∗ = a(r)∗ = ea(r)∗. corollary 2.5. let t be a pid which is not a field. let m ∈ max(t). let n ≥ 2 and let r = t mn . then i(r)∗ == a(r)∗ = ea(r)∗. proof. let m ∈ m be such that m = tm. observe that r is a principal ideal ring. note that m mn ∈ spec(r). let p ∈ spec(r). then p = p mn for some p ∈ spec(t) with p ⊇ mn. this implies that p ⊇ m and so, p = m. therefore, p = m mn . thus r is a principal ideal ring with spec(r) = { m mn }. hence, (r, m mn ) is a spir. therefore, we obtain from lemma 2.4 that i(r)∗ = a(r)∗ = ea(r)∗. we provide some more examples in example 2.8 to illustrate corollary 2.2. we use lemmas 2.6 and 2.7 in the verification of example 2.8. lemma 2.6. let n ≥ 2 and let ri be a ring for each i ∈ {1, 2, . . . ,n}. let r = r1 × r2 ×···× rn. suppose that for any i ∈ {1, 2, . . . ,n} and any ideal ii of ri, annri (annri (ii)) = ii. then i(r)∗ = a(r)∗ = ea(r)∗. proof. let i ∈ i(r)∗. then for each i ∈ {1, 2, . . . ,n}, there exists an ideal ii of ri such that i = i1 × i2 × ···× in. since i 6= r, it follows that ii 6= ri for at least one i ∈ {1, 2, . . . ,n}. if ii = (0), then i(rei) = (0) × (0) × ···× (0), where ei is the element of r whose i-th coordinate equals 1 and whose j-th coordinate equals 0 for all j ∈ {1, 2, . . . ,n}\{i}. as rei is a non-zero ideal of r and i(rei) equals the zero ideal of r, it follows that i ∈ a(r)∗. suppose that ii 6= (0). then from the hypothesis, annri (annri (ii)) = ii, we get that ii ∈ a(ri)∗ and so, i ∈ a(r)∗. this shows that i(r)∗ ⊆ a(r)∗ and so, i(r)∗ = a(r)∗. let i = i1 × i2 × ···× in ∈ i(r)∗ = a(r)∗. observe that ann(ann(i)) = annr1 (annr1 (i1))×annr2 (annr2 (i2))×···×annrn (annrn (in)) = i1×i2×···×in = i. thus for each i ∈ a(r)∗, ann(ann(i)) = i. hence, we obtain from (3) ⇒ (1) of corollary 2.2 that a(r)∗ = ea(r)∗. therefore, i(r)∗ = a(r)∗ = ea(r)∗. lemma 2.7. let r be a ring and let m ∈ max(r). let q be a m-primary ideal of r. then r q ∼= rmqm as rings. 123 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. this is well-known. we provide a proof of this lemma for the sake of completeness. let f : r → rm be the usual homomorphism of rings defined by f(r) = r1. using the hypothesis that q is m-primary, it can be shown that f−1(qm) = q. hence, f induces an injective ring homomorphism f : r q → rm qm defined by f(r + q) = f(r) + qm. we verify that f is onto. let y be any element of rm qm . then there exist r ∈ r,s ∈ r\m such that y = r s + qm. since s ∈ r\m and m ∈ max(r), we get that m + rs = r. hence, √ q + √ rs = r and so, we obtain from [4, proposition 1.16] that q + rs = r. therefore, there exist x ∈ r and q ∈ q such that q + xs = 1. hence, r = rq + rxs and so, y = r s + qm = rsx+rq s + qm = rx 1 + qm, since rq s ∈ qm. thus y = f(rx + q). this shows that f is onto. hence, f : r q → rm qm is an isomorphism of rings. therefore, r q ∼= rmqm as rings. example 2.8. (1) let n ≥ 2. let r = f1 ×f2 ×···×fn, where fi is a field for each i ∈{1, 2, . . . ,n}. then i(r)∗ = a(r)∗ = ea(r)∗. (2) let t be a dedekind domain and let i be a non-zero proper ideal of t such that i /∈ max(t). let r = t i . then i(r)∗ = a(r)∗ = ea(r)∗. (3) let t be a principal ideal domain. let i be a non-zero proper ideal of t such that i /∈ max(t). let r = t i . then i(r)∗ = a(r)∗ = ea(r)∗. proof. (1) let i ∈{1, 2, . . . ,n}. observe that fi and (0) are the only ideals of fi and for each ideal ii of fi, annfi (annfi (ii)) = ii. now, it follows from lemma 2.6 that i(r) ∗ = a(r)∗ = ea(r)∗. (2) since t is a dedekind domain, t is noetherian, dimt = 1, and t is integrally closed. thus any non-zero prime ideal of t is maximal. it follows from [4, corollary 9.4] that there exist distinct maximal ideals m1, . . . ,mn of t and positive integers k1, . . . ,kn such that i = n∏ i=1 mkii . observe that for each i ∈{1, . . . ,n}, √ mkii = mi ∈ max(t) and so, we obtain from [4, proposition 4.2] that m ki i is a mi-primary ideal of t. we know from lemma 2.7 that t m ki i ∼= tmi (m ki i )mi as rings. we know from (i) ⇒ (iii) of [4, theorem 9.3] that tmi is a discrete valuation ring and so, it is a pid. now, for all distinct i,j ∈{1, . . . ,n},√ mkii + √ m kj j = t and so by [4, proposition 1.16] that m ki i + m kj j = t . hence, we obtain from [4, proposition 1.10] that t i ∼= t m k1 1 ×···× t m kn n . note that for each i ∈{1, . . . ,n}, (mi)mi is the unique maximal ideal of tmi and (m ki i )mi = ((mi)mi ) ki. therefore, we obtain that t i ∼= tm1 ((m1)m1 ) k1 ×···× tmn ((mn)mn ) kn as rings. let i ∈{1, . . . ,n} and let us denote the ring tmi ((mi)mi ) ki by ri and the unique maximal ideal (mi)mi of tmi by ni. if ki = 1, then ri is a field. if ki ≥ 2, then as tmi is a pid, we obtain from the proof of corollary 2.5 that (ri, ni n ki i ) is a spir. from the proof of lemma 2.4, we know that annri (annri (ii)) = ii for each ideal ii of ri. now, r ∼= r1 ×···× rn as rings. suppose that n = 1. since i /∈ max(t) by hypothesis, k1 ≥ 2 and so, it follows from lemma 2.4 that i(r)∗ = a(r)∗ = ea(r)∗. suppose that n ≥ 2. as annri (annri (ii)) = ii for each ideal ii of ri and for each i ∈ {1, 2, . . . ,n}, we obtain from lemma 2.6 that i(r)∗ = a(r)∗ = ea(r)∗. (3) if t is a pid which is not a field, then we know from [4, example (1), page 96] that t is a dedekind domain. therefore, the conclusion of (3) follows immediately from (2). let r be a ring such that a(r)∗ 6= ∅. it was shown in [7, corollary 2.9(a)] that a(r)∗ = {i} if and only if (r,i) is a spir with i2 = (0) (see also, [19, lemma 2.6]). let r be a ring and let i ∈ a(r)∗. in theorem 2.9, we determine necessary and suffcient conditions on i such that ea(r)∗ = {i}. theorem 2.9. let r be a ring and let i be a non-zero ideal of r. the following statements are equivalent: (1) ea(r)∗ = {i}. (2) i ∈ spec(r),i2 = (0), and z(r) = i. 124 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. (1) ⇒ (2) as i ∈ ea(r)∗, i ∈ a(r)∗ and ann(i) ∈ a(r)∗. hence, we obtain from (2) ⇒ (1) of lemma 2.1 that ann(i) ∈ ea(r)∗ = {i} and so, ann(i) = i. this proves that i2 = (0). we next verify that i ∈ spec(r). it is clear that i 6= r. if b is any non-zero ideal of r with ann(b) 6= (0), then ann(b) ∈ a(r)∗ and it follows from (2) ⇒ (1) of lemma 2.1 that ann(b) ∈ ea(r)∗ = {i}. hence, ann(b) = i. let a,b ∈ r be such that ab ∈ i = ann(i). this implies that iab = (0). suppose that a /∈ i = ann(i). then ia 6= (0) and from i(ia) = (0), it follows that ann(ia) 6= (0). hence, ann(ia) = i. from b ∈ ann(ia), we get that b ∈ i. this proves that i ∈ spec(r). as any member of a(r) is a subset of z(r), it follows that i ⊆ z(r). let r ∈ z(r)∗. then there exists s ∈ r\{0} such that rs = 0. as rs 6= (0) and ann(rs) 6= (0), it follows that ann(rs) = i. from rs = 0, we obtain that r ∈ i. this shows that z(r) ⊆ i and so, z(r) = i. (2) ⇒ (1) we claim that for any non-zero ideal j of r with j ⊆ i, ann(j) = i. let r ∈ ann(j). then rj = (0). from j 6= (0), it follows that r ∈ z(r) = i. this shows that ann(j) ⊆ i. from i2 = (0) and j ⊆ i, we get that ji ⊆ i2 = (0). this shows that i ⊆ ann(j). therefore, ann(j) = i and so, in particular, ann(i) = i. hence, i ∈ ea(r)∗. let a ∈ ea(r)∗. then there exists a non-zero ideal b of r such that ann(a) = b and ann(b) = a. this implies that ab = (0) and so, a∪b ⊆ z(r) = i. hence, ann(a) = ann(b) = i. from ann(b) = a, we obtain that a = i. this proves that ea(r)∗ = {i}. let r be a ring such that it admits p ∈ spec(r) with p = rp is principal, p 6= (0) but p is nilpotent. let n ≥ 2 be least with the property that pn = (0). if z(r) = p, then we prove in proposition 2.10 that ea(r)∗ = {pi | i ∈{1, . . . ,n− 1}}. proposition 2.10. let r be a ring. let p ∈ spec(r) be such that p = rp is principal, p 6= (0) but p is nilpotent, and z(r) = p. let n ≥ 2 be least with the property that pn = (0). then ea(r)∗ = {pi | i ∈ {1, . . . ,n− 1}}. moreover, a(r)∗ = ea(r)∗ if and only if p ∈ max(r). proof. let i ∈ {1, . . . ,n − 1}. as p = rp, we get that pi = rpi. from pn = 0, it follows that pn−i ∈ ann(rpi). hence, rpn−i ⊆ ann(rpi). let r ∈ ann(rpi). then rpi = 0. as pi 6= 0, we obtain that r ∈ z(r) = rp. we claim that r ∈ rpn−i. this is clear if r = 0. suppose that r 6= 0. it is possible to find j ∈ {1, . . . ,n− 1} such that r ∈ rpj\rpj+1. hence, there exists s ∈ r\z(r) such that r = pjs. from rpi = 0, we get that spi+j = 0. as s ∈ r\z(r), it follows that pi+j = 0. since n is least with the property that pn = 0, we obtain that i + j ≥ n and so, j ≥ n − i. therefore, r ∈ rpj ⊆ rpn−i. this proves that ann(rpi) ⊆ rpn−i and so, we obtain that ann(rpi) = rpn−i. as n− i ∈{1, . . . ,n− 1}, it follows that ann(rpn−i) = rpi. thus for any i ∈ {1, . . . ,n− 1}, ann(pi) = pn−i and ann(pn−i) = pi. this proves that {pi | i ∈{1, . . . ,n−1}}⊆ ea(r)∗. let a ∈ ea(r)∗. then there exists a non-zero ideal b of r such that ann(a) = b and ann(b) = a. from ab = (0), we get that a∪b ⊆ z(r) = rp. it is possible to find j ∈{1, . . . ,n− 1} such that b ⊆ rpj but b 6⊆ rpj+1. note that rpn−j ⊆ ann(b) = a. let b ∈ b\rpj+1. as b ⊆ rpj, it follows that b = spj for some s ∈ r\z(r). from ab = (0), we obtain that for any a ∈ a, a(spj) = 0 and so, apj = 0. this implies that a ∈ ann(rpj) = rpn−j. this shows that a ⊆ rpn−j and so, a = rpn−j. hence, ea(r)∗ ⊆ {pi | i ∈ {1, . . . ,n− 1}}. therefore, we obtain that ea(r)∗ = {pi | i ∈{1, . . . ,n− 1}}. we next verify the moreover part of this proposition. assume that a(r)∗ = ea(r)∗. as ea(r)∗ = {pi | i ∈ {1, . . . ,n − 1}}, we obtain that a(r)∗ is finite. hence, r satisfies descending chain condition (d.c.c.) on a(r)∗. therefore, it follows from [7, theorem 1.1] that r is artinian and so, we obtain from [4, proposition 8.1] that p ∈ max(r). we also include a direct argument to show that p ∈ max(r). let m ∈ max(r) be such that p ⊆ m. let m ∈ m. if pm = 0, then m ∈ z(r) = p. suppose that pm 6= 0. note that rpm ∈ a(r)∗. therefore, rpm = pi = rpi for some i ∈{1, . . . ,n−1}. if i = 1, then p = rpm for some r ∈ r. hence, p(1 −rm) = 0. this implies that 1 −rm ∈ z(r) = p ⊆ m and so, 1 ∈ m. this is impossible and therefore, i ≥ 2. from pn = 0 and rpm = rpi, it follows that pn−i+1m = 0. as i ≥ 2, it follows that pn−i+1 6= 0. hence, m ∈ z(r) = p. this proves that m ⊆ p and so, p = m ∈ max(r). conversely, assume that p ∈ max(r). let p ∈ spec(r). now, p ⊇ (0) = pn. this implies that p ⊇ p. since p ∈ max(r), it follows that p = p. therefore, spec(r) = max(r) = {p}. now, p = rp is principal and n ≥ 2 is least with the property that pn = (0). hence, we obtain from the proof of (iii) ⇒ (i) 125 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 of [4, proposition 8.8] that {pi = rpi | i ∈ {1, . . . ,n− 1}} is the set of all non-zero proper ideals of r. therefore, it follows that (r,p) is a spir and so, i(r)∗ = a(r)∗ = ea(r)∗ = {pi | i ∈{1, . . . ,n−1}}. we provide example 2.12 to illustrate theorem 2.9 and proposition 2.10. we use lemma 2.11 in the verification of example 2.12. lemma 2.11. let p be a prime element of an integral domain t. let n ≥ 2. let r = t t pn . let p = t p t pn . then ea(r)∗ = {pi | i ∈{1, . . . ,n− 1}}. proof. by hypothesis, p is a prime element of t. hence, tp ∈ spec(t) and so, p = t p t pn ∈ spec(r = t t pn ). it is clear that p = r(p + tpn) is principal. observe that n ≥ 2 is least with the property that pn = (0 + tpn). note that tpn is a tp-primary ideal of t . hence, the zero ideal (0 + tpn) of r is a p-primary ideal of r. therefore, we obtain from [4, proposition 4.7] that z(r) = p. now, it follows from proposition 2.10 that ea(r)∗ = {pi | i ∈{1, . . . ,n− 1}}. example 2.12. let t = z[x] be the polynomial ring in one variable x over z. let n ≥ 2 and let r = t t xn . let p = t x t xn . then ea(r)∗ = {pi | i ∈ {1, . . . ,n− 1}}, a(r)∗ = {i ∈ i(r)∗ | i ⊆ p}, and a((r)∗ 6= ea(r)∗. proof. note that t is an integral domain. indeed, t is a unique factorization domain and x is a prime element of t . therefore, we obtain from lemma 2.11 that ea(r)∗ = {pi | i ∈{1, . . . ,n−1}}. let i ∈ i(r)∗ be such that i ⊆ p. note that pn−1 6= (0 + txn) and ipn−1 = (0 + txn). hence, i ∈ a(r)∗. let a ∈ a(r)∗. as any annihilating ideal of a ring is contained in its set of zero-divisors, we get that a ⊆ z(r) = p. this proves that a(r)∗ = {i ∈ i(r)∗ | i ⊆ p}. observe that i = r(2x + txn) ⊆ p, i 6= (0 + txn), and i /∈ {pi | i ∈ {1, . . . ,n − 1}}. hence, i ∈ a(r)∗\ea(r)∗. therefore, a(r)∗ 6= ea(r)∗. in example 2.13, we illustrate that (2) ⇒ (1) of theorem 2.9 can fail to hold if the assumption that i2 = (0) is omitted. example 2.13. let r be as in example 2.3. in the notation of example 2.3, r is a local artinian ring with unique maximal ideal m = rx + ry + rz, m3 = (0), and z(r) = m. it is already verified in the verification of example 2.3 that r has 16 non-zero proper ideals and i(r)∗ = a(r)∗ = ea(r)∗. in example 2.14, we illustrate that (2) ⇒ (1) of theorem 2.9 can fail to hold if the assumption that z(r) = i is omitted. example 2.14. let t = k[x,y ] be the polynomial ring in two variables x,y over a field k. let a = tx2 + txy . let r = t a . let p = t x a . then p = r(x + a) is principal, p2 = (0 + a), and |ea(r)∗| = 2. proof. as x is a prime element of t , it follows that tx ∈ spec(t) and so, p = t x a ∈ spec( t a = r). it is clear that p = r(x + a) is principal and from x2 ∈ a, we obtain that p2 = (0 + a). observe that t t x+t y ∼= k as rings. from k is a field, it follows that tx + ty ∈ max(t) and so, m = t x+t ya ∈ max(r). it is convenient to denote x + a by x and y + a by y. note that m = rx + ry. observe that a = tx ∩ (tx2 + ty ). as √ tx2 + ty = tx + ty ∈ max(t), we obtain from [4, proposition 4.2] that tx2 + ty is a tx + ty -primary ideal of t . hence, a = tx ∩ (tx2 + ty ) is a minimal primary decomposition of a with tx is a tx-primary ideal of t and tx2 + ty is a tx + ty -primary ideal of t. therefore, (0 + a) = t x a ∩ t x 2+t y a is a minimal primary decomposition of the zero ideal of r with t x a is a p-primary ideal of r and t x 2+t y a is a m-primary ideal of r. hence, it follows from [4, proposition 4.7] that z(r) = p ∪ m = m, since p ⊂ m. note that mp = (0 + a) and so, m ⊆ ann(p). from m ∈ max(r) and ann(p) 6= r, it follows that ann(p) = m. from pm = (0 + a), we get that 126 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 p ⊆ ann(m). let r ∈ ann(m). then rm = (0 + a) ⊂ p. as p ∈ spec(r) and m 6⊆ p, we obtain that r ∈ p. this proves that ann(m) ⊆ p and so, ann(m) = p. thus the non-zero ideals m,p of r are such that ann(p) = m and ann(m) = p. hence, p,m ∈ ea(r)∗ and so, |ea(r)∗| ≥ 2. let c ∈ ea(r)∗. we claim that c ∈ {p,m}. as c ∈ ea(r)∗, there exists a non-zero ideal d of r such that ann(c) = d and ann(d) = c. observe that cd = (0 + a) ⊂ p. from p ∈ spec(r), it follows that either c ⊆ p or d ⊆ p. suppose that c ⊆ p. then m = ann(p) ⊆ ann(c) = d. from d 6= r, we get that d = m and so, c = ann(d) = p. if d ⊆ p, then it follows similarly that c = m. therefore, c ∈{p,m}. this proves that ea(r)∗ = {p,m} and so, |ea(r)∗| = 2. let r be a ring such that ea(r)∗ 6= ∅. inspired by theorem 2.9 and example 2.14, we try to characterize ideals i,j of a ring r such that ea(r)∗ = {i,j}. in theorem 2.16, we are able to characterize ideals i,j of a reduced ring r such that ea(r)∗ = {i,j}. we use proposition 2.15 in the proof of (3) ⇒ (1) of theorem 2.16. let r be a reduced ring which is not an integral domain. suppose that |ea(r)∗| < ∞. let a ⊆ z(r)∗ be such that xy = 0 for all distinct x,y ∈ a. note that for each x ∈ a, x 6= 0, ann(x) ∈ a(r)∗, and it follows from (2) ⇒ (1) of lemma 2.1 that ann(x) ∈ ea(r)∗. let x,y ∈ a be such that x 6= y. observe that y ∈ ann(x). since r is reduced and y 6= 0, it follows that y /∈ ann(y). hence, ann(x) 6= ann(y). from the assumption that |ea(r)∗| < ∞, we get that a is finite. hence, we obtain from (4) ⇒ (3) of [6, theorem 3.7] that min(r) is finite. this shows that if |ea(r)∗| < ∞ for a reduced ring r, then |min(r)| < ∞. let r be a reduced ring with |min(r)| = n ≥ 2. then we prove in proposition 2.15 that |ea(r)∗| = 2n − 2. proposition 2.15. let r be a reduced ring which is not an integral domain. let |min(r)| = n and let min(r) = {p1, . . . ,pn}. then |ea(r)∗| = 2n − 2. moreover, a(r)∗ = ea(r)∗ if and only if pi ∈ max(r) for each i ∈{1, 2, . . . ,n}. proof. it is known that any prime ideal p of a ring t contains a minimal prime ideal of t [14, theorem 10]. since r is reduced, nil(r) = (0). we know from [4, proposition 1.8] that (0) = nil(r) = ⋂ p∈spec(r) p. since any prime ideal of r contains at least one minimal prime ideal of r, we obtain that ⋂ p∈min(r) p = (0). as min(r) = {p1, . . . ,pn}, we obtain that n⋂ i=1 pi = (0). it is clear that n ≥ 2, since r is not an integral domain. note that distinct minimal prime ideals of a ring r are not comparable under the inclusion relation and hence, it follows from [4, proposition 1.11(ii)] that for any proper non-empty subset a of {1, 2, . . . ,n}, ⋂ i∈a pi 6= (0). let a ⊂ {1, 2, . . . ,n} with a 6= ∅. let us denote ⋂ i∈a pi by ia. observe that for any a ⊂{1, 2, . . . ,n} with a 6= ∅, ac ⊂{1, 2, . . . ,n} and ac 6= ∅, where ac = {1, 2, . . . ,n}\a and it is easy to verify that ann(ia) = iac. hence, ia ∈ a(r)∗ and note that ann(ann(ia)) = ia. therefore, we obtain from (3) ⇒ (1) of lemma 2.1 that ia ∈ ea(r)∗. this proves that {ia |a ⊂{1, 2, . . . ,n},a 6= ∅}⊆ ea(r)∗. let i ∈ ea(r)∗. as i ∈ a(r)∗, ir = (0) for some r ∈ r\{0}. since n⋂ i=1 pi = (0), r /∈ pi for at least one i ∈ {1, 2, . . . ,n}. from ir = (0) ⊂ pi ∈ spec(r), we get that i ⊆ pi. since i 6= (0), there exists at least one j ∈{1, 2, . . . ,n} such that i 6⊆ pj. thus there exists a ⊂{1, 2, . . . ,n},a 6= ∅ such that i ⊆ pi for each i ∈ a and i 6⊆ pj for any j ∈ {1, 2, . . . ,n}\a. from iann(i) = (0) ⊆ pj for any j ∈ ac, we obtain that ann(i) ⊆ pj for each j ∈ ac. thus i ⊆ ia and ann(i) ⊆ iac. from ann(i) ⊆ iac, it follows that ia = ann(iac ) ⊆ ann(ann(i)). since i ∈ ea(r)∗, we obtain from (1) ⇒ (3) of lemma 2.1 that ann(ann(i)) = i and so, ia ⊆ i. hence, i = ia for some a ⊂ {1, 2, . . . ,n} with a 6= ∅. this proves that ea(r)∗ ⊆{ia | a ⊂{1, 2, . . . ,n},a 6= ∅} and so, ea(r)∗ = {ia | a ⊂{1, 2, . . . ,n},a 6= ∅}. if a1,a2 are distinct non-empty proper subsets of {1, 2, . . . ,n}, then it is clear that ia1 6= ia2. since |{a ⊂{1, 2, . . . ,n},a 6= ∅}| = 2n − 2, it follows that |ea(r)∗| = 2n − 2. we next verify the moreover part of this proposition. assume that a(r)∗ = ea(r)∗. hence, |a(r)∗| = 2n − 2 < ∞. therefore, r satisfies d.c.c. on a(r)∗ and so, we obtain from [7, theorem 1.1] that r is artinian. we know from [4, proposition 8.1] that spec(r) = max(r). therefore, pi ∈ max(r) 127 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 for each i ∈ {1, 2, . . . ,n}. we also include a direct argument to show that pi ∈ max(r) for each i ∈ {1, 2, . . . ,n}. first, we show that p1 ∈ max(r). let m ∈ max(r) be such that p1 ⊆ m. since distinct minimal prime ideals of a ring are not comparable under the inclusion relation, it follows from [4, proposition 1.11(ii)] that there exists x ∈ ( n⋂ j=2 pj)\p1. let m ∈ m. suppose that xm 6= 0. as x ∈ z(r)∗, it follows that xm ∈ z(r)∗, and so, rxm ∈ a(r)∗. it is shown in the previous paragraph that ea(r)∗ = {ia | a ⊂{1, 2, . . . ,n},a 6= ∅}, where for a non-empty proper subset a of {1, 2, . . . ,n}, ia = ⋂ i∈a pi. from the assumption a(r)∗ = ea(r)∗, it follows that rxm = ia for some non-empty proper subset a of {1, 2, . . . ,n}. from xm ∈ ( n⋂ j=2 pj)\{0} and n⋂ i=1 pi = (0), we get that 1 /∈ a. hence, a ⊆ {1, 2, . . . ,n}\{1}. it follows from the choice of x that x ∈ ia. therefore, x ∈ rxm. this implies that x(1 − rm) = 0 for some r ∈ r. as x /∈ p1, we obtain that 1 − rm ∈ p1 ⊆ m. this implies that 1 ∈ m and this is impossible. therefore, xm = 0. hence, m ∈ p1, since p1 ∈ spec(r) and x /∈ p1. this proves that m ⊆ p1. therefore, p1 = m ∈ max(r). similarly, it can be shown that pj ∈ max(r) for each j ∈{2, . . . ,n}. conversely, assume that pi ∈ max(r) for each i ∈{1, 2, . . . ,n}. note that pi +pj = r for all distinct i,j ∈ {1, 2, . . . ,n} and n⋂ i=1 pi = (0). hence, we obtain from [4, proposition 1.10(ii) and (iii)] that the mapping f : r → r p1 × r p2 ×···× r pn defined by f(r) = (r + p1,r + p2, . . . ,r + pn) is an isomorphism of rings. let i ∈ {1, 2, . . . ,n}. since pi ∈ max(r), it follows that rpi is a field. let us denote the ring r p1 × r p2 ×···× r pn by t. it follows from example 2.8(1) that i(t)∗ = a(t)∗ = ea(t)∗. since r ∼= t as rings, we obtain that i(r)∗ = a(r)∗ = ea(r)∗. theorem 2.16. let r be a reduced ring which is not an integral domain. the following statements are equivalent: (1) ea(r)∗ = {i,j}. (2) j = ann(i) and i,j ∈ spec(r). (3) min(r) = {i,j}. proof. (1) ⇒ (2) as i ∈ ea(r)∗, it follows that i ∈ a(r)∗ and so, ann(i) 6= (0). it is clear that ann(i) ∈ a(r)∗. observe that we obtain from (2) ⇒ (1) of lemma 2.1 that ann(i) ∈ ea(r)∗ = {i,j}. since r is reduced, i2 6= (0) and so, ann(i) 6= i and therefore, ann(i) = j. let b ∈ a(r)∗. then ann(b) ∈ a(r)∗. therefore, we obtain from (2) ⇒ (1) of lemma 2.1 that ann(b) ∈ ea(r)∗. from the hypothesis ea(r)∗ = {i,ann(i)}, it follows that if b ∈ a(r)∗, then either ann(b) = i or ann(b) = ann(i). we next verify that i,j = ann(i) ∈ spec(r). let a,b ∈ r be such that ab ∈ i. then abann(i) = (0). we know from (1) ⇒ (3) of lemma 2.1 that ann(ann(i)) = i. if aann(i) = (0), then a ∈ ann(ann(i)) = i. similarly, if bann(i) = (0), then b ∈ i. hence, we can assume that aann(i) 6= (0) and bann(i) 6= (0). now, aann(i) 6= (0), ann(aann(i)) 6= (0), bann(i) 6= (0), and ann(bann(i)) 6= (0). therefore, ann(aann(i)),ann(bann(i)) ∈ ea(r)∗ = {i,ann(i)}. observe that ann(aann(i)) 6= ann(bann(i)). for if ann(aann(i)) = ann(bann(i)), then from abann(i) = (0), it follows that b2ann(i) = (0). since r is reduced, we get that bann(i) = (0) and this contradicts our assumption. hence, ann(aann(i)) 6= ann(bann(i)). therefore, either ann(aann(i)) = i or ann(bann(i)) = i. if ann(aann(i)) = i, then b ∈ i. if ann(bann(i)) = i, then a ∈ i. this proves that i ∈ spec(r). similarly, it can be shown that ann(i) ∈ spec(r). (2) ⇒ (3) we are assuming that the ideals i,j of r are such that j = ann(i), and i,j ∈ spec(r). by hypothesis, r is not an integral domain. hence, i 6= (0) and j 6= (0). as r is reduced, i2 6= (0), and so, it follows that i 6= ann(i) = j. note that ij = (0). if r ∈ i ∩ j, then r2 ∈ ij = (0) and since r is reduced, we obtain that r = 0 and so, i ∩j = (0). it is convenient to denote i by p1 and ann(i) by p2. note that p1 ∩ p2 = (0). we claim that min(r) = {p1,p2}. if p ∈ spec(r), then from p1 ∩ p2 = (0), it follows that p ⊇ pi for some i ∈{1, 2}. since r is not an integral domain, we obtain that p1 and p2 are not comparable under the inclusion relation. the above arguments imply that min(r) = {p1 = i,p2 = j}. 128 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 (3) ⇒ (1) we are assuming that min(r) = {i,j}. it now follows from the proof of proposition 2.15 that ea(r)∗ = {i,j}. let t be a ufd. if a( t t p2 )∗ = ea( t t p2 )∗ for every prime element p of t , then we prove in theorem 2.17 that t is a pid. theorem 2.17. let t be a ufd which is not a field. the following statements are equivalent: (1) for any prime element p of t, a( t t p2 )∗ = ea( t t p2 )∗. (2) t is a pid. (3) for any non-zero proper ideal i of t with i /∈ max(t), i( t i )∗ = a( t i )∗ = ea( t i )∗. proof. (1) ⇒ (2) let p be a prime element of t . we claim that tp ∈ max(t). for the sake of convenience, let us denote t t p2 by r. observe that tp ∈ spec(t) and let us denote t p t p2 by p. note that p ∈ spec(r), p = r(p + tp2) is principal, p 6= (0 + tp2) but p2 = (0 + tp2) and we know from the proof of lemma 2.11 that z(r) = p. we are assuming that that a(r)∗ = ea(r)∗. therefore, we obtain from the moreover part of proposition 2.10 that p ∈ max(r). as p = t p t p2 , we get that tp ∈ max(t). this is true for any prime element p of t. let p ∈ spec(t)\{(0)}. since any non-zero non-unit of t can be expressed as the product of a finite number of prime elements of t, it follows that p ⊇ tp for some prime element p of t . as tp ∈ max(t), we obtain that p = tp ∈ max(t). this shows that dimt = 1. hence, any prime ideal of t is principal. therefore, we obtain from [14, exercise 10, page 8] that any ideal of t is principal. therefore, t is a pid. (2) ⇒ (3) this follows from example 2.8(3). (3) ⇒ (1) this is clear, since for any prime element p of t, tp2 /∈ max(t). let t be a ufd which is not a field. if for every pair of non-associate prime elements p1,p2 of t , ea( t t p1p2 )∗ = a( t t p1p2 )∗, then we prove in theorem 2.18 that t is a pid. suppose that t has a prime element p such that any prime element of t is an associate of p in t. let a be any nonzero non-unit of t . then a = upn for some u ∈ u(t) and n ≥ 1. hence, ta ⊆ tp. therefore, max(t) = spec(t)\{(0)} = {tp}. let i be any non-zero proper ideal of t . then i ⊆ tp. from ∞⋂ n=1 tpn = (0), we get that there exists n ∈ n such that i ⊆ tpn but i 6⊆ tpn+1. let x ∈ i\tpn+1. then x = upn for some u ∈ u(t). this implies that pn = u−1x ∈ i. this proves that tpn ⊆ i and so, i = tpn. thus any ideal of t is principal and so, t is a pid. hence, in proving theorem 2.18, we assume that t has at least two non-associate prime elements. theorem 2.18. let t be a ufd such that t has at least two non-associate prime elements. the following statements are equivalent: (1) for any non-associate prime elements p1,p2 of t, ea( tt p1p2 ) ∗ = a( t t p1p2 )∗. (2) t is a pid. (3) for any non-zero proper ideal i of t with i /∈ max(t) , ea( t i )∗ = a( t i )∗. proof. (1) ⇒ (2) we are assuming that for any two non-associate prime elements p1,p2 of t , ea(r)∗ = a(r)∗ with r = t t p1p2 . let p be any prime element of t . by assumption, t has at least two nonassociate prime elements. let q be a prime element of t such p and q are non-associates in t. by (1), ea( t t pq )∗ = a( t t pq )∗. observe that t t pq is a reduced ring with min( t t pq ) = { t p t pq , t q t pq }. from ea( t t pq )∗ = a( t t pq )∗, we obtain from the moreover part of proposition 2.15 that t p t pq , t q t pq ∈ max(r) 129 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 and so, tp,tq ∈ max(t). thus for any prime element p of t , tp ∈ max(t). now, it follows as in the proof of (1) ⇒ (2) of theorem 2.17 that t is a pid. (2) ⇒ (3) this follows from example 2.8(3). (3) ⇒ (1) this is clear, since for any non-associate prime elements p1,p2 of t , tp1p2 /∈ max(t). recall from [10, exercise 16, page 111] that a ring t is von neumann regular if given a ∈ t , there exists b ∈ t such that a = a2b. if a is a non-zero non-unit of a von neumann regular ring t , then from a = a2b, it follows that e = ab = a2b2 = e2. hence, e is an idempotent element of t with e /∈ {0, 1}. it is known from (a) ⇔ (d) of [10, exercise 16, page 111] that a ring t is von neumann regular if and only if dimt = 0 and t is reduced. an idempotent element e of r with e /∈ {0, 1} is referred to as a non-trivial idempotent element. let r be a von neumann regular ring which is not a field. we verify in corollary 2.19 that |ea(r)∗| < ∞ if and only if there exist n ≥ 2 and fields f1,f2, . . . ,fn such that r ∼= f1 ×f2 ×···×fn as rings. corollary 2.19. let r be a von neumann regular ring which is not a field. the following statements are equivalent: (1) |ea(r)∗| < ∞. (2) there exist n ≥ 2 and fields f1,f2, . . . ,fn such that r ∼= f1 ×f2 ×···×fn as rings. proof. (1) ⇒ (2) since r is von neumann regular, we obtain that spec(r) = max(r) = min(r). since r is reduced, we get that ⋂ m∈max(r) m = (0). from r is not a field, it follows that |max(r)| ≥ 2. we are assuming that |ea(r)∗| < ∞. hence, we obtain from the remark which appears just preceding the statement of proposition 2.15 that |min(r) = max(r)| < ∞. let max(r) = {mi|i ∈{1, 2, . . . ,n}}. now, it follows as in the proof of the moreover part of proposition 2.15 that r ∼= n∏ i=1 r mi as rings. let i ∈ {1, 2, . . . ,n} and let us denote the field r mi by fi. thus there exist n ≥ 2 and fields f1,f2, . . . ,fn such that r ∼= f1 ×f2 ×···×fn as rings. (2) ⇒ (1) we are assuming that there exist n ≥ 2 and fields f1,f2, . . . ,fn such that r ∼= f1×f2×···×fn as rings. let us denote the ring n∏ i=1 fi by t . we know from example 2.8(1) that i(t)∗ = a(t)∗ = ea(t)∗. therefore, |ea(t)∗| = |i(t)∗| = 2n − 2. hence, |ea(r)∗| = 2n − 2 < ∞. 3. some results on eag(r) let g = (v,e) be a graph. g is said to be connected if for distinct vertices a,b ∈ v , there exists at least one path in g between a and b. let g = (v,e) be a connected graph. let a,b ∈ v with a 6= b. recall from [5] that the distance between a and b, denoted by d(a,b) is defined as the length of a shortest path in g between a and b. we define d(a,a) = 0 and define the diameter of g, denoted by diam(g) as diam(g) = sup{d(a,b) | a,b ∈ v}. a simple graph g is said to be complete if every pair of distinct vertices of g are adjacent in g. let n ≥ 1. a complete graph with n vertices is denoted by kn [5, definition 1.1.11]. let r be a ring such that ea(r)∗ 6= ∅. the aim of this section is to discuss some results on eag(r). first, we prove some results regarding the connectedness of eag(r). proposition 3.1. let r be a ring such that ea(r)∗ 6= ∅. let i,j ∈ ea(r)∗ be such that there is a path in eag(r) between i and j. then i and j are adjacent in eag(r). in particular, if eag(r) is connected and if |ea(r)∗| ≥ 2, then diam(eag(r)) = 1. 130 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. let i,j ∈ ea(r)∗ be such that there is a path in eag(r) between i and j. we claim thati and j are adjacent in eag(r). suppose that i and j are not adjacent in eag(r). let i0 = i − i1 − ···− in = j be a shortest path in eag(r) between i and j. it is clear that n ≥ 2. note that for all i ∈ {0, 1, . . . ,n − 1}, ii and ii+1 are adjacent in eag(r). hence, ann(ii) = ii+1 and ann(ii+1) = ii. if a ∈ ea(r)∗, then we know from (1) ⇒ (3) of lemma 2.1 that a = ann(ann(a)). therefore, i = i0 = ann(ann(i0)) = ann(i1) = i2. this is a contradiction. therefore, i and j are adjacent in eag(r). we now verify the in particular statement of this proposition. suppose that eag(r) is connected and |ea(r)∗| ≥ 2. let i,j ∈ ea(r)∗ be such that i 6= j. since eag(r) is connected, there exists a path in eag(r) between i and j. hence, we obtain from what is shown in the previous paragraph that i and j are adjacent in eag(r). therefore, it follows that diam(eag(r)) = 1. let g = (v,e) be a graph. recall from [9, page 21] that a maximal connected subgraph of g is called a component of g. let r be a ring such that ea(r)∗ 6= ∅. we prove in corollary 3.3 that each component of eag(r) is a complete graph with at most two vertices. we use lemma 3.2 in the proof of corollary 3.3. lemma 3.2. let r be a ring such that ea(r)∗ 6= ∅. let i − j be an edge of eag(r). let a ∈ ea(r)∗\{i,j}. then i and a are not adjacent in eag(r) and j and a are not adjacent in eag(r). proof. since i − j is an edge of eag(r), we obtain that ann(i) = j and ann(j) = i. as a ∈ ea(r)∗, we know from (1) ⇒ (3) of lemma 2.1 that ann(ann(a)) = a. as a /∈ {i,j}, it follows that ann(a) /∈ {i,j}. therefore, we obtain that i and a are not adjacent in eag(r) and j and a are not adjacent in eag(r). corollary 3.3. let r be a ring such that ea(r)∗ 6= ∅. if g is any component of eag(r), then g is a complete graph with at most two vertices. in particular, if eag(r) is connected, then eag(r) is a complete graph with at most two vertices. proof. let g be any component of eag(r). suppose that |v (g)| ≥ 2. let i,j ∈ v (g) with i 6= j. then there exists a path in eag(r) between i and j. hence, we obtain from proposition 3.1 that i and j are adjacent in eag(r) and so, they are adjacent in g. let a ∈ ea(r)∗\{i,j}. we know from lemma 3.2 that i and a are not adjacent in eag(r) and j and a are not adjacent in eag(r). therefore, a /∈ v (g) and so, v (g) = {i,j}. this proves that any component g of eag(r) is a complete graph with at most two vertices. we next verify the in particular statement of this corollary. suppose that eag(r) is connected. then eag(r) is the only component of eag(r) and so, eag(r) is a complete graph with at most two vertices. next, we assume that (r,m) is a spir and try to determine the structure of eag(r). proposition 3.4. let r be a ring. let p ∈ spec(r) be such that p 6= (0) but p2 = (0). if p = z(r), then eag(r) is a graph with v (eag(r)) = {p}. in particular, if (r,m) is a spir with m 6= (0) but m2 = (0), then eag(r) is a graph with v (eag(r)) = {m}. proof. we know from (2) ⇒ (1) of theorem 2.9 that ea(r)∗ = {p}. as v (eag(r)) = ea(r)∗, we obtain that v (eag(r)) = {p}. we next verify the in particular statement of this proposition. let (r,m) be a spir with m 6= (0) but m2 = (0). as z(r) = m, it follows that v (eag(r)) = {m}. proposition 3.5. let r be a ring. let p ∈ spec(r) be such that p = rp is principal, n ≥ 3 is least with the property that pn = (0), and z(r) = p. then the following statements hold: 131 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 (1) if n is odd, then eag(r) has exactly [ n 2 ] components and each component is a complete graph with two vertices. (2) if n is even, then eag(r) has exactly n 2 components g1,g2, . . . ,gn 2 such that gj is a complete graph with two vertices for each j ∈{1, . . . , n 2 − 1} and gn 2 is a complete graph on a single vertex. in particular, if (r,m) is a spir and n ≥ 3 is least with the property that mn = (0), then the statements (1) and (2) hold for eag(r). proof. note that v (eag(r)) = ea(r)∗ and we know from proposition 2.10 that ea(r)∗ = {pi | i ∈ {1, 2, . . . ,n − 1}}. we know from the proof of proposition 2.10 that for each i ∈ {1, 2, . . . ,n − 1}, ann(pi) = pn−i and ann(pn−i) = pi. suppose that n ≥ 4. let j ∈ {1, . . . , [ n 2 ] − 1}. as 2j < n and n is least with the property that pn = (0), it follows that pj 6= pn−j. observe that pj and pn−j are adjacent in eag(r). let gj be the subgraph of eag(r) induced by {pj,pn−j}. then gj is a complete graph with two vertices and it follows from corollary 3.3 that gj is necessarily a component of eag(r). (1) assume that n is odd. if n = 3, then v (eag(r)) = {p,p2} and eag(r) is a complete graph with two vertices. let n ≥ 5. note that p n−1 2 6= p n+1 2 . observe that ann(p n−1 2 ) = p n+1 2 and ann(p n+1 2 ) = p n−1 2 . hence, p n−1 2 and p n+1 2 are adjacent in eag(r). let g[ n 2 ] be the subgraph of eag(r) induced by {p n−1 2 ,p n+1 2 }. note that g[ n 2 ] is a complete graph with two vertices and it follows from corollary 3.3 that g[ n 2 ] is necessarily a component of eag(r). observe that v (eag(r)) = ea(r)∗ = [ n 2 ]⋃ j=1 {pj,pn−j} = [ n 2 ]⋃ j=1 v (gj). it is clear that for any distinct j1,j2 ∈ {1, 2, . . . , [ n2 ]}, v (gj1 ) ∩ v (gj2 ) = ∅. from the above arguments, it is clear that eag(r) has exactly [ n 2 ] components and each component is a complete graph with two vertices. (2) assume that n is even. it is clear that n ≥ 4. observe that v (eag(r)) = ea(r)∗ = ( n 2 −1⋃ j=1 {pj,pn−j}) ∪{p n 2 } = ( n 2 −1⋃ j=1 v (gj)) ∪{p n 2 }. let gn 2 be the subgraph of eag(r) induced by {p n 2 }. it is clear that for all distinct j1,j2 ∈{1, 2, . . . , n2}, v (gj1 )∩v (gj2 ) = ∅. from the above given arguments, it follows that eag(r) has exactly n 2 components g1,g2, . . . ,gn 2 such that gj is a complete graph with two vertices for each j ∈{1, . . . , n 2 − 1} and gn 2 is a complete graph on a single vertex. we next verify the in particular statement of this proposition. now, (r,m) is a spir and n ≥ 3 is least with the property that mn = (0). let m ∈ m be such that m = rm. observe that z(r) = m. hence, the hypotheses of this proposition are satisfied and therefore, the statements (1) and (2) hold for eag(r). remark 3.6. let r be a ring. let p ∈ spec(r) be such that p = rp is principal, p 6= (0) but p is nilpotent. let n ≥ 2 be least with the property that pn = (0). then the following hold: (1) (r,p) is a spir if and only if p ∈ max(r). (2) suppose that z(r) = p. then eag(r) is connected if and only if n ∈{2, 3}. proof. (1) assume that (r,p) is a spir. then p ∈ max(r) and indeed, it is the only prime ideal of r. conversely, assume that p ∈ max(r). then it is shown in the proof of the moreover part of proposition 2.10 that (r,p) is a spir. (2) if n ≥ 4, then [ n 2 ] ≥ 2. we know from proposition 3.5 that eag(r) has exactly [ n 2 ] components. thus if eag(r) is connected, then n ∈ {2, 3}. assume that n ∈ {2, 3}. if n = 2, then we know from proposition 3.4 that v (eag(r)) = {p}. if n = 3, then we know from the proof of proposition 3.5(1) that eag(r) is a complete graph with two vertices. therefore, if n ∈{2, 3}, then eag(r) is connected. 132 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 let r be a ring. let p ∈ spec(r) be such that p satisfies the hypotheses mentioned in the statement of remark 3.6. if z(r) = p, then in theorem 3.7, we characterize r such that eag(r) = ag(r). theorem 3.7. let r be a ring. let p ∈ spec(r) be such that p = rp is principal. let n ≥ 2 be least with the property that pn = (0). if z(r) = p, then the following statements are equivalent: (1) eag(r) = ag(r). (2) (r,p) is a spir and n ∈{2, 3}. proof. (1) ⇒ (2) from the assumption eag(r) = ag(r), we get that ea(r)∗ = v (eag(r)) = v (ag(r)) = a(r)∗. we know from proposition 2.10 that ea(r)∗ = {pi | i ∈ {1, . . . ,n − 1}}. hence, a(r)∗ = {pi | i ∈ {1, . . . ,n− 1}}. we first verify that (r,p) is a spir. in view of the statement (1) of remark 3.6, it is enough to prove that p ∈ max(r). as a(r)∗ = ea(r)∗, we obtain from the moreover part of proposition 2.10 that p ∈ max(r). therefore, (r,p) is a spir. it is known that ag(r) is connected and diam(ag(r)) ≤ 3 [7, theorem 2.1]. therefore, from eag(r) = ag(r), we get that eag(r) is connected. hence, we obtain from remark 3.6(2) that n ∈{2, 3}. (2) ⇒ (1) we are assuming that (r,p) is a spir and n ∈{2, 3}. if n = 2, then we know from the proof of lemma 2.4 that ea(r)∗ = a(r)∗ = {p}. hence, eag(r) = ag(r) in this case. if n = 3, then we know from the proof of lemma 2.4 that ea(r)∗ = a(r)∗ = {p,p2}. thus v (eag(r)) = v (ag(r)) = {p,p2}. we know from the proof of the statement (1) of proposition 3.5 that eag(r) is a complete graph with vertex set {p,p2}. for any ring t , eag(t) is a subgraph of ag(t). hence, ag(r) is a complete graph with vertex set {p,p2}. therefore, eag(r) = ag(r) in this case also. therefore, if (r,p) is a spir and n ∈{2, 3}, then eag(r) = ag(r). let r be a ring. let p ∈ spec(r) be such that p = rp is principal, p2 6= (0) but p3 = (0), and z(r) = p. then we know from the proof of proposition 3.5(1) that eag(r) is a complete graph with v (eag(r)) = {p,p2}. we provide example 3.8 to illustrate that in the above result, if the hypothesis that p is principal is omitted, then the conclusion can fail to hold. example 3.8. let r be the ring considered by d.d. anderson and m. naseer in [3, page 501]. then i(r)∗ = a(r)∗ = ea(r)∗, ag(r) is connected with diam(ag(r)) = 2, and eag(r) has exactly eight components and each component is a complete graph with two vertices. proof. the ring r is also considered in example 2.3 of this article. in the notation of example 2.3, r = t i , where t = z4[x,y,z], the polynomial ring in three variables x,y,z over z4, and i is the ideal of t generated by {x2 − 2,y 2 − 2,z2,xy,xz,y z − 2, 2x, 2y, 2z}. let us denote x + i by x, y + i by y, and z + i by z. observe that r is a local artinian ring with m = rx + ry + rz as its unique maximal ideal, m2 = {0 + i, 2 + i}, m3 = (0 + i), and |r| = 32. it is already noted in example 2.3 that i(r)∗ = {m,m2,rx,ry,rz,r(x + y),r(y + z),r(x + z),r(x + y + z),rx + ry,ry + rz,rx + rz,rx + r(y + z),ry + r(z + x),rz + r(x + y),r(x + y) + r(y + z)}. it is verified in example 2.3 that i(r)∗ = a(r)∗ = ea(r)∗. let a,b ∈ a(r)∗ with a 6= b. suppose that a and b are not adjacent in ag(r). from m3 = (0), we obtain that a−m2 −b is a path of length two between a and b in ag(r). this proves that diam(ag(r)) ≤ 2. observe that (ry)(rz) 6= (0) and so, ry and rz are not adjacent in ag(r). this shows that diam(ag(r)) ≥ 2 and so, diam(ag(r)) = 2. we next verify that eag(r) has exactly eight components and each component is a complete graph with two vertices. let a1 = {a11 = m,a12 = m2},a2 = {a21 = rx,a22 = ry + rz},a3 = {a31 = ry,a32 = rx + r(y + z)},a4 = {a41 = rz,a42 = rx + rz},a5 = {a51 = r(x + y),a52 = r(y + z) + r(z + x)},a6 = {a61 = r(y + z),a62 = rx + ry},a7 = {a71 = r(z + x),a72 = rz + r(x + y)}, and a8 = {a81 = r(x + y + z),a82 = ry + r(x + z)}. let gi be the subgraph of eag(r) induced by ai for each i ∈ {1, 2, . . . , 8}. we know from the proof of example 2.3, that ann(ai1) = ai2 and ann(ai2) = ai1 for each i ∈ {1, 2, 3, . . . , 8}. it is clear that gi is a complete graph with two vertices for each i ∈ {1, 2, 3, . . . , 8} and it follows from corollary 3.3 that each gi is a component of eag(r). as 133 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 ai = v (gi) for each i ∈{1, 2, 3, . . . , 8}, ea(r)∗ = 8⋃ i=1 ai, ai∩aj = ∅ for all distinct i,j ∈{1, 2, 3, . . . , 8}, it follows that {gi | i ∈{1, 2, 3, . . . , 8}} is the set of all components of eag(r). let r be a reduced ring which is not an integral domain. if t is a ring which is not an integral domain, then it is already noted in the paragraph which appears just preceding the statement of corollary 2.2 that ea(t)∗ 6= ∅. hence, ea(r)∗ 6= ∅. in corollary 3.10, we answer when eag(r) is connected. in corollary 3.11, we prove that eag(r) = ag(r) if and only if r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2}. we use lemma 3.9 in the proof of corollary 3.10. lemma 3.9. let r be a reduced ring which is not an integral domain. then ea(r)∗ 6= ∅ and any component g of eag(r) is a k2. proof. if r is not an integral domain (whether it is reduced or not), then it is already noted that ea(r)∗ 6= ∅. let g be a component of eag(r). let i ∈ v (g). it follows from (1) ⇒ (3) of lemma 2.1 that ann(ann(i)) = i. since r is reduced, i2 6= (0) and so, i 6= ann(i). with j = ann(i), it follows that ann(j) = i. hence, i and j are adjacent in eag(r). therefore, j ∈ v (g). also, i and j are adjacent in g. it follows from corollary 3.3 that g is a complete graph with two vertices. corollary 3.10. let r be a reduced ring which is not an integral domain. the following statements are equivalent: (1) eag(r) is connected. (2) |min(r)| = 2. proof. (1) ⇒ (2) assume that eag(r) is connected. we know from lemma 3.9 that eag(r) is a complete graph with two vertices. hence, |eag(r)| = 2. as v (eag(r)) = ea(r)∗, we get that |ea(r)∗| = 2. hence, it follows from (1) ⇒ (3) of theorem 2.16 that |min(r)| = 2. (2) ⇒ (1) let min(r) = {pi | i ∈ {1, 2}}. we know from the proof of proposition 2.15 that ea(r)∗ = {pi | i ∈ {1, 2}}, ann(p1) = p2, and ann(p2) = p1. therefore, p1 and p2 are adjacent in eag(r). this shows that eag(r) is a complete graph with two vertices and so, we obtain that eag(r) is connected. corollary 3.11. let r be a reduced ring which is not an integral domain. the following statements are equivalent: (1) eag(r) = ag(r). (2) r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2}. proof. (1) ⇒ (2) we are assuming that eag(r) = ag(r). we know from [7, theorem 2.1] that ag(r) is connected. therefore, eag(r) is connected. hence, we obtain from the proof of (1) ⇒ (2) of corollary 3.10 that |min(r)| = 2 and ea(r)∗ = min(r). let min(r) = {p1,p2}. now, a(r)∗ = v (ag(r)) = v (eag(r)) = ea(r)∗. in such a case, we obtain from the proof of moreover part of proposition 2.15 that r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2}. (2) ⇒ (1) assume that r ∼= f1 × f2 as rings, where fi is a field for each i ∈ {1, 2}. let us denote the ring f1 × f2 by t. we know from example 2.8(1) that i(t)∗ = a(t)∗ = ea(t)∗. note that i(t)∗ = {m1 = (0) × f2,m2 = f1 × (0)}. observe that min(t) = {mi | i ∈ {1, 2}}. now, it follows from the proof of (2) ⇒ (1) of corollary 3.10 that eag(t) is a complete graph with two vertices. since eag(t) is a subgraph of ag(t), we get that eag(t) = ag(t) is a complete graph with two vertices. since r ∼= t as rings, we obtain that eag(r) = ag(r). corollary 3.12. let r be a reduced ring with |min(r)| = n ≥ 2. then eag(r) has exactly 2n−1 − 1 components and each component is a k2. 134 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. we know from proposition 2.15 that |v (eag(r)) = ea(r)∗| = 2n − 2. let t be the number of components of eag(r). let {gi | i ∈ {1, . . . , t}} be the set of all components of eag(r). we know from lemma 3.9 that gi is a k2 for each i ∈ {1, . . . , t}. now, ea(r)∗ = t⋃ i=1 v (gi), |v (gi)| = 2 for each i ∈ {1, . . . , t}, v (gi) ∩v (gj) = ∅ for all distinct i,j ∈ {1, . . . , t}. therefore, 2n − 2 = |ea(r)∗| = 2t and so, t = 2n−1 − 1. this proves that eag(r) has exactly 2n−1 − 1 components and each component is a k2. corollary 3.13. let n ≥ 2 and let ri be an integral domain for each i ∈ {1, 2, . . . ,n}. let r = r1 ×r2 ×···×rn. then eag(r) has exactly 2n−1 − 1 components and each component is a k2. proof. note that r is a reduced ring and {p1,p2, . . . ,pn} is the set of all minimal prime ideals of r, where for each i ∈ {1, 2, . . . ,n}, pi = i1 × ··· × ii × ··· × in with ii = (0) and ij = rj for all j ∈ {1, 2, . . . ,n}\{i}. thus |min(r)| = n and so, we obtain from corollary 3.12 that eag(r) has exactly 2n−1 − 1 components and each component is a complete graph with two vertices. let r be a ring which is not reduced. we are not able to determine i,j ∈ ea(r)∗ such that ea(r)∗ = {i,j}. however, as a consequence of corollary 3.3 and [7, theorem 2.7], we characterize in theorem 3.14 rings r with ea(r)∗ 6= ∅ such that eag(r) = ag(r). theorem 3.14. let r be a ring such that ea(r)∗ 6= ∅. the following statements are equivalent: (1) eag(r) = ag(r). (2) either r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2} or (r,m) is a spir satisfying the property that if n ∈ n is least such that mn = (0), then n ∈{2, 3}. proof. (1) ⇒ (2) we are assuming that eag(r) = ag(r). we know from [7, theorem 2.1] that ag(r) is connected. therefore, eag(r) is connected. hence, we obtain from corollary 3.3 that eag(r) is complete. therefore, ag(r) is complete and so, it follows from [7, theorem 2.7] that one of the following holds: (a) r ∼= f1 × f2 as rings, where fi is a field for each i ∈ {1, 2}; (b) z(r) is an ideal of r with (z(r))2 = (0); (c) (r,m) is a spir with m3 = (0) but m2 6= (0). assume that (b) holds. as z(r) is an ideal of r, z(r) is necessarily a prime ideal of r. let us denote z(r) by p. now, p 6= (0), p2 = (0), and p = z(r). therefore, we obtain from (2) ⇒ (1) of theorem 2.9 that ea(r)∗ = {p}. from eag(r) = ag(r), it follows that a(r)∗ = ea(r)∗ = {p}. hence, we obtain from [7, corollary 2.9(a)] that (r,m) (with m = p) is a spir with m2 = (0). therefore, we obtain that either r ∼= f1 × f2 as rings, where fi is a field for each i ∈ {1, 2} or (r,m) is a spir satisfying the property that if n ∈ n is least such that mn = (0), then n ∈{2, 3}. (2) ⇒ (1) suppose that r ∼= f1 × f2 as rings. then we obtain from (2) ⇒ (1) of corollary 3.11 that eag(r) = ag(r). suppose that (r,m) is a spir satisfying the property that if n ∈ n is least such that mn = (0), then n ∈{2, 3}. then we know from (2) ⇒ (1) of theorem 3.7 that eag(r) = ag(r). let n ≥ 2 and let t = k[x1,x2, . . . ,xn] be the polynomial ring in n variables x1,x2, . . . ,xn over a field k. let r = t t x1x2 . observe that r is a reduced ring with min(r) = { t x1 t x1x2 , t x2 t x1x2 } and thus |min(r)| = 2. hence, we obtain from (2) ⇒ (1) of corollary 3.10 that eag(r) is connected. we know from [18, theorem 3, page 281] that each maximal ideal of t is of height n and hence, it follows that dimr = n − 1. it follows from [18, corollary 1, page 279] that tx1 is the intersection of all maximal ideals of t that contain tx1. therefore, it follows that max(r) is infinite. if a ring is zero-dimensional which admits at least one non-zero exact annihilating ideal and if its exact annihilating-ideal graph is connected, then we prove in proposition 3.16 that there is a bound on the number of its maximal ideals. we use lemma 3.15 in the proof of proposition 3.16. lemma 3.15. let r be a ring such that dimr = 0. let n ≥ 2. if |max(r)| ≥ n, then there exist zero-dimensional rings r1,r2, . . . ,rn such that r ∼= r1 ×r2 ×···×rn as rings. 135 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. we prove this lemma using induction on n. suppose that |max(r)| ≥ 2. let us denote the ring r nil(r) by t . from |max(r)| ≥ 2, it follows that |max(t)| ≥ 2. note that dimt = 0 and we know from [4, proposition 1.7] that t is reduced. therefore, t is a von neumann regular ring which is not a field. hence, t admits a non-trivial idempotent element a + nil(r). since nil(r) is a nil ideal of r, we obtain from [13, proposition 7.14, page 405] that there exists an idempotent element e of r such that a + nil(r) = e + nil(r). it is clear that e /∈{0, 1}. note that the mapping f : r → re×r(1−e) defined by f(r) = (re,r(1−e)) is an isomorphism of rings. let us denote the ring re by r1 and r(1−e) by r2. let i ∈{1, 2}. since ri is a homomorphic image of r, it follows that dimri = 0. thus there exist zerodimensional rings r1,r2 such that r ∼= r1 ×r2 as rings. let n ≥ 3 and assume by induction that the lemma is true for n−1. now, |max(r)| ≥ n > n−1. by induction hypothesis, there exist zero-dimensional rings r′1, . . . ,r ′ n−1 such that r ∼= r′1 ×···×r′n−1 as rings. since |max(r)| ≥ n, |max(r′i)| > 1 for at least one i ∈ {1, . . . ,n− 1}. without loss of generality, we can assume that |max(r′1)| > 1. hence, by the case n = 2, there exist zero-dimensional rings r′11,r ′ 12 such that r ′ 1 ∼= r′11×r′12 as rings. therefore, r ∼= r′11 × r′12 × r′2 ×···× r′n−1 as rings. let r1 = r′11,r2 = r′12,r3 = r′2, . . . ,rn = r′n−1. then dimri = 0 for each i ∈{1, 2, . . . ,n} and r ∼= r1 ×r2 ×r3 ×···×rn as rings. proposition 3.16. let r be a zero-dimensional ring such that ea(r)∗ 6= ∅. if eag(r) is connected, then |max(r)| ≤ 2. proof. we are assuming that dimr = 0, ea(r)∗ 6= ∅, and eag(r) is connected. suppose that |max(r)| ≥ 3. then it follows from lemma 3.15 that there exist zero-dimensional rings r1,r2, and r3 such that r ∼= r1 ×r2 ×r3 as rings. let us denote the ring r1 ×r2 ×r3 by t . since r ∼= t as rings, we obtain that eag(t) is connected. hence, we obtain from proposition 3.1 that for any i.j ∈ ea(t)∗ with i 6= j, i and j are adjacent in eag(t). let e1 = (1, 0, 0),e2 = (0, 1, 0), and e3 = (0, 0, 1). observe that for all distinct i,j ∈ {1, 2, 3}, eiej = (0, 0, 0) and so, tei ∈ a(t)∗. moreover, for all i ∈ {1, 2, 3}, ann(ann(tei)) = tei and so, we obtain from (3) ⇒ (1) of lemma 2.1 that tei ∈ ea(t)∗. observe that ann(te1) = te2 + te3 6= te2 and so, te1 and te2 are not adjacent in eag(t). this is a contradiction. therefore, |max(r)| ≤ 2. let r be a ring such that dimr = 0 and |max(r)| = 2. in corollary 3.17, we characterize r such that eag(r) is connected. corollary 3.17. let r be a ring such that dimr = 0 and |max(r)| = 2. the following statements are equivalent: (1) eag(r) is connected. (2) r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2}. proof. (1) ⇒ (2) by hypothesis, dimr = 0 and |max(r)| = 2. we know from lemma 3.15 that there exist rings r1,r2 such that dimri = 0 for each i ∈ {1, 2} and r ∼= r1 × r2 as rings. since |max(r)| = 2, it follows that |max(ri)| = 1 for each i ∈ {1, 2}. let mi denote the unique maximal ideal of ri for each i ∈ {1, 2}. let us denote the ring r1 × r2 by t. as eag(r) is connected, it follows that eag(t) is connected. note that {(0) ×r2,r1 × (0)} ⊆ ea(t)∗. we know from corollary 3.3 that eag(t) is a complete graph with two vertices. therefore, ea(t)∗ = {(0) × r2,r1 × (0)}. we next verify that ri is a field for each i ∈ {1, 2}. suppose that r1 is not a field. then m1 6= (0). since spec(r1) = {m1}, it follows from [4, proposition 1.8] that nil(r1) = m1. let x ∈ m1,x 6= 0. let n ≥ 2 be least with the property that xn = 0. then xn−1 ∈ ann(x) and xn−1 6= 0. observe that ann(x) × (0) ∈ ea(t)∗ = {(0) ×r2,r1 × (0)}. this is impossible. therefore, r1 is a field. similarly, it can be shown that r2 is a field. hence, r ∼= f1 ×f2 as rings, where fi = ri is a field for each i ∈{1, 2}. (2) ⇒ (1) we are assuming that r ∼= f1 ×f2 as rings, where fi is a field for each i ∈ {1, 2}. note that r is reduced and |min(r)| = 2. hence, we obtain from (2) ⇒ (1) of corollary 3.10 that eag(r) is connected. 136 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 let t be a dedekind domain. let i be a non-zero proper ideal of t i /∈ max(t) and let r = t i . it is shown in example 2.8(2) that i(r)∗ = a(r)∗ = ea(r)∗. in corollary 3.18, we characterize r such that eag(r) is connected. corollary 3.18. let i be a non-zero proper ideal of a dedekind domain t such that i /∈ max(t). let r = t i . the following statements are equivalent: (1) eag(r) is connected. (2) |max(r)| ≤ 2 and eag(r) is complete. (3) either r ∼= f1 ×f2 as rings, where fi is a field for each i ∈{1, 2} or (r,m) is a spir and if k ≥ 2 is least with the property that mk = (0 + i), then k ∈{2, 3}. proof. (1) ⇒ (2) since dimt = 1, it follows that dimr = 0. hence, we obtain from proposition 3.16 that |max(r)| ≤ 2 and we obtain from corollary 3.3 that eag(r) is a complete graph with at most two vertices. (2) ⇒ (3) suppose that |max(r)| = 2. since dimr = 0 and eag(r) is connected, we obtain from (1) ⇒ (2) of corollary 3.17 that r ∼= f1 ×f2 as rings, where fi is a field for each i ∈ {1, 2} and in this case, i = m1m2 for some distinct m1,m2 ∈ max(t). suppose that |max(r)| = 1. we know from the proof of example 2.8(2) that i = mk for some k ≥ 2 and (r,m = m mk ) is a spir. it follows from [4, corollary 9.4] that mi 6= mj for all distinct i,j ∈ n. hence, k is least with the property that mk = (0 +i). from the assumption that eag(r) is connected, we obtain from remark 3.6(2) that k ∈{2, 3}. (3) ⇒ (1) if r ∼= f1 ×f2 as rings, where fi is a field for each i ∈ {1, 2}, then it follows from the proof of (2) ⇒ (1) of corollary 3.11 that eag(r) is a complete graph with two vertices. suppose that (r,m) is a spir and if k ≥ 2 is least with the property that mk = (0 + i), then k ∈ {2, 3}. then it follows from the proof of remark 3.6(2) that eag(r) is a complete graph with at most two vertices. therefore, eag(r) is connected. let n ≥ 2 be such that n is not a prime number. since z is a pid and hence, a dedekind domain, and zn ∼= znz as rings, the following corollary is an immediate consequence of corollary 3.18. corollary 3.19. let n ≥ 2 be not a prime number. let r = zn. then eag(r) is connected if and only if either n = p1p2 for some distinct prime numbers p1,p2 or n ∈{p2,p3} for some prime number p. let g = (v,e) be a graph. suppose that g admits a cycle. recall from [5, page 159] that the girth of g, denoted by girth(g) is defined as the length of a shortest cycle in g. if g does not contain any cycle, then we define girth(g) = ∞. recall from [5, definition 1.2.2] that a clique of g is a complete subgraph of g. suppose that there exists k ∈ n such that any clique of g is a clique on at most k vertices. then the clique number of g, denoted by ω(g) is defined as the largest integer n ≥ 1 such that g contains a clique on n vertices [5, definition, page 185]. we set ω(g) = ∞ if g contains a clique on n vertices for all n ≥ 1. let g = (v,e) be a graph. recall from [5, page 129] that a vertex coloring of g is a map f : v → s, where s is a set of distinct colors. a vertex coloring f : v → s is said to be proper if adjacent vertices of g receive distinct colors of s; that is, if a and b are adjacent in g, then f(a) 6= f(b). the chromatic number of g, denoted by χ(g) is the minimum number of colors needed for a proper vertex coloring of g [5, definition 7.1.2]. it is well-known that for any graph g, ω(g) ≤ χ(g). recall from [5] that a graph g is said to be weakly perfect if χ(g) = ω(g). a graph g is said to be perfect if any induced subgraph h of g is weakly perfect; that is, for any induced subgraph h of g, χ(h) = ω(h). corollary 3.20. let r be a ring such that ea(r)∗ 6= ∅. then the following hold: (1) girth(eag(r)) = ∞. (2) eag(r) is perfect. 137 s. visweswaran, p. t. lalchandani / j. algebra comb. discrete appl. 8(2) (2021) 119–138 proof. (1) if g is any component of eag(r), then we know from corollary 3.3 then g is a complete graph with at most two vertices. therefore, eag(r) does not contain any cycle and so, girth(eag(r)) = ∞. (2) let h be any induced subgraph of eag(r). let h be any component of h. suppose that |v (h)| ≥ 2. then it follows from proposition 3.1 and lemma 3.2 that h is a complete graph with two vertices. hence, it follows that χ(h) = ω(h) ∈{1, 2}. therefore, we obtain that eag(r) is perfect. acknowledgment: we are very much thankful to the referees for their useful and helpful suggestions. we are also very much thankful to professor irfan siap and professor daniela ferrero for their support. references [1] g. aalipour, s. akbari, m. behboodi, r. nikandish, m. j. nikmehr, f. shaveisi, the classification of annihilating-ideal graphs of commutative rings, algebra colloq. 21(2) (2014) 249–256. [2] g. aalipour, s. akbari, r. nikandish, m. j. nikmehr, f. shaveisi, on the coloring of the annihilatingideal graph of a commutative ring, discrete math. 312 (2012) 2620–2626. [3] d. d. anderson, m. naseer, beck’s coloring of a commutative ring, j. algebra 159 (1993) 500–514. 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[19] s. visweswaran and p. sarman, on the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring, discrete math. algorithms appl. 8(3) (2016) article id: 1650043 22 pages. 138 https://doi.org/10.1142/s1005386714000200 https://doi.org/10.1142/s1005386714000200 https://doi.org/10.1016/j.disc.2011.10.020 https://doi.org/10.1016/j.disc.2011.10.020 https://doi.org/10.1006/jabr.1993.1171 https://doi.org/10.1016/0021-8693(88)90202-5 https://doi.org/10.1142/s0219498811004896 https://doi.org/10.1142/s0219498811004896 https://doi.org/10.1142/s0219498811004902 https://doi.org/10.1142/s0219498811004902 https://doi.org/10.1080/00927872.2011.587489 https://doi.org/10.1080/00927872.2011.587489 https://doi.org/10.1007/s00209-009-0639-z https://doi.org/10.1007/s00209-009-0639-z https://dx.doi.org/10.22124/jart.2017.2400 https://dx.doi.org/10.22124/jart.2017.2400 https://doi.org/10.1142/s1793830916500439 https://doi.org/10.1142/s1793830916500439 https://doi.org/10.1142/s1793830916500439 introduction some basic properties of ea(r)* some results on eag(r) references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.784999 j. algebra comb. discrete appl. 7(3) • 237–245 received: 23 may 2019 accepted: 25 march 2020 journal of algebra combinatorics discrete structures and applications generating generalized necklaces and new quasi-cyclic codes∗ research article rumen daskalov, elena metodieva abstract: in many cases there is a need of exhaustive lists of combinatorial objects of a given type. we consider generation of all inequivalent polynomials from which defining polynomials for constructing quasicyclic (qc) codes are to be chosen. using these defining polynomials we construct 34 new good qc codes over gf(11) and 36 such codes over gf(13). 2010 msc: 94b05, 94b65 keywords: finite field, quasi-cyclic linear codes, necklaces 1. introduction let gf(q) denote the galois field of q elements and let v(n,q) denote the vector space of all ordered n-tuples over gf(q). the hamming weight of a vector x, denoted by wt(x), is the number of nonzero entries in x. a linear code c of length n and dimension k over gf(q) is a k-dimensional subspace of v(n,q). such a code is called [n,k,d]q code if its minimum hamming distance is d. for linear codes, the minimum distance is equal to the smallest of the weights of the nonzero codewords. a k ×n matrix g having as rows the vectors of a basis of a linear code c is called a generator matrix for c. let ai denote the number of codewords of c with weight i. the weight distribution of c is the list of numbers ai. the weight distribution a0 = 1, ad = α, . . . , an = γ is expressed as 01dα . . .nγ also. in order to obtain a q-ary linear code which is capable of correcting most errors for given values of n, k, and q, it is sufficient to obtain an [n,k,d]q code c with maximum minimum distance d among all such codes or for given values of k, d, and q, to obtain an [n,k,d]q code c whose length n is the smallest one. the respective codes in these two cases are called optimal. ∗ this work was partially supported by the bulgarian ministry of education and science under contract in tu– gabrovo. rumen daskalov (corresponding author), elena metodieva; department of mathematics and informatics, technical university of gabrovo, bulgaria (email: daskalov@tugab.bg, metodieva@tugab.bg). 237 https://orcid.org/0000-0001-7441-4757 https://orcid.org/0000-0001-5360-4762 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 the problem of determining optimal code parameters, known as the main linear coding theory problem, has two aspects. one is the construction of codes which optimize minimum distance and the other is proving non-existence of codes of certain parameters ([14], [21]). in the former one often uses computers, but this approach becomes ineffective when the dimension of the codes is large, because, as we know, computing the minimum distance of linear codes is an np-hard problem [30]. thus, it becomes expedient to use classes of codes that have a rich mathematical structure. in recent years it has been shown that qc and qt codes form such nice classes. being generalizations of cyclic and consta-cyclic codes, they contain many good, record-breaking codes [1–6, 10, 11, 15–20, 24, 27]. markus grassl and eric chen maintain online tables of linear codes. the grassl’s tables [28] contain lower and upper bounds on minimum distances for linear codes over small finite fields (q ≤ 9). many of the best-known codes in these tables are qc and qt codes. the chen’s table [9] contains only good and best-known qc and qt codes (q ≤ 13). these two databases are updated when new codes are discovered. in recent years there has been an increased interest in codes over gf(11) and gf(13). in [25] and [26] gulliver constructed qt codes over gf(11) for k ≤ 7 and qc codes over gf(13) for k ≤ 6. venkaiah and gulliver [31] constructed quasi-cyclic [pk,k,d]13 codes of dimensions k ≤ 6 and n ≤ 150. in [12] and [13] e. chen and n. aydin constructed 45 new oc and qt codes over gf(11), 38 such codes over gf(13) and presented databases for small dimensions and n ≤ 150. new qt codes over gf(11) and gf(13) are also presented in [7]. three-dimensional projective codes are closely related to (n,r)−arcs in projective finite planes, and a lot of research has been done over finite fields of size up to 19 [8]. recently an optimal (78, 8)-arc in pg(2,11) was constructed in [25] as a [78, 3, 70]11 qc code. in this paper we present 34 new qc codes over gf(11) and 36 new qc codes over gf(13). 2. qc codes a code c is said to be p-qc if a cyclic shift of any codeword by p positions results in another codeword. suppose that c is a p-qc [pm,k] code (m ≥ k). it is convenient to take the co-ordinate places of c in the following order 1,p + 1, 2p + 1, . . . , (m− 1)p + 1, 2,p + 2, . . . , (m− 1)p + 2, p, 2p,. . . ,mp. then c will be generated by a matrix of the form [b1,b2, . . . ,bp] where each bi is a circulant matrix, i.e. a matrix of the form b =   b0 b1 b2 · · · bm−1 bm−1 b0 b1 · · · bm−2 bm−2 bm−1 b0 · · · bm−3 ... ... ... ... b1 b2 b3 · · · b0   if the row vector (b0b1 · · ·bm−1) is identified with the polynomial d(x) = b0 + b1x + ... + bm−1xm−1, then we may write 238 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 b =   d(x) xd(x) x2d(x) ... xm−1d(x)   where each polynomial is reduced modulo xm − 1. denote the polynomials associated with the matrices b1, b2, b3, . . . , bp by d1(x), d2(x), d3(x), . . . , dp(x). these polynomials are called defining polynomials of c. taking the polynomials axldi(x) instead of di(x) we make a cyclic shift of the columns of bi and multiply them by a nonzero element of the field. this leads to a generator matrix of an equivalent code. so, the defining polynomials of a qc-code can be chosen from a fixed set of representatives of the equivalence classes of polynomials of degree less than m under the following relation: ci(x) ≈ cj(x) ⇐⇒ ci(x) ≡ axlcj(x) mod (xm − 1) (1) it stands to reason that we need an efficient algorithm to produce such a set of polynomials. 3. necklaces we identify the polynomials with the strings of their coefficients. in terms of strings the relation (1) is a composition of two actions on strings, namely, rotating the string and multiplying its entries by nonzero elements of the field (scaling the string). efficient algorithms are known for generating the equivalence classes of the first of the actions. by efficient we mean that the amount of computation used in generating the objects is proportional to the number of objects generated. let σq be the alphabet {0, 1, 2, . . . ,q − 1} and σmq be the set of q-ary strings of length m. denote by αi, 1 ≤ i ≤ m, the entries of the string α ∈ σmq , and at is the string of length t whose entries are all equal to a. the symbol � is used for lexicographic order in σmq . call two strings equivalent if one is a cyclic shift of the other. an equivalence class of strings under this relation is called a necklace of m beads in q colors. we identify each necklace with the lexicographically smallest representative in the equivalence class. thus we call a string α = α1α2 . . .αm a necklace if, for 1 ≤ i ≤ m, α1 . . .αm � αi . . .αmα1 . . .αi−1. for given m and q the number of necklaces is well known to be nq(m) = 1 m ∑ d|m φ(d)q m d where φ is the euler totient function. a simple and elegant algorithm was proposed by fredricksen, kessler and maiorana [22], [23] to generate all necklaces in σmq . we will refer to this as the fkm algorithm. for given m and q, the fkm algorithm creates a list, fkm(q,m), consisting of a certain subset of σmq in lexicographic order. the list begins with the string 0 m and ends with (q − 1)m. for a given α 239 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 on fkm(q,m), the successor of α, succ(α), is obtained as follows: for α = α1α2 . . .αm ≺ (q − 1)m, succ(α) = (α1 . . .αi−1(αi + 1))tα1 . . .αj, where i is the largest integer 1 ≤ i ≤ m such that αi < q − 1 and t,j satisfy ti + j = m, j < i. it is shown in [22] that there is no necklace between two elements of fkm(q,m), so that the list contains all necklaces. thus, discarding non necklaces of fkm(q,m) would result in a list of all necklaces in increasing order. in [29] ruskey, savage and wang proved that succ(α) is a necklace if and only if the ”i” from the definition of the successor of α is a divisor of m. including this test, the entire algorithm can be represented by the following pascal code: for i:=0 to m do a[i]:=0; br:=0; i:=m; repeat a[i]:=a[i]+1; for j:=1 to m-i do a[j+i]:=a[j]; if m mod i = 0 then begin br:=br+1; for j:=1 to m do write(a[j]); writeln; end; i:=m; while a[i]=q-1 do i:=i-1; until i=0; 4. generalized necklaces in its turn, scaling the necklaces partitions the set of all necklaces into new equivalence classes. we call them generalized necklaces. to generate their representatives we know no more efficient algorithm than rejecting those necklaces that are not the smallest representatives. a naive approach to testing whether a length m necklace is the smallest representative of an equivalence class according to (1) is to compare the necklace with all of its scaled rotations. however, by taking into consideration some facts we can decrease the number of comparisons that have to be made. first of all we generate only necklaces with first non-zero element 1. it follows from the fkm algorithm that any necklace has the form α = (α1α2 . . .αi) t, t ≥ 1 any rotation of α has the form α′ = αs+1 . . .αi(α1α2 . . .αi) t−1α1 . . .αs = = (αs+1 . . .αiα1 . . .αs) t for s = 1, 2, . . . , i− 1. thus comparing the strings α and bα′, b ∈ gf(q) \ {0, 1} it suffices to compare the substring β = α1α2 . . .αi with its rotations, multiplied by an element of gf(q) \{0, 1}. if p is the index of the first non-zero element of β, than scaled cyclic shifts with starting positions 2, 3, . . . ,p will obviously follow β in lexicographic order. the cyclic shifts with starting positions i−p + 2, i−p + 3, . . . , i 240 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 will have αi 6= 0 in position with number less than p and therefore they will also follow β = α1 . . .αi. so, the only comparisons that have to be made are with cyclic shifts starting from positions l = p + 1,p + 2, . . . , i−p + 1 if i−p + 1 ≥ p + 1, i.e. i ≥ 2p, having αl+p−1 > 1, and multiplied only by the inverse of αl+p−1. the implementation of these considerations yields the following pascal code, which produces the desired set of strings (or polynomials): for i:=0 to n-1 do a[i]:=0; a[n]:=1; br:=1; i:=n; p:=n; repeat if i=p then begin i:=i-1; p:=p-1; end; a[i]:=a[i]+1; for j:=1 to n-i do a[j+i]:=a[j]; if n mod i = 0 then begin for l:=p+1 to i-p+1 do if a[l+p-1]>1 then begin m:=inv[a[l+p-1]]; k1:=1; k2:=l; while (k1<=i) and (a[k1]=mul[a[k2],m]) do begin k1:=k1+1; k2:=(k2 mod i)+1; end; if k1<= i then if a[k1] > mul[a[k2],m] then goto 10; end; begin br:=br+1; for j:=1 to n do write(a[j]); end; end; 10: i:=n; while a[i]=q-1 do i:=i-1; until (i=i1) and (i=1); in the table below the number k11(m) and k13(m) of generated objects is given for m = 6, 7, 8, 9. the respective number n11(m) and n13(m) of necklaces is also given for comparison. from some seconds up to some minutes are needed for the generation of the objects (cpu, intel i3, 2.2ghz). table 1: nq(m) and kq(m) m n11(m) k11(m) m n13(m) k13(m) 7 2783891 278389 6 804895 67116 8 26796726 2679859 7 8964085 747007 9 261994491 26199449 8 101969959 8497806 241 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 the next explicit formula for the number of generalized necklaces (defining polynomials for qc codes) was given recently in [31] (for qt codes see [32]) kq(m) = 1 (q − 1)m ∑ d|m φ(d) ( q m d − 1 ) gcd(d,q − 1) 5. new qc codes over gf(11) and gf(13) in this section we present 34 new qc codes over gf(11) in dimensions k = 8, 9 and 36 new qc codes over gf(13) in dimensions k = 7, 8. the new codes are obtained by non-exhaustive local computer search. by reason of space the defining polynomials and weight distributions only of some of the codes are given. for the rest of the codes, the respective information is available on request from the authors. all constructed codes will be send to e. chen to be included in the respective table[9]. the elements of the fields are denoted by 0, 1, 2, ... , 9, 10 = a, 11 = b, 12 = c. codes over gf(11) there exist quasi-cyclic codes with parameters: [16, 8, 8]11, [24, 8, 14]11, [32, 8, 20]11, [40, 8, 26]11, [48, 8, 33]11, [56, 8, 39]11, [64, 8, 46]11, [72, 8, 53]11, [80, 8, 59]11, [88, 8, 66]11, [96, 8, 73]11, [104, 8, 80]11, [112, 8, 87]11, [120, 8, 94]11, [128, 8, 100]11, [136, 8, 107]11, [144, 8, 114]11, [152, 8, 121]11. a [16, 8, 8]11 optimal code: aaaaaaa9, aaa9a362; 01 87400 955200 10367360 112031680 128526000 1326073600 1456016800 1574628160 1646652680 a [24, 8, 14]11 code: aaa986a2, aaaa9a68, aaaa7022; 01 144880 1529840 16159390 17753120 182924120 199254640 2023128500 2144045200 2260063480 2352233280 2421762430 a [32, 8, 20]11 code: aaa98558, aaaa6960, aaaa9784, a9073007; 01 202540 2113360 2266000 23280960 241077200 253415120 269184520 2720465280 2836499940 2950328800 3050421000 3132438240 3210165920 a [40, 8, 26]11 code: aaa98503, aaaa7006, aaaa9a28, a9074154, aaaa9232; 01 261360 274880 2824300 29113760 30406400 311287440 323650050 338810720 3418211080 3531126480 3643291000 3746825520 3836935880 3918923760 404746250 a [48, 8, 33]11 code: a9074154, aaa986a2, aaaa9232, aaaa7022, aaaa9a68, a92a7a64; 01 332400 3410200 3544800 36152880 37501200 381448040 393704240 408343150 4116214080 4227158960 4337882960 4442926260 4538227600 4624916320 4710623840 482201950 a [56, 8, 39]11 code: a9074154, aaa98449, aaaa7011, aaaa9a28, aaaa9162, aa461627, aaa80096; 01 391120 403720 4116160 4263720 43192000 44575440 451534240 463666320 477828560 4814643050 4923845680 5033507360 5139412640 5237813820 5328560000 5415893160 555777200 561024690 a [64, 8, 46]11 code: aaa98449, aa606690, aaaa97a4, a9072196, aaaa9149, aa461627, aaa80091, aa2a9a89; 01 462000 476640 4823420 4974960 50234280 51628720 521577940 533584880 547284360 5513222240 5621322560 5729816640 5836078240 5936648480 6030601420 6119981760 629712480 633080160 64477700 a [72, 8, 53]11 code: aaa986a2, aaaa7022, aaaa9a68, a9074154, aaaa9232, aa461651, aaa800a5, aa301291, aa6a5912; 01 532480 549760 5529760 5693600 57258960 58665280 591602880 603442340 616787200 6212060720 6319044240 6426783290 6533069280 6635145400 6731353600 6823078000 6913339120 705749880 711620720 72222370 a [80, 8, 59]11 code: a8606780, aaaa6a63, aa8896a9, a9074154, aaa25520, aa461651, aaa7a862, aa2a9a89, aa6a5384, aa3a2803; 01 591200 603520 6112080 6240120 63102640 64283920 65691600 661585440 673298320 686291980 6910963040 7017246760 7124241840 242 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 7230352890 7333273280 7431397080 7525181760 7616544120 778602400 783299480 79839040 80106370 a [88, 8, 66]11 code: a8606759, aaaa6a60, aa8896a6, a9074150, aaa255a5, aa461649, aaa7a857, aa2a9a86, aa6a5379, aa3a2803, aaa97887; 01 661440 674960 6816000 6942400 70118120 71296160 72698300 731567200 743124800 755893520 7610031240 7715640320 7821953920 7927887520 8031397440 8131029840 8226457160 8319099600 8411413680 855327520 861884000 87424960 8848780 there exist quasi-cyclic codes with parameters: [9p, 9, 7p− 6]11 for p = 2, 3, ..., 7, [9p, 9, 15p/2 − 9]11 for p = 8, 10, 12, 14, 16 and [9p, 9, 15(p− 1)/2 − 2]11 for p = 9, 11, 13, 15, 17. a [18, 9, 8]11 code: aaaa98708, aaaaa7023; 01 81440 922320 10187740 111340640 127871640 1336363600 14129741480 15346074720 16648874890 17763370460 18424098760 a [45, 9, 29]11 code: aaaaa7023, aaaaa9949, aaaa98708, aa98113a9, aaa9a9a68; 01 292880 3011160 3153550 32236160 33930270 343294720 3510311930 3628639260 3769744510 38146759040 39263601600 40395301510 41481704840 42459168780 43320269140 44145588050 4532330290 a [108, 9, 81]11 code: aaaa98708, aaaaa7023, aaaaa9949, aa98113a9, aaa9a9a59, aaa520467, aaa249519, aa9a9a937, aaa721a86, aa9a9a823, aaa9a8973, aa9a89a82 01 811800 825400 8317280 8450580 85142650 86394740 87976260 882353500 895258880 9011138730 9121897720 9240764510 9369821070 94111537090 95164458260 96222387870 97275444100 98309390300 99312102790 100280789380 101222478830 102152968320 10388963020 10442785460 10516267290 1064608720 107864810 10878330 codes over gf(13) there exist quasi-cyclic codes with parameters: [14, 7, 7]13, [21, 7, 13]13, [28, 7, 18]13, [7p, 7, 6p − 6]13 for p = 5, 6, ..., 15 and [7p, 7, 6p− 5]13 for p = 16, 17, 18, 19. a [14, 7, 7]13 code: ccccccb, cccbca7; 01 72100 821336 9164220 10983556 114319196 1212922308 1323875068 1420460732 a [21, 7, 13]13 code: cc484c1, ca650c9, ccb0b70; 01 136636 1439396 15210336 16957684 173356220 189025884 1917011344 2020458452 2111682564 a [28, 7, 18]13 code: cc48499, ca650c7, cc78aca, cccca17; 01 181848 197896 2049560 21219168 22847980 232627268 246584004 2512663000 2617505348 2715570492 286671952 a [35, 7, 24]13 code: cc4849a, ca650c9, cc78acb, cccca15, cb04650; 01 242772 2511172 2652500 27209160 28713412 292061696 304989516 319623040 3214413560 3315735888 3411134032 353801768 a [42, 7, 30]13 code: cc48499, cc78b00, cccca18, cb04647, ccbb381, ca650c9; 01 302352 3112768 3251912 33194376 34599592 351619184 363827880 377441812 3811757564 3914433048 4013019076 417613844 422175108 a [112, 7, 91]13 code: cccca24, ca650c9, cc484c1, ccc76a8, ccbc4b2, ccbb385, cccbcb4, cc78b01, ccb9656, ccbc163, ccbc962, ccbac74, ccbab1a, ccbab23, ccccbc1, cb38697; 01 914200 9214448 9335784 9491056 95196224 96407736 97825636 981507632 992539824 1003945984 1015683272 1027303128 1038580348 1048870568 1058095248 1066441624 1074324320 1082388792 1091068228 110336672 11179716 1128076 there exist quasi-cyclic codes with parameters: [16, 8, 8]13, [24, 8, 14]13, [32, 8, 20]13, [40, 8, 27]13, [48, 8, 33]13, [56, 8, 40]13, [64, 8, 47]13, [72, 8, 53]13 and [8p, 8, 7p− 10]13 for p = 10, 11, ..., 19. a [16, 8, 8]13 code: cccccc77, cc82b0ca; 01 87116 980352 10583632 114004448 1219892040 1373427424 14188896848 15302188704 16226650156 243 r. daskalov, e. metodieva / j. algebra comb. discrete appl. 7(3) (2020) 237–245 a [24, 8, 14]13 code: ccccccc6, ccb50a09, ccc538ca; 01 143312 1532544 16205248 171154400 185379504 1920390304 2061267056 21139904640 22228977184 23238950528 24119466000 a [64, 8, 47]13 code: ccccccca, ccb50a15, ccc7b8b6, ccb75441, cc629b0b, cc847138, ccc538ca, cc6c9aa0; 01 473168 4812156 4950016 50189360 51593184 521788792 534855392 5411944992 5525823712 5650130300 5784293952 58122167440 59148750176 60149087808 61117161664 6268071872 6325970304 644836432 a [80, 8, 60]13 code: cccccc56, 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department of mathematics, deenbandhu chhotu ram university of science and technology, murthal-131039, sonepat, india (email: manjitsingh.math@gmail.com). 21 http://orcid.org/0000-0003-3351-7287 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 for an [n,k]q-linear code c, the dual code c⊥ is also a linear code of length n and dimension n−k over fq. the code c is self-orthogonal if c ⊆ c⊥ and self-dual if c = c⊥. the length n of a self-dual code is even and dimension is n/2. further, if q is odd, then there is no self-dual cyclic code of length 2n over fq and, if q is even, then there is a unique self-dual cyclic code of dimension 2n−1 of length 2n (see [16, section 5]). blackford [4] obtained the necessary and sufficient conditions for the existence of simple-root self-dual negacyclic codes. bakshi and raka [3] obtained all the self-dual negacyclic codes of length 2n over fpm, where p is an odd prime and integer m ≥ 1. dinh [8, corollary 3.3] characterized all repeated-root self-dual negacyclic codes of length 2ps over fpm. remarkably note that constacyclic codes, except negacyclic codes or binary cyclic codes, do not include an important class of codes, that is, self-dual codes, however these codes contain formally self-dual and isodual codes. a linear code c is called isodual if c is equivalent to its dual code c⊥. a code c is called formally self-dual (f.s.d.) code if c and c⊥ have the same weight distribution. automatically, self-dual codes are isodual codes and isodual codes are formally self-dual codes. huffman and pless [9] studied formally self-dual binary codes, however little work is known about formally self-dual codes over non-binary fields. these codes are important due to their applications in lattices [2], designs [10], code based cryptography, particularly in determining a minimal access set [14] and coding theory [5]. in this paper, motivated by the numerous practical applications of isodual codes and formally selfdual codes, we investigate a class of isodual and formally self-dual constacyclic codes which are permutation and monomially equivalent to each other. further, we study the form of codewords, generator polynomials and the weight distribution of a class of µkirreducible constacyclic codes of length 2`n over fq, where ` is odd prime such that `k|(q−1) and µk is an element of order `k in f∗q \{1,−1}. this paper also presents a recursive factorization of x2 m`n −µk ∈ fq[x], where integer m ≥ 1. the structure of the paper is as follows: some background of constacyclic codes, the necessary notation and some known results are to be used presented in section 2. in section 3, the explicit factorization of x2` n −µk over fq is obtained. further, we introduce a recursive factorization of x2 m`n−µk. the form of codewords of all irreducible µk-constacyclic codes of length `n and 2`n are also obtained. moreover, the weight distributions of these codes are determined by simply observing the weight of each message word. in section 4, we construct a family of isodual and hence formally self-dual codes of length 2`n over fq in a very special case. in the end of this section, we illustrate a class of isodual codes and formally self-dual codes by means of some examples. 2. preliminaries throughout this paper fq denotes a finite field with q elements, where q is a power of a prime. for a fixed µ ∈ f∗q, where f∗q = fq \{0}, a linear code c of length n over fq is called a µ-constacyclic code if (c1,c2, . . . ,cn−1,µc0) ∈ c for each (c0,c1,c2, . . . ,cn−1) ∈ c. the code c is called cyclic if µ = 1 and negacyclic if µ = −1. if gcd(n,q) = 1, a µ-constacyclic code of length n is called simple-root code; otherwise it is called repeated-root code. identifying the vector (a0,a1, . . . ,an−1) ∈ fnq with the polynomial a0xn−1 + a1xn−2 + · · · + an−1 ∈ fq[x], a simple-root µconstacyclic code c can be represented as an ideal of the quotient ring fq[x]/〈xn − µ〉. each ideal in fq[x]/〈xn −µ〉 is of the form 〈g(x)〉, where g(x) is a monic divisor of xn −µ in fq[x]. the polynomial g(x) is known as the generator polynomial of the code c. a µ-constacyclic code c is called irreducible or minimal over fq if h(x) = (xn −µ)/g(x) is an irreducible polynomial over fq, where the polynomial h(x) is known as the parity check polynomial of the code c. the structure of constacyclic codes have been extensively studied and can be found in [1, 3, 5–8, 11]. for any two vectors a = (a1,a2, . . . ,an) and b = (b1,b2, . . . ,bn) in fnq , the (euclidean) inner product or dot product of a and b is defined as follows: 22 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 a ·b = n∑ i=1 aibi ∈ fq, where ai,bi ∈ fq for each i = 1,2, . . . ,n. the two vectors a and b are said to be orthogonal if a · b = 0. the dual code of c, denoted by c⊥, is defined as c⊥ = {b ∈ fnq : a ·b = 0 for all a ∈ c}. for any polynomial f(x) = r∑ i=0 aix r−i, where a0 6= 0, over fq, the reciprocal polynomial of f(x), denoted by f∗(x), is given by: f∗(x) = xrf(x−1) = r∑ i=0 aix i. for a µ-constacyclic [n,k]q-code c with parity check polynomial h(x) of degree k over fq, the generator polynomial of c⊥ is given by h∗(x) over fq. note that h∗(x) divides xn −µ−1 over fq if and only if h(x) divides xn −µ over fq. lemma 2.1. [8, proposition 2.2] the dual of µ-constacyclic code is a µ−1-constacyclic code. remark 2.2. the dual code c⊥ of a cyclic (negacyclic) code c is a cyclic (negacyclic) code, however, from lemma 2.1, we notice that the dual code of µ-constacyclic code is not a µ-constacyclic code for every µ ∈ f∗q \{1,−1}. definition 2.3 (see [13]). two (n,m)-codes, where m is the size of the code, over fq are equivalent if one can be obtained from the other by a combination of operations of the following type: (i) permutation of the n digits of the codewords; (ii) multiplication of the symbols appearing in a fixed position by a nonzero scalar. lemma 2.4. [13, p.72] equivalent linear codes have the same length, dimension and distance. the following result contains a criterion on irreducible non-linear binomials over fq, which was given by serret in 1866. lemma 2.5. [12, theorem 3.75] let t ≥ 2 be an integer and a ∈ f∗q. then the binomial xt − a is irreducible in fq[x] if and only if the following two conditions are satisfied: (i) each prime factor of t divides the order e of a in f∗q, but does not divide (q −1)/e; (ii) q ≡ 1 (mod 4) if t ≡ 0 (mod 4). for this section and the rest of this work, we set the following notation: let ` be a prime such that gcd(`,q) = 1 and v = max{k : `k|(q−1)}. obviously, v = 0 if and only if ` (q−1). let µk be a primitive `kth root of unity in f∗q with µ0 = 1. for a fixed k, where 1 ≤ k ≤ v, we rephrase the factorization of x` n − µk over fq in the following form: lemma 2.6. [6, theorem 4.1(ii)] let n be a positive integer, gcd(q,`) = 1, r = min{n,v−k}, 1 ≤ k ≤ v, and v ≥ 2 if ` = 2. then the irreducible factorization of x` n −µk is given by x` n −µk = `r∏ i=1 (x` n−r −µ` ki+1 k+r ), where µk+r is an element of order `k+r in f∗q. 23 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 in particular, by taking ` = 2 in lemma 2.6, a factorization of x2 n −µk into the product of 2r factors is given as follows: lemma 2.7. let n be a positive integer, q be odd prime power, r = min{n,v−k} and 1 ≤ k ≤ v. then a factorization of x2 n −µk over fq is given by: x2 n −µk = 2r∏ i=1 (x2 n−r −µ2 ki+1 k+r ), where µk+r is an element of order 2k+r in f∗q. further, if v ≥ 2, the factorization is irreducible over fq, and if v = 1, the irreducible factorization of x2 n −µv is given in lemma [16, lemma 2.6]. in order to give a detailed explanation of our examples, we consider the following notions and results given in [15]. let sq = {a2 : a ∈ f∗q} and oq = {a ∈ f∗q : |a| is odd}, where |a| denotes the order of a ∈ f∗q. readily note that oq and sq are subgroups of the multiplicative group f∗q of the nonzero elements of fq satisfying oq ⊆sq. remarkably note that q ≡ 3 (mod 4) if and only if oq = sq. lemma 2.8. [15, theorem 4.2] let q and t be odd primes such that q = 2t + 1. then sq is generated by 4. lemma 2.9. [15, theorem 4.4] let q and t be odd primes such that q = 4t+1. then the following holds: (i) oq = 〈t〉 = {ti : 0 ≤ i ≤ t−1}. (ii) sq = 〈4〉 and oq = 〈16〉 for q > 13. 3. a class of irreducible constacyclic codes this section studies a class of µkirreducible constacyclic code over fq. the weight distribution of this class of codes is followed by the form of codewords without putting much effort. first of all, by using the same notation introduced in lemma 2.6, let ck,i denote a µk-constacyclic [`n,`n−r]q code with the parity check polynomial hk,i(x) = x` n−r −µ` ki+1 k+r . then the generator polynomial gk,i(x) of ck,i is given by: gk,i(x) = x` n −µk x` n−r −µ` ki+1 k+r . theorem 3.1. let n be a positive integer, ` be a prime such that gcd(`,q) = 1, `k|(q−1) for 1 ≤ k ≤ v and r = min{n,v −k}. then the generator polynomial of ck,i is given by: gk,i(x) = `r−1∑ u=0 µ (`ki+1)u k+r x `n−r(`r−u−1). further, if a = (a0,a1, . . . ,a`n−r−1) ∈ f` n−r q be any message word, then ck,i = {(a,aµ` ki+1 k+r , · · · ,aµ (`ki+1)(`r−1) k+r )}. proof. let r = min{n,v −k} and 1 ≤ k ≤ v. since µk = µ (`ki+1)`r k+r for 1 ≤ i ≤ ` r, so x` n −µk = (x` n−r )` r −µ(` ki+1)`r k+r = (x `n−r −µ` ki+1 k+r ) (`r−1∑ u=0 µ (`ki+1)u k+r x `n−r(`r−u−1)). 24 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 therefore, the generator polynomial gk,i(x) of a µk-constacyclic [`n,`n−r]q code ck,i is given by: gk,i(x) = x` n −µk x` n−r −µ` ki+1 k+r = `r−1∑ u=0 µ (`ki+1)u k+r x `n−r(`r−u−1). let a = (a0,a1, · · · ,a`n−r−1) ∈ f` n−r q be any message word. then the corresponding message polynomial can be expressed as a(x) = `n−r−1∑ j=0 ajx `n−r−j−1. therefore, the code polynomial of ck,i is given as follows: a(x)gk,i(x) =  `n−r−1∑ j=0 ajx `n−r−j−1  (`r−1∑ u=0 µ (`ki+1)u k+r x `n−r(`r−u−1) ) = `r−1∑ u=0 µ (`ki+1)u k+r  `n−r−1∑ j=0 ajx `n−r−j−1  x`n−r(`r−u−1) = `r−1∑ u=0 `n−r−1∑ j=0 ajµ (`ki+1)u k+r x `n−r(`r−u)−j−1. further, the uth component of codeword (c∗0,c ∗ 1, · · · ,c∗`r−1) is c∗u = (a0µ (`ki+1)u k+r ,a1µ (`ki+1)u k+r , · · · ,a`n−r−1µ (`ki+1)u k+r ). by using notation aθ for the vector (a0θ,a1θ, · · · ,a`n−r−1θ), where θ ∈ f∗q, the code ck,i is given by: ck,i = {(a,aµ` ki+1 k+r , · · · ,aµ (`ki+1)(`r−1) k+r ) : a ∈ f `n−r q }. this completes the proof. remark 3.2. by lemma 2.5, x` n−r − µ` ki+1 k+r is an irreducible factor of x `n − µk over fq for every 1 ≤ i ≤ `r provided that v ≥ 2 if ` = 2. let us consider the exceptional case v = 1 if ` = 2. then k = 1, i = 1 and hence gk,i = 1. thus, the code ck,i becomes the whole space f2 n q , a trivial negacyclic code. for any n ≥ 2, x2 n + 1 is reducible over fq, where q ≡ 3 (mod 4). therefore, there are non-trivial negacyclic equivalent codes of length 2n. these negacyclic codes of length 2n over fq and their weight distribution have been studied in [17, theorem 4.6]. lemma 3.3. for any 1 ≤ i,j ≤ `r, ck,i is equivalent to ck,j. proof. there are `r irreducible codes ck,i for 1 ≤ i ≤ `r for each fixed k, where 1 ≤ k ≤ v and r = min{n,v −k}. for every pair (i,j), where 1 ≤ i ≤ j ≤ `r, there exists a unique l = j − i (mod `r) such that µ` kj+1 k+r = µ l rµ `ki+1 k+r . using the form of codewords of ck,i and ck,j given in theorem 3.1, it follows that ck,j = µj−ir ck,i. thus, the code ck,j can be obtained form the code ck,i by multiplying µj−ir and hence they are equivalent. by lemma 3.3, in particular, the code ck,i is equivalent to ck,`r . for convenience point of view, we denote ck,`r by ck. 25 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 theorem 3.4. by using the assumptions of theorem 3.1, the weight distribution of µk-constacyclic [`n,`n−r]q code ck is a`rj = ( `n−r j ) (q −1)j for 0 ≤ j ≤ `n−r. proof. direct from theorem 3.1. keeping in mind the previous notation, we now present the following result, which will be important in order to study of isodual and formally self-dual constacyclic codes that we are interested in. theorem 3.5. let q, `, r and k be as before, and additionally, let ` be odd. then x2` n −µk = `r∏ i=1 (x` n−r ±θ` ki+1 k+r ), where θk+r is an element of order `k+rand θ2k = µk. proof. for each 1 ≤ k ≤ v, the order of µk is `k. clearly `k is odd and µ` k+1 k = µk. let θk = µ (`k+1)/2 k for 1 ≤ k ≤ v. then the order of θk is `k and θ2k = µk. it follows that x 2`n − µk = x2` k − θ2k = (x` k −θk)(x` k + θk). by lemma 2.6, the factorization of x` n −θk over fq is given by: x` n −θk = `r∏ i=1 (x` n−r −θ` ki+1 k+r ). by changing x to −x in the above factorization of x` n −θk, we obtain the factorization of x` n + θk over fq as follows: x` n + θk = `r∏ i=1 (x` n−r + θ` ki+1 k+r ). this completes the proof. theorem 3.6. let q, `, r, k ` be as in theorem 3.5, and additionally, let m be a positive integer. then a recursive factorization of x2 m`n −µk over fq is given by x2 m`n −µk = `r∏ i=1 (x2 m−1`n−r ±θ` ki+1 k+r ), where θk+r is an element of order `k+rand θ2k = µk. proof. the required form of factorization follows directly by using the substitution x to x2 m−1 in theorem 3.5. by lemma 2.5, binomials x2 m−1`n ±θ` ki+1 k+r is reducible over fq if and only if ∓θ `ki+1 k+r ∈sq, where sq is the set of all square elements in f∗q. for each 1 ≤ k ≤ v, in view of theorem 3.5, there are 2`r distinct irreducible factors of x2` n −µk over fq. let c′k,i and c ′′ k,i be µk-constacyclic [2` n,`n−r]q codes such that c′k,i = 〈g ′ k,i(x)〉 and c ′′ k,i = 〈g ′′ k,i(x)〉, where 1 ≤ i ≤ `r, 1 ≤ k ≤ v and g′k,i(x) = x2` n −µk x` n−r −θ` ki+1 k+r = ( x` n −θk x` n−r −θ` ki+1 k+r ) (x` n + θk) and g′′k,i(x) = x2` n −µk x` n−r + θ` ki+1 k+r = ( x` n + θk x` n−r + θ` ki+1 k+r ) (x` n −θk). 26 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 theorem 3.7. let ` be a prime. 1 ≤ i ≤ `r and 1 ≤ k ≤ v. then the generator polynomial g′k,i(x) of c′k,i is given by g′k,i(x) = `r−1∑ u=0 θ (`ki+1)u k+r x `n−r(`r−u−1)(x` n + θk). further, if a ∈ f` n−r q is any message word, then c′k,i = {(a,aθ `ki+1 k+r , · · · ,aθ (`ki+1)(2`r−1) k+r )}. proof. using the argument of theorem 3.1, we obtain x` n −θk = (x` n−r −θ` ki+1 k+r ) (`r−1∑ u=0 θ (`ki+1)u k+r x `n−r(`r−u−1)). it follows that x2` n −µk = (x` n −θk)(x` n + θk) = (x` n−r −θ` ki+1 k+r ) (`r−1∑ u=0 θ (`ki+1)u k+r x `n−r(`r−u−1))(x`n + θk) = (x` n−r −θ` ki+1 k+r ) (2`r−1∑ u=0 θ (`ki+1)u k+r x `n−r(2`r−u−1)). thus the generator polynomial g′k,i(x) of a µk-constacyclic [2` n,`n−r]q code c′k,i is g′k,i(x) = x2` n −µk x` n−r −θ` ki+1 k+r = 2`r−1∑ u=0 θ (`ki+1)u k+r x `n−r(2`r−u−1). let a = (a0,a1, · · · ,a`n−r−1) ∈ f` n−r q be any message word. then the corresponding message polynomial can be expressed as a(x) = `n−r−1∑ j=0 ajx `n−r−j−1. it follows that the code polynomial of c′k,i is a(x)g′k,i(x) =  `n−r−1∑ j=0 ajx `n−r−j−1  (2`r−1∑ u=0 θ (`ki+1)u k+r x `n−r(2`r−u−1) ) = `r−1∑ u=0 θ (`ki+1)u k+r  `n−r−1∑ j=0 ajx `n−r−j−1  x`n−r(2`r−u−1) = 2`r−1∑ u=0 `n−r−1∑ j=0 ajθ (`ki+1)u k+r x `n−r(2`r−u)−j−1. further, the uth component of codeword (c∗∗0 ,c ∗∗ 1 , · · · ,c∗∗`r−1) is c∗∗u = (a0θ (`ki+1)u k+r ,a1θ (`ki+1)u k+r , · · · ,a`n−r−1θ (`ki+1)u k+r ). if we denote aθ = (a0θ,a1θ, · · · ,a`n−r−1θ) for some θ ∈ f∗q, then c∗∗u = aθ (`ki+1)u k+r for each 0 ≤ u ≤ 2` r−1. therefore 27 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 c′k,i = {(a,aθ `ki+1 k+r , · · · ,aθ (`ki+1)(2`r−1) k+r ) : a ∈ f `n−r q }. this completes the proof. using the argument discussed in lemma 3.3, observe that codes c′k,i and c ′′ k,i are equivalent to codes c′k,`r and c ′′ k,`r respectively. denote c ′ k,`r by c ′ k, and c ′′ k,`r by c ′′ k . theorem 3.8. with our notation and assumption, c′k is equivalent to c ′′ k . proof. let g′k(x) and g ′′ k(x) be generator polynomials of c ′ k and c ′′ k respectively. now the equivalence of these codes is asserted by the following identity: g′′k(−x) = x2` n −µk −x`n−r + θk+r = − ( x2` n −µk x` n−r −θk+r ) = −g′k(x). remark 3.9. for each fixed 1 ≤ k ≤ v and r = min{n,v −k}, there are 2`r irreducible codes c′k,i and c′′k,i for 1 ≤ i ≤ ` r. by theorem 3.8, these codes are equivalent to c′k. by lemma 2.4, since equivalent linear codes share the same weight distributions, so it is sufficient to determine the weight distribution of c′k, a µk-constacyclic code of length 2` n with the parity check polynomial x` n−r −θk+r. theorem 3.10. let ` be a prime, r = min{n,v − k} and 1 ≤ k ≤ v. then the weight distribution of µk-constacyclic [2`n,`n−r]q code c′k is a2`rj = ( `n−r j ) (q −1)j for 0 ≤ j ≤ `n−r. proof. from theorem 3.7, the explicit form of codewords of c′k is given by: c′k = {(a,aθk+r, · · · ,aθ 2`r−1 k+r ) : a ∈ f `n−r q }. now, the weight distribution of c′k follows directly by using the above form of the code. in end of this section, we present two examples as follows. example 3.11. with our notation, let q = 163, ` = 3. then v = 4, 1 ≤ k ≤ 3 and r = min{n,4−k}≥ 1. since sq(= oq), a subgroup of order 81, has φ(3i) = 3i−1·2 elements of order 3i for 1 ≤ i ≤ 4, and 4 ∈sq, so its order is either 9, 27 or 81. using simple modular arithmetic, we obtain the order of 4 is 81. thus one of the value of µ4 is 4. take µ4 = −18 = 461, µ3 = 36 = 421, µ2 = 38 = 463, µ1 = 104 = 427. it is important to note that the choice of µ4 determines µi uniquely as follows µ4−i = µ3 i 4 for 0 ≤ i ≤ 3, however if we choose µ1 first, then µ2 has more than one choices, for example if µ1 = 104, then µ2 ∈ {40,38}. by lemma 2.6, the factorization of x3 n −µk is given by: x3 n −µk = 3r∏ i=1 (x3 n−r −µ3 ki+1 k+r ). in view of table 1, the explicit factorization of x81 −38 over f163 is given by: x81 −38 = (x9 + 32)(x9 + 75)(x9 + 79)(x9 + 68)(x9 −24) (x9 + 66)(x9 + 63)(x9 −51)(x9 + 18). further, by theorem 3.4, the weight distribution of 38-constacyclic [81,9,9]163 code c is a9j = ( 9 j ) (162)j for 0 ≤ j ≤ 9. 28 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 table 1. parameters of the factorization x3 n −µk; 1 ≤ k ≤ 3 n k r = min{n,4−k} µk degree coefficients in f∗163 3n−r {(−18)3 ki+1 : 1 ≤ i ≤ 3r} 5 1 3 104 9 {(36)i(−18) : 1 ≤ i ≤ 27} 5 2 2 38 27 {(38)i(−18) : 1 ≤ i ≤ 9} 5 3 1 36 81 {(104)i(−18) : 1 ≤ i ≤ 3} 4 1 3 104 3 {(36)i(−18) : 1 ≤ i ≤ 27} 4 2 2 38 9 {(38)i(−18) : 1 ≤ i ≤ 9} 4 3 1 36 27 {(104)i(−18) : 1 ≤ i ≤ 3} 3 1 3 104 1 {(36)i(−18) : 1 ≤ i ≤ 27} 3 2 2 38 3 {(38)i(−18) : 1 ≤ i ≤ 9} 3 3 1 36 9 {(104)i(−18) : 1 ≤ i ≤ 3} 2 1 2 104 3 {(36)i(−18) : 1 ≤ i ≤ 9} 2 2 2 38 1 {(38)i(−18) : 1 ≤ i ≤ 9} 2 3 1 36 3 {(104)i(−18) : 1 ≤ i ≤ 3} 1 1 1 104 1 {(36)i(−18) : 1 ≤ i ≤ 3} 1 2 1 38 1 {(38)i(−18) : 1 ≤ i ≤ 3} 1 3 1 36 1 {(104)i(−18) : 1 ≤ i ≤ 3} the weight distribution of ck, where a3 n−r j = ( 3n−r j ) (162)j for 0 ≤ j ≤ 3n−r n k r = 4−k µk weight no. of codewords of weight i i = 3rj ai = a 3n−r j 5 1 3 104 27j a9j 5 2 2 38 9j a27j 5 3 1 36 3j a81j 4 1 3 104 27j a3j 4 2 2 38 9j a9j 4 3 1 36 3j a27j 3 1 3 104 27j a1j 3 2 2 38 9j a3j 3 3 1 36 3j a9j table 1 provides the weight distributions of all irreducible µk-constacyclic codes of length 3n for n = 3,4,5 and 1 ≤ k ≤ 3 over f163. example 3.12. with our notation, let q = 251, ` = 5. then v = 3 and r = min{n,3 −k}. the set of all square elements sq contains 125 elements as follows: φ(125) = 100 elements of order 125, φ(25) = 20 elements of order 25, 4 elements of order 5 and one element of order 1. since 4,9 ∈ sq are of order 25 and 125 respectively. take µ3 = 9, thereby µ2 = µ53 = 64 and µ 5 2 = µ1 = −32 = 219. now, µ2 = 64, µ3 = 9, θ2 = 6413 = −8 and θ3 = 913 = 88. by theorem 3.5, the factorization of x2·5 n −µk for n ≥ 1 is given by: x2·5 n −µk = 5r∏ i=1 (x5 n−r ±θ5 ki+1 k+r ). now for n = 2, k = 2, r = 1, µ2 = 64, θ3 = 88, the explicit factorization of x50 − 64 over f251 is given 29 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 by: x50 −64 = 5∏ i=1 (x5 ±8825i+1) = (x5 + 96)(x5 + 55)(x5 −60)(x5 −3)(x5 −88) (x5 −96)(x5 −55)(x5 + 60)(x5 + 3)(x5 + 88). also by theorem 3.10, the weight distribution of 64-constacyclic [50,5,10]251 code is a10j = ( 5 j ) (250)j for 0 ≤ j ≤ 5. further, by theorem 3.6, a reducible factorization of x100 −64 is given by: x100 −64 = 5∏ i=1 (x10 ±8825i+1) = (x10 + 96)(x10 + 55)(x10 −60)(x10 −3)(x10 −88) (x10 −96)(x10 −55)(x10 + 60)(x10 + 3)(x10 + 88). since −1 /∈sq and 88 ∈sq, so there are 5 reducible binomials, for example x10−88 = (x5−313)(x5+313) = (x5 + 29)(x5 − 29), and the remaining 5 are irreducible over fq. moreover all binomials x10 − 8825i+1 for 1 ≤ i ≤ 5 are reducible over fq such as x10 − 8825i+1 = (x5 − 313(25i+1))(x5 + 313(25i+1)) = (x5 + 2925i+1)(x5 −2925i+1) for 1 ≤ i ≤ 5. this process can be carried forward for a reducible factorization of x200 −64 from the explicit factorization of x100 −64 with 15 irreducible factors over fq. 4. formally self-dual and isodual codes a code is formally self-dual (f.s.d.) if the code and its dual have the same weight distribution. a code is isodual if it is equivalent to its dual. since two equivalent codes have the same weight distribution, so the class of formally self-dual codes automatically contains the class of isodual codes, however, a formally self-dual code need not be isodual (see [9, p. 378]). in this section, we construct isodual and formally self-dual linear codes. theorem 4.1. let ` be an odd prime such that `|(q−1), but `2 (q−1) and n be a positive integer. let η be a primitive `th root of unity in f∗q. then there exists an element θ = η (`+1)/2 ∈ f∗q of the order ` such that x2` n −η = (x` n −θ)(x` n + θ). if c1 = 〈x` n + θ)〉 and c2 = 〈x` n −θ)〉. then c1 = {(a,θa)} , c⊥1 = {(θa,−a)}, c2 = {(a,−θa) and c⊥2 = {(θa,a) where a = (a0,a1, . . . ,a`n−1) ∈ f` n q . further, c1 and c2 are isodual codes, and ci, c⊥i for i = 1,2 are formally self-dual codes. proof. on substituting v = 1, r = 0, k = 1, η = µk+r = µ1 in theorem 3.5, the factorization of x2` n −µk reduces in the following form: x2` n −η = x2` n −η`+1 = (x` n −θ)(x` n + θ), where θ = η(`+1)/2. clearly the order of θ ∈ f∗q is `. further, by theorem 3.7, we find the desired form of c1 and c2. furthermore, c⊥1 is a η −1-constacyclic code. since x2` n −η−1 = 1 η (θx` n − 1)(θx` n + 1) and c1 is generated by x` n + θ, so c⊥1 = 〈θx` n −1〉 = 〈x` n −θ−1〉. similarly c⊥2 = 〈θx` n + 1〉 = 〈x` n + θ−1〉. obviously, c1, c2, c⊥1 and c ⊥ 2 are all equivalent via a permutation of coordinates and multiplying certain coordinates by constants. since the relation of equivalence of codes is an equivalence relation, so c1 is equivalent to c⊥1 and c2 is equivalent to c ⊥ 2 . this proves that c1 and c2 are isodual codes. further, by theorem 3.10, the weight distribution of ci and its dual c⊥i is a2j = ( ` j ) (q − 1)j, where 0 ≤ j ≤ ` for i = 1,2. therefore c1 and c2 are formally self-dual codes. 30 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 remark 4.2. in view of theorem 3.5, θk ∈ f∗q presents one of the solution of the equation x2 = µk. obviously, −θk is another solution of this equation. since the order of θk is `k, then θk is a square element in f∗q. note that −1 is a square in fq if and only if 4|(q − 1). in fact, the order of −θk is 2`k when q ≡ 3 (mod 4) and, `k when q ≡ 1 (mod 4). from theorem 3.6, a reducible factorization of x4` n −µk over fq is given recursively as follows: x4` n −µk = `r∏ i=1 (x2` n−r ±θ` ki+1 k+r ). (i) if q ≡ 1 (mod 4), for each i, since ±θ` ki+1 k+r is a square element of order ` k+r, where r and k are as usual, and hence by applying theorem 3.5, one has the explicit factorization of x2` n−r ± θ` ki+1 k+r into the product of 2`r1 + 2`r1 irreducible factors over fq, where r1 = min{n − r,v − k}. in this case when q ≡ 1 (mod 8), this process of factorization allows us to select the generator polynomials of isodual and formally self-dual codes of length 8`n and dimension 4`n over a finite field from the given factorization of x8` n −µk ∈ fq[x] (see example 4.6). (ii) if q ≡ 3 (mod 4), then −1 /∈ sq, and one of the element ±θ` ki+1 k+r is a square element and hence either of the binomial x2` n−r ±θ` ki+1 k+r can be expressed as a product of 2` r1 irreducible factors over fq, where r1 = min{n − r,v − k}. in this case, the generator polynomials of isodual codes and formally self-dual codes are given by theorem 4.1. 4.1. worked examples the following are examples of isodual codes and formally self-dual codes by means of theorem 4.1 and theorem 3.6. example 4.3. for q = 13, we have ` = 3, v = 1, r = 0. by lemma 2.9, the order of 3 is 3 modulo 13, it follows that µ1 = 3 θ1 = 9. by lemma 2.5, x3 − 3 is an irreducible over f13. by theorem 3.5, x6 −3 = (x3 −9)(x3 + 9). let c1 = 〈x3 −9〉 = {(a0,a1,a2,4a0,4a1,4a2) : a0,a1,a2 ∈ f13}. then c⊥1 is 9-constacyclic code and is given by: c⊥1 = 〈x3 + 3〉 = {(a0,a1,a2,3a0,3a1,3a2) : a0,a1,a2 ∈ f13}. the weight distribution of c1 and c⊥1 is a2j = ( 3 j ) (12)j for 0 ≤ j ≤ 3. thus the weight distribution of this code is same as the weight of the code c1. denote by c2 = 〈x3 +9〉, c3 = 〈x3−3〉 and c4 = 〈x3 +3〉. then these codes are formally self-dual codes, while pairs (c1,c4) and (c2,c3) are of isodual codes. moreover, using theorem 3.6, we obtain x12 −3 = (x3 −3)(x3 + 3)(x3 −2)(x3 + 2). now, let c = 〈(x3 + 10)(x3 + 11)〉, then c⊥ = 〈(x3 + 4)(x3 + 6)〉. here c and c⊥ are respectively, a 3-constacyclic [12,6,3]13-code and a 9-constacyclic [12,6,3]13-code. it is quite easy to see that these codes are equivalent and hence c is isodual. example 4.4. for q = 11, we obtain ` = 5, v = 1, r = 0. by lemma 2.8, µ1 ∈ 〈4〉 is an element of order 5. all φ(5) = 4 elements of order 5 are given in {3,4,5,9}. consider µ1 = 3. by lemma 2.5, x5−3 is irreducible and x10−3 is reducible over f11. note that x10−3 = x10−36 = (x5−33)(x5 +33) = (x5 − 5)(x5 + 5). by theorem 3.6, x20 − 3 = (x10 − 5)(x10 + 5). since 42 = 5, so θ1 = 4, x10 − 5 = (x5 − 4)(x5 + 4) and hence x20 − 3 = (x5 − 4)(x5 + 4)(x10 − 6). let c = 〈x5 + 5〉 = {(a,5a) : a ∈ f511}. 31 m. singh / j. algebra comb. discrete appl. 7(1) (2020) 21–33 then c is a 3-constacyclic code of length 10 over f11 with the parity check polynomial x5 − 5. then c⊥ = 〈5x5 −1〉 = 〈x5 + 2〉 = {(a,2a) : a ∈ f511} is a 4-constacyclic code of length 10 over f11. since c⊥ is equivalent to c, so c is isodual and hence formally self-dual codes. example 4.5. for q = 29, we obtain ` = 7, v = 1, r = 0. by lemma 2.9, 7 is an element of order 7 modulo 29. take µ1 = 7. by lemma 2.5, x7 − 7 is irreducible and x14 − 7 is reducible over f29. note that x14 −7 = x14 −78 = (x7 −74)(x7 + 74) = (x7 + 6)(x7 −6). let c = 〈x7 −6〉 = {(a,−6a) : a ∈ f729}. then c is a 7-constacyclic code of length 14 over f29 with the parity check polynomial x7 + 6. then c⊥ = 〈6x7 + 1〉 = 〈x7 + 5〉 = {(a,5a) : a ∈ f729} is a 4-constacyclic code of length 14 over f29. since c⊥ = 4c and hence c is isodual and hence formally self-dual codes. by theorem 3.6, x28 − 7 = (x14 − 6)(x14 − 23) = (x7 ± 8)(x7 ± 9) and x28 − 25 = (x14 − 5)(x14 + 5) = (x7 ± 11)(x7 ± 13). now let c1 = 〈(x7 − 8)(x7 + 9)〉,c⊥1 = 〈(x7 − 11)(x7 + 13)〉,c2 = 〈(x7 − 8)(x7 − 9)〉,c⊥2 = 〈(x7 − 11)(x7 − 13)〉, c3 = 〈(x7 +8)(x7−9)〉, c⊥3 = 〈(x7 +11)(x7−13)〉 and c4 = 〈(x7 +8)(x7 +9)〉, c⊥4 = 〈(x7 +11)(x7 +13)〉 can be used for describing equivalent isodual and formally self-dual 23-constacyclic codes. example 4.6. for q = 41, we obtain ` = 5, v = 1, r = 0. note that 16 is an element of order 5 modulo 41. take µ1 = 16, we have µ −1 1 = 18. by lemma 2.5, x 5 − 16 is irreducible and x40 − 16 is reducible over f41. the factorization of x40 −16 over f41 is given by x40 −16 = (x5 ±17)(x5 ±11)(x5 ±10)(x5 ±8). also x40 −18 = (x5 ±12)(x5 ±15)(x5 ±4)(x5 ±5). let c = 〈f1(x)f2(x)g1(x)g2(x)〉, where fi(x) = x5 −ηi and gi(x) = x5 +ηi, where ηi ∈{8,10,11,17} and 1 ≤ i ≤ 2. then c is a 16-constacyclic code of length 40 over f41 and then c⊥ = 〈 x40 −18 f∗1 (x)f ∗ 2 (x)g ∗ 1(x)g ∗ 2(x) 〉 is an 18-constacyclic code of length 40 over f41. this represents a class of isodual codes. acknowledgment: the author would like to sincerely thank the anonymous referees who gave many helpful comments and suggestions to create an improved version. references [1] n. aydin, i. siap, d. k. ray–chaudhuri, the structure of 1–generator quasi–twisted codes and new linear codes, des. codes cryptogr. 24(3) (2001) 313–326. 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[12] r. lidl, h. niederreiter, introduction to finite fields and their applications, cambridge university press, 1986. [13] s. ling, c. xing, coding theory: a first course, cambridge university press, 2004. [14] j. l. massey, minimal codewords and secret sharing, proc. 6th joint swedish–russian workshop on information theory, mölle, sweden, (1993) 276–279. [15] m. singh, some subgroups of f∗q and explicit factors of x 2nd −1 ∈ fq[x], transactions on combinatorics (2019) doi: 10.22108/toc.2019.114742.1612. [16] m. singh, s. batra, some special cyclic codes of length 2n, j. algebra appl. 16(1) (2017) 17 pages. [17] m. singh, s. batra, weight distribution of a class of cyclic codes of length 2n, j. algebra comb. discrete appl. 6(1) (2018) 1–11. [18] x. zhu, q. yue, l. hu, weight distributions of cyclic codes of length lm, finite fields appl. 31 (2015) 241–257. 33 https://doi.org/10.1007/bf01388559 https://doi.org/10.1007/bf01388559 https://doi.org/10.1007/s10623-017-0356-2 https://doi.org/10.1007/s10623-017-0356-2 http://dx.doi.org/10.22108/toc.2019.114742.1612 http://dx.doi.org/10.22108/toc.2019.114742.1612 https://doi.org/10.1142/s0219498817500025 http://dx.doi.org/10.13069/jacodesmath.505364 http://dx.doi.org/10.13069/jacodesmath.505364 https://doi.org/10.1016/j.ffa.2014.07.005 https://doi.org/10.1016/j.ffa.2014.07.005 introduction preliminaries a class of irreducible constacyclic codes formally self-dual and isodual codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.671815 j. algebra comb. discrete appl. 7(1) • 85–101 received: 30 june 2019 accepted: 16 november 2019 journal of algebra combinatorics discrete structures and applications zq(zq + uzq)− linear skew constacyclic codes research article ahlem melakhessou, nuh aydin, zineb hebbache, kenza guenda abstract: in this paper, we study skew constacyclic codes over the ring zqr where r = zq + uzq, q = ps for a prime p and u2 = 0. we give the definition of these codes as subsets of the ring zαq rβ. some structural properties of the skew polynomial ring r[x, θ] are discussed, where θ is an automorphism of r. we describe the generator polynomials of skew constacyclic codes over zqr, also we determine their minimal spanning sets and their sizes. further, by using the gray images of skew constacyclic codes over zqr we obtained some new linear codes over z4. finally, we have generalized these codes to double skew constacyclic codes over zqr. 2010 msc: 94b15, 94b60 keywords: linear codes, skew constacyclic codes, zqzq[u]− linear skew constacyclic codes, bounds 1. introduction codes over finite rings have been known for several decades, but interest in these codes increased substantially after the discovery that good non-linear binary codes can be constructed from codes over rings. several methods have been introduced to produce certain types of linear codes with good algebraic structures and parameters. cyclic codes and their various generalizations such as constacyclic codes and quasi-cyclic (qc) codes have played a key role in this quest. one particularly useful generalization of cyclic codes has been the class of quasi-twisted (qt) codes that produced hundreds of new codes with best known parameters [4, 8, 9, 11, 12, 16, 17] recorded in the database [25]. yet another generalization of cyclic codes, called skew cyclic codes, were introduced in [15] and they have been the subject of an increasing research activity over the past decade. this is due to their algebraic structure and their applications to dna codes and quantum codes [14, 19, 20]. skew constacyclic codes over various rings ahlem melakhessou; department of mathematics, mostefa ben boulaïd university (batna2), batna, algeria (email: a.melakhessou@univ-batna2.dz). nuh aydin; department of mathematics and statistics, kenyon college, usa (email: aydinn@kenyon.edu). zineb hebbache, kenza guenda (corresponding author); faculty of mathematics, usthb, laboratory of algebra and number theory, bp 32 el alia, bab ezzouar, algeria (email: zinebhebache@gmail.com, ken.guenda@gmail.com). 85 https://orcid.org/0000-0002-5618-2427 https://orcid.org/0000-0002-1482-7565 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 have been studied in [1, 2, 5, 13, 21, 23, 26, 30, 32, 33] as a generalization of skew cyclic codes over finite fields. recently, p. li et al. [28] gave the structure of (1 + u)-constacyclic codes over the ring z2z2[u] and aydogdu et al. [6] studied z2z2[u]-cyclic and constacyclic codes. further, jitman et al. [27] considered the structure of skew constacyclic codes over finite chain rings. more recently a. sharma and m. bhaintwal studied skew cyclic codes over ring z4 + uz4, where u2 = 0. the aim of this paper is to introduce and study skew constacyclic codes over the ring zq(zq + uzq), where q is a prime power and u2 = 0. some structural properties of the skew polynomial ring r[x, θ] are discussed, where θ is an automorphism of r. we describe the generator polynomials of skew constacyclic codes over r and zqr. using gray images of skew constacyclic codes over zqr we obtained some new linear codes over z4. further, we generalize these codes to double skew constacyclic codes over zqr. the paper is organized as follows. we first give some basic results about the ring r = zq + uzq, where q = ps, p is a prime and u2 = 0, and linear codes over zqr, we construct the non-commutative ring r[x, θ], where the structure of this ring depends on the elements of the commutative ring r and an automorphism θ of r. we give some results on skew constacyclic codes over the ring r. in section 3, we study the algebraic structure of skew constacyclic codes over the ring zqr, section 4 includes the work on the generator polynomials of these codes, their minimal spanning sets and their sizes. in section 5, we determine the gray images of skew constacyclic codes over r and zqr. these codes are then further generalized to double skew constacyclic codes in the next section. finally. in section 7, we use the gray images of skew constacyclic codes over zqr to obtain some new linear codes over z4. 2. preliminaries let (α,β) denote n = α + 2β where α and β are positive integers. consider the ring r = zq + uzq, where q = ps, p is a prime and u2 = 0. the ring r is isomorphic to the quotient ring zq[u]/ 〈 u2 〉 . the ring r is not a chain ring, whereas it is a local ring with the maximal ideal 〈u,p〉. each element r of r can be expressed uniquely as r = a + ub, where a,b ∈ zq. 2.1. skew polynomial ring over r in this subsection we construct the non-commutative ring r[x, θ]. the structure of this ring depends on the elements of the commutative ring r and an automorphism θ of r. note that an automorphism θ in r must fix every element of zq, hence it satisfies θ(a + ub) = a + δ(u)b. therefore, it is determined by its action on u. let δ(u) = k + ud, where k is a non-unit in zq, k2 ≡ 0 mod q and 2kd ≡ 0 mod q. then, θ(a + ub) = a + δ(u)b = (a + kb) + udb, (1) for all a + ub ∈ r. further, let θ an automorphism of r and let m be its order. the skew polynomial ring r[x, θ] is the set of polynomials over r in which the addition is defined as the usual addition of polynomials and the multiplication is defined by the rule xa = θ(a)x. the multiplication is extended to all elements in r[x, θ] by associativity and distributivity. the ring r[x, θ] is called a skew polynomial ring over r and an element in r[x, θ] is called a skew polynomial. further, an element g(x) ∈ r[x, θ] is said to be a right divisor (resp. left divisor) of f(x) if there exists q(x) ∈ r[x, θ] such that f(x) = q(x)g(x) ( resp. f(x) = g(x)q(x)). in this case, f(x) is called a left multiple (resp. right multiple) of g(x). 86 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 lemma 2.1. [31, lemma 1] let f(x), g(x) ∈ r[x, θ] be such that the leading coefficient of g(x) is a unit. then there exist q(x), r(x) ∈ r[x, θ] such that f(x) = q(x)g(x) + r(x), where r(x) = 0 or deg(r(x)) < deg(g(x)). definition 2.2. [31, definition 3.2] a polynomial f(x) ∈ r[x, θ] is said to be a central polynomial if f(x)r(x) = r(x)f(x) for all r(x) ∈ r[x, θ]. theorem 2.3. the center z(r[x, θ]) of r[x, θ] is rθ[xm], where m is the order of θ and rθ is the subring of r fixed by θ. proof. we know r = zq + uzq is the fixed ring of θ. since order of θ is m, for any non-negative integer i, we have xmia = (θm)i(a)xmi = axmi for all a ∈ r. it gives xmi ∈ z(r[x, θ]), and hence all polynomials of the form f = a0 + a1x m + a2x 2m + · · · + alxlm with ai ∈ r are in the center. conversely, let f = f0 + f1x + f2x2 + · · · + fkxk ∈ z(r[x, θ]) we have fx = xf which gives that all fi are fixed by θ, so that fi ∈ r. further, choose a ∈ r such that θ(a) 6= a. then it follows from the relation af = fa that fi = 0 for all indices i not divides m. thus f(x) = a0 + a1x m + a2x 2m + · · · + alxlm ∈ rθ[xm]. corollary 2.4. let f(x) = xβ − 1. then f(x) ∈ z(r[x, θ]) if and only if m | β. further, xβ − λ ∈ z(r[x, θ]) if and only if m | β and λ is fixed by θ. 2.2. skew constacyclic codes over r in this section we generalize the structure and properties from [31] to codes over zq + uzq. hence the proofs of many of the theorems will be omitted. we start with some structural properties of r[x, θ]/〈xβ − λ〉. the corollary 2.4, shows that the polynomial (xβ −λ) is in the center z(r[x, θ]) of the ring r[x, θ], hence generates a two-sided ideal if and only if m | β and λ is fixed by θ. therefore, in this case r[x, θ]/〈xβ −λ〉 is a well-defined residue class ring. if m β, then the quotient space r[x, θ]/〈xβ − λ〉 which is not necessarily a ring is a left r[x, θ]-module with multiplication defined by r(x)(f(x) + (xβ −λ)) = r(x)f(x) + (xβ −λ), for any r(x), f(x) ∈ r[x, θ]. next we define the skew λ−constacyclic codes over the ring r. a code of length β over r is a nonempty subset of rβ. a code c is said to be linear if it is a submodule of the r−module rβ. in this paper, all codes are assumed to be linear unless otherwise stated. given an automorphism θ of r and a unit λ in r, a code c is said to be skew constacyclic, or specifically, θ −λ−constacyclic if c is closed under the θ −λ−constacyclic shift: ρθ,λ : r β → rβ 87 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 defined by ρθ,λ((a0,a1, . . . ,aβ−1)) = (λθ(aβ−1), θ(a0), . . . , θ(aβ−2)). (2) in particular, such codes are called skew cyclic and skew negacyclic codes when λ is 1 and −1, respectively. when θ is the identity automorphism, he become classical constacyclic and we denote ρλ the constacyclic shift. in the rest of paper, we restrict our study to the case where the length β of codes is a multiple of the order of θ and λ is a unit in rθ, where rθ denotes the subring of r fixed by θ. the proofs of the next theorems are analogous to the proofs of [31] given for the ring z4 + uz4, therefore we omit them. theorem 2.5. [31, theorem 3] a code cβ of length β in rβ = r[x, θ]/〈xβ−λ〉 is a θ−λ−constacyclic code if and only if cβ is a left r[x, θ]−submodule of the left r[x, θ]−module rβ. corollary 2.6. [31, corollary 2] a code c of length β over r is θ −λ−constacyclic code if and only if the skew polynomial representation of c is a left ideal in r[x, θ]/〈xβ −λ〉. the following theorem is the generalization of the theorems 4 and 5 of [31]. theorem 2.7. let cβ be a skew contacyclic code of length β over r. then, cβ is a free principally generated skew constayclic code if and only if there exists a minimal degree polynomial gβ(x) ∈ cβ having its leading coefficient a unit such that cβ = 〈gβ(x)〉 and gβ(x) | xβ − λ. moreover, cβ has a basis {gβ(x),xgβ(x), . . . ,xβ−deg(gβ(x))−1} and |cβ| = |r|β−deg(gβ(x)). in this next, we study duals of θ−λ−constacyclic codes over r. further, the euclidean inner product defined by 〈v′,w′〉 = β−1∑ i=0 v′iw ′ i, for v′ = (v′0,v ′ 1, . . . ,v ′ β−1) and w ′ = (w′0,w ′ 1, . . . ,w ′ β−1) in r β. definition 2.8. let cβ be a θ−λ−constacyclic code of length β over r. then its dual c⊥β is defined as c⊥β = {v ′ ∈ rβ; 〈v′,w′〉 = 0 for all w′ ∈ cβ} lemma 2.9. let cβ be a code of length β over r, where β is a multiple of the order of the automorphism θ and λ is fixed by θ. then cβ is θ −λ−constacyclic if and only if c⊥β is θ −λ −1−constacyclic. in particular, if λ2 = 1, then cβ is θ −λ−constacyclic if and only if c⊥β is θ −λ−constacyclic. proof. note that, for each unit λ in r,λ ∈ rθ if and only if λ−1 ∈ rθ, since λ ∈ rθ, so is λ−1. let v′ = (v′0,v ′ 1, . . . ,v ′ β−1) ∈ cβ and w ′ = (w′0,w ′ 1, . . . ,w ′ β−1) ∈ c ⊥ β be two arbitrary elements. since cβ is θ −λ−constacyclic code, ρ β−1 θ,λ (v ′) = ( θβ−1(λv′1), θ β−1(λv′2), . . . , θ β−1(λv′β−1), θ β−1(v′0) ) ∈ cβ. then, we have 0 = 〈ρβ−1θ,λ (v ′),w′〉 = 〈(θβ−1(λv′1), θβ−1(λv′2), . . . , θβ−1(λv′β−1), θ β−1(v′0)), (w ′ 0, . . . ,w ′ β−1)〉 = λ〈(θβ−1(v′1), θβ−1(v′2), . . . , θβ−1(v′β−1), θ β−1(λ−1v′0)), (w ′ 0, . . . ,w ′ β−1)〉 = λ ( θβ−1(λ−1v′0)w ′ β−1 + β−1∑ j=1 θβ−1(v′j)w ′ j−1 ) . 88 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 as β is a multiple of the order of θ and λ−1 is fixed by θ, it follows that 0 = θ(0) = θ(λθβ−1(λ−1v′0)w ′ β−1 + β−1∑ j=1 θβ−1(v′j)w ′ j−1) = λ(v′0θ(λ −1w′β−1) + β−1∑ j=1 v′jθ(w ′ j−1)) = λ〈ρθ,λ−1 (w′),v′〉. this implies that, ρθ,λ−1 (w′) ∈ c⊥β . in addition, assume that λ 2 = 1. then λ = λ−1. therefore cβ is a θ −λ−constacyclic code. the converse follows from the fact that (c⊥β ) ⊥ = cβ. 3. zqr−linear skew constacyclic codes in this section, we study skew λ−constacyclic codes over the ring zqr. we known that the ring zq is a subring of the ring r. we construct the ring zqr = {(e,r); e ∈ zq,r ∈ r}. the ring zqr is not an r−module under the operation of standard multiplication. to make zqr an r−module, we follow the approach in [2] and define the map η : r → zq a + ub 7→ a. it is clear that the mapping η is a ring homomorphism. now, for any d ∈ r, we define the multiplication ∗ by d∗ (e,r) = (η(d)e,dr). this multiplication can be naturally generalized to the ring zαq r β as follows. for any d ∈ r and v = (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ zαq rβ define dv = (η(d)e0,η(d)e1, . . . ,η(d)eα−1,dr0,dr1, . . . ,drβ−1), where (e0,e1, . . . ,eα−1) ∈ zαq and (r0,r1, . . . ,rβ−1) ∈ rβ. the following results are analogous to the ones obtained in [2, 5] for the ring z2(z2 + uz2). lemma 3.1. the ring zαq r β is an r-module under the above definition. the above lemma allows us to give the next definition. definition 3.2. a non-empty subset c of zαq r β is called a zqr-linear code if it is an r−submodule of zαq r β. we note that the ring r is isomorphic to zq as an additive group. hence, for some positive integers k0, k1 and k2, any zqr-linear code c is isomorphic to a group of the form zk0q ×z 2k1 q ×z k2 q . definition 3.3. if c ⊆ zαq rβ is a zqr-linear code, group isomorphic to zk0q × z2k1q × zk2q , then c is called a zqr-additive code of type (α,β,k0,k1,k2), where k0, k1, and k2 are as defined above. 89 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 the following results and definitions are analogous to the ones obtained in [6]. let c be a zqr-linear code and let cα (respectively cβ) be the canonical projection of c on the first α (respectively on the last β) coordinates. since the canonical projection is a linear map, cα and cβ are linear codes over zq and over r of length α and β, respectively. a code c is called separable if c is the direct product of cα and cβ, i.e., c = cα ×cβ. we introduce an inner product on zαq r β. for any two vectors v = (v0, . . . ,vα−1,v ′ 0, . . . ,v ′ β−1),w = (w0, . . . ,wα−1,w ′ 0, . . . ,w ′ β−1) ∈ z α q ×r β let 〈v,w〉 = u α−1∑ i=0 viwi + β−1∑ j=0 v́jẃj. let c be a zqr-linear code. the dual of c is defined by c⊥ = {w ∈ zαq ×r β, 〈v,w〉 = 0,∀v ∈ c}. if c = cα ×cβ is separable, then c⊥ = c⊥α ×c ⊥ β . (3) now we are ready to define the skew constacyclic codes over zαq r β. we start by the following lemma. lemma 3.4. let r = zq + uzq, where zq is a subring of r. then an element λ is unit in r if and only if η(λ) is unit in zq. proof. assume that λ is unit in r; where λ = λ1 + uλ2 and λ1,λ2 ∈ zq, then we have λ.v = v.λ = 1 and since η is a ring homomorphism, then we have η(λ.v) = η(v.λ) = η(1) thus η(λ).v′ = v′.η(λ) = 1 which means that η(λ) is unit in zq, where v′ = η(v) ∈ zq. conversely, suppose that η(λ) = λ1 is unit in zq we should prove that λ = λ1 + uλ2 is unit in r. the fact that λ is unit in r means that λ.λ−1 = 1, therefore λ.λ−1 = (λ1 + uλ2)(λ1 + uλ2)−1 = (λ1 + uλ2)(λ −1 1 + uλ3) = λ1λ −1 1 + u(λ2λ −1 1 + λ1λ3), then we denote λ3 = −λ2λ−11 λ1 = −λ2(λ−11 ) 2 and since λ1 is unit in zq, then λ1λ−11 = 1 which implies that λ.λ −1 = 1, so λ is unit in r. definition 3.5. let θ be an automorphism of r. a linear code c over zαq r β is called skew constacyclic code if c satisfies the following two conditions. (i) c is an r−submodule of zαq rβ, (ii) (η(λ)θ(eα−1), θ(e0), . . . , θ(eα−2),λθ(rβ−1), θ(r0), . . . , θ(rβ−2)) ∈ c whenever (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ c remark 3.6. θ(ei) = ei for 0 ≤ i ≤ α− 1, as ei ∈ zq (the fixed ring of θ). 90 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 in polynomial representation, each codeword c = (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) of a skew constacyclic code can be represented by a pair of polynomials c(x) = ( e0 + e1x + · · · + eα−1xα−1,r0 + r1x + · · · + rβ−1xβ−1 ) = (e(x),r(x)) ∈ zq[x]/〈xα −η(λ)〉×r[x, θ]/〈xβ −λ〉. let h(x) = h0 +h1x+· · ·+htxt ∈ r [x, θ] and let (f(x),g(x)) ∈ zq[x]/〈xα−η(λ)〉×r[x, θ]/〈xβ−λ〉. the multiplication is defined by the basic rule h(x)(f(x),g(x)) = (η(h(x))f(x),h(x)g(x)), where η(h(x)) = η(h0) + η(h1)x + · · · + η(ht)xt. lemma 3.7. a code c of length (α,β) over zqr is a θ −λ−constacyclic code if and only if c is left r[x, θ]−submodule of zq[x]/〈xα −η(λ)〉×r[x, θ]/〈xβ −λ〉. proof. assume that c is a skew constacyclic code and let c ∈ c. we denote by c(x) = (e(x),r(x)) the associated polynomial of c. as xc(x) is a skew constacyclic shift of c, xc(x) ∈ c. then, by linearity of c, r(x)c(x) ∈ c for any r(x) ∈ r[x, θ]. thus c is left r[x, θ]−submodule of zq[x]/〈xα−η(λ)〉×r[x, θ]/〈xβ− λ〉. conversely, suppose that c is a left r[x, θ]−submodule of zq[x]/〈xα−η(λ)〉×r[x, θ]/〈xβ −λ〉, then we have that xc(x) ∈ c. thus, c is a θ −λ−constacyclic code. the converse is straightforward. theorem 3.8. let c be a linear code over zqr of length (α,β), and let c = cα×cβ, where cα is linear code over zq of length α and cβ is linear code over r of length β. then c is a skew λ−constacyclic code if and only if cα is a η(λ)−constacyclic code over zq and cβ is a skew λ−constacyclic code over r. proof. let (e0,e1, . . . ,eα−1) ∈ cα and let (r0,r1, . . . ,rβ−1) ∈ cβ. if c = cα×cβ is a skew constacyclic code, then (η(λ)θ(eα−1), θ(e0), . . . , θ(eα−2),λθ(rβ−1), θ(r0), . . . , θ(rβ−2)) ∈ c, which implies that (η(λ)θ(eα−1), θ(e0), . . . , θ(eα−2)) ∈ cα as θ is fixed by zq, then (η(λ)eα−1,e0, . . . ,eα−2) ∈ cα and (λθ(rβ−1), θ(r0), . . . , θ(rβ−2)) ∈ cβ. hence, cα is a constacyclic code over zq and cβ is a θ −λ−constacyclic code over r. on the other hand, suppose that cα is a constacyclic code over zq and cβ is a θ −λ−constacyclic code over r. note that (η(λ)eα−1,e0, . . . ,eα−2) ∈ cα and (λθ(rβ−1), θ(r0), . . . , θ(rβ−2)) ∈ cβ. since c = cα ×cβ and θ(ei) = ei, then (η(λ)θ(eα−1), θ(e0), . . . , θ(eα−2),λθ(rβ−1), θ(r0), . . . , θ(rβ−2)) ∈ c, so c is a skew constacyclic code over zqr. 91 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 corollary 3.9. let c = cα ×cβ be a skew λ−constacyclic code over zqr, where β is a multiple of the order θ and λ−1 is fixed by θ. then the dual code c⊥ = c⊥α ×c⊥β of c is a skew λ −1-constacyclic code over zqr. proof. from equation (3), we have c⊥ = c⊥α × c⊥β . clearly, if cα is a constacyclic code over zq then c⊥α is also a constacyclic code over zq. moreover, from lemma (2.9), we have c ⊥ β is a skew λ−constacyclic code over r. hence the dual code c⊥ is skew λ−1−constacyclic over zqr. 4. the generators and the spanning sets for zqr−skew constacyclic codes in this section, we find a set of generators for zqr−skew constacyclic codes as a left r[x, θ]−submodules of zq[x]〈xα −η(λ)〉×r[x, θ]/〈xβ −λ〉. let c be a zqr−skew constacyclic codes, c and r[x, θ]/〈xβ −λ〉 are r[x, θ]−modules and w define the following mapping: ψ : c → r[x, θ]/〈xβ −λ〉 where ψ(f1(x),f2(x)) = f2(x). it is clear that ψ is a module homomorphism whose image is a r[x, θ]−submodule of r[x, θ]/〈xβ −λ〉 and ker(ψ) is a submodule of c. proposition 4.1. let c be a skew constacyclic code of length n over zqr. then c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉, where f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ). proof. assume that β is a positive integer coprime to the characteristic of r, by similarly theory of cyclic codes over z2z4 (see. [3]) we have that ψ(c) = (a(x) + ug(x)) with a(x),g(x) ∈ r[x, θ] and g(x) | a(x) | (xβ −λ). note that: ker(ψ) = {(f(x), 0) ∈ c : f(x) ∈ zq[x]/〈xα −η(λ)〉}. define the set i to be i = {f(x) ∈ zq[x]/〈xα −η(λ)〉 : (f(x), 0) ∈ ker(ψ)}. clearly, i is an ideal of zq[x]/〈xα −η(λ)〉. therefore, there exist a polynomial f(x) ∈ zq[x]/〈xα −η(λ)〉, such that i = 〈f(x)〉. now, for any element (c1(x), 0) ∈ ker(ψ), we have c1(x) ∈ i = 〈f(x)〉 and there exists some polynomials m(x) ∈ zq[x]/〈xα −η(λ)〉 such that c1(x) = m(x)f(x). thus (c1(x), 0) = m(x)∗(f(x), 0), which implies that ker(ψ) is a left submodule of c generated by one element of the form (f(x), 0) where f(x) | (xα −η(λ)). thus, by the first isomorphism theorem, we have c/ker(ψ) ∼= 〈a(x) + ug(x)〉. let (l(x),a(x) + ug(x)) ∈ c, with ψ(l(x),a(x) + ug(x)) = 〈a(x) + ug(x))〉. 92 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 then any zqr−skew constacyclic code of length (α,β) can be generated as left r[x, θ]−submodule of zq[x]/〈xα −η(λ)〉×r[x, θ]/〈xβ −λ〉 by two elements of the form (f(x), 0) and (l(x),a(x) + ug(x)), in other word, any element in the code c can be described as d1(x) ∗ (f(x), 0) + d2(x) ∗ (l(x),a(x) + ug(x)), where d1(x) and d2(x) are polynomials in the ring r[x, θ]. in fact, the element d1(x) can be restricted to be an element in the ring zq[x]. we will write this as: c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉, where, f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ). lemma 4.2. if c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉 is a zqr−skew constacyclic code, then we may assume that deg (l(x)) ≤ deg (f(x)). proof. suppose that deg (l(x)) ≥ deg (f(x)) with deg (l(x)) = i. consider an other zqr−skew constacyclic code of length (α,β) with generators of the form d = 〈(f(x), 0), (l(x),a(x) + ug(x)) + xi ∗ (f(x), 0)〉 = 〈(f(x), 0), (l(x) + xif(x),a(x) + ug(x))〉. clearly, d ⊆ c. however, we also have that: (l(x),a(x) + ug(x)) = (l(x) + xif(x),a(x) + ug(x)) −xi ∗ (f(x), 0), which implies that (l(x),a(x) + ug(x)) ∈ c. therefore, c ⊆ d implying c = d. lemma 4.3. if c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉 is a zqr−skew constacyclic code, then we may assume that f(x) | x β−λ g(x) l(x). proof. since x β−λ g(x) ∗(l(x),a(x)+ug(x)) = (x β−λ g(x) l(x), 0), it follow that ψ(x β−λ g(x) ∗(l(x),a(x)+ug(x))) = 0. therefore, (x β−λ g(x) l(x), 0) ∈ ker(ψ) ⊆ c and f(x) | (x β−λ g(x) )l(x). the above lemma shows that if the zqr−skew constacyclic code c has only one generator of the form c = 〈l(x),a(x) + ug(x)〉 then, (xα − η(λ)) | x β−λ g(x) l(x) with g(x) | a(x) | (xβ − λ). thus from this discussion and lemma 4.2 and 4.3, we have the following results. theorem 4.4. let c be a skew constacyclic code of length n over zqr. then c can be identified uniquely as c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉, where f(x) | (xα −η(λ)) and g(x) | a(x) | (xβ −λ). and l(x) is a skew polynomial satisfying deg (l(x)) ≤ deg (f(x)) and f(x) | x β−λ g(x) l(x). proof. following from proposition 4.1, lemma 4.2 and 4.3, we can easily see that c = 〈(f(x), 0), (l(x), a(x) + ug(x))〉,where the polynomials f(x), l(x),a(x) and g(x) are stated in the theorem. now, we will prove the uniqueness of the generators. since 〈f(x)〉 and 〈a(x) + ug(x)〉 are skew constacyclic codes over zq and r respectively, then, the skew polynomials f(x),a(x) and g(x) are unique. now, suppose that c = 〈(f(x), 0), (l1(x),a(x) + ug(x))〉 = 〈(f(x), 0), (l2(x),a(x) + ug(x))〉, 93 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 then, we have ((l1(x) − l2(x)), 0) ∈ ker (ψ) = 〈f(x), 0〉, which implies that l1(x) − l2(x) = f(x)j(x), for some skew polynomial j(x), and since deg (l1(x) − l2(x)) ≤ deg (l1(x)) ≤ deg (f(x)) then j(x) = 0 and l1(x) = l2(x). definition 4.5. let a be an r−module. a linearly independent subset b of a that spans a is called a basis of a. if an r−module has a basis, then it is called a free r−module. note that if c is a zqr−skew constacyclic code of the form c = 〈(f(x), 0), (l(x),a(x) + ug(x))〉, with g(x) 6= 0, then c is a free r−module. if c is not of this form then it is not a free r−module. but we still present a minimal spanning set for the code. the following theorem gives us a spanning minimal set for zqr−skew constacyclic codes. theorem 4.6. let c be a skew constacyclic code of length n over zqr, where f(x), l(x),a(x) and g(x) are as in theorem 4.4 and f(x)hf (x) = xα −η(λ),a(x)ha(x) = xβ −λ,a(x) = g(x)m(x). let s1 = deg (hf )−1⋃ i=0 {xi ∗ (f(x), 0)}, s2 = deg (ha)−1⋃ i=0 {xi ∗ (l(x),a(x) + ug(x))}, and s3 = deg (m)−1⋃ i=0 {xi ∗ (η(ha(x))l(x),uha(x)g(x))}. then s = s1 ∪s2 ∪s3, forms a minimal spanning set for c and c has qdeg (hf )q2deg (ha)qdeg (m) codewords. proof. let c(x) = η(d(x))(f(x), 0) + e(x)(l(x),a(x) + ug(x)) ∈ zq[x]/〈xα −η(λ)〉×r[x, θ]/〈xβ −λ〉 be a codeword in c where d(x) and e(x) are skew polynomials in r[x, θ]. now, if deg (η(d(x))) ≤ deg (hf (x)) − 1, then η(d(x))(f(x), 0) ∈ span(s1). otherwise, by using right division algorithm we have η(d(x)) = hf (x)η(q1(x)) + η(r1(x)), where q1(x),r1(x) ∈ r[x, θ] and η(r1(x)) = 0 or deg (η(r1(x))) ≤ deg (hf (x)) − 1. therefore, η(d(x))(f(x), 0) = (hf (x)η(q1(x)) + η(r1(x)))(f(x), 0) = η(r1(x))(f(x), 0). hence, we can assume that η(d(x))(f(x), 0) ∈ span(s1). 94 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 now, if deg (η(e(x))) ≤ deg (ha(x)) − 1, then η(e(x))(l(x),a(x) + ug(x)) ∈ span(s2). otherwise, again by the right division algorithm, we get polynomials q2(x) and r2(x) such that: e(x) = q2(x)ha(x) + r2(x), where r2(x) = 0 or deg (r2(x)) ≤ deg (ha(x)) − 1. so, we have e(x)(l(x),a(x) + ug(x)) = (q2(x)ha(x) + r2(x))(l(x),a(x) + ug(x)) = q2(x)(η(ha(x))l(x),uha(x)g(x)) + r2(x)(l(x),a(x) + ug(x)). since r2(x) = 0 or deg (r2(x)) ≤ deg (ha(x)) − 1, then r2(x)(l(x),a(x) + ug(x)) ∈ span(s2). let us consider q2(x)(η(ha(x))l(x),uha(x)g(x)) ∈ span(s), we know that xβ − λ = a(x)ha(x) = g(x)m(x)ha(x) and also we have f(x) | x β−λ g(x) l(x). therefore, xβ−λ g(x) l(x) = f(x)k(x). again, if deg (q2(x)) ≤ deg (m(x)) − 1 then q2(x)(η(ha(x))l(x),uha(x)g(x)) ∈ span(s3). otherwise, q2(x) = x β−λ ha(x)g(x) q3(x) + r3(x) with r3(x) = 0 or deg (r3(x)) ≤ deg (m(x)) − 1. so, q2(x)(η(ha(x))l(x),uha(x)g(x)) = ( xβ −λ ha(x)g(x) q3(x)η(ha(x))l(x), xβ −λ ha(x)g(x) q3(x)uha(x)g(x) ) +r3(x)(η(ha(x))l(x),uha(x)g(x)) = ( xβ −λ ha(x)g(x) q3(x)η(ha(x))l(x), 0 ) + r3(x)(η(ha(x))l(x),uha(x)g(x)). since x β−λ g(x) l(x) = f(x)k(x), then ( xβ−λ ha(x)g(x) q3(x)η(ha(x))l(x), 0 ) ∈ span(s1) and hence r3(x)(η(ha(x))l(x),uha(x)g(x)) ∈ span(s3). consequently, s = s1 ∪s2 ∪s3 forms a minimal spanning set for c. 5. gray images of skew constacyclic codes over zqr in this section, we define a gray map on zqr, and then extend it to zαq r β. we discuss the gray images of zqr−skew constacyclic codes where λ is fixed by θ. we start by recall some results which we will need its in the next. from [24, definition 2] we have the following definition definition 5.1. let cβ be a linear code over r of length β = n` and let λ be unit in r. if for any codeword ( c0,0,c0,1, . . . ,c0,`−1,c1,0,c1,1, . . . ,c1,`−1, . . . , cn−1,0,cn−1,1, . . . ,cn−1,`−1 ) ∈ cβ, then ( λθ(cn−1,0),λθ(cn−1,1), . . . ,λθ(cn−1,`−1), θ(c0,0), θ(c0,1), . . . , θ(c0,`−1), . . . , θ(cn−2,0), θ(cn−2,1), . . . , θ(cn−2,`−1) ) ∈ cβ. then we say that cβ is a θ −λ−quasi-twisted code of length β. if ` is the least positive integer satisfies that β = n`, then cβ is said to be a θ − λ−quasi-twisted code with index `. furthermore, if θ is the identity map, we call cβ a quasi-twisted code of index l over r. 95 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 according to [31], we define a gray map φ over r by φ : rβ → z2βq φ(a + ub) = (b,a + b), where a,b ∈ zβq furthermore, for r = a + ub ∈ r, we define a map φ : zqr 7→ z3q by φ(e,r) = (e,φ(r)) = (e,b,a + b) and it can be extended componentwise zαq r β to znq as φ(e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) = (e0,e1, . . . ,eα−1,φ(r0),φ(r1), . . . ,φ(rβ−1)), for all (e0,e1, . . . ,eα−1) ∈ zαq and (r0,r1, . . . ,rβ−1) ∈ rβ, where n = α + 2β. φ is known as the gray map on zαq r β. let a ∈ z2βq with a = (a0,a1) = (a(0) | a(1)), a(i) ∈ zβq , for i = 0, 1. let σ⊗2 be a map from z2βq to z2βq given by σ⊗2(a) = (σλ(a (0)) | σλ(a(1))), where σλ is a constacyclic shift from zβq to z β q given by σλ(a (i)) = (λai,β−1,a(i,0), . . . ,a(i,β−2)), for every a(i) = (a(i,0),a(i,1), . . . ,a(i,β−1)) where a(i,j) ∈ zq, for j = 0, 1, . . . ,β − 1. a linear code cβ of length 2β over zq is said to be a quasi-twisted of index 2 if σ⊗2(cβ) = cβ. in addition, for each θ ∈ aut(r), let tθ : rβ 7→ rβ be a linear transformation given by tθ(a0,a1 . . . ,aβ−1) = (θ(a0), θ(a1) . . . , θ(aβ−1)). remark 5.2. cβ is a skew constacyclic code if and only if tθ ◦ρλ(cβ) = cβ. proposition 5.3. with the previous notation, we have tθ ◦φ◦ρλ = σ⊗2 ◦φ. proof. let ri = ai + ubi be the elements of r for i = 0, 1, . . . ,β − 1, we have ρλ(r0,r1, . . . ,rβ−1) = (λrβ−1,r0,r1, . . . ,rβ−2). if we apply φ, we have φ(ρλ(r)) = φ(λrβ−1,r0, . . . ,rβ−2) = (λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),a0 + b0, . . . ,aβ−2 + bβ−2). where φi(r) = (bi,ai + ubi), now we apply tθ in the above equation we get, tθ ◦φ(ρλ(r)) = tθ(λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1),a0 + b0, . . . ,aβ−2 + bβ−2) = ( θ(λbβ−1), θ(b0), . . . , θ(bβ−2),λθ(aβ−1 + bβ−1), θ(a0 + b0), . . . , θ(aβ−2 + bβ−2) ) , since λ is fixed by θ and by (1), for any a ∈ zq, we have θ(a) = a. so, we have tθ ◦φ◦ρλ(r) = ( λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1), (a0 + b0), . . . , (aβ−2 + bβ−2) ) . 96 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 for the other direction, σ⊗2(φ(r)) = σ⊗2(b0,b1, . . . ,bβ−1,a0 + b0,a1 + b1, . . . ,aβ−1 + bβ−1) = ( λbβ−1,b0, . . . ,bβ−2,λ(aβ−1 + bβ−1), (a0 + b0), . . . , (aβ−2 + bβ−2) ) , and the result follows. as a consequence of the above proposition, we have the following theorem. theorem 5.4. let cβ be a code of length β over r. then cβ is a skew λ−constacyclic code of length β over r if and only if φ(cβ) is a quasi-twisted code of length 2β over zq of index 2. proof. the necessary part follows from proposition 5.3, i.e., σ⊗2 ◦φ(cβ) = tθ ◦φ◦ρλ(cβ) = φ(cβ). for the sufficient part, assume that φ(cβ) is a quasi-twisted code of index 2, then φ(cβ) = σ ⊗2 ◦φ(cβ) = tθ ◦φ◦ρλ(cβ). the injectivity of φ implies that tθ (ρλ(cβ)) = cβ, i.e., cβ is a skew constacyclic code over r. theorem 5.5. let c = cα × cβ be θ − λ−constacyclic code of length n = α + 2β over zq[x]/〈xα − η(λ)〉×r[x, θ]/〈xβ −λ〉. (i) if α = β, then φ(c) is a quasi-twisted code of index 3 and length 3α. (ii) if α 6= β and λ = 1, then φ(c) is a generalized quasi cyclic code of index 3. proof. assume that c = cα ×cβ is a skew λ−constacyclic code over zqr then by theorem 3.8, we have that cα is a constacyclic codes over zq and cβ is skew constacyclic codes over r. further, from theorem 5.4, we have that, if cβ is skew constacyclic code over r then φ(cβ) is a quasi twisted code of length 2β over zq of index 2. which implies that φ(e,r) = ( λeα−1,e0, . . . ,eα−2,λbβ−1,b0, . . . ,bβ−2, λ(aβ−1 + bβ−1), (a0 + b0), . . . , (aβ−2 + bβ−2) ) , for any (e0,e1, . . . ,eα−1,r0,r1, . . . ,rβ−1) ∈ c. therefore, 1. if α = β, then φ(c) is a quasi-twisted code of length 3α over zq of index 3. 2. if α 6= β and λ = 1, then according to [22], φ(c) is a generalized quasi-cyclic code of index 3. 6. double skew constacyclic codes over zqr in this subsection, we study double skew constacyclic codes over zqr. let ń = ά+2β́ and ´́n = ´́α+2 ´́ β be integers such that n = ń + ´́n. we consider a partition of the set of the n coordinates into two subsets of ń and ´́n coordinates, respectively, so that c is a subset of zάq r β́ ×z´́αq r ´́ β. definition 6.1. a linear code c of length n over zqr is called a double skew constacyclic code if c satisfies the following conditions. 97 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 (i) c is an r−submodule of zά+ ´́αq rβ́+ ´́ β. (ii) (η(λ)θ(éά−1), θ(é0), . . . , θ(éά−2),λθ(ŕβ́−1), θ(ŕ0), . . . , θ(ŕβ́−2) | η(λ)θ(´́e´́α−1), θ( ´́e0), . . . , θ(´́e´́α−2),λθ( ´́r ´́ β−1 ), θ(´́r0), . . . , θ(´́r ´́ β−2 )) ∈ c. whenever (é0, . . . , éά−1, ŕ0, . . . , ŕβ́−1 | ´́e0, . . . , ´́e´́α−1, ´́r0, . . . , ´́r ´́β−1) ∈ c. remark 6.2. θ(éi) = éi and θ(´́ei) = ´́ei for 0 ≤ i ≤ α− 1, as éi, ´́ei ∈ zq (the fixed ring of θ). denote by r ά,β́,´́α, ´́ β the ring: zq[x]/〈xά −η(λ)〉×r[x, θ]/〈xβ́ −λ〉×zq[x]/〈x ´́α −η(λ)〉×r[x, θ]/〈x ´́ β −λ〉. in polynomial representation, each codeword c = (é0, é1, . . . , éά−1, ŕ0, . . . , ŕβ́−1 | ´́e0, ´́e1, . . . , ´́e´́α−1, ´́r0, . . . , ´́r ´́β−1) of a skew constacyclic code can be represented by four polynomials c(x) =   é0 + é1x + · · · + éά−1xά−1, ŕ0 + ŕ1x + · · · + ŕβ́−1x β́−1, ´́e0 + ´́e1x + · · · + ´́e´́α−1x ´́α−1, ´́r0 + ´́r1x + · · · + ´́r ´́ β−1 x ´́ β−1   = (é(x), ŕ(x) | ´́e(x), ´́r(x)) ∈ rά,β́,´́α, ´́β. let h(x) = h0 + h1x + · · · + htxt ∈ r [x, θ] and let (f́(x), ǵ(x) | ´́f(x), ´́g(x)) ∈ r ά,β́,´́α, ´́ β . we define the multiplication of h(x) and (f́(x), ǵ(x) | ´́f(x), ´́g(x)) by h(x)(f́(x), ǵ(x) | ´́f(x), ´́g(x)) = (η(h(x))f́(x),h(x)ǵ(x) | η(h(x)) ´́f(x),h(x)´́g(x)), where η(h(x)) = η(h0) + η(h1)x + · · · + η(ht)xt. this gives us the following theorem. but before that, we need to give the following remark. remark 6.3. if c(x) = (é(x), ŕ(x) | ´́e(x), ´́r(x)) represents the code word c, then xc(x) represents the ń´́n−skew constacyclic shift of c. theorem 6.4. a linear code c is a double skew constacyclic code if and only if it is a left r[x, θ]−sub -module of the left-module r ά,β́,´́α, ´́ β . proof. assume that c is a double skew constacyclic code. let c ∈ c, and let the associated polynomial of c be c(x) = (é(x), ŕ(x) | ´́e(x), ´́r(x)). since xc(x) is an ń´́n−skew constacyclic shift of c. (see remark 6.3), then xc(x) ∈ c. further, by the linearity of c, it follows that h(x)c(x) ∈ c, for any h(x) ∈ r[x, θ]. therefore c is a left r[x, θ]−submodule of r ά,β́,´́α, ´́ β . converse is straightforward. 98 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 7. new linear codes over z4 codes over z4, sometimes called quaternary codes as well, have a special place in coding theory. due to their importance, a database of quaternary codes was introduced in [7] and it is available online [18]. hence we consider the case q = 4 to possibly obtain quaternary codes with good parameters. we conducted a computer search using magma software [29] to find skew cyclic codes over z4(z4 + uz4) whose gray images are quaternary linear codes with better parameters than the currently best known codes. we have found ten such codes which are listed in the table below. the automorphism of r = z4 + uz4 that we used is θ(a + bu) = a + 3bu = a− bu. in addition to the gray map given in section 4.1, there are many other possible linear maps from z4 + uz4 to z`4 for various values of `. for example, the following map was used in [10] a + bu → (b, 2a + 3b,a + 3b) which triples the length of the code. we used both of these gray maps in our computations, and obtained new codes from each map. we first chose a cyclic code cα over z4 generated by gα(x). the coefficients of this polynomial is given in ascending order of the terms in the table. therefore, the entry 31212201, for example, represents the polynomial 3 + x + 2x2 + x3 + 2x4 + 2x5 + x7. then we searched for divisors of xβ − 1 in the skew polynomial ring r[x, θ] where r = z4 + uz4 and θ(a + bu) = a − bu. for each such divisor gβ(x) we constructed the skew cyclic code over z4r generated by (gα(x),gβ(x)) and its z4-images under each gray map described above. as a result of the search, we obtained ten new linear codes over z4. they are now added to the database ([18]) of quaternary codes. in the table below, which gray map is used to obtain each new code is not explicitly stated, but it can be inferred from the values of α,β and n, the length of the z4 image. if n = α + 2β, then it is the map given in section 4.1 and if n = α + 3β it is the map described in this section. for example, the second code in the table has length 57 = 15 + 3 · 14. this means that the gray map that triples the length of a code over r is used to obtain this code. when xβ − 1 = g(x)h(x) we can use either the generator polynomial g(x) or the parity check polynomial h(x) to define the skew cyclic code over r. for the codes given in the table below we used the parity check polynomial because it has smaller degree. in general a linear code c over z4 has parameters [n, 4k1 2k2 ], and when k2 = 0, c is a free code. in this case c has a basis with k vectors just like a linear code over a field. all of the codes in the table below are free codes, hence we will simply denote their parameters by [n,k,d] where d is the lee weight over z4. our computational results suggest that considering skew cyclic and skew constacyclic codes over zq(zq + uzq) is promising to obtain codes with good parameters over zq. table 1. new quaternary codes α β gα gβ z4 parameters 15 14 31212201 x4 + (u + 1)x3 + x2 + (3u + 2)x + 3u + 3 [43, 8, 26] 15 14 31212201 x4 + (u + 1)x3 + x2 + (3u + 2)x + 3u + 3 [57, 8, 38] 15 14 3021310231 x3 + 2ux2 + (3u + 3)x + 2u + 3 [43, 6, 30] 15 14 3021310231 x3 + 3x2 + (3u + 2)x + 1 [57, 6, 42] 7 14 3121 x4 + (3u + 3)x3 + 3x2 + (u + 2)x + 3u + 3 [35, 8, 20] 7 14 3121 x4 + (u + 3)x3 + (u + 1)x2 + (u + 2)x + 3u + 3 [49, 8, 32] 7 14 12311 x3 + (2u + 1)x2 + 3ux + 3u + 3 [35, 6, 22] 7 14 12311 x3 + (2u + 1)x2 + ux + u + 1 [35, 6, 24] 7 14 12311 x3 + ux2 + (3u + 3)x + 1 [49, 6, 35] 7 14 12311 x3 + (u + 2)x2 + x + 1 [49, 6, 36] 99 a. melakhessou et al. / j. algebra comb. discrete appl. 7(1) (2020) 85–101 acknowledgment: the authors wish to express their thanks to the anonymous reviewers for their careful checking and valuable remarks that improved the presentation and the content of the paper. references [1] t. abualrub, i. siap, cyclic codes over the rings z2 + uz2 and z2 + uz2 + u2z2, designs, codes and cryptography, 42(3) (2007) 273–287. 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2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729465 j. algebra comb. discrete appl. 7(2) • 183–193 received: 9 july 2018 accepted: 10 october 2019 journal of algebra combinatorics discrete structures and applications computing the zero forcing number for generalized petersen graphs∗ research article saeedeh rashidi, nosratollah shajareh poursalavati, maryam tavakkoli abstract: let g be a simple undirected graph with each vertex colored either white or black, u be a black vertex of g, and exactly one neighbor v of u be white. then change the color of v to black. when this rule is applied, we say u forces v, and write u → v. a zero forcing set of a graph g is a subset z of vertices such that if initially the vertices in z are colored black and remaining vertices are colored white, the entire graph g may be colored black by repeatedly applying the color-change rule. the zero forcing number of g, denoted z(g), is the minimum size of a zero forcing set. in this paper, we investigate the zero forcing number for the generalized petersen graphs (it is denoted by p(n, k)). we obtain upper and lower bounds for the zero forcing number for p(n, k). we show that z(p(n, 2)) = 6 for n ≥ 10, z(p(n, 3)) = 8 for n ≥ 12 and z(p(2k + 1, k)) = 6 for k ≥ 5. 2010 msc: 05c83, 05c10 keywords: zero forcing number, generalized petersen graph, colin de verdière parameter 1. introduction let g = (v,e) be a simple undirected graph. each vertex is colored either white or black. in such a case we say that g has a coloring and the set of all black vertices is called an initial coloring of g. the color-change rule is defined as follows: if u is a black vertex of g and exactly one neighbor v of u is white, then the color of v changes to black. given a coloring of g, let a be the set of all black vertices of g. the derived coloring of a, denoted der(a), is the result of applying the color-change rule until no more changes are possible. the zero forcing set for a graph g (zfs) is an initial coloring z of g such that der(z) = g. the zero forcing number z(g) is the minimum size of all zero forcing sets of g. the concept of zero forcing set indicates one ∗ this work was supported by mahani mathematical research center, shahid bahonar university of kerman, kerman, iran. saeedeh rashidi (corresponding author), nosratollah shajareh poursalavati, maryam tavakkoli; department of applied mathematics, faculty of mathematics and computer, shahid bahonar university of kerman, kerman, iran (email: saeedeh.rashidi@uk.ac.ir, salavati@uk.ac.ir, mtavakkoli1362@gmail.com). 183 https://orcid.org/0000-0002-8262-0910 https://orcid.org/0000-0003-0046-0325 https://orcid.org/0000-0002-2863-4799 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 model of propagation in general networks. it was introduced in [4]. the associated terminology has been extended in [5, 7, 11, 12]. for example according to [4] if g is a path, an endpoint of g is the zero forcing set for g. if g is a cycle, each set of two adjacent vertices is a zero forcing set. a contraction of a graph g is the graph obtained by identifying two adjacent vertices of g, and ignoring any loops or multiple edges occurred. a minor of g is a graph obtained by applying a sequence of deletions of edges, deletions of isolated vertices, and contraction of edges. a graph parameter ζ is called minor monotone if for any minor h of g, ζ(h) ≤ ζ(g). definition 1.1 ([9]). the generalized petersen graph p(n,k) is defined to be the graph with the vertex set and edge set respectively as follows v (p(n,k)) = {u1, . . . ,un,v1, . . . ,vn} e(p(n,k)) = {uiui+1, uivi, vivi+k : 1 ≤ i ≤ n}. here, the subscripts are assumed as integers modulo n such that n ≥ 5. note that, p(n,k) ∼= p(n,n−k). so, we assume n ≥ 2k + 1. p(n,k) is a 3-regular graph with 2n vertices. the generalized petersen graph has been studied from several points of view, such as: hamiltonicity [1, 3, 15], crossing numbers [13, 14], spectrum [10] and vertex domination [9]. in section 2, we turn to the zero forcing number of the generalized petersen graphs. we present an upper bound for z(p(n,k)). we show that z(p(n,2)) = 6 for n ≥ 10, z(p(n,3)) = 8 for n ≥ 12 and z(p(2k + 1,k)) = 6 for k ≥ 5. in section 3, we show that kk,[ n k ] is a minor of p(n,k) (where [x] is the maximum integer not greater than x). using this, we conclude that: min{k, [ n k ]}+ 1 = µ ( kk,[ n k ] ) ≤ µ(p(n,k)) ≤ z(p(n,k)). the graph parameter µ is introduced by colin de verdiere in 1990 [6]. it is equal to the maximum nullity among all matrices satisfying several conditions. this conditions are stated in section 3. it is the first parameter of colin de verdiere type parameters. there exist a relation between this parameter and the zero forcing number that we apply it for achieving the upper bound. there exists a comparison between the zero forcing sets and dynamic monopolies in the last section. note that, in all figures of this paper the vertex vi(ui) is indicated by [i]v([i]u). 2. upper bounds and equalities for z(p(n, k)) in the following theorem, we obtain an upper bound for z(p(n,k)), where k n. theorem 2.1. if n = rk + s, then z(p(n,k)) ≤ r(s + 2), where 1 ≤ s ≤ k −1, r,s ∈ n. proof. let a = {u1,u2, · · · ,us+2,u1+k,u2+k, · · · ,us+2+k, · · · ,u1+(r−1)k,u2+(r−1)k, · · · ,us+2+(r−1)k} be an initial coloring of p(n,k). the following vertices change to black by the color-change rule: {v2, . . . ,vs+1,v2+k, . . . ,vs+1+k, . . . ,v2+(r−1)k, . . . ,vs+1+(r−1)k}. since, two neighbors of the vertices uj, 2 + ik ≤ j ≤ s + 1 + ik for 0 ≤ i ≤ r − 1 are black and the only white neighbor of them is vj. we also show that the vertices {v1,v1+k, . . . ,v1+(r−1)k} are in der(a). note that s + 1 + (r −1)k = s + rk + 1−k = n + 1−k ≡ 1−k (mod(n)). we have the following adjacency: 184 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 the vertex v1 is the only white neighbor of the black vertex vs+1+(r−1)k and the vertex vs+1+(r−1)k forces it. the vertex vk+1 is the only white neighbor of the black vertex v1 and the vertex v1 forces it. the vertex v1+(r−1)k is the only white neighbor of the black vertex v1+(r−2)k and the vertex v1+(r−2)k forces it. also, the color of the vertices un,uk, . . . ,u(r−1)k,vn,vk, . . . ,v(r−1)k change to black. the vertex un is the only white neighbor of the black vertex u1 and is forced by it. for i = 1, · · · ,r−1, the vertex uik is the only white neighbor of the black vertex uik+1 and is forced by it. the vertex vn is the only white neighbor of the black vertex v(r−1)k+s and is forced by it. the vertex vk is the only white neighbor of the black vertex vn and is forced by it. for i = 2, · · · ,r−1, the vertex vik is the only white neighbor of the black vertex v(i−1)k and is forced by it. now, for i = 1, . . . ,r: the vertex uik−1 is the only white neighbor of the black vertex uik and is forced by it. then the vertex vik−1 is the only white neighbor of the black vertex uik−1 and is forced by it. also for each t < k, it is one counter, we have: the vertex uik−t is the only white neighbor of the black vertex uik−t+1 and is forced by it. the vertex vik−t is the only white neighbor of the black vertex uik−t and is forced by it. this process continues until ik−t = i(k −1) + s + 3 and ui(k−1)+s+3 forced by uik+s+4. then vi(k−1)+s+2 is the only white neighbor of the black vertex ui(k−1)+s+3 and is forced by it. so, all vertives of graph became black. in the next theorem we obtain an upper bound for z(p(n,k)). this bound does not depend on n. theorem 2.2. z(p(n,k)) ≤ 2k + 2. proof. let a = {u1,u2, . . . ,u2k+2} be an initial coloring of p(n,k) (see figure 1). the vertex vj is the only white neighbor of the black vertex uj for 2 ≤ j ≤ 2k + 1. it is forced by uj. so, vj ∈ der(a). now the vertex v1 is the only white neighbor of the vertex u1 and is forced by it. therefore v2k+2 is the only white neighbor of the black vertex vk+2. hence, the vertices v2, v3,. . . ,v2k+2 are in der(a). we continue by induction. let m ≥ 2k + 2 and the color of vertices{ um, . . . ,u2,u1 vm, . . . ,v2 have been changed to black. it suffices to show that the color of the vertices um+1 and vm+1 change to black. note that m ≥ 2k + 2 hence m ≥ m + 1 − 2k ≥ 3. therefore, the vertex vm+1 is the only white neighbor of the black vertex vm+1−k and um+1 is the only white neighbor of the black vertex um. corollary 2.3. if n = rk + s, then z(p(n,k)) ≤ min{r(s + 2),2k + 2}, where 1 ≤ s ≤ k −1. remark 2.4. in [2] the authors proved that z(p(15r,2)) = 6 and z(p(24r,5)) = 12 for all r ≥ 1. also they proved theorem 2.2 another way. they obtain the upper bound z(p(2k + 1,k)) ≤ 6 that we conclude this upper bound from theorem 2.1 and obtain the equality in theorem 2.6. theorem 2.5. if n ≥ 10, then z(p(n,2)) = 6. proof. by theorem 2.2, we have z(p(n,2)) ≤ 6. hence it suffices to show that no initial coloring of the graph with five vertices can be a zero forcing set. let a be such an initial coloring. by checking all of possible cases we show that |der(a)| ≤ 10 < 2n = |p(n,2)|. we have illustrated all cases, unless the trivial or similar ones, in the following figures. in each figure the white vertices are the vertices that will change to black by a. the set a can consist some vertices of type ui or vi. therefore, the following division is considered. note that, r vertices can be belong to the inner cycle of generalized peterson graph 185 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 figure 1. an initial coloring of p(n, k) by 2k + 2 vertices and 5−r vertices must be belong to the outer cycle of it. 1. the set a consists of five v-vertices (r = 5): note that, the vertex vi and vi+8 are adjacent for n = 10. 2. the set a consists of five u-vertices (r = 0): 186 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 3. the set a consists of four u-vertices and one v-vertex (r = 1): 4. the set a consists of four v-vertices and one u-vertex (r = 4): 5. the set a consists of three u-vertices and two v-vertices (r = 2): 187 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 6. the set a consists of two u-vertices and three v-vertices (r = 3): 188 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 theorem 2.6. if k ≥ 5, then z(p(2k + 1,k)) = 6. proof. by theorem 2.1, we have z(p(2k+1,k)) ≤ 6. by the same argument of theorem 2.5, we show that, no initial coloring a with 5 vertices can be a zero forcing set for p(n,k). the cases are essentially same as theorem 2.5. 1. the set a consists of five u-vertices: 2. the set a consists of five v-vertices: 3. the set a consists of four u-vertices and one v-vertex: 189 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 4. the set a consists of four v-vertices and one u-vertex: 5. the set a consists of three u-vertices and two v-vertices: 6. the set a consists of two u-vertices and three v-vertices: 190 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 3. lower bound for z(p(n, k)) in this section, we obtain a lower bound for z(p(n,k)). for this aim, we use the graph parameter µ(g). it has a monotonicity property, which proved first by colin de verdière in [8]. definition 3.1. [16] let g = (v,e) be an undirected graph, assuming (without loss of generality) that v = {1, . . . ,n}. then parameter µ(g) is the largest corank of any matrix m = (mi,j) ∈ rn such that: (m1) for all i,j with i 6= j: mi,j < 0 if i and j are adjacent and mi,j = 0 if i and j are nonadjacent; (m2) m has exactly one negative eigenvalue, of multiplicity 1; (m3) there is no nonzero matrix x = (xi,j) ∈ rn such that mx = 0 and such that xi,j = 0 whenever i = j or mi,j 6= 0. in [6] it is stated that µ(g) ≤ z(g). theorem 3.2. [8] if h is a minor of a graph g, then µ(h) ≤ µ(g). this property sometimes described as µ minor-monotone. for instance µ(kn) = n−1 and for p ≤ q µ(kp,q) = { p if q ≤ 2 p + 1 if q ≥ 3 see [16] for more details. definition 3.3. let g = (v,e) be a simple graph and a,b be none-empty subsets of v (g). we say that a and b are adjacent if there exist vertices x ∈ a and y ∈ b such that xy ∈ e. in such a case we write a ∼ b. theorem 3.4. if k ≥ 3, then the graph kk,[ n k ] is a minor of p(n,k). proof. let ai = {u(1+(i−1)k),u2+(i−1)k, . . . ,uk+(i−1)k} for each 1 ≤ i ≤ r = nk . put bj = {vj,vj+k, . . . ,vj+(r−1)k} for each 1 ≤ j ≤ k. it is easy to see that each ai is adjacent to each bj and bi is not adjacent to bj for i 6= j. now proceed as follows: 1. delete the edges between the vertices 1 + mku and mku for each 1 ≤ m ≤ r. 2. contract all the vertices of ai in the vertex u1+(i−1)k (starting with the vertex uk+(i−1)k and contracting successively). 3. contract all the vertices of bj in the vertex vj. 4. delete all the remaining vertices and their edges. finally, we achieve the complete bipartite graph now, we obtain the following lower bound. corollary 3.5. if k ≥ 3, then min{k, n k }+ 1 ≤ z(p(n,k)). theorem 3.6. if n ≥ 12, then z(p(n,3)) = 8. proof. let a be a set of initial black vertices of the graph p(n,3). by corollary 3.5, 4 ≤ |a|. let ui be one white vertex on the outer cycle. the color of it can be forced by the vertex ui−1, vertex ui+1 or the vertex vi. 191 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 assume the vertices u1, · · · ,ui−1 ∈ a, then: 1) if the vertex ui−1 wants to force the vertex ui, then it is necessary that vi−1 ∈ a. 2) if the vertex ui+1 wants to force the vertex ui, then it is necessary that vi+1,ui+1,ui+2 ∈ a. 3) if the vertex vi wants to force the vertex ui, then it is necessary that vi,vi−k,vi+k ∈ a. therefor the best case for the color-change processing in the vertices of the outer cycle is that the vertex ui−1 forces the vertex ui. so, suppose {u1,u2,u3,u4} ⊆ a. this set can not change the color of all vertices. by a simple argument, we conclude that the set a be {u1,u2,u3 · · · ,u8}. 4. a comparison between zero forcing sets and dynamic monopolies in the last section, we compare the zero forcing sets with another propagation concept of graph theory. this concept is dynamic monopoly. it is studied by zaker in [17]. definition 4.1. [17] by a threshold assignment for the vertices of g we mean any function τ : v (g) → n ∪{0}. a subset of vertices d is said to be a τ-dynamic monopoly of g or simply τ-dynamo of g, if for some nonnegative integer k, the vertices of g can be partitioned into subsets d0,d1, ...,dk such that d0 = d and for any i,1 ≤ i ≤ k, the set di consists of all vertices v which has at least τ(v) neighbors in d0 ∪ . . .∪di−1. denote the smallest size of any τ-dynamo of g by dyn(g). it is obvious that each zfs is a 1-dynamo. for τ = 1, there does not exist any resistant subgraph. so, each subgraph can be a candidate for a dynamo of graph. [17] a resistant subgraph of g means each subgraph k such that for each vertex v ∈ k one has dk(v) ≥ dg(v)t(v) + 1, where dg(v) is the degree of v in g. zaker proved that each dynamo of graph does not contain any resistant subgraph of it [17]. so, it is satisfy for the zfs. example 4.2. we know z(kn) = n− 1 and z(pn) = 1. the zfs of complete graph kn and path pn are 1-dynamo too. for the complete graph k4, dyn(k4) 6= z(k4). it is an interesting question that for what graphs there exist this equality. in this example the subsets d0 and d1 are zfs. figure 2. z(k4) = 3 and dyn(k4) = 1 if we consider d0 = {a} and d1 = {b,c,d}. then the subset d0 is a dynamo and it is not a zfs. there exists another question. a dynamo under what condition is a zfs? the following lemma states this conditions. lemma 4.3. consider one dynamo as d0,d1, ...,dk. if for each vertex u ∈ di+1 there exist a vertex v ∈ di such that n(v)−{u}⊆ (d0 ∪d1 ∪·· ·∪di), then that dynamo is a zfs. there exists one lower bound for z(g) which is obtained from the following results about dynamos. theorem 4.4. [17] let d be a dynamic monopoly of size k in g. set h = g\d and let tmax be the maximum threshold among the vertices of h. then: 1) ∑ v∈h t(v) ≤ |e(g)|− |e(g[d])|− δ(g) + tmax. 2) ∑ v∈h t(v) ≤ |e(g)| provided that t(v) ≤ dg(v) for any vertex v ∈ h. 192 s. rashidi et al. / j. algebra comb. discrete appl. 7(2) (2020) 183–193 we know that each zfs is a 1-dynamo. so, we have the following corollary. corollary 4.5. let g be a graph with 1 ≤ δ(g), then: 1) |g|− |e(g)|+ |e(g[zfs])|+ δ −1 ≤ z(g). 2) |g|− |e(g)| ≤ z(g) also, corollary 2 from [17] confirms the second inequality. this first bound is equality for some graphs. for example, let g be a complete graph kn. so |g| − |e(g)| + |e(g[zfs])| + δ − 1 = n − (n)(n−1) 2 + (n−1)(n−2) 2 + (n−1)−1 = n−1 and z(kn) = n−1. also, it is equality for path (z(pt) = 1). the characterization of all graphs that satisfy this bound will be interesting. references [1] b. alspach, the classification of hamiltonian of generalized petersen graphs, j. combin. theory ser. b 34(3) (1983) 293–312. 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[12] l. h. huang, g. j. chang, h. g. yeh, on minimum rank and zero forcing sets of a graph, linear algebra appl. 432(11)( (2010) 2961–2973. [13] d. mcquillan, r. b. richter, on the crossing numbers of certain generalized petersen graphs, discrete math. 104(3) (1992) 311–320. [14] g. salazar, on the crossing numbers of loop networks and generalized petersen graphs, discrete math. 302(1–3) (2005) 243–253. [15] a. j. schwenk, enumeration of hamiltonian cycles in certain generalized petersen graphs, j. combin. theory ser. b 47(1) (1989) 53–59. [16] h. van der holst, l. lovász, a. schrijver, the colin de verdière graph parameter, graph theory and combinatorial biology (balatonlelle, 1996) volume 7 of bolyai soc. math. stud., pages 29–85. jános bolyai math. soc., budapest, 1999. [17] m. zaker, on dynamic monopolies of graphs with general thresholds, discrete math. 312(6) (2012) 1136–1143. 193 https://doi.org/10.1016/0095-8956(83)90042-4 https://doi.org/10.1016/0095-8956(83)90042-4 https://doi.org/10.1515/spma-2018-0006 https://doi.org/10.1515/spma-2018-0006 https://doi.org/10.1016/0095-8956(78)90019-9 https://doi.org/10.1016/0095-8956(78)90019-9 https://doi.org/10.1016/j.laa.2007.10.009 https://doi.org/10.1016/j.laa.2007.10.009 https://doi.org/10.1016/j.laa.2007.10.009 https://doi.org/10.1016/j.laa.2007.10.009 https://doi.org/10.1016/j.laa.2010.03.008 https://doi.org/10.1016/j.laa.2010.03.008 https://doi.org/10.1002/jgt.21637 https://doi.org/10.1002/jgt.21637 https://doi.org/10.1002/jgt.21637 https://doi.org/10.13001/1081-3810.1300 https://doi.org/10.13001/1081-3810.1300 https://doi.org/10.13001/1081-3810.1300 https://doi.org/10.1016/0095-8956(90)90093-f https://doi.org/10.1016/0095-8956(90)90093-f https://doi.org/10.1016/j.disc.2009.01.018 https://doi.org/10.1016/j.disc.2009.01.018 https://doi.org/10.1016/j.laa.2009.05.003 https://doi.org/10.1016/j.laa.2010.01.001 https://doi.org/10.1016/j.laa.2010.01.001 https://doi.org/10.1016/0012-365x(92)90453-m https://doi.org/10.1016/0012-365x(92)90453-m https://doi.org/10.1016/j.disc.2004.07.036 https://doi.org/10.1016/j.disc.2004.07.036 https://doi.org/10.1016/0095-8956(89)90064-6 https://doi.org/10.1016/0095-8956(89)90064-6 https://doi.org/10.1016/j.disc.2011.11.038 https://doi.org/10.1016/j.disc.2011.11.038 introduction upper bounds and equalities for z(p(n,k)) lower bound for z(p(n,k)) a comparison between zero forcing sets and dynamic monopolies references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1111379 j. algebra comb. discrete appl. 9(2) • 79–84 received: 14 september 2021 accepted: 6 january 2022 journal of algebra combinatorics discrete structures and applications a new formula for the minimum distance of an expander code research article sudipta mallik abstract: an expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. we provide a new formula for the minimum distance of such codes. we also provide a new proof of the result that 2(1 − ε)γn is a lower bound of the minimum distance of the expander code given by an (m,n,d,γ,1 − ε) expander bipartite graph. 2010 msc: 94b05, 94b25 keywords: linear code, minimum distance, expander graph, adjacency matrix 1. introduction binary linear codes can be constructed from graphs. one such construction was given from bipartite graphs by tanner in [7]. sipser and spielman constructed expander codes from bipartite expander graphs in [6]. one of the goals of all these constructions was to have linear codes with relatively large minimum distance for efficient error correction. for more details on the literature of linear codes and bipartite graphs, see [1, 2, 6, 8]. in this article we provide a new formula for the minimum distance of expander codes. we also provide a new proof of the result that 2(1−ε)γn is a lower bound of the minimum distance of the expander code given by an (m,n,d,γ,1−ε) expander bipartite graph. now we present a brief introduction to coding theory: a binary linear code c of length n and dimension k is a k dimensional subspace of fn2 where f2 is the binary field. the code c is called an [n,k]-code. the support of a codeword ∈ c is the set of indices i such that ith entry of x is 1. the hamming weight wh(x) of a vector x ∈ fn2 is the size of the support of x. the hamming distance, denoted by dh(x,y), between two codewords x and y in c is dh(x,y) = wh(x − y). the minimum distance of c, denoted by d(c), is the minimum distance between distinct codewords in c. note that d(c) is the minimum hamming weight of a nonzero codeword in c. we call c to be an [n,k,d] code when d(c) = d. a binary matrix h is called the parity-check matrix of c if c the null space of h, i.e., c = {c ∈ fn2 |hc t = 0}. sudipta mallik; department of mathematics and statistics, northern arizona university, 801 s. osborne dr. po box: 5717, flagstaff, az 86011, usa (email: sudipta.mallik@nau.edu). 79 https://orcid.org/0000-0001-7496-2147 s. mallik / j. algebra comb. discrete appl. 9(2) (2022) 79–84 1 2 3 4 5 6 7 8 9 l r figure 1. a ( 5,4,2, 1 2 , 2 3 ) expander graph. the minimum distance d(c) can be expressed as the minimum number of linear dependent columns of the parity-check matrix of c as follows: theorem 1.1. [5, theorem 2.2] let c be a linear code and h its parity-check matrix. then c has minimum distance d if and only if any d−1 columns of h are linearly independent and some d columns of h are linearly dependent. for a vertex v of a graph g, the set of all vertices in g adjacent to v is called the neighbor of v, denoted by n(v). for a set s of vertices of g, n(s) denotes the union of neighbors of vertices in s. now we define a bipartite expander graph based on its definition in [6, 7] with the roles of left and right set of vertices switched: definition 1.2. suppose g is a bipartite graph with vertex set l∪̇r such that |l| = m, |r| = n, each edge of g joins a vertex of l with a vertex of r, and each vertex of r is adjacent to exactly d vertices of l. for positive γ and α, g is called an (m,n,d,γ,α) expander graph if for each set s ⊆ r satisfying |s| ≤ γn, we have |n(s)| ≥ dα|s|. example 1.3. the bipartite graph in figure 1 is a (5,4,2, 1 2 , 2 3 ) expander graph. each vertex in r has degree d = 2. if s ⊆ r satisfies |s| ≤ γn = 2, then |s| = 1 or 2. for |s| = 1, |n(s)| = 2 ≥ 4 3 = dα|s|. also for |s| = 2, |n(s)| ≥ 8 3 = dα|s|. definition 1.4. suppose g is an (m,n,d,γ,α) expander graph and b is the m×n bi-adjacency matrix of g, i.e., a = [ om b bt on ] is the adjacency matrix of g. the binary linear code whose parity-check matrix is b is called the expander code of g, denoted by c(g). in other words, c(g) = {c ∈ fn2 | bc t = 0 in f2}. 80 s. mallik / j. algebra comb. discrete appl. 9(2) (2022) 79–84 example 1.5. the bi-adjacency matrix of the (5,4,2, 1 2 , 2 3 ) expander graph g in figure 1 is given by b =   1 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0   . the expander code c(g) of g is given by c(g) = {c ∈ f42 | bc t = 0 in f2}. 2. main results we start with the following notation and definition of von from [3, 4]. definition 2.1. for a nonempty subset s of vertices of a graph g, the set of vertices of g with odd number of neighbors in s is denoted by von(s), i.e., von(s) = {v ∈ v (g) : |n(v)∩s| is odd}. example 2.2. consider the (5,4,2, 1 2 , 2 3 ) expander graph g in figure 1. for v = 6,7,8,9, von({v}) = n(v). for s = {6,7,8}, von(s) = {1,3,4,5}. for s = {6,8,9}, von(s) = ∅. now we proceed to the main results of this article which give a new formula of the minimum distance of an expander code. theorem 2.3. suppose g is a bipartite graph with vertex set l∪̇r such that |l| = m, |r| = n, and b is the m×n bi-adjacency matrix of g. let s be a nonempty subset of r. if von(s) = ∅, then the columns of b indexed by s are linearly dependent. conversely if the columns of b indexed by s are minimally linearly dependent, then von(s) = ∅. proof. suppose von(s) = ∅ where s = {i1, i2, . . . , it}⊆ r. then bi1 + bi2 + · · ·+ bit ≡ 0 (mod 2) which implies columns bi1,bi2, . . . ,bit of b are linearly dependent. conversely suppose s = {1,2, . . . ,k} ⊆ r and b1,b2, . . . ,bk are minimally linearly dependent columns of b. then b1 + b2 + · · · + bk ≡ 0 (mod 2). we claim von(s) = ∅. otherwise let i ∈ von(s). then (b1 + b2 + · · ·+ bk)i ≡ 1 (mod 2) , which contradicts b1 + b2 + · · ·+ bk ≡ 0 (mod 2). thus von(s) = ∅. theorem 2.4. suppose g is a bipartite graph with vertex set l∪̇r such that |l| = m, |r| = n, and b is the m×n bi-adjacency matrix of g. suppose c is the binary linear code whose parity-check matrix is b. then the minimum distance d(c) of c is given by d(c) = min{|s| : ∅ 6= s ⊆ r, von(s) = ∅}. proof. first note that b is the parity-check matrix of c. by theorem 1.1, the support of a code word in c with weight d(c) is the set of indices of some minimally dependent columns of b, say indexed by t for some nonempty subset t of r. by theorem 2.3, von(t) = ∅. then d(c) = |t | ≥ min{|s| : ∅ 6= s ⊆ r, von(s) = ∅}. 81 s. mallik / j. algebra comb. discrete appl. 9(2) (2022) 79–84 to show the equality, on the contrary suppose there is a nonempty subset s of r for which d(c) > |s| and von(s) = ∅. then by theorem 2.3, we find |s| linearly dependent columns of b giving a codeword of c with weight less than d(c), a contradiction. example 2.5. consider the (5,4,2, 1 2 , 2 3 ) expander graph g in figure 1. suppose c is the binary linear code whose parity-check matrix is the bi-adjacency matrix of g. we can verify that for any nonempty set s ⊆ r with |s| ≤ 2, we have von(s) 6= ∅. now for s = {6,8,9}, von(s) = ∅. thus by theorem 2.4, d(c) = min{|s| : ∅ 6= s ⊆ r, von(s) = ∅} = |{6,8,9}| = 3. the preceding theorem results in a new formula for the minimum distance of expander codes. theorem 2.6. suppose g is an (m,n,d,γ,α) expander graph with vertex set l∪̇r such that |l| = m and |r| = n. then the minimum distance d(c) of the expander code c of g is given by d(c) = min{|s| : ∅ 6= s ⊆ r, von(s) = ∅}. using the minimum distance formula given in theorem 2.6, we provide a new proof of the following known result which gives a lower bound of the minimum distance of an expander code. theorem 2.7. let 0 < ε < 1 2 and γ > 0 such that γn is a positive integer. suppose g is an (m,n,d,γ,1− ε) expander graph with vertex set l∪̇r such that |l| = m and |r| = n. then the minimum distance d(c) of the expander code c of g has the following lower bound: d(c) ≥ 2(1−ε)γn. to prove theorem 2.7, we first prove the following lemmas: lemma 2.8. let 0 < ε < 1 2 . suppose g is an (m,n,d,γ,1−ε) expander graph with vertex set l∪̇r such that |l| = m and |r| = n. for each set s ⊆ r satisfying |s| ≤ γn, we have d(1−2ε)|s| ≤ |von(s)| ≤ |n(s)|. proof. suppose s ⊆ r satisfies |s| ≤ γn. the second inequality follows from the fact von(s) ⊆ n(s) ⊆ l by definition. to show the first inequality, note that there are d|s| edges between vertices in s and vertices in n(s) ⊆ l and each vertex in von(s) has at least one neighbor in s. also each vertex in n(s)\von(s) has even number (at least 2) of neighbors in s. thus d|s| ≥ |von(s)|+ 2|n(s)\von(s)| = 2|n(s)|− |von(s)| which implies |von(s)| ≥ 2|n(s)|−d|s|. since |s| ≤ γn and g is an (m,n,d,γ,1−ε) expander graph, |n(s)| ≥ d(1−ε)|s|. thus |von(s)| ≥ 2|n(s)|−d|s| ≥ 2d(1−ε)|s|−d|s| = d(1−2ε)|s|. lemma 2.9. suppose g is a bipartite graph with vertex set l∪̇r. let a and b be nonempty disjoint subsets of s ⊆ r such that s = a∪b. if von(s) = ∅, then von(a) = von(b). proof. let von(s) = ∅. to show von(a) ⊆ von(b), suppose x ∈ von(a). we claim x ∈ von(b). otherwise x /∈ von(b), i.e., x is adjacent to an even number of vertices in b. since x ∈ von(a), x is adjacent to an odd number of vertices in a. thus x is adjacent to an odd number of vertices in s = a∪b. therefore von(s) 6= ∅, a contradiction. thus von(a) ⊆ von(b). similarly we can show that von(b) ⊆ von(a). 82 s. mallik / j. algebra comb. discrete appl. 9(2) (2022) 79–84 using the above lemmas, we prove theorem 2.7. proof of theorem 2.7. by theorem 2.6, consider a nonempty set s ⊆ r such that d(c) = |s| and von(s) = ∅. to prove by contradiction, suppose 2(1−ε)γn > d(c) = |s|. case 1. |s| ≤ γn by lemma 2.8, d(1−2ε)|s| ≤ |von(s)|. since ε < 1 2 , we have 0 < d(1−2ε)|s| ≤ |von(s)|, which implies von(s) 6= ∅, a contradiction. case 2. |s| > γn in this case 2(1−ε)γn > |s| > γn. choose a nonempty subset t of s ⊆ r such that |t | = γn. then by lemma 2.8, d(1−2ε)γn = d(1−2ε)|t| ≤ |von(t)| ≤ |n(t)|. (1) note that |s \t| = |s|− |t | < 2(1−ε)γn−γn = (1−2ε)γn. since each vertex in s \t has d neighbors in l, by lemma 2.8, |von(s \t)| ≤ |n(s \t)| ≤ d|s \t | < d(1−2ε)γn. (2) combining (1) and (2), we have |von(s \t)| < d(1−2ε)γn ≤ |von(t)|, which implies von(s \ t) 6= von(t). since s = s ∪ (s \ t) and von(s) = ∅, by lemma 2.9, we have von(s \t) = von(t), a contradiction. observation 2.10. if we like to find the minimum distance d(c) of the expander code c of an (m,n,d,γ,1 − ε) expander graph g with vertex set l∪̇r by brute force using theorem 2.6, then we need to consider all possible subset s ⊆ r such that von(s) = ∅. but because of theorem 2.7, we need to look at only s ⊆ r satisfying |s| > 2(1−ε)γn. example 2.11. consider the expander code c(g) of the (5,4,2, 1 2 , 2 3 ) expander graph g in figure 1. note that 1 − ε = 2 3 . by theorem 2.7, we need to look at only s ⊆ r satisfying |s| > 2(1 − ε)γn = 8 3 . so we look at nonempty sets s ⊆ r satisfying |s| ≥ 3 and verify whether von(s) = ∅. for s = {6,8,9}, von(s) = ∅. thus by theorem 2.6, d(c(g)) = min{|s| : ∅ 6= s ⊆ r, von(s) = ∅} = |{6,8,9}| = 3. acknowledgment: the author would like to thank his colleague dr. bahattin yildiz for his valuable suggestions. the author would also like to thank the anonymous referee for the quick review. 83 s. mallik / j. algebra comb. discrete appl. 9(2) (2022) 79–84 references [1] n. alon, j. bruck, j. naor, m. naor, r. roth, construction of asymptotically good low-rate errorcorrecting codes through pseudo-random graphs, ieee trans. inf. theory 38(2) (1992) 509–516. 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[8] g. zemor, on expander codes, ieee trans. inf. theory 47(2) (2001) 835-837. 84 https://doi.org/10.1109/18.119713 https://doi.org/10.1109/18.119713 https://doi.org/10.1145/509907.510003 https://doi.org/10.1145/509907.510003 https://mathscinet.ams.org/mathscinet-getitem?mr=4224400 https://mathscinet.ams.org/mathscinet-getitem?mr=4224400 http://dx.doi.org/10.12958/adm1645 http://dx.doi.org/10.12958/adm1645 https://doi.org/10.1017/cbo9780511808968 https://doi.org/10.1109/18.556667 https://doi.org/10.1109/tit.1981.1056404 https://doi.org/10.1109/tit.1981.1056404 https://doi.org/10.1109/18.910593 introduction main results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.645026 j. algebra comb. discrete appl. 7(1) • 55–71 received: 17 may 2019 accepted: 14 october 2019 journal of algebra combinatorics discrete structures and applications g-codes over formal power series rings and finite chain rings research article steven t. dougherty, joe gildea, adrian korban abstract: in this work, we define g-codes over the infinite ring r∞ as ideals in the group ring r∞g. we show that the dual of a g-code is again a g-code in this setting. we study the projections and lifts of g-codes over the finite chain rings and over the formal power series rings respectively. we extend known results of constructing γ-adic codes over r∞ to γ-adic g-codes over the same ring. we also study g-codes over principal ideal rings. 2010 msc: 94b05, 16s34 keywords: g-codes, finite chain rings, formal power series rings, γ-adic codes 1. introduction one of the most widely studied families of codes is the family of cyclic codes. one reason for this, is that cyclic codes over a frobenius ring r have an algebraic description as ideals in the polynomial ring r[x]/〈xn − 1〉 where n is the length of the code. to classify cyclic codes, it is simply a matter of finding ideals in this ring via a factorization of xn−1 over r. another very important reason is that cyclic codes are held invariant by the action of the cyclic shift. this, in turn, means that their automorphism group must contain the cyclic group of order n as a subgroup. this gives a structure to these codes which is highly useful in applications both in and out of mathematics. cyclic codes were first studied over finite fields and later were studied over frobenius rings, especially for chain rings and principal ideal rings. in [1], calderbank and sloane made a more unified approach to studying cyclic codes over the rings zpe by studying cyclic codes over over the p-adic numbers. this approach implied results for cyclic codes over zpe for all e > 0 by considering these rings as projections steven t. dougherty; department of mathematics, university of scranton, scranton, pa 18510, usa (email: prof.steven.dougherty@gmail.com). joe gildea, adrian korban (corresponding author); department of mathematical and physical sciences, university of chester, thornton science park, pool ln, chester ch2 4nu, england (email: j.gildea@chester.ac.uk, adrian3@windowslive.com). 55 https://orcid.org/0000-0003-4877-1923 https://orcid.org/0000-0001-7242-779x https://orcid.org/0000-0001-5206-6480 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 of the p-adic ring. this work has been generalized to study codes over arbitrary chain rings by s.t. dougherty and y.h. park in [6]. in [4], γ-adic codes are defined over a formal power series ring which are then used to study codes over finite chain rings. also, cyclic codes over formal power series rings are studied in [5]. recently, g-codes have been defined as codes that are ideals in the group ring rg, where r is a finite commutative frobenius ring and g is a finite group of order n. this gives an alternative view of cyclic codes as ideals in the group ring rcn where cn is the cyclic group of order n. moreover, it generalizes the notion of cyclic codes by considering codes whose automorphism group contains the arbitrary group g as a subgroup. in [3], parallels between cyclic codes and g-codes are drawn. for example, it is shown that the dual of a g-code is also a g-code, as in the case of cyclic codes, namely that the dual of a cyclic codes is also a cyclic code. moreover, constructions of these codes are given as well as algebraic properties of their structure. in this work, we generalize these ideas to study g-codes in a very broad sense. namely, we study g-codes over formal power series rings and use that canonical projection to study g-codes over finite chain rings. this allows for a construction of infinite families of g-codes from a single code and helps to determine their minimum weight and structural properties. 2. preliminaries we begin by recalling some standard definitions from the theory of rings and the theory of codes. 2.1. codes we shall give the definitions for codes over rings. for a complete description of algebraic coding theory in this setting, see [2]. let r be a commutative ring. note that we are not necessarily assuming that the ring is finite. a code of length n over r is a subset of rn and a code is linear if it is a submodule of the ambient space rn. we assume that all finite rings we use as alphabets are frobenius, where a frobenius ring is characterized by the following. let r̂ be the character module of the ring r. for a finite ring r the following are equivalent: • r is a frobenius ring. • as a left module, r̂ ∼= rr. • as a right module, r̂ ∼= rr. the hamming weight of a vector is the number of non-zero coordinates in that vector and the minimum weight of a code is the smallest weight of all non-zero vectors in the code. we define the standard inner-product on the ambient space, namely [v,w] = ∑ viwi. we define the orthogonal with respect to this inner-product as: c⊥ = {v ∈ rn | [v,w] = 0,∀ w ∈c}. the code c⊥ is linear, whether or not c is. if r is a finite frobenius ring, then we have that (c⊥)⊥ = c for all linear codes c over r. however, if r is infinite this is not always true, which prompts the following definition. definition 2.1. a linear code c over an infinite ring r is called basic if c = (c⊥)⊥. 56 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 not all linear codes are basic. for example, consider the code over the p-adic integers of length 2 given by c = 〈(p,p)〉. here c⊥ = 〈(1,−1)〉. however, 〈(1,−1)〉⊥ = 〈(1, 1)〉 which strictly contains the code c. therefore, this code is not basic. 2.2. finite chain rings and formal power series rings we recall the definitions and properties of a finite chain ring r and the formal power series ring r∞. we refer the reader to [4] and [5] for details and further explanations. in this paper, we assume that all rings have a multiplicative identity and that all rings are commutative. a ring is called a chain ring if its ideals are linearly ordered by inclusion. in particular, this means that any finite chain ring has a unique maximal ideal. let r be a finite chain ring. denote the unique maximal ideal of r by m, and let γ̃ be the generator of the unique maximal ideal m. this gives that m = 〈γ̃〉 = rγ̃, where rγ̃ = 〈γ̃〉 = {βγ̃ | β ∈ r}. we have the following chain of ideals: r = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃i〉⊇ ·· · . (1) the chain in (1) can not be infinite, since r is finite. therefore, there exists i such that 〈γ̃i〉 = {0}. let e be the minimal number such that 〈γ̃e〉 = {0}. the number e is called the nilpotency index of γ̃. this gives that for a finite chain ring we have the following: r = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃e〉. (2) if the ring r is infinite then the chain in equation 1 is also infinite. consider, for example, the infinite chain in the p-adic integers: 〈1〉⊇ 〈p〉⊇ 〈p2〉⊇ 〈p3〉 · · · . (3) let r× denote the multiplicative group of all units in the ring r. let f = r/m = r/〈γ̃〉 be the residue field with characteristic p, where p is a prime number, then |f| = q = pr for some integers q and r. we know that |f×| = pr −1. we now state two well-known lemmas for which the proofs can be found in [10]. lemma 2.2. for any 0 6= r ∈ r there is a unique integer i, 0 ≤ i < e such that r = µγ̃i, with µ a unit. the unit µ is unique modulo γ̃e−i. lemma 2.3. let r be a finite chain ring with maximal ideal m = 〈γ̃〉, where γ̃ is a generator of m with nilpotency index e. let v ⊆ r be a set of representatives for the equivalence classes of r under congruence modulo γ̃. then (i) for all r ∈ r there are unique r0, · · · ,re−1 ∈ v such that r = ∑e−1 i=0 riγ̃ i; (ii) |v | = |f|; (iii) |〈γ̃j〉| = |f|r−j for 0 ≤ j ≤ e− 1. from lemma 2.3, we know that any element ã of r can be written uniquely as ã = a0 + a1γ̃ + · · · + ae−1γ̃e−1, where the ai can be viewed as elements in the field f. in the next definitions, which can be found in [4], γ will indicate the generator of the ideal of a chain ring, not necessarily the maximal ideal. definition 2.4. the ring r∞ is defined as a formal power series ring: r∞ = f[[γ]] = { ∞∑ l=0 alγ l|al ∈ f}. 57 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 let i be an arbitrary positive integer. the rings ri are defined as follows: ri = {a0 + a1γ + · · · + ai−1γi−1|ai ∈ f}, where γi−1 6= 0, but γi = 0 in ri. if i is finite or infinite then the operations over ri are defined as follows: i−1∑ l=0 alγ l + i−1∑ l=0 blγ l = i−1∑ l=0 (al + bl)γ l (4) i−1∑ l=0 alγ l · i−1∑ l′=0 bl′γ l = i−1∑ s=0 ( ∑ l+l′=s albl′)γ s. (5) we note that if i = 1 then r1 = f and if i = e then re ∼= r. the following results can be found in [4]. 1. the ring ri is a chain ring with the maximal ideal 〈γ〉 for all i < ∞. 2. the multiplicative group r×∞ = { ∑∞ j=0 ajγ j|a0 6= 0}. 3. the ring r∞ is a principal ideal domain. we note that the ring r∞ is an infinite ring whereas each ri is a finite ring. the fact that the ring r∞ is a principal ideal domain makes the situation quite different than it is for codes over finite rings ri. for example, assume i > 1 so that ri is not a field. then the ideal in ri generated by γ is a non-trivial code c of length 1, where c⊥ = 〈γi−1〉. note here that (c⊥)⊥ = c. however, the ideal in r∞ generated by γ is a non-trivial code c of length 1, and its orthogonal is {0} as the ring is a domain. but, {0}⊥ = r∞. in other words, while there are non-trivial codes of length 1 corresponding to ideals in the rings, their orthogonals act quite differently than they do in the finite ring since there are no zero divisors. it is well-known that the generator matrix for a code c over a finite chain ring ri, where i < ∞ is permutation equivalent to a matrix of the following form: g =   ik0 a0,1 a0,2 a0,3 a0,e γik1 γa1,2 γa1,3 γa1,e γ2ik2 γ 2a2,3 γ 2a2,e ... ... ... ... γe−1ike−1 γ e−1ae−1,e   , (6) where e is the nilpotency index of γ. this matrix g is called the standard generator matrix of the code c. in this case, the code c is said to have type 1k0γk1 (γ2)k2 . . . (γe−1)ke−1. (7) for linear codes over r∞, the situation is a little different. let c be a finitely generated linear code over r∞. then the generator matrix of code c is permutation equivalent to the following standard form generator matrix (see [4] for more details). 58 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 let c be a finitely generated, nonzero linear code over r∞ of length n, then any generator matrix of c is permutation equivalent to a matrix of the following form: g =   γm0ik0 γ m0a0,1 γ m0a0,2 γ m0a0,3 γ m0a0,r γm1ik1 γ m1a1,2 γ m1a1,3 γ m1a1,r γm2ik2 γ m2a2,3 γ m2a2,r ... ... ... ... γmr−1ikr−1 γ mr−1ar−1,r   , (8) where 0 ≤ m0 < m1 < · · · < mr−1 for some integer r. the column blocks have sizes k0,k1, . . . ,kr and ki are nonnegative integers adding to n. definition 2.5. a code c with generator matrix of the form given in equation 8 is said to be of type (γm0 )k0 (γm1 )k1 . . . (γmr−1 )kr−1, where k = k0 + k1 + · · · + kr−1 is called its rank and kr = n−k. a code c of length n with rank k over r∞ is called a γ-adic [n,k] code. we call k the dimension of c and denote the dimension by dim c = k. let i,j be two integers with i ≤ j, we define a map ψ j i : rj → ri, (9) j−1∑ l=0 alγ l 7→ i−1∑ l=0 alγ l. (10) if we replace rj with r∞ then we obtain a map ψ∞i . for convenience, we denote it by ψi. since both, ψji and ψi are projection maps, it is easy to show that ψ j i and ψi are ring homomorphisms. let a,b be two arbitrary elements in rj. it is easy to get that ψ j i (a + b) = ψ j i (a) + ψ j i (b), ψ j i (ab) = ψ j i (a)ψ j i (b). (11) if a,b ∈ r∞, we have that ψi(a + b) = ψi(a) + ψi(b), ψi(ab) = ψi(a)ψi(b). (12) note that the map ψji and ψi can be extended naturally from r n j to r n i and r n ∞ to r n i . the construction method above gives a chain of rings where ri is a finite ring for all finite i and r∞ is an infinite principal ideal domain. this gives the following diagram: r f ‖ ‖ r∞ → ··· → re → re−1 → ··· → r1 59 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 we note that in the above diagram, r is a finite chain ring with maximal ideal m = 〈γ̃〉, where γ̃ is a generator of m with nilpotency index e. 2.3. g-codes we begin by defining a circulant matrix, a reverse circulant matrix and a block circulant matrix before we introduce group rings. definition 2.6. 1. a circulant matrix over a ring r is a square n×n matrix, which takes the form circ(a1,a2, . . . ,an) =   a1 a2 a3 . . . an an a1 a2 . . . an−1 an−1 an a1 . . . an−2 ... ... ... ... ... a2 a3 a4 . . . a1   where ai ∈ r. 2. a reverse circulant matrix over a ring r is a square n×n matrix, which takes the form rcirc(a1,a2, . . . ,an) =   a1 a2 a3 . . . an a2 a3 a4 . . . a1 a3 a4 a5 . . . a2 ... ... ... ... ... an a1 a2 . . . an−1   where ai ∈ r. 3. a block circulant matrix over a ring r is a square kn×kn matrix, which takes the form circ(a1,a2, . . . ,an) =   a1 a2 a3 . . . an an a1 a2 . . . an−1 an−1 an a1 . . . an−2 ... ... ... ... ... a2 a3 a4 . . . a1   where each ai is a k ×k matrix over r. we shall now give the necessary definitions for group rings. let g be a finite group of order n and let r be a ring, then the group ring rg consists of ∑n i=1 αigi, αi ∈ r, gi ∈ g. addition in the group ring is done by coordinate addition, namely n∑ i=1 αigi + n∑ i=1 βigi = n∑ i=1 (αi + βi)gi. (13) the product of two elements in a group ring is given by ( n∑ i=1 αigi)( n∑ j=1 βjgj) = ∑ i,j αiβjgigj. (14) it follows that the coefficient of gk in the product is ∑ gigj=gk αiβj. 60 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 the following construction, first given by hurly in [8], produces codes in rn from elements in the group ring rg. let r be a ring and let g = {g1,g2, . . . ,gn} be a group of order n. let v = αg1g1 + αg2g2 + · · · + αgngn ∈ rg. define the matrix σ(v) ∈ mn(r) to be σ(v) =   αg−11 g1 αg−11 g2 αg−11 g3 · · · αg−11 gn αg−12 g1 αg−12 g2 αg−12 g3 · · · αg−12 gn ... ... ... ... ... αg−1n g1 αg−1n g2 αg−1n g3 · · · αg−1n gn   . (15) we note that the elements g−11 ,g −1 2 , . . . ,g −1 n are simply the elements of the group g in a given order. for a given v ∈ rg, the code c(v) is defined as follows: c(v) = 〈σ(v)〉. (16) therefore, the code is formed by taking the row space of σ(v) over the ring r. in [3], it is shown that such codes are ideals in the group ring rg, and are held invariant by the action of the elements of g. such codes are referred to as g-codes. we note that these codes necessarily have the group g as a subgroup of their automorphism group. namely, there may be other automorphism of the code but the code must be held invariant by the actions of the group g on the coordinates of the code. it is precisely this property that makes these codes interesting. for example, many classical constructions of codes force the code to have a certain automorphism group simply by the form of their generator matrix. consider how many self-dual codes are generated by matrices of the form (i | m) where m is a circulant matrix. this construction means that self-dual codes formed in this manner will have a certain form to its automorphism group, see [3] for a complete description. then constructing self-dual codes by the group ring construction can give self-dual codes with different automorphism groups thus enabling the discovery of self-dual codes that would not be found using the classical techniques. hence, part of the motivation for using this technique is to discover codes which the usual techniques fail to produce. in previous work relating group rings and codes, it has always been assumed that the ring is finite. we shall consider here group rings with the infinite ring r∞. of course, the theory of group rings always allowed for the study of infinite rings. 3. g-codes and ideals in the group ring r∞g we shall extend the results from [3], where it is shown that the g-codes are ideals in rg and that the dual of a g-code is also a g-code in rg when r was a finite frobenius ring. here, we extend the results to r∞g, where g is an arbitrary finite group. the proofs are very similar to the ones in [3], with the difference that each nonzero element in r∞ is an infinite sum, rather than a finite sum. for simplicity, we write each non zero element in r∞ in the form γia where a = a0 + a1γ + · · ·+ · · · with a0 6= 0 and i ≥ 0, which means that a is a unit in r∞. we note that if v = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ r∞g, then the rows of σ(v) consist precisely of the vectors that correspond to the elements hv in r∞g where h is any element of the group g. then we take the row space of the matrix σ(v) over r∞ to get the corresponding g-code, namely c(v). theorem 3.1. let r∞ be the formal power series ring and g a finite group of order n. let v ∈ r∞g and let c(v) be the corresponding code in rn∞. let i(v) be the set of elements of r∞g such that ∑ γliaigi ∈ i(v) if and only if (γl1a1,γl2a2, . . . ,γlnan) ∈c(v). then i(v) is a left ideal in r∞g. proof. the rows of σ(v) consist precisely of the vectors that correspond to the elements hv in r∞g where h is any element of g. let a = ∑ γliaigi and b = ∑ γljbjgi be two elements in i(v), then a + b = ∑ (γliai + γ ljbj)gi, which corresponds to the sum of the corresponding elements in c(v). this implies that i(v) is closed under addition. 61 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 let w1 = ∑ γlibigi ∈ r∞g. then if w2 corresponds to a vector in c(v), it is of the form ∑ (γljαj)hjv. then w1w2 = ∑ γlibigi ∑ (γljαj)hjv = ∑ γlibiγ ljαjgihjv which corresponds to an element in c(v) and gives that the element is in i(v). therefore i(v) is a left ideal of r∞g. it is well known that cyclic codes can be viewed as ideals in the ring r[x]/〈xn − 1〉, and that the reciprocal polynomial of the check polynomial h(x), is used to generate the ideal in r[x]/〈xn − 1〉 corresponding to the dual code. in [3], the authors apply a similar approach to show that the dual of a g-code is also a g-code over a commutative frobenius ring. they define an element in the group ring rg which is an ideal in that group ring, and also corresponds to the dual code. we now extend this result to g-codes over r∞. let i be an ideal in a group ring r∞g. define r(c) = {w | vw = 0, ∀v ∈ i}. it is immediate that r(i) is an ideal of r∞g. let v = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ r∞g and c(v) be the corresponding code. let ω : r∞g → rn∞ be the canonical map that sends γl1ag1g1 + γl2ag2g2 + · · · + γlnagngn to (γl1ag1,γ l2ag2, · · · ,γlnagn ). let i be the ideal ω−1(c). let w = (w1,w2, . . . ,wn) ∈c⊥. then any row of the matrix σ(v) dot product w should equal zero: [(γl1ag−1 j g1 ,γl2ag−1 j g2 , . . . ,γlnag−1 j gn ), (w1,w2, . . . ,wn)] = 0, ∀j. (17) which gives n∑ i=1 γliag−1 j gi wi = 0, ∀j. (18) let w = ω−1(w) = ∑ γkiwgigi and define w ∈ r∞g to be w = γk1bg1g1 +γk2bg2g2 +· · ·+γknbgngn, where γkibgi = γ kiwg−1 i . (19) then n∑ i=1 γliag−1 j gi wi = 0 =⇒ n∑ i=1 γliag−1 j gi γkibg−1 i = 0. (20) here, g−1j gig −1 i = g −1 j , thus this is the coefficient of g −1 j in the product of w and g −1 j v, where g −1 j v is a row of the matrix σ(v). this gives that w ∈r(i) if and only if w ∈c⊥. let φ : rn∞ → r∞g by φ(w) = w, then this map is a bijection between c⊥ and r(ω−1(c)) = r(i). now we have the following result. theorem 3.2. let c = c(v) be a code in r∞g formed from the vector v ∈ r∞g. then ω−1(c⊥) is an ideal of r∞g. proof. the composite mapping ω(φ(c⊥)) is permutation equivalent to c⊥ and φ(c⊥) is an ideal of r∞g. we know that φ is a bijection between c⊥ and r(ω−1(c)), and we also know that ω−1(c) is an ideal of r∞g as well. this proves that the dual of a g-code is also a g-code, over the formal power series ring. in cyclic codes, the coefficients of the reciprocal polynomial are those of the check polynomial but in reverse order. in w above, the elements wg−1 i correspond to the elements wgi in w, but in different order, and this order will depend on the choice of the group. therefore, we have used the permutation equivalence property in the proof. 62 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 4. projections and lifts of g-codes we begin by showing that if v ∈ r∞g then σ(v) is permutation equivalent to the matrix defined in equation 8. for simplicity, we write each non-zero element in r∞ in the form γia where a = a0 + a1γ + · · · + · · · with a0 6= 0 and i ≥ 0, which means that a is a unit in r∞. theorem 4.1. let v = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ r∞g, where agi are units in r∞. let c = σ(v) be a finitely generated code over r∞. then σ(v) =   γl1ag−11 g1 γl1ag−11 g2 γl1ag−11 g3 . . . γl1ag−11 gn γl2ag−12 g1 γl2ag−12 g2 γl2ag−12 g3 . . . γlnag−12 gn ... ... ... ... ... γlnag−1n g1 γ lnag−1n g2 γ lnag−1n g3 . . . γ lnag−1n gn   , is permutation equivalent to the standard generator given in (8). proof. first, we take one non-zero element with form γm0agi, where m0 is the minimal nonnegative integer. by applying column and row permutations and by dividing a row by a unit, the element that corresponds to the first row and column of σ(v) can be replaced by γm0. the elements in the first column of matrix σ(v) have the form γljagj with lj ≥ m0 and agj a unit, thus, these can be replaced by zero when they are added to the first row multiplied by −γlj−m0 (agj )−1. continuing the process using elementary operations, we obtain the standard generator matrix of the code c given in equation 8. example 4.2. let v = γ2 + γ2(1 + γ)yx + γ2(1 + γ + γ2)yx2 + γ2yx3 ∈ r∞d8 where 〈x,y〉∼= d8. then σ(v) = ( a b b a ) , where a = circ(γ2, 0, 0, 0),b = rcirc(0,γ2(1 + γ),γ2(1 + γ + γ2),γ2). then σ(v) is equivalent to g =   γ2 0 0 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 γ2 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 0 0 γ2 0 γ2(1 + γ + γ2) γ2 0 γ2(1 + γ) 0 0 0 γ2 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2)   . clearly, g = c(v) = 〈σ(v)〉 is the [8, 4, 4] extended hamming code. let v1 = γ2 +γ2(1 +γ)yx+γ2(1 +γ + γ2)yx2 + γ2yx3 ∈ r∞d8, v2 = γ2 + γ2(1 + γ)y + γ2(1 + γ + γ2)yx + γ2yx2 ∈ r∞d8, v3 = γ2 + γ2(1 + γ + γ2)y +γ2yx+γ2(1 +γ)yx3 ∈ r∞d8 and v4 = γ2 +γ2y +γ2(1 +γ)yx2 +γ2(1 +γ +γ2)yx3 ∈ r∞d8 where vi are the group ring elements corresponding to the rows of g. let i(v) = { ∑4 i=1 γ l1aivi|γl1ai ∈ r∞}. then i(v) is a left ideal of r∞d8. we now examine the projection of codes with a given type. proposition 4.3. let c be a g-code over r∞ of type {(γm0 )k0, (γm1 )k1, . . . , (γmr−1 )kr−1} with generator matrix σ(v). then the code generated by ψi(σ(v)) is a code over ri of type {(γm0 )k0, (γm1 )k1, . . . , (γms−1 )ks−1} where s is the largest mj that is less than i. moreover, the code generated by ψi(σ(v)) is equal to {(ψi(c1), ψi(c2), . . . , ψi(cn)) | (c1,c2, . . . ,cn) ∈c}. (21) 63 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 proof. if mj > i − 1 then ψi sends γm1m′, where m′ is a matrix, to a zero matrix which gives the first statement. the code c is formed by taking the row space of σ(v) over the ring r∞, i.e. γl1a1v1 +γl2a2v2 +· · ·+ γlnanvn where γliai ∈ r∞ and vj are the rows of σ(v). if w = γljajvj, then ψi(w) = ψi(γljaj)ψi(vj) by the equation given in (12) where ψi(vj) applies the map coordinate-wise. this gives the second statement. example 4.4. let v = ∑3 j=0 γ lj+1aj+1x j +γlj+5aj+5x jy ∈ r∞(c2×c4) where (c2×c4) = 〈x1,x2 | x4 = y2 = 1,xy = yx〉. then, σ(v) = ( a b b a ) , where a = circ(γl1a1,γl2a2,γl3a3,γl4a4),b = circ(γl5a5,γl6a6,γl7a7,γl8a8) and γljaj ∈ r∞. let σ(v) be a generator matrix of a (c2 × c4)-code c. we know from theorem 4.1 that c is a type {(γm0 )k0, (γm1 )k1, . . . , (γmr−1 )kr−1} code, since it is permutation equivalent to the standard generator matrix. each row of σ(v) has the elements γl1a1,γl2a2, . . . ,γl8a8 in some specific order. now, ψi(σ(v)) = ( a b b a ) , where a = circ(ψi(γl1a1), ψi(γl2a2), ψi(γl3a3), ψi(γl4a4)),b = circ(ψi(γl5a5), ψi(γl6a6), ψi(γ l7a7), ψi(γ l8a8)) and ψi(γljagj ) ∈ ri. it follows that ψi(σ(v)) is a code over ri of type {(γm0 )k0, (γm1 )k1, . . . , (γms−1 )ks−1} where s is the largest mj that is less than i. we note that ψi may send a non-zero coordinate to 0. this means that the hamming weight of a code may decrease by applying ψi, i.e. the minimum weight of ψ(c) may be less than the minimum weight of c. the minimum weight cannot increase with the application of this map. lemma 4.5. if c is a g-code over r∞, then c⊥ has type 1m for some m. proof. first we notice that c is a linear code. from the matrix σ(v), and the fact that it is permutation equivalent to the standard generator matrix in equation 8, we know that all the codewords in c⊥ are of the form γlv for some nonnegative integer l. this gives that [γlv,w] = 0 ∀ w ∈ c⊥, hence, [γlv,w] = γl ∑n l=1 vlwl = γ l[v,w] = 0, which gives that [v,w] = 0, since 0 6= γl ∈ r∞ and r∞ is a domain. so if γlv ∈c⊥ then v ∈c⊥. therefore the code c⊥ has the type 1m for some m. proposition 4.6. let c be a g-code over r∞. then c = (c⊥)⊥ if and only if c has type 1k for some k. proof. first we note that c is linear. secondly, we note that (c⊥)⊥ ⊆c. since c is a linear code then by lemma 4.5, the code c⊥ is a linear code with type 1k for some k. this gives that (c⊥)⊥ has type 1n−(n−k) = 1k. the above two are extensions of the results from [4]. the following result, which can also be found in [4], is very useful when it comes to finding the generator matrix of the dual code c⊥ of c, given that c has a standard generator matrix g as in equation 8. we extend this result to g-codes over r∞ but omit the proof as it is exactly the same as in [4]. theorem 4.7. let c be a g-code of length n over r∞. if c has a standard generator matrix g as in equation (8), then we have (i) the dual code c⊥ of c has a generator matrix h = ( b0,r b0,r−1 . . . b0,2 b0,1 ikr ) , (22) where b0,j = − ∑j−1 l=1 b0,la t r−j,r−l −a t r−j,r for all 1 ≤ j ≤ r; 64 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 (ii) rank(c)+rank(c⊥)=n. example 4.8. if we take the generator matrix g of a code c from example 4.2, we can see that g =  γ2   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1   γ2   0 1 + γ 1 + γ + γ2 1 1 + γ 1 + γ + γ2 1 0 1 + γ + γ2 1 0 1 + γ 1 0 1 + γ 1 + γ + γ2     , which is the standard generator matrixhere, a0,1 =   0 1 + γ 1 + γ + γ2 1 1 + γ 1 + γ + γ2 1 0 1 + γ + γ2 1 0 1 + γ 1 0 1 + γ 1 + γ + γ2   . in this case the generator matrix of the dual code c⊥ of c has the form: h = ( b0,1 ik1 ) . now, from theorem 4.7 b0,1 = −at0,1, thus h =   0 −(1 + γ) −(1 + γ + γ2) −1 1 0 0 0 −(1 + γ) −(1 + γ + γ2) −1 0 0 1 0 0 −(1 + γ + γ2) −1 0 −(1 + γ) 0 0 1 0 −1 0 −(1 + γ) −(1 + γ + γ2) 0 0 0 1   . we also have rank(c) + rank(c⊥) = 4 + 4 = 8 = n. proposition 4.9. let c be a self-orthogonal g-code over r∞. then the code ψi(c) is a self-orthogonal g-code over ri for all i < ∞. proof. we first show that ψi(c) is self-orthogonal. let v ∈ r∞g and 〈σ(v)〉 = c(v) be the corresponding self-orthogonal g-code. this implies that [v,w] = 0 for all v,w ∈ 〈σ(v)〉 = c(v). this gives that n∑ l=1 vlwl ≡ n∑ l=1 ψi(vl)ψi(wl)(mod γi) ≡ ψi([v,w])(mod γi) ≡ 0 (mod γi). hence ψi(c) is a self-orthogonal code over ri. to show that ψi(c) is also a g-code, we notice that when taking ψi(c) = ψi(〈σ(v)〉), it corresponds to ψi(v) = ψi(γliagi )g1 + ψi(γl2ag2 )g2 + · · · + ψi(γlnagn )gn, then ψi(c) ∈ rig. thus ψi(c) is also a g-code. definition 4.10. let i,j be two integers such that 1 ≤ i ≤ j < ∞. we say that an [n,k] code c1 over ri lifts to an [n,k] code c2 over rj, denoted by c1 �c2, if c2 has a generator matrix g2 such that ψ j i (g2) is a generator matrix of c1. we also denote c1 by ψ j i (c2). if c is a [n,k] γ-adic code, then for any i < ∞, we call ψi(c) a projection of c. we denote ψi(c) by ci. lemma 4.11. let c be a g-code over r∞ with type 1k. if σ(v) is a standard form of c, then for any positive integer, i, ψi(σ(v)) is a standard form of ψi(c). 65 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 proof. we know from theorem 4.1 that σ(v) is permutation equivalent to a standard form matrix defined in equation 8. we also have that c has type 1k, hence ψi(c) has type 1k. the rest of the proof is the same as in [4]. in the following, to avoid confusion, we let v∞ and v be elements of the group rings r∞g and rig respectively. let v∞ = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ r∞g, and c(v∞) = 〈σ(v∞)〉 be the corresponding g-code. we now define the following map: σ1 : r∞g →c(v∞), (γl1ag1g1 + γ l2ag2g2 + · · · + γ lnagngn) 7→ m(r∞g,v∞). we define a projection of g-codes from r∞g to rig. let ψi : r∞g → rig (23) γia 7→ ψ(γia). (24) the projection is a homomorphism which means that if i is an ideal of r∞g, then ψi(i) is an ideal of rig. we have the following commutative diagram: r∞g σ1−→ c(v∞) ψi ↓ ↓ ψi rig −→σ1 c(v) . this gives that ψiσ1 = σ1ψi, which gives the following theorem. theorem 4.12. if c is a g-code over r∞, then ψi(c) is a g-code over ri for all i < ∞. proof. let v∞ ∈ r∞g and c(v∞) be the corresponding g-code over r∞. then σ1(v∞) = c(v∞) is an ideal of r∞g. by the homomorphism in equation 23 and the commutative diagram above, we know that ψi(σ1(v∞)) = σ1(ψi(v∞)) is an ideal of the group ring rig. this implies that ψi(c) is a g-code over ri for all i < ∞. this gives that if we take any element v ∈ r∞g, for a finite group g, and form σ(v), then we get a family of infinitely many g-code by taking ψi(c(v)) for all i. in the same way, if we take any element v ∈ r1g, then we get a family of infinitely many g-code by taking the lifts of the code c(v). hence, each g code over a finite chain ring is part of an infinite family of g-codes. theorem 4.13. let c be a g-code over ri, then the lift of c, c̃ over rj, where j > i, is also a g-code. proof. let v1 = αg1g1 + αg2g2 + · · · + αgngn ∈ rig and c = 〈σ(v1)〉 be the corresponding g-code. let v2 = βg1g1 + βg2g2 + · · · + βgngn ∈ rjg and c̃ = 〈σ(v2)〉 be the corresponding g-code. we can say that v1 and v2 act as generators of c and c̃ respectively. we can clearly see that ψ j i (v2) = ψ j i (βg1 )g1 + ψ j i (βg2 )g2 + · · ·+ ψ j i (βgn )gn = αg1g1 + αg2g2 + · · ·+ αgngn ∈ rig, thus ψ j i (v2) is a generator matrix of c. this implies that the g-code c(v1) over ri lifts to a g-code over rj, for all j > i. example 4.14. here we construct an infinite family of a g-code. if we take v∞ = γ2+γ2(1+γ)yx+γ2(1+ γ +γ2)yx2 +γ2yx3 ∈ r∞d8 where 〈x,y〉∼= d8, then we saw in example 4.2 that c(v∞) = 〈σ(v∞)〉 is the 66 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 [8, 4, 4] extended hamming code. if we take ψ1(c(v∞)) so that the elements {γ2,γ2(1+γ),γ2(1+γ+γ2)}∈ r∞ all get mapped to 1 ∈ r1 = f2, we get the [8, 4, 4] extended binary hamming code, i.e., v∞ = γ 2 + γ2(1 + γ)yx + γ2(1 + γ + γ2)yx2 + γ2yx3 ∈ r∞d8 σ1−→ c(v∞) ψ1 ↓ ↓ ψ1 v = 1 + yx + yx2 + yx3 ∈ r1d8 = f2d8 −→σ1 c(v) . if we now take v = 1 + yx + yx2 + yx3 ∈ r1d8 = f2d8 then c(v) = 〈σ(v)〉 is equivalent to the [8, 4, 4] extended binary hamming code. next we take v∞ = γ2 +γ2(1 +γ)yx+γ2(1 +γ +γ2)yx2 +γ2yx3 ∈ r∞d8 where c(v∞) = 〈σ(v∞)〉 is also the [8, 4, 4] extended hamming code. we can then have ψ1(v∞) = ψ1(γ 2)1 + ψ1(γ 2(1 + γ)yx + ψ1(γ 2(1 + γ + γ2)yx2 + ψ1(γ 2)yx3 = 1 + yx + yx2 + yx3 ∈ r1d8 = f2d8. thus, ψ1(v∞) is a generator matrix of c(v). this implies that the g-code c(v) over r1 = f2 lifts to a g-code over r∞. hence we have constructed an infinite family of g-codes. the following is an extension of codes over chain rings that are projections of γ-adic codes (see [4] for details), to g-codes. by lemma 4.11 and theorem 4.12, we know that for an [n,k] g-code c over r∞ of type 1k, ci = ψi(c) is an [n,k] g-code of type 1k over ri. we also have ci � ci+1 for all i. thus if a g-code c over r∞ of type 1k is given, then we obtain a series of lifts of g-codes as follows: ci �c2 � ···� ci � . . . conversely, let c be an [n,k] g-code over f = re/〈γ〉 = r1, and let g = g1 be its generator matrix. it is clear that we can define a series of generator matrices, gi ∈ mk×n(ri) such that ψi+1i (gi+1) = gi, where mk×n(ri) denotes all the matrices with k rows and n columns over ri. this defines a series of lifts ci of c to ri for all i. then this series of lifts determines a unique code c that ci = ci. let c be an [n,k] g-code of type 1k, and g,h be a generator and parity-check matrices of c. let gi = ψi(g) and hi = ψi(h). then gi and hi are generator and parity check matrices of ci respectively. the following results are well known and can be applied to g-codes over r∞ since these are also γ-adic codes. proofs can be found in [4]. lemma 4.15. let i < j < ∞ be two positive integers, then (i) γj−igi ≡ γj−igj (mod γj); (ii) γj−ihi ≡ γj−ihj (mod γj). (iii) γj−1ci ⊆cj; (iv) v = γiv0 ∈cj if and only if v0 ∈cj−i; (v) ker(ψji )=γ icj−i. lemma 4.15 (v) shows that the hamming weight enumerator of the kernel ker(ψji ) is equal to the hamming weight enumerator of cj−i. in [4], the authors study the weights of codewords in lifts of a code. we state the result with an extension to g-codes over r∞. we omit the proof since it is the same as in [4] as g-codes over r∞ are just a special type of γ-adic codes. theorem 4.16. let c be a g-code over r∞. then the following two results hold. (i) the minimum hamming distance dh (ci) of ci is equal to d = dh (c1) for all i < ∞; (ii) the minimum hamming distance d∞ = dh (c) of c is at least d = dh (c1). 67 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 in the remainder of this section, we extend the two results from [4] on mds and mdr codes over r∞ to the same type of codes which are also g-codes over the same ring. it is known (see [9]) that for codes c of length n over any alphabet of size m dh (c) ≤ n− logm(|c|) + 1 (25) codes meeting this bound are called mds (maximal distance separable) codes. for a code c of length n over an finite quasi-frobenius ring r, we have (see [7]) rc = min{l | there exists a monomorphism c → rl as r−modules}. if c is linear, then we have (see [7]) dh (c) ≤ n−rc + 1. (26) codes meeting this bound are called mdr (maximal distance with respect to rank) codes. a linear code c over r is called free if c is isomorphic as a module to rt for some t. this implies that if c is free then rc = rank(c). theorem 4.17. let c be a g-code over r∞. if c is an mdr or mds code then c⊥ is an mds code. proof. we know that c is linear. assume that c is of length n, rank k, with dh (c) = n − k + 1. we know from lemma 4.5 that if c is of rank k then c⊥ is of type 1n−k. since r∞ is a domain, any n−k columns of the generator matrix (equivalently a parity check matrix) of c⊥ are linearly independent giving that the minimum hamming weight of c⊥ is n− (n−k) + 1 = k + 1. theorem 4.18. let c be a g-code over ri, and c̃ be a lift g-code of c over rj, where j > i. if c is an mds code over ri then the code c̃ is an mds code over rj. proof. since c is a linear code, the proof is the same as in [4]. 5. self-dual γ-adic g-codes one of the most significant uses of g-codes so far has been their use in constructing good self-dual codes. in this section, we extend some results for self-dual γ-adic codes to g-codes over r∞. we therefore fix the ring r∞ with r∞ →···→ ri →···→ r2 → r1 and r1 = fq where q = pr for some prime p and nonnegative integer r. the field fq is said to be the underlying field of the rings. we now state two very well known results for self-dual codes over fq and self-dual codes over r∞. these can be found in [11] and [4] respectively. theorem 5.1. (i) if p = 2 or p ≡ 1 (mod 4), then a self-dual code of length n exists over fq if and only if n ≡ 0 (mod 2); (ii) if p ≡ 3 (mod 4), then a self-dual code of length n exists over fq if and only if n ≡ 0 (mod 4). corollary 5.2. let c be a self-dual code of length n over r∞. recall that p is the characteristic of the underlying field f. we have (i) if p = 2 or p ≡ 1 (mod 4), then n ≡ 0 (mod 2); (ii) if p ≡ 3 (mod 4), then n ≡ 0 (mod 4). 68 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 in [4], the authors also prove that if i is even, then self-dual codes of length n exist over ri for all n. this can be easily extended to self-dual g-codes as these are a special type of self-dual codes over ri. we now look at two theorems from [4] where one considers self-dual codes over ri with a specific type and one considers projections of self-dual codes over r∞. we extend these to self-dual g-codes over ri and r∞ respectively. theorem 5.3. let i be odd and c be a g-code over ri with type 1k0 (γ)k1 (γ2)k2 . . . (γi−1)ki−1. then c is a self-dual code if and only if c is self-orthogonal and kj = ki−j for all j. proof. it is enough to show that σ(v) where v ∈ rig and g is a finite group, is permutation equivalent to the matrix (6). the rest of the proof is the same as in [4]. theorem 5.4. if c is a self-dual g-code of length n over r∞ then ψi(c) is a self-dual g-code of length n and type 1k over ri for all i < ∞. proof. we first show that ψi(c) is self-dual. since c is self-dual, c = c⊥ which gives that c = c⊥ = (c⊥)⊥. we also know from proposition 4.6 that the code c has type 1k for some k. hence, k = n − k, which implies that k = n 2 . then ψi(c) also has type 1k, with k = n2 giving the desired size condition. we also know from proposition 4.9 that ψi(c) is self-orthogonal. therefore ψi(c) is a self-dual code. to show that ψi(c) is also a g-code, we notice that when taking ψi(c) = ψi(〈σ(v)〉), it corresponds to ψi(v) = ψi(γliagi )g1 + ψi(γ l2ag2 )g2 + · · · + ψi(γlnagn )gn, then ψi(c) ∈ rig. thus ψi(c) is also a g-code. in the remainder of this section, we extend two more results from [4]. the first one describes a method to construct a self-dual code over f from a self-dual code over ri. we extend this to self-dual g-codes. theorem 5.5. let i be odd. a self-dual g-code of length n over ri induces a self-dual g-code of length n over fq. proof. the proof is similar to the one in [4] but with two extra things added. first, we notice that σ(v) where v ∈ rig which generates the self-dual code c(v), is permutation equivalent to a standard generator matrix g of the form: g =   ik0 a0,1 a0,2 a0,3 a0,i γik1 γa1,2 γa1,3 γa1,i γ2ik2 γ 2a2,3 γ 2a2,i ... ... ... ... γi−1iki−1 γ i−1ai−1,i   . then the code c over ri is of type 1k0 (γ)k1 (γ2)k2 . . . (γi−1)ki−1. secondly, when the map ψi1(g̃) is used in [4], we notice that in our case the map will correspond to ψi1(g̃) = ψ i 1(v) = ψ i 1(γ liagi )g1 + ψ i 1(γ l2ag2 )g2 + · · · + ψi1(γlnagn )gn, assuming that g̃ is the generator matrix of a g-code and v ∈ rig. then ψi1(g̃) is the generator matrix of a g-code over fq. the last result from [4] which we extend to g-codes is the one which considers lifts of self-dual codes over f to self-dual codes over r∞. the authors prove the result by starting with a generator matrix of the code c over r1(= f) which has the form g1 = (i | a1), and then using the induction to show that there exist matrices gi = (i | ai) such that ψi+1i (gi+1) = gi. to extend this to self-dual g-codes, we can define the matrix g1 by taking the row space of σ(v) over f and perform row or column permutations to get a self-dual code of the form (i | a1) where a1 is a matrix over f = r1. a similar approach can be found in [3], where examples of the [16, 5, 8] reed-muller code, the [8, 4, 4] extended hamming code or [24, 12, 8] golay code over f2 are constructed from group rings and σ(v). 69 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 theorem 5.6. let r = re be a finite chain ring, f = r/〈γ〉, where |f| = q = pr, 2 6= p is a prime. then any self-dual g-code c over f can be lifted to a self-dual g-code over r∞. proof. we know by theorem 4.13 that a g-code over ri can be lifted to a g-code over rj, where j > i. to show that a self-dual g-code over f lifts to a self-dual g-code over r∞, it is enough to follow the proof in [4], but with the generator matrix g1 being defined as above. 6. g-codes over principal ideal rings let r1e1,r 2 e2 , . . . ,rses be chain rings, where r j ej has unique maximal ideal 〈γj〉 and the nilpotency index of γj is ej. let fj = rjej/〈γj〉. let a = crt(r1e1, . . . ,r j ej , . . . ,rses ). we know that a is a principal ideal ring. for any 1 ≤ i < ∞, let a j i = crt(r 1 e1 , . . . ,r j i , . . . ,r s es ). this gives that all the rings aji are principal ideal rings. in particular, a j ej = a. we denote crt(r1e1 . . . ,r j ∞, . . . ,r s es ) by aj∞. for 1 ≤ i < ∞, let cji be a code over r j i . let cji = crt(c 1 e1 , . . . ,cji , . . . ,c s es ) be the associated code over aji. let cj∞ = crt(c 1 e1 , . . . ,cj∞, . . . ,c s es ) be associated code over aj∞. we can now prove the following. theorem 6.1. let cjej be a g-code over the chain ring r j ej . then cj∞ =crt(c1e1, . . . ,c j ∞, . . . ,cses ) is a g-code over aj∞. proof. let g ∈ g and vi ∈ cjej. each row of c j ej is of the form gvi, for all i. now, if v = crt(v1,v2, . . . ,vs), then gv = crt(gv1,gv2, . . . ,gvs) and so gv ∈ cj∞, giving that cj∞ is an ideal in aj∞g. 7. conclusion in this work, we studied g-codes over the formal power series rings and finite chain rings. we generalized many known results of codes over these rings to g-codes. we showed that the dual of a gcode is also a g-code and we studied projections and lifts of g-codes with a given type in this setting. we also extended known methods of constructing a self-dual code over f from a self-dual code over ri and a self-dual code over f to a self-dual code over r∞, to self-dual g-codes. we lastly considered g-codes over principal ideal rings. throughout the paper, we constructed examples in which the codes have generator matrices that consist of blocks which are either circulant or reverse circulant matrices. we do not know if this is a general feature for any finite group ring, we therefore suggest that this is studied in the future work. 70 s.t. dougherty et al. / j. algebra comb. discrete appl. 7(1) (2020) 55–71 references [1] a. r. calderbank, n. j. a. sloane, modular and p–adic cyclic codes, des. codes cryptogr. 6(1) (1995) 21–35. 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[9] f. j. macwilliams, n. j. a. sloane, the theory of error–correcting codes, north–holland, amsterdam, 1977. [10] b. r. mcdonald, finite rings with identity , new york: marcel dekker, inc, 1974. [11] e. rains, n. j. a. sloane, self–dual codes, in the handbook of coding theory , pless v.s. and huffman w.c., eds., elsevier, amsterdam, 177–294, 1998. 71 https://doi.org/10.1007/bf01390768 https://doi.org/10.1007/bf01390768 https://doi.org/10.1007/s10623-017-0440-7 https://doi.org/10.1007/s10623-017-0440-7 https://doi.org/10.1016/s0252-9602(11)60233-6 https://doi.org/10.1016/s0252-9602(11)60233-6 https://doi.org/10.1007/s10623-005-2542-x https://doi.org/10.1007/s10623-005-2542-x introduction preliminaries g-codes and ideals in the group ring rg projections and lifts of g-codes self-dual -adic g-codes g-codes over principal ideal rings conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.867617 j. algebra comb. discrete appl. 8(1) • 41–51 received: 31 march 2020 accepted: 25 september 2020 journal of algebra combinatorics discrete structures and applications decomposition of product graphs into sunlet graphs of order eight∗ research article kaliappan sowndhariya, appu muthusamy abstract: for any integer k ≥ 3 , we define sunlet graph of order 2k, denoted by l2k, as the graph consisting of a cycle of length k together with k pendant vertices, each adjacent to exactly one vertex of the cycle. in this paper, we give necessary and sufficient conditions for the existence of l8-decomposition of tensor product and wreath product of complete graphs. 2010 msc: 05c51 keywords: graph decomposition, product graphs, corona graph, sunlet graph 1. introduction all graphs considered here are finite, simple and undirected. for the standard graph-theoretic terminology the readers are referred to [7]. a cycle of length k is called k-cycle and it is denoted by ck. let km denotes the complete graph on m vertices and km,n denotes the complete bipartite graph with m and n vertices in the parts. we denote the complete m-partite graph with n1,n2, . . . ,nm vertices in the parts by kn1,n2,...,nm. for any integer λ > 0, λg denotes the graph consisting of λ edge-disjoint copies of g. the complement of the graph g is denoted by g. the subgraph of g induced by s ⊆ v (g) is denoted as 〈s〉. for any two graphs g and h of orders m and n, respectively, the corona product g�h is the graph obtained by taking one copy of g, m copies of h and then joining the ith vertex of g to every vertex in the ith copy of h. we define the sunlet graph l2k with v (l2k) = {x1,x2, . . . ,xk,xk+1,xk+2, . . . ,x2k} and e(l2k) = {xixi+1,xixk+i | i = 1,2, ...,k and subscripts of the first term is taken addition modulo k}. we denote it by l2k = ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) . clearly, ck �k1 ∼= l2k. for two graphs g and h, their tensor product g×h and lexicographic or wreath product g⊗h have ∗ this work was supported by department of science and technology, university grant commission, government of india. kaliappan sowndhariya(corresponding author), appu muthusamy; department of mathematics, periyar university, salem, tamil nadu, india (email: sowndhariyak@gmail.com, ambdu@yahoo.com). 41 https://orcid.org/0000-0001-9014-6916 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 the same vertex set v (g)×v (h) = {(g,h) : g ∈ v (g) and h ∈ v (h)} and their edge sets are defined as follows: e(g×h) = {(g,h)(g′,h′) : gg′ ∈ e(g) and hh′ ∈ e(h)}, e(g ⊗ h) = {(g,h)(g′,h′) : gg′ ∈ e(g) or g = g′ and hh′ ∈ e(h)}. it is well known that the above products are associative and distributive over edge-disjoint unions of graphs and the tensor product is commutative. it is easy to observe that km ⊗kn ∼= kn,n,...n(m times). we shall use the following notation throughout the paper. let g and h be simple graphs with vertex sets v (g) = {x1,x2, . . . ,xn} and v (h) = {y1,y2, . . . ,ym}. then for our convenience, we write v (g) × v (h) = ⋃n i=1 xi, where xi stands for xi × v (h). further, in the sequel, we shall denote the vertices of xi as { x j i|1 ≤ j ≤ m } , where xji stands for the vertex (xi,yj) ∈ v (g)×v (h). a labeling of a graph g with n edges is an injection ρ from v (g), the vertex set of g, into a subset s ⊆ z2n+1, the additive group z2n+1. the length of an edge e = xy with end vertices x and y is defined as l(xy) = min{ρ(x)−ρ(y),ρ(y)−ρ(x)}. note that the subtraction is performed in z2n+1 and hence 1 ≤ l(e) ≤ n. if the length of the n edges are distinct and is equal to {1,2, . . . ,n}, then ρ is a rosy labeling; moreover, if s ⊆ {1,2, . . . ,n}, then ρ is a graceful labeling. a graceful labeling is said to be an α-labeling if there exists a number α0 with the property that for every edge e = xy in g with α(x) < α(y) it holds that α(x) ≤ α0 < α(y). by a decomposition of a graph g, we mean a list of edge-disjoint subgraphs of g whose union is g. for a graph g, if e(g) can be partitioned into e1,e2, ...,ek such that the subgraph induced by ei is hi, for all i, 1 ≤ i ≤ k, then we say that h1,h2, ...,hk decompose g and we write g = h1 ⊕h2 ⊕ ...⊕hk, since h1,h2, ...,hk are edge-disjoint subgraphs of g. for 1 ≤ i ≤ k, if hi ∼= h, we say that g has a hdecomposition. study of h-decomposition of graphs is not new. many authors around the world are working in the field of cycle decomposition [4, 8, 9, 21, 22], path decomposition [24, 25], star decompositon [19, 23, 26, 27] and hamilton cycle decomposition [2, 3, 15, 16] problems in graphs. here we consider the sunlet decomposition of product graphs. anitha and lekshmi [5, 6] proved that n-sun decomposition of complete graph, complete bipartite graph and the harary graphs. liang and guo [17, 18] gave the existence spectrum of a k-sun system of order v as k = 2,4,5,6,8. fu et. al. [12, 13] obtained that 3-sun decompositions of kp,p,r, knand embed a cyclic steiner triple system of order n into a 3-sun system of order 2n − 1, for n = 1(mod 6). further they obtained k-sun system when k = 6,10,14,2t, for t > 1. fu et. al. [11] obtained the existence of a 5-sun system of order v. gionfriddo et.al. [14] obtained the spectrum for uniformly resolvable decompositions of kv into 1-factor and h-suns. akwu and ajayi [1] obtained the necessary and sufficient conditions for the existence of decomposition of kn ⊗km and (kn − i) ⊗km, where i denote the 1-factor of a complete graph into sunlet graph of order twice the prime. in this paper, we obtain the decomposition of some product graphs into sunlet graphs of order eight which is the least even order not proved so far for product graphs, which motivate us to consider this problem. in section 2, we obtain the necessary and sufficient conditions for the existence of l8-decomposition of complete bipartite graphs with part size m and n. in section 3, we obtain the necessary and sufficient conditions for the existence of l8-decomposition of tensor product of complete graphs. in section 4, we obtain the necessary and sufficient conditions for the existence of l8-decomposition of complete multipartite graphs with uniform part size. to prove our results, we state the following: theorem 1.1. [20] for all n ≥ 3, cn �k1 is an α-labeling. theorem 1.2. [10] let g be a graph with n edges. if g admits a rosy labeling, then it decomposes k2n+1; if g admits an α-labeling, then it decomposes k2np+1 for every p > 0. theorem 1.3. [13] let t ≥ 2 be an integer. an l2.2t-decomposition of kn exists if and only if n ≡ 0 (or) 1 (mod 2t+2). remark 1.4. if n ≡ 0 (mod 4), then k4,n can be decomposed as copies of k4,4 and l8decomposition of k4,4 is shown in below figure. therefore l8-decomposition exists in k4,n for n is a multiple of 4. 42 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 figure 1. l8decomposition of k4,4. 2. l8decomposition of km,n now we obtain the necessary and sufficient conditions for the existence of an l8-decomposition of km,n as follows. let the vertices of km,n be {x1,x2, ...,xm,y1,y2, ...,yn}. lemma 2.1. there exists an l8decomposition of k8,6. proof. we exhibit the l8decomposition of k8,6 as follows:( x1 y1 x2 y2 y3 x5 y4 x6 ) , ( x3 y3 x4 y4 y5 x2 y6 x1 ) , ( x5 y5 x6 y6 y3 x1 y4 x2 ) , ( x7 y5 x8 y6 y1 x2 y2 x1 ) ,( x7 y3 x8 y4 y2 x6 y1 x5 ) , ( x3 y1 x4 y2 y6 x6 y5 x5 ) . lemma 2.2. there exists an l8decomposition of k8,7. proof. we exhibit the l8decomposition of k8,7 as follows:( x1 y1 x2 y2 y5 x4 y7 x7 ) , ( x3 y3 x4 y4 y2 x1 y5 x2 ) , ( x5 y5 x6 y6 y4 x3 y7 x1 ) , ( x7 y1 x8 y7 y3 x6 y6 x5 ) ,( x7 y4 x8 y5 y6 x1 y3 x2 ) , ( x3 y6 x4 y7 y1 x2 y2 x1 ) , ( x5 y2 x6 y3 y1 x8 y4 x2 ) . lemma 2.3. there exists an l8decomposition of k8,9. proof. we exhibit the l8decomposition of k8,9 as follows:( x1 y1 x2 y2 y3 x6 y4 x7 ) , ( x3 y3 x4 y4 y8 x5 y5 x6 ) , ( x5 y5 x6 y6 y1 x3 y7 x4 ) , ( x7 y7 x8 y8 y5 x4 y6 x5 ) ,( x3 y1 x4 y9 y6 x8 y2 x7 ) , ( x5 y2 x6 y9 y7 x3 y8 x8 ) , ( x7 y3 x8 y4 y1 x6 y2 x5 ) , ( x1 y5 x2 y6 y4 x8 y9 x7 ) ,( x1 y7 x2 y8 y9 x3 y3 x4 ) . lemma 2.4. there exists an l8decomposition of k12,6. proof. we exhibit the l8decomposition of k12,6 as follows:( x1 y1 x2 y2 y3 x12 y4 x11 ) , ( x3 y3 x4 y4 y5 x2 y6 x1 ) , ( x5 y5 x6 y6 y1 x1 y2 x2 ) , ( x7 y1 x8 y2 y3 x9 y4 x10 ) ,( x9 y3 x10 y4 y6 x11 y5 x12 ) , ( x11 y5 x12 y6 y1 x2 y2 x1 ) , ( x3 y1 x4 y2 y6 x10 y5 x9 ) , ( x5 y3 x6 y4 y2 x12 y1 x11 ) ,( x7 y5 x8 y6 y4 x9 y3 x10 ) . lemma 2.5. there is no l8decomposition of k8,5. 43 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 proof. let a and b be the partite set of k8,5 such that |a| = 8 and |b| = 5. in l8, four vertices are of degree 3 and four vertices are of degree 1. since k8,5 is a bipartite graph, then l8 has two vertices of degree 3 and two vertices of degree 1 in one partite set a and similarly in b. total number of edges in k8,5 is 40, then we have 5l8 in k8,5. first we pull out 4l8 from k8,5(as shown in fig.2). since each vertices in a has degree 5, the remaining degree of each vertices of k8,5\4l8in the set a is 1. here we can’t find a l8 in k8,5\4l8, since we need atleast two vertices of degree 3. hence we conclude that l8-decomposition does not exists in k8,5. figure 2. 4l8 in k8,5 lemma 2.6. there is no l8decomposition of k4,n for n ≡ 2 (mod 4). proof. let n = 4s + 2 for some s > 0. suppose that k4,n has an l8-decomposition, then it has (2s + 1)l8. let a = {x1,x2,x3,x4} and b = {y1,y2, ...,y4s+1,y4s+2} be the partite sets of k4,n. consider the (2s)l8 = { l18,l 2 8, ...,l 2s 8 } which exists in k4,n−2. now we have to find the last l8 i.e., l2s+18 . let (x1y4s+1x2y4s+2) be a cycle in k4,n. then join y4s+1 to x3 and y4s+2 to x4. now we have to find pendant edges to the vertices x1 and x2. suppose there are vertices ya and yb which are joined to x1 and x2, resp, as the pendant edges in l t1 8 for some t1 ∈{1,2, ...,2s}. then we can join these edges to the vertices x1 and x2 in l 2s+1 8 . suppose ya = y4s+1 and yb = y4s+2 or viceversa, then we can join the remaining edges x3y4s+2,x4y4s+1 in k4,n to ya and yb, resp. therefore deglt18 (y4s+1) = deglt18 (y4s+2) = 3 and degl2s+18 (y4s+1) = degl2s+18 (y4s+2) = 3. this implies deg(y4s+1) = deg(y4s+2) = 6,which is a contradiction. therefore ya 6= y4s+1 and yb 6= y4s+2 or viceversa. then we find the pendant edges to ya and yb. there exist vertices xi and xj, i,j ∈{1,2,3,4} which are joined to ya and yb,resp, as the pendant edges in l t2 8 for some t1 6= t2 ∈ {1,2, ...,2s}. then we can join these edges to the vertices ya and yb in l t1 8 . now xi 6= x3 and xj 6= x4 or viceversa, since x3ya,x3yb,x4ya and x4yb are the edges of the cycle in l t1 8 . then xi,xj must be x1,x2. again we have to find the pendant edges to x1,x2. repeat the above procedure cyclically we get to find the pendant edges of x1,x2. therefore we can’t find the pendant edges to x1,x2. hence the proof. theorem 2.7. for any m,n ≥ 4, km,n has an l8decomposition if and only if mn ≡ 0 (mod 8) except (m,n) = (4,2 (mod 4)) & (8,5). proof. necessity. we first observe that km,n has m + n vertices and mn edges. assume that km,n admits an l8decomposition. then the number of edges in the graph must be divisible by 8 i.e., 8|mn and hence mn ≡ 0 (mod 8). further, (m,n) 6= (4,2 (mod 4)) & (8,5) follows from lemmas 2.6 and 2.5. sufficiency. we construct the required decomposition in two cases. case(1) m (or) n ≡ 0 (mod 8). suppose we take m ≡ 0 (mod 8). further we divide the proof into four subcases. subcase(i) m ≡ 0 (mod 8)and n ≡ 0 (mod 4). let m = 8s and n = 4t for some s,t > 0. then we can write km,n = 2stk4,4. we know that k4,4 has an l8-decomposition(see fig.2). 44 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 subcase(ii) m ≡ 0 (mod 8)and n ≡ 1 (mod 4). let m = 8s and n = 4t + 1 for some s,t > 1, since for s = t = 1, km,n has no l8decomposition by lemma 2.5. then we can write km,n = 2s(t − 2)k4,4 ⊕ sk8,9. then by lemma 2.3, we get an l8decomposition of km,n. subcase(iii) m ≡ 0 (mod 8)and n ≡ 2 (mod 4). let m = 8s and n = 4t + 2 for some s,t > 0. then we can write km,n = 2s(t−1)k4,4 ⊕sk8,6. then by lemma 2.1, we get an l8decomposition of km,n. subcase(iv) m ≡ 0 (mod 8)and n ≡ 3 (mod 4). let m = 8s and n = 4t + 3 for some s,t > 0. then we can write km,n = 2s(t−1)k4,4 ⊕sk8,7. then by lemma 2.2, we get an l8decomposition of km,n. case(2) m ≡ 0 (mod 4)and n ≡ 0 (mod 2). subcase(i) m ≡ 0 (mod 4)and n ≡ 0 (mod 4). let m = 4s and n = 4t for some s,t > 0. then we can write km,n = stk4,4. we know that k4,4 has an l8-decomposition. subcase(ii) m ≡ 0 (mod 4)and n ≡ 2 (mod 4). let m = 4s and n = 4t + 2 for some s,t > 0. for s = 1, k4,n has no l8decomposition by lemma 2.6. consider s ≥ 2. for even s, m must be the multiple of 8. then by case(1), result is proved for even s. it is sufficient to prove the case for odd s. consider s is odd and s ≥ 3. then we can write km,n = s(t− 1)k4,4 ⊕k4(s−3),6 ⊕k12,6. since s is odd, s− 3 is even. hence the results follows by the above cases and by the lemma 2.4. 3. l8decomposition of km ×kn in this section we investigate the existence of l8decomposition of the tensor product of complete graphs. lemma 3.1. for an even integer k > 2 and any graph g, there exists an l2kdecomposition of l2k×g. proof. let l2k be ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) and yj1yj2 be any edge in g, then the induced subgraph 〈l2k ×{yj1yj2}〉 of l2k ×g gives two l2k’s as follows:( x j1 1 x j2 2 x j1 3 . . . x j2 k x j2 k+1 x j1 k+2 x j2 k+3 . . . x j1 2k ) , ( x j2 1 x j1 2 x j2 3 . . . x j1 k x j1 k+1 x j2 k+2 x j1 k+3 . . . x j2 2k ) . so, for each edge in g there are two l2k’s in l2k ×g, and hence we have 2|e(g)| l2k’s in l2k ×g. lemma 3.2. there exists an l8decomposition of k4 ×k4. proof. the l8decomposition of k4 ×k4 is given below.( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1 4 x j2 1 x j1 2 ) for j1 < j2 ∈ {1,2,3,4}, ( x j+2 1 x j 2 x j+2 3 x j 4 x j+1 4 x j+1 1 x j+1 2 x j+1 3 ) for j = 1,2, ( x41 x 1 2 x 4 3 x 1 4 x32 x 2 3 x 3 4 x 2 1 ) . lemma 3.3. there exists an l8decomposition of k4 ×k5. proof. the l8decomposition of k4 ×k5 is given as follows:( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1 4 x j2 1 x j1 2 ) for j1 < j2 ∈{1,2,3,4,5} except (j1,j2) = (2,4),(3,4),( x31 x 4 2 x 3 3 x 4 4 x43 x 5 1 x 4 1 x 5 3 ) , ( x21 x 4 2 x 2 3 x 4 4 x43 x 5 3 x 4 1 x 5 1 ) , ( x31 x 1 2 x 3 3 x 1 4 x22 x 2 1 x 2 4 x 2 3 ) , ( x41 x 1 2 x 4 3 x 1 4 x32 x 2 3 x 3 4 x 2 1 ) ,( x41 x 2 2 x 4 3 x 2 4 x34 x 3 3 x 3 2 x 3 1 ) , ( x51 x 2 2 x 5 3 x 2 4 x12 x 4 4 x 1 4 x 4 2 ) , ( x51 x 3 2 x 5 3 x 3 4 x14 x 4 4 x 1 2 x 4 2 ) . lemma 3.4. there exists an l8decomposition of k4 ×k4,5. 45 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 proof. we can write k4,5 = k4,4 ⊕k4,1. now k4 ×k4,5 = (k4 ×k4,4)⊕ (k4 ×k4,1). by theorem 2.7 and lemma 3.1, it is sufficient to prove the existence of l8decomposition of k4 × k4,1. the l8-decomposition of k4 ×k4,1 shown in fig.3 gives the required decomposition. figure 3. l8 decomposition of k4 × k4,1. lemma 3.5. there exists an l8decomposition of k8 ×k7. proof. we can write k8 = 3l8⊕c4 (see fig.4). now k8×k7 = 3 (l8 ×k7)⊕(c4 ×k7). to complete the proof, by lemma 3.1, it is sufficient to prove the existence of l8decomposition of c4 ×k7, which is given as follows:( x11 x 2 2 x 1 3 x 2 4 x74 x 7 3 x 7 2 x 7 1 ) , ( x21 x 1 2 x 2 3 x 1 4 x74 x 7 1 x 7 2 x 7 3 ) , ( x21 x 3 2 x 2 3 x 3 4 x54 x 6 1 x 5 2 x 6 3 ) , ( x31 x 2 2 x 3 3 x 2 4 x62 x 7 1 x 6 4 x 7 3 ) ,( x31 x 4 2 x 3 3 x 4 4 x64 x 7 1 x 6 2 x 7 3 ) , ( x41 x 3 2 x 4 3 x 3 4 x72 x 6 3 x 7 4 x 6 1 ) , ( x41 x 5 2 x 4 3 x 5 4 x74 x 7 3 x 7 2 x 7 1 ) , ( x51 x 4 2 x 5 3 x 4 4 x12 x 7 3 x 1 4 x 7 1 ) ,( x51 x 6 2 x 5 3 x 6 4 x14 x 1 1 x 1 2 x 1 3 ) , ( x61 x 5 2 x 6 3 x 5 4 x12 x 1 3 x 1 4 x 1 1 ) , ( x61 x 7 2 x 6 3 x 7 4 x14 x 1 1 x 1 2 x 1 3 ) , ( x71 x 6 2 x 7 3 x 6 4 x52 x 1 3 x 5 4 x 1 1 ) ,( x11 x 3 2 x 1 3 x 3 4 x52 x 7 1 x 5 4 x 7 3 ) , ( x31 x 1 2 x 3 3 x 1 4 x72 x 7 3 x 7 4 x 7 1 ) , ( x21 x 4 2 x 2 3 x 4 4 x62 x 1 1 x 6 4 x 1 3 ) , ( x41 x 2 2 x 4 3 x 2 4 x12 x 6 1 x 1 4 x 6 3 ) ,( x31 x 5 2 x 3 3 x 5 4 x74 x 2 1 x 7 2 x 2 3 ) , ( x51 x 3 2 x 5 3 x 3 4 x24 x 7 3 x 2 2 x 7 1 ) , ( x41 x 6 2 x 4 3 x 6 4 x14 x 2 3 x 1 2 x 2 1 ) , ( x61 x 4 2 x 6 3 x 4 4 x24 x 1 3 x 2 2 x 1 1 ) ,( x51 x 7 2 x 5 3 x 7 4 x22 x 2 1 x 2 4 x 2 3 ) . figure 4. k8 = 3l8 ⊕ c4. lemma 3.6. there exists an l8-decomposition of k8 ×k4,7. 46 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 proof. we can write k8 ×k4,7 = 2(k4 ×k4,4) ⊕ 2(k4 ×k4,3) ⊕ (k4,4 ×k4,4) ⊕ (k4,4 ×k4,3). by theorem 2.7 and lemma 3.1, it is sufficient to prove the existence of l8decomposition of k4 × k4,3. the l8decomposition of k4 ×k4,3 is given below.( x j1 1 x j2 2 x 1 3 x j2 4 x j2 3 x j1+1 4 x j2 1 x j1+1 2 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+1 3 x j2 4 x j1+1 1 x j2 2 ) for j1 = 3, j2 ∈{5,6,7}( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1+2 3 x j2 1 x j1+2 1 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+2 4 x j2 4 x j1+2 2 x j2 2 ) for j1 = 2, j2 ∈{5,6,7}( x j1 1 x j2 2 x j1 3 x j2 4 x j2 3 x j1+3 1 x j2 1 x j1+3 3 ) , ( x j2 1 x j1 2 x j2 3 x j1 4 x j1+3 2 x j2 4 x j1+3 4 x j2 2 ) for j1 = 1, j2 ∈{5,6,7}. lemma 3.7. there exists an l8-decomposition of k5 ×k5. proof. we exhibit the l8decomposition of k5 ×k5 as follows:( x11 x 2 3 x 1 4 x 2 5 x52 x 1 2 x 2 1 x 3 2 ) , ( x21 x 1 3 x 2 4 x 1 5 x12 x 2 2 x 4 2 x 5 3 ) , ( x21 x 3 3 x 2 4 x 3 5 x32 x 2 2 x 5 2 x 1 3 ) , ( x31 x 2 3 x 3 4 x 2 5 x12 x 3 2 x 5 2 x 4 2 ) ,( x31 x 4 3 x 3 4 x 4 5 x52 x 3 2 x 1 2 x 5 3 ) , ( x41 x 3 3 x 4 4 x 3 5 x32 x 4 2 x 5 2 x 5 3 ) , ( x41 x 5 3 x 4 4 x 5 5 x52 x 4 2 x 2 2 x 1 2 ) , ( x51 x 4 3 x 5 4 x 4 5 x42 x 5 2 x 1 2 x 3 2 ) ,( x11 x 5 3 x 1 4 x 5 5 x32 x 1 2 x 5 1 x 4 2 ) , ( x51 x 1 3 x 5 4 x 1 5 x12 x 5 2 x 4 1 x 4 2 ) , ( x11 x 3 3 x 1 4 x 3 5 x44 x 1 2 x 5 2 x 2 2 ) , ( x31 x 1 3 x 3 4 x 1 5 x14 x 3 2 x 2 1 x 2 2 ) ,( x31 x 5 3 x 3 4 x 5 5 x54 x 3 2 x 4 1 x 2 3 ) , ( x51 x 3 3 x 5 4 x 3 5 x32 x 5 5 x 2 1 x 5 2 ) , ( x21 x 4 3 x 2 4 x 4 5 x42 x 2 2 x 5 1 x 5 2 ) , ( x41 x 2 3 x 4 4 x 2 5 x12 x 4 2 x 5 1 x 5 2 ) ,( x11 x 4 3 x 1 4 x 4 5 x54 x 1 2 x 4 1 x 2 2 ) , ( x41 x 1 3 x 4 4 x 1 5 x22 x 4 2 x 2 1 x 5 2 ) , ( x21 x 5 3 x 2 4 x 5 5 x52 x 2 2 x 4 1 x 3 2 ) , ( x51 x 2 3 x 5 4 x 2 5 x22 x 5 2 x 3 2 x 4 3 ) ,( x12 x 2 4 x 3 2 x 4 4 x25 x 1 1 x 1 5 x 3 1 ) , ( x14 x 2 2 x 3 4 x 4 2 x32 x 5 5 x 5 1 x 5 4 ) , ( x13 x 2 5 x 3 3 x 4 5 x55 x 5 3 x 5 2 x 1 2 ) , ( x15 x 2 3 x 3 5 x 4 3 x33 x 4 5 x 1 2 x 5 5 ) ,( x11 x 2 2 x 3 1 x 4 2 x34 x 5 4 x 2 4 x 3 5 ) . theorem 3.8. km ×kn has an l8decomposition if and only if mn(m−1)(n−1) ≡ 0 (mod 16). proof. necessity. assume that km ×kn admits an l8decomposition. then the number of edges in the graph km × kn is mn(m−1)(n−1) 2 which should be divisible by 8, the number of edges in l8 i.e., 16|mn(m−1)(n−1) and hence mn(m−1)(n−1) ≡ 0 (mod 16). sufficiency. we construct the required decomposition in the following cases. case(1) m,n ≡ 0 (mod 4). let m = 4s and n = 4t for some s,t > 0. then we can write km × kn = st(k4 ×k4) ⊕ 2st(s − 1)(t − 1)k4,16 ⊕ 2st(s + t − 2)k4,12. by lemma 3.2 and theorem 2.7 the graph km × kn has the desired decomposition. case(2) m ≡ 0 (mod 4), n ≡ 1 (mod 4). let m = 4s and n = 4t + 1 for some s,t > 0. then we can write km × kn = (k4s × k4(t−1)) ⊕ s(k4 ×k5) ⊕ ( 5s(s−1) 2 ) k4,16 ⊕ s(t− 1) (k4 ×k4,5) ⊕ 4s(s− 1)(t− 1)k4,20. by the case (1) above, lemmas 3.3, 3.4 and theorem 2.7 the graph km ×kn has the desired decomposition. case(3) m ≡ 0 (mod 8), n ≡ 3 (mod 4). subcase(i) m = 8 and n ≡ 3 (mod 4) let n = 4t + 3 for some t > 0. then we can write k8 × kn = ( k8 ×k4(t−1) ) ⊕ (k8 ×k7) ⊕ (t−1) (k8 ×k4,7). the l8decomposition of all the three terms follows from case (1) and the lemmas 3.5, 3.6. subcase(ii) m ≡ 0 (mod 8), m > 8 and n ≡ 3 (mod 4) now, let m = 8s for some s > 1. then we can write km×kn = s(k8 ×kn) ⊕ ( s(s−1) 2 ) (k8,8 ×kn). km ×kn has the desired decomposition, by the theorem 2.7, lemma 3.1 and subcase 3(i) above. case(4) m ≡ 0 (mod 16). 47 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 let m = 16s, for some s > 0. we can write km ×kn = s(k16 ×kn) ⊕ ( s(s−1) 2 ) (k16,16 ×kn). the l8decomposition of the terms in rhs follows from lemma 3.1 and theorems 1.3, 2.7. case(5) m ≡ 1 (mod 16). let m = 16s + 1, for some s > 0. then by theorem 1.3, we have l8decomposition of k16s+1, for any s > 0. then by lemma 3.1, km ×kn has an l8decomposition. case(6) m ≡ 1 (mod 4), n ≡ 1 (mod 4). let m = 4s + 1 and n = 4t + 1 for some s,t > 0. then we can write km ×kn = ( k4(s−1) ×k4(t−1) ) ⊕ ( k4(s−1) ×k5 ) ⊕ ( k5 ×k4(t−1) ) ⊕ 2(s− 1)(t− 1) (k4 ×k4,5) ⊕ 4(s− 1)(t− 1)(s + t− 4)k4,20 ⊕ k5 ×k5 ⊕ 5 (s + t−2)k4,20 ⊕ 2(s−1)(t−1)k16,25. then by the cases (1), (2) above and by lemmas 3.4, 3.7 and theorem 2.7, the graph km ×kn has the desired decomposition. 4. l8decomposition of km ⊗kn in this section we investigate the existence of l8decomposition of wreath product of complete graphs. lemma 4.1. if the graph g has an l2k-decomposition, then g⊗kn has an l2k-decomposition for any n > 0 and even k > 2. proof. let g has an l2kdecomposition. for each l2k, ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) , in g, we exhibit the l2kdecomposition of l2k ⊗ kn as follows: ( x j1 1 x j2 2 . . . x j2 k x j2 k+1 x j1 k+2 . . . x j1 2k ) for j1 ≤ j2 ∈ {1,2, ...,n},( x j2 1 x j1 2 . . . x j1 k x j1 k+1 x j2 k+2 . . . x j2 2k ) for j1 < j2 ∈{1,2, ...,n}. lemma 4.2. there exists an l8decomposition of k4,5 ⊗k6. proof. the l8decomposition of k4,5 ⊗k6 is given as follows:( x j1 1 x j2 5 x j1 2 x j2 6 x j2 7 x j1 3 x j2 8 x j1 4 ) for j1 ≤ j2 ∈{1,2,3,4,5,6} except (j1,j2) = (4,4),(2,3), (2,4),(3,4),( x j2 1 x j1 5 x j2 2 x j1 6 x j1 7 x j2 3 x j1 8 x j2 4 ) for j1 < j2 ∈{1,2,3,4,5,6} except (j1,j2) = (3,4), (3,5),(3,6),(4,5),(4,6),( x j1 3 x j2 7 x j1 4 x j2 8 x j2 6 x j1 2 x j2 5 x j1 1 ) for j1 ≤ j2 ∈{1,2,3,4,5,6} except (j1,j2) = (2,4),(2,5),( x j2 3 x j1 7 x j2 4 x j1 8 x j1 6 x j2 2 x j1 5 x j2 1 ) for j1 < j2 ∈{1,2,3,4,5,6} except (j1,j2) = (3,5),(4,5),( x2i1 x 1 i2 x4i1 x 3 i2 x5i2 x 3 i1 x4i2 x 1 i1 ) , ( x1i1 x 2 i2 x3i1 x 4 i2 x5i2 x 6 i1 x3i2 x 2 i1 ) , ( x4i1 x 2 i2 x5i1 x 6 i2 x5i2 x 2 i1 x3i2 x 6 i1 ) for i1 ∈{1,2} & i2 ∈{9},( x5i1 x 1 i2 x6i1 x 5 i2 x4i2 x 1 i1 x3i2 x 3 i1 ) for i1 ∈{1,2,3,4} & i2 ∈{9},( x41 x 4 5 x 4 2 x 4 6 x47 x 4 3 x 4 8 x 5 3 ) , ( x21 x 6 9 x 2 2 x 3 6 x37 x 1 2 x 3 8 x 2 4 ) , ( x21 x 4 5 x 2 2 x 4 6 x47 x 2 3 x 4 8 x 3 4 ) , ( x31 x 4 5 x 3 2 x 6 9 x47 x 3 3 x 4 8 x 1 1 ) ,( x41 x 3 5 x 4 2 x 3 6 x37 x 2 1 x 3 8 x 4 4 ) , ( x51 x 3 5 x 5 2 x 3 6 x37 x 2 2 x 3 8 x 5 4 ) , ( x61 x 3 5 x 6 2 x 3 6 x37 x 5 4 x 3 8 x 6 4 ) , ( x51 x 4 5 x 5 2 x 4 6 x47 x 5 3 x 4 8 x 3 1 ) ,( x61 x 4 5 x 6 2 x 4 6 x47 x 6 3 x 4 8 x 3 2 ) , ( x23 x 4 7 x 2 4 x 4 8 x35 x 2 2 x 4 5 x 2 1 ) , ( x23 x 5 7 x 2 4 x 5 8 x56 x 2 2 x 4 6 x 2 1 ) , ( x53 x 3 7 x 5 4 x 3 8 x36 x 5 2 x 4 6 x 5 1 ) , 48 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 ( x53 x 4 7 x 5 4 x 4 8 x35 x 5 2 x 4 5 x 5 1 ) , ( x23 x 1 9 x 4 3 x 3 9 x59 x 3 3 x 3 5 x 1 3 ) , ( x24 x 1 9 x 4 4 x 3 9 x59 x 3 4 x 4 6 x 1 4 ) , ( x13 x 2 9 x 3 3 x 4 9 x59 x 6 3 x 3 9 x 6 1 ) ,( x14 x 2 9 x 3 4 x 4 9 x59 x 6 4 x 3 9 x 6 2 ) , ( x43 x 2 9 x 5 3 x 6 9 x59 x 2 3 x 3 9 x 3 3 ) , ( x44 x 2 9 x 5 4 x 6 9 x59 x 2 4 x 3 9 x 3 4 ) , ( x23 x 4 9 x 6 3 x 6 9 x46 x 4 3 x 3 5 x 1 3 ) ,( x24 x 4 9 x 6 4 x 6 9 x55 x 4 4 x 4 6 x 1 4 ) . lemma 4.3. there exists an l8-decomposition of k4,5 ⊗k10. proof. we exhibit the l8decomposition of k4,5 ⊗k10 as follows:( x1i1 x 1 i2 x2i1 x 2 i2 x7i2 x 5 i1 x8i2 x 6 i1 ) , ( x1i1 x 3 i2 x2i1 x 4 i2 x8i2 x 6 i1 x7i2 x 5 i1 ) , for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x3i1 x 1 i2 x4i1 x 2 i2 x9i2 x 7 i1 x10i2 x 8 i1 ) , ( x1i1 x 5 i2 x2i1 x 6 i2 x9i2 x 9 i1 x10i2 x 10 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x3i1 x 5 i2 x4i1 x 6 i2 x8i2 x 8 i1 x7i2 x 7 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9},( x9i1 x 1 i2 x10i1 x 2 i2 x3i2 x 6 i1 x4i2 x 5 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,9)( x3i1 x 3 i2 x4i1 x 4 i2 x7i2 x 5 i1 x8i2 x 6 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x5i1 x 5 i2 x6i1 x 6 i2 x9i2 x 7 i1 x10i2 x 8 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,5),(4,6)( x7i1 x 7 i2 x8i1 x 8 i2 x2i2 x 9 i1 x1i2 x 10 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x9i1 x 9 i2 x10i1 x 10 i2 x6i2 x 2 i1 x5i2 x 1 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,5),(1,6)( x5i1 x 7 i2 x6i1 x 8 i2 x10i2 x 10 i1 x9i2 x 9 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (3,7),(4,8)( x7i1 x 9 i2 x8i1 x 10 i2 x3i2 x 4 i1 x4i2 x 3 i1 ) for i1 ∈{1,2,3,4}& i2 ∈{5,6,7,8,9} except (i1, i2) = (1,9),(3,7), (4,8)( x91 x 1 9 x 10 1 x 2 9 x46 x 6 1 x 4 5 x 5 1 ) , ( x33 x 3 7 x 4 3 x 4 7 x107 x 8 1 x 9 7 x 7 1 ) , ( x34 x 3 8 x 4 4 x 4 8 x108 x 8 1 x 9 8 x 7 1 ) , ( x53 x 5 5 x 6 3 x 6 5 x95 x 10 1 x 10 5 x 9 1 ) ,( x54 x 5 6 x 6 4 x 6 6 x96 x 10 1 x 10 6 x 9 1 ) , ( x73 x 7 7 x 8 3 x 8 7 x27 x 3 3 x 1 7 x 4 3 ) , ( x74 x 7 8 x 8 4 x 8 8 x28 x 3 4 x 1 8 x 4 4 ) , ( x91 x 9 5 x 10 1 x 10 5 x47 x 2 1 x 3 7 x 1 1 ) ,( x91 x 9 6 x 10 1 x 10 6 x48 x 2 1 x 3 8 x 1 1 ) , ( x53 x 7 7 x 6 3 x 8 7 x37 x 10 3 x 4 7 x 9 3 ) , ( x54 x 7 8 x 6 4 x 8 8 x38 x 10 4 x 4 8 x 9 4 ) , ( x71 x 9 9 x 8 1 x 10 9 x36 x 4 1 x 3 5 x 3 1 ) ,( x73 x 9 7 x 8 3 x 10 7 x37 x 6 3 x 4 7 x 5 3 ) , ( x74 x 9 8 x 8 4 x 10 8 x38 x 6 4 x 4 8 x 5 4 ) , ( x83 x 3 5 x 10 3 x 3 6 x39 x 8 2 x 8 7 x 10 2 ) , ( x73 x 4 5 x 9 3 x 4 6 x49 x 7 2 x 7 7 x 9 2 ) ,( x83 x 3 7 x 10 3 x 3 8 x65 x 8 2 x 3 9 x 10 2 ) , ( x73 x 4 7 x 9 3 x 4 8 x55 x 7 2 x 4 9 x 9 2 ) , ( x84 x 3 5 x 10 4 x 3 6 x39 x 10 2 x 8 8 x 8 2 ) , ( x74 x 4 5 x 9 4 x 4 6 x49 x 9 2 x 7 8 x 7 2 ) ,( x84 x 3 7 x 10 4 x 3 8 x66 x 10 2 x 3 9 x 8 2 ) , ( x74 x 4 7 x 9 4 x 4 8 x56 x 9 2 x 4 9 x 7 2 ) , ( x81 x 3 9 x 10 1 x 4 9 x35 x 8 2 x 3 6 x 7 2 ) , ( x71 x 3 9 x 9 1 x 4 9 x45 x 10 2 x 4 6 x 9 2 ) . theorem 4.4. km ⊗kn has an l8decomposition if and only if mn2(m−1) ≡ 0 (mod 16). proof. necessity. assume that km ⊗kn admits an l8decomposition. then the number of edges in the graph km ⊗ kn is mn2(m−1) 2 which should be divisible by 8, the number of edges in l8 i.e., 16|mn2(m−1) and hence mn2(m−1) ≡ 0 (mod 16). sufficiency. we construct the required decomposition in five cases. case(1) n ≡ 0 (mod 4). 49 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 8(1) (2021) 41–51 let n = 4s, for some s > 0. then we can write km ⊗kn = ( m(m−1) 2 ) k4s,4s. now, we get the desired decomposition by theorem 2.7. case(2) m ≡ 0 (mod 4), n ≡ 0 (mod 2). subcase(i) m ≡ 0 (mod 4), n ≡ 0 (mod 4). proof follows from case (1). subcase(ii) m ≡ 0 (mod 4), n ≡ 2 (mod 4). let m = 4s and n = 6,10 for some s > 0. now we can write km⊗kn = s(k4⊗kn) ⊕ s(s−1) 2 (k4,4⊗kn) and the graphs k4 ⊗k6,k4 ⊗k10 can be viewed as 3k6,12,3k10,20, respectively. therefore we get the desired decomposition, by lemma 4.1 and the theorem 2.7. let m = 4s and n = 4t + 2 for some s > 0, t > 2. now we can write km ⊗kn = ( k4s ⊗k4(t−1) ) ⊕ ( s(s−1) 2 )( k4,4 ⊗k6 ) ⊕ 4s(4s−1)k4(t−1),6 ⊕ 3sk6,12. we get the desired decomposition, by the case (1), lemma 4.1 and theorem 2.7. case(3) m ≡ 1 (mod 4), n ≡ 0 (mod 2). subcase(i) m ≡ 1 (mod 4), n ≡ 0 (mod 4). proof follows from case (1). subcase(ii) m ≡ 1 (mod 4), n ≡ 2 (mod 4). let m = 4s + 1 and n = 6,10 for some s > 0. now we can write km ⊗kn = (k4(s−1) ⊗kn) ⊕ k5 ⊗kn ⊕ (s− 1)(k4,5 ⊗kn)and the graphs k5 ⊗k6,k5 ⊗k10 can be viewed as 5k6,12,5k10,20, respectively. therefore we get the desired decomposition, by the case(2), lemmas 4.2, 4.3 and theorem 2.7. further, let m = 4s + 1 and n = 4t + 2 for some s > 0, t > 2. now we can write km ⊗kn = k4(s−1) ⊗k4t+2 ⊕ k5 ⊗ k4(t−1) ⊕ (s−1) ( k4,5 ⊗k4(t−1) ) ⊕ (s−1) ( k4,5 ⊗k6 ) ⊕ 20sk4(t−1),6 ⊕ 5k6,12. the l8decomposition of 1st term of the above sum follows from case (2), 2nd and 3rd term follows from case (1) and the remaining terms of the above sum follows from the lemma 4.2 and theorem 2.7. hence we get the desired decomposition. case(4) m ≡ 0 (mod 16). let m = 16s, for some s > 0. we can write km ⊗kn = s ( k16 ⊗kn ) ⊕ ( s(s−1) 2 )( k16,16 ⊗kn ) . the desired decomposition follows from lemmma 4.1 and theorems 1.3, 2.7. case(5) m ≡ 1 (mod 16). desired decomposition follows from theorem 1.3 and 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research centre in natural and applied sciences (department of computer science), raja n. l. khan women’s college (autonomous), india (email: amitasamanta1@gmail.com). sukumar mondal; department of mathematics, raja n. l. khan women’s college (autonomous), india (email: smnlkhan@gmail.com). sambhu charan barman (corresponding author); department of mathematics, shahid matangini hazra government general degree college for women, india (email: barman.sambhu@gmail.com). jonecis a. dayap; department of mathematics and sciences, university of san jose-recoletos, cebu, philippines (email: jdayap@usjr.edu.ph). 133 https://orcid.org/0000-0001-8982-6260 https://orcid.org/0000-0002-5639-8439 https://orcid.org/0000-0002-6662-5172 https://orcid.org/0000-0003-1047-1490 https://orcid.org/0000-0003-1047-1490 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 and every edge of the tree is generated either directly or indirectly from the root. the root is always drawn at the top of the tree as shown in figure 1(b). the eccentricity e(z) of a vertex z in tree t is the maximum distance from vertex z to any other vertex y of t that is, e(z) = max{d(z,y) : y ∈ v}. maximum eccentricity is called the diameter of the tree t, i.e., diameter(t) = max{e(z) : z ∈ v}. a vertex with minimum eccentricity is the central vertex. every tree has either one or two central vertices. a tree is called central tree if it has single central vertex and bi-central tree if it has two central vertices. some applications of trees are counting of saturated-hydrocarbons, cryptographic application in modern computer science and formation of electrical circuits, etc. [20]. a tree is one of the popular and useful data structure in modern computer science. there are different types of trees for different purposes. for storing image data effectively, some hierarchical data structures such as quadtrees, octrees are utilized [35]. for fast key searching in database systems, we use self-balancing binary search trees named avl trees. by the graph structure g = (v (g),e(g)), we mean that v is the node set or vertex set of g and e is the edge set of g, wherein |v (g)| = n, |e(g)| = m. for an arbitrary graph g, if each node of v is dominated by at least one node of a node set b ⊆ v , then b is a dominating-set (d-set) of g. a d-set with minimum cardinality is called a minimum d-set. the cardinality of a minimum d-set is called the domination number (d-number) and it is denoted by γ(g). a node set dhd ⊆ v is a d-hop dominatingset (d-hop d-set) of graph g if every node t ∈ v is situated at most d-distances from at least one node z ∈ dhd, that is, d(t,z) ≤ d, where d is a fixed positive integer. now, a node set dhd ⊆ v is a d-hop connected dominating-set (d-hcds) of graph g if every node t ∈ v is situated at most d-distances from at least one node z ∈ dhd, that is, d(t,z) ≤ d, and the subgraph of g induced by dhd is connected. a d-hcds with minimum cardinality is called a minimum d-hop connected dominating-set (mdhcds). the d-hop domination number of a graph g is the cardinality of a minimum d-hop connected dominating-set of g, and it is denoted by γhk(g). 1.1. survey of the related works domination is always an interesting and vital topic to researchers. claude berge, in his book, [8] defined the basic idea of the d-number γ(g) of a graph for the first time. the term d-set and d-number were first used by ore [28]. about twenty years ago, haynes et al. [21] wrote a revolutionary book on fundamentals of domination, and they recorded over twelve thousand research articles on domination and its different parameters in graphs. sampathkumar et al. [31] introduced the term “connected domination number” first. later, dorbec et al. [17] established independent-domination in cubic graphs. variations of domination like connected domination [15], edge domination, k-tuple domination and k-hop domination [16, 24, 33], weighted domination, paired-domination, perfect domination, secured domination, independent-domination, roman domination have been briefly discussed in the literature [10, 21, 23]. slater renames a k-hop dominating set as a k-basis [33]. besides these, many researchers studied various properties of graphs with respect to domination ([11, 13, 21, 22, 34, 36–38]). like most of the general graph problems, mdhcds problem is np-complete for random graphs ([39]), even in unit disc graphs (in short, udg), this problem is np-complete ([27]). on planar graphs, demaine et al. [16] proposed an algorithm that takes o(n4) time to determine k-hop dominating set with the fewest members. natarajan et al.[26] have done some fundamental works on hop domination number on some special class of graphs. also, on permutation-graphs rana et al. [29] set up an effective algorithm to find a distance-k dominating set. later, ayyaswamy et al. [3] worked on the upper and lower limits of the hop d-number on trees. besides these, on udgs and random graphs many researchers set up some algorithms for solving k-hop connected d-set problem [14, 19, 30, 41]. also, basuchowdhuri et al. [5] determined influential nodes for traditional communication networks using k-hop d-set. kundu et al. [24] set up an optimal algorithm for determining a k-hop d-set on trees. it is a general work of the prior result to find an 1-hop d-set (with the fewest cardinality) of a tree [13]. favaron et al. [18] showed that the diameter of the domination k-critical graphs is at most 2(k−1) when k ≥ 2. later, ramy s. shaheen [32] showed the bounds for the 2-domination number of toroidal grid graphs. also, barman et al. [4] designed an o(n) time algorithm 134 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 to find minimum d-hop d-set on interval graphs. recently adhya et al. [1] presented an algorithm for computing minimum k-hop connected d-set in o(n) time on permutation graphs. 1.2. applications to the best of our knowledge, domination is one of the important and fast-developing research problems in the core graph theory. some useful applications [2, 6, 7] of domination are found in many vital areas, such as computer-based communication networks, wireless-radio stations, kernels of games, coding-theory, modelling of biological-networks, land-surveying and problems of finding radar-stations. besides these, d-hop domination has lots of applications in facility location problems(flp). domination appears in flp for fixed number of facilities(like relief centres, disaster control centres [2]) and if somebody efforts to shorten the distance to travel so that he can get the service from the closest facility. the concept of d-hcds is used in ad-hoc networks [40, 42] to enhance the performance of efficiency in communication. ad-hoc networks have no fixed infrastructure. these networks are used in applications like search and rescue, mobile commerce and military battlefields. 1.3. main outcome here, we present two o(n) time algorithms to determine an mdhcds on trees, where n = |v |. the concept of the problem is the same with computing a connected d-set, only it differs concerning the number of steps/hops which is required for reaching all the members of v . actually we generalize minimum connected hop domination problem as minimum d-hop connected d-set problem of trees. 1.4. organization of our paper section 2 describes the formation of bfs-tree t ′(u) and t(v) of tree t . here we also present some notations which are used to solve our proposed problem. we state and prove some essential properties related with mdhcds of trees in section 3. besides these, we also present two algorithms: one for co-ordinates generation and other for finding central vertex(s) of tree t . in section 4, we present two complete optimal algorithms for computing mdhcds on trees. in this section, we also calculate time and space complexity of our main algorithms. 2. construction of bfs-trees t ′(u) and t(v) in graph theory, there are several graph traversal technique, one of them is bfs technique. it can form a bfs-tree. many researchers used bfs to solve different problems. a polynomial time algorithm which runs in o(n + m) time is available for construction of a bfs-tree on an arbitrary graph. on tree [12] chen et al. presented an efficient algorithm for creating a bfs-tree. in this paper, first we make a bfs-tree t ′(u) taking u as root, where u is an arbitrary vertex of a tree t . after then, we make another bfs-tree t(v) taking v as root, where v is any leaf node at last level(highest level) of t ′(u). these two bfs-trees can be made separately in o(m + n) time ≈ o(n) time, because m = n− 1. the bfs-tree t(v), taking root at v = 1 of the tree of figure 1(a) is shown in figure 2(a), taking u = 11. we denote the level of a node x belongs to any bfs-tree rooted at y by the symbol level(x) and defined by level(x) = d(x,y), assuming that level(v) = 0. lemma 2.1. if w be any vertex at the last level of t(v) then diameter(t) = d(v,w). 135 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 proof. let t(v,e) be an arbitrary tree. now, we construct a bfs-tree t ′(u) taking u, an arbitrary vertex of the tree t as root. after then, we make another bfs-tree t(v) taking v, any leaf node at last level(highest level) of t ′(u) as root. since the tree t ′(u) is a bfs-tree and level of v is maximum on it, then v is a farthest vertex from u, i.e., d(u,v) = max{d(u,x) : x ∈ v (t)}, i.e., e(u) = d(u,v). again, t(v) is also a bfs-tree, so, w, a vertex at highest level on t(v) is a farthest vertex from v, i.e., d(v,w) = max{d(v,y) : y ∈ v (t)}, i.e., e(v) = d(v,w). now, there are three cases may arise. case 1: if level(w) = level(u) on t(v), then u,w are both farthest vertices from v, i.e., d(v,u) = d(v,w) = max{d(x,y) : x,y ∈ v (t)} = diameter(t). case 2: if the shortest path between v and w passes through u, then d(v,w) = d(v,u) + d(u,w) = max{d(x,y) : x,y ∈ v (t)} = diameter(t). case 3: if level(w) > level(u) on t(v) and the shortest path between v and w does not pass through u, then d(v,w) = d(v,u) + d(u,w)−2×length of the common part of the shortest paths from u to v and u to w. also, d(v,w) = max{d(x,y) : x,y ∈ v (t)} = diameter(t). hence the result is proved. corollary 2.2. if h is the height of the bfs-tree t(v) then diameter(t) = diameter(t(v)) = diameter(t ′(u)) = h. u u pu v u u v ur v v @ @ � � u u u u up vu v v !!! � � v u v 1 2 5 4 6 uu 8 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 3 (a) v vu u uu u v u uu up vp up��� v u v up up u u u u u��� ttt \\\ 11 12 13 14 6 7 8 4 10 15 3 5 2 1 17 16 18 21 19 22 23 9 24 20 1 2 level 0 3 (b) figure 1. (a) a tree t and (b) bfs-tree t ′(u) of tree t . 2.1. identification of central-path lying on bfs-tree t(v) we consider h as the height of the formed bfs-tree t(v). also we presume that vlast is an arbitrary node at level h on t(v). now we consider the shortest path vlast → parent(vlast) → parent(parent(vlast)) →···→ v between v and vlast as the central-path between v and vlast. we also consider that if a node lying on the central-path at level j, then we denote it by v∗j . 2.2. notations in this subsection, we present some useful notations that are needed throughout this paper. 136 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 w wq wp v vpw vv v v v v v ww v v w w v v v v root 1 4 3 6 2 7 9 11 10 8 12 13 14 15 17 16 18 21 19 22 23 24 20 5 w wq wpw vv v v v vw v w w v v v root 1 4 3 6 2 7 9 v∗3 8 12 13 14 15 (1,0) 17 21 19 23 24 (1,2) (2,1) (3,0) 0 1 2 3 4 5 6 level v∗0 v∗1 w v @ @ @@ 5 w10 v 16 (1,0) v∗5vp 18 v 20 v∗6 v (b)(a) 22 v v∗4 v∗2 11 (0,3) figure 2. (a) bfs-tree t(v) and (b) bfs-tree t(v) after 2-tuple weight assignment. v∗r : the node lying on the central-path of bfs-tree t(v) at level r. ts(v ∗ r) : rooted sub-tree ( v ∗ r as root) of t(v) contains all the vertices imitate from : v∗r (excluding the nodes lying on the central-path except v ∗ r). here v ∗ r is the : source vertex of any other vertex of that sub-tree. mj : mj is the set of vertices at level j on the bfs-tree t(v). xj : xj is any member of the set mj −{v∗j}. h : height of the bfs-tree t(v). d : d is a fixed positive integer. dhd : dhd is mdhcds. u(r,j) : vertex u with 2-tuple weight assignment, where r,j are respectively 1st and : 2nd weight components on t(v). parent(z) : parent node of the vertex z. x ↪→ y : shortest path between x and y. 2.3. weight assignment on vertices here we describe the weight assignment process to some vertices, when d < b(h/2)c. for the bfs-tree t(v), we assign a new kind of 2-tuple weights (of the form (x,y), where x,y are respectively the 1st and 2nd weight components) on the vertices of rooted sub-trees ts(v∗d+1),ts(v ∗ d+2), . . . ,ts(v ∗ h−d−1) rooted at respectively v ∗ d+1,v ∗ d+2, . . . ,v ∗ h−d−1. first, select the vertex v ∗ d+1 and assign its 1st weight component as 0 and then move to child nodes and assign their first weight components as 1. after that for each child nodes of v∗d+1, we move to its child nodes and place their first weight components as 2. this process will be continued until all the vertices of the rooted sub-trees rooted at v∗d+1 get their first weight components. then we start back tracing from the last level (highest level) of the sub-tree ts(v∗d+1), selecting (one by one) a vertex z having no 2nd weight component and assign its 2nd weight component as 0. after that we assign the 2nd weight component of the parent vertex of z as 1 and for the parent of the parent vertex of z as 2 and continue until v∗d+1 gets 2nd weight component. again we start back tracing from the 2nd last level (2nd highest level) of the sub-tree ts(v∗d+1), selecting 137 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 (one by one) a vertex y (if exits) having no 2nd weight component and assign its 2nd weight component as 0. after that we assign the 2nd weight component (if it’s absent) of the parent vertex of y as 1 and for the parent of the parent vertex of y as 2 and so on. this process will be continued until all the members of the sub-rooted trees rooted at v∗d+1 get their second weight components. it is to be keep in mind that during this weight assignment process, if a vertex of ts(v∗d+1) has already got 2nd weight component then we skip it. we apply the same procedure to assign the 2-tuple weight assignment on the vertices of rooted sub-trees ts(v∗d+2),ts(v ∗ d+3), . . . ,ts(v ∗ h−d−1) separately. in this way, we assign the 2-tuple weight assignment (as shown in figure 2(a)) on all the vertices of the rooted sub-trees, rooted at v∗r, where r ∈{d + 1,d + 2, . . . ,h −d−1}. 2.4. algorithm for 2-tuple weight assignment algorithm 2tup-wt input : tree t(v). output: 2-tuple weight assignment of ∪h−d−1r=d+1 v (ts(v ∗ r)). initially d(< b(h/2)c) ∈ n. step 1: identify the nodes v∗i , i = 0,1, ...,h, lying on the central-path of t(v). step 2: set r = d + 1. step 3: do { step 3.1: select a vertex v∗r and assign its 1st weight component as 0. step 3.2: move to the child nodes of v∗r on ts(v ∗ r) and assign their 1st weight components as 1. step 3.3: for each child nodes of v∗r, move to its child nodes and assign their 1st weight components as 2. step 3.4: continue till all the members of the set v (ts(v∗r)) get their 1st weight components. step 3.5: compute the level of any vertex placed at last level (highest level) of ts(v∗d+1) and assign it with l. step 3.6: do{ step 3.6.1: assign the 2nd weight components of all the vertices of ts(v∗d+1) at lth level as 0, if unavailable; else skip. step 3.6.2: for each vertex z at lth level of ts(v∗d+1) move to its parent vertex and assign the 2nd weight component of the parent(z) as 1, if unavailable; else skip. step 3.6.3: similarly assign 2nd weight component of parent(parent(z)) as 2 and continue until all internal nodes (including v∗d+1) of the path between z and v∗d+1 get 2nd weight components. step 3.6.4: set l = l−1. } while (l > r). step 3.7: r = r + 1. } while (r ≤ (h −d−1)). end 2tup-wt. 2.5. illustrative example of 2-tuple weight assignment to describe the procedures of the above algorithm (algorithm 2tup-wt), we first consider the tree t(v) of figure 2(a). so, h = 6. initially, we assume that d = 2 ≤ b(h/2)c. in step 1, we identify the members on the central-path (considering vlast = 22) as v∗0 = 1,v ∗ 1 = 4,v ∗ 2 = 6,v ∗ 3 = 11,v ∗ 4 = 17,v ∗ 5 = 21,v∗6 = 22. in step 2, we set r = 2+1 = 3. in step 3, we enter in the do-while loop for (r ≤ (h−d−1)) to assign 2-tuple weight on vertices. in step 3.1, we select the root v∗3 = 11 of the rooted sub-tree ts(v ∗ d+1) = ts(v ∗ 3) and assign its 1st weight component as 0 and write like 11(0,−). after that, in step 138 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 3.2, we move to its child nodes and assign their 1st weight components as 1, 12(1,−),15(1,−),16(1,−). in step 3.3, for each child nodes of 11, we move to their child nodes and assign their 1st 1st weight components as 2, i.e, 13(2,−). similarly vertex 14 get its 1st weight component as 14(3,−) in step 3.4. thus all vertices of the tree ts(v∗3) are assigned with their 1st weight components. in step 3.5, we compute l = 6. in step 3.6, we enter into another do-while loop for l > r. in this loop we will the 2nd weight components of the vertices of ts(v∗3). at first, in step 3.6.1, we move to vertices placed at the last level (6th) of ts(v∗3) and see that there is only one vertex which is 14 and it has no 2nd weight component. so we assign 2nd weight component of the vertex 14 as 0, i.e., 14(3,0). in step 3.6.2, we select the parent node of 14, which is 13 and put its 2nd weigh component as 13(2,1). similarly, in step 3.6.3, the vertices 12 and 11 get their 2nd weight components as 2 and 3 respectively and write like 12(1,2) and 11(0,3). next in step 3.6.4, we reset l = l − 1 = 5 and move to the step 3.6.1 as l > r. since, all vertices of ts(v∗3) at level 5 have 2nd weight components, so move to the step 3.6.4, and reset again, we move to the step 3.6.1 (as l > r) and see that there are two vertices 15 and 16 which have no 2nd weight components. so we assign 2nd weight components of 15 and 16 as 0. since parent(15) = parent(16) = 11 and vertex 11 has already got 2nd weight component, so, we move to the step 3.6.4, and reset l = l− 1 = 3. since l < r, so the present do-while loop for l > r is finished and move to the step 3.7. here we reset r = r + 1 = 4. since, r > h −d− 1 = 3, so the do-while loop for r ≤ h − d − 1 is finished. thus the execution process of algorithm 2tup-wt is finished. the final 2-tuple weight assignment of the vertices of t(v) is shown in figure 2(b). theorem 2.3. the run time of algorithm 2tup-wt is o(n). proof. in step 1, the time needed for identifying the vertices v∗i , i = 0,1,2, . . . ,h, on the central path of the tree t(v) is o(n). step 2 takes constant time. since, v (ts(v∗d+1)),v (ts(v ∗ d+2)), . . . , v (ts(v ∗ h−d−1)) are mutually disjoint, so, in step 3, the time for assigning 1st weight components of the vertices ∪h−d−1r=d+1 v (ts(v ∗ r)) is o(n+m) ≈ o(n) time (in a worst case) and the time required for assigning 2nd weight components of the same vertices is also o(n) time. so, step 3 can be finished in o(n) time. hence, the overall time complexity of algorithm 2tup-wt is o(n). 2.6. algorithm for computation of central node(s) of t algorithm t-center input : tree t . output: central node(s) of tree t. step 1: make bfs-tree t ′(u) taking u as root, where u is an arbitrary vertex. step 2: make bfs-tree t(v) taking v as root, where v is an arbitrary pendant vertex at highest level of t ′(u). step 3: identify the central-path and determine the height h. step 4: if h is even then central node of t is v∗bh/2c else central nodes of t are v∗bh/2c and v ∗ bh/2c+1. end if end t-center. the algorithm t-center provides that the central vertex of the tree t shown in the figure 1(a) is v ∗3 = 11. theorem 2.4. the run time of algorithm t-center for determining the central node(s) of tree t is o(n), where |v | = n. proof. at step 1 and step 2, two bfs-trees t ′(u) and t(v) can be made separately in o(n)-time. in step 3, the members of the central-path and the height h can be determined in o(n)-time, because there 139 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 is only (h + 1)(≤ n) nodes located on the central-path. again, we can finish step 4 in constant time. so, overall time complexity of the algorithm t-center is o(n). 3. important properties related to mdhcds dhd of trees. here, we have proved some emergent results related to mdhcds dhd of trees. lemma 3.1. if r ≤bh/2c and zj is any leaf node at jth level on ts(v∗r) then d(v∗r,xj) ≤ r. proof. we know, by corollary 2.2, that h = diameter(t) = diameter(t(v)), i.e., v∗0 → v∗1 → ··· → v∗r → v∗r+1 →···→ v∗h is the longest shortest path on both trees t and t(v). now, if r ≤bh/2c and zj be any leaf node at jth level on ts(v∗r), then r < j ≤ h. again, d(v∗0,v∗r) = r and d(v∗r,v∗h) = (h − r). so, d(v∗r,v ∗ h) ≥ d(v ∗ 0,v ∗ r) as r ≤ bh/2c. let us assume that d(v∗r,zj) > r. so (j − r) > r ⇒ j > 2r ⇒ (j − 2r) > 0. therefore, d(v∗h,zj) = d(v ∗ h,v ∗ r) + d(v ∗ r,zj) = (h − r) + (j − r) = h + (j − 2r) > h as (j − 2r) > 0. so, v∗h ↪→ v ∗ r ↪→ zj is the longest shortest path on t(v) which is impossible as diameter is h. so, our assumption is wrong. therefore, d(v∗r,zj) ≤ r. lemma 3.2. if bh/2c < r and zj is any leaf node at jth level on ts(v∗r) then d(v∗r,zj) ≤ h −r. proof. we know, by corollary 2.2, that h = diameter(t) = diameter(t(v)), i.e., v∗0 → v∗1 → ··· → v∗r → v∗r+1 →···→ v∗h is the longest shortest path on both trees t and t(v). now, if r > bh/2c and zj be any leaf node at jth level on ts(v∗r), then bh/2c < r < j. also, d(v∗0,v∗r) = r and d(v∗r,v∗h) = (h−r). since, r > bh/2c, d(v∗0,v∗r) > d(v∗r,v∗h). let us assume that d(v ∗ r,zj) > h −r. now, d(v∗r,zj) = (j −r). so, (j−r) > (h−r) that implies j > h, i.e., d(v∗0,zj) = j > h which is impossible, as h is the diameter of t . so, our assumption is wrong. therefore, d(v∗r,zj) ≤ (h −r). corollary 3.3. if zr be any member at even level r > 1 on t(v), then its source vertex will be any one of {v∗r−1,∗ ,v∗r−2, . . . ,v∗r/2}. corollary 3.4. if zr be any member at odd level r > 2 on t(v), then its source vertex will be any one of {v∗r−1,∗ ,v∗r−2, . . . ,v∗(r+1)/2}. lemma 3.5. the tree t(v) with at least three nodes has only one internal node at level 1. proof. let t be any tree having at least three nodes. now, for constructing t(v), at first we make a bfs-tree t ′(u) where u is any leaf node on t . then we construct t(v), where v is any leaf node at the highest level(last level) on t ′(u). this implies that the vertex v is incident only one edge i.e., v is the parent of only one internal vertex on t(v). corollary 3.6. there is no leaf nodes at level 1 on t(v). lemma 3.7. for each z ∈∪2dr=0mr, d(v∗d,z) ≤ d. proof. let zr be any member of {mr − v∗r}, for r = 2,3, . . . ,2d. so, for r = 0,1, . . . ,d, d(v∗d,v ∗ r) ≤ (d − r). now if r is even and 2 ≤ r ≤ d then, using corollary 3.3, d + 2 − r ≤ d(zr,v∗d) ≤ d(as zr → v∗r−1 → v∗r → v∗r+1 → ··· → v∗d or zr → zr−1 → v ∗ r−2 → v∗r−1 → v∗r → v∗r+1 → ··· → v∗d or . . .zr → zr−1 → ··· → z(r/2)+1 → v∗r/2 → v ∗ (r/2)+1 → ··· → v∗d). again if r is odd and 3 ≤ r ≤ d then, using corollary 3.4, d + 2 − r ≤ d(zr,v∗d) ≤ (d − 1)(as zr → v ∗ r−1 → v∗r → v∗r+1 → ··· → v∗d or zr → zr−1 → v∗r−2 → v∗r−1 → v∗r → v∗r+1 → ··· → v∗d or . . .zr → zr−1 → ··· → z((r+1)/2)+1 → v∗ (r+1)/2 → v∗ ((r+1)/2)+1 →···→ v∗d). therefore, d(v ∗ d,z) ≤ d, ∀ z ∈∪ d r=0mr. again, d(v ∗ d,v ∗ r) ≤ (r −d), for r = d + 1,d + 2, . . . ,2d. also if r is even and d + 1 ≤ r ≤ 2d − 2, then r − d ≤ d(v∗d,zr) ≤ d(as zr → v∗r−1 → v∗r−2 → ··· → v∗d or zr → zr−1 → v ∗ r−2 → v∗r−3 → ··· → v∗d or . . . or zr → zr−1 → 140 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 zr−2 → ··· → zd+1 → v∗d or zr → zr−1 → ··· → zd → v ∗ d−1 → v ∗ d or zr → zr−1 → ··· → zd → zd−1 → v∗d−2 → v ∗ d−1 → v ∗ d or . . . or zr → zr−1 → ··· → z(r/2)+1 → v ∗ r/2 → v∗ (r/2)+1 → ··· → v∗d). again if r is odd and d + 1 ≤ r < 2d − 2, then r − d ≤ d(v∗d,zr) ≤ d − 1(as zr → v ∗ r−1 → v∗r−2 → ··· → v∗d or zr → zr−1 → v∗r−2 → v∗r−3 → ··· → v∗d or . . . or zr → zr−1 → zr−2 → ··· → zd+1 → v ∗ d or zr → zr−1 → ··· → zd → v∗d−1 → v ∗ d or zr → zr−1 → ··· → zd → zd−1 → v ∗ d−2 → v ∗ d−1 → v ∗ d or . . . or zr → zr−1 → ··· → z((r+1)/2)+1 → v∗(r+1)/2 → v ∗ ((r+1)/2)+1 → ··· → v∗d). also for r = 2d − 1,2d d(v∗d,zr) = d (or d− 1) if r is even (or odd) (as zr → v ∗ r−1 → v∗r−2 → ··· → v∗d or zr → zr−1 → v ∗ r−2 → v∗r−3 → ··· → v∗d or . . . or zr → zr−1 → zr−2 → ··· → zd+1 → v ∗ d). so, d(v ∗ d,z) ≤ d, ∀ z ∈ ∪ 2d r=d+1mr. therefore d(v∗d,z) ≤ d, ∀ z ∈∪ 2d r=0mr. corollary 3.8. v∗d is a possible member of dhd. corollary 3.9. all the members of ∪dr=1v (ts(v∗r)) are within d distances from v∗d. lemma 3.10. for each z ∈∪dr=1v (ts(v∗h−r)), d(v ∗ h−d,z) ≤ d. proof. let zr be any node at level r on t(v) and (h−d) < r ≤ h. also, let zr ∈∪dr=1v (ts(v∗h−r)), i.e., source vertex of zr be one of the set {v∗h−d,v ∗ h−d+1, ...,v ∗ h−1}. now, d(v ∗ h−d,v ∗ h) = d and d(v ∗ h−d,zr) = (r −h + d) ≤ d (as zr → v∗r−1 → v∗r−2 → ··· → v∗h−d or zr → zr−1 → v ∗ r−2 → v∗r−3 → ··· → v∗h−d or . . . or zr → v∗h−d). so, the result is proved. now, by observing the last result, we present the following result. corollary 3.11. v∗h−d is a possible member of dhd. lemma 3.12. if u(r,j) ∈ ts(v∗l (0,r + j)), j ≥ d and v ∗ l (0,r + j) ∈ dhd, then u(r + j −d,d),u(r + j − d−1,d + 1), . . . ,u(1,r + j −1) are the possible members of dhd. proof. let u(r,j) ∈ ts(v∗l (0,r + j)), j ≥ d and v ∗ l (0,r + j) ∈ dhd, that is there exists a path of length (r + j) imitate from v∗l (0,r + j) and u(r + j,0) passing through u(r,j) of the form v ∗ l (0,r + j) → u(1,r + j − 1) → u(2,r + j − 2) → ··· → u(r,j) → u(r + 1,j − 1) → ··· → u(r + j − d − 1,d + 1) → u(r+j−d,d) → u(r+j−d+1,d−1) →···→ u(r+j,0). so, d(v∗l (0,r+j),u(r+j,0)) = r+j ≥ r+k and d(u(r+j−d,d),u(r+j,0)) = d. therefore, u(r+j−d,d) is a possible member of dhd. also, v∗l (0,r+j) ∈ dhd. so, for computing d-hcds, we have to select u(r+j−d,d),u(r+j−d−1,d+1), . . . ,u(1,r+j−1) as the possible member of dhd. lemma 3.13. if h ≤ d, then dhd = {v∗r}, where r refers any single member of {0, 1, 2,3,...,h}. proof. let zr be a member of mr −{v∗r}, for r = 2,3, . . . ,h. now, d(v∗0,v∗h) = h ≤ d. we know (corollary 3.6) that there is no leaf nodes at level 1 on t(v). so v∗0 be a probable member of dhd because d(v∗0,zr) = r ≤ d (as v∗0 → v∗1 → . . . → v∗r−1 → zr or v∗0 → v∗1 → . . . → v∗r−2 → zr−1 → zr or v∗0 → v∗1 → . . . → v∗i−3 → zr−2 → zr−1 → zr or. . . or v ∗ 0 → v∗1 → z2 → z3 → . . . → zr−3 → zr−2 → zr−1 → zr), for r = 2,3, . . . ,h. furthermore, if r is even and 2 ≤ r ≤ h then d(zr,v∗h) ≤ h ≤ d (as zr → v∗r−1 → v∗r → . . . → v∗h or zr → zr−1 → v ∗ r−2 → v∗r−1 → v∗r . . . → v∗h or . . . or zr → zr−1 → . . . → z(r−(r/2)+1) → v∗(r−(r/2) → v ∗ (r−(r/2)+1) → . . . → v ∗ h). again, if r is odd and 3 ≤ r ≤ h then d(zr,v ∗ h) ≤ h ≤ d−1 < d (as zr → v ∗ r−1 → v∗r → . . . → v∗h or zr → zr−1 → v ∗ r−2 → v∗r−1 → v∗r . . . → v∗h or . . . or zr → zr−1 → . . . → z(r−((r+1)/2)+1) → v∗r−((r+1)/2) → v ∗ (r−((r+1)/2)+1) → . . . → v ∗ h). hence, v∗h is a possible member of dhd. similarly, other members of the central path on t(v) may be possible members of dhd. so, if h ≤ d, then dhd = {v∗r}, where r refers any single member of {0, 1, 2,...,h}. lemma 3.14. d-hds dhd of a tree t is dhd = { {z : z is the central node(s) of t} ifbh/2c = d, {z : z any single central vertices of t} if bh/2c < d. 141 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 proof. let diameter(t) = diameter(t(v)) = h. now, two cases may appear — case 1: d > bh/2c and case 2: d = bh/2c. case 1: in this case, d(v∗0,v ∗ h) = h. now two sub-cases may appear here. sub-case 1.1: when h is even. in this sub-case, t is mono-centric and the central vertex is v∗ (h/2) . so, d(v∗ (h/2) ,x) ≤ h 2 < d, ∀ z ∈ v (t). therefore, dhd = {v∗h/2}. sub-case 1.2: when h is odd. in this subcase t is bi-centric, and the central vertices are v∗bh/2c and v ∗ bh/2c+1. so, d(v ∗ bh/2c,z) ≤br/2c+ 1 ≤ d and d(v ∗ bh/2c+1,z) ≤bh/2c+ 1 ≤ d, ∀ z ∈ v (t). therefore, dhd = {v∗bh/2c} or dhd = {v ∗ bh/2c+1}. case 2: in this case, two sub-cases may appear. sub-case 2.1: when h is even. in this sub-case, t has single central vertex v∗ h/2 . so, d(v∗ h/2 ,x) ≤ h/2 = d, ∀ z ∈ v (t). therefore, dhd = {v∗h/2}. sub-case 2.2: when h is odd. in this sub-case t has two central vertices v∗bh/2c and v ∗ bh/2c+1. so, d(v ∗ bh/2c,z) ≤ br/2c = d ∀ z ∈ ∪h−1r=0 mr but d(v ∗ bh/2c,zh) = br/2c + 1 = d + 1 > d, where zh ∈ mh. for instance d(v∗bh/2c,v ∗ h) = br/2c + 1 = d + 1 (as v ∗ h → v ∗ h−1 → v ∗ h−2 → . . . → v ∗ bh/2c). again, d(v ∗ bh/2c+1,z) ≤ br/2c = d ∀ z ∈ ∪hr=1mr but d(v∗bh/2c+1,v ∗ 0) = d + 1 > d (as v ∗ h → v ∗ h−1 → v ∗ h−2 → . . . → v ∗ 0). so, all the vertices of t are within d distances from the members of the set dhd = {v∗bh/2c,v ∗ bh/2c+1}. hence the result. 4. complete algorithms and their complexities here, we have presented two optimal algorithms for determining minimum d-hcds dhd of tree t . 4.1. first complete algorithm from the results discussed in section 3, it is observed that when d ≤ bh/2c and if h is even, then v∗ h/2 is a possible member of dhd, and if h is odd, then v∗bh/2c and v ∗ bh/2c+1 are two possible members of dhd. two possible cases for selecting the members of dhd are discussed in lemma 3.14. the first complete algorithm mdhcdst1 is designed for computing a mdhcds dhd of tree t , presented below. algorithm mdhcdst1 input: a tree t(v,e). output: mdhcds dhd of tree t. step 1: make the bfs-tree t ′(u) taking u as root, where u is any arbitrary vertex. step 2: make bfs-tree t(v) taking v as root, where v is any leaf node at highest level of t ′(u). step 3: identify central path and compute height h of t and set h= diameter(t). step 4: find the center(s) of t.//by algorithm t-center// step 5: if d ≤bh/2c, then step 5.1: construct a subtree t1(v) of t(v), by removing the set v1 of all pendant vertices from t(v). step 5.2: for r = 2,3, ...,d, construct the subtree tr(v) of tr−1(v), by removing the set vr of all pendant vertices from tr−1(v). step 5.3: assign dhd = {v (td(v))}. else dhd= {z : z is any one central node of t.}} //lemma 3.14// end if end mdhcdst1. if we apply the algorithm mdhcdst1 for d = 2 on the tree t of figure 1 (a), then we get a minimum 2-hop connected dominating set dh2 = {6,11,17,12}. 142 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 figure 3. flowchart of algorithm mdhcdst1. lemma 4.1. the algorithm mdhcdst1 gives a mdhcds dhd of trees. proof. in algorithm mdhcdst1, if d ≤bh/2c (step 4), then we set dhd = v (tbh/2c(v)). in this case, dhd is a minimum because if we eliminate anyone pendent vertex from tbh/2c(v), then d(x,y) ≥ d, for all x ∈ v (t) and y ∈ v (tbh/2c(v)). again dhd is connected as tbh/2c(v) is a connected sub-graph of t. moreover, by lemma 3.13, when h is even and d > bh/2c, then t is mono-centric and the central node is v∗ h/2 . for this reason we set dhd = {v∗h/2}. again, when h is odd and d > bh/2c, then t is bi-centric and the central nodes are v∗bh/2c and v ∗ bh/2c+1. also each of these vertices can dominate each vertex of t by d or less steps. therefore, we set dhd = {v∗bh/2c} or {v ∗ bh/2c+1}. therefore, dhd is a mdhcds of trees. theorem 4.2. the run time of the algorithm mdhcdst1 is o(n), where |v | = n. proof. step 1 and step 2 need o(n) time, respectively, for building the bfs-trees t ′(u) and t(v). in step 3, we can compute the members of the central path and the diameter in o(n) time as h= diameter(t) and there are only h +1 ≤ n vertices in the central path on t(v). in step 4, central node(s) 143 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 of t(v) can be identified in o(n) time (theorem 2.4). at first iteration of step 5, we find, and then delete the set v1 of pendent vertices of tree t(v) and it can be done in |v1| time. similarly, other iterations can be finished. hence, step 5 takes o(∪bh/2cr=1 |vr|) ≈ o(n) time as vr, r = 1,2,3, . . . ,bh/2c are mutually disjoint. therefore, the algorithm mdhcdst1 runs in o(n)-time. 4.2. second complete algorithm in section 3, we proved that if h ≤ d, then v∗l can be accepted as an element of dhd, where l is only one arbitrary element of {0,1, . . . ,h} (lemma 3.12). also, if h > d, then by corollary 3.6, corollary 3.9 and lemma 3.10, we can take the vertices of the set {v∗d,v ∗ d+1, . . . ,v ∗ h−d}∪{u(r,j) ∈ t ∗(v) : j ≥ d} as members of dhd. all probable cases for choosing the elements of dhd are already discussed in section 3. the second complete algorithm mdhcdst2 for determining a mdhcds dhd of tree t is presented below. algorithm mdhcdst2 input: a tree t . output: mdhcds dhd of tree t. initially dhd = φ. step 1: make bfs-tree t ′(u) taking u as root, where u is any vertex of v (t). step 2: make bfs-tree t(v) taking v as root, where v is any leaf node at highest level of t ′(u). step 3: identify the central-path and determine the vertices lying on that central-path of t(v) and denote them by v∗r,r = 0,1, . . . ,h. step 4: do 2-tuple weight assignment of v (ts(v∗r)), r = d + 1,d + 2, . . . ,h −d−1. step 5: if h ≤ d, then dhd = dhd ∪{v∗l }, where l is only one arbitrary element of {0,1, . . . ,h}. //lemma 3.12// else dhd = dhd ∪{v∗d,v ∗ d+1, . . . ,v ∗ h−d}∪{u(r,j) ∈ t ∗(v) : j ≥ d andr 6= 0} // (lemma 3.10, corollary 3.6 and corollary 3.9)// end if end mdhcdst2. if we apply the algorithm mdhcdst2 for d = 2 on the tree t of figure 1 (a), then we get a m2hcds dh2 = {6,11,17,12}. lemma 4.3. the algorithm mdhcdst2 gives a mdhcds dhd of trees. proof. we have noticed in algorithm mdhcdst2 that if h ≤ d then, using the result of lemma 3.12, we find dhd ={v∗l }, where l is only one arbitrary element of {0,1,2,3, . . . ,h}. so, in that case, dhd is mdhds. if h > d, we select first member of dhd as v∗d, by corollary 3.6, which dominates | ∪dr=0 mr| vertices & d(v∗h−d,z) = d, and choose second element of dhd as v ∗ h−d (by corollary 3.9) because d(v∗h−d,z) = d, ∀ z ∈ mh. in step 4, we do 2-tuple weight assignment of the vertices of the sub-trees ts(v∗d+1),ts(v ∗ d+2), . . . ,ts(v ∗ h−d−1) rooted at, respectively, v ∗ d+1,v ∗ d+2, . . . ,v ∗ h−d−1. these roots are consecutively adjacent and lie on the central path on t(v). so these are essential for making connectedness of dhd. besides these, we select some vertices whose second weight components are greater or equal to d (lemma 3.10). during this selection process, we always keep in mind that selected members cover the maximum number of nodes of v and every node of v is located within d distances from at least one element of dhd. so, dhd is mdhds. now, we have to prove that dhd is connected. since some members of dhd stay consecutively on the central path of bfs-tree t(v), so they form a sub-central path, so they are connected. the remaining members of dhd which are imitated from some vertices lie on the central-path, they are connected with the previous sub-central path (lemma 3.10). hence, dhd is connected. therefore, dhd is a mdhcds of trees. 144 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 figure 4. flowchart of algorithm mdhcdst2. theorem 4.4. the algorithm mdhcdst2 runs in o(n) time, where n = |v |. proof. step 1 takes o(n) time for constructing the bfs-tree t ′(u). similarly o(n) time is required for building the bfs-tree t(v) in step 2. in step 3, the vertices on the central path can be identified in o(n) time, as there is only h + 1 ≤ n vertices in the central path. for 2-tuple weight assignment of the vertices of v (ts(v∗r),r = d + 1,d + 2, . . . ,h − d − 1, o(n) time is needed, in step 4, as v (ts(v∗r)),r = d + 1,d + 2, . . . ,h −d−1 are mutually disjoint. if h ≤ d, then we can compute dhd in constant time as |dhd| = 1. on the other hand if h > d, the time needed to find the members of dhd is o(n). therefore, overall time complexity of algorithm mdhcdst2 is o(n) time. 5. conclusion domination is always an attractive and crucial problem to the researchers who work in graph theory. within the various types of dominations, we can apply connected domination in many practical-life problems, and many researchers have done much research on this in the past. nowadays, in advanced 145 a. s. adhya et. al. / j. algebra comb. discrete appl. 9(3) (2022) 133–147 graph 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[42] j. yu, n. wang, g. wang, d. yu, connected dominating sets in wireless ad hok and sensor networks-a comprehensive survey, comput. commun. 36(2) (2013) 121–134. 147 https://doi.org/10.1002/jgt.21855 https://doi.org/10.1002/jgt.21855 https://doi.org/10.1002/(sici)1097-0118(200005)34:1<9::aid-jgt2>3.0.co;2-o https://doi.org/10.1002/(sici)1097-0118(200005)34:1<9::aid-jgt2>3.0.co;2-o https://doi.org/10.1504/ijsnet.2012.050447 https://doi.org/10.1504/ijsnet.2012.050447 https://www.worldcat.org/title/graph-theory/oclc/64676 https://doi.org/10.1201/9781482246582 https://doi.org/10.1201/9781482246582 https://doi.org/10.1201/9781315141428 https://doi.org/10.1201/9781315141428 https://www.elsevier.com/books/topics-on-domination/hedetniemi/978-0-444-89006-1 https://doi.org/10.1016/j.ipl.2015.07.014 https://doi.org/10.1016/j.ipl.2015.07.014 https://mathscinet.ams.org/mathscinet-getitem?mr=543669 https://doi.org/10.1515/auom-2015-0036 https://doi.org/10.1515/auom-2015-0036 https://doi.org/10.1109/wiopt.2006.1666454 https://doi.org/10.1109/wiopt.2006.1666454 https://doi.org/10.1109/wiopt.2006.1666454 https://mathscinet.ams.org/mathscinet-getitem?mr=0244094 https://doi.org/10.1080/09720529.2014.986906 https://doi.org/10.1080/09720529.2014.986906 https://doi.org/10.1016/j.comnet.2004.09.005 https://doi.org/10.1016/j.comnet.2004.09.005 https://mathscinet.ams.org/mathscinet-getitem?mr=575817 https://mathscinet.ams.org/mathscinet-getitem?mr=575817 https://doi.org/10.1080/00207160701690284 https://doi.org/10.1080/00207160701690284 https://doi.org/10.1145/321958.321964 https://doi.org/10.1016/0095-8956(83)90007-2 https://doi.org/10.1142/s0129626494000417 https://doi.org/10.1142/s0129626494000417 https://mathscinet.ams.org/mathscinet-getitem?mr=1676474 https://mathscinet.ams.org/mathscinet-getitem?mr=1676474 https://mathscinet.ams.org/mathscinet-getitem?mr=1676474 https://mathscinet.ams.org/mathscinet-getitem?mr=1658130 https://mathscinet.ams.org/mathscinet-getitem?mr=1658130 https://doi.org/10.7151/dmgt.1143 https://doi.org/10.7151/dmgt.1143 https://doi.org/10.1109/icccn.2000.885513 https://doi.org/10.1109/icccn.2000.885513 https://doi.org/10.1504/ijesms.2015.066127 https://doi.org/10.1504/ijesms.2015.066127 https://doi.org/10.1007/s11036-008-0039-3 https://doi.org/10.1007/s11036-008-0039-3 https://doi.org/10.1016/j.comcom.2012.10.005 https://doi.org/10.1016/j.comcom.2012.10.005 introduction construction of bfs-trees t'(u) and t(v) important properties related to mdhcds dhd of trees. complete algorithms and their complexities conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.83854 j. algebra comb. discrete appl. 3(1) • 1–6 received: 5 august 2015 accepted: 20 october 2015 journal of algebra combinatorics discrete structures and applications the unit group of group algebra fqsl(2, z3) research article swati maheshwari, r. k. sharma abstract: let fq be a finite field of characteristic p having q elements, where q = pk and p ≥ 5. let sl(2, z3) be the special linear group of 2 × 2 matrices with determinant 1 over z3. in this note we establish the structure of the unit group of fqsl(2, z3). 2010 msc: 16u60, 20c05 keywords: group algebra, unit group, finite field 1. introduction let fg be a group algebra of a finite group g over a field f and u(fg) be the group of units in fg. it is a classical problem to study units and their properties in group ring theory. the case, when g is a finite abelian group, the structure of fg is studied by perlis and walker in [14]. in 2006, t. hurley introduced a correspondence between group ring and certain ring of matrices (see [6]). as an application of units of a group ring, t. hurley gave a method to construct convolutional codes from units in group ring (see [7]). a lot of work has been done for finding the algebraic structure of the unit group u(fg) of a group algebra fg, when g is a finite non-abelian group. here we are providing some literature survey for the same. for dihedral groups, the structure of the unit group u(fg) over a finite field f is discussed in [1, 4, 10, 12]. j. gildea et.al. (see [3]) and r. k. sharma et.al. (see [15]) have given the structure of the unit group u(fg), where g is alternating group a4. unit group of algebra of circulant matrix has been discussed in [11, 17]. the unit group of group algebras of some non-abelian groups with small orders are established in [16, 18, 19]). in this article, we are interested in studying the structure of the unit group of fqsl(2,z3) over a finite field of characteristic greater than 3. this work was supported by iit delhi, india through gate senior research fellowship. swati maheshwari (corresponding author), r. k. sharma; department of mathematics, indian institute of technology delhi, india (email: swatimahesh88@gmail.com, rksharmaiitd@gmail.com). 1 s. maheshwari, r. k. sharma / j. algebra comb. discrete appl. 3(1) (2016) 1–6 2. preliminaries the following results provide useful information about the decomposition of a/j(a), where a = fg, j(a) be its jacobson radical and f being a field of characteristic p. for basic definitions and results, we refer to [13]. we briefly introduce some definitions and notations those will be needed subsequently. definition 2.1. an element g ∈ g is said to be p-regular if p o(g). let s be the l.c.m. of the orders of the p-regular elements of g, ζ be a primitive s-th root of unity over f. then tg,f be the multiplicative group consisting of those integers t, taken modulo s, for which ζ 7→ ζt defines an automorphism of f(ζ) over f. that is, tg,f is gal(f(ζ)/f) seen as a subgroup of u(zs). note that if u is a power of a prime such that (u,s) = 1 and c = ords (u) is the multiplicative order of u modulo s, then tg,fu = {1,u, . . . ,u c−1} mod s and fu(ζ) ∼= fuc follow using [8, theorem 2.21]. definition 2.2. if g ∈ g is a p-regular element, then the sum of all conjugates of g ∈ g is denoted by γg and the cyclotomic f-class of g is defined to be the set sf(γg) = {γgt | t ∈ tg,f}. proposition 2.3. [2, theorem 1.2] the number of simple components of fg/j(fg) is equal to the number of cyclotomic f-classes in g. theorem 2.4. [2, theorem 1.3] suppose that gal(f(ζ)/f) is cyclic. let w be the number of cyclotomic f-classes in g. if k1,k2, . . . ,kw are the simple components of z(fg/j(fg)) and s1,s2, . . . ,sw are the cyclotomic f-classes of g, then with a suitable re-ordering of indices, | si |= [ki : f]. lemma 2.5. [9, observation 2.2.1, p.22] let b1,b2 be two finite dimensional f-algebras such that b2 is semisimple. if f : b1 → b2 is an onto homomorphism of f-algebras, then there exists a semisimple f-algebra ` such that b1/j(b1) ∼= `⊕b2. throughout this article, g = sl(2,z3). fq is a field of characteristic p, where q = pk and k is a positive integer. the conjugacy class of g ∈ g is denoted by [g]. 3. main result we shall use the presentation of g given in [5], 〈a,b | a3,b4,(ab)3 = b2,(a2b)6〉 where a = [ 1 0 1 1 ] and b = [ 0 1 −1 0 ] . we can see that g has 7 conjugacy classes as follows: 2 s. maheshwari, r. k. sharma / j. algebra comb. discrete appl. 3(1) (2016) 1–6 representative elements in the class order of element [a] a,(ba)4,(ab)4,b−1ab 3 [a−1] a−1,(ba)2,(ab)2,aba 3 [b] b,b−1,a2ba,aba2,ab−1a2,a2b−1a 4 [b2] b2 2 [ab] ab,ba,a2ba2,ab2 6 [(ab)−1] (ab)−1,a2b−1,ab−1a,a2b2 6 we have (p, |g|) = 1 and so j(fpkg) = 0. further, we discuss the decomposition of fpkg. theorem 3.1. let fq be a finite field of characteristic p, where p ≥ 5. then the wedderburn decomposition of fqg is given by condition on k fqg k is even f3q ⊕m(2,fq)3 ⊕m(3,fq) k is odd p ≡ 1 mod 3 and p ≡±1mod 4 f3q ⊕m(2,fq)3 ⊕m(3,fq) k is odd p ≡−1 mod 3 and p ≡±1mod 4 fq ⊕fq2 ⊕m(2,fq)⊕m(2,fq2)⊕m(3,fq) proof. since fqg is semisimple, so it has the wedderburn decomposition which is given by fqg ∼= ⊕ri=1m(ni,fi), where for each i,ni ≥ 1and fi is a finite extension of fq. by using lemma 2.5, we have fqg ∼= fq ⊕r−1i=1 m(ni,fi). (1) further, we find ni’s and fi’s. since | g |= 24, hence any element g ∈ g is a pregular element. for finding cyclotomic fq classes of g, first we assume that k is even. we have pk ≡ 1 mod 4 and pk ≡ 1 mod 3. then by chinese remainder theorem pk ≡ 1 mod 12. by using above observation, we have sfq (γg) = {γg} and | sfq (γg) |= 1. therefore by using equation (1), proposition 2.3 and theorem 2.4, we have fqg ∼= fq ⊕6i=1 m(ni,fq) 3 s. maheshwari, r. k. sharma / j. algebra comb. discrete appl. 3(1) (2016) 1–6 for some ni ≥ 1. as dimension of fqg is 24, we get 6∑ i=1 n2i = 23. using above equality, 1 ≤ ni ≤ 3. clearly any ni = nj = 3 for 1 ≤ i 6= j ≤ 3 not possible. so the only possible choice for ni’s is n1 = n2 = 1,n3 = n4 = n5 = 2 and n6 = 3. therefore the decomposition fqg is given by fqg ∼= f3q ⊕m(2,fq)3 ⊕m(3,fq). now we consider the case when k is odd. we shall discuss this case into two parts 1. p ≡ 1 mod 3 and p ≡±1 mod 4 2. p ≡−1 mod 3 and p ≡±1 mod 4 case 1. suppose k is odd with p ≡ 1 mod 3 and p ≡±1 mod 4. observe that pk ≡ p mod 4 and pk ≡ p mod 3. then by chinese remainder theorem pk ≡ p mod 12. since [b] = [b−1]. we have sfq (γg) = {γg}. hence ni’s and fi’s are same as above. so the decomposition of fqg is given by fqg ∼= f3q ⊕m(2,fq) 3 ⊕m(3,fq). case 2. suppose k is odd with p ≡−1 mod 3 and p ≡±1 mod 4. using the observation in case 1, we have pk ≡ p mod 12. sfq(γb) = {γb},sfq(γb2) = {γb2}, sfq(γa) = {γa,γa−1} and sfq(γab) = {γab,γ(ab)−1}. therefore by using equation (1), proposition 2.3 and theorem 2.4, we have fqg ∼= fq ⊕m(n1,fq)⊕m(n2,fq)⊕m(n3,fq2)⊕m(n4,fq2) for some ni ≥ 1. as dimension of fqg is 24, we get n21 + n 2 2 + 2n 2 3 + 2n 2 4 = 23 and hence, 1 ≤ ni ≤ 3, ∀1 ≤ i ≤ 4. clearly n3 and n4 can not be equal to 3. so the only possible choice for ni’s is n1 = 2,n2 = 3,n3 = 1,n4 = 2. therefore the decomposition of fqg is given by fqg ∼= fq ⊕fq2 ⊕m(2,fq)⊕m(2,fq2)⊕m(3,fq). 4 s. maheshwari, r. k. sharma / j. algebra comb. discrete appl. 3(1) (2016) 1–6 corollary 3.2. let q = pk,where p ≥ 5 is a prime. then the structure of u(fqg) is given by condition on k u(fqg) k is even c3q−1 ⊕gl(2,fq)3 ⊕gl(3,fq) k is odd p ≡ 1 mod 3 and p ≡±1mod 4 c3q−1 ⊕gl(2,fq)3 ⊕gl(3,fq) k is odd p ≡−1 mod 3,±1mod 4 cq−1 ⊕cq2−1 ⊕gl(2,fq)⊕gl(2,fq2)⊕gl(3,fq) proof. it follows by the fact that, if r and s are two rings then u(r⊕s) = u(r)⊕u(s). references [1] l. creedon, j. gildea, the structure of the unit group of the group algebra f2kd8, canad. math. bull. 54(2) (2011) 237–243. [2] r. a. ferraz, simple components of the center of fg/j(fg), comm. algebra 36(9) (2008) 3191–3199. [3] j. gildea, the structure of the unit group of the group algebra fk2a4, czechoslovak math. j. 61(2) (2011) 531–539. [4] j. gildea, f. monaghan, units of some group algebras of groups of order 12 over any finite field of characteristic 3, algebra discrete math. 11(1) (2011) 46–58. [5] p. r. helm, a presentation for sl(2,zpr ), comm. algebra 10(15) (1982) 1683–1688. [6] t. hurley, group rings and ring of matrices, int. j. pure appl. math. 31(3) (2006) 319–335. [7] t. hurley, convolutional codes from units in matrix and group rings, int. j. pure appl. math. 50(3) (2009) 431–463. [8] r. lidl, h. niederreiter, introduction to finite fields and their applications, cambridge university press, new york, 2000. [9] n. makhijani, units in finite group algebras, iit delhi, 2014. [10] n. makhijani, r. k. sharma, j. b. srivastava, a note on units in fpmd2pn, acta math. acad. paedagog. nyházi. 30(1) (2014) 17–25. [11] n. makhijani, r. k. sharma, j. b. srivastava, the unit group of algebra of circulant matrices, int. j. group theory. 3(4) (2014) 13–16. [12] n. makhijani, r. k. sharma, j. b. srivastava, the unit group of fq[d30], serdica math. j. 41(2-3) (2015) 185–198. [13] c. p. milies, s. k. sehgal, an introduction to group rings, kluwer academic publishers, 2002. [14] s. perlis, g. l. walker, abelian group algebras of finite order, trans. amer. math. soc. 68(3) (1950) 420–426. [15] r. k. sharma, j. b. srivastava, m. khan, the unit group of fa4, publ. math. debrecen 71(1-2) (2007) 21–26. [16] r. k. sharma, j. b. srivastava, m. khan, the unit group of fs3, acta math. acad. paedagog. nyházi. 23(2) (2007) 129–142. [17] r. k. sharma, p. yadav, unit group of algebra of circulant matrices, int. j. group theory. 2(4) (2013) 1–6. 5 http://dx.doi.org/10.4153/cmb-2010-098-5 http://dx.doi.org/10.4153/cmb-2010-098-5 http://dx.doi.org/10.1080/00927870802103503 http://dx.doi.org/10.1007/s10587-011-0071-5 http://dx.doi.org/10.1007/s10587-011-0071-5 http://www.ams.org/mathscinet-getitem?mr=2868359 http://www.ams.org/mathscinet-getitem?mr=2868359 http://dx.doi.org/10.1080/00927878208822796 http://www.ams.org/mathscinet-getitem?mr=2266951 http://www.ams.org/mathscinet-getitem?mr=2490664 http://www.ams.org/mathscinet-getitem?mr=2490664 http://www.ams.org/mathscinet-getitem?mr=3285078 http://www.ams.org/mathscinet-getitem?mr=3285078 http://www.ams.org/mathscinet-getitem?mr=3181770 http://www.ams.org/mathscinet-getitem?mr=3181770 http://www.ams.org/mathscinet-getitem?mr=3363601 http://www.ams.org/mathscinet-getitem?mr=3363601 http://dx.doi.org/10.2307/1990406 http://dx.doi.org/10.2307/1990406 http://www.ams.org/mathscinet-getitem?mr=2340031 http://www.ams.org/mathscinet-getitem?mr=2340031 http://www.ams.org/mathscinet-getitem?mr=2368934 http://www.ams.org/mathscinet-getitem?mr=2368934 http://www.ams.org/mathscinet-getitem?mr=3053357 http://www.ams.org/mathscinet-getitem?mr=3053357 s. maheshwari, r. k. sharma / j. algebra comb. discrete appl. 3(1) (2016) 1–6 [18] r. k. sharma, p. yadav, the unit group of zpq8, algebras groups geom. 25(4) (2008) 425–429. [19] g. tang, y. wei, nanning, y. li, units group of group algebras of some small groups, czechoslovak math. j. 64(1) (2014) 149–157. 6 http://www.ams.org/mathscinet-getitem?mr=2526898 http://dx.doi.org/10.1007/s10587-014-0090-0 http://dx.doi.org/10.1007/s10587-014-0090-0 introduction preliminaries main result references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(3) • 211-216 received: 29 may 2015; accepted: 29 august 2015 doi 10.13069/jacodesmath.66269 journal of algebra combinatorics discrete structures and applications some new quasi-twisted ternary linear codes∗ research article rumen daskalov1∗∗, plamen hristov1∗ ∗ ∗ 1. department of mathematics, technical university of gabrovo, bulgaria abstract: let [n, k, d]q code be a linear code of length n, dimension k and minimum hamming distance d over gf(q). one of the basic and most important problems in coding theory is to construct codes with best possible minimum distances. in this paper seven quasi-twisted ternary linear codes are constructed. these codes are new and improve the best known lower bounds on the minimum distance in [6]. 2010 msc: 94b05, 94b65 keywords: ternary linear codes, quasi-twisted codes 1. introduction let gf(q) denote the galois field of q elements and let v (n,q) denote the vector space of all ordered n-tuples over gf(q). a q-ary linear code c of length n and dimension k, or an [n,k]q code, is a kdimensional subspace of v (n,q). an inner product (x,y) of vectors x,y ∈ v (n,q) defines orthogonality in the space two vectors are said to be orthogonal if their inner product is 0. the set of all vectors of v (n,q) orthogonal to all codewords from c is called the orthogonal code c⊥ to c. it is well-known that the code c⊥ is a linear [n,n−k]q code. a k ×n matrix g whose rows form a basis of c is called a generator matrix of c. the number of nonzero coordinates of a vector x ∈ v (n,q) is called its hamming weight wt(x). the hamming distance d(x,y) between two vectors is defined by d(x,y) = wt(x−y). the minimum distance of a linear code c is d(c) = min {d(x,y) | x,y ∈ c,x 6= y} = min {wt(c) | c ∈ c,c 6= 0}. a q-ary linear code of length n, dimension k and minimum distance d is said to be an [n,k,d]q code. let ai denote the number of codewords of c with weight i. the weight distribution of c is the list of numbers ai. the weight distribution a0 = 1, ad = α, . . . , an = γ is expressed as 01dα . . .nγ also. ∗ this work was partially supported by the bulgarian ministry of education and science under contract in tu– gabrovo. ∗∗ e-mail: daskalov@tugab.bg (corresponding author) ∗ ∗ ∗ e-mail: plhristov9@gmail.com 211 qt ternary linear codes if c ⊆ c⊥, then the code c is called self-orthogonal. for a ternary linear code c the next theorem is well-known every codeword of c has weight divisible by three if and only if c is self-orthogonal. a central problem in coding theory is that of optimizing one of the parameters n,k and d for given values of the other two and q-fixed. two are the basic versions: problem 1: find dq(n,k), the largest value of d for which there exists an [n,k,d]q code. problem 2: find nq(k,d), the smallest value of n for which there exists an [n,k,d]q code. a code which achieves one of these two values is called d-optimal or n-optimal respectively. both distance-optimal and length-optimal codes are called optimal codes. the problem of finding the parameters of the optimal codes is very difficult one (see [15], [9]) and has two aspects one is the construction of new codes with better minimum distances and the other is to prove the nonexistence of codes with given parameters. it is entirely solved only for small finite fields and dimensions (see the following table and [10], [2], [6]). q 2 3 4 5,7,8,9 k ≤ 8 5 4 3 many optimal linear codes are constructed when n−k is also small (see [6]). for the first aspect computer search is often used but it is well known fact that computing the minimum distance of a linear code is an np-hard problem [16]. since it is not possible to conduct exhaustive searches for linear codes with large dimension, a natural way is to focus our efforts on subclasses of linear codes, having rich mathematical structures. quasi-twisted (qt) codes are known to have this structure and it has been shown in recent years that this subclass contains many new good linear codes [1,3-8,11-14]. grassl [6] maintains a table with lower and upper bounds on minimum distances over small finite fields. for any n and k there are two numbers in the table dl and du. the former, dl, is the best known minimum distance of an [n,k,d]q code constructed to date, whereas the latter, du, is the theoretical upper bound on the minimum distance of an [n,k]q code. when dl = du = d then the [n,k,d]q code is optimal. many of the best-known codes in grassl’s tables are qt codes. a code that attains a lower bound in the table is called a good code. a code that improves a lower bound in the table we will call a new code. chen also maintains online tables of linear codes. chen’s table [3] contains only good and best-known qc and qt codes (q ≤ 13). these two databases are updated when new codes are discovered. 2. quasi-twisted codes a code c is said to be quasi-twisted if a constacyclic shift of a codeword by p positions results in another codeword. a constacyclic shift of an m-tuple (x0,x1, . . . ,xm−1) is the m-tuple (αxm−1,x0, . . . ,xm−2),α ∈ gf(q) \ {0}. the blocklength, n, of a qt code is a multiple of p, so that n = pm. a matrix b of the form b =   b0 b1 b2 · · · bm−2 bm−1 αbm−1 b0 b1 · · · bm−3 bm−2 αbm−2 αbm−1 b0 · · · bm−4 bm−3 ... ... ... ... ... αb1 αb2 αb3 · · · αbm−1 b0   , (1) where α ∈ gf(q)\{0} is called a twistulant matrix. a class of qt codes can be constructed from m×m twistulant matrices. in this case, the generator matrix, g , can be represented as g = [b1, b2, ... , bp] , (2) 212 r. daskalov, p. hristov where bi is a twistulant matrix [14]. the algebra of m×m twistulant matrices over gf(q) is isomorphic to the algebra of polynomials in the ring gf(q)[x]/(xm−α) if b is mapped onto the polynomial, b(x) = b0 +b1x+b2x2 +· · ·+bm−1xm−1, formed from the entries in the first row of b. the bi(x) associated with a qt code are called the defining polynomials. if the defining polynomials bi(x) contain a common factor which is also a factor of xm −α, then the qt code is called degenerate. the dimension k of the qt code is equal to the degree of h(x), where [14] h(x) = xm −α gcd{xm −α,b1(x),b2(x), · · · ,bp(x)} . (3) if the polynomial h(x) has degree m, the dimension of the code is m, and (2) is a generator matrix. if deg(h(x)) = k < m, a generator matrix for the code can be constructed by deleting m−k rows of (2). let the defining polynomials of the code c be in the next form d1(x) = g(x), d2(x) = f2(x)g(x), · · · , dp(x) = fp(x)g(x), (4) where g(x)|(xm −α),g(x),fi(x) ∈ gf(q)[x]/(xm −α), (fi(x),(xm −α)/g(x)) = 1 and deg fi(x) < m− deg g(x) for all 1 ≤ i ≤ p. then c is a degenerate qt code, which is a one-generator qt code and for this code n = mp, and k = m−deg g(x). a p -qt code over gf(q) of length n = pm can be viewed as a gf(q)[x]/(xm −α) submodule of (gf(q)[x]/(xm−α))p [14]. then an r-generator qt code is spanned by r elements of (gf(q)[x]/(xm− α))p. a well-known result regarding the one-generator qt codes is given in the next theorem. theorem 2.1. [14]: let c be a one-generator qt code over gf(q) of length n = pm. then, a generator g(x) ∈ (gf(q)[x]/(xm −α))p of c has the following form g(x) = (f1(x)g1(x),f2(x)g2(x), · · · ,fp(x)gp(x)) where gi(x)|(xm −1) and (fi(x),(xm −α)/gi(x)) = 1 for all 1 ≤ i ≤ p. in this paper seven new one-generator qt codes (p ≥ 2) are constructed, using an algebraiccombinatorial computer search similar to that in [14]. all constructed codes are self-orthogonal. 3. the new codes we have restricted our search to one-generator qt codes with a generator of the form (4). example 3.1. let q = 3, m = 52 and α = 2. the factorization of the polynomial x52 + 1 over gf(3) is x52 + 1 = 10∏ i=1 pi(x), where p1(x) = x 6 + x5 + 2x4 + 2x3 + x2 + 2x + 2 p6(x) = x 6 + x5 + 2x4 + 2x3 + 2 p2(x) = x 6 + 2x5 + 2x4 + x3 + x2 + x + 2 p7(x) = x 6 + 2x5 + 2x4 + x3 + 2 p3(x) = x 6 + 2x5 + 2x4 + 2x3 + x2 + x + 2 p8(x) = x 6 + 2x3 + x2 + x + 2 p4(x) = x 6 + x5 + 2x4 + x3 + x2 + 2x + 2 p9(x) = x 2 + x + 2 p5(x) = x 6 + x3 + x2 + 2x + 2 p10(x) = x 2 + 2x + 2. 213 qt ternary linear codes let the dimension k be 20. then the degree of the polynomials g(x) have to be 32. there are 2. ( 8 3 ) = 112 possibilities to obtain a polynomial g(x) of degree 32. we check these possibilities consecutively, using non-exhaustive search. when g(x) = x32 + x31 + 2x30 + x28 + x27 + 2x26 + x25 + 2x24 + 2x23 + x22 +x19 + 2x17 + x15 + 2x14 + x13 + x12 + 2x11 + 2x9 + 2x9 + 2x9 + 1 and f2(x) = x 8 + x6 + 2x5 + 2x4 + x3 + 2x2 + x + 2, a good [104,20,45]3 quasi-twisted code is obtained. afterwards, we search for f3(x) and f4(x) in succession. the polynomial f3(x) = x8 + x7 + x6 + x5 + 2x3 + x + 1 yields a new [156,20,75]3 code and the polynomial f4(x) = x8 + 2x2 + x + 1 leads to a new [208,20,105]3 code. we conducted a similar search for other lengths. the obtained new results are presented in the next theorem. theorem 3.2. there exist self-orthogonal one-generator quasi-twisted codes (α = 2) with parameters: [156,14,84]3, [136,18,66]3, [156,18,78]3, [80,20,33]3 [156,20,75]3, [208,20,105]3, [164,24,75]3. proof. the coefficients of the defining polynomials and the weight distributions of the codes are as follows: a [156,14,84]3 code (m = 52,p = 3): 2001001112200021222210122020112222120010000000000000, 1010111122102011212102201120021100000012122100000000, 1011020111220020110122112100022000222102122001000000; 01 844056 8717992 9060112 93173576 96384072 99661024 102904280 105963976 108789672 111484120 114233376 11781648 12020384 1233952 126624 129104 a [136,18,66]3 code (m = 34,p = 4): 1100212000222002100000000000000000,2012120222200201222012010000000000, 1110101101210221012212200010000000,1012120022211202011001001111100000; 01 664420 6956712 72332656 751654168 786230092 8118084804 8439732400 8766373440 9083286876 9377889716 9653790516 9927168584 1029809960 1052509268 108439892 11152496 1144284 117204 a [156,18,78]3 code (m = 52,p = 3): 2112221111101000202210102100010000100000000000000000, 2001021122201101100112210000012012000212211000000000, 1111210012122200102120211111102202002010200100000000; 01 789568 8152104 84304304 871384448 904931680 9313906776 9630902040 9953999712 10273411312 10577915448 10863630008 11139595816 11418649384 1176627296 1201732224 123321048 12643056 1294160 132104 a [80,20,33]3 code (m = 40,p = 2): 2001100012102210110010000000000000000000, 1202220122210221220201202022102010000000; 01 3318960 36359360 394063760 4230350480 45144118432 48436419120 51831804960 54979844960 57696755760 60288531648 6366176080 667894000 69437440 729440 214 r. daskalov, p. hristov a [156,20,75]3 code (m = 52,p = 3): 1000100110021121020100122121102110000000000000000000, 1221120201102202120000202210021101112110000000000000, 1102121020011222010112010002201010101012110000000000; 01 758216 7870200 81490568 842755480 8712400856 9044366920 93125212568 96278102448 99485583176 102661878672 105700633856 108572121472 111356398640 114168530440 11759392328 12015466360 1232948400 126387088 12933904 1322704 135104 a [208,20,105]3 code (m = 52,p = 4): 1000100110021121020100122121102110000000000000000000, 2121100002110110121022011010211200121001100000000000, 1102121102022101002111201220111021001112100000000000, 1120112100221020012120010010002221111021100000000000; 01 1056968 10836816 111225368 1141084720 1174490824 12015554344 12344887960 126109041296 129221353392 132374073232 135524057352 138608355904 141582839608 144458328208 147294288384 150153923432 15364968592 15621913224 1595855616 1621267136 165203736 16825792 1712392 174104 a [164,24,75]3 code (m = 82,p = 2): 1121222111221010212020202121202222202020222010112212212222100000000000000000000000, 2011121210102200210211202120010122002121000011212011011012000102010000000000000000; 01 7515744 78172200 811673128 8412794460 8779656112 90396419160 931584181452 965070823256 9912959375940 10226376357636 10542571391912 10854230847436 11154217062252 11442255118404 11725471036636 12011763022344 1234119948304 1261079104584 129208391028 13229038660 1352892960 138201720 14110988 144164 from the constructed codes, by trivial constructions as shortening, puncturing and extension, 29 improvements on [6] are obtained. for example from [80,20,33]3 code it follows that there exist [79,19,33]3, [78,19,32]3, [77,19,31]3, [79,20,32]3, [78,20,31]3 and [81,20,33]3 codes. references [1] r. ackerman and n. aydin, new quinary linear codes from quasi-twisted codes and their duals, appl. math. lett., 24(4), 512–515, 2011. [2] s. ball, three-dimensional linear codes, online table, http://www-ma4.upc.edu/∼simeon/. [3] e. z. chen, database of quasi-twisted codes, available at http://moodle.tec.hkr.se/ chen/research/codes/searchqt.htm [4] e. z. chen, a new iterative computer search algorithm for good quasi-twisted codes, des. codes cryptogr, 76(2), 307-323, 2014. [5] r. daskalov and p. hristov, new quasi-twisted degenerate ternary linear codes, ieee trans. inform. theory, 49(9), 2259–2263, 2003. [6] m. grassl, linear code bound, [electronic table; online], http://www.codetables.de. [7] p. p. greenough and r. hill, optimal ternary quasi-cyclic codes, des. codes cryptogr., 2(1), 81–91, 1992. [8] t. a. gulliver and p. r. j. ostergard, improved bounds for ternary linear codes of dimension 7, ieee trans. inform. theory, 43, 1377–1388, 1997. [9] r. hill, a first course in coding theory, oxford applied mathematics and computing sciences series, 1992. [10] t. maruta, griesmer bound for linear codes over finite fields, online table, http://www.mi.s.osakafuu.ac.jp/~maruta/griesmer.htm. [11] t. maruta, m. shinohara and m. takenaka, constructing linear codes from some orbits of projectivities, discrete math., 308(5-6), 832–841, 2008. 215 ~ qt ternary linear codes [12] e. metodieva and n. daskalova, generating generalized necklaces and new quasi-cyclic codes, problemi peredachi informatsii, (submitted). [13] i. siap, n. aydin and d. ray-chaudhury, new ternary quasi-cyclic codes with better minimum distances, ieee trans. inform. theory, 46(4), 1554–1558, 2000. [14] i. siap, n. aydin and d. ray-chaudhury, the structure of 1-generator quasi-twisted codes and new linear codes, des. codes cryptogr., 24, 313–326, 2001. [15] s. dougherty, j. kim and p. solé, open problems in coding theory, contemporary mathematics, 634, http://dx.doi.org/10.1090/conm/634/12692, 2015. [16] a. vardy, the intractability of computing the minimum distance of a code, ieee trans. inform. theory, 43, 1757–1766, 1997. 216 introduction quasi-twisted codes the new codes references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1112177 j. algebra comb. discrete appl. 9(2) • 123–131 received: 8 october 2021 accepted: 5 january 2022 journal of algebra combinatorics discrete structures and applications the bicyclic semigroup as the quotient inverse semigroup by any gauge inverse submonoid research article emil daniel schwab abstract: every gauge inverse submonoid (including jones-lawson’s gauge inverse submonoid of the polycyclic monoid pn) is a normal submonoid. in 2018, alyamani and gilbert introduced an equivalence relation on an inverse semigroup associated to a normal inverse subsemigroup. the corresponding quotient set leads to an ordered groupoid. in this note we shall show that this ordered groupoid is inductive if the normal inverse subsemigroup is a gauge inverse submonoid and the corresponding quotient inverse semigroup by any guage inverse submonoid is isomorphic either to the bicyclic semigroup or to the bicyclic semigroup with adjoined zero. 2010 msc: 20m18, 20l05 keywords: inverse semigroup, ordered groupoid, gauge inverse submonoid, bicyclic semigroup 1. introduction an equivalence relation 'n on an inverse semigroup s associated to a normal inverse subsemigroup n is introduced in [1]. usually, it is not a congruence on s. following [1] the quotient set s/ 'n (also denoted by s//n) leads to an ordered groupoid [1, theorem 3.6]. if this ordered groupoid is inductive then the set of all morphisms, that is s//n, equipped with the "pseudoproduct" ⊗ ([3, page 112]) forms an inverse semigroup (see [3, proposition 4.1.7 (1)]), and we say, by abuse of language (since 'n is not necessary a congruence), that this inverse semigroup (s//n,⊗) is the quotient inverse semigroup of s by the normal inverse subsemigroup n. the gauge inverse monoid gm is a special submonoid of such a combinatorial bisimple (0-bisimple) inverse monoid s(m) for which the submonoid m of right units is an `-rill monoid (see [5]). any gauge inverse submonoid is normal ([5, proposition 5.6]). jones-lawson’s gauge inverse monoid is the gauge inverse submonoid (denoted by gn) of the polycyclic monoid pn ([2, section 3]). emil daniel schwab; department of mathematical sciences, university of texas at el paso, 500 w. university ave, el paso, texas 79968-0514, usa (email: eschwab@utep.edu). 123 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 the case of the polycyclic monoid pn is examined in example 3.11 from [1]. the conclusion of this examination is that pn//gn is isomorphic to the brandt semigroup on the set of non-negative integers. in fact the product ”[(u,v)]gn[(s,t)]gn = [(u,t)]gn” considered at the end of section 3 in [1] is the composition of two morphisms (if it is defined) in the corresponding ordered groupoid and it is not the pseudoproduct ⊗ which defines the quotient inverse semigroup pn//gn. the aim of this note is to show that for any gauge inverse submonoid gm, the quotient inverse semigroup (s(m)//gm,⊗) is isomorphic either to the bicyclic semigroup b or to the bicyclic semigroup with adjoined zero b0. in the next section, we will survey the background results, particularly from [3] (subsection 2.1), [1] (subsection 2.2) and [5] (subsection 2.3), needed to understand this paper. the symbol ◦ is used only for composition (from right to left) of two morphisms. 2. background. ordered groupoids, normal inverse subsemigroups and gauge inverse submonoids 2.1. ordered groupoids a groupoid g is a small category in which every morphism is an isomorphism, meaning that for any morphism f : x → y there is a morphism f−1 : y → x such that f−1 ◦ f = 1x and f ◦ f−1 = 1y , where 1x and 1y are the identity morphisms of x and y , respectively. a groupoid gx is said to be connected simple system on the set x (or simplicial groupoid on x) if the set of objects obgx = x and there is exactly one morphism between any two objects. we call the groupoid g0x obtained from gx by adjoining an extra object 0 such that the set of morphisms from x to y is empty if either x = 0,y 6= 0 or x 6= 0,y = 0 and it is a singleton if x = y = 0, the connected simple system with adjoined 0. a groupoid g is said to be ordered if the set of all morphisms mor(g) of g is equipped with a partial order � such that: (o1) f � g implies f−1 � g−1; (o2) if f � g, f′ � g′ and f ◦f′ and g ◦g′ are defined then f ◦f′ � g ◦g′; (o3) if 1z � 1x and f : x → y then there exists a unique morphism f|z : z →• called the restriction of f to z such that f|z � f; (o4) if 1z � 1y and f : x → y then there exists a unique morphism f|z : •→ z called the corestriction of f to z such that f|z � f; the axiom (o4) is a consequence of axioms (o1)− (o3). an inverse semigroup s (i.e. a semigroup s in which every element s ∈ s has a unique inverse s−1 ∈ s in the sense that s = ss−1s and s−1 = s−1ss−1) can be considered as an ordered groupoid g(s) in which the set of objects is the set of idempotents e(s) of s, the set of morphisms from e to f is the set {s ∈ s|s−1s = e and ss−1 = f} and the composition s◦ t of two morphisms s and t t−1t t−→ tt−1 = s−1s s−→ ss−1 is the usual product st in s (i.e., the composition is just the restriction of the multiplication of s to composable pairs). the partial order on the set of all morphisms of g(s) is the natural partial order ≤ on the inverse semigroup s, i.e. s ≤ t ⇔ s = ss−1t (or equivalently s = ts−1s). in the ordered groupoid g(s) the partially ordered set of identities forms a meet-semilattice. if s is the brandt semigroup bω whose set of elelements is {(m,n) | m,n ∈ ω = {0,1,2, · · ·}}∪{0} with the multiplication defined by: (m,n) · (m′,n′) = { (m,n′) if n = m′ 0 if n 6= m′ and 0 · (m,n) = (m,n) ·0 = 0 ·0 = 0, 124 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 then g(bω) is category isomorphic to the connected simple system with adjoined 0: g0ω. but g(bω) is an ordered groupoid and the order ≤bω on g(bω) (that is the natural partial order on bω) induces a partial order ≤bω on mor(g0ω) given by: 10 ≤bω f for all f ∈ mor(g0ω), and f ≤bω g iff f = g, otherwise. note that g0ω (and gω) can be equipped as an ordered groupoid in many other way. now, an ordered groupoid in which the set of identities forms a meet-semilattice (like in the case of the ordered groupoids g(s)) is called inductive. if f : x → y and f′ : x′ → y ′ are two morphisms of an inductive groupoid g and 1x ∧1y ′ = 1z then the pseudoproduct ⊗: f ⊗f′ = f|z ◦f′|z defines a binary operation on the set mor(g) such that (mor(g),⊗) is an inverse semigroup ([3, proposition 4.1.7 (1)]). note that if we denote this semigroup by s(g), then s(g(s)) = s ([3, proposition 4.1.7 (3)]), g(s(g,�)) = (g,�) ([3, proposition 4.1.7 (2)]), and s(g0ω,≤) ∼= bω only if ≤ is the induced order ≤bω on mor(g0ω) considered above. 2.2. normal inverse subsemigroup and the corresponding ordered groupoid an inverse subsemigroup n of an inverse semigroup s is called normal if e(s) = e(n) and if s−1ns ⊆ n for all s ∈ s. a normal inverse subsemigroup n of an inverse semigroup s together with the defining concepts (≤ and ◦) of the ordered groupoid g(s) determine a preorder ≤n on s = morg(s), as follows: s ≤n t ⇔ there exist two morphisms a,b of g(s) such that a,b ∈ n, the compositions a ◦ s and s ◦ b are both defined, and a◦s◦ b ≤ t. since ≤n is a preorder on the set s then it defines an equivalence relation 'n on s by s 'n t ⇔ s ≤n t and t ≤n s, and a partial order on the set of equivalence classes s/ 'n. in [1] this quotient set is denoted by s//n and the 'nclass of s ∈ s by [s]n. the equivalence relation 'n needs not be a congruence on s. however, the quotient set s//n leads us to an ordered groupoid g(s//n): the objects are the classes [e]n where e ∈ e(s), and mor(g(s//n) = s//n with [s]n being a morphism from [s−1s]n to [ss−1]n. the composition of two morphisms [s]n ◦ [t]n (if [s−1s]n = [tt−1]n) is given by [s]n ◦ [t]n = [sat]n, where a ∈ n such that a−1a = tt−1 and aa−1 ≤ s−1s; and [s]n �n [t]n ⇔ s ≤n t, is the partial order of g(s//n). now, if this ordered groupoid g(s//n) is inductive then s//n = mor(g(s//n)) forms an inverse semigroup (s//n,⊗) (where ⊗ is the pseudoproduct) called here the quotient inverse semigroup of s by the normal inverse subsemigroup n. 2.3. gauge inverse submonoids following [5], a nontrivial right cancellative monoid m is a rill monoid if 1m is indecomposable and any two elements s,t ∈ m that admit a common left multiple admit a least common left multiple s ∨ t. in the rill monoid m, we shall denote s � t if t is a left multiple of s, t = rs, and by tbs the "left quotient" r. since m is right cancellative and 1 is indecomposable, the "right divisibility" relation � is a partial order on m. a length function on the rill monoid m is a monoid homomorphism ` : m → (n,+) such that `−1(0) = 1m. a non-trivial monoid with a length function is atomic (every non-units element is a product of finitely many atoms). a length function ` is said to be normalized if `(s) = 1 ⇔ s is an atom. an `-rill monoid is a rill monoid equipped with a normalized length function `. if m is an `-rill monoid then the set s(m) = { m ×m if ms∩mt 6= ∅ for any s,t ∈ m (m ×m)∪{θ} if there exist s,t ∈ m such that ms∩mt = ∅ 125 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 (that is m ×m, adjoining an extra element θ if necessary), together with the product � defined by (s,t)� (s′, t′) = { (t∨s ′ bt s, t∨s′ bs′ t ′) if t and s′ admit a common left multiple θ otherwise and θ � (s,t) = (s,t)�θ = θ �θ = θ (if necessary), is an inverse monoid (the inverse of (s,t) is (t,s); the element (s,t) is an idempotent if and only if s = t, and (1m,1m) is the identity element). the submonoid of s(m): gm = { {(s,t) ∈ m ×m| `(s) = `(t)} if s(m) = m ×m {(s,t) ∈ m ×m| `(s) = `(t)}∪{θ} if s(m) = (m ×m)∪{θ} is the gauge inverse submonoid of s(m) induced by the `-rill monoid m. this submonoid of s(m) is a normal submonoid ([5, proposition 5.6]). in [5] the first example of a gauge inverse submonoid is the submonoid of idempotents e(b) of the bicyclic semigroup b. the bicyclic semigroup b is the monoid of all pairs of non-negative integers equipped with the multiplication defined by: (m,n) · (m′,n′) = { (m,n−m′ + n′) if n ≥ m′ (m−n + m′,n′) if n ≤ m′. in this paper (b0, ·) denotes the bicyclic semigroup with adjoined zero 0. 3. main results. the quotient inverse monoid s(m)//gm let m be an `-rill monoid and (s(m),�) the corresponding inverse monoid. proposition 3.1. the natural partial order ≤, the preorder ≤gm and the equivalence relation 'gm on s(m) are given by: (i) (s,t) ≤ (s′, t′) ⇔ s′ � s, t′ � t and sbs′ = t bt′ ([4, proposition 2.6 (1)]) (θ ≤ x for any x ∈ s(m) if s(m) = (m ×m)∪{θ}); (ii) (s,t) ≤gm (s′, t′) ⇔ there exists (p,q) ∈ s(m) such that `(p) = `(s), `(q) = `(t) and (p,q) ≤ (s′, t′) (θ ≤gm x for any x ∈ s(m) if s(m) = (m ×m)∪{θ}); (iii) (s,t) 'gm (s′, t′) ⇔ `(s) = `(s′) and `(t) = `(t′) (if s(m) = (m ×m)∪{θ} then the 'gm -class [θ]gm is a singleton). proof. (i). we have (s,t) ≤ (s′, t′) ⇔ (s,t) = (s,t)� (t,s)� (s′, t′) ⇔ (s,t) = (s,s)� (s′, t′) ⇔ (s,t) = ( s∨s′ bs s, s∨s′ bs′ t′) ⇔ (s,t) = (s∨s′, s∨s′ bs′ t′) ⇔ s′ � s and s bs′ t′ = t ⇔ s′ � s, t′ � t and s bs′ = t bt′ . 126 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 (ii). we have (s,t) ≤gm (s ′, t′) ⇔ there exist (p,u),(v,q) ∈ gm such that (p,u)−1 � (p,u) = (s,t)� (s,t)−1, (s,t)−1 � (s,t) = (v,q)� (v,q)−1 and (p,u)� (s,t)� (v,q) ≤ (s′, t′). since (p,u)−1 � (p,u) = (u,u) and (s,t)� (s,t)−1 = (s,s), it follows u = s. analogously, v = t. now, we have: (p,u)� (s,t)� (v,q) = (p,s)� (s,t)� (t,q) = (p,q) and taking into account that (p,s),(t,q) ∈ gm we obtain: (s,t) ≤gm (s ′, t′) ⇔ there exist p,q ∈ m such that `(p) = `(s), `(q) = `(t) and (p,q) ≤ (s′, t′). (iii). the assertion follows from (i) and (ii). remark 3.2. the equivalence relation 'gm is not necessarily a congruence on s(m). for example, if m is the multiplicative `-rill monoid of positive integers (z+, ·) ([5, example 4.2]), where `(1) = 0 and `(n) =the total number of prime divisors of n counted with their multiplicities if n > 1, then s(z+) is the multiplicative analogue of the bicyclic semigroup: s(z+) = z+ ×z+; (m,n) · (m′,n′) = ( [n,m′] n m, [n,m′] m′ n′), [n,m′] being the least common multiple of n and m′. now, if p and q are two distinct primes then (p,q) 'gm (p,q) and (p,q) 'gm (q,p) (since `(p) = `(q) = 1), but (p,q) · (p,q) = (p2,q2) and (p,q) · (q,p) = (p,p), that is (p,q) ·(p,q) 6'gm (p,q) ·(q,p). thus 'gm is not a congruence on the multiplicative analogue of the bicyclic semigroup. the 'gm -class [(s,t)]gm = {(u,v) ∈ s(m)| `(u) = `(s) and `(v) = `(t)} is a morphism in the ordered groupoid g(s(m)//gm) from [(t,t)]gm to [(s,s)]gm . if [(s,t)]gm and [(s′, t′)]gm are two morphisms of g(s(m)//gm) such that `(s′) = `(t) (that is [(s′,s′)]gm = [(t,t)]gm ), [(t′, t′)]gm [(s′,t′)]gm−→ [(s′,s′)]gm = [(t,t)]gm [(s,t)]gm−→ [(s,s)]gm , then the composition of these two morphisms, [(s,t)]gm ◦ [(s′, t′)]gm is given by [(s,t)]gm ◦ [(s ′, t′)]gm = [(s,t)� (a,b)� (s ′, t′)]gm , where (a,b) ∈ gm such that (a,b)−1 � (a,b) = (s′, t′) � (s′, t′)−1 and (a,b) � (a,b)−1 ≤ (s,t)−1 � (s,t). we choose (a,b) = (t,s′) which is an element of gm since `(t) = `(s′). thus the composition [(s,t)]gm ◦ [(s′, t′)]gm in g(s(m)//gm) such that `(t) = `(s′) is given by: [(s,t)]gm ◦ [(s ′, t′)]gm = [(s,t)� (t,s ′)� (s′, t′)]gm = [(s,t ′)]gm . 127 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 the ordering �gm of 'gm -classes in the ordered groupoid g(s(m)//gm) is given by: [(s,t)]gm �gm [(s ′, t′)]gm ⇔ there exists (p,q) ∈ [(s,t)]gm such that s′ � p, t′ � q and p bs′ = q bt′ . and [θ]gm �gm [x]gm for any morphism [x]gm of g(s(m)//gm) if s(m) = (m ×m)∪{θ}. remark 3.3. the objects of g(s(m)//gm) other than [θ]gm (that is the 'gm -classes [(s,s)]gm) can be indexed by non-negative integers (namely [(s,s)]gm by `(s)), then the set of morphisms from m to n is a singleton (for any pair (m,n) of non-negative integers) and, it goes without saying the composition of two morphisms. it follows: theorem 3.4. the (ordered) groupoid g(s(m)//gm) is category isomorphic either to the connected simple system gn (if s(m) = m ×m) or to the connected simple system with adjoined 0: g0n (if s(m) = m ×m ∪{θ}). theorem 3.5. the ordered groupoid g(s(m)//gm) is inductive. proof. it is straightforward to see that in the set of identities of g(s(m)//gm) we have: [(s,s)]gm �gm [(t,t)]gm ⇔ `(t) ≤ `(s). it follows that the partially ordered set of identities of g(s(m)//gm) forms a meet-semilattice: [(s,s)]gm ∧ [(t,t)]gm = { [(s,s)]gm if `(s) ≥ `(t) [(t,t)]gm if `(s) ≤ `(t) and [θ]gm ∧ [x]gm = [θ]gm for any identity morphism [x]gm of g(s(m)//gm) if s(m) = (m ×m)∪{θ}. therefore the ordered groupoid g(s(m)//gm) is inductive. theorem 3.6. the corresponding inverse semigroup (s(m)//gm,⊗) is isomorphic either to the bicyclic semigroup (b, ·) (if s(m) = m ×m) or to the bicyclic semigroup with adjoined zero (b0, ·) (if s(m) = (m ×m)∪{θ}). proof. let [(s,t)]gm , [(s ′, t′)]gm ∈ s(m)//gm. as morphisms of g(s(m)//gm), we have: [(s,t)]gm : [(t,t)]gm → [s,s]gm and [(s ′, t′)]gm : [(t ′, t′)]gm → [s ′,s′]gm . since [(t,t)]gm ∧ [(s ′,s′)]gm = { [(t,t)]gm if `(t) ≥ `(s′) [(s′,s′)]gm if `(t) ≤ `(s′). we shall consider two cases: 128 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 1) [(t,t)]gm ∧[(s′,s′)]gm = [(t,t)]gm . then the restriction [(s,t)]gm |[(t,t)]gm of [(s,t)]gm to [(t,t)]gm is just [(s,t)]gm . the corestriction [(s ′, t′)]gm | [(t,t)]gm of [(s′, t′)]gm to [(t,t)]gm is the morphism [(t,y)]gm : [y,y]gm → [t,t]gm , where y ∈ m such that `(y) = `(t)− `(s′) + `(t′), since [(t,y)]gm �gm [(s′, t′)]gm . in this case, [(s,t)]gm ⊗ [(s ′, t′)]gm = [(s,t)]gm |[(t,t)]gm ◦ [(s ′, t′)]gm|[(t,t)]gm = [(s,t)]gm ◦ [(t,y)]gm = [(s,y)]gm . 2) [(t,t)]gm ∧ [(s′,s′)]gm = [(s′,s′)]gm . then the restriction [(s,t)]gm |[(s′,s′)]gm of [(s,t)]gm to [(s′,s′)]gm is the morphism [(x,s ′)]gm : [(s ′,s′)]gm → [(x,x)]gm , where x ∈ m such that `(x) = `(s′)− `(t) + `(s) since [x,s′]gm �gm [(s,t)]gm . the corestriction [(s′, t′)]gm | [(s′,s′)]gm of [(s′, t′)]gm to [(s ′,s′)]gm is just [(s′, t′)]gm . so, in this case, the product [(s,t)]gm ⊗ [(s′, t′)]gm is given by: [(s,t)]gm ⊗ [(s ′, t′)]gm = [(s,t)]gm |[(s′,s′)]gm ◦ [(s ′, t′)]gm|[(s ′,s′)]gm = [(x,s′)]gm ◦ [(s ′, t′)]gm = [(x,t ′)]gm . (if s(m) = (m × m) ∪ {θ}) then it is straightforward to check that [θ]gm is the zero element of (s(m)//gm,⊗).) now, a careful examination shows that ` : (s(m)//gm,⊗) → (b, ·) if s(m) = m ×m (` : (s(m)//gm,⊗) → (b0, ·) if s(m) = (m ×m)∪{θ}) defined by `([(s,t)]gm ) = (`(s),`(t)) (and `([θ]gm ) = 0 if s(m) = (m ×m)∪{θ}) is a monoid isomorphism. remark 3.7. what is happening if the `-rill monoid m is the additive monoid of non-negative integers ? (that is if the monoid (s(m),�) is the bicyclic semigroup b ?) the gauge inverse submonoid of b is the semilattice of idempotents e(b) ([5, example 4.1]). it is straightforward to check that 'e(b) is the trivial relation (the equality) on b and of course b//e(b) = b (and g(b//e(b)) = g(b)). now, since for any inverse semigroup s the relation 'e(s) is the trivial relation on s ([1, proposition 3.4 (g)]), it follows that corollary 3.8. the bicyclic semigroup is the only combinatorial bisimple inverse monoid for which the gauge inverse submonoid is the semilattice of idempotents. remark 3.9. the ordered groupoid g(s(m)//gm) is isomorphic either to the ordered groupoid g(b) or to the ordered groupoid g(b0). of course, the groupoids g(pn//gn) and g(bω) are also isomorphic as two categories (since both are category isomorphic to the connected simple system with adjoined 0: g0ω), but they are not isomorphic as two ordered groupoids due to the two partial orders �gn and ≤bω on g(pn//gn) and g(bω), respectively. 129 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 4. supplements. the quotient group s(m)/gm if ρ is a relation on an inverse semigroup s, the kernel kerρ is the set kerρ = {s ∈ s | sρe for some e ∈ e(s)}. if ρ is a group congruence on s then we agree to write s/kerρ for the quotient group s/ρ. in what follows assume that s(m) = m ×m (that is, ms∩mt 6= ∅ for any s,t ∈ m). we have: proposition 4.1. the relation ≈m on s(m) defined by (x,y) ≈m (x′,y′) if and only if `(x)− `(y) = `(x′)− `(y′), is a group congruence on s(m). the gauge inverse submonoid gm is the kernel of ≈m, and it is the identity element of the quotient group s(m)/gm (= s(m)/ ≈m). this quotient group is isomorphic to the additive group of integers (z,+). proof. the relation ≈m is an equivalence relation on s(m). obviously, gm is the kernel of ≈m . if (s,t) ≈m (s′, t′) and (u,v) ≈m (u′,v′), then (s,t) � (u,v) = (t∨u.t s, t∨u .u v), (s′, t′) � (u′,v′) = (t ′∨u′ .t′ s′, t ′∨u′ .u′ v′) and `( t∨u .t s)− `( t∨u .u v) = `(t∨u)− `(t) + `(s)− (`(t∨u)− `(u) + `(v)) = `(s)− `(t) + `(u)− `(v) = `(s′)− `(t′) + `(u′)− `(v′) = `( t′ ∨u′ .t′ s′)− `( t′ ∨u′ .u′ v′). it follows that ≈m is a congruence relation on s(m). the quotient monoid s(m)/ ≈m is again an inverse monoid. since gm is the only idempotent of s(m)/ ≈m it follows that this inverse monoid is a group (the quotient group s(m)/gm). the map ` : s(m)/gm → z defined by ([x,y]≈m ∈ s(m)/ ≈m) `([x,y]≈m ) = `(x)− `(y) is an isomorphism from the group s(m)/gm onto the additive group of integers (z,+). remark 4.2. it is straightforward to see that the kernel of 'gm is also the gauge inverse submonoid gm. however, the differences between the relations 'gm and ≈m are significant: (a) in general, the equivalence relation 'gm is not a congruence on s(m) (remark 3.2), but ≈m is a group congruence on s(m); (b) the gauge inverse submonoid gm is not a 'gm -equivalence class in s(m), but it is an ≈mequivalence class in s(m); (c) there is not a 'gm -equivalence class [(s,t)]gm such that e(s(m)) ⊆ [(s,t)]gm , but the ≈mequivalence class gm contains the set of all idempotents of s(m); (d) the group s(m)/gm is equipped with the product � via the inverse monoid s(m); the product in the inverse monoid s(m)//gm is the pseudoproduct ⊗ via the inductive groupoid g(s(m)//gm); (e) the following inclusion holds: 'gm ⊂ ≈m . acknowledgment: the author would like to thank the referee for helpful suggestions. 130 e. d. schwab / j. algebra comb. discrete appl. 9(2) (2022) 123–131 references [1] n. alyamani, n. d. gilbert, ordered groupoid quotients and congruences on inverse semigroups, semigroup forum 96 (2018) 506–522. [2] d. g. jones, m. v. lawson, strong representations of the polycyclic inverse monoids: cycles and atoms, period. math. hung. 64 (2012) 54–87. [3] m. v. lawson, inverse semigroups: the theory of partial symmetries, world scientific, singapore (1998). [4] e. d. schwab, möbius monoids and their connection to inverse monoids, semigroup forum 90 (2015) 694–720. [5] e. d. schwab, gauge inverse monoids, algebra colloq. 27(2) (2020) 181–192. 131 https://doi.org/10.1007/s00233-017-9891-4 https://doi.org/10.1007/s00233-017-9891-4 https://doi.org/10.1007/s10998-012-9053-0 https://doi.org/10.1007/s10998-012-9053-0 https://doi.org/10.1142/3645 https://doi.org/10.1142/3645 https://doi.org/10.1007/s00233-014-9666-0 https://doi.org/10.1007/s00233-014-9666-0 https://doi.org/10.1142/s1005386720000152 introduction background. ordered groupoids, normal inverse subsemigroups and gauge inverse submonoids main results. the quotient inverse monoid s(m)//gm supplements. the quotient group s(m)/gm references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1111746 j. algebra comb. discrete appl. 9(2) • 115–122 received: 18 december 2021 accepted: 8 january 2022 journal of algebra combinatorics discrete structures and applications on the isomorphism of unitary subgroups of noncommutative group algebras∗ research article zsolt adam balogh abstract: let fg be the group algebra of a finite p-group g over a field f of characteristic p. let ~ be an involution of the group algebra fg which arises form the group basis g. the upper bound for the number of non-isomorphic ~-unitary subgroups is the number of conjugacy classes of the automorphism group g with all the elements of order two. the upper bound is not always reached in the case when g is an abelian group, but for non-abelian case the question is open. in this paper we present a non-abelian p-group g whose group algebra fg has sharply less number of non-isomorphic ~-unitary subgroups than the given upper bound. 2010 msc: 16s34, 16u60 keywords: group ring, group of units, unitary subgroup 1. introduction let fg be the group algebra of the group g over a field f. let ~ be an involution of the group algebra fg. we say that the algebra involution ~ arises from the group g when ~ is an antiautomorphism on g. this antiautomorphism of g may also be called involution (for more details see in [19]). in this case the algebra involution ~ is the linear extension of the group involution ~ defined on g. a group algebra is always an algebra with involution, because the canonical ∗-involution of fg (the linear extension of the involution on g which sends each element of g to its inverse) exists for every f and g. the canonical involution ∗ on fg is a simple example of an algebra involution that arises from the group basis g. let v (fg) denote the normalized unit group of fg, that is, the subgroup of the unit group of fg containing all units with augmentation 1. an element u ∈ v (fg) is called ~-unitary if u−1 = u~. the set of all ~-unitary units of fg forms a subgroup of v (fg), which is called ~-unitary subgroup and is denoted by v~(fg). interest in the unitary subgroups arose in algebraic topology and unitary k-theory ∗ this work was supported by uaeu research start-up grant no. g00002968. zsolt adam balogh; department of mathematical sciences, united arab emirates university, united arab emirates (baloghzsa@gmail.com). 115 https://orcid.org/0000-0001-5908-3443 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 introduced by novikov [20]. the ∗-unitary subgroup is an actively investigated subgroup and it plays an important role of studying the structure of v (fg) for more details we refere the reader to bovdi’s paper [9]). let l be a finite galois extension of f with galois group g, where f is a finite field of characteristic two. a relation between the self-dual normal basis of l over f and the ∗-unitary subgroup of fg was discovered by serre [21]. it was shown in [2] that the ∗-unitary subgroup of a group algebra determines the group basis g when it is a finite abelian p-group and f is a finite field of characteristic p. the structure of the unitary subgroups was studied in several papers (see [3], [4], [5], [12], [14], [15], [16], [17] and [22]). let f be a field of characteristic p and g a nonabelian locally finite p-group. the groups g when v∗(fg) is normal in v (fg) are listed in [12]. bovdi and szakács [10] described the structure of the group v∗(fg) when g is a finite abelian p-group and f is a finite field of characteristic p. they also constructed a basis for v∗(fg) in [11]. the order of the unitary subgroup v∗(fg) is determined for finite p-groups and finite fields of characteristic p, if p is an odd prime (see in [13]). the order of v∗(fg) when p = 2 is an open question. it was determined only for some group classes (see in [1], [8] and [13]). the structure of v∗(f2g), where g is a 2-group of maximal class of order 8 or 16 and f2 is the field of two elements has been established in [6]. additionally, the structures of v∗(fq8) and v∗(fd8) are established in [16] and [18] respectively, where f is a finite field of characteristic 2, q8 is the quaternion group of order 8 and d8 is the dihedral group of order 8. in the case when f is a homomorphism of g into the multiplicative group of the commutative ring k all the groups g whose f-unitary subgroup coincides with the unit group of kg are established in [9]. in [8] the invariants of the ~-unitary subgroup of fg are presented, when g is a finite abelian p-group, f is a field of p elements (p is an odd prime) and ~ is an involutory automorphism of g. in [3] an upper bound for the non-isomorphic ~-unitary subgroups is given, when ~ arises from g. the upper bound coincides the number of conjugacy classes of the automorphism group g with all the elements of order two including the identity map. in the case, when g is an abelian p-group the upper bound is not always sharp. a counterexample can be found in [4]. for non-abelian groups this question is open. in this paper we gave an example for a non-abelian p-group whose group algebra fg has less non-isomorphic ~-unitary subgroups than the given upper bound. 2. involutions and unitary subgroups let f be a finite field and g is either the dihedral group of order 8 or the quaternion group of order 8. in this section we show that the number of non-isomorphic ~-unitary subgroups of fg with respect to the involutions which arise from g is equals to the upper bound mentioned in the introduction. let aut g{2} be the set of all automorphism of g with the identity map. the composition of two antiautomorphisms ~ and ∗ of the group g is an automorphism of order two. therefore, ~ can be considered as a composition of an automorphism of order two and the canonical involution, that is, ~ = φ◦∗, where φ ∈ aut g{2}. we say that the involutions ~1 = φ1 ◦∗ and ~2 = φ2 ◦∗ are similar if φ1 is conjugate to φ2 in aut g. we need the following lemma. lemma 2.1. [3, proposition 7] let g be a group and f a field and let ~1 and ~2 be involutions of fg which arise from g. if ~1 is similar to ~2, then v~1 (fg) ∼= v~2 (fg). let g be a finite group and let λ2 denote the number of all distinct conjugacy classes of aut g{2}. as a consequence of the previous lemma we have the following corollary. corollary 2.2. [3, corollary 8] let g be a finite group and f a field. the number of non-isomorphic unitary subgroups of v (fg) with respect to the involutions which arise from g is at most λ2. 116 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 in this section we show that the upper bound λ2 is sharp for all the non-abelian groups of order 8. moreover, we establish the structure of all non-isomorphic ~-unitary subgroups for these groups. first, let us consider the dihedral group d8 of order 8. it is well known that d8 ∼= aut d8 and aut g{2} is the union of four distinct conjugacy classes, that is, λ2 = 4. throughout this section we will use lemma 2.4 in [1] free. lemma 2.3. the number of non-isomorphic unitary subgroups of fd8 with respect to the involutions which arise from d8 is equals to λ2, where |f| = 2n ≥ 2. proof. it was shown in [18] that v∗(fd8) ∼= c25n o c2n. according to lemma 2.1 it is enough to establish the structure of v~(fd8) when the involution ~ links to different conjugacy classes in aut g{2}. then cσ1 = {σ1}, cσ2, cσ3 and cσ4 are the distinct conjugacy classes of aut g{2}, where σ1 is the identity map and σ2 : { a 7→ a3 b 7→ ab σ3 : { a 7→ a3 b 7→ a2b σ4 : { a 7→ a b 7→ a2b . case σ2. let α = ∑3 i=0 a i(αi + βib) ∈ fd8 where αi,βj ∈ f. then α is ~-unitary if and only if αα~ = 1. a straightforward computation shows that αα~ equals to (α0 + α2) 2+(β1 + β3) 2a + (α1 + α3) 2a2+ (β0 + β2) 2a3 + δ1(1 + a)b + δ2(a 2 + a3)b, where δ1 = α0(β0 + β1) + α1(β1 + β2) + α2(β2 + β3) + α3(β0 + β3) and δ2 = α0(β2 + β3) + α1(β3 + β0) + α2(β0 + β1) + α3(β1 + β2). clearly αα~ = 1 if and only if α0 + α2 = 1, β0 = β2, α1 = α3 and β1 = β3. therefore δ1 = δ2 = β0 + β1 = 0, that is, β0 = β1 and every ~-unitary element can be written as α0 + α1a + (1 + α0)a 2 + α1a 3 + β0b + β0ab + β0a 2b + β0a 3b. therefore v~(fd8) ∼= c23n. case σ3. let α = ∑3 i=0 a i(αi + βib) ∈ fd8, where αi,βj ∈ f. then α~α = (α0 + α2 + β1 + β3) 2 + (α1 + α3 + β0 + β2) 2a2 + δ(1 + a2)b, where δ = (α0 + α2)(β0 + β2) + (α1 + α3)(β1 + β3). clearly α~α = 1 if and only if α0 + α2 + β1 + β3 = 1, β0 = β2 and α1 = α3. therefore every element of v~(fd8) is central or it can be written in the form either ab + x1 or a3b + x2, where x1,x2 ∈ ζ(v (fd8)). since the exponent of ζ(v (fd8)) is two we have proved that v~(fd8) ∼= c25n. case σ4. let α = ∑3 i=0 a i(αi + βib) ∈ fd8, where αi,βj ∈ f. then αα~ = (α0 + α1 + α2 + α3) 2 + (δ0 + δ1)(a + a 3) + (β0 + β1 + β2 + β3) 2a2 + (δ2 + δ3)(1 + a 2)b + (δ4 + δ5)(1 + a 2)ab, where δ0 = (α0 + α2)(α1 + α3), δ1 = (β0 + β2)(β1 + β3), δ2 = (α0 + α2)(β0 + β2), δ3 = (α1 + α3)(β1 + β3), δ4 = (α0 + α2)(β1 + β3), δ5 = (α1 + α3)(β0 + β2). therefore, α0 + α1 + α2 + α3 = 1, β0 + β1 + β2 + β3 = 0, δ0 + δ1 = 0, δ2 + δ3 = 0 and δ4 + δ5 = 0. since β1 + β3 = β0 + β2 we conclude that δ4 = δ2 and δ5 = δ3. moreover, 0 = δ2 + δ3 = (α0 + α1 + α2 + 117 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 α3)(β0 + β2) = β0 + β2 and we have that β0 = β2, β1 = β3, δ1 = 0 and δ0 = 0. thus, every ~-unitary element can be written as either a3 + α0ĉ + α1ĉa + β0ĉb + β1ĉab, if α2 = α0, or a2 + α0ĉ + α1ĉa + β0ĉb + β1ĉab, if α2 = 1 + α0. let us denote by n the central elementary abelian subgroup 〈 1+α0ĉ+α1ĉa+β0ĉb+β1ĉab | αi,βi ∈ f 〉. evidently, a2 ∈ n. since a,a3 belong to the ~-unitary subgroup we have proved that v~(fd8) ∼= c4 ×c24n−1. it is well-known that aut q8 ∼= s4, where s4 is the symmetric group of order 24. it follows that λ2 = 3. lemma 2.4. the number of non-isomorphic unitary subgroups of fq8 with respect to the involutions which arise from d8 equals λ2, where |f| = 2n ≥ 2. proof. let σ1 be the identity automorphism of q8. a straightforward computation shows that aut g{2} = cσ1 ∪cσ2 ∪cσ3, where σ2 : { a 7→ b b 7→ a σ3 : { a 7→ a3 b 7→ b . it was shown in [16] that v~(fq8) ∼= q8 ×c4n−12 . let us consider the following two cases. case i = 2. let α = ∑3 i=0 a i(αi + βib) ∈ fq8, where αi,βj ∈ f. then α~α = (α0 + α2) 2 + δ1a + (β0 + β2) 2a2 + δ2a 3 + (β1 + β3) 2b + δ1ab+ (α1 + α3) 2a2b + δ2a 3b, where δ1 = α0(α1 + β1) + α2(α3 + β3) + β0(α1 + β3) + β2(α3 + β1), δ2 = α0(α3 + β3) + α2(α1 + β1) + β0(α3 + β1) + β2(α1 + β3). evidently, α~α = 1 if and only if α0 + α2 = 1, β0 = β2, α1 = α3 and β1 = β3. they imply that δ1 = δ2 = α1 + β1 = 0, and so α1 = β1. therefore every ~-unitary element can be written as a2 + α0ĉa 2 + α1ĉa + β0ĉb + α1ĉab. thus v~(fq8) ∼= c23n. case i = 3. let α = ∑3 i=0 a i(αi + βib) ∈ fq8, where αi,βj ∈ f. then αα~ = (α0 + α2 + β0 + β2) 2 + (α1 + α3 + β1 + β3) 2a2 + δ(1 + a2)b, where δ = (α0 + α2)(β0 + β2) + (α1 + α3)(β1 + β3). let s~ = {αα~ |α ∈ v (fq8)}. clearly, s~ is a subgroup of ζ(v (fq8)), therefore ψ : v (fq8) → s~ (given by x 7→ xx~) is a homomorphism with kernel v~(fq8). thus |v~(fq8)| = |v (fq8)| |s~| = 27n 22n = 25n. 118 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 let n = 1 and g~ = {g ∈ g |g~ = g−1}. it is easy to see that g~ = 〈 b 〉 and v~(fq8) is a subgroup of g~ ·n, where n is an elementary abelian group. since g~ ∼= c4, we get that v~(fq8) ∼= c4 ×c32. suppose that n > 1 and let ω1 and ω2 be elements of the unit group of f satisfying that ω1 6= 1 and ω1 + ω2 = 1. it is easy to see that b and ω1 + a + ω2b + ab are elements of v~(fq8), but they are not commute. therefore v~(fq8) is not an abelian group. according to theorem 2 in [7], the exponent of v (fq8) is 4. since b is a ~-unitary element with exponent 4 it follows that the exponent of v~(fq8) is 4. since |ζ(v (fq8))| = 24n and x2 ∈ ζ(v (fq8)) for all x ∈ v (fq8) we have proved that v~(fq8)/ζ(v (fq8)) ∼= cn2 . therefore v~(fq8) is a central extension of cn2 by c2 4n. 3. isomorphic unitary subgroups of noncommutative group algebra with non similar involutions in this section we present a non-abelian group whose group algebra has sharply less number of non-isomorphic ~-unitary subgroups than the given upper bound given in corollary 2.2. let h16 = 〈 a,c | a4 = b2 = c2 = 1, (a,b) = 1, (a,c) = b, (b,c) = 1 〉 be and let f be a finite field with |f| = 2n. the automorphism group of h16 is isomorphic to the following group 〈 σ1,σ2,σ3 | σ21 = σ 2 2 = σ 2 3 = σ 2 4 = σ 2 5 = 1, (σ1,σ2) = σ4, (σ1,σ3) = 1, (σ2,σ3) = σ5 〉, where σ1 =:   a 7→ a b 7→ a2b c 7→ c σ2 :   a 7→ a b 7→ bc c 7→ c σ3 :   a 7→ ab b 7→ b c 7→ c. let us consider the following two automorphisms of order two in aut h16 τ1 = σ1σ2σ5 :   a 7→ ac b 7→ a2bc c 7→ c and τ2 = (σ1,σ2) :   a 7→ a3c b 7→ bc c 7→ c. the conjugacy class of τ1 is cτ1 = {σ1σ2σ5, σ1σ2σ4} and τ2 is a central element of the automorphism group. theorem 3.1. let ~1 = τ1 ◦ ∗ and ~2 = τ2 ◦ ∗ be involutions of h16 and let f be a finite field with |f | = 2n (n ≥ 1). then ~1 is not similar to ~2 and v~1 (fh16) ∼= v~2 (fh16). proof. first, we establish the structure of v~1 (fh16). since every element of fh16 can be written as x =α0 + α1a + α2a 2 + α3a 3 + α4b + α5ab + α6a 2b + α7a 3b+ (α8 + α9a + α10a 2 + α11a 3 + α12b + α13ab + α14a 2b + α15a 3b)c (1) we have xx~ =(α0 + α2 + α8 + α10) 2 + (α5 + α7 + α13 + α15) 2a2 + δ1(a + a 3c)+ δ2(a 3 + ac) + δ3(b + a 2bc) + δ4(ab + abc) + δ5(a 2b + bc) + δ6(a 3b + a3bc)+ (α1 + α3 + α9 + α11) 2c + (α4 + α6 + α12 + α14) 2a2c, 119 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 where δ1 = (α0 + α10)(α1 + α11) + (α2 + α8)(α3 + α9) + (α4 + α14)(α5 + α15) + (α6 + α12)(α7 + α13) δ2 = (α0 + α10)(α3 + α9) + (α2 + α8)(α1 + α11) + (α4 + α14)(α7 + α13) + (α6 + α12)(α5 + α15) δ3 = (α0 + α10)(α4 + α14) + (α1 + α11)(α5 + α15) + (α2 + α8)(α6 + α12) + (α3 + α9)(α7 + α13) δ4 = (α0 + α8)(α5 + α13) + (α2 + α10)(α7 + α15) + (α4 + α12)(α3 + α11) + (α6 + α14)(α1 + α9) δ5 = (α0 + α10)(α6 + α12) + (α1 + α11)(α7 + α13) + (α3 + α9)(α5 + α15) + (α4 + α14)(α2 + α8) δ6 = (α0 + α8)(α7 + α15) + (α2 + α10)(α5 + α13) + (α1 + α9)(α4 + α12) + (α3 + α11)(α6 + α14). evidently, x belongs to v~1 (fh16) if and only if xx ~1 = 1. therefore α0 + α2 + α8 + α10 = 1, α5 + α7 + α13 + α15 = 0, α1 + α3 + α9 + α11 = 0, α4 + α6 + α12 + α14 = 0 and δ1 = δ2 = δ3 = δ4 = δ5 = δ6 = 0. since α2 + α8 = 1 + α0 + α10 and α4 + α14 = α6 + α12 we have that δ1 = (α3 + α9) + (α0 + α10)(α1 + α11 + α3 + α9) + (α4 + α14)(α5 + α15 + α7 + α13) = α3 + α9, δ2 = (α1 + α11) + (α0 + α10)(α3 + α9 + α1 + α11) + (α4 + α14)(α7 + α13 + α5 + α15) = α1 + α11, δ3 = (α6 + α12) + (α0 + α10)(α4 + α14 + α6 + α12) + (α1 + α11)(α5 + α15 + α7 + α13) = α6 + α12, δ4 = (α7 + α15) + (α0 + α8)(α5 + α13 + α7 + α15) + (α4 + α12)(α3 + α11 + α1 + α9) = α7 + α15, δ5 = (α4 + α14) + α0 + α10)(α6 + α12 + α4 + α14) + (α1 + α11)(α7 + α13 + α5 + α15) = α4 + α14, δ6 = (α5 + α13) + (α0 + α8)(α7 + α15 + α5 + α13) + (α1 + α9)(α4 + α12 + α6 + α14) = α5 + α13. therefore x = α0 + α2a 2 + α8c + α10a 2c + α1ĉa + α4ĉb + α5ĉab, where ĉ = 1 + a2 + c + a2c. as a consequence v~1 (fh16) is a central subgroup of v (fh16). let n = 〈 1 + β1ĉa, 1 + β2ĉb, 1 + β3ĉab | βi ∈ f 〉 be. evidently, n ∼= c3n2 . since a2ĉ = cĉ = a2cĉ = ĉ we conclude that n ∼= a2n ∼= cn ∼= a2cn. since a2n · cn = a2cn and the pairwise intersections of n,a2n,a2cn are {1} we have proved that v~(fg) ∼= n ×a2n ×cn. thus v~1 (fg) ∼= c9n2 . now, we establish the structure of v~2 (fh16). let x ∈ fh16 be. using formula (1) we can compute the product xx~ = (α0 + α2 + α8 + α10) 2 + (α5 + α7 + α13 + α15) 2a2 + δ1(a + ac)+ δ2(a 3 + a3c) + δ3(b + bc) + δ4(ab + abc) + δ5(a 2b + a2bc) + δ6(a 3b + a3bc)+ (α4 + α6 + α12 + α14) 2c + (α1 + α3 + α9 + α11) 2a2c, where δ1 = (α0 + α8)(α9 + α1) + (α2 + α10)(α11 + α3) + (α4 + α12)(α5 + α13) + (α6 + α14)(α7 + α15), δ2 = (α0 + α8)(α11 + α3) + (α2 + α10)(α9 + α1) + (α4 + α12)(α7 + α15) + (α6 + α12)(α5 + α15), δ3 = (α0 + α8)(α12 + α4) + (α2 + α10)(α14 + α6) + (α1 + α9)(α15 + α7) + (α3 + α11)(α13 + α5), δ4 = (α0 + α8)(α13 + α5) + (α2 + α10)(α15 + α7) + (α4 + α12)(α1 + α9) + (α6 + α14)(α3 + α11), δ5 = (α0 + α8)(α14 + α6) + (α1 + α9)(α13 + α5) + (α2 + α10)(α12 + α4) + (α3 + α11)(α15 + α7), δ6 = (α0 + α8)(α15 + α7) + (α2 + α10)(α13 + α5) + (α1 + α9)(α14 + α6) + (α3 + α11)(α12 + α4). keeping in mind that x belongs to v~2 (fh16), it follows that xx ~2 = 1. therefore α0 + α2 + α8 + α10 = 1, α5 + α7 + α13 + α15 = 0, α1 + α3 + α9 + α11 = 0, α4 + α6 + α12 + α14 = 0 and δ1 = δ2 = δ3 = δ4 = δ5 = δ6 = 0. since α0 + α8 = 1 + α2 + α10 and α4 + α12 = α6 + α14 we have that δ1 = α3 + α11, δ2 = α1 + α9, δ3 = α6 + α14, δ4 = α7 + α15, δ5 = α4 + α12 and δ6 = α5 + α13. therefore α1 = α3 = α9 = α11, α6 = α4 = α12 = α14 and α5 = α7 = α13 = α15. 120 z. a. balogh / j. algebra comb. discrete appl. 9(2) (2022) 115–122 according to the above calculations we get that every x ∈ v~2 (fh16) can be written as x = α0 + α2a 2 + α8c + α10a 2c + α1ĉa + α4ĉb + α5ĉab, where ĉ = 1 + a2 + c + a2c, so v~2 (fh16) is a central subgroup of v (fh16). let n = 〈 1+β1ĉa, 1+β2ĉb, 1+β3ĉab | βi ∈ f 〉 be. clearly, n ∼= c3n2 and n ∼= a2n ∼= cn ∼= a2cn because a2ĉ = cĉ = a2cĉ = ĉ. since a2n · cn = a2cn and the pairwise intersections of n,a2n,a2cn are {1} we have proved that v~(fg) ∼= n ×a2n 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(2021). 122 https://doi.org/10.1112/s002460930601873x https://doi.org/10.1112/s002460930601873x https://mathscinet.ams.org/mathscinet-getitem?mr=0292913 https://mathscinet.ams.org/mathscinet-getitem?mr=0292913 https://mathscinet.ams.org/mathscinet-getitem?mr=0292913 https://mathscinet.ams.org/mathscinet-getitem?mr=0292913 https://doi.org/10.1007/s00031-014-9269-6 https://doi.org/10.1007/s00031-014-9269-6 https://doi.org/10.1142/s0219498823500433 https://doi.org/10.1142/s0219498823500433 introduction involutions and unitary subgroups isomorphic unitary subgroups of noncommutative group algebra with non similar involutions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.82415 j. algebra comb. discrete appl. 3(2) • 91–96 received: 18 november 2015 accepted: 2 feburary 2016 journal of algebra combinatorics discrete structures and applications matrix rings over a principal ideal domain in which elements are nil-clean research article somayeh hadjirezaei, somayeh karimzadeh abstract: an element of a ring r is called nil-clean if it is the sum of an idempotent and a nilpotent element. a ring is called nil-clean if each of its elements is nil-clean. s. breaz et al. in [1] proved their main result that the matrix ring mn(f) over a field f is nil-clean if and only if f ∼= f2, where f2 is the field of two elements. m. t. koşan et al. generalized this result to a division ring. in this paper, we show that the n × n matrix ring over a principal ideal domain r is a nil-clean ring if and only if r is isomorphic to f2. also, we show that the same result is true for the 2×2 matrix ring over an integral domain r. as a consequence, we show that for a commutative ring r, if m2(r) is a nil-clean ring, then dimr = 0 and charr/j(r) = 2. 2010 msc: 15a23, 15b33, 16s50 keywords: nil-clean matrix, idempotent matrix, nilpotent matrix, principal ideal domain 1. introduction throughout this paper, all rings are associative with identity. an element in a ring r is said to be (strongly) clean if it is the sum of an idempotent and a unit element(and these commute ). a (strongly) clean ring is one in which every element is (strongly) clean. local rings are obviously strongly clean. strongly clean rings were introduced by nicholson [8]. an element in a ring r is said to be (strongly) nil-clean if it is the sum of an idempotent and a nilpotent element(and these commute). a (strongly) nil-clean ring is one in which every element is (strongly) nil-clean. it is easy to see that every strongly nil-clean element is strongly clean and that every nil-clean ring is clean ([3, proposition 3.1.3]). nil-clean rings were extensively investigated by diesl in [3] and [4]. s. breaz et al. in [1] proved their main result that the matrix ring mn(f) over a field f is nil-clean if and only if f ∼= f2, where f2 is the field of two elements. m. t. koşan et al. in [6], generalized this result to a division ring. that is, the matrix ring mn(d) over a division ring d is nil-clean if and only if d ∼= f2. we show that this is true for a principal ideal domain (pid). somayeh hadjirezaei (corresponding author), somayeh karimzadeh; vali-e-asr university of rafsanjan (email: s.hajirezaei@vru.ac.ir, karimzadeh@vru.ac.ir). 91 s. hadjirezaei, s. karimzadeh / j. algebra comb. discrete appl. 3(2) (2016) 91–96 throughout this paper an integral domain is a commutative ring without zero divisors and the jacobson radical of a ring is denoted by j(r). we write mn(r) for the n×n matrix ring over r, in for the n×n identity matrix. 2. main results first, we recall from [5, proposition vii.2.11 ], the following proposition. proposition 2.1. if a is an n × m matrix of rank r > 0 over a principal ideal domain r, then a is equivalent to a matrix of the form ( lr 0 0 0 ) , where lr is an r×r diagonal matrix with nonzero diagonal entries d1, ..., dr such that d1 | ... | dr. the ideals (d1), ..., (dr) in r are uniquely determined by the equivalence class of a. further, we use the following lemmas. lemma 2.2. (see [4, proposition 3.14]) let r be a nil-clean ring. then the element 2 is (central) nilpotent and, as such, is always contained in j(r). lemma 2.3. (see [9, corollary 5]) let a be an n×n idempotent matrix over a ring r. if a is equivalent to a diagonal matrix, then a is similar to a diagonal matrix. next lemmas are the main results of [1] and [6]. lemma 2.4. (see [1, theorem 3]) let f be a field and let n ≥ 1. then mn(f) is a nil-clean ring if and only if f ∼= f2. lemma 2.5. (see [6, theorem 3]) let d be a division ring and let n ≥ 1. then mn(d) is a nil-clean ring if and only if d ∼= f2. theorem 2.6. let r be a principal ideal domain and let n ≥ 1. then mn(r) is a nil-clean ring if and only if r ∼= f2. proof. if r ∼= f2, then by lemma 2.4, mn(r) is a nil-clean ring. now, assume that mn(r) is a nil-clean ring. by lemma 2.2, 2in is a nilpotent element. thus 2 = 0 in r, because r is an integral domain. proof in the case n = 1 is obvious, so assume that n > 1. take a ∈ r \{0, 1} and put a =   a 0 . . . 0 0 0 . . . 0 ... ... ... ... 0 0 . . . 0   = e + n, where e is an idempotent element and n is a nilpotent element of mn(r). by proposition 2.1, e is equivalent to a diagonal matrix. thus by lemma 2.3, e is similar to a diagonal matrix where it’s entries are 0 and 1. hence u−1eu = ( ik 0 0 0 ) , for some invertible matrix u = (uij) ∈ mn(r). therefore u−1au = ( ik 0 0 0 ) + n′, (1) where n′ = u−1nu is a nilpotent element. since a is not nilpotent, hence u−1au is not nilpotent, so k ≥ 1. if k = n, then a = in +n is invertible, a contradiction because det a = 0. thus 1 ≤ k < n. since in + n ′ is invertible, u(in + n′) is invertible. we have u(in + n ′) = u ( ik 0 0 0 ) + un′ + u ( 0 0 0 in−k ) = au + u ( 0 0 0 in−k ) 92 s. hadjirezaei, s. karimzadeh / j. algebra comb. discrete appl. 3(2) (2016) 91–96 =   au11 . . . au1n 0 . . . 0 ... ... ... 0 . . . 0   +   0 . . . 0 u1(k+1) . . . u1n 0 . . . 0 u2(k+1) . . . u2n ... ... ... ... ... 0 . . . 0 un(k+1) . . . unn   =   au11 . . . au1k (1 + a)u1(k+1) . . . (1 + a)u1n 0 . . . 0 u2(k+1) . . . u2n ... ... ... ... ... ... 0 . . . 0 un(k+1) . . . unn   . we imply that k = 1 and u11 6= 0. thus u(in + n ′) =   au11 (1 + a)u12 . . . (1 + a)u1n 0 u22 . . . u2n ... ... ... ... 0 un2 . . . unn   . put u1 :=   u22 . . . u2n... ... ... un2 . . . unn   . since det(u(in + n′)) = au11 det u1, u1 is invertible in mn−1(r) and u11 is invertible in r, hence (1) implies that ( a 0 0 0 ) u = u ( 1 0 0 0 ) + un′. this implies that ( u−111 0 0 u−11 )( a 0 0 0 )( u11 0 0 u1 )( u−111 0 0 u−11 ) u = ( u−111 0 0 u−11 ) u ( 1 0 0 0 ) + ( u−111 0 0 u−11 ) un′, i.e., ( a 0 0 0 ) v = v ( 1 0 0 0 ) + v n′, (2) where v = ( u−111 0 0 u−11 ) u = ( 1 x y in−1 ) . let v −1 = ( c x′ y ′ c1 ) . from v v −1 = v −1v = in, it follows that 1 = c + xy ′ = c + x′y in−1 = y x ′ + c1 = y ′x + c1 0 = x′ + xc1 = cx + x ′ 93 s. hadjirezaei, s. karimzadeh / j. algebra comb. discrete appl. 3(2) (2016) 91–96 0 = cy + y ′ = y ′ + c1y. since 2 = 0 in r (by lemma 2.2, and since r is an integral domain) hence, we have 1 = −1 in r, so c = −c. therefore x′ = −cx = cx, y ′ = cy and c1 = in−1 + y x′ = in−1 + cy x. also, 1 = c + xy ′ = c + cxy = c(1 + xy ), so c is a unit element of r and xy = 1 + c−1. (3) hence v −1 = ( c cx cy in−1 + cy x ) . if xy = 0, then c = 1 and v −1 = ( 1 x y in−1 + y x ) . then by (2), n ′′ := v nv −1 = ( a 0 0 0 ) + v ( 1 0 0 0 ) v −1 = ( 1 + a x y y x ) , and, for k ≥ 1, n ′′ k+1 = ( (1 + a)k+1 (1 + a)k+1x (1 + a)k+1y (1 + a)k+1y x ) 6= 0 (as (1 + a) 6= 0). this is a contradiction because n ′′ is a nilpotent matrix. therefore xy 6= 0. from (2) it follows that( 1 x 0 in−1 )( a 0 0 0 )( 1 x 0 in−1 )( 1 x 0 in−1 ) v = ( 1 x 0 in−1 ) v ( 1 0 0 0 ) + ( 1 x 0 in−1 ) v n′, i.e., ( a x 0 0 ) p = p ( 1 0 0 0 ) + pn′, (4) where p = ( 1 x 0 in−1 ) v = ( 1 x 0 in−1 )( 1 x y in−1 ) = ( 1 + xy x + x y in−1 ) . since 2 = 0 in r, hence x + x = 2x = 0. also by (3), we have xy − 1 = xy + 1 = c−1. hence p = ( c−1 0 y in−1 ) and p−1 = ( c 0 cy in−1 ) . it follows from (4) that 4 := pnp−1 = ( a ax 0 0 ) + p ( 1 0 0 0 ) p−1 = ( 1 + a ax cy 0 ) . if q is an n×n matrix, then we will write q in block form q = ( q11 q12 q21 q22 ) , where q11, q12, q21, q22 have size 1×1, 1× (n−1), (n−1)×1 and (n−1)× (n−1), respectively. for k ≥ 1 we have 4k+1 = 4k4 = ( (4k)11 (4)k12 (4k)21 (4k)22 )( 1 + a ax cy 0 ) = ( (4k)11(1 + a) + (4)k12cy a(4k)11x (4k)21(1 + a) + (4k)22cy a(4k)21x ) . (5) 94 s. hadjirezaei, s. karimzadeh / j. algebra comb. discrete appl. 3(2) (2016) 91–96 an easy induction shows that there exist ak, bk, ck ∈ r such that for k ≥ 1 we have (4k)12 = bkx, (4k)21 = cky, (4k)22 = aky x. (6) since 4 is a nilpotent matrix and 421 = cy 6= 0, there exists a positive integer s such that (4s+1)21 = 0 but (4s)21 6= 0. then by (5) and (6), 4s+1 = ( (4s+1)11 (4s+1)12 0 csay x ) , where csa 6= 0. for r ∈ r, it is easily seen that ry x = 0 if and only if r = 0. we have (4s+1)k22 = (csa) k(xy )k−1y x. since csa 6= 0 and xy 6= 0, hence (4s+1)k22 6= 0, for k ≥ 2. it is a contradiction because 4 is nilpotent. theorem 2.7. let r be an integral domain . if mn(r) is a nil-clean ring, then r is a field. proof. let q be the field of fractions of r and 0 6= a ∈ r. we know that ain is nil-clean. so, ain = e + n with e idempotent and n nilpotent. we have in = a−1e + a−1n, in mn(q) . thus a−1e ( and consequently e) is invertible in mn(q). since e is idempotent, so e = in. therefore ain is invertible, hence r is a field. lemma 2.8. let r be an integral domain and 0, i2 6= a ∈ m2(r). then a is idempotent if and only if rank(a) = 1 and tr(a) = 1. proof. by [2, lemma 1.5]. lemma 2.9. let r be an integral domain. if a ∈ mn(r) be a nilpotent matrix, then det(a) = 0. proof. let a be a nonzero nilpotent matrix. thus there exists some k ∈ n such that ak = 0. thus adj(a)ak = 0. hence det(a)ak−1 = 0. so det(a) adj(a)ak−1 = 0. therefore (det(a))2ak−2 = 0. continuing this process we have (det(a))k−1a = 0. since r is an integral domain and a 6= 0, hence det(a) = 0 theorem 2.10. let r be an integral domain. then m2(r) is a nil-clean ring if and only if r ∼= f2. proof. ⇐=)this is by theorem 2.6. =⇒) assume that r is not isomorphic to f2. so, there exists a ∈ r\{0, 1}. put a = ( a 0 0 0 ) = e + n, where e is idempotent and n is a nilpotent matrix. if e = i2, then a is invertible, a contradiction. if e = 0, then a is nilpotent. hence a = 0, a contradiction. so by lemma 2.8, e = ( e b c 1−e ) , where e, b, c ∈ r and e(1 − e) = bc. hence n = ( n −b −c −(1−e) ) , for some n ∈ r. by lemma 2.9, −n(1 − e) = bc. therefore e(1 − e) = −n(1 − e). if e 6= 1, then e = −n. so n = −e, a contradiction. thus e = 1 and bc = 0. hence b = 0 or c = 0. we consider two cases. case 1) let b = 0. so n = ( n 0 0 0 ) . since n is nilpotent, hence there exists a positive integer k such that nk = 0. so n = 0. therefore a = 1. case 2) let c = 0. thus n = ( n −b 0 0 ) . since n is nilpotent, hence there exists a positive integer k such that nk = 0. so n = 0. therefore a = 1. let r be a commutative ring with identity. by a chain of prime ideals of r we mean a finite strictly increasing sequence of prime ideals of r of the type po $ p1 $ p2 $ ... $ pn. the integer n is called the length of the chain. 95 s. hadjirezaei, s. karimzadeh / j. algebra comb. discrete appl. 3(2) (2016) 91–96 definition 2.11. the krull dimension of r is the supremum of all lengths of chains of prime ideals of r. krull dimension of r is denoted by dimr. corollary 2.12. let r be a commutative ring. if m2(r) is a nil-clean ring, then dimr = 0 and charr/j(r) = 2. proof. let p be a prime ideal of r. we have m2(r/p) = m2(r)/m2(p) is nil-clean. hence by theorem 2.10, r/p ∼= f2. so p is a maximal ideal of r and 2 ∈ j(r). therefore charr/j(r) = 2. remark 2.13. note that all of these results can also be obtained as some consequences of [7, theorem 6.1]. acknowledgment: the authors are grateful to the referees’ invaluable comments, which helped to improve our study. references [1] s. breaz, g. călugăreanu, p. danchev, t. micu, nil-clean matrix rings, linear algebra appl. 439(10) (2013) 3115-3119. [2] j. chen, x. yang, y. zhou, on strongly clean matrix and triangular matrix rings, comm. algebra. 34(10) (2006) 3659–3674. [3] a. j. diesl, classes of strongly clean rings, ph. d. thesis, university of california, berkeley, 2006. [4] a. j. diesl, nil clean rings, j. algebra. 383 (2013) 197–211. [5] t. w. hungerford, algebra, springer-verlag, 1980. [6] m.t. koşan, t. k. lee, y. zhou, when is every matrix over a division ring a sum of an idempotent and a nilpotent?, linear algebra appl. 450 (2014) 7–12. [7] t. koşan, z. wang, y. zhou, nil-clean and strongly nil-clean rings, j. pure appl. algebra. 220(2) (2016) 633–646. [8] w. k. nicholson, strongly clean rings and fitting’s lemma, comm. algebra. 27(8) (1999) 3583–3592. [9] g. song, x. guo, diagonability of idempotent matrices over noncommutative rings, linear algebra appl. 297(1-3) (1999) 1–7. 96 http://dx.doi.org/10.1016/j.laa.2013.08.027 http://dx.doi.org/10.1016/j.laa.2013.08.027 http://dx.doi.org/10.1080/00927870600860791 http://dx.doi.org/10.1080/00927870600860791 http://search.proquest.com/docview/305347901?accountid=17384 http://dx.doi.org/10.1016/j.jalgebra.2013.02.020 http://dx.doi.org/10.1016/j.laa.2014.02.047 http://dx.doi.org/10.1016/j.laa.2014.02.047 http://dx.doi.org/10.1016/j.jpaa.2015.07.009 http://dx.doi.org/10.1016/j.jpaa.2015.07.009 http://dx.doi.org/10.1080/00927879908826649 http://dx.doi.org/10.1016/s0024-3795(99)00059-2 http://dx.doi.org/10.1016/s0024-3795(99)00059-2 introduction main results references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 1(1) • 13-17 received: 31 may 2014; accepted: 3 august 2014 doi 10.13069/jacodesmath.25090 journal of algebra combinatorics discrete structures and applications the existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codes research article vladimir d. tonchev ∗ michigan technological university, houghton, michigan 49931-1295, usa abstract: the existence of a quantum [[28, 12, 6]] code was one of the few cases for codes of length n ≤ 30 that was left open in the seminal paper by calderbank, rains, shor, and sloane [2]. the main result of this paper is the construction of the first optimal linear quaternary [28, 20, 6] code which contains its hermitian dual code and yields the first optimal quantum [[28, 12, 6]] code. 2010 msc: 94b05, 94b65, 81-99 keywords: quantum code, optimal code 1. introduction we assume familiarity with the basics of classical and quantum error-correcting codes [2], [5]. the hermitian inner product in gf(4)n is defined as (x, y)h = n∑ i=1 xiy 2 i , (1) while the trace inner product in gf(4)n is defined as (x, y)t = n∑ i=1 (xiy 2 i + x 2 i yi). (2) a code c is self-orthogonal if c ⊆ c⊥, and self-dual if c = c⊥. a linear code c ⊆ gf(4)n is selforthogonal with respect to the trace product (2) if and only if it is self-orthogonal with respect to the hermitian product (1) [2]. ∗ e-mail: tonchev@mtu.edu 13 the existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codes an additive (n, 2k) code c over gf(4) is a subset of gf(4)n consisting of 2k vectors which is closed under addition. an additive code is even if the weight of every codeword is even, and otherwise odd. note that an even additive code is trace self-orthogonal, and a linear self-orthogonal code is even [2]. if c is an (n, 2k) additive code with weight enumerator w(x, y) = n∑ j=0 ajx n−jyj, (3) the weight enumerator of the trace-dual code c⊥ is given by w⊥ = 2−kw(x + 3y, x−y) (4) in their seminal paper [2], calderbank, rains, shor and sloane described a method for the construction of quantum error-correcting codes from additive codes that are self-orthogonal with respect to the trace product (2). theorem 1.1. [2] an additive trace self-orthogonal (n, 2n−k) code c such that there are no vectors of weight < d in c⊥ \c yields a quantum code with parameters [[n, k, d]]. a quantum code associated with an additive code c is pure if the minimum distance of c⊥ is d; otherwise, the code is called impure. a quantum code is called linear if the associated additive code c is linear. a table with lower and upper bounds on the minimum distance d for quantum [[n, k, d]] codes of length n ≤ 30 is given in the paper by calderbank, rains, shor and sloane [2]. in particular, according to table iii on page 1382 in [2], the largest minimum distance d of a known quantum [[28, 12]] code is d = 5, while the best upper bound is d ≤ 6. in the next section, we describe a simple construction of quaternary hermitian self-orthogonal codes with parameters [2n + 1, k + 1] and [2n + 2, k + 2] from a given pair of hermitian self-orthogonal [n, k] codes. as an application of this construction, we find the first optimal quaternary linear [28, 20, 6] which contains its dual code and hence yields the first optimal [[28, 12, 6]] quantum code. an extended version of calderbank-rains-shor-sloane table for quantum codes [2, table iii], as well as tables with bounds on the minimum distance of linear codes, was compiled by grassl [4]. 2. a doubling construction lemma 2.1. suppose that ci (i = 1, 2) is a linear hermitian self-orthogonal [n, k] code over gf(4) with generator matrix gi, and x(i) ∈ c⊥i is a vector of odd weight. (a) the code c′ with generator matrix g′ =   0 g1 g2 . . . 0 x(1) 0 . . . 0 1   (5) is a hermitian self-orthogonal [2n + 1, k + 1] code with dual distance d(c′)⊥ ≤ min(d(c⊥11), d(c ⊥ 2 )), (6) where c11 is the code spanned by the rows of g11 given by (7): g11 =   0 g1 . . . 0 x(1) 1   . (7) 14 v. d. tonchev (b) the code c′′ with generator matrix g′′ =   0 0 g1 g2 . . . 0 0 x(1) 0 . . . 0 1 0 0 . . . 0 x(2) 0 1   (8) is a hermitian self-orthogonal [2n + 2, k + 2] code with dual distance d(c′′)⊥ ≤ min(d(c⊥11), d(c ⊥ 22)), (9) where c22 is the code spanned by the rows of g22 given by (10): g22 =   0 g2 . . . 0 x(2) 1   . (10) proof. the self-orthogonality of c′ and c′′ follows from the fact that all rows of g′ and g′′ have even weights, and every pair of rows of g′, as well as every pair of rows of g′′, are pairwise orthogonal. since the weight of x(1) (resp. x(2)) is odd, x(1) does not belong to c1, and x(2) does not belong to c2, and that implies the dimensions of c′ and c′′. the bounds (6), (9) on the dual distance follow trivially by the observation that every codeword of c⊥11 (resp. c ⊥ 22) extends to a codeword of (c ′)⊥ (resp (c′′)⊥) by filling in all remaining coordinates with zeros. � it is worth mentioning that since c1 and c2 are self-orthogonal, their minimum distances are trivial upper bounds on the minimum dual distances d(c′)⊥ and d(c′′)⊥. for example, if d(c1) = 2 then d(c′)⊥ ≤ 2. we note also that using codes c1, c2 with large minimum distances is a necessary, but not always sufficient condition for large dual distances d(c′)⊥ and d(c′′)⊥. for example, if g1 = g2 and x(1) is the all-one vector, then for every 1 ≤ i ≤ n, the columns of (5) with indices i, i + n and 2n + 1 determine a codeword of weight 3 in (c′)⊥. these simple observations illustrate that one cannot expect a good general lower bound on d(c′)⊥ or d(c′′)⊥, and finding codes c1, c2 with appropriate generator matrices g1, g2 and vectors x(1), x(2) which lead to optimal dual distances d(c′)⊥ and d(c′′)⊥ is not a trivial task. using the connection to quantum codes described in theorem 1.1, lemma 2.1 implies the following. corollary 2.2. the existence of quaternary hermitian self-orthogonal [n, k] codes ci (i = 1, 2) satisfying the assumptions of lemma 2.1 implies the existence of a pure quantum linear [[2n + 1, 2n − 2k − 1, d′]] code with d′ ≤ min(d(c⊥11), d(c⊥2 )), and a pure quantum linear [[2n + 2, 2n − 2k − 2, d′′]] code with d′′ ≤ min(d(c⊥11), d(c⊥22)). we will apply lemma 2.1 and corollary 2.2 to some self-orthogonal codes of length n = 2k + 1 being shortened codes of extremal self-dual [2k +2, k +1] codes, that is, self-dual codes having maximum possible minimum distance for the given code length. example 2.3. the matrix g1 = ( 1 0 1 ω ω 0 1 ω ω 1 ) is the generator matrix of a self-orthogonal [5, 2, 4] code c1 over gf(4) = {0, 1, ω, ω2}. the code c1 is a shortened code of the unique (up to equivalence) self-dual [6, 3, 4] code. applying lemma 2.1 with 15 the existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codes c2 = c1, g2 = g1, and x(1) = x(2) being the all-one vector of length 5, gives a self-orthogonal [11, 3] code c′ with dual distance 3 and a self-orthogonal [12, 4] code c′′ with dual distance 4, which gives optimal quantum [[11, 5, 3]] and [[12, 4, 4]] codes respectively via corollary 2.2. example 2.4. a pair of self-orthogonal [7, 3] codes obtained as shortened codes of the unique (up to equivalence) self-dual [8, 4, 4] code can be used to obtain optimal quantum [[15, 7, 3]] and [[16, 6, 4]] codes. 3. an optimal quantum [[28, 12, 6]] code the smallest parameters of a self-dual quaternary linear code that yields a quantum code with minimum distance d ≥ 5 via corollary 2.2 are [14, 7, 6]. the only such code, up to equivalence, is the quaternary extended quadratic residue code q14 [6, page 340]. we apply lemma 2.1 using the pair of self-orthogonal [13, 6] codes c1, c2 generated by the following matrices: g1 =   0000100210233 3000010021023 3300001002102 2330000100210 0233000010021 1023300001002   , g2 =   0000113023002 2000011302300 0200001130230 0020000113023 3002000011302 2300200001130   , where for convenience, the elements ω and ω2 of gf(4) are written as 2 and 3 respectively. the matrices g1, g2 are circulant. the codes c1, c2 are cyclic and equivalent to a shortened code of q14. choosing x(1) = x(2) to be the all-one vector of length 13, we obtain the generator matrix g′ (5) of a self-orthogonal [27, 7] code c′ with dual distance 5, and the generator matrix g′′ (8) of a self-orthogonal [28, 8] code with dual distance 6. the matrix g′′ is available on line at http://www.math.mtu.edu/~tonchev/gm28-8.html by corollary 2.2, c′ gives a pure optimal quantum [[27, 13, 5]] code, while c′′ gives a pure optimal quantum [[28, 12, 6]] code. an alternative geometric construction of a quantum code with the first parameters, [[27, 13, 5]], was given by the author in [7]. to the best of our knowledge, the quantum code with the second parameters, [[28, 12, 6]], is the first known quantum code with these parameters (a quantum [[28, 12, 5]] code was listed in [2]). the weight distribution of the [28, 8] code c′′ is given in table 1. the weight enumerator of the dual [28, 20] code (c′′)⊥ is 1 + 6240y6 + 37128y7 + 314223y8 + 2044848y9 + 11883768y10 + . . . we note that the code (c′′)⊥ is an optimal linear [28, 20, 6] quaternary code: 6 is the best theoretical upper bound on the minimum distance of a quaternary linear [28, 20] code. the largest minimum distance of any previously known [28, 20] code was 5 [3], [4]. acknowledgments magma [1] was used for some of the computations. this research was partially supported by nsa grant h98230-12-0213. the author wishes to thank the unknown reviewers for noticing some typos and suggesting several improvements of the text layout and notation. 16 v. d. tonchev table 1. w aw 12 39 14 6 16 3198 18 9204 20 18213 22 22854 24 10569 26 1248 28 204 references [1] w. bosma, j. cannon, j, handbook of magma functions, department of mathematics, university of sydney, 1994. [2] a. r. calderbank, e. m. rains, p. w. shor, and n. j. a. sloane, quantum error correction via codes over gf(4), ieee trans. information theory, 44(4), 1369-1387, 1998. [3] a. e. brouwer, tables of linear codes, http://www.win.tue.nl/ aeb/. [4] m. grassl, http://www.codetables.de. [5] f. j. macwilliams and n. j. a. sloane, the theory of error-correcting codes, north-holland, amsterdam 1977. [6] g. nebe, e. m. rains, n. j. a. sloane, self-dual codes and invariant theory, springer, berlin, 2006. [7] v. d. tonchev, quantum codes from caps, discrete math., 308, 6368-6372, 2008. 17 introduction a doubling construction an optimal quantum [[28,12,6]] code acknowledgments references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056624 j. algebra comb. discrete appl. 9(1) • 57–70 received: 11 july 2021 accepted: 8 october 2021 journal of algebra combinatorics discrete structures and applications 1-generator two-dimensional quasi-cyclic codes over z4[u]/〈u2 −1〉 research article arazgol ghajari, kazem khashyarmanesh, zohreh rajabi abstract: in this paper, we obtain generating set of polynomials of two-dimensional cyclic codes over the ring r = z4[u]/〈u2−1〉, where u2 = 1. moreover, we find generator polynomials for two-dimensional quasicyclic codes and two-dimensional generalized quasi-cyclic codes over r and specify a lower bound on minimum distance of free 1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over r. 2010 msc: 12e20, 94b05, 94b15, 94b60 keywords: two-dimensional cyclic codes, two-dimensional quasi-cyclic codes, two-dimensional generalized quasi-cyclic codes 1. introduction there are many generalizations of cyclic codes. one of them is two-dimensional cyclic codes. a lot of works on two-dimensional cyclic codes has done. ikai et al. first introduced the concept of common zeros for characterizing two-dimensional codes [6], and showed the existence of two-dimensional codes that can be characterized by the common zeros. after that the researchers have studied with different concepts in these codes. the reader can find some of such studies in the papers [11–13]. moreover, lalason et al. [7] construct a basis of an s-dimensional cyclic code over a finite field. on the other hand, quasi-cyclic codes are another natural generalizations of cyclic codes. the study of quasi-cyclic codes over finite rings has provided useful information in coding theory. we shall use the phrase ‘qc code’ as an abbreviation for ‘quasi-cyclic code’ and ‘gqc code’ for ‘generalized quasi-cyclic codes’. qc codes form an important class of linear codes which also include cyclic codes (when we consider the case ` = 1). ling and solé studied the algebraic structure of qc codes over finite fields and provided a new algebraic approach to qc codes (see also [8]). there have been a lot of investigations of qc codes and gqc codes over arazgol ghajari, kazem khashyarmanesh (corresponding author), zohreh rajabi; department of pure mathematics, ferdowsi university of mashhad, mashhad, iran (email: ghajari3051@gmail.com, khashyar@ipm.ir, rajabi261@yahoo.com). 57 https://orcid.org/0000-0002-3675-2832 https://orcid.org/0000-0003-3314-7298 https://orcid.org/0000-0001-7857-2672 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 the rings, for example [1–3, 5, 9, 14]. in [10], the authors studied the constacyclic codes over the finite non-chain ring z4 + uz4 with u2 = 1 and obtained some new z4-linear codes. finally, gao et al. [4] have generalized qc codes and gqc codes over the finite non-chain ring r = z4[u]/〈u2 − 1〉 with u2 = 1. they have determined the structure of the generators and the minimal generating sets of 1-generator qc codes and gqc codes. they also have given a lower bound on minimum distance of free 1-generator qc codes and gqc codes over r. furthermore, in [4], some new z4-linear codes were constructed by 1-generated qc codes and gqc codes over r. hence, there are many examples of cyclic codes and qc codes over r. there exist many researches of two-dimensional cyclic codes over finite fields. however, the research of two-dimensional cyclic codes over r has not been considered by any coding scientist. moreover, quasi-cyclic codes perform very well on the codes have great lengths. therefore, these codes are the important and most intensively studies classes of linear codes. the ring r = z4[u]/〈u2 − 1〉 with u2 = 1 is a frobenius non-chain ring with 16 elements. there are some examples of cyclic codes over r whose z4 gray images have better parameters than previous best-known z4-linear codes were presented (see for example [4] and [10]). the main purpose of this paper is to obtain sets of generator polynomials of two-dimensional cyclic codes over r. we also determine the structure of the generators and the minimal generating sets of 1-generator two-dimensional qc codes and two-dimensional gqc codes. this method probably helps to decode two-dimensional cyclic codes and two-dimensional qc codes as it has done for cyclic codes and qc codes. this paper is organized as follows: at first, we find the generator polynomials corresponding to two-dimensional cyclic codes over r. then, by using these polynomials, we obtain generator polynomials for two-dimensional qc codes over r. moreover, we study the structure of generators two-dimensional qc codes. the last part of the paper is devoted to obtain 1-generator polynomial two-dimensional gqc codes and determine a lower bound for the minimum distance of free 1-generator gqc codes. 2. generator polynomials as was mentioned in the introduction, the purpose of this section is to obtain a generating set of polynomials for two-dimensional qc codes over the ring r = z4[u]/ < u2 − 1 > with u2 = 1. assume that s := z4[x]/ < xm − 1 >, r′ := r[x,y]/ < xm − 1,y3 − 1 >, where y3 = 1, xm = 1 and m is an odd positive integer. suppose that n = 3m` and r := r ′`. as before, fq denotes a finite field with q elements. recall that a linear code c′ of length ms over a finite field f is a two-dimensional cyclic code, if it is closed under row shift and column shift of codewords, whose codewords are viewed as ms arrays. this means that for every codeword c of the form c =   c0,0 c0,1 · · · c0,s−1 c1,0 c1,1 · · · c1,s−1 ... ... ... cm−1,0 cm−1,1 · · · cm−1,s−1   in c′, the codewords   cm−1,0 cm−1,1 · · · cm−1,s−1 c0,0 c0,1 · · · c0,s−1 ... ... ... cm−2,0 cm−2,1 · · · cm−2,s−1   and   c0,s−1 c0,0 · · · c0,s−2 c1,s−1 c1,0 · · · c1,s−2 ... ... ... cm−1,s−1 cm−1,01 · · · cm−1,s−2   58 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 also belong to c′. it is well known that these codes are the ideals of the quotient ring f [x,y]/ < xm − 1,ys − 1 >. similarly, we consider the above definition for a two-dimensional cyclic code c′ of length ms over the ring r. so we define a two-dimensional qc codes over r as follows. definition 2.1. let c be a linear code of length n. if there exists a least positive integer ` such that c is closed the `th composition under the row shift and the column shift, then we call c is a two-dimensional qc code over r. clearly, ` is a divisor of n. if ` = 1, then c is a two-dimensional cyclic code over r. an r-generator two-dimensional qc code is an ideal of c with r generators. in the rest of this section, we shall focus on 1-generator two-dimensional qc code over r. according to gao et al.[3], 1-generator two-dimensional qc code c over r can be generated by element (b1(x,y), . . . ,b`(x,y)) ∈r, and so c = {f(x,y)(b1(x,y), . . . ,b`(x,y))|f(x,y) ∈ r[x,y]} = {(f(x,y)b1(x,y), . . . ,f(x,y)b`(x,y))|f(x,y) ∈ r[x,y]}. özen et al.[10] have studied cyclic codes over r. in fact, they determined a generators of the cyclic codes over r. in [10], it was proved that if m is odd, then s is a principal ideal ring. now, by using a method similar that used for two-dimensional cyclic codes over a field in [13], we obtain a generator polynomials for two-dimensional cyclic codes over r. our generating set has an important role in determining generator polynomials two-dimensional qc codes over r. note that r is isomorphic to z4 + uz4. we begin with the following lemma. lemma 2.2. suppose that c′ is a two-dimensional cyclic code of length 3m over r. then {pi(x,y) | i = 1, . . . , 6 } is a generating set of c′, where p1(x,y) =α01(x) + (u + 1)α11(x) + (β01(x) + (u + 1)β11(x))y + (γ01(x) + (u + 1)γ11(x))y 2, p2(x,y) =(u + 1)α12(x) + (β02(x) + (u + 1)β12(x))y + (γ02(x) + (u + 1)γ12(x))y 2, p3(x,y) =(β03(x) + (u + 1)β13(x))y + (γ03(x) + (u + 1)γ13(x))y 2, p4(x,y) =(u + 1)β14(x)y + (γ04(x) + (u + 1)γ14(x))y 2, p5(x,y) =(γ05(x) + (u + 1)γ15(x))y 2, p6(x,y) =(u + 1)γ16(x)y 2, and α0i(x), α1i(x), β0i(x), β1i(x),γ0i(x) and γ1i(x) are generator polynomials of cyclic codes over z4 for each i = 1, . . . , 6. proof. suppose that i is an ideal of r′ and that f(x,y) is an arbitrary element of i. so it can be written uniquely as the following form f(x,y) = f0(x) + (u + 1)f1(x) + (f ′ 0(x) + (u + 1)f ′ 1(x))y + (f ′′ 0 (x) + (u + 1)f ′′ 1 (x))y 2, where f0(x), f′0(x), f ′′ 0 (x), f1(x), f ′ 1(x) and f ′′ 1 (x) are polynomials in s. the main strategy employed in our proof is to introduce six auxiliary ideals in s. to achieve this, we break our proof into six steps as follows: step 1: set i0 := {g0(x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) =g0(x) + (u + 1)g1(x) + (g ′ 0(x) + (u + 1)g ′ 1(x))y + (g ′′ 0 (x) + (u + 1)g ′′ 1 (x))y 2, where g1(x),g ′ 0(x),g ′ 1(x),g ′′ 0 (x),g ′′ 1 (x) ∈ s}. 59 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 it is not hard to see that the set i0 is an ideal of s. since m is odd, s is a principal ideal ring. thus, there exists a polynomial α01(x) in s such that i0 = 〈α01(x)〉. since α01 is an element of i0, according to the definition of i0, there exists p1(x,y) ∈ i with p1(x,y) = α01(x) + (u + 1)α11(x) + (β01(x) + (u + 1)β11(x))y + (γ01(x) + (u + 1)γ11(x))y 2, where α01(x), α11(x), β01(x), β11(x), γ01(x), γ11(x) ∈ s. it is clear that f0(x) ∈ i0. hence there exists t0(x) ∈ z4[x] such that f0(x) = α01(x)t0(x). set h1(x,y) :=f(x,y) −p1(x,y)t0(x) =(u + 1)h01(x) + (h ′ 01(x) + (u + 1)h ′ 11(x))y + (h ′′ 01(x) + (u + 1)h ′′ 11(x))y 2, where h01(x), h′01(x), h ′ 11(x), h ′′ 01(x), h ′′ 11(x) ∈ s. since f(x,y) and p1(x,y) are in i and i is an ideal of r, h1(x,y) is again in i. step 2: put i′0 := {g1(x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) =(u + 1)g1(x) + (g ′ 0(x) + (u + 1)g ′ 1(x))y + (g ′′ 0 (x) + (u + 1)g ′′ 1 (x))y 2, where g′0(x),g ′ 1(x),g ′′ 0 (x),g ′′ 1 (x) ∈ s}. clearly, i′0 is an ideal of s. thus, there exists a polynomial α12(x) ∈ s such that i′0 = 〈α12(x)〉. there exists p2(x,y) ∈ i such that p2(x,y) = (u + 1)α12(x) + (β02(x) + (u + 1)β12(x))y + (γ02(x) + (u + 1)γ12(x))y 2, where β02(x), β12(x), γ02(x), γ12(x) ∈ s. according to the definition of i′0, h01(x) ∈ i′0, and so h01(x) = α12(x)t1(x) for some t1(x) ∈ z4[x]. set h2(x,y) :=h1(x,y) −p2(x,y)t1(x) =(h′02(x) + (u + 1)h ′ 12(x))y + (h ′′ 02(x) + (u + 1)h ′′ 12(x))y 2, where h′02(x), h ′ 12(x), h ′′ 02(x), h ′′ 12(x) ∈ s. since h1(x,y) and p2(x,y) are polynomials in i we have that h2(x,y) ∈ i. step 3: set i1 := {g′0(x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) =(g′0(x) + (u + 1)g ′ 1(x))y + (g ′′ 0 (x) + (u + 1)g ′′ 1 (x))y 2, where g′1(x),g ′′ 0 (x),g ′′ 1 (x) ∈ s}. obviously, i1 is an ideal of s, and so there exists a polynomial β03(x) in s such that i1 = 〈β03(x)〉. there exists a polynomial p3(x,y) ∈ i such that p3(x,y) = (β03(x) + (u + 1)β13(x))y + (γ03(x) + (u + 1)γ13(x))y 2, where β13(x), γ03(x), γ13(x) ∈ s. according to the definition of i1, h′02(x) in i1. hence h′02(x) = β03(x)t2(x) for some t2(x) ∈ z4[x]. put h3(x,y) :=h2(x,y) −p3(x,y)t2(x) =(u + 1)h′13(x)y + (h ′′ 03(x) + (u + 1)h ′′ 13(x))y 2, where h′13(x), h ′′ 03(x), h ′′ 13(x) ∈ s. similar to the previous discussion h3(x,y) ∈ i. 60 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 step 4: set i′1 := {g′1(x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) = (u + 1)g′1(x)y + (g ′′ 0 (x) + (u + 1)g ′′ 1 (x))y 2, where g′′0 (x),g ′′ 1 (x) ∈ s}. it is clear that i′1 is an ideal of s. thus, there exists a polynomial β13(x) ∈ s such that i′1 = 〈β13〉. since β13(x) in i′1, according to definition i ′ 1, we have a polynomial p4(x,y) ∈ i, where p4(x,y) = (u + 1)β13(x)y + (γ03(x) + (u + 1)γ13(x))y 2, and γ03(x), γ13(x) ∈ s. obviously, h′13(x) ∈ i′1. so, we get h′13(x) = β13(x)t3(x) for some t3(x) ∈ z4[x]. put h4(x,y) := h3(x,y) −p4(x,y)t3(x) = (h04(x) + (u + 1)h14(x))y2, where h04(x), h14(x) ∈ s. it is clear that h4(x,y) ∈ i. step 5: set i2 := {g′′0 (x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) = (g′′0 (x) + (u + 1)g ′′ 1 (x))y 2, where g′′1 (x) ∈ s}. clearly, i2 is an ideal of s. therefore, there exists γ05(x) ∈ s such that i2 = 〈γ05(x)〉. besides, γ05(x) in i2, and so we have a polynomial p5(x,y) ∈ i, where p5(x,y) = (γ05(x) + (u + 1)γ15(x))y 2, where γ15(x) ∈ s. obviously, h04(x) ∈ i2, and so we obtain that h04(x) = γ05(x)t4(x) for some t4(x) ∈ z4[x]. put h5(x,y) := h4(x,y)−p5(x,y)t4(x) = (u + 1)h05(x)y2, where h05(x) ∈ s. similarly, h5(x,y) in i. step 6: put i′2 := {g ′′ 1 (x) ∈ s : there exists g(x,y) ∈ i such that g(x,y) = (u + 1)g ′′ 1 (x)y 2}. it is clear that i′2 is an ideal of s. thus there exists γ15(x) ∈ s such that i′2 =< γ15(x) >. also there exists a polynomial p6(x,y) ∈ i such that p6(x,y) = (u + 1)γ15(x)y2. now, since h05 ∈ i′2, there exists t5(x) ∈ s such that h05(x) = γ15(x)t5(x). therefore, h5(x,y) = (u + 1)γ15(x)t5(x)y2 = p6(x,y)t5(x). now, we get f(x,y) = h1(x,y) + p1(x,y)t0(x), h1(x,y) = h2(x,y) + p2(x,y)t1(x), h2(x,y) = h3(x,y) + p3(x,y)t2(x), h3(x,y) = h4(x,y) + p4(x,y)t3(x), h4(x,y) = h5(x,y) + p5(x,y)t4(x), h5(x,y) = p6(x,y)t5(x). these equality imply that f(x,y) = p1(x,y)t0(x) + p2(x,y)t1(x) + p3(x,y)t2(x) + p4(x,y)t3(x) + p5(x,y)t4(x) + p6(x,y)t5(x). thus i = 〈p1(x,y),p2(x,y),p3(x,y),p4(x,y),p5(x,y),p6(x,y)〉. 61 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 now, we state an important lemma. lemma 2.3. let c be a 1-generator two-dimensional qc code of length n = 3m` which is generated by g(x,y) = (g1(x,y),g2(x,y), . . . ,g`(x,y)) ∈ r, where gi(x,y) ∈ r′ for all i with 1 ≤ i ≤ `. then gi(x,y) ∈ ci, where ci is a two-dimensional cyclic code of length m over r. furthermore, if m is odd, then gi(x,y) can be selected as the form gi(x,y) = ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y 2, where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in r[x] for all i with 1 ≤ i ≤ ` and moreover, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all i = 1, · · · ,`. proof. consider the projection map ψi : r→ r′ given by ψi(k1(x,y), . . . ,k`(x,y)) = ki(x,y), where ki(x,y) ∈ r′ for all i = 1, · · · ,`. it is clear that the set ψi(c) is a two-dimensional cyclic code over r for all i with 1 ≤ i ≤ `. now, in view of lemma 2.2, for all 1 ≤ i ≤ `, one can obtain a generator for ψi(c) as follows ψi(c) = 〈p1i(x,y),p2i(x,y),p3i(x,y),p4i(x,y),p5i(x,y),p6i(x,y)〉, where, for each j = 1, . . . , 6, pji(x,y) are polynomials as described in lemma 2.2 . since gi(x,y) ∈ ψi(c) for all 1 ≤ i ≤ `, there exists a polynomial fi(x,y) ∈ r[x,y] such that gi(x,y) = fi(x,y)(α i 01(x) + (u + 1)α i 11(x) + (β i 01(x) + (u + 1)β i 11(x))y + (γi01(x) + (u + 1)γ i 11(x))y 2) = ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y 2, where αi01(x), α i 11(x), β i 01(x), β i 11(x), γ i 01(x) and γ i 11(x) are generator polynomials of cyclic codes over z4 and, for all 1 ≤ i ≤ `, ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in r[x]. moreover, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all i = 1, · · · ,`. in the light of the above two lemmas, we will obtain the minimal generating sets for 1-generator two-dimensional qc codes. theorem 2.4. let c be a 1-generator two-dimensional qc code of length n = 3m` over r which is generated by g = (g1(x,y),g2(x,y), . . . ,g`(x,y)), with m is odd and gi(x,y) = ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y 2, where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in r[x] for all i with 1 ≤ i ≤ ` and moreover, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all 1 ≤ i ≤ `. assume that deg(ϕ0i(x)) >deg(ϕ1i(x)), deg(ψ0i(x)) >deg(ψ1i(x)) and deg(θ0i(x)) >deg(θ1i(x)), for all 1 ≤ i ≤ `, and that the polynomials ϕ0i(x) + (u + 1)ϕ1i(x), (ψ0i(x) + (u + 1)ψ1i(x))y and (θ0i(x) + (u + 1)θ1i(x))y2 are not zero divisor in r′. assume that g0(x) = gcd{ϕ01(x),ϕ02(x), . . . ,ϕ0`(x)}, q0(x) = gcd{ϕ11(x),ϕ12(x), . . . ,ϕ1`(x)}, g1(x) = gcd{ψ01(x),ψ02(x), . . . ,ψ0`(x)}, q1(x) = gcd{ψ11(x),ψ12(x), . . . ,ψ1`(x)}, g2(x) = gcd{θ01(x),θ02(x), . . . ,θ0`(x)}, q2(x) = gcd{θ11(x),θ12(x), . . . ,θ1`(x)} 62 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 and, for k = 0, 1, 2, gk(x)|xm − 1 and qk(x)|xm − 1. let s1 = r0−1⋃ j=0 {xj(ϕ01(x) + (u + 1)ϕ11(x), . . . ,ϕ0`(x) + (u + 1)ϕ1`(x))}, s2 = r1−1⋃ j=0 {xj((ψ01(x) + (u + 1)ψ11(x))y, . . . , (ψ0`(x) + (u + 1)ψ1`(x))y)}, s3 = r2−1⋃ j=0 {xj((θ01(x) + (u + 1)θ11(x))y2, . . . , (θ0`(x) + (u + 1)θ1`(x))y2)}, s4 = t0−1⋃ j=0 {xj((u + 1)h0ϕ11(x), . . . , (u + 1)h0ϕ1`(x))}, s5 = t1−1⋃ j=0 {xj((u + 1)h1ψ11(x)y, . . . , (u + 1)h1ψ1`(x)y)}, s6 = t2−1⋃ j=0 {xj((u + 1)h2θ11(x)y2, . . . , (u + 1)h2θ1`(x)y2)}, where, for all k = 0, 1, 2, we set rk := deg(x m−1 gk(x) ) and tk := deg(x m−1 qk(x) ). then s1 ∪s2 ∪s3 ∪s4 ∪s5 ∪s6 is a minimal generating set for c. moreover, | c |= 16r0+r1+r2 4t0+t1+t2 for all 1 ≤ i ≤ `. proof. for k = 0, 1, 2, put hk(x) := x m−1 gk(x) and δk(x) := x m−1 qk(x) , and let c(x,y) = f(x,y)g be a codeword in c, where f(x,y) = f0(x) + f1(x)y + f2(x)y2, with fi(x) ∈ r[x] for all 1 ≤ i ≤ 3. for simplicity of presentation, in our proof, we will use the notion f instead of f(x). by the division algorithm, we get the unique polynomials q0(x), q1(x), q2(x), r0(x), r1(x), r2(x) in r[x] such that f0 = h0q0 + r0, where r0 = 0 or deg(r0) < r0, f1 = h1q1 + r1, where r1 = 0 or deg(r1) < r1, f2 = h2q2 + r2, where r2 = 0 or deg(r2) < r2. there exist polynomials ai, a′i, a ′′ i ∈ z4[x] such that h0ϕ0i = h0g0ai = 0, h1ψ0i = h1g1a ′ i = 0, h2θ0i = h2g2a ′′ i = 0 for all 1 ≤ i ≤ `. we have c(x,y) = f(x,y)g = (h0q0 + r0)(ϕ01 + (u + 1)ϕ11, . . . ,ϕ1` + (u + 1)ϕ1`) + (h1q1 + r1)((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y) + (h2q2 + r2)((θ01 + (u + 1)θ11)y 2, . . . , (θ0` + (u + 1)θ1`)y 2) = q0h0((u + 1)ϕ11, . . . , (u + 1)ϕ1`) + r0(ϕ01 + (u + 1)ϕ11, . . . ,ϕ0` + (u + 1)ϕ1`) + q1h1((u + 1)ψ11y, . . . , (u + 1)ψ1`y) + r1((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y) + q2h2((u + 1)θ11y 2, . . . , (u + 1)θ1`y 2) + r2((θ01 + (u + 1)θ11)y 2, . . . , (θ0` + (u + 1)θ1`)y 2). they are not difficult to verify that r0(ϕ01 + (u + 1)ϕ11, . . . ,ϕ0` + (u + 1)ϕ1`) ∈span(s0), r1((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y) ∈span(s1), and r2((θ01 + (u + 1)θ11)y 2, . . . , (θ0` + (u + 1)θ1`)y 2) ∈span(s2). 63 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 again, using the division algorithm, we get the unique polynomials q′0(x),q ′ 1(x), q′2(x),r ′ 0(x),r ′ 1(x), r ′ 2(x) ∈ r[x] such that q0 = δ0q ′ 0 + r ′ 0, where r ′ 0 = 0 or deg(r ′ 0) < t0, q1 = δ1q ′ 1 + r ′ 1, where r ′ 1 = 0 or deg(r ′ 1) < t1, and q2 = δ2q ′ 2 + r ′ 2, where r ′ 2 = 0 or deg(r ′ 2) < t2. there exist polynomials bi, b′i, b ′′ i ∈ z4[x] such that δ0q ′ 0(u + 1)h0ϕ1i = (u + 1)q ′ 0q0δ0bi = 0, δ1q ′ 1(u + 1)h1ψ1i = (u + 1)q ′ 1q1δ1b ′ i = 0, and δ2q ′ 2(u + 1)h2θ1i = (u + 1)q ′ 2q2δ2b ′′ i = 0 in r′ for all 1 ≤ i ≤ `. it is not hard to see that q0((u + 1)h0ϕ11, . . . , (u + 1)h0ϕ1`) = r ′ 0((u + 1)h0ϕ11, . . . , (u + 1)h0ϕ1`) ∈ span(s4), q1((u + 1)h1ψ11y, . . . , (u + 1)h1ψ1`y) = r ′ 1((u + 1)h1ψ11y, . . . , (u + 1)h1ψ1`y) ∈ span(s5), and q2((u + 1)h2θ11y 2, . . . , (u + 1)h2θ1`y 2) = r′2((u + 1)h2θ11y 2, . . . , (u + 1)h2θ1`y 2) ∈ span(s6). thus s1∪s2∪s3∪s4∪s5∪s6 is a spanning set for c. also, it is clear s1∩s2∩s3∩s4∩s5∩s6 = {0}. with the aid of the above theorem, we obtain the following corollary. corollary 2.5. if ` is a positive integer and ϕ1i(x) = xm −1, ψ1i(x) = xm −1 and θ1i(x) = xm −1 are polynomials over r, for all i with 1 ≤ i ≤ `, then c is a free two-dimensional qc code of rank r0 +r1 +r2 over r and its minimal generating set is s1 ∪s2 ∪s3 such that s1 = r0−1⋃ j=0 {xj(ϕ01(x), . . . ,ϕ0`(x))}, s2 = r1−1⋃ j=0 {xj(ψ01(x)y, . . . ,ψ0`(x)y)}, s3 = r2−1⋃ j=0 {xj(θ01(x)y2, . . . ,θ0`(x)y2)}, where, for all k = 0, 1, 2, we set rk := deg(x m−1 gk(x) ). furthermore, | c |= 16r0+r1+r2 . proof. by theorem 2.4, if ϕ1i(x) = xm − 1, ψ1i(x) = xm − 1 and θ1i(x) = xm − 1 are polynomials over r, for all i with 1 ≤ i ≤ `, then q0 =gcd{ϕ11(x), . . . ,ϕ1`(x),xm − 1} = xm − 1, q1 =gcd{ψ11(x), . . . ,ψ1`(x),xm − 1} = xm − 1, and q2 =gcd{θ11(x), . . . ,θ1`(x),xm − 1} = xm − 1. hence δ0 = 1, δ1 = 1 and δ2 = 1. clearly, s1 ∩ s2 ∩ s3 = {0}. therefore, its minimal generating set is s1 ∪ s2 ∪ s3. this means that c is a free two-dimensional qc code of rank r0 + r1 + r2. thus | c |= 16r0+r1+r2. in the next theorem, we provide a lower bound on minimum distance of the free 1-generator twodimensional qc codes over r. 64 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 theorem 2.6. let c be a free 1-generator two-dimensional qc code of length n = 3m` over r as in corollary 2.5. suppose that h0i =(x m − 1)/ϕ0i(x), h1i = (xm − 1)/ψ0i(x), h2i =(x m − 1)/θ0i(x), h0 = lcm{h01, . . . ,h0`}, h1 =lcm{h11, . . . ,h1`} and h2 = lcm{h21, . . . ,h2`} for all i with 1 ≤ i ≤ `. then we have the following statements. (i) dmin(c) ≥ ∑ i/∈a d0i + ∑ j/∈b d1j + ∑ t/∈d d2t for all 1 ≤ i,j,t ≤ `, where a,b,c ⊆{1, 2, . . . ,`} are sets from maximum size for which lcm{h0i, i ∈ a} 6= h0, lcm{h1j,j ∈ b} 6= h1 and lcm{h2t, t ∈ d} 6= h2. (ii) if h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then dmin(c) ≥ ∑̀ i=1 d0i + ∑̀ i=1 d1i + ∑̀ i=1 d2i. proof. (i) consider the projection map ψi : r→ r′ given by ψi(k1(x,y), . . . ,k`(x,y)) = ki(x,y), where ki(x,y) ∈ r′ for all i with 1 ≤ i ≤ `. it is easy to show that ψi(c) is a two-dimensional code over r. let c(x,y) = f(x,y)g be a nonzero codeword in c, where f(x,y) ∈ r[x,y]. since c is a free 1-generator two-dimensional qc code, we have that ϕ1i(x) = xm − 1, ψ1i(x) = xm − 1, θ1i(x) = xm − 1 for all 1 ≤ i ≤ `. so, the i-th component is zero if and only if (xm − 1) | f(x,y)g. this means that (xm−1) | f0(x)ϕ0i(x), (xm−1) | f1(x)ψ0i(x) and (xm−1) | f2(x)θ0i(x), that is, if and only if h0i | f0(x), h1i | f1(x), h2i | f2(x) for all 1 ≤ i ≤ `. thus c(x,y) = 0 if and only if h0 | f0(x), h1 | f1(x) and h2 | f2(x). therefore, c(x,y) 6= 0 if and only if h0 f0(x) or h1 f1(x) or h2 f2(x). thus, c(x,y) 6= 0 have the most number of zero blocks whenever h0 6= lcm{h0i, i ∈ a}, where lcm{h0i, i ∈ a} | f0(x), h1 6= lcm{h1j,j ∈ b}, where lcm{h1j,j ∈ b} | f1(x), h2 6= lcm{h2t, t ∈ d}, where lcm{h2t, t ∈ d} | f2(x), where a, b and d are a maximal subset of {1, 2, . . . ,`} having this property. thus dmin(c) ≥ ∑ i/∈a d0i + ∑ i/∈b d1i + ∑ i/∈d d2i. (ii) now, we know that a = ∅ if and only if h01 = h02 = . . . = h0` and also, b = ∅ if and only if h11 = h12 = . . . = h1`. moreover, d = ∅ if and only if h21 = h22 = . . . = h2`. thus, dmin(c) ≥ ∑` i=1 d0i + ∑` i=1 d1i + ∑` i=1 d2i. corollary 2.7. let c be a 1-generator two-dimensional qc code of length n = 3m` over r which is generated by g = (ϕ01(x) + (u + 1)ϕ11(x) + (ψ01(x) + (u + 1)ψ11(x))y+ (θ01(x) + (u + 1)θ11(x))y 2, . . . ,ϕ0`(x) + (u + 1)ϕ1`(x)+ (ψ0`(x) + (u + 1)ψ1`)y + (θ0`(x) + (u + 1)θ1`(x))y 2), 65 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 where m is odd. assume that ϕ1i(x) = xm−1, ψ1i(x) = xm−1 and θ1i(x) = xm−1 for each i = 1, 2, . . . ,`. let h0i = (xm − 1)/ϕ0i(x), h1i = (xm − 1)/ψ0i(x) and h2i = (xm − 1)/θ0i(x), for all i with 1 ≤ i ≤ `, and that h0 = lcm{h01,h02, . . . ,h0`}, h1 = lcm{h11,h12, . . . ,h1`}, and h2 = lcm{h21,h22, . . . ,h2`}. then (i) c is a free two-dimensional qc code from deg(h0)+ deg(h1)+ deg(h2). moreover, | c |= 16deg(h0)+deg(h1)+deg(h2). (ii) dmin(c) ≥ ∑ i/∈a d0i + ∑ i/∈b d1i + ∑ i/∈d d2i, where a,b,d ⊆{1, 2, . . . ,`} are set from maximum size for which lcm{h0i, i ∈ a} 6= h0, lcm{h1j,j ∈ b} 6= h1 and lcm{h2t, t ∈ d} 6= h2. (iii) if h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then dmin(c) ≥ ∑̀ i=1 d0i + ∑̀ i=1 d1i + ∑̀ i=1 d2i. proof. let c(x,y) = f(x,y)g be a codeword in c such that f(x,y) = f0(x) + f1(x)y + f2(x)y2, where fi(x) ∈ r[x] for i = 0, 1, 2. by the division algorithm, we can find unique polynomials q1(x),q2(x),q3(x),r1(x),r2(x), r3(x) ∈ r[x] such that f0(x) = h0q1(x) + r1(x), where r1(x) = 0, or degr1(x) < deg(h0), f1(x) = h1q2(x) + r2(x), where r2(x) = 0, or degr2(x) < deg(h1), f2(x) = h2q2(x) + r3(x), where r3(x) = 0, or degr3(x) < deg(h2). now, we have c(x,y) = f(x,y)g = (h0q1(x) + r1(x))(ϕ01(x), . . . ,ϕ0`(x)) + (h1q2(x) + r2(x))(ψ01(x)y, . . . ,ψ0`(x)y) + (h2q3(x) + r3(x))(θ01(x)y 2, . . . ,θ0`(x)y 2). we know that h0ϕ0i(x) = h1ψ0i(x) = h2θ0i(x) = xm − 1. therefore, we obtain r1(x)(ϕ01(x), . . . ,ϕ0`(x)) ∈ span(s1), r2(x)(ψ01(x)y, . . . ,ψ0`(x)y) ∈ span(s2) and r3(x)(θ01(x)y 2, . . . ,θ0`(x)y 2 ∈ span(s3). thus t0 = 0, t1 = 0 and t2 = 0 which implies that s4 = s5 = s6 = ∅. using the definition of free module, we obtain c is a free two-dimensional qc code of rank deg(h0) + deg(h1) + deg(h2). therefore, | c |= 16deg(h0)+deg(h1)+deg(h2). the statements (ii) and (iii) follow from theorem 2.6. 66 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 3. 1-generator two-dimensional gqc codes in this section, we study two-dimensional gqc codes over r. at first, we recall the definition of 1-generator two-dimensional gqc codes over r. definition 3.1. let m1, m2, . . . ,m` be positive integers and ri = r[x,y]/〈xmi − 1,y3 − 1〉 for all i with 1 ≤ i ≤ `. any ideal of r = r1 ×r2 × . . .×r` is called a two-dimensional gqc code of length (m1,m2, . . . ,m`) with index ` over r. if c is a two-dimensional gqc code of length (m1,m2, . . . ,m`) with m = m1 = . . . = m`, then c is a two-dimensional qc code with length n = 3m`. lemma 3.2. let c be a 1-generator two-dimensional gqc code of length (m1, . . . ,m`) and g′(x,y) = (g′1(x,y),g ′ 2(x,y), . . . ,g ′ `(x,y)) ∈ r be a generator of c, where g ′ i(x,y) ∈ ri for all i with 1 ≤ i ≤ `. then g′i(x,y) ∈ ci, where ci is a two-dimensional cyclic code of length mi over r for i = 1, · · · ,`. also, if mi is odd, then g′i(x,y) can be selected to be of the form g ′ i(x,y) = (ϕ0i(x) + (u+ 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y 2) where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in r[x] for all i with 1 ≤ i ≤ `. furthermore, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all 1 ≤ i ≤ `. by using a method similar to that we used in the proof of theorem 2.4, one can obtain the next theorem which gives the minimal generating set of 1-generator two-dimensional gqc codes over r. theorem 3.3. let c be a 1-generator two-dimensional gqc code of length (m1,m2, . . . ,m`) over r which is generated by g′(x,y) = (g′1(x,y),g ′ 2(x,y), . . . ,g ′ `(x,y)), where mi is odd for all i with 1 ≤ i ≤ `. then g′i(x,y) = ϕ0i(x) + (u+ 1)ϕ1i(x) + (ψ0i(x) + (u+ 1)ψ1i)y + (θ0i(x) + (u+ 1)θ1i(x)y 2), where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in r[x] for all i with 1 ≤ i ≤ `. furthermore, ϕ0i(x), ψ0i(x) and θ1i(x) are monic polynomials for all 1 ≤ i ≤ `. assume that deg(ϕ0i(x)) ≥ deg(ϕ1i(x)), deg(ψ0i(x)) ≥ deg(ψ1i(x)) and that deg(θ0i(x)) ≥ deg(θ1i(x)). suppose that polynomials ϕ0i(x) + (u + 1)ϕ1i(x), (ψ0i(x) + (u + 1)ψ1i)y, (θ0i(x) + (u + 1)θ1i(x))y 2 are not zero-divisor of ri. let h0i = (x mi − 1)/gcd(ϕ0i(x),xmi − 1), h0 = lcm(h01, . . . ,h0l), deg(h0) = r0, δ0,i = (x mi − 1)/gcd(h0ϕ1i(x),xmi − 1), δ0 = lcm(δ01, . . . ,δ0`) and deg(δ0) = t0. let h1i = (x mi − 1)/gcd(ψ0i(x),xmi − 1), h1 = lcm(h11,h12, . . . ,h1`), deg(h1) = r1, δ1i = (x mi − 1)/gcd(h1ψ1i(x),xmi − 1), δ1 = lcm(δ11,δ12, . . . ,δ1`) and deg(δ1) = t1. suppose that h2i = (x mi − 1)/gcd(θ0i(x),xmi − 1), h2 = lcm(h21, . . . ,h2`), deg(h2) = r2, δ2i = (x mi − 1)/gcd(h2θ1i(x),xmi − 1), δ2 = lcm(δ21,δ22, . . . ,δ2`) and deg(δ2) = t2. 67 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 then the minimal generating set of c is s1 ∪s2 ∪s3 ∪s4 ∪s5 ∪s6, where s1 = r0−1⋃ j=0 {xj(ϕ01(x) + (u + 1)ϕ11(x), . . . ,ϕ0`(x) + (u + 1)ϕ1`(x))}, s2 = r1−1⋃ j=0 {xj((ψ01(x) + (u + 1)ψ1i(x))y, . . . , (ψ0`(x) + (u + 1)ψ1`(x))y)}, s3 = r2−1⋃ j=0 {xj((θ01(x) + (u + 1)θ11(x))y2, . . . , (θ0`(x) + (u + 1)θ1`(x))y2)}, s4 = t0−1⋃ j=0 {xj((u + 1)h0ϕ11(x), . . . , (u + 1)h0ϕ1`(x))}, s5 = t1−1⋃ j=0 {xj((u + 1)h1ψ11(x)y, . . . , (u + 1)h1ψ1`(x)y)}, s6 = t2−1⋃ j=0 {xj((u + 1)h2θ11(x)y2, . . . , (u + 1)h2θ1`(x)y2)}. thus | c |= 16r0+r1+r2 4t0+t1+t2 . according to theorem 3.3, we have the following corollary. corollary 3.4. if ` is a positive integer and ϕ1i(x) = xmi −1, ψ1i(x) = xmi −1 and θ1i(x) = xmi −1 are polynomials over r for all i with 1 ≤ i ≤ `, then c is a free two-dimensional gqc code of rank r0 +r1 +r2 over r and its minimal generating set is s1 ∪s2 ∪s3. furthermore, c has 16r0+r1+r2 codewords. in the following theorem, we give a lower bound on the minimum distance of free 1-generator twodimensional gqc codes over r. its proof is exactly the same as the proof of theorem 2.6, so we delete it. theorem 3.5. let c be a free 1-generator two-dimensional gqc code of length (m1,m2, . . . ,m`) over r as in corollary 3.4. let h0i = (x mi − 1)/ϕ0i(x), h0 = lcm{h01, . . . ,h0`}, h1i = (x mi − 1)/ψ0i(x), h1 = lcm{h11, . . . ,h1`}, h2i = (x mi − 1)/θ0i(x) and h2 = lcm{h21, . . . ,h2`}. then (i) dmin(c) ≥ ∑ i/∈a d0i + ∑ i/∈b d1i + ∑ i/∈d d2i, where a,b,d ⊆ {1, 2, . . . ,`} are sets of maximum size for which lcm{h0i, i ∈ a} 6= h0, lcm{h1j,j ∈ b} 6= h1 and lcm{h2t, t ∈ d} 6= h2. (ii) if h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then dmin(c) ≥ ∑̀ i=1 d0i + ∑̀ i=1 d1i + ∑̀ i=1 d2i. according to corollary 3.4 and theorem 3.5 we obtain the following corollary. 68 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 corollary 3.6. assume that ` is a positive integer and ϕ1i(x) = xmi − 1, ψ1i(x) = xmi − 1 and θ1i(x) = x mi − 1 are polynomials over r for all i with 1 ≤ i ≤ `. let h0i = (x mi − 1)/ϕ0i(x), h0 = lcm{hoi, . . . ,h0`}, h1i = (x mi − 1)/ψ0i(x), h1 = lcm{h11, . . . ,h1`}, h2i = (x mi − 1)/θ0i(x) and h2 = lcm{h21, . . . ,h2`} for all 1 ≤ i ≤ `. then we have the following statements. (i) c is a free two-dimensional code of rank deg(h0) + deg(h1) + deg(h2). moreover, | c |= 16deg(h0)+deg(h1)+deg(h2), (ii) dmin(c) ≥ ∑ i/∈a d0i + ∑ j/∈b d1j + ∑ k/∈d d2k, where a, b, d ⊆{1, . . . ,`}, (iii) let h01 = . . . = h0` = h0, h11 = . . . = h1` = h1 and h21 = . . . = h2` = h2. then we have dmin(c) ≥ ∑` i=1 d0i + ∑` i=1 d1i + ∑` i=1 d2i. 4. conclusion this paper is devoted to the study of two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes of length 3m` which are a natural generalization of quasi-cyclic codes and generalized quasi-cyclic codes over the ring r = z4[u]/〈u2 − 1〉 with u2 = 1. we first determine the generator polynomials of two-dimensional cyclic codes over r. then we find the generator polynomials of two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over r and give their minimal generating sets. moreover, we study the minimum distances of the family of the free 1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes. references [1] y. cao, structural properties and enumeration of 1-generator generalized quasi-cyclic codes, des. codes cryptogr. 60(1) (2011) 67–79. [2] y. cao, j. gao, constructing quasi-cyclic codes from linear algebra theory, des. codes cryptogr. 67(1) (2013) 59–75. [3] m. esmaeili, s. yari, generalized quasi-cyclic codes: structural properties and code construction, appl. algebra engrg. comm. comput 20(2) (2009) 159–173. [4] y. gao, j. gao, t. wu, f. w. fu, 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring z4[u]〈u2−1〉, appl. algebra engrg. comm. comput. 28(6) (2017) 457–467. [5] c. güneri, f. özbudak, b. özkaya, e. saãğäśkara, z. sepasdar, p. solé, structure and performance of generalized quasi-cyclic codes, finite fields appl. 47 (2017) 183–202. [6] t. ikai, h. kosako, y. kojima, two-dimensional cyclic codes, electron. comm. japan 57(4) (1974/75) 27–35. [7] r. m. lalasoa, r. andriamifidisoa, t. j. rabeherimanana, basis of a multicyclic code as an ideal in fq[x1, . . . ,xs]/〈x ρ1 1 − 1, . . . ,x ρs s − 1〉, j. algebra relat. topics 6(2) (2018) 63–78. [8] s. ling, ch. xing, coding theory: a first course, cambridge university press, new york (2004). [9] s. ling, p. solé, on the algebraic structure of quasi-cyclic codes. i. finite fields, ieee trans. inform. theory 47(7) (2001) 2751–2760. [10] m. özen, f. z. uzekmek, n. aydin, n.t. özzaim, cyclic and some constacyclic codes over the ring z4[u] 〈u2−1〉, finite fields appl. 38 (2016) 27–39. 69 https://doi.org/10.1007/s10623-010-9417-5 https://doi.org/10.1007/s10623-010-9417-5 https://doi.org/10.1007/s10623-011-9586-x https://doi.org/10.1007/s10623-011-9586-x https://doi.org/10.1007/s00200-009-0095-3 https://doi.org/10.1007/s00200-009-0095-3 https://doi.org/10.1007/s00200-017-0315-1 https://doi.org/10.1007/s00200-017-0315-1 https://doi.org/10.1016/j.ffa.2017.06.005 https://doi.org/10.1016/j.ffa.2017.06.005 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://mathscinet.ams.org/mathscinet-getitem?mr=3938616 https://mathscinet.ams.org/mathscinet-getitem?mr=3938616 https://doi.org/10.1017/cbo9780511755279 https://doi.org/10.1109/18.959257 https://doi.org/10.1109/18.959257 https://doi.org/10.1016/j.ffa.2015.12.003 https://doi.org/10.1016/j.ffa.2015.12.003 a. ghajari et. al. / j. algebra comb. discrete appl. 9(1) (2022) 57–70 [11] z. sepasdar, generator matrix for two-dimensional cyclic codes of arbitrary length, arxiv:1704.08070v1 [math.ac] 26 apr (2017). [12] z. sepasdar, some notes on the characterization of two dimensional skew cyclic codes, j. algebra relat. topics 4(2) (2016) 1–8. [13] z. sepasdar, k. khashyarmanesh, characterizations of some two-dimensional cyclic codes correspond to the ideals of f[x,y]/〈xs − 1,y2 k − 1〉, finite fields appl. 41 (2016) 97–112. [14] i. siap, t. abualrub, b. yildiz, one generator quasi-cyclic codes over f2 + uf2, j. frankl. inst. 349(1) (2012) 284–292. 70 https://arxiv.org/abs/1704.08070 https://arxiv.org/abs/1704.08070 https://mathscinet.ams.org/mathscinet-getitem?mr=3597268 https://mathscinet.ams.org/mathscinet-getitem?mr=3597268 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1016/j.jfranklin.2011.10.020 https://doi.org/10.1016/j.jfranklin.2011.10.020 introduction generator polynomials 1-generator two-dimensional gqc codes conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.560406 j. algebra comb. discrete appl. 6(2) • 75–94 received: 2 january 2019 accepted: 16 april 2019 journal of algebra combinatorics discrete structures and applications self-dual and complementary dual abelian codes over galois rings∗ research article somphong jitman, san ling abstract: self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. in this paper, abelian codes over galois rings are studied in terms of the ideals in the group ring gr(pr, s)[g], where g is a finite abelian group and gr(pr, s) is a galois ring. characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in gr(pr, s)[g]. a general formula for the number of such selfdual codes is established. in the case where gcd(|g|, p) = 1, the number of self-dual abelian codes in gr(pr, s)[g] is completely and explicitly determined. applying known results on cyclic codes of length pa over gr(p2, s), an explicit formula for the number of self-dual abelian codes in gr(p2, s)[g] are given, where the sylow p-subgroup of g is cyclic. subsequently, the characterization and enumeration of complementary dual abelian codes in gr(pr, s)[g] are established. the analogous results for self-dual and complementary dual cyclic codes over galois rings are therefore obtained as corollaries. 2010 msc: 94b15, 94b05, 16a26 keywords: abelian codes, galois rings, self-dual codes, complementary dual codes, codes over rings 1. introduction algebraically structured codes over finite fields with self-duality and complementary duality are important families of linear codes that have been extensively studied for both theoretical and practical reasons (see [1], [3], [11], [13], [15], [21], [26], [27], and references therein). codes over finite rings have been interesting since it was proven that some binary non-linear codes such as the kerdock, preparata, and goethal codes are the gray images of linear codes over z4 [10]. algebraically structured codes such ∗ s. jitman was supported by the thailand research fund and silpakorn university under research grant rsa6280042. s. ling was supported by nanyang technological university research grant m4080456. somphong jitman(corresponding author); department of mathematics, faculty of science, silpakorn university, nakhon pathom 73000, thailand (email: sjitman@gmail.com). san ling; division of mathematical sciences, school of physical and mathematical sciences, nanyang technological university, singapore 637371, republic of singapore (email: lingsan@ntu.edu.sg). 75 https://orcid.org/0000-0003-1076-0866 https://orcid.org/0000-0002-1978-3557 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 as cyclic, constacyclic, and abelian codes have extensively been studied over zpr, galois rings, and finite chain rings in general (see [7],[18], and references therein). the characterization and enumeration of euclidean self-dual cyclic codes over finite fields have been established in [11] and generalized to euclidean and hermitian self-dual abelian codes over finite fields in [13] and [15], respectively. over some finite rings, a characterization of self-dual cyclic, constacyclic and abelian codes has been done (see, for example, [1], [7],[16], [17], [24], and [26]). in [1], [5], [4] and [23], characterization and enumeration of euclidean and hermitian self-dual cyclic codes over finite chain rings have been discussed. euclidean complementary dual cyclic codes over finite fields have been studied in [27]. recently, they have been generalized to euclidean and hermitian complementary dual abelian codes over finite fields in [3]. the complete characterization and enumeration of complementary dual abelian codes over finite fields have been established in the said paper. in this paper, we focus on abelian codes over galois rings gr(pr,s), i.e., ideals in the group ring gr(pr,s)[g] of an abelian group g over a galios ring gr(pr,s). specifically, we study self-dual and complementary dual abelian codes in gr(pr,s)[g] with respect to both the euclidean and hermitian inner products. we characterize such self-dual abelian codes and determine necessary and sufficient conditions for the existence of a self-dual abelian code in gr(pr,s)[g]. we give a formula for the number of self-dual abelian codes in gr(pr,s)[g]. under the restriction i) gcd(|g|,p) = 1; or ii) r = 2 and the sylow p-subgroup of g is cyclic, the numbers of self-dual abelian codes in gr(pr,s)[g] are explicitly determined. subsequently, the characterization and enumeration of complementary dual abelian codes in gr(pr,s)[g] are given. the number of complementary dual abelian codes in gr(pr,s)[g] is shown to be independent of r and the sylow p-subgroup of g. we note that the hermitian duality is meaningful only when s is even. since we study euclidean and hermitian self-dual codes in parallel, the assumption that s is even is included whenever we refer to the hermitian duality. the paper is organized as follows. in section 2, we recall and prove some basic results for group rings, abelian codes, and their duals. in section 3, we present the characterization and a general set up for the enumeration of self-dual abelian codes in gr(pr,s)[g]. the complete enumeration of euclidean and hermitian self-dual abelian codes in gr(pr,s)[g] is given in the special cases where i) gcd(p, |g|) = 1; and ii) r = 2 and the sylow p-subgroup of g is cyclic. in section 4, the characterization and enumeration of complementary dual abelian codes in gr(pr,s)[g] are given. 2. preliminaries in this section, we recall some definitions and basic properties of abelian codes and prove some results on their euclidean and hermitian duals. 2.1. abelian codes for a finite commutative ring r with identity and a finite abelian group g, written additively, let r[g] denote the group ring of g over r. the elements in r[g] will be written as ∑ g∈g αgy g, where αg ∈ r. the addition and the multiplication in r[g] are given as in the usual polynomial rings over r with the indeterminate y , where the indices are computed additively in g. by convention, y 0 = 1 is the identity of r, where 0 is the additive identity of g. an abelian code in r[g] is defined to be an ideal of r[g]. if g = {g1,g2, . . . ,gn} is an abelian group of order n, it is not dificult to see that the map π : r[g] → rn defined by ∑n i=1 αgiy gi 7→ (αg1,αg2, . . . ,αgn) is an r-module isomorphism. hence, an abelian code c in r[g] can be viewed as an r-submodule π(c) in rn. precisely, π(c) is a linear code of length n over r. if g is cyclic of order n, an abelian code in r[g] becomes a classical cyclic code of length n over r. in this case, an abelian code will be referred to as a cyclic code. it is well known that a cyclic code of length n over r can also be regarded as an ideal 76 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 in the quotient polynomial ring r[x]/〈xn − 1〉. from now on, we focus on the case where the ring is a galois ring gr(pr,s), a galois extension of degree s of an integer residue ring zpr. let ξ be an element in gr(pr,s) that generates a teichmüller set ts of gr(pr,s). in other words, ts = {0, 1,ξ,ξ2, . . . ,ξp s−2}. then every element in gr(pr,s) has a unique p-adic expansion of the form α = α0 + α1p + · · · + αr−1pr−1, where αi ∈ ts for all i = 0, 1, . . . ,r − 1. in the case where s is even, the map ¯ : gr(pr,s) → gr(pr,s) defined by α = α ps/2 0 + α ps/2 1 p + · · · + α ps/2 r−1 p r−1 (1) is a ring automorphism on gr(pr,s). for more details on galois rings, we refer the readers to [25]. let p be the sylow p-subgroup of g and let a be a complementary subgroup of p in g. then g ∼= a×p . let r := gr(pr,s)[a]. then the map φ : gr(pr,s)[g] →r[p ] given by φ( ∑ a∈a ∑ b∈p αa+by a+b) = ∑ b∈p αb(y )y b, where αb(y ) = ∑ a∈a αa+by a ∈r, is a well-known ring isomorphism (see [19, section 3.2, exercise 4]). then φ induces a one-to-one correspondence between the ideals in gr(pr,s)[g] and the ideals in r[p ]. since an abelian code is an ideal in a group ring, the above discussion can be interpreted in terms of abelian codes as follows. lemma 2.1. the map φ induces a one-to-one correspondence between the abelian codes in gr(pr,s)[g] and the abelian codes in r[p]. an abelian code c in gr(pr,s)[g] is said to be euclidean self-dual (resp., euclidean complementary dual) if c = c⊥e (resp., c∩c⊥e = {0}), where c⊥e is the dual of c with respect to the form 〈u,v〉e := ∑ g∈g αgβg, where u = ∑ g∈g αgy g and v = ∑ g∈g βgy g. define an involution ̂ on r to be the gr(pr,s)-module homomorphism that fixes gr(pr,s) and sends y a to y −a for all a ∈ a. an abelian code d in r[p ] is said to be -̂self-dual if d = d ⊥̂, where d ⊥̂ is the dual of d with respect to the form 〈x,y〉̂ := ∑ b∈p xb(y )ŷb(y ), where x = ∑ b∈p xb(y )y b and y = ∑ b∈p yb(y )y b. in addition, if s is even, an abelian code c in gr(pr,s)[g] is said to be hermitian self-dual (resp., hermitian complementary dual) if c = c⊥h (resp., c ∩c⊥h = {0}), where c⊥h is the dual of c with respect to the form 〈u,v〉h := ∑ g∈g αgβg, where u = ∑ g∈g αgy g and v = ∑ g∈g βgy g. 77 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 define an involution ˜ on r to be the gr(pr,s)-module homomorphism that sends α to α for all α ∈ gr(pr,s) and sends y a to y −a for all a ∈ a. an abelian code d in r[p ] is said to be ∼-self-dual if d = d⊥∼, where d⊥∼ is the dual of d with respect to the form 〈x,y〉∼ := ∑ b∈p xb(y )ỹb(y ), where x = ∑ b∈p xb(y )y b and y = ∑ b∈p yb(y )y b. similarly to the finite field case, the following relations among the above forms can be verified using arguments similar to those in [13, proposition 2.4] and [15, proposition 2.4]. lemma 2.2. let r and s be positive integers and let p be a prime number. let g ∼= a×p be as above and u,v ∈ gr(pr,s)[g]. then the following statements hold. i) 〈y gu,v〉e = 0 for all g ∈ g if and only if 〈y bφ(u), φ(v)〉̂ = 0 for all b ∈ p. ii) if s is even, then 〈y gu,v〉h = 0 for all g ∈ g if and only if 〈y bφ(u), φ(v)〉∼ = 0 for all b ∈ p. the next corollary follows immediately. corollary 2.3. let r and s be positive integers and let p be a prime number. let c be an abelian code in gr(pr,s)[g]. then the following statements hold. i) φ(c) ⊥̂ = φ(c⊥e ). in particular, c is euclidean self-dual if and only if φ(c) is ̂-self-dual. ii) if s is even, then φ(c)⊥∼ = φ(c⊥h ). in particular, c is hermitian self-dual if and only if φ(c) is ∼-self-dual. therefore, to study euclidean and hermitian self-dual abelian codes in gr(pr,s)[g], it is sufficient to consider ̂-self-dual and ∼-self-dual abelian codes in r[p ], respectively. 2.2. decomposition and dualities recall that p represents a prime number, s is a positive integer, and a is a finite abelian group such that gcd(p, |a|) = 1. for coprime positive integers i,j, let ordi(j) denote the multiplicative order of j modulo i. fora ∈ a, denote by ord(a) the additive order ofa ina. for each a ∈ a, a ps-cyclotomic class of a containing a is defined to be the set sps(a) :={psi ·a | i = 0, 1, . . .} = {psi ·a | 0 ≤ i < ordord(a)(ps)}, where psi · a := ∑psi j=1 a in a. a p s-cyclotomic class sps(a) is said to be of type i if a = −a, type ii if sps(a) = sps(−a) and a 6= −a, or type iii if sps(−a) 6= sps(a). if s is even, a ps-cyclotomic class sps(a) is said to be of type ii′ if sps(a) = sps(−ps/2 ·a) or type iii′ if sps(−ps/2 ·a) 6= sq(a), where −ps/2 ·a denotes ps/2 · (−a). remark 2.4. we have the following facts for the ps-cyclotomic classes of a (see [13, remark 2.5] and [15, remark 2.6]). 1. a ps-cyclotomic class of type i has cardinality one. 2. sps(0) is a ps-cyclotomic class of both types i and ii ′. 3. if a ps-cyclotomic class of type ii exists, then its cardinality is even. moreover, if sps(a) is a ps-cyclotomic class of type ii of cardinality 2ν, then −a = psν ·a. 78 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 4. a ps-cyclotomic class of a of type ii′ has odd cardinality. moreover, if sps(a) is a ps-cyclotomic class of type ii′ of cardinality ν, then −a = psν/2 ·a and −ps/2 ·a = ps(ν+1)/2 ·a. assume that a has cardinality m and exponent m. by the fundamental theorem of finite abelian groups, a can be written as a direct product of finite cyclic groups a = ∏n i=1 zmi, where zmi = {0, 1, . . . ,mi − 1} denotes the additive cyclic group of order mi ≥ 2 for all 1 ≤ i ≤ n. then an element b ∈ a can be written as b = (b1,b2, . . . ,bn ), where bi ∈ zmi. for each h ∈ a, let γh : a → z be defined by γh(b) = n∑ i=1 bihi(m/mi), (2) where the sum is a rational sum. let µ be the order of ps modulo m. denote by ζ a primitive mth root of unity in gr(pr,sµ). for a given c = ∑ a∈a cay a ∈r := gr(pr,s)[a], its discrete fourier transform (dft) is c̆ = ∑ h∈a c̆hy h, where c̆h = ∑ a∈a caζ γh(a) ∈ gr(pr,sµ). (3) moreover, if sps(h) has cardinality ν, then it is not difficult to verify that c̆h is contained in a subring of gr(pr,sµ) which is isomorphic to gr(pr,sν). using this dft, the decomposition of r := gr(pr,s)[a], where gcd(p, |a|) = 1, has been given in [18] in terms of the mix-radix representation of the elements in a. in order to utilize the decomposition in [18] for characterizing self-dual codes, we need to consider a suitable rearrangement of the terms in the decomposition. 2.2.1. euclidean case for the euclidean self-duality, we consider the rearrangement based on the ps-cyclotomic classes of types i − iii as follows. assume that a contains l ps-cyclotomic classes. without loss of generality, let {a1,a2, . . . ,al} be a set of representatives of the ps-cyclotomic classes such that {ai | i = 1, 2, . . . , ti}, {ati+j | j = 1, 2, . . . , tii} and {ati+tii+k,ati+tii+tiii+k = −ati+tii+k | k = 1, 2, . . . , tiii} are sets of representatives of ps-cyclotomic classes of types i,ii, and iii, respectively, where l = ti + tii + 2tiii. from the definition, |sps(ai)| = 1 for all i = 1, 2, . . . , ti. from remark 2.4, the order of the ps-cyclotomic classes of type ii is even order. for j = 1, 2, . . . , tii, let 2ej denote the cardinality of sps(ati+j). for k = 1, 2, . . . , tiii, sps(ati+tii+k) and sps(ati+tii+tiii+k) have the same cardinality and denote it by fk. rearranging the terms in the decomposition in [18] based on the ps-cyclotomic classes of a of types i− iii, we have r∼= ( ti∏ i=1 gr(pr,s) ) ×   tii∏ j=1 gr(pr, 2sej)  × ( tiii∏ k=1 (gr(pr,sfk) × gr(pr,sfk)) ) , (e1) where gr(pr, 2sej) is induced by sps(ati+j) for all j = 1, 2, . . . , tii and gr(p r,sfk) × gr(pr,sfk) is induced by (sps(ati+tii+k), sps(−ati+tii+k)) for all k = 1, 2, . . . , tiii. for more details and the explicit isomorphism, the readers may refer to [18, section ii]. it follows that r[p ] ∼= ( ti∏ i=1 gr(pr,s)[p ] ) ×   tii∏ j=1 gr(pr, 2sej)[p ]  × ( tiii∏ k=1 (gr(pr,sfk)[p] × gr(pr,sfk)[p ]) ) . (e2) 79 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 therefore, by lemma 2.1, every abelian code in gr(pr,s)[g] ∼= r[p ] can be written in the form c ∼= ( ti∏ i=1 ui ) ×   tii∏ j=1 vj  × ( tiii∏ k=1 (wk ×w ′k) ) , (e3) where ui is an abelian code in gr(pr,s)[p ], vj is an abelian code in gr(pr, 2sej)[p], and wk,w ′k are abelian codes in gr(pr,sfk)[p ] for all i = 1, 2, . . . , ti, j = 1, 2, . . . , tii, and k = 1, 2, . . . , tiii. the euclidean dual of c in (e3) can be viewed to be of the form c⊥e ∼= ( ti∏ i=1 u⊥ei ) ×   tii∏ j=1 v ⊥hj  × ( tiii∏ k=1 ( (w ′k) ⊥e ×w⊥ek )) . (e4) the detailed justification for (e4) is provided in appendix a.1. 2.2.2. hermitian case in the case where s is even, we consider the other rearrangement of the decomposition of r in terms of the ps-cyclotomic classes of a of types ii′ and iii′. let {b1 = 0,b2, . . . ,bl} denote a set of representatives of the ps-cyclotomic classes such that {bj | j = 1, 2, . . . , tii′} and {btii′+k,btii′+tiii′+k = −ps/2 · btii′+k | k = 1, 2, . . . , tiii′} represent p s-cyclotomic classes of types ii′ and iii′, respectively, where l = tii′ + 2tiii′. for j = 1, 2, . . . , tii′, let éj denote the cardinality of sps(bj). for k = 1, 2, . . . , tiii′, sps(btii′+k) and sps(btii′+tiii′+k) have the same cardinality and denote it by f́k. rearranging the terms in the decomposition in [18] based on the ps-cyclotomic classes of a of types ii′ and iii′, we have r∼=   tii′∏ j=1 gr(pr,séj)  × ( tiii′∏ k=1 ( gr(pr,sf́k) × gr(pr,sf́k) )) , (h1) where gr(pr,séj) is induced by sps(bj) for all j = 1, 2, . . . , tii′ and gr(pr,sf́k)×gr(pr,sf́k) is induced by ( sps(btii′+k),sps(−p s/2 · btii′+k) ) for all k = 1, 2, . . . , tiii′. consequently, r[p ] ∼=   tii′∏ j=1 gr(pr,séj)[p ]  × ( tiii′∏ k=1 ( gr(pr,sf́k)[p ] × gr(pr,sf́k)[p ] )) , (h2) and, by lemma 2.1, every abelian code in gr(pr,s)[g] ∼= r[p ] can be viewed as c ∼=   tii′∏ j=1 ej  × ( tiii′∏ k=1 (fk ×f ′k) ) , (h3) where ej is an abelian code in gr(pr,séj)[p ] and fk,f ′k are abelian codes in gr(p r,sf́k)[p ] for all j = 1, 2, . . . , tii′ and k = 1, 2, . . . , tiii′. then the hermitian dual of c in (h3) has the form c⊥h ∼=   tii′∏ j=1 e⊥hj  × ( tiii′∏ k=1 ( (f ′k) ⊥e ×f⊥ek )) . (h4) the detailed discussion for (h4) is provided in appendix a.2. 80 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 3. self-dual abelian codes in gr(pr, s)[g] in this section, we characterize and enumerate euclidean and hermitian self-dual abelian codes in gr(pr,s)[g]. we determine necessary and sufficient conditions for the existence of self-dual abelian codes in gr(pr,s)[g] in subsection 3.1 and followed by general results for the enumeration of such self-dual codes in subsection 3.2. some special cases will be discussed in subsections 3.3 and 3.4. 3.1. the existence of self-dual abelian codes the characterizations of euclidean and hermitian self-dual abelian codes in gr(pr,s)[g] are given as follows. from (e3) and (e4), the characterization of euclidean self-dual abelian codes in gr(pr,s)[g] is given in the next proposition. proposition 3.1. let r and s be positive integers and let p be a prime number. an abelian code c in gr(pr,s)[g] is euclidean self-dual if and only if, in the decomposition (e3), i) ui is euclidean self-dual for all i = 1, 2, . . . , ti, ii) vj is hermitian self-dual for all j = 1, 2, . . . , tii, and iii) w ′k = w ⊥e k for all k = 1, 2, . . . , tiii. the characterization of hermitian self-dual abelian codes in gr(pr,s)[g] follows immediately from (h3) and (h4). proposition 3.2. let r and s be positive integers such that s is even and let p be a prime number. then an abelian code c in gr(pr,s)[g] is hermitian self-dual if and only if, in the decomposition (h3), i) ej is hermitian self-dual for all j = 1, 2, . . . , tii′, and ii) f ′k = f ⊥e k for all k = 1, 2, . . . , tiii′. necessary and sufficient conditions for the existence of euclidean and hermitian self-dual abelian codes in gr(pr,s)[g] are given as follows. the conditions for the euclidean case have been proven in [26, theorem 1.1]. here, we provide an alternative constructive proof. proposition 3.3. let r and s be positive integers and let p be a prime number. let g be a finite abelian group. then there exists a euclidean self-dual abelian code in gr(pr,s)[g] if and only if one of the following statements holds, i) r is even, or ii) p = 2 and |g| is even. in addition, if s is even, then the conditions are equivalent to the existence of a hermitian self-dual abelian code in gr(pr,s)[g]. proof. assume that g is decomposed as g = a⊕p , where p |a| and p is the sylow p-subgroup of g of order pa, where a ≥ 0. from (e3), assume that the code c ∼= ( ti∏ i=1 ui ) ×   tii∏ j=1 vj  × ( tiii∏ k=1 (wk ×w ′k) ) 81 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 is euclidean self-dual in gr(pr,s)[g]. then, by proposition 3.1, u1 is euclidean self-dual in gr(pr,s)[p]. it follows that |u1| = (ps)rp a/2 and rpa/2 is an integer. hence, r is even, or p = 2 and a ≥ 1. for the converse, if r is even, then pr/2gr(pr,s)[g] is euclidean self-dual. assume that r is odd, p = 2 and |g| is even. let r′ = dr 2 e. then |p | = 2a with a ≥ 1 and r = 2r′ − 1. since the order of p is even, p contains an element x of order 2. define c ∼= ( ti∏ i=1 ( 2r ′ gr(2r,s)[p ] + 2r ′−1gr(2r,s)[p ](y x + 1) )) ×   rii∏ j=1 ( 2r ′ gr(2r, 2sej)[p] + 2 r′−1gr(2r, 2sej)[p](y x + 1) ) × ( tiii∏ k=1 (gr(2r,sfk)[p ] ×{0}) ) . we prove that c is euclidean self-dual. by proposition 3.1, it is sufficient to show that u := 2r ′ gr(2r,s)[p ] + 2r ′−1gr(2r,s)[p ](y x + 1) is euclidean self-dual and vj := 2 r′gr(2r, 2sej)[p ] + 2 r′−1gr(2r, 2sej)[p ](y x + 1) is hermitian self-dual for all j = 1, 2 . . . , tii. let u = 2r ′ e + 2r ′−1e′(y x + 1) and v = 2r ′ f + 2r ′−1f′(y x + 1) be elements in u, where e, e′, f, and f′ are in gr(2r,s)[p ]. since r = 2r′ − 1 and x = −x, we have 〈u,v〉e = 〈2r ′ e, 2r ′ f〉e + 〈2r ′ e, 2r ′−1f′(y x + 1)〉e + 〈2r ′−1e′(y x + 1), 2r ′ f〉e + 〈2r ′−1e′(y x + 1), 2r ′−1f′(y x + 1)〉e = 2r−1〈e′(y x + 1),f′(y x + 1)〉e = 2r−1 ( 〈e′y x,f′y x〉e + 〈e′y x,f′〉e + 〈e′,f′y x〉e + 〈e′,f′〉e ) = 2r−1 ( 2〈e′,f′〉e + 2〈e′y x,f′〉e ) = 0. it is not difficult to verify that |u| = |2r ′ gr(2r,s)[p ]||2r ′−1gr(2r,s)[p ](y x + 1)| |(2r′gr(2r,s)[p ]) ∩ (2r′−1gr(2r,s)[p ](y x + 1))| = (2s)(r ′−1)2a(2s)r ′2a/2 (2s)(r ′−1)2a/2 = (2 s)r2 a−1 . therefore, u is euclidean self-dual. using arguments similar to the above, we can see that vj is hermitian self-dual for all j = 1, 2 . . . , tii. for the hermitian case, we assume that s is even. the proof of the sufficiency is similar to the euclidean case, except that (h3) and proposition 3.2 are applied instead of (e3) and proposition 3.1 . for the converse, if r is even, then pr/2gr(pr,s)[g] is hermitian self-dual. assume that r is odd, p = 2 and |g| is even. then p contains an element x of order 2. let r′ = dr 2 e and define c ∼=  rii′∏ j=1 ( 2r ′ gr(2r,séj)[p ] + 2 r′−1gr(2r,séj)[p](y x + 1) )× ( tiii′∏ k=1 ( gr(2r,sf́k)[p ] ×{0} )) . by arguments similar to those in the proof of the euclidean case, we can verity that 2r ′ gr(2r,séj)[p ] + 2r ′−1gr(2r,séj)[p ](y x + 1) is hermitian self-dual for all j = 1, 2, . . . , tii′. therefore, c is hermitian self-dual by proposition 3.2. 82 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 3.2. enumeration of self-dual abelian codes we aim to characterize and enumerate euclidean and hermitian self-dual cyclic and abelian codes over galois rings. for convenience, we fix the following notations. • nc(gr(pr,s),n) – the number of cyclic codes of length n over gr(pr,s), • nec(gr(pr,s),n) – the number of euclidean self-dual cyclic codes of length n over gr(pr,s), • nhc(gr(pr,s),n) – the number of hermitian self-dual cyclic codes of length n over gr(pr,s), • na(gr(pr,s)[g]) – the number of abelian codes in gr(pr,s)[g], • nea(gr(pr,s)[g]) – the number of euclidean self-dual abelian codes in gr(pr,s)[g], • nha(gr(pr,s)[g]) – the number of hermitian self-dual abelian codes in gr(pr,s)[g], where s is assumed to be even in cases of nhc(gr(pr,s),n) and nha(gr(pr,s)[g]). to determine the numbers of euclidean and hermitian self-dual abelian codes, we need some grouptheoretic and number-theoretic results. for completeness, we recall the following results. for a finite group a and a positive integer d, let na(d) denote the number of elements in a of order d. the explicit expression of na(d) is completely determined in [2]. let q be a prime power and let j be a positive integer. the pair (j,q) is said to be oddly good if j divides qt + 1 for some odd integer t ≥ 1, and evenly good if j divides qt + 1 for some even integer t ≥ 2. it is said to be good if it is oddly good or evenly good, and bad otherwise. the characterization of good and oddly-good pairs of integers can be found in [12], [11], [13], [15], and [20]. let χ and λ be functions defined on the pair (j,q), where j is a positive integer, as follows. χ(j,q) = { 0 if (j,q) is good, 1 otherwise, (4) and λ(j,q) = { 0 if (j,q) is oddly good, 1 otherwise. (5) the following two lemmas are extended from the case where q is a power of 2 in [13] and [15] and the proof is omitted. the readers may refer to [13, lemma 4.5] and [15, lemma 7] for the idea of the proofs. lemma 3.4. let s be a positive integer and let p be a prime number. let a be a finite abelian group such that gcd(|a|,p) = 1 and let h ∈ a. then sps(h) is of type iii if and only if (ord(h),ps) is bad. lemma 3.5. let s be an even positive integer and let p be a prime number. let a be a finite abelian group such that gcd(|a|,p) = 1 and let h ∈ a\{0}. then sps(h) is of type iii′ if and only if (ord(h),ps/2) is evenly good or bad. utilizing the decomposition in section 2 and the discussion above, we obtain the following formulas for the numbers of euclidean and hermitian self-dual abelian codes in gr(pr,s)[g], where g is an arbitrary finite abelian group. without loss of generality, we assume that g = a⊕p , where p is a finite abelian p-group and a is a finite abelian group such that p |a|. theorem 3.6. let p be a prime and let s,r be integers such that 1 ≤ s and 1 ≤ r. let a be a finite abelian group of exponent m such that p m and let p be a finite abelian p-group. then nea(gr(pr,s)[a⊕p ]) = (nea(gr(pr,s)[p ])) ∑ d|m,ordd(p s)=1 (1−χ(d,ps))na(d) × ∏ d|m ordd(p s) 6=1 (nha(gr(pr,s · ordd(ps))[p ])) (1−χ(d,ps)) na(d) ordd(p s) × ∏ d|m (na(gr(pr,s · ordd(ps))[p ])) χ(d,ps) na(d) 2ordd(p s) . 83 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 in addition, if s is even, then nha(gr(pr,s)[a⊕p ]) = ∏ d|m (nha(gr(pr,s · ordd(ps))[p])) (1−λ(d,p s 2 )) na(d) ordd(p s) × ∏ d|m (na(gr(pr,s · ordd(ps))[p ])) λ(d,p s 2 ) na(d) 2ordd(p s) . proof. first, we consider the euclidean case. from (e3) and proposition 3.1, it is sufficient to count the numbers of euclidean self-dual abelian codes ui’s, the numbers of hermitian self-dual abelian codes vi’s, and the numbers of abelian codes wi’s which correspond to the ps-cyclotomic classes of types i, ii, and iii, respectively. from [14, remark 2.5], we note that the elements in a of the same order are partitioned into pscyclotomic classes of the same type. for each divisor d of m, a ps-cyclotomic class containing an element of order d has cardinality ordd(ps), and hence, the number of such ps-cyclotomic classes is na(d) ordd(ps) . for each divisor d of m, we consider the following 3 cases. case 1. χ(d,ps) = 0 and ordd(ps) = 1. by lemma 3.4, every ps-cyclotomic class of a containing an element of order d is of type i. since there are na(d) ordd(ps) such ps-cyclotomic classes, the number of euclidean self-dual abelian codes ui’s corresponding to d is (nea(gr(pr,s · ordd(ps))[p ])) na(d) ordd(p s) = (nea(gr(pr,s)[p])) (1−χ(d,ps))na(d) . case 2. χ(d,ps) = 0 and ordd(ps) 6= 1. by lemma 3.4, every ps-cyclotomic class of a containing an element of order d is of type ii. since there are na(d) ordd(ps) such ps-cyclotomic classes, the number of hermitian self-dual abelian codes vi’s corresponding to d is (nha(gr(pr,s · ordd(ps))[p ])) na(d) ordd(p s) = (nha(gr(pr,s · ordd(ps))[p ])) (1−χ(d,ps)) na(d) ordd(p s) . case 3. χ(d,ps) = 1. by lemma 3.4, every ps-cyclotomic class of a containing an element of order d is of type iii. since there are na(d) ordd(ps) such ps-cyclotomic classes, the number of abelian codes wi’s corresponding to d is (na(gr(pr,s · ordd(ps))[p ])) na(d) ordd(p s) (na(gr(pr,s · ordd(ps))[p ])) χ(d,ps) na(d) 2ordd(p s) . since d runs over all divisors of m, we conclude the desired result. for the hermitian case, by proposition 3.2, it suffices to count the numbers of hermitian self-dual abelian codes ei’s and the numbers of abelian codes fi’s in (h3) which correspond to the ps-cyclotomic classes of types ii′ and iii′, respectively. considering the cases where λ(d,p s 2 ) = 1 and where λ(d,p s 2 ) = 0, the desired result can be obtained similarly to the euclidean case, where lemma 3.5 is applied instead of lemma 3.4. note that, if a is a cyclic group, the exponent m is just the cardinality of a and na(d) is just φ(d), where φ is an euler’s totient function. in theorem 3.6, if p is cyclic of order pa, then the values na,nea and nha may be replaced by nc,nec, and nhc, respectively. in general, these values are not known in the literature. some special cases where i) gcd(p, |g|) = 1; and ii) r = 2 and the sylow p-subgroup of g is cyclic are discussed in the following subsections. 84 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 3.3. self-dual abelian codes in gr(pr, s)[a], gcd(p, |a|) = 1 in this subsection, we complete the enumeration of euclidean and hermitian self-dual abelian codes in gr(pr,s)[a], where gcd(p, |a|) = 1, or equivalently, gr(pr,s)[a] is a principal ideal group ring (see proposition 3.7). if a is cyclic, this case is identical with that of simple root cyclic codes. proposition 3.7. let p be a prime number and let r,s be positive integers. let g be a finite abelian group. then one of the following statements holds. i) if r = 1, then gr(pr,s)[g] ∼= fps[g] is a principal ideal ring if and only if the sylow p-subgroup of g is cyclic. ii) if r ≥ 2, then gr(pr,s)[g] is a principal ideal ring if and only if gcd(p, |g|) = 1. for r = 1, the statement has been proven in [9]. for r ≥ 2, using notion of morphic rings (see the definition in [6]), it has been shown that zpr [g] is principal ideal if and only if gcd(p, |g|) = 1 (see [8, theorem 1.2] and [6, theorem 3.12 and corollary 3.13]). the statements can be extended naturally to the case of gr(pr,s)[g]. the enumerations of euclidean and hermitian self-dual abelian codes in a principal ideal group ring gr(pr,s)[a] is given as follows. theorem 3.8. let p be a prime and let s,r be positive integers. let a be a finite abelian group of exponent m such that gcd(p, |a|) = 1. then nea(gr(pr,s)[a]) =  (1 + r) ∑ d|m χ(d,ps) na(d) 2ordd(p s) if r is even, 0 if r is odd. in addition, if s is even, then nha(gr(pr,s)[a]) =  (1 + r) ∑ d|m λ(d,ps/2) na(d) 2ordd(p s) if r is even, 0 if r is odd. proof. in gr(pr,s), every ideal can be regarded as an abelian code in gr(pr,s)[g] with g = {0}, and we have the following facts. i) the number of abelian codes in gr(pr,s) is r + 1. ii) if r is odd, then there are neither euclidean self-dual abelian codes nor hermitian self-dual abelian codes in gr(pr,s). iii) if r is even, then rr/2gr(pr,s) is the only euclidean self-dual abelian code and it is the only hermitian self-dual abelian code if s is even. the above results hold true for any galois extension of gr(pr,s). by considering p = {0} in theorem 3.6, the result follows immediately. note that, if a is cyclic, m and na(d) can be replaced by the cardinality of a and φ(d), respectively, where φ is the euler’s totient function. if a is cyclic of order n with gcd(n,p) = 1, then the number of euclidean self-dual cyclic codes of length n over gr(pr,s) obtained in theorem 3.8 is a special case of [1, theorem 5.7] by viewing gr(pr,s) as a finite chain ring of depth r. 85 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 3.4. self-dual abelian codes in gr(p2, s)[a⊕cpa] in this section, we restrict our study to the case where r = 2 and p = cpa, a cyclic group of order pa. the enumerations of euclidean and hermitian self-dual abelian codes in gr(p2,s)[a⊕cpa] can be obtained as an application of theorem 3.6 and some known results on cyclic codes of length pa over gr(p2,s). we recall some results on cyclic codes of length pa over gr(p2,s). the next lemma follows immediately from [16, corollary 3.9] and [16, theorem 3.6]. lemma 3.9. the number of cyclic codes of length pa over gr(p2,s) is nc(gr(p2,s),pa) = 2 pa−1∑ d=0 ps(min{b d 2 c,pa−1}+1) − 1 ps − 1 + ps(p a−1+1) − 1 ps − 1 . (6) proposition 3.10 ([17, corollary 3.5]). the number of euclidean self-dual cyclic codes of length 2a over gr(22,s) is nec(gr(22,s), 2a) =   1 if a = 1, 1 + 2s if a = 2, 1 + 2s + 22s+1 ( (2s)(2 a−2−1)−1 2s−1 ) if a ≥ 3. if p is an odd prime, then the number of euclidean self-dual cyclic codes of length pa over gr(p2,s) is nec(gr(p2,s),pa) = 2 ( (ps) (pa−1+1) /2 − 1 ps − 1 ) . proposition 3.11 ([14, theorem 3.5]). let p be a prime and let s,a be positive integers such that s is even. then the number of hermitian self-dual cyclic codes of length pa over gr(p2,s) is nhc(gr(p2,s),pa) = pa−1∑ i1=0 psi1/2 = ps(p a−1+1)/2 − 1 ps/2 − 1 . remark 3.12. for cyclic codes of length pa over gr(p2,s), the numbers nc, nec, and nhc have already been determined in lemma 3.9, proposition 3.10, and proposition 3.11, respectively. combining these results and theorem 3.6, the numbers nea(gr(p2,s)[a ⊕ cpa]) and nha(gr(p2,s)[a ⊕ cpa]) are explicitly determined. the numbers of euclidean and hermitian self-dual cyclic codes of arbitrary length n over gr(p2,s) can be obtained as a corollary of remark 3.12. some parts of the formulas can be simplified as in the next corollary. corollary 3.13. let p be a prime and let s,n be positive integers. write n = mpa, where a ≥ 0 and p m. then nec(gr(p2,s),n) = ( nec(gr(p2,s),pa) )η(m) × ∏ d|m d6∈{1,2} ( nhc(gr(p2,s · ordd(ps)),pa) )(1−χ(d,ps)) φ(d) ordd(p s) × ∏ d|m ( nc(gr(p2,s · ordd(ps)),pa) )χ(d,ps) φ(d) 2ordd(p s) , 86 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 where η(m) = { 1 if m is odd, 2 if m is even. in addition, if s is even, then nhc(gr(p2,s),n) = ∏ d|m ( nhc(gr(p2,s · ordd(ps)),pa) )(1−λ(d,ps2 )) φ(d) ordd(p s) × ∏ d|m ( nc(gr(p2,s · ordd(ps)),pa) )λ(d,ps2 ) φ(d) 2ordd(p s) . proof. setting r = 2 and a a cyclic group of order m in theorem 3.6, the exponent of a is m and na(d) is just φ(d), where φ is the euler’s function. note that sps(0) is the only ps-cyclotomic class of a of type i if m is odd, and sps(0) and sps(m2 ) are the only ps-cyclotomic classes of a of type i if m is even. therefore, the values of η(m) follows. 4. complementary dual abelian codes in gr(pr, s)[g] in this section, the characterization and enumeration of complementary dual abelian codes in the group ring gr(pr,s)[g] are given based on the decomposition in section 2 and the theory of local group rings. 4.1. characterization and enumeration of complementary dual abelian codes in gr(pr, s)[p ] in this subsection, we focus on complementary dual abelian codes and direct summand ideals in each component of gr(pr,s)[p ] in the decompositions (e1) and (h1), where p is a finite abelian p-group. first, we recall some useful definitions and properties in ring theory. for a finite commutative ring r with identity, the jacobson radical of r, denoted by jac(r), is defined to be the intersection of all maximal ideals of r. the ring r is said to be local if it has a unique maximal ideal. a local group ring has been characterized in the following lemma. lemma 4.1 ([22, theorem]). let r be a commutative ring with identity and let g be a finite abelian group. then r[g] is local if and only if r is local, g is a p-group and p ∈ jac(r). proposition 4.2. let p be a prime number and let r,s be positive integers. let p be a finite abelian p-group. then gr(pr,s)[p ] is a local group ring. proof. since the ideal 〈p〉 is the unique maximal ideal of gr(pr,s), the ring gr(pr,s) is local. moreover, p ∈ 〈p〉 = jac(gr(pr,s). by lemma 4.1, gr(pr,s)[p ] is a local group ring. by proposition 4.2, gr(pr,s)[p ] is local. denote by m the maximal ideal of gr(pr,s)[p]. the characterizations of the euclidean and hermitian complementary dual abelian codes and the direct summands in a local group ring gr(pr,s)[p ] are given in the following theorems. theorem 4.3. let p be a prime number and let r,s be positive integers. let p be a finite abelian p-group. then {0} and gr(pr,s)[p ] are the only euclidean complementary dual abelian codes in gr(pr,s)[p ]. 87 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 proof. clearly, {0} and gr(pr,s)[p ] are euclidean complementary dual abelian codes in gr(pr,s)[p]. let c be an abelian code in gr(pr,s)[p ] such that {0} ( c ( gr(pr,s)[p ]. then c ⊆ m. it follows that m⊥e ⊆ c⊥e ⊆ m which implies m⊥e ⊆ c ⊆ m. hence, {0} 6= m⊥e ⊆ c∩c⊥e ⊆ m. consequently, c is not euclidean complementary dual. therefore, the ideals {0} and gr(pr,s)[p ] are the only euclidean complementary dual abelian codes in gr(pr,s)[p ]. it is not difficult to see that the proof of theorem 4.3 is independent of the inner product. hence, we have the following corollary. corollary 4.4. let p be a prime number and let r,s be positive integers such that s is even. let p be a finite abelian p-group. then {0} and gr(pr,s)[p ] are the only hermitian complementary dual abelian codes in gr(pr,s)[p ]. theorem 4.5. let p be a prime number and let r,s be positive integers. let p be a finite abelian p-group. then ideals {0} and gr(pr,s)[p ] are the only direct summands in gr(pr,s)[p ]. proof. clearly, {0} and gr(pr,s)[p ] are direct summands in gr(pr,s)[p]. let {0} ( c ( gr(pr,s)[p ] be an ideal in gr(pr,s)[p ]. suppose that c is a direct summand. then there exists an ideal c′ in gr(pr,s)[p] such that c ∩ c′ = {0} and c + c′ = gr(pr,s)[p ]. since m is the maximal ideal in gr(pr,s)[p], we have c ⊆ m and c′ ⊆ m. hence, c + c′ ⊆ m ( gr(pr,s)[p], a contradiction. therefore, the ideals {0} and gr(pr,s)[p ] are the only direct summands in gr(pr,s)[p ]. the above results can be summarized as follows. corollary 4.6. let p be a prime number and let r,s be positive integers. let p be a finite abelian p-group. then the following statements hold. 1. the number of euclidean complementary dual abelian codes in gr(pr,s)[p ] is 2. 2. if s is even, the number of hermitian complementary dual abelian codes in gr(pr,s)[p ] is 2. 3. the number of direct summand ideals in gr(pr,s)[p ] is 2. 4.2. characterization and enumeration of complementary dual abelian codes in gr(pr, s)[g] in this subsection, we focus on the characterization and enumeration of complementary dual abelian codes in gr(pr,s)[g], where g is an arbitrary finite abelian group. using the decompositions in section 2 and results in the previous subsection, the characterization and enumeration of such complementary dual codes are given independent of r and the sylow p-subgroup of g. recall that g ∼= a×p , where p is the sylow p-subgroup of g and a is its complement subgroup. the group ring gr(pr,s)[g] is viewed as gr(pr,s)[g] ∼= r[p ], where r = gr(pr,s)[a]. using the decomposition of gr(pr,s)[g] in (e2), the characterization of a euclidean complementary dual abelian code in gr(pr,s)[g] can be concluded via (e3) and (e4) as follows. proposition 4.7. let p be a prime number and let r,s be positive integers. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. then an abelian code c in gr(pr,s)[a × p ] decomposed as in (e3) is euclidean complementary dual if and only if the following statements hold. 1. ui is euclidean complementary dual for all i = 1, 2, . . . ,ri. 2. vj is hermitian complementary dual for all j = 1, 2, . . . ,rii. 3. wk ∩ (w ′k) ⊥e = {0} and w ′k ∩w ⊥e k = {0} for all k = 1, 2, . . . ,riii. 88 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 the next corollary follows immediately from theorem 4.3, proposition 4.7, and corollary 4.4. corollary 4.8. let p be a prime number and let r,s be positive integers. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. then an abelian code c in gr(pr,s)[a × p ] decomposed as in (e3) is euclidean complementary dual if and only if the following statements hold. 1. ui ∈{{0}, gr(pr,s)[p ]} for all i = 1, 2, . . . ,ri. 2. vj ∈{{0}, gr(pr, 2sej)[p ]} for all j = 1, 2, . . . ,rii. 3. (wk,w ′k) ∈{({0},{0}), (gr(p r,sfk)[p], gr(p r,sfk)[p])} for all k = 1, 2, . . . ,riii. from corollary 4.8, it is not difficult to see that the number of euclidean complementary dual abelian codes in gr(pr,s)[a×p ] is independent of r and the group p and it can be determined in the following corollary. corollary 4.9. let p be a prime number and let r,s be positive integers. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. if the exponent of a is m and gr(pr,s)[a×p ] is decomposed as in (e2), then the number of euclidean complementary dual abelian codes in gr(pr,s)[a× p ] is 2ri+rii+riii = 2 ∑ d|m (1−χ(d,ps)) na(d) ordd(p s) + ∑ d|m χ(d,ps) na(d) 2ordd(p s) , where na(d) denote the number of elements in a of order d. proof. the first part follows from corollary 4.8. the equality can be derived similar to the proof of theorem 3.6. using the decomposition of gr(pr,s)[g] in (h2), the characterization of a hermitian complementary dual abelian code in gr(pr,s)[g] can be concluded via (h3) and (h4) in the following proposition. proposition 4.10. let p be a prime number and let r,s be positive integers such that s is even. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. then an abelian code c in gr(pr,s)[a × p] decomposed as in (h3) is hermitian complementary dual if and only if the following statements hold. 1. ej is hermitian complementary dual for all j = 1, 2, . . . ,r′ii. 2. fk ∩ (f ′k) ⊥e = {0} and f ′k ∩f ⊥e k = {0} for all j = 1, 2, . . . ,riii′. the following result follows directly from proposition 4.10 and corollary 4.4. corollary 4.11. let p be a prime number and let r,s be positive integers such that s is even. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. then an abelian code c in gr(pr,s)[a × p] decomposed as in (h3) is hermitian complementary dual if and only if the following statements hold. 1. ej ∈{{0}, gr(pr,séj)[p]} for all j = 1, 2, . . . ,r′ii. 2. (fk,f ′k) ∈{({0},{0}), (gr(p r,sf́k)[p ], gr(p r,sf́k)[p ])} for all k = 1, 2, . . . ,riii′. from corollary 4.11, the number of hermitian complementary dual abelian codes in gr(pr,s)[a×p ] is independent of r and the group p and it is given in the following corollary. 89 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 corollary 4.12. let p be a prime number and let r,s be positive integers such that s is even. let a be finite abelian group such that p |a| and let p be a finite abelian p-group. if the exponent of a is m and gr(pr,s)[a×p ] is decomposed as in (h2), then the number of hermitian complementary dual abelian codes in gr(pr,s)[a×p ] is 2ri′+rii′ = 2 ∑ d|m (1−λ(d,p s 2 )) na(d) ordd(p s) + ∑ d|m λ(d,p s 2 ) na(d) 2ordd(p s) , where na(d) denote the number of elements in a of order d. proof. the first part follows from corollary 4.11. the equality can be derived similar to the proof of theorem 3.6. 5. conclusion self-dual and complementary dual abelian codes in gr(pr,s)[g], a group ring of a finite abelian group g over a galois ring gr(pr,s), have been studied with respect to the euclidean and hermitian inner products. we have characterized such self-dual codes as well as determined necessary and sufficient conditions for gr(pr,s)[g] to contain a euclidean (resp, hermitian) self-dual abelian code. for any finite abelian group g and galois ring gr(pr,s), the enumerations of such self-dual codes have been given. in the case where gcd(|g|,p) = 1, the enumeration has been completed by restricting the sylow p-subgroup to be {0}. applying some known results on cyclic codes of length pa over gr(p2,s), we have determined explicitly the numbers of euclidean and hermitian self-dual abelian codes in gr(p2,s)[g] if the sylow p-subgroup of g is cyclic. as corollaries, analogous results on euclidean and hermitian self-dual cyclic codes over gr(pr,s) have been concluded. subsequently, the characterization and enumeration of complementary dual abelian codes in gr(pr,s)[g] have been given. the number of complementary dual abelian codes in gr(pr,s)[g] has been shown to be independent of r and the sylow p-subgroup of g. it would be interesting to study the unknown terms in theorem 3.6 and extend the results to abelian codes over finite chain rings or the case where the sylow p-subgroup of the group is not cyclic. appendix a in this appendix, we discuss the euclidean and hermitian duals of abelian codes in gr(pr,s)[g]. first, we recall that g ∼= a×p , where p is the sylow p-subgroup of g and a is a complementary subgroup of p in g. the group ring r := gr(pr,s)[a] is decomposed as in (e1) or (h1), and gr(pr,s)[g] ∼= r[p]. a.1. euclidean duality let ψ denote the isomorphism in (e1). for each element x ∈r, we can write ψ(x) = (x1, . . . ,xri,y1, . . . ,yrii,z1,z ′ 1, . . . ,zriii,z ′ riii ), (7) where xi ∈ gr(pr,s), yj ∈ gr(pr, 2sej), and zk,z′k ∈ gr(p r,sfk) for all i = 1, 2, . . . ,ri, j = 1, 2, . . . ,rii, and k = 1, 2, . . . ,riii. we are going to view x̂ defined in section 2 in terms of (7). we note that, for c = ∑ a∈a cay a ∈ gr(p2,s)[a], we have ĉ = ∑ a∈a cay −a = ∑ a∈a c−ay a. 90 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 then ˘̂c = ∑ a∈a d̆ay a, where d̆a = ∑ h∈a c−hζ γa(h). from (3), we can see that, if sps(h) is of type i, then d̆h = c̆h, (8) and if sps(h) is of type ii with cardinality 2ν, then −h = psν ·h by remark 2.4. it follows that d̆h = ∑ a∈a c−aζ γh(a) = ∑ a∈a caζ γ−h(a) = ∑ a∈a caζ γpsν·h(a) = ∑ a∈a ca ( ζγh(a) )psν = θ(c̆h), (9) where θ(α) = αp sν 0 + α psν 1 p + · · · + α psν r−1p r−1 for all α = α0 + α1p + · · · + αr−1pr−1. therefore, by the isomorphism ψ (see also [18]), the following properties are obtained. 1. from (8), the involution ̂ induces the identity automorphism on gr(pr,s). 2. from (9), the involution ̂ induces the ring automorphism ¯ on gr(pr, 2sej) as defined in (1), i.e., α = α p sej 0 + α p sej 1 p + · · · + α p sej r−1 p r−1 for all α = α0 + α1p + · · · + αr−1pr−1 in gr(pr, 2sej), where αi ∈t2sej for all i = 0, 1, . . . ,r − 1. 3. for each pair (z,z′) ∈ gr(pr,sfk) × gr(pr,sfk), we have ̂ψ−1(z,z′) = ψ−1(z′,z). from the discussion, we have ψ(x̂) = (x1, . . . ,xri,y1, . . . ,yrii,z ′ 1,z1, . . . ,z ′ riii ,zriii ), where ¯ is induced as above in an appropriate galois extension. proposition a.1. let x = ∑ b∈p xby b and u = ∑ b∈p uby b be elements in r[p]. decomposing xb,ub using (7), we have ψ(xb) = (xb,1, . . . ,xb,ri,yb,1, . . . ,yb,rii,zb,1,z ′ b,1, . . . ,zb,riii,z ′ b,riii ) and ψ(ub) = (ub,1, . . . ,ub,ri,vb,1, . . . ,vb,rii,wb,1,w ′ b,1, . . . ,wb,riii,w ′ b,riii ). then ψ(〈x,u〉̂) = ψ (∑ b∈p xbûb ) = ∑ b∈p ψ(xb)ψ(ûb) = (∑ b∈p xb,1ub,1, . . . , ∑ b∈p xb,riub,ri, ∑ b∈p yb,1vb,1, . . . , ∑ b∈p yb,riivb,rii, ∑ b∈p zb,1w ′ b,1, ∑ b∈p z′b,1wb,1, . . . , ∑ b∈p zb,riiiw ′ b,riii , ∑ b∈p z′b,riiiwb,riii ) . 91 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 in particular, 〈x,u〉̂ = 0 if and only if ψ(〈x,u〉̂) = 0, or equivalently,∑ b∈p xb,jub,j = 0 for all j = 1, 2, . . . ,ri, ∑ b∈p yb,jṽb,j = 0 for all j = 1, 2, . . . ,rii, and ∑ b∈p zb,jw ′ b,j = 0 = ∑ b∈p z′b,jwb,j for all j = 1, 2, . . . ,riii. using the orthogonality in proposition a.1, the euclidean dual of c in (e3) can be viewed to be of the form c⊥e ∼= ( ti∏ i=1 u⊥ei ) ×   tii∏ j=1 v ⊥hj  × ( tiii∏ k=1 ( (w ′k) ⊥e ×w⊥ek )) . (10) a.2. hermitian duality let ψ denote the isomorphism in (h1). then each element x ∈r, we can write ψ(x) = (x1, . . . ,xtii′ ,y1,y ′ 1, . . . ,ytiii′ ,y ′ tiii′ ), (11) where xj ∈ gr(pr,séj) and yk,y′k ∈ gr(p r,sf́k) for all j = 1, 2, . . . , tii′ and k = 1, 2, . . . , tiii′. we note that, for c = ∑ a∈a cay a ∈ gr(pr,s)[a], we have c̃ = ∑ a∈a cay −a = ∑ a∈a c−ay a, where α0 + pα1 + · · · + pr−1αr−1 = α ps/2 0 + pα ps/2 1 + · · · + p r−1α ps/2 r−1 . then ˘̃c = ∑ a∈a w̆ay a, where w̆a = ∑ h∈a c−hζ γa(h). from (3), if sps(h) is of type ii ′ with cardinality ν, then −a = psν/2 · a by remark 2.4. since ν is odd, we have w̆h = ∑ a∈a c−aζ γh(a) = ∑ a∈a caζ γ−h(a) = ∑ a∈a caζ γ psν/2·h (a) = ∑ a∈a ca ( ζγh(a) )psν/2 = θ(c̆h), (12) where θ(α) = αp sν/2 0 + α psν/2 1 p + · · · + α psν/2 r−1 p r−1 for all α = α0 + α1p + · · · + αr−1pr−1. by the isomorphism ψ (see also [18]), we have the following properties. 1. by (12), the involution ˜ induces the ring automorphism ¯ on gr(pr,séj) as defined in (1), i.e., α = α p séj/2 0 + α p séj/2 1 p + · · · + α p séj/2 r−1 p r−1 for all α = α0 + α1p + · · · + αr−1pr−1 in gr(pr,séj), where ai ∈tséj for all i = 0, 1, . . . ,r − 1. 2. for each pair (z,z′) ∈ gr(pr,sf́k) × gr(pr,sf́k), we have ˜ψ−1(z,z′) = ψ−1(z′,z). hence, x̃ defined in section 2 can be viewed in terms of (11) as ψ(x̃) = (x1, . . . ,xtii′ ,y ′ 1,y1, . . . ,y ′ rii ,ytiii′ ). where ¯ is induced as above in an appropriate galois extension. 92 s. jitman, s. ling / j. algebra comb. discrete appl. 6(2) (2019) 75–94 proposition a.2. let x = ∑ b∈p xby b and u = ∑ b∈p uby b be elements in r[p]. decomposing xb,ub using (11), we have ψ(xb) = (xb,1, . . . ,xb,tii′ ,yb,1,y ′ b,1, . . . ,yb,tiii′ ,y ′ b,tiii′ ) and ψ(ub) = (ub,1, . . . ,ub,tii′ ,vb,1,v ′ b,1, . . . ,vb,tiii′ ,v ′ b,tiii′ ). then ψ(〈x,u〉∼) = ψ (∑ b∈p xbũb ) = ∑ b∈p ψ(xb)ψ(ũb) = (∑ b∈p xb,1ub,1, . . . , ∑ b∈p xb,tii′ub,tii′ , ∑ b∈p yb,1v ′ b,1, ∑ b∈p y′b,1vb,1, . . . , ∑ b∈p yb,tiii′v ′ b,tiii′ , ∑ b∈p y′b,tiii′vb,tiii′ ) . in particular, 〈x,u〉∼ = 0 if and only if ψ(〈x,u〉∼) = 0, or equivalently,∑ b∈p xb,jub,j = 0 for all j = 1, 2, . . . , tii′ and ∑ b∈p yb,kv ′ b,k = 0 = ∑ b∈p y′b,kvb,k for all k = 1, 2, . . . , tiii′. using the orthogonality in proposition a.2, the hermitian dual of c in (h3) can be viewed of the form c⊥h ∼=   tii′∏ j=1 e⊥hj  × ( tiii′∏ k=1 ( (f ′k) ⊥e ×f⊥ek )) . 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[27] x. yang, j. l. massey, the condition for a cyclic code to have a complementary dual, discrete math. 126(1–3) (1994) 391–393. 94 https://doi.org/10.1109/tit.2010.2092415 https://doi.org/10.1109/tit.2010.2092415 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1007/s12095-018-0314-5 https://doi.org/10.1109/tit.2012.2236383 https://doi.org/10.1109/tit.2012.2236383 http://dx.doi.org/10.3934/amc.2016004 http://dx.doi.org/10.3934/amc.2016004 https://doi.org/10.1109/tit.2013.2296495 https://doi.org/10.1109/tit.2013.2296495 https://doi.org/10.1016/j.ffa.2008.02.003 https://doi.org/10.1016/j.ffa.2008.02.003 https://doi.org/10.1007/s10623-011-9538-5 https://doi.org/10.1007/s10623-011-9538-5 https://doi.org/10.1109/tit.2003.815816 https://doi.org/10.1109/tit.2003.815816 https://mathscinet.ams.org/mathscinet-getitem?mr=1896125 https://mathscinet.ams.org/mathscinet-getitem?mr=1896125 https://doi.org/10.4064/aa-80-3-197-212 https://mathscinet.ams.org/mathscinet-getitem?mr=2209183 https://mathscinet.ams.org/mathscinet-getitem?mr=2209183 https://doi.org/10.4153/cmb-1972-025-1 https://doi.org/10.1016/j.dam.2005.03.016 https://doi.org/10.1016/j.dam.2005.03.016 https://doi.org/10.1016/j.ffa.2009.01.004 https://doi.org/10.1016/j.ffa.2009.01.004 https://doi.org/10.1142/5350 https://doi.org/10.1109/tit.2002.805076 https://doi.org/10.1016/0012-365x(94)90283-6 https://doi.org/10.1016/0012-365x(94)90283-6 introduction preliminaries self-dual abelian codes in gr(pr,s)[g] complementary dual abelian codes in gr(pr,s)[g] conclusion appendix a references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.90728 j. algebra comb. discrete appl. 3(3) • 155–158 received: 30 october 2015 accepted: 06 january 2016 journal of algebra combinatorics discrete structures and applications the covering number of m24 research article michael epstein, spyros s. magliveras abstract: a finite cover c of a group g is a finite collection of proper subgroups of g such that g is equal to the union of all of the members of c. such a cover is called minimal if it has the smallest cardinality among all finite covers of g. the covering number of g, denoted by σ(g), is the number of subgroups in a minimal cover of g. in this paper the covering number of the mathieu group m24 is shown to be 3336. 2010 msc: 20b40, 20d05, 20e07, 20e32 keywords: group theory, group coverings, finite simple groups 1. introduction a finite collection c of proper subgroups of a group g is said to be a finite cover of g if ⋃ h∈c h = g. of course if g is cyclic then g does not admit such a cover, but any group with a finite noncyclic homomorphic image has a finite cover. the covering number of such a group g is denoted by σ(g), and is defined by σ(g) = min{|c| : c is a finite cover of g}. any cover satisfying |c| = σ(g) is called minimal. in [3] j. h. e. cohn proved that if g is a finite noncyclic supersolvable group then σ(g) = p + 1, where p is the least prime such that g has more than one subgroup of index p, and conjectured that if g is a finite noncyclic solvable group, then σ(g) = pa + 1, where pa is the order of the smallest chief factor of g with more than one complement in g. this conjecture was proven by tomkinson in [11], who suggested investigating the covering numbers of simple groups. in [2], r. bryce, v. fedri, and l. serena determined the covering numbers of some linear groups. the covering numbers of the suzuki groups were investigated by m. s. lucido in [9]. a. maróti considers alternating and symmetric groups in [10], wherein it is shown that σ(sn) = 2n−1 if n is odd and not equal to 9, that σ(sn) ≤ 2n−2 if n is even, and that if n is not equal to 7 or 9 then σ(an) ≥ 2n−2 with equality if and only if n ≡ 2 (mod 4). further results on the covering numbers of small alternating and symmetric groups can be found in [3, 5, 7, 8]. michael epstein (corresponding author), spyros s. magliveras; department of mathematical sciences, florida atlantic university, boca raton, fl 33431 (email: mepstein2012@fau.edu, spyros@fau.edu). 155 m. epstein, s. magliveras / j. algebra comb. discrete appl. 3(3) (2016) 155–158 in [6], p. e. holmes determined the covering numbers of the mathieu groups m11, m22, and m23, as well as the lyons group and the o’nan group, and gave upper and lower bounds for the covering numbers of the janko group j1 and the mclaughlin group. the covering number of m12 was determined by l. c. kappe, d. nikolova-popova, and e. swartz in [8]. the aim of this paper is to show that σ(m24) = 3336. 2. preliminaries throughout we use standard terminology and notation from group theory. we will write n ·h and n\h to denote a split extension of n by h and a general extension of n by h respectively. if π is an element of a permutation group and the disjoint cycle decomposition of π has ki cycles of length mi, 1 ≤ i ≤ r, with m1 > m2 > ... > mr, we will write the cycle type of π as mk11 m k2 2 ...m kr r . let g be a group and x ∈ g. if 〈x〉 is maximal among cyclic subgroups of g then we call x a principal element and 〈x〉 a principal subgroup of g. it is easy to see that a collection c of proper subgroups of g is a cover if and only if every principal subgroup is contained in a member of c. if g is a finite noncyclic group and c is a finite cover of g, then by replacing each subgroup h ∈c with a maximal subgroup m of g such that h ≤ m, we can obtain a cover c′ of g consisting of maximal subgroups with |c′| ≤ |c|. so, for the purpose of determining the covering number of such a group it suffices to consider covers consisting solely of maximal subgroups. 3. the mathieu group m24 in light of the discussion in section 2, we begin with the maximal subgroups and the principal elements of m24. as seen in [4], there are 9 conjugacy classes of maximal subgroups of m24, which we denote by mi, 1 ≤ i ≤ 9 ordered such that |m1| ≤ |m2| ≤ ... ≤ |m9|. the sizes of these conjugacy classes of maximal subgroups are given by (|m1|, ..., |m9|) = (24,276,759,1288,1771,2024,3795,40320, 1457280). if hi ∈ mi for i = 1, ...,9 then the isomorphism types of the hi are as follows: h1 ∼= m23, h2 ∼= m22 ·z2, h3 ∼= z42 ·a8, h4 ∼= m12 ·z2, h5 ∼= z62 · (z3\s6), h6 ∼= l3(4) ·s3, h7 ∼= z62 · (l3(2)×s3), h8 ∼= l2(23), and h9 ∼= l2(7). let x = {j ∈ z | 1 ≤ j ≤ 24}, and for a positive integer k let ( x k ) denote the set of all subsets of x with cardinality k. we note that h1, h2, and h6 are stabilizers in the actions of m24 on x, ( x 2 ) , and ( x 3 ) respectively. the principal elements of m24 (represented on 24 points) have cycle types 82412112, 10222, 11212, 121614121, 122, 141712111, 151513111, 21131, and 23111. we will denote the sets of principal elements with these cycle types by t1, . . . ,t9 respectively. we remark that t6, t7, t8, and t9 are each the union of two conjugacy classes of principal elements with the same cycle type, while the remaining ti consist of a single conjugacy class of elements. the cardinalities of these sets are given by (|t1|, . . . , |t9|) = (15301440,12241152,22256640,20401920,20401920,34974720,32643072,23316480,21288960). we describe the incidence between the sets t1, . . . ,t9 and the classes m1, . . . ,m9 of maximal subgroups with a matrix a = (ai,j) where the entry ai,j in row ti and column mj is the number of elements from ti contained in each maximal subgroup from class mj. the entries of this matrix were computed using the magma algebra system [1], and are given in table 1. observe that the elements from t1, t3, t6, t7, and t9 each fix a point of x and therefore are contained within the subgroups from class m1. each element from t8 has a single cycle of length 3 and is therefore contained within a unique member of class m6. from table 1 we can see that the subgroups from class m4 contain elements from each of t2, t4, and t5, and since each of these sets of principal elements consists of a single conjugacy class, every element from t2 ∪t4 ∪t5 is contained within some member of m4. consequently, m1 ∪m4 ∪m6 is a cover of m24 by 24 + 1288 + 2024 = 3336 maximal subgroups, and σ(m24) ≤ 3336. 156 m. epstein, s. magliveras / j. algebra comb. discrete appl. 3(3) (2016) 155–158 table 1. the incidence matrix a ti\mj m1 m2 m3 m4 m5 m6 m7 m8 m9 t1 1275120 110880 20160 23760 8640 15120 4032 0 0 t2 0 88704 0 28512 6912 0 0 0 0 t3 1854720 80640 0 17280 0 0 0 2760 0 t4 0 73920 26880 31680 23040 0 5376 0 0 t5 0 0 0 15840 11520 0 5376 1012 0 t6 1457280 126720 46080 0 0 17280 9216 0 0 t7 1360128 0 43008 0 18432 16128 0 0 0 t8 0 0 0 0 0 11520 6144 0 0 t9 887040 0 0 0 0 0 0 528 0 now suppose that c is a cover of m24 which consists of maximal subgroups. for 1 ≤ i ≤ 9, let xi = |c∩mi|. since the subgroups from class m9 contain no principal elements, we may assume without loss of generality that x9 = 0. then since c is a cover of m24 we must have 8∑ j=1 ai,jxj ≥ |ti|, 1 ≤ i ≤ 9. (1) the reader can verify (by integer linear programming, for example) that if (x1, ...,x8) is a tuple of nonnegative integers with xj ≤ |mj| for 1 ≤ j ≤ 8 which satisfies the system of inequalities given by (1), then ∑8 j=1 xj ≥ 3336. thus for any such cover c we have |c| ≥ 3336, and so we conclude that σ(m24) = 3336. acknowledgment: the authors would like to express their thanks to dr. igor kliakhandler whose generous support made possible a most significant conference on algebraic combinatorics and applications at michigan technical university in august, 2015. the authors also wish to thank prof. vladimir tonchev for his work to secure extra funding, a superbly organized conference, and the wonderful hospitality. references [1] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symbolic comput. 24(3-4) (1997) 235–265. [2] r. a. bryce, v. fedri, l. serena, subgroup coverings of some linear groups, bull. austral. math. soc. 60(2) (1999) 227–238. [3] j. h. e. cohn, on n−sum groups, math. scand. 75(1) (1994) 44–58. [4] j. h. conway, r. t. curtis, s. p. norton, r. a. parker, r. a. wilson, atlas of finite groups, clarendon press, oxford, 1985. [5] m. epstein, s.s. magliveras, d. nikolova-popova, the covering numbers of a9 and a11, to appear in the j. combin. math. combin. comput. [6] p. e. holmes, subgroup coverings of some sporadic groups, j. combin. theory ser. a 113(6) (2006) 1204–1213. 157 http://dx.doi.org/10.1006/jsco.1996.0125 http://dx.doi.org/10.1006/jsco.1996.0125 http://dx.doi.org/10.1017/s0004972700036364 http://dx.doi.org/10.1017/s0004972700036364 http://www.ams.org/mathscinet-getitem?mr=1308936 http://dx.doi.org/10.1016/j.jcta.2005.09.006 http://dx.doi.org/10.1016/j.jcta.2005.09.006 m. epstein, s. magliveras / j. algebra comb. discrete appl. 3(3) (2016) 155–158 [7] l. c. kappe, j. l. redden, on the covering number of small alternating groups, contemp. math. 511 (2010) 109–125. [8] l. c. kappe, d. nikolova-popova, e. swartz, on the covering number of small symmetric groups and some sporadic simple groups, arxiv:1409.2292v1 [math.gr]. [9] m. s. lucido, on the covers of finite groups, in: c. m. campbell, e. f. robertson, g. c. smith (eds), groups st. andrews 2001, in oxford, vol ii, in : london math. soc. lecture note ser. 305, 2003, 395–399. [10] a. maróti, covering the symmetric groups with proper subgroups, j. combin. theory ser. a 110(1) (2005) 97–111. [11] m. j. tomkinson, groups as the union of proper subgroups, math. scand. 81(2) (1997) 191–198. 158 http://dx.doi.org/10.1090/conm/511/10045 http://dx.doi.org/10.1090/conm/511/10045 http://dx.doi.org/10.1016/j.jcta.2004.10.003 http://dx.doi.org/10.1016/j.jcta.2004.10.003 http://www.ams.org/mathscinet-getitem?mr=1613772 introduction preliminaries the mathieu group m24 references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.729440 j. algebra comb. discrete appl. 7(2) • 141–160 received: 7 october 2019 accepted: 4 december 2019 journal of algebra combinatorics discrete structures and applications trace forms of certain subfields of cyclotomic fields and applications∗ research article agnaldo josé ferrari, antonio aparecido de andrade, robson ricardo de araujo, josé carmelo interlando abstract: in this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in euclidean space with optimal center density. we also obtain a closed formula for the gram matrix of algebraic lattices obtained from these subfields. the obtained lattices are rotated versions of the lattices λ9, λ10 and λ11 and they are images of z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. we also obtain algebraic lattices in odd dimensions up to 7 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity. 2010 msc: 11h06, 11h31, 11r80, 97n70 keywords: cyclotomic fields, algebraic lattices, twisted homomorphism, signal design 1. introduction lattices have been considered in different areas, especially in coding theory and more recently in cryptography. algebraic lattices are lattices obtained via the ring of integers of a number field and they have been studied in several papers and from different points of view [1–7, 10–13, 15, 16, 18]. ∗ this work was supported by fapesp 2013/25977-7 and cnpq 429346/2018-2. agnaldo josé ferrari; department of mathematics, school of sciences, são paulo state university (unesp), bauru-sp, brazil (email: agnaldo.ferrari@unesp.br). antonio aparecido de andrade (corresponding author); department of mathematics, institute of biosciences, humanities and exact sciences (ibilce), são paulo state university (unesp), são josé do rio preto-sp, brazil (email: antonio.andrade@unesp.br). robson ricardo de araujo; são paulo federal institute at cubatão, são paulo, brazil (email: dearaujorobisonricardo@gmail.com). josé carmelo interlando; department of mathematics & statistics, san diego university, san diego, california, usa (email: interlan@sdsu.edu). 141 https://orcid.org/0000-0002-1422-1416 http://orcid.org/0000-0001-6452-2236 https://orcid.org/0000-0002-1357-9926 https://orcid.org/0000-0002-1357-9926 https://orcid.org/0000-0003-4928-043x a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 the classical sphere packing problem is to determine how densely a large number of identical spheres can be packed together in the euclidean space. the packing density of a lattice λ is the proportion of the space covered by the non-overlapping spheres of maximum radius centered at the points of λ. the densest known lattice packings in dimensions 1 through 8 and 24 are also the optimal ones, see [12, p. 12] for n = 1, 2, . . . , 8, and [11] for n = 24. those lattice packings are unique. for all other dimensions, it is not known whether the current records are optimal. a lattice λ has diversity equal to k if k is the maximum number such that any non-zero vector y ∈ λ has at least k non-zero coordinates. given an n-dimensional lattice λ ⊆ rn of full diversity, the minimum product distance of λ is defined as dp,min(λ) = min{ ∏n i=1 |yi|; y = (y1,y2, . . . ,yn) ∈ λ, y 6= 0}. usually the problem of finding good signal constellations for a gaussian channel is associated with the search for lattices with high packing density, see [12, chapter 3]. on the other hand, for a rayleigh fading channel, the efficiency of the signal constellation, measured by the error probability in the transmission, is strongly related to the lattice diversity and its minimum product distance, see [10, section iii]. for this purpose the lattice parameters we consider here are packing density, diversity, and minimum product distance. the approach in this work, following [2, 3] is the use of algebraic number theory to construct lattices which have good performance on both channels. for general lattices the packing density and the minimum product distance are usually hard to estimate [17]. those parameters can be calculated in certain cases of lattices associated to number fields through algebraic properties. in [4–6] some families of rotated znlattices of full diversity and high minimum product distance are studied for transmission over rayleigh fading channels. in [13] some families of rotated zn-lattices of full diversity in dimensions power of 3 are studied and lower bounds for the minimum product distances of such construction are also presented. in [7] the lattices ap−1, p prime, e6, e8, k12 and λ24 were realized as full diversity ideal lattices via some subfields of cyclotomic fields. in this work we construct the lattices d3,e5,e7, λ9, λ10 and λ11, calculate (or estimate) their minimum product distance and compare the obtained values with those known in literature, mainly zn-lattices given in [5, 18]. in [14, theorem 1] a trace form for cyclotomic fields q(ζn) via the minkowski homomorphism is derived. in this work, we generalize the result for the maximal real subfields q(ζn + ζ−1n ) via the twisted homomorphism. in [6, proposition 4.1(a)] and [8] the gram matrix of the algebraic lattice constructed via the minkowski homomorphism over q(ζpr +ζ−1pr ) is determined, but in this work we use a different aproach. in this work, we generalize the result to the case q(ζn + ζ−1n ) considering the twisted homomorphism. trace forms are used to calculate the packing radius of algebraic lattices. as an application, we present constructions of algebraic lattices with optimal center density in dimensions 3, 5, 7, 9, 10 and 11. the paper is organized as follows. in sections 2, we collect some results on number fields and algebraic lattices. in section 3, we present a explicit trace form for the maximal real subfields via the twisted homomorphism. we also present a closed formula of gram matrix for the lattice σα(ok), where k = q(ζn + ζ−1n ). in section 4, we construct algebraic lattices in euclidean space with optimal center density in dimensions 3, 5, 7, 9, 10 and 11 and calculate (estimate) their minimum product distance. finally, in section 5, we draw our conclusions. 2. background on number fields and algebraic lattices if l is a number field of degree n, that is, a field that is a finite degree extension of q, then l = q(α), where α ∈ c is a root of a monic irreducible polynomial p(x) ∈ z[x]. the n distinct roots of p(x), namely, α1,α2, . . . ,αn, are the conjugates of α. if σ : l → c is a q-homomorphism, then σ(α) = αi for some i = 1, 2, . . . ,n. furthermore, there are exactly n q-homomorphisms σi, for i = 1, 2, . . . ,n, of l in c, where r1 are real monomorphisms and 2r2 are complex monomorphisms with n = r1 + 2r2. an element α ∈ l is called an algebraic integer if there is a monic polynomial f(x) with integer coefficients such that f(α) = 0. the set ol = {α ∈ l : α is an algebraic integer} 142 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 is a ring, called ring of algebraic integers of l [20, 21]. it can be shown that ol, as a z-module, has a basis {α1,α2, . . . ,αn} over z, called integral basis, where n is the degree of l. in other words, every element α ∈ol can be uniquely written as α = n∑ i=1 aiαi, where ai ∈ z for all i = 1, 2, . . . ,n. the trace and the norm of an element α ∈ l over q are defined as the rational numbers trl/q(α) = n∑ i=1 σi(α) and nl/q(α) = n∏ i=1 σi(α). if α ∈ ol, then trl/q(α) and nl/q(α) are algebraic integers. the discriminant of l over q is defined by dl = d(α1,α2, . . . ,αn) = det 1≤i,j≤n (σi(αj)) 2, where {α1,α2, . . . ,αn} is an integral basis of l. a lattice λ is a discrete additive subgroup of rn considered as the standard real vector space, that is, λ ⊆ rn is a lattice if there are linearly independent vectors α1,α2, . . . ,αm ∈ rn such that λ = { m∑ i=1 aiαi; ai ∈ z, i = 1, 2, . . . ,m } . the classical sphere packing problem is to find out how densely a large number of identical spheres can be packed together in the euclidean space. the packing density, ∆(λ), of a lattice λ is the proportion of the space rn covered by the non-overlapping spheres of maximum radius centered at the points of λ. the densest possible lattice packings have only be determined in dimensions 1 to 8 and 24 [12, p. 12]. it is also known that these densest lattice packings are unique. let {α1,α2 . . . ,αm} be a set of linearly independent vectors in rn and λ = { ∑m i=1 aiαi; ai ∈ z} a lattice. the set {α1,α2, . . . ,αm} is called a basis for λ. a matrix m whose rows are these vectors is said to be a generator matrix for λ whereas g = mmt = (〈αi,αj〉) m i,j=1 is called the gram matrix of λ. the determinant of λ, denoted by det λ, is equal to det g and it is an invariant under change of basis. the volume of λ is equal to √ det(λ). the packing density of λ is the proportion occupied by the spheres centered in the points of the lattice and having radius min{||x−y||; x,y ∈ λ, x 6= y}/2 relative to the entire space rn. if ∆(λ) is the packing density of λ, then δ(λ) = ∆(λ)/vn is the center density of the lattice, where vn is the volume of an n-dimensional sphere of radius 1 [12, p. 9]. let α ∈ l such that αi = σi(α) > 0 for all i = 1, . . . ,n. if r(x) and i(x) denote, respectively, the real part and the imaginary part of x, the homomorphism σα : l −→ rn defined by σα(x) = (√ α1σ1(x), . . . , √ αr1σr1 (x), . . . , √ αr1+r2r(σr1+r2 (x)), √ αr1+r2i(σr1+r2 (x)) ) , for every x ∈ l, is called twisted homomorphism [2, 3]. when α = 1 the twisted homomorphism is the minkowski homomorphism. if m is a z-module in l of rank n with z-basis {w1,w2, . . . ,wn}, then the set λ = σα(m) is a complete lattice in rn with basis {σα(w1),σα(w2), . . . ,σα(wn)}. if l is a totally real number field then g = ( trl/q(αwiwj) )n i,j=1 is a gram matrix for σα(m). from [2], det(λ) = [ok : m]2nl/q(α)|dl|, so the center density of λ is given by δ(λ) = ρn√ det(λ) = tn/2 2n[ol : m] √ nl/q(α)|dl| , 143 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 where dl denotes the discriminant of the number field l, [ol : m] denotes the index of m and t = min { trl/q(αx 2) : x ∈m, x 6= 0 } . if m is a z-module in l of rank m, m < n, with z-basis {w1,w2, . . . ,wm}, then the set λ = σα(m) is a lattice of rank m in rn with basis {σα(w1),σα(w2), . . . ,σα(wm)} and q = ( trl/q(αwiwj) )n i,j=1 is a gram matrix for σα(m). the center density of λ is given by δ(λ) =   tm/2 2m √ det(q) , if l is totally real. tm/2 23m/2 √ det(q) , if l is totally complex. (1) if k is a totally real field number with [k : q] = n and m⊆ k a free z-module of rank n, then the minimum product distance of λ = σα(m) is defined as dp,min(λ) = √ nk/q(α) min 06=y∈m |nk/q(y)|. (2) in particular, by [5], if m⊆ k is a principal ideal then dp,min(λ) = √ det(λ) |dk| . (3) the relative minimum product distance of λ, denoted by dp,rel(λ), is the minimum product distance of a scaled version of λ with unitary minimum norm vector. thus, if λ1 is a scaled version of λ of dimension n with scale factor √ k, i. e., λ1 = √ kλ and the minimum norm of λ is µ, then the relative minimum product distance of λ1 is given by dp,rel(λ1) = ( 1 √ kµ )n dp,min(λ1). (4) 3. trace forms for cyclotomic fields in this section, we present explicit trace forms for maximal real subfiel k = q(ζn + ζ−1n ) via twisted homomorphism. we also present a closed formula of gram matrix for the lattice σα(ok). a related result, through a different approach, can be found in [6] and [8], where the authors use abelian fields of odd prime power conductor. a cyclotomic field is a number field l such that l = q(ζn), where ζn is a primitive n-th root of unity. it can be shown that [l : q] = ϕ(n), where ϕ is the euler function, ol = z[ζn] is the ring of algebraic integers of z[ζn], {1,ζn,ζ2n, . . . ,ζ ϕ(n)−1 n } is an integral basis of l. let k = q(ζn + ζ−1n ) be the maximal real subfield of a cyclotomic field q(ζn). in this case, [k : q] = ϕ(n)/2, ok = z[ζn + ζ−1n ] and {1,ζn + ζ−1n ,ζ2n + ζ−2n , . . . ,ζ ϕ(n) 2 −1 n + ζ −ϕ(n) 2 +1 n } is an integral basis of ok. [21]. lemma 3.1. [14] let j,n be integers. if gcd(j,n) = d, then trq(ζn)/q(ζ j n) = ϕ(n) ϕ(n/d) trq(ζn/d)/q(ζ j/d n/d ). lemma 3.2. [14] if j,ai are integers, ai ≥ 1 and pi is a prime number such that gcd(j,piai) = 1, then trq(ζpiai )/q (ζ j pi ai ) = { −1, if ai = 1. 0, if ai > 1. 144 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 lemma 3.3. [14] let n = pa11 · · ·p as s , where ak ≥ 1, for k = 1, . . . ,s. if j is a prime number and gcd(j,n) = d, then trq(ζn)/q(ζ j n) = ϕ(n) ϕ(n/d) µ(n/d), where µ is the moebius function. lemma 3.4. let n = pa11 · · ·p as s , where aj ≥ 1, for j = 1, . . .s. if i is an integer such that i < ϕ(n) and d = gcd(i,n), then trq(ζn)/q(ζ i n) 6= 0 ⇔ d = (n/p)tk and i = (n/p)k, where p = p1 · · ·ps, tk = gcd(k,p) and k = 0, 1, 2, . . . ,ϕ(p) − 1. proof. if d = (n/p)tk, where tk = gcd(k,p), then the values that tk can assume are 1 and pα1 · · ·pαt, where 1 ≤ αr ≤ s, for r = 1, 2, . . . , t and αr 6= αl if r 6= l and 1 ≤ t < s. so, d = pa1−11 · · ·p as−1 s or d = pa1−11 · · ·p aα1 α1 · · ·p aαt αt · · ·pas−1s . thus, n/d = p1 · · ·ps or n/d = p1 · · ·pα1−1pα1+1 · · ·pαt−1pαt+1 · · ·ps, and therefore, µ(n/d) = ±1 6= 0. but, from lemma 3.3, it follows that trq(ζn)/q(ζ i n) = ϕ(n) ϕ(n/d) µ(n/d) 6= 0. now, if i = (n/p)k, then gcd(i,n) = d, for k = 0, 1, 2, . . . ,ϕ(p) − 1. in fact, if gcd(i,n) = d′, then gcd((n/p)k,n) = d′. thus, gcd(((n/p)k)/(n/p)n/(n/p)) = d′/(n/p), that is, tk = gcd(k,p) = (p/n)d′. so, d′ = (n/p)tk = d and k = 0, 1, 2, . . . ,ϕ(p) − 1. from the euler function, it follows that ϕ(n) ϕ(p) = ϕ(p a1 1 ···p as s ) ϕ(p1...ps) = ϕ(p a1 1 )···ϕ(p as s ) ϕ(p1)···ϕ(ps) = [p1 a1−1(p1−1)]···[psas−1(ps−1)] (p1−1)···(ps−1) = pa1−11 · · ·p as−1 s = n p . thus, if k ≥ ϕ(p), then i = (n/p)k = ((ϕ(n))/(ϕ(p))k ≥ ((ϕ(n))/(ϕ(p))ϕ(p) = ϕ(n), which is a contradiction. therefore, k = 0, 1, 2, . . . ,ϕ(p) − 1. furthermore, d = (n/p)tk if and only if i = (n/p)k. reciprocically, suppose that trq(ζn)/q(ζ i n) 6= 0 com d 6= (n/p)tk. thus, n/d is not square free. so, from definition of euler function, it follows that µ(n/d) = 0, and therefore, trq(ζn)/q(ζ i n) = 0, which is a contradiction. lemma 3.5. let n = pa11 · · ·p as s , where ar ≥ 1, for r = 1, . . .s. if i and j are integers such that i,j < ϕ(n) and d = gcd(i− j,n), then trq(ζn)/q(ζ i−j n ) 6= 0 ⇔ d = (n/p)tk and |i− j| = (n/p)k, where p = p1 · · ·ps, tk = gcd(k,p) and k = 0, 1, 2, . . . ,ϕ(p) − 1. proof. it is enough to observe that trq(ζn)/q(ζ i−j n ) = trq(ζn)/q(ζ j−i n ), gcd(i − j,n) = gcd(j − i,n) and as i,j < ϕ(n) then |i− j| < ϕ(n). therefore, by lemma 3.4, it follow the result. lemma 3.6. let n = pa11 · · ·p as s , where ar ≥ 1, for r = 1, . . .s. if i and j are integers such that i,j < ϕ(n) and d = gcd(i + j,n), then trq(ζn)/q(ζ i+j n ) 6= 0 ⇔ d = (n/p)tk and i + j = (n/p)k, where p = p1 · · ·ps, tk = gcd(k,p) and k = 0, 1, . . . , 2ϕ(p) − 2 if n = p and k = 0, 1, . . . , 2ϕ(p) − 1, otherwise. 145 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 proof. for n = p , if k ≥ 2ϕ(p) − 1, then i + j = (n/p)k = k ≥ 2ϕ(p) − 1 = 2ϕ(n) − 1, which is a contradiction, since i,j ≤ ϕ(n) − 1 implies i + j ≤ 2ϕ(n) − 2. thus, k = 0, 1, . . . , 2ϕ(p) − 2. for n > p, if k ≤ 2ϕ(p) − 1, then i + j = (n/p)k ≤ (ϕ(n)/ϕ(p))(2ϕ(p) − 1) = 2ϕ(n) − (n/p) < 2ϕ(n)−1. so, i + j ≤ 2ϕ(n) − 2. if k ≥ 2ϕ(p), then i + j = (n/p)k ≥ (ϕ(n)/ϕ(p))2ϕ(p) = 2ϕ(n), which is a contradiction, since i + j ≤ 2ϕ(n) − 2. therefore, k = 0, 1, . . . , 2ϕ(p) − 1. lemma 3.7. let n = pa11 · · ·p as s , where ar ≥ 1, for r = 1, 2, . . .s. if i and j are integers such i,j ≤ ϕ(n)/2 − 1 and d = gcd(i + 2j,n), then trq(ζn)/q(ζ i+2j n ) 6= 0 ⇔ d = (n/p)tk and i + 2j = (n/p)k, where p = p1 · · ·ps, tk = gcd(k,p) and k = 0, 1, . . . ,b3ϕ(p)/2 − 3p/nc, where byc is the greater integer less than or equal to y. proof. if k > 3ϕ(p)/2 − 3p/n, then i + 2j = (n/p)k > (n/p) (3ϕ(p)/2 − 3p/n) = 3ϕ(n)/2 − 3, which is a contradiction, since i,j ≤ ϕ(n)/2 − 1 implies i + 2j ≤ 3ϕ(n)/2 − 3. thus, k = 0, 1, . . . ,b3ϕ(p)/2 − 3p/nc. lemma 3.8. let n = pa11 · · ·p as s , where ar ≥ 1, for r = 1, 2, . . .s. if i and j are integers such i,j ≤ ϕ(n)/2 − 1 and d = gcd(−i + 2j,n), then trq(ζn)/q(ζ −i+2j n ) 6= 0 ⇔ d = (n/p)tk and |− i + 2j| = (n/p)k, where p = p1 · · ·ps, tk = gcd(k,p) and k = 0, 1, 2, . . . ,bϕ(p) − 3p/nc. proof. it is enough to observe that trq(ζn)/q(ζ −i+2j n ) = trq(ζn)/q(ζ i−2j n ), gcd(−i + 2j,n) = gcd(i−2j,n). if k > ϕ(p)−3p/n, then |−i+2j| = (n/p)k > (n/p) (ϕ(p) − 3p/n) = ϕ(n)−3, which is a contradiction, since i,j ≤ ϕ(n)/2−1 implies |−i+2j| ≤ ϕ(n)−3. thus, k = 0, 1, . . . ,bϕ(p) − 3p/nc. proposition 3.9. let l = q(ζn) and k = q(ζn +ζ−1n ) be its maximal real subfield, where n = p a1 1 . . .p as s , with aj ≥ 1, for j = 1, 2, . . .s, m = ϕ(n). let α = α0+α1(ζn+ζ−1n )+α2(ζ2n+ζ−2n )+· · ·+αm/2−1(ζ m/2−1 n + ζ −m/2+1 n ) be a totally positive element of z[ζn + ζ−1n ], i. e., σi(α) > 0, for all i = 1, 2, . . . ,m/2, where σi are the m/2 distinct q-homomorphisms from k to c. if x = a0 + a1(ζn + ζ−1n ) + a2(ζ 2 n + ζ −2 n ) + · · · + 146 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 am 2 −1(ζ m/2−1 n + ζ −m/2+1 n ) is an element of z[ζn + ζ−1n ], then trk/q(αx 2) = n p  ϕ(p)2 α0a20 + α0 u1∑ k=l1 n p k:even a2nk 2p ρ(tk) + 2α0a0 u2∑ k=l2 ank p ρ(tk) + 2α0 u3∑ k=l3 bnk p ρ(tk) + ϕ(p)α0 m/2−1∑ j=1 a2j + 2α0 u4∑ k=l2 ank p ρ(tk) + a 2 0 u2∑ k=l2 αnk p ρ(tk) + ∑ n6=i+2j= nk p l3≤k≤u5 1≤i≤m 2 −1 1≤j≤m 2 −1 2∑ i αia 2 j ρ(tk) + ϕ(p) 2∑ i+2j=n 1≤i≤m 2 −1 1≤j≤m 2 −1 2∑ i αia 2 j + ∑ 0≤|−i+2j|= nk p 0≤k≤u3 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αia 2 j ρ(tk) + 2a0 ∑ i+j= nk p l1≤k≤u1 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αiaj ρ(tk) + +2a0 ∑ 0≤|i−j|= nk p 0≤k≤u4 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αiaj ρ(tk) + 2 ∑ n6=i+j= nk p l4≤k≤u7 1≤i≤m/2−1 3≤j≤m−3 2∑ i αibj ρ(tk) + 2ϕ(p) ∑ i+j=n 1≤i≤m/2−1 3≤j≤m−3 2∑ i αibj + 2 ∑ 0≤|i−j|= nk p 0≤k≤u8 1≤i≤m/2−1 3≤j≤m−3 αibj ρ(tk) + 2 u2∑ k=l2 m/2−1∑ j=1 αnk p a2jρ(tk) + 2 ∑ i+j= nk p l1≤k≤u3 1≤i≤m/2−1 1≤j≤m 2 −2 2∑ i αiaj ρ(tk) + 2 ∑ 0≤|i−j|= nk p 0≤k≤u4 1≤i≤m/2−1 1≤j≤m 2 −2 2∑ i αiaj ρ(tk) u1∑ k=l1 n p k:even   . where p = p1 · · ·ps, tk = gcd(k,p), dye is the smaller integer greater than or equal to y, byc is the greater integer less than or equal to y, l1 = d2p/ne, l2 = dp/ne, l3 = d3p/ne, l4 = d4p/ne, u1 = bϕ(p) − 2p/nc, u2 = bϕ(p)/2 −p/nc, u3 = bϕ(p) − 3p/nc, u4 = bϕ(p)/2 − 2p/nc, u5 = b3ϕ(p)/2 − 3p/nc, u6 = b(m− 2)/4c, u7 = b3ϕ(p)/2 − 4p/nc, u8 = bϕ(p) − 4p/nc, ρ(tk) = µ( ptk )ϕ(tk) with µ the mobius function and ϕ the euler function, aj = a1aj+1+a2aj+2+· · ·+am 2 −1−jam/2−1, bj = ∑ k≥1 k<j−k≤m/2−1 akaj−k, and any sum of the trace form must be disregarded if the lower bound of k is greater than the upper bound. proof. let x = a0 + a1(ζn + ζ−1n ) + a2(ζ 2 n + ζ −2 n ) + · · · + am/2−1(ζ m/2−1 n + ζ −m/2+1 n ) be an element of ok = z[ζn + ζ−1n ], with ai ∈ z, for i = 0, 1, 2, . . .m/2 − 1. thus, x2 = (a0 + a1ζn + a1ζ −1 n + · · · + am/2−1ζ m/2−1 n + am/2−1ζ −m 2 +1 n ) 2 =  m/2−1∑ j=0 ajζ j n + m/2−1∑ j=1 ajζ −j n  2 =  m/2−1∑ j=0 ajζ j n  2 +  m/2−1∑ j=1 ajζ −j n  2 + 2  m/2−1∑ j=0 ajζ j n    m/2−1∑ j=1 ajζ −j n   = = m/2−1∑ j=0 a2jζ 2j n + 2 ∑ 0≤i<j≤m/2−1 aiajζ i+j n + m/2−1∑ j=1 a2jζ −2j n + 2 ∑ 1≤i<j≤m/2−1 aiajζ −i−j n 147 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 + 2  m/2−1∑ j=0 ajζ j n    m/2−1∑ j=1 ajζ −j n   = = a20 + m/2−1∑ j=1 a2j(ζ 2j n + ζ −2j n ) + 2 m/2−1∑ j=1 a0aj(ζ j n + ζ −j n ) + 2 m−3∑ j=3 bj(ζ j n + ζ −j n ) + 2 m/2−1∑ j=1 a2jtttttttttt + 2 m/2−2∑ j=1 aj(ζ j n + ζ −j n ), where aj = a1aj+1 + a2aj+2 + · · · + am/2−1−jam/2−1 and bj = ∑ k≥1 k<j−k≤m/2−1 akaj−k. now, let α = α0 + α1(ζn + ζ −1 n ) + α2(ζ 2 n + ζ −2 n ) + · · · + αm/2−1(ζ m/2−1 n + ζ −m/2+1 n ) be a totally positive element of ok = z[ζn + ζ−1n ], with αi ∈ z, for i = 0, 1, 2, . . .m/2 − 1. thus, αx2 =  α0 + m/2−1∑ i=1 αj(ζ i n + ζ −i n )    a20 + m/2−1∑ j=1 a2j(ζ 2j n + ζ −2j n ) + 2 m/2−1∑ j=1 a0aj(ζ j n + ζ −j n ) + 2 m−3∑ j=3 bj(ζ j n + ζ −j n ) + 2 m/2−1∑ j=1 a2j + 2 m 2 −2∑ j=1 aj(ζ j n + ζ −j n )   = = α0a 2 0 + α0 m/2−1∑ j=1 a2j(ζ 2j n + ζ −2j n ) + 2α0a0 m/2−1∑ j=1 aj(ζ j n + ζ −j n ) + 2α0 m−3∑ j=3 bj(ζ j n + ζ −j n ) +2α0 m/2−1∑ j=1 a2j + 2α0 m 2 −2∑ j=1 aj(ζ j n + ζ −j n ) + a 2 0 m/2−1∑ i=1 αi(ζ i n + ζ −i n ) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ i+2j n + ζ −i−2j n ) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ −i+2j n + ζ i−2j n ) +2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiaj(ζ i+j n + ζ −i−j n ) + 2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiaj(ζ i−j n + ζ −i+j n ) + 2 m/2−1∑ i=1 m−3∑ j=3 αibj(ζ i+j n + ζ −i−j n ) + 2 m/2−1∑ i=1 m−3∑ j=3 αibj(ζ i−j n + ζ −i+j n ) +2 m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ i n + ζ −i n ) + 2 m/2−1∑ i=1 m/2−2∑ j=1 αiaj(ζ i+j n + ζ −i−j n ) + 2 m/2−1∑ i=1 m/2−2∑ j=1 αiaj(ζ i−j n + ζ −i+j n ). since ζkn and ζ −k n are conjugates, it follows that they have the same trace, i. e., trl/q(ζ k n + ζ −k n ) = 2trl/q(ζ k n) and as trl/q(αx 2) = [l : k] trk/q(αx2), it follows that trk/q(αx 2) = 1 2 trl/q(αx 2). 148 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 thus, trk/q(αx 2) = 1 2 trl/q  α0a20 + α0 m/2−1∑ j=1 a2j(ζ 2j n + ζ −2j n ) + 2α0a0 m/2−1∑ j=1 aj(ζ j n + ζ −j n ) +2α0 m−3∑ j=3 bj(ζ j n + ζ −j n ) + 2α0 m/2−1∑ j=1 a2j + 2α0 m/2−2∑ j=1 aj(ζ j n + ζ −j n ) +a20 m/2−1∑ i=1 αi(ζ i n + ζ −i n ) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ i+2j n + ζ −i−2j n ) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ −i+2j n + ζ i−2j n ) + 2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiaj(ζ i+j n + ζ −i−j n ) +2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiaj(ζ i−j n + ζ −i+j n ) + 2 m/2−1∑ i=1 m−3∑ j=3 αibj(ζ i+j n + ζ −i−j n ) +2 m/2−1∑ i=1 m−3∑ j=3 αibj(ζ i−j n + ζ −i+j n ) + 2 m/2−1∑ i=1 m/2−1∑ j=1 αia 2 j(ζ i n + ζ −i n ) +2 m/2−1∑ i=1 m/2−2∑ j=1 αiaj(ζ i+j n + ζ −i−j n ) + 2 m/2−1∑ i=1 m/2−2∑ j=1 αiaj(ζ i−j n + ζ −i+j n )   = m 2 α0a 2 0 + α0 m/2−1∑ j=1 a2jtrl/q(ζ 2j n ) + 2α0a0 m/2−1∑ j=1 ajtrl/q(ζ j n) +2α0 m−3∑ j=3 bjtrl/q(ζ j n) + mα0 m/2−1∑ j=1 a2j + 2α0 m/2−2∑ j=1 ajtrl/q(ζ j n) +a20 m/2−1∑ j=1 αjtrl/q(ζ j n) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ i+2j n ) + m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ −i+2j n ) + 2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiajtrl/q(ζ i+j n ) +2a0 m/2−1∑ i=1 m/2−1∑ j=1 αiajtrl/q(ζ i−j n ) + 2 m/2−1∑ i=1 m−3∑ j=3 αibjtrl/q(ζ i+j n ) +2 m/2−1∑ i=1 m−3∑ j=3 αibjtrl/q(ζ i−j n ) + 2 m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ i n) +2 m/2−1∑ i=1 m/2−2∑ j=1 αiajtrl/q(ζ i+j n ) + 2 m/2−1∑ i=1 m/2−2∑ j=1 αiajtrl/q(ζ i−j n ). 149 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 from lemmas 3.3 to 3.8, it follows that m/2−1∑ j=1 a2jtrl/q(ζ 2j n ) = u1∑ k=l1 n p k:even a2nk 2p ρ(tk), m/2−1∑ j=1 ajtrl/q(ζ j n) = u2∑ k=l2 ank p ρ(tk), m−3∑ j=3 bjtrl/q(ζ j n) = u3∑ k=l3 bnk p ρ(tk), m/2−2∑ j=1 ajtrl/q(ζ j n) = u4∑ k=l2 ank p ρ(tk), m/2−1∑ j=1 αjtrl/q(ζ j n) = u2∑ k=l2 αnk p ρ(tk), m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ i+2j n ) = ∑ n 6=i+2j= nk p l3≤k≤u5 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αia 2 j ρ(tk) + 2∑ i+2j=n 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αia 2 j, m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ −i+2j n ) = ∑ 0≤|−i+2j|= nk p 0≤k≤u3 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αia 2 j ρ(tk), m/2−1∑ i=1 m/2−1∑ j=1 αiajtrl/q(ζ i+j n ) = ∑ i+j= nk p l1≤k≤u1 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αiaj ρ(tk), m/2−1∑ i=1 m/2−1∑ j=1 αiajtrl/q(ζ i−j n ) = ∑ 0≤|i−j|= nk p 0≤k≤u4 1≤i≤m/2−1 1≤j≤m/2−1 2∑ i αiaj ρ(tk), m/2−1∑ i=1 m−3∑ j=3 αibjtrl/q(ζ i+j n ) = ∑ n 6=i+j= nk p l4≤k≤u7 1≤i≤m/2−1 3≤j≤m−3 ∑ 2 iαibj ρ(tk) + ϕ(p) ∑ i+j=n 1≤i≤m/2−1 3≤j≤m−3 ∑ 2 iαibj, m/2−1∑ i=1 m−3∑ j=3 αibjtrl/q(ζ i−j n ) = ∑ 0≤|i−j|= nk p 0≤k≤u8 1≤i≤m/2−1 3≤j≤m−3 αibj ρ(tk), m/2−1∑ i=1 m/2−1∑ j=1 αia 2 jtrl/q(ζ i n) = u2∑ k=l2 m/2−1∑ j=1 αnk p a2jρ(tk), m/2−1∑ i=1 m/2−2∑ j=1 αiajtrl/q(ζ i+j n ) = ∑ i+j= nk p l1≤k≤u3 1≤i≤m/2−1 1≤j≤m 2 −2 ∑ 2 iαiaj ρ(tk), m/2−1∑ i=1 m/2−2∑ j=1 αiajtrl/q(ζ i−j n ) = ∑ 0≤|i−j|= nk p 0≤k≤u4 1≤i≤m/2−1 1≤j≤m 2 −2 2∑ i αiaj ρ(tk), where l1 = ⌈ 2p n ⌉ , l2 = ⌈ p n ⌉ , l3 = ⌈ 3p n ⌉ , l4 = ⌈ 4p n ⌉ , u1 = ⌊ ϕ(p) − 2p n ⌋ , u2 = ⌊ ϕ(p) 2 − p n ⌋ , 150 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 u3 = ⌊ ϕ(p) − 3p n ⌋ , u4 = ⌊ ϕ(p) 2 − 2p n ⌋ , u5 = ⌊ 3ϕ(p) 2 − 3p n ⌋ , u6 = ⌊ m−2 4 ⌋ , u7 = ⌊ 3ϕ(p) 2 − 4p n ⌋ , u8 = ⌊ ϕ(p) − 4p n ⌋ , ρ(tk) = µ( ptk )ϕ(tk) with µ the mobius function and ϕ the euler function. therefore, the result follows. as a corollary of the last theorem, we present an explicit gram matrix of a lattice via the maximal real subfield of k = q(ζn + ζ−1n ), when the z-module is the ring ok. corollary 3.10. let be k = q(ζn + ζ−1n ), e0 = 1 and ej = ζ j n + ζ −j n for j = 1, 2, . . . ,m/2 − 1, where n = pa11 · · ·p as s , m = ϕ(n) and α = α0 + α1(ζn + ζ −1 n ) + α2(ζ 2 n + ζ −2 n ) + · · · + αm/2−1(ζ m/2−1 n + ζ −m/2+1 n ) is a totally positive element of z[ζn + ζ−1n ], i. e., σi(α) > 0, for all i = 1, 2, . . . ,m/2, where σi are the m/2 distinct q-homomorphisms from k to c. a gram matrix for the lattice λ = σα(ok) is given by g = ( trk/q(αeiej) )m/2−1 i,j=0 , where (a) trk/q(αe0e0) = mα0 2 + n p u1∑ k=l1 αnk p ρ(tk), (b) for j ≥ 1, trk/q(αe0ej) = ϕ(n) ϕ(n/d1) µ(n/d1)α0 + n p ∑ r+j= nk p l2≤k≤u2 1≤r≤m/2−1 αrρ(tk) + n p ∑ |r−j|= nk p 0≤k≤u3 1≤r≤m/2−1 αrρ(tk), (c) for i,j ≥ 1, trk/q(αeiej) = ϕ(n) ϕ(n/d2) µ(n/d2)α0 + ϕ(n) ϕ(n/d3) µ(n/d3)α0 + n p ∑ r+i+j= nk p l3≤k≤u4 1≤r≤m/2−1 αrρ(tk) + n p ∑ |r−(i+j)|= nk p 0≤k≤u5 1≤r≤m/2−1 αrρ(tk) + n p ∑ |r+(i−j)|= nk p 0≤k≤u5 1≤r≤m/2−1 αrρ(tk) + n p ∑ |r+(j−i)|= nk p 0≤k≤u5 1≤r≤m/2−1 αrρ(tk), where p = p1 · · ·ps, tk = gcd(k,p), dye is the smaller integer greater than or equal to y, byc is the greater integer less than or equal to y, l1 = dp/ne, l2 = d2p/ne, l3 = d3p/ne, u1 = bϕ(p)/2 −p/nc, u2 = bϕ(p) − 2p/nc, u3 = bϕ(p)/2c, u4 = b3ϕ(p)/2 − 3p/nc, u5 = bϕ(p) − 3p/nc, d1 = gcd(j,n), d2 = gcd(i + j,n), d3 = gcd(i − j,n), ρ(tk) = µ( ptk )ϕ(tk) with µ the mobius function and ϕ the euler function, and any sum must be disregarded if the lower bound of k is greater than the upper bound. proof. from lemmas 3 to 6 and following the same steps of the proof of the proposition 3.9, the result follows. the next proposition, stated and proved in [19, corollary 2.3], will be used in the next section. before recalling it, we need a few assumptions. let l/q be a galois extension of prime degree p such that p is unramifed in ol, the ring of integers of l. denote the conductor of l by n, that is, n is the smallest positive integer such that l ⊆ q(ζn). then {σ(i)(θ)} p−1 i=0 is an integral basis for l where θ = trq(ζn)/l(ζn) and σ is any generator of gal(l/q). proposition 3.11. let x = p−1∑ i=0 aiσ (i)(θ) be any element in ol. then trl/q(x 2) = n p−1∑ i=0 a2i − n− 1 p ( p−1∑ i=0 ai )2 . 151 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 4. constructions of algebraic lattices in this section, we construct algebraic lattices in euclidean space with optimal center density in dimensions 9, 10 and 11 which are rotated versions of the lattices λ9, λ10 and λ11 via twisted embeddings applied to z-modules of the ring of integers of a number field k. we believe these constructions, as algebraic lattices, are new in the literature. constructions of rotated d3, d5 and e7-lattices via ideals and free z-modules that are not ideals are also presented. the same lattices are also constructed in [15, 16], through a different approach, where the authors construct these lattices by shifting ideal lattices constructed over cyclotomic fields via ideal or module in the maximal totally real subfields of cyclotomic fields. 4.1. construction of the d3-lattice if k = q(ζ9 + ζ−19 ), then [k : q] = 3 and dk = 3 4. if α = 1, then α is a totally positive element of z[ζ9 + ζ−19 ] and n(α) = 1. if m is a submodule of ok generated by {2 + e1 + e2,−e1 + e2, 2 − e2}, where ej = ζ j 9 + ζ −j 9 , for j = 1, 2, then m is a submodule of ok of index 6 and δ(σα(m)) = tn/2 2n[ol : m] √ n(α)|dk| = 183/2 236 √ |34| = 1 4 √ 2 , i.e., with the same center density of the lattice d3. the norm equation |nk/q(y)| = 2 has no solution in ok [9], however |nk/q(y)| = 3 when y = 2 − e2 ∈ m. thus, min 06=y∈m |nk/q(y)| = 3. and the minimum norm in d3 is µ = 2. as nk/q(α) = 1 and σα(m) is a scaled version of d3 with scale factor √ 9, from equation (4), it follows that dp,rel(σα(m)) = ( 1 √ 18 )3 √ 1 × 3 = 0.03928, and therefore, 3 √ dp,rel(σα(m)) = 0.33994. 4.2. construction of the d5-lattice if k = q(ζ11 + ζ−111 ), then [k : q] = 5 and dk = 11 4. if α = 2 − e1, where e1 = ζ11 + ζ−111 , then α is a totally positive element of z[ζ11 + ζ−111 ] and n(α) = 11. if m is a submodule of ok generated by {2 +e1,−e1,−e2,−e3,−e4}, where ej = ζ j 11 +ζ −j 11 , for j = 1, 2, 3, 4, then σα(m) is a lattice of rank 5 and δ(σα(m)) = tn/2 2n[ol : m] √ n(α)|dk| = 225/2 26 √ 115 = 1 8 √ 2 , i.e., with the same center density of the lattice d5. in this case, min 06=y∈m |nk/q(y)| = 1, because nk/q(y) = 1, where y = 2 + e1 ∈m. the minimum norm in d5 is µ = 2. as nk/q(α) = 11 and σα(m) is a scaled version of d5 with scale factor √ 11, by equation (4), it follows that dp,rel(σα(m)) = ( 1 √ 22 )5 √ 11.1 = 0.00146, 152 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 and therefore, 5 √ dp,rel(σα(m)) = 0.27097. 4.3. construction of the e7-lattice if k is a subfield of l = q(ζ29) such that [k : q] = 7 and 〈σ〉 = gal(k/q), where σ : ζ29 7→ ζ729, then k = q(θ), where θ = trl/k(ζ29) = ζ29 + ζ −1 29 + ζ 12 29 + ζ −12 29 . if α = nl/k(1 − ζ29) = ∏ i∈{1,−1,12,−12} (1 − ζi29) = = −(6θ + 4σ(θ) + 4σ2(θ) + 4σ3(θ) + 4σ4(θ) + 3σ5(θ) + 4σ6(θ)), then trk/q(α) = nk/q(α) = 29. if m = {a0θ + a1σ(θ) + · · · + a6σ6(θ) ∈ok; aj ≡ 0 (mod 2), for j = 0, 1, 2, 4}, then m is a z-submodule of ok and m is not an ideal of ok, because for x = σ(θ) ∈ ok and y = σ3(θ) ∈ m it follows that xy = −3θ − 2σ(θ) − 4σ2(θ) − 2σ3(θ) − 3σ4(θ) − 3σ5(θ) − 4σ6(θ) /∈ m. since dk = 29 6, it follows that δ(σα(m)) = tn/2 2n[ok : m] √ n(α)|dk| = (22.29)7/2 27.16 √ 29|296| = 1 16 , i.e., with the same center density of the lattice e7. as e7 is the only lattice with such center density in r7, it follows that σα(m) is a rotated version of e7. in this case, min 06=y∈m |nk/q(y)| = 1, since for y = −2θ − 2σ2(θ) + σ3(θ) − σ6(θ) ∈ m, nk/q(y) = 1. now, the minimum norm in e7 is µ = 2. as nk/q(α) = 29 and σα(m) is a scaled version of e7 with scale factor √ 58, by equation (4), it follows that dp,rel(σα(m)) = ( 1 √ 116 )7 √ 29.1 = 3.203 × 10−7, and consequently, 7 √ dp,rel(σα(m)) = 0.11809. 4.4. construction of the λ9 and λ10-lattice let k be a number field such that k = q(ζ180 + ζ−1180). in this case, [k : q] = 24, taking e0 = 1 and ej = ζ j 180 + ζ −j 180, for j = 1, 2, . . . , 23, we have that {1,e1,e2, . . . ,e22,e23} is a basis of k. let α = 165 + 129e2 + 153e4 + 120e6 + 119e8 + 105e10 + 67e12 + 82e14 + 25e16 + 49e18 + 3e20 + 17e22 be a 153 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 totally positive element of z[ζ180 + ζ−1180] and let m1 be a submodule of ok generated by the linearly independent vectors {w1,w2, · · · ,w10}, where w = mv,w = (w1,w2, · · · ,w10)t ,v = (1,e1,e2, · · · ,e23)t and the matrix m is given by   8 −24 8 8 0 8 16 −16 24 0 −16 0 8 −24 16 8 −32 32 −8 8 −8 0 −24 8 8 −16 −8 24 −16 0 8 −72 8 24 −16 64 8 −56 16 8 −16 40 −8 −48 16 8 8 32 −8 −16 8 24 16 16 −8 −8 −8 24 16 16 −8 −40 −16 −8 16 24 8 −16 −16 −24 −8 0 0 −16 −8 24 0 8 8 −40 0 24 8 40 8 −48 −8 0 0 32 −24 −32 0 −8 16 16 −8 48 0 −32 16 −8 0 96 −8 −16 24 −56 0 88 −16 −24 8 −88 −16 32 −16 −16 8 −48 −16 −8 −8 24 16 16 −16 0 −32 32 40 24 −8 −32 −32 −16 40 8 −8 −24 −8 −32 32 −8 −56 −16 −16 16 56 8 −56 −24 −96 16 128 24 −32 −32 −104 0 136 24 −8 −16 −40 −8 80 8 0 8 −16 0 −16 8 −8 40 −16 8 0 −16 0 16 0 −16 16 −24 0 8 24 −24 32 −24 24 48 16 −52 −8 8 24 84 44 −36 −32 −48 12 88 24 −20 −56 −72 0 40 −8 0 −48 −36 −16 52 0 −60 8 −4 −24 104 −16 −36 20 −56 −12 96 −12 −28 28 −80 48 0 4 −8 20 −44   . in this case, σα(m1) is a lattice of rank 10 in r24 and forall x ∈m1 we have that σα(x) = γta, where γ = (a1,a2,a3,a4,a5,a6,a7,a8,a9,a10), t = 4   2 −6 8 2 0 2 4 −4 6 0 −4 0 2 −6 4 2 −8 8 −2 2 −2 0 −6 2 2 −4 −2 6 −4 0 2 −18 2 6 −4 16 2 −14 4 2 −4 10 −2 −12 4 2 2 8 −2 −4 2 6 4 4 −2 −2 −2 6 4 4 −2 −10 −4 −2 4 6 2 −4 −4 −6 −2 0 0 −4 −2 6 0 2 2 −10 0 6 2 10 2 −12 0 0 −2 8 −6 −8 0 −2 4 4 −2 12 0 −8 4 −2 0 24 −2 −4 6 −14 0 22 −4 −6 2 −22 −4 8 −4 −4 2 −12 −4 −2 −2 6 4 4 −4 0 −8 8 10 6 −2 −8 −8 −4 10 2 −2 −6 −2 −8 8 −2 −14 −4 −4 4 14 2 −14 −6 −24 4 32 6 −8 −8 −26 0 34 6 −2 −4 −10 −2 20 2 0 2 −4 0 −4 2 −2 10 −4 2 0 −4 0 4 0 −4 4 −6 0 2 6 −6 8 −6 6 12 4 −13 −2 −2 6 21 11 −9 −8 −12 3 22 6 −5 −14 −18 0 10 −2 0 −12 −9 −4 13 0 −15 2 −1 −6 26 −4 −9 5 −14 −3 24 −3 −7 7 −20 0 12 1 −2 5 −11   and a =   √ σ1(α)σ1(1) · · · √ σ24(α)σ24(1)√ σ1(α)σ1(e1) · · · √ σ24(α)σ24(e1) ... ... ...√ σ1(α)σ1(e23) · · · √ σ24(α)σ24(e23)   , where a is a generator matrix of lattice σα(ok), since {1,e1,e2, . . . ,e22,e23} is a z-basis for ok. we have that b1 = ta is a generator matrix for σα(m1), and as b1 has rank 10, it follows that the lattice σα(m1) has rank 10. a gram matrix for the lattice σα(m1) is q = tgtt, where g = ( trk|q(αeiej) )23 i,j=0 is a 154 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 gram matrix for the lattice σα(ok). from proposition 3.10, it follows that the matrix g is given by 90   47 0 68 0 89 0 72 0 80 0 76 0 69 0 76 0 62 0 71 0 55 0 64 0 0 162 0 157 0 161 0 152 0 156 0 145 0 145 0 138 0 133 0 126 0 119 0 , 112 68 0 183 0 140 0 169 0 148 0 149 0 152 0 131 0 147 0 117 0 135 0 103 0 0 157 0 166 0 148 0 165 0 141 0 156 0 138 0 140 0 131 0 126 0 119 0 111 89 0 140 0 174 0 144 0 158 0 148 0 142 0 147 0 124 0 140 0 110 0 127 0 0 161 0 148 0 170 0 137 0 165 0 134 0 151 0 131 0 133 0 124 0 , 118 0 109 72 0 169 0 144 0 163 0 144 0 151 0 143 0 135 0 140 0 117 0 132 0 100 0 0 152 0 165 0 137 0 170 0 130 0 160 0 127 0 144 0 124 0 125 0 114 0 109 80 0 148 0 158 0 144 0 156 0 139 0 144 0 136 0 128 0 132 0 107 0 123 0 0 156 0 141 0 165 0 130 0 165 0 123 0 153 0 120 0 136 0 114 0 116 0 106 76 0 149 0 148 0 151 0 139 0 149 0 132 0 137 0 128 0 118 0 123 0 99 0 0 145 0 156 0 134 0 160 0 123 0 158 0 116 0 145 0 110 0 127 0 106 0 102 69 0 152 0 142 0 143 0 144 0 132 0 142 0 124 0 127 0 119 0 110 0 109 0 0 145 0 138 0 151 0 127 0 153 0 116 0 150 0 106 0 136 0 102 0 113 0 100 76 0 131 0 147 0 135 0 136 0 137 0 124 0 132 0 115 0 119 0 105 0 104 0 0 138 0 140 0 131 0 144 0 120 0 145 0 106 0 141 0 98 0 122 0 96 0 96 62 0 147 0 124 0 140 0 128 0 128 0 127 0 115 0 124 0 101 0 113 0 88 0 0 133 0 131 0 133 0 124 0 136 0 110 0 136 0 98 0 127 0 92 0 105 0 93 71 0 117 0 140 0 117 0 132 0 118 0 119 0 119 0 101 0 118 0 84 0 110 0 0 126 0 126 0 124 0 125 0 114 0 127 0 102 0 122 0 92 0 110 0 89 0 87 55 0 135 0 110 0 132 0 107 0 123 0 110 0 105 0 113 0 84 0 115 0 66 0 0 119 0 119 0 118 0 114 0 116 0 106 0 113 0 96 0 105 0 89 0 92 0 82 64 0 103 0 127 0 100 0 123 0 99 0 109 0 104 0 88 0 110 0 66 0 108 0 0 112 0 111 0 109 0 109 0 106 0 102 0 100 0 96 0 93 0 87 0 82 0 80   , and consequently, q = tgtt = 5760   4 −2 0 0 0 0 0 0 0 0 −2 4 −2 2 0 0 0 0 0 0 0 −2 4 0 0 2 0 0 0 0 0 2 0 4 2 2 0 0 0 0 0 0 0 2 4 2 0 0 2 1 0 0 2 2 2 4 2 2 1 2 0 0 0 0 0 2 4 2 0 2 0 0 0 0 0 2 2 4 0 2 0 0 0 0 2 1 0 0 4 2 0 0 0 0 1 2 2 2 2 4   , such that det(q) = 278321510. by proposition 3.9, the trace form of x ∈m1 is given by trk/q(αx 2) = 23040a21 − 23040a1a2 + 23040a22 − 23040a2a3 + 23040a23 + 23040a2a4 + 23040a24 +11520a10a5 +23040a4a5 +23040a 2 5 +23040a10a6 +23040a3a6 +23040a4a6 +23040a5a6 + 23040a26 + 23040a10a7 + 23040a6a7 + 23040a 2 7 + 23040a10a8 + 23040a6a8 + 23040a7a8 + 23040a28 + 23040a10a9 + 23040a5a9 + 11520a6a9 + 23040a 2 9 + 23040a 2 10. thus, t = min{trk/q(αx2); x ∈ m1, x 6= 0} = 23040 with a1 = 1 and aj = 0, for j 6= 1. by equation (1) it follows that the center density of lattice σα(m1) is given by δ(σα(m1)) = tm/2 2m √ det(q) = (23040)10/2 210 √ 278321510 = 1 16 √ 3 . therefore, σα(m1) is a lattice of rank 10 with the same center density of λ10. as 1 5760 q is a standard gram matrix of λ10 [12], it follows that 1 √ 5760 σα(m1) is a rotated version of λ10. using a computer, 155 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 we observed that min 0 6=y∈m |nk/q(y)| ≤ 4.13 × 1020. now, the minimum norm in λ10 is µ = 4. as nk/q(α) = 8.3 × 109 and σα(m) is a scaled version of λ10 with scale factor √ 5760, by equation (4) dp,rel(σα(m)) ≤ ( 1 √ 23040 )24 √ 8.3 × 109 × 4.13 × 1020 = 1.68 × 10−27, and consequently, 10 √ dp,rel(σα(m)) ≤ 0.00210. now, let m2 be a submodule of ok generated by the linearly independent vectors {w1,w2, . . . ,w9}, following the same steps as the construction of λ10, it follows that t = 4   2 −6 8 2 0 2 4 −4 6 0 −4 0 2 −6 4 2 −8 8 −2 2 −2 0 −6 2 2 −4 −2 6 −4 0 2 −18 2 6 −4 16 2 −14 4 2 −4 10 −2 −12 4 2 2 8 −2 −4 2 6 4 4 −2 −2 −2 6 4 4 −2 −10 −4 −2 4 6 2 −4 −4 −6 −2 0 0 −4 −2 6 0 2 2 −10 0 6 2 10 2 −12 0 0 −2 8 −6 −8 0 −2 4 4 −2 12 0 −8 4 −2 0 24 −2 −4 6 −14 0 22 −4 −6 2 −22 −4 8 −4 −4 2 −12 −4 −2 −2 6 4 4 −4 0 −8 8 10 6 −2 −8 −8 −4 10 2 −2 −6 −2 −8 8 −2 −14 −4 −4 4 14 2 −14 −6 −24 4 32 6 −8 −8 −26 0 34 6 −2 −4 −10 −2 20 2 0 2 −4 0 −4 2 −2 10 −4 2 0 −4 0 4 0 −4 4 −6 0 2 6 −6 8 −6 6 12 4 −13 −2 −2 6 21 11 −9 −8 −12 3 22 6 −5 −14 −18 0 10 −2 0 −12 −9   , and consequently a gram matrix for the lattice σα(m2) is given by q = tgtt = 5760   4 −2 0 0 0 0 0 0 0 −2 4 −2 2 0 0 0 0 0 0 −2 4 0 0 2 0 0 0 0 2 0 4 2 2 0 0 0 0 0 0 2 4 2 0 0 2 0 0 2 2 2 4 2 2 1 0 0 0 0 0 2 4 2 0 0 0 0 0 0 2 2 4 0 0 0 0 0 2 1 0 0 4   , such that det(q) = 27231859. by proposition 3.9 the trace form of x ∈m2 is given by trk/q(αx 2) = 23040a21 − 23040a1a2 + 23040a22 − 23040a2a3 + 23040a23 + 23040a2a4 + 23040a24 + 23040a4a5 + 23040a 2 5 + 23040a3a6 + 23040a4a6 + 23040a5a6 + 23040a 2 6 + 23040a6a7 + 23040a27 + 23040a6a8 + 23040a7a8 + 23040a 2 8 + 23040a5a9 + 11520a6a9 + 23040a 2 9. thus, t = min{trk/q(αx2); x ∈ m2, x 6= 0} = 23040 with a1 = 1 and aj = 0, for j 6= 1. by equation (1) it follows that the center density of lattice σα(m2) is given by δ(σα(m2)) = tm/2 2m √ det(q) = (23040)9/2 29 √ 27231859 = 1 16 √ 2 . therefore, σα(m2) is a lattice of rank 9 with the same center density of λ9. as 1 5760 q is a standard gram matrix of λ9 [12], it follows that 1 √ 5760 σα(m2) is a rotated version of λ9. using a computer, we observed 156 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 that min 06=y∈m |nk/q(y)| ≤ 2.28 × 1021. now, the minimum norm in λ9 is µ = 4. as nk/q(α) = 8.3 × 109 and σα(m) is a scaled version of λ9 with scale factor √ 5760, by equation (4) dp,rel(σα(m)) ≤ ( 1 √ 23040 )24 √ 8.3 × 109 × 2.28 × 1021 = 9.29 × 10−27, and consequently, 9 √ dp,rel(σα(m)) ≤ 0.00128. 4.5. construction of the λ11-lattice let l = q(ζ23), k the subfield of l such that [k : q] = 11, and 〈σ〉 = gal(k/q), where σ : ζ23 7→ ζ223. then k = q(θ) where θ = trl/k(ζ23) = ζ23 + ζ −1 23 . a z-basis for ok, the ring of integers of k, is {σi(θ)}10i=0. by proposition 3.11, the trace form of k is given by trk/q(x 2) = 23(a20 + a 2 1 + · · · + a 2 10) − 2(a0 + a1 + · · · + a10) 2, where x = a0θ + a1σ(θ) + · · · + a10σ10(θ). let γ = nl/k(1 − ζ23) = (1 − ζ23)(1 − ζ−123 ) = −θ − 2 10∑ i=0 σi(θ). it follows that trk/q(γ) = nk/q(γ) = 23. furthermore, let � = 6 10∑ i=0 σi(θ) + 4σ6(θ) + σ7(θ) be a unit in ok. the element α = �γ is totally positive and nk/q(α) = 23. let λ0 denote the lattice σα(ok), where σα : k → r11 is the twisted homomorphism given by σα(x) = ( √ σ0(α)σ0(x), √ σ(α)σ(x), . . . , √ σ10(α)σ10(x)) and m be the z-submodule of ok defined by m = {a0θ + a1σ(θ) + · · · + a10σ10(θ) ∈ok; aj ≡ 0 (mod 2), for j = 2, 4, 6, 8, 10}, then the gram matrix of σα(m) is given by g = 23   4 −2 0 2 −2 −2 0 2 0 0 −2 −2 4 0 −1 4 0 0 −1 −4 −1 2 0 0 12 0 0 −2 0 4 −8 −2 −4 2 −1 0 4 −6 −2 −2 0 0 2 0 −2 4 0 −6 24 0 0 0 −4 −6 0 −2 0 −2 −2 0 4 4 −2 0 −1 0 0 0 0 −2 0 4 12 −2 −4 −4 0 2 −1 4 0 0 −2 −2 4 −2 0 −4 0 −4 −8 0 −4 0 −4 −2 16 4 4 0 −1 −2 2 −6 −1 −4 0 4 4 0 −2 2 −4 0 0 0 0 −4 4 0 8   . 157 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 we have that t = min{trk/q(αx2) : x ∈m,x 6= 0} = 22 × 23, with a0 = 1, a1 = a2 = · · · = a10 = 0 and [ok : m] = 32. since dk = 2310, the center density of the lattice σα(m) is given by δ(σα(m)) = tn/2 2n[ok : m] √ n(α)|dk| = (22.23)11/2 211 × 32 √ 23|2310| = 1 32 , i.e., the same as that of the λmax11 lattice [12, chapter 6, section 4]. the above lattice is a rotated version of λmax11 . indeed, g ′ = 1 23 ugut, where g′ =   4 −2 0 0 0 0 0 0 0 0 0 −2 4 −2 2 0 0 0 0 0 0 0 0 −2 4 0 0 2 0 0 0 0 0 0 2 0 4 2 2 0 0 0 0 0 0 0 0 2 4 2 0 0 2 1 0 0 0 2 2 2 4 2 2 1 2 0 0 0 0 0 0 2 4 2 0 2 0 0 0 0 0 0 2 2 4 0 2 0 0 0 0 0 2 1 0 0 4 2 0 0 0 0 0 1 2 2 2 2 4 2 0 0 0 0 0 0 0 0 0 2 4   is a gram matrix of λmax11 and u =   −1 0 0 3 1 2 1 3 0 2 1 0 0 0 −2 −1 −2 −1 −2 0 −2 −1 −1 −2 −1 −1 0 −2 0 −1 −1 0 0 1 0 1 1 1 2 0 1 0 2 1 3 2 2 1 1 3 0 −1 1 2 0 0 0 0 −1 0 −2 0 −2 0 0 −1 −2 0 −1 −2 −1 −4 0 −2 0 −2 −2 −1 0 −1 −2 −1 −4 0 −2 0 −2 −2 −1 1 0 2 0 0 0 0 1 −2 −1 −2 0 −1 −1 −1 −4 0 −2 0 −3 −2 −2 −2 −1 −2 −1 −4 0 −2 −1 −2 −1   is an element of gl(11,z). thus, min 06=y∈m |nk/q(y)| = 1, since for y = −θ ∈m, nk/q(y) = 1. now, the minimum norm in λ11 is µ = 4. as nk/q(α) = 23 and σα(m) is a scaled version of λ11 with scale factor√ 23, by equation (4) dp,rel(σα(m)) = ( 1 √ 92 )11 √ 23.1 = 7.58 × 10−11, and consequently, 11 √ dp,rel(σα(m)) = 0.12022. 5. conclusions in this section, we assess the performance of the lattices presented in this paper in terms of center density and relative minimum product distance. these parameters are associated with the efficiency in signal transmission over gaussian and rayleigh fading channels, respectively. table 1 shows a comparison between the best known relative product distance of rotated zn-lattices, obtained via cyclotomic, cyclic and mixed constructions [5] and via krüskemper method [18] (first column), the densest lattices λ0 158 a. j. ferrari et al. / j. algebra comb. discrete appl. 7(2) (2020) 141–160 (d3, d5 and e7) obtained in [15, 16] (second column), and the densest lattices λn (λ3 = d3, λ5 = d5, λ7 = e7 and λ12 = k12) from our construction (third column). the center density δ of these lattices are also displayed. for the lattice d3 the minimum product distance 0.36964 presented in [16, table 3] is higher than that from our construction, while for dimension 5 our construction yields the same value. in dimension 7 (see [15, remark 4.13]) the authors obtained a lower bound on the minimum product distance whereas for our construction the bound holds. for dimensions 9 and 10, we constructed the lattices λ9 and λ10 with an upper bound for their minimum product distance. although the bounds are not very high, the lattices in those two dimensions and in dimension 11 are new and are not known in the literature (as algebraic lattices). in particular, the value that we obtained in dimension 11 is half of the one obtained for zn (cyclotomic and cyclic constructions in [5]). a broader question to be investigated is whether algebraic constructions of lattices, mainly in dimensions 9, 10 and 11, as the ones approached here, can provide greater relative minimum product distance for rotated densest lattices. it is noticed that the relative minimum product distances dp,rel(λn) of the rotated lattices obtained in the present paper are smaller than the relative minimum product distances dp,rel(zn) of rotated znlattices constructed for the rayleigh channels in [5, 18]. nevertheless, if the goal is to construct lattices which have good performance on both gaussian and rayleigh channels, were may assert that taking into account the trade-off center density versus product distance, there are some advantages in considering the rotated λn-lattices instead of rotated zn-lattices. table 1. relative minimum product distance versus center density (from [5, 15, 16, 18] and the results presented here) n n √ dp,rel(zn) n √ dp,rel(λ0) n √ dp,rel(λn) δ(zn) δ(λ) 3 0.52275 ≥ 0.36964 ≥ 0.33994 0.12500 0.17677 5 0.38321 ≥ 0.27097 ≥ 0.27097 0.03125 0.08838 7 0.30080 ≥ 0.11809 ≥ 0.11809 0.00781 0.06250 9 0.27018 ≤ 0.00128 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https://doi.org/10.1216/rmj-2017-47-4-1075 https://doi.org/10.1216/rmj-2017-47-4-1075 introduction background on number fields and algebraic lattices trace forms for cyclotomic fields constructions of algebraic lattices conclusions references issn 2148-838x j. algebra comb. discrete appl. 9(3) • 175–184 received: 4 july 2021 accepted: 25 november 2021 journal of algebra combinatorics discrete structures and applications on mignotte secret sharing schemes over gaussian integers research article diego munera-merayo abstract: secret sharing schemes (sss) are methods for distributing a secret among a set of participants. one of the first secret sharing schemes was proposed by m. mignotte, based on the chinese remainder theorem over the ring of integers. in this article we extend the mignotte’s scheme to the ring of gaussian integers and study some of its properties. while doing this we aim to solve a gap in a previous construction of such extension. in addition we show that any access structure can be made through a sss over z[i]. 2010 msc: 94a62, 13f07 keywords: secret sharing, mignotte sharing scheme, gaussian integer, chinese remainder theorem 1. introduction secret sharing schemes (sss’s) are methods for distributing a secret among a set of participants. among their applications we may highlight key management, threshold and visual cryptography, evoting, multiparty computation and more. given the increasing importance of information security in today’s society, it is not surprising that sss’s have become an active field of research within public key cryptography. in a sss, a secret s is broken into n pieces or shares. each share is given to one out of n participants. this shares are designed ensuring that some authorized coalitions of participants can reconstruct the secret by pooling the shares of its members, while non-authorized coalitions cannot do it. this family of authorized coalitions is called access structure of the scheme. the problem of secret sharing was firstly introduced by a. shamir in [11]. in that article he also suggested a method to perform it: the so-called shamir threshold secret sharing schemes, based on lagrangian interpolation. since the publication of shamir’s work, several other methods have been proposed. for our purposes within this article, we may highlight the ones by m. mignotte [9] and c. diego munera-merayo; university of valladolid (email:diego.munuera@alumnos.uva.es). 175 https://orcid.org/0000-0002-7735-6203 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 asmuth and j. bloom [1], both based on the chinese remainder theorem for coprime modules over the ring of integers z. this three methods belong to the category of threshold (t,n) schemes. this means that their access structures are formed by all the coalitions with at least t out of n participants, for a certain t. on the other hand, the chinese remainder theorem has many known applications in cryptography, [3]. it is based on a well known property of integers, the so-called euclidean division: for any two integers n,m with m 6= 0, there exists integers q (quotient) and r (remainder) such that m = nq +r with 0 ≤ r < |m|. of course, euclidean division is not exclusive to z. as there exist other rings that admit it, the called euclidean domains. among these we may highlight the ring of univariate polynomials over a field k[x], and the ring of gaussian integers z[i]. as a consequence a chinese remainder theorem holds over both k[x] and z[i], and therefore we can extend the mignotte secret sharing scheme to these rings. in the literature we can find successive generalizations of mignotte’s scheme. firstly by s. iftene to not necessarily coprime integers [7]. later t. galibus and g. matveev [5] provided a version over polynomial rings of one variable. in that paper the authors also showed the remarkable property that any access structure can be realized by a mignotte polynomial sss. finally, in [10] an extension of mignotte threshold scheme to the ring of gaussian integers is proposed, and subsequently applied to the problem of image sharing.recall that gaussian integers play a role in digital signal processing, cryptography, coding, and many other fields of science and technology, see [2]. and that mignotte’s threshold scheme has already been proposed for image sharing, [13]. however, the extension of mignotte sss to gaussian integers proposed in [10] contains a crucial gap. contrary to what occurs over z and k[x], in the euclidean division over z[i], quotient and remainder are not unique. thus, given v,z ∈ z[i], the expression v (mod z) does not uniquely identify one element of z[i]. this causes that the solution of a system of simultaneous congruence equations is not unique over z[i] and, finally, that mignotte’s scheme based on it may provide wrong recovered secrets. in this article we propose a new version of mignotte’s sss over z[i] that works properly and investigate some of its properties. the paper is organized as follows: in section 2 we recall some basic facts on sss’s and the arithmetic of z[i]. the proposed scheme is developed in section 3, where we also study its main properties. in particular, following [5] we show that any access structure can be realized by our scheme. also we characterize those access structures which are obtained from coprime modules. some examples are included in order to illustrate the obtained results. 2. some background on sss’s and gaussian integers first, we recall some known facts about secret sharing schemes and gaussian integers. a complete study of most of this subjects can be found in [12] for secret sharing and [4] for gaussian integers. 2.1. secret sharing schemes let p = {1, . . . ,n} be a set of n participants and s be a finite and non-empty set of secrets. we want to share a secret s ∈s among the participants in p. for that purpose, each participant will receive a data si about the secret, which we will call its share. a dealer computes the s1, . . . ,sn from s and assigns si to each participant i. a secret sharing scheme is a method r of calculating the shares si so that certain groups of participants, previously determined, can recover s by pooling the shares of their members; while making it impossible for any other coalition of participants to recover the secret. we will call access structure of the scheme (denoted by a) to the family of all coalitions authorized to recover the secret. note that this family is monotone increasing. we say that a scheme is perfect if unauthorized coalitions cannot deduce any information about the secret. another important feature to take into account in a scheme r is the size of the shares distributed 176 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 to the participants. let us denote by si the set of all possible values that si can take when s runs over s. it can be proved that when no unauthorized coalition can discard any element of s to be the shared secret, it holds that |si| ≥ |s|. we define the information rate of r as ρ(r) = min { log |si| log |s| : i = 1, . . . ,n } . if ρ(r) = 1 then the scheme r is called ideal. 2.2. the chinese remainder theorem in its classical version, over the ring of integers, the chinese remainder theorem states the following. theorem 2.1. let a1, . . . ,an ∈ z, and let m1, . . . ,mn be relatively pairwise coprime integers such that m1, . . . ,mn ≥ 2. the system of simultaneous congruence equations x ≡ ai (mod mi), i = 1, . . . ,n has a unique solution modulo lcm(m1, . . . ,mn). furthermore, this solution can be explicitly given as x = a1q1r1 + · · ·+ anqnrn. where qi = m1 · · ·mn/mi and ri = q−1i in z/miz, for i = 1, . . . ,n. this theorem can be generalized, as the integers need not to be relatively pairwise coprime. in this case there exists a solution if and only if ai ≡ aj(mod lcm(mi,mj)) for all 1 ≤ i,j ≤ n. furthermore, it can be stated over euclidean rings, such as gaussian integers. 2.3. mignotte sss over z the mignotte’s original secret sharing scheme allows the construction of threshold (t,n) schemes over a set p = {1, . . . ,n} of n participants as follows [9]: let m : m1 < m2 < · · · < mn, be a sequence of n pairwise relatively coprime integers such that mn−t+2 · · ·mn < m1 · · ·mt. the integer mi is assigned to participant i. let us write m− = mn−t+2 · · ·mn, m+ = m1 · · ·mt and for a coalition c ⊆ p, let m(c) = ∏ i∈c mi. the above condition on the mi’s implies that m(c) ≤ m − when |c| < t and m(c) ≥ m+ when |c| ≥ t. the scheme operates as follows: let s = {s ∈ z : m− < s < m+} be the set of secrets to be shared. given a secret s, the share of participant i is si = s (mod mi). an authorized coalition a with |a| ≥ t may recover the secret by solving the system (sa) x ≡ si (mod mi), i ∈ a whose solution x is unique modulo m(a) as guaranteed by the chinese remainder theorem. since s < m+ ≤ m(a), it holds that x = s. an unauthorized coalition b with |b| < t, may try to recover the secret by solving the system (sb) x ≡ si (mod mi), i ∈ b whose solution x is unique modulo m(b). but since x < m(b) ≤ m− ≤ s, it holds that x 6= s and so b does not recover the legitimate secret. 2.4. extensions of mignotte sss in [7] iftene suggested an extension of mignotte sss over the integers as follows. let m : m1,m2, · · · ,mn, be a sequence of n (not necessarily coprime) integers. for a coalition c ⊆ p let us denote by lcm(c) the least common multiple of {mi : i ∈ c}. take two integers m− < m+ such that the 177 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 interval (m−,m+) does not contain the lcm(c) of any coalition c ⊆ p. furthermore, we impose that π(m+−4m−) 4m− > 1. in this setting let us consider the family a = a(m,m+) = {a ⊆ p : lcm(a) ≥ m+}. this is a monotonous increasing family, so we may consider it as an access structure over p. the set of secrets is s = {s ∈ z : m− ≤ s < m+}. sharing and reconstruction of secrets is carried out as in the original mignotte scheme. another extension of mignotte’s scheme to the ring of polynomials in one variable over a finite field, fq[x], was given in [5]. following this idea, other extensions of the method to euclidean rings have been made, such as [10] or [13]. from now on we will focus on the extension to gaussian integers. 2.5. the ring of gaussian integers a gaussian integer is a complex number z = a + bi, where both a and b are integers. the set of gaussian integers is thus z[i]. over this ring we may consider the euclidean function norm of z, n(z) = a2 + b2 = zz (where z denotes the complex conjugated of z). thus the norm is multiplicative, n(vz) = n(v)n(z) for all v,z ∈ z[i] and satisfies n(v) ≤ n(vz) if z 6= 0. the most interesting arithmetic property of z[i] is the existence of an euclidean division with respect to the norm: given v,z ∈ z[i] with z 6= 0, there exist q,r ∈ z[i] such that v = zq + r with n(r) < n(z). we say that z[i] is an euclidean domain. thus z[i] is also a principal ideal domain [4], and therefore a chinese remainder theorem similar to theorem 2.1, holds over it. we observe, however, that the quotient and remainder of the euclidean division are not uniquely determined, unlike what we have over fq[x] or what we can do over z, see [8]. for example we have 10i = (5 + 4i)(1 +i) + (i−1) = (5 + 4i)(1 + 2i) + (3−4i). thus there is no a consistent way to define the expression v (mod z), as it does not uniquely identify any element. as a result, the solution of a system of congruence equations is not uniquely determined. 2.6. the mignotte sss over z[i] of [10] in [10] the authors propose an extension of mignotte’s scheme to a threshold (t,n) sss over z[i], and they use this extension to give a method for sharing secret images. however they did not take into account the lack of uniqueness of euclidean division. this causes that the proposed sss may lead to reconstruct the wrong secrets. the method goes as expected: in order to share a secret among a set p of n participants, we begin from a sequence m : m1, . . . ,mn of pairwise coprime gaussian integers such that n(m1) < · · · < n(mn) and n(mn−t+2 · · ·mn) < n(m1 · · ·mt). the set of secrets is s = {s ∈ z[i] : n(mn−t+2 · · ·mn) ≤ n(s) < n(m1 · · ·mt)}. the sharing of a secret s ∈ s and its subsequent reconstruction are performed in the usual way: given a secret s, the share of participant i is si = s (mod mi). a coalition a with |a| ≥ t, may recover the secret by solving the system (sa) x ≡ si (mod mi), i ∈ a. we will now show an example of how it operates and how it may lead to mistakes. example 2.2. (example 3.3 of [10]). let n = 3, m : 7 + 4i,−3 − 13i,11 + 8i and take t = 2. the set of secrets is then s = {s ∈ z[i] : 185 ≤ n(s) < 11570}. let s = 70 − 70i. note that n(s) = 9800 so s is a valid secret. the shares are s1 = 1 + 2i,s2 = 4,s3 = 3 − i. the authorized coalition {2,3} wants to recover the secret. to that end they solve the system{ x ≡ 4 (mod −3−13i) x ≡ 3− i (mod 11 + 8i) whose solution is x = 70−70i, but also x = −1 + 97i, and both are valid secrets. most computer systems choose the solution of smaller norm. since n(−1 + 97i) = 9410 < n(70 + 70i) = 9800, then the coalition {2,3} recovers s = −1 + 97i, which is a wrong secret. 178 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 3. a mignotte sss over gaussian integers we now aim to develop an extension of mignotte sss to gaussian integers, making sure that the remainders are uniquely determined. that is, to ensure that the expression v (mod z) refers to a unique element of z[i]. on top of that, we will not impose the condition that the modules m1, · · · ,mn are coprime, so that we find more general schemes. 3.1. fundamental domains in order to ensure the uniqueness of the remainder, we will operate in restrictions of z[i]. let z ∈ z[i], z 6= 0.we may define the fundamental domain of z is the subset of c f(z) = {z(α + βi) : α,β ∈ r,−1 2 < α,β ≤ 1 2 }. geometrically f(z) is a semi-open square in c ∼ r2, with vertices z(1 2 + 1 2 i), z(1 2 − 1 2 i), z(−1 2 − 1 2 i), z(−1 2 + 1 2 i). figure 1. fundamental domain of z. as seen in the above figure f(z) is centered at 0 and has side length |z| = √ n(z) . note that for every gaussian integer v ∈f(z) it holds that n(v) ≤ n(z)/2. on this space, we find the following. proposition 3.1. given two gaussian integers v,z ∈ z[i] with z 6= 0, there exists q,r ∈ z[i] such that r ∈f(z) and v = zq + r. moreover such q and r are unique and n(r) ≤ n(z)/2. proof. let us consider the complex number v/z = x + yi. let a, b be the integers a = dx − 1 2 e and b = dy− 1 2 e. thus −1 2 < x−a ≤ 1 2 , −1 2 < y−b ≤ 1 2 . let q = a+bi. then v = zq + (z(v/z−q)) = zq +r where r = z(v/z − q) = z((x−a) + (y − b)i) ∈f(z). to see the uniqueness, if zq1 + r1 = zq2 + r2 with r1,r2 ∈ f(z), then write r1 = z(α1 + β1i), r2 = z(α2 + β2i) with −12 < α1,β1,α2,β2 ≤ 1 2 . we have z(q2 −q1) = r1 −r2 = z((α1 −α2) + (β1 −β2)i), hence α1 −α2 and β1 −β2 are both integers. so α1 = α2 and β1 = β2. the last statement, n(r) ≤ n(z)/2, is a consequence of the fact that r ∈f(z). 179 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 given v,z ∈ z[i] with z 6= 0 as above, we say that the unique remainder r ∈ f(z) of the division v = zq + r, is the principal value of v (mod z). we will write this as r = v (mod z). let us note that the above proof gives a procedure to explicitly compute this number. besides f(z) we shall also consider the strict fundamental domain of z, which is the set fo(z) of points in f(z) but not on its border fo(z) = {z(α + βi) : α,β ∈ r,−1 2 < α,β < 1 2 )} and the closure of f(z) f(z) = {z(α + βi) : α,β ∈ r,−1 2 ≤ α,β ≤ 1 2 )}. clearly fo(z) ⊂f(z) ⊂f(z). now we will expose some properties of these domains, that will allow us to successfully develop the scheme. proposition 3.2. let v,z ∈ z[i], z 6= 0. (a) if n(v) < n(z)/2, then f(v) ⊂f(z). (b) {u ∈ z[i] : n(u) < n(z/2)}⊂fo(z) ⊂{u ∈ z[i] : n(u) ≤ n(z)/2}. proof. (a) since both f(v) and f(z) are squares centered at 0, it suffices to see that the highest norm of an element in f(v) is less than the smallest norm of an element on the border of f(z). as these are reached at v(1 2 + 1 2 i) and z(1 2 ) respectively, the inclusion is a consequence of the chain of inequalities n ( v(1 2 + 1 2 i) ) = 1 2 n(v) < 1 4 n(z) = n ( z(1 2 ) ) . (b) the left hand inclusion holds because the element with the lowest norm on the border of f(z) has norm n(z)/4. the right hand one is due to the property of the normalized remainder r in the euclidean division that n(r) ≤ n(z)/2. the units (that is, the invertible elements) of z[i] are precisely the elements with norm 1, that is ±1,±i. the associates of a gaussian integer z are the products uz, where u is a unit. thus, the associates of z are ±z,±iz. clearly two associated numbers z and uz generate the same ideal in z[i]. so given z1, . . . ,zn ∈ z[i], the expressions gcd(z1, . . . ,zn) and lcm(z1, . . . ,zn) are defined up to associates. note that in general f(z) 6= f(uz) and thus, for v ∈ z[i], we may have v (mod z) 6= v (mod uz). for example, z/2 ∈f(z) but z/2 /∈f(−z) and thus, if z/2 ∈ z[i] then we have z/2 (mod z) = z/2,z/2 (mod −z) = −z/2. proposition 3.3. let u,v,z be three gaussian integers such that u is a unit. the following properties hold. (a) fo(z) = fo(uz) and f(z) = f(uz). (b) if v (mod z) ∈fo(z), then v (mod uz) = v (mod z). proof. the proof of (a) is straightforward from the definitions of fo(z) and f(z). (b) if v = qz + r then also v = (q/u)uz + r, so the result is a direct consequence of (a). as a consequence of this result we find that if v (mod lcm(z1, . . . ,zn)) ∈ fo(lcm(z1, . . . ,zn)) for some choice of lcm(z1, . . . ,zn) and v (mod lcm(z1, . . . ,zn)), then the gaussian integer given by the expression v (mod lcm(z1, . . . ,zn)) is uniquely determined, and moreover v (mod lcm(z1, . . . ,zn)) ∈ fo(lcm(z1, . . . ,zn)). proposition 3.4. let v1,v2,z be gaussian integers such that v1,v2 ∈ f(z). if v1 ≡ v2 (mod z) and v1 ∈fo(z), then v1 = v2. proof. if v1 ≡ v2 (mod z) with v1 ∈ fo(z),v2 ∈ f(z), then there exist a unit u such that v1,v2 ∈ f(uz). the result follows from the uniqueness of the remainder in a fundamental domain stated in proposition 3.1. 180 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 3.2. a mignotte sss over z[i] now that we have found a way to guarantee the uniqueness of a remainder, we may expose the desired extension of the scheme. let p = {1, . . . ,n} be a set of n participants and let m : m1,m2, . . . ,mn, be a sequence of n (not necessarily pairwise coprime) non-zero gaussian integers. we assign the gaussian integer mi to each participant i. to abbreviate, given a coalition c ⊆ p we write lcm(c) = lcm{mi : i ∈ c}. we say that n(lcm(c)) is the norm of c. let m−,m+ be two integers such that 4m− < m+ and the norm of no coalition lies within the interval (m−,m+) , that is, for all c ⊆ p we have either n(lcm(c)) ≤ m− or n(lcm(c)) ≥ m+. furthermore, we impose that π(m +−4m−) 4m− > 1. we may now consider the access structure over p a = {a ⊆p : n(lcm(a)) ≥ m+}. let us now develop a sss r realizing the structure a. the set of secrets to be shared is s = {s ∈ z[i] : m− ≤ n(s) < m+ 4 }. the procedure goes as usual. given a secret s ∈s, the shares si are si = s (mod mi). if an authorized coalition a wants to recover the secret, they may solve the system of congruence equations (sa) x ≡ si (mod mi) i ∈ a whose solution is unique modulo lcm(a). the secret is x (mod lcm(a)). please note that for any coalition c the corresponding system (sc) has a solution, since s (mod lcm(c)) is so. thus we only need the chinese theorem to find the solution and to guarantee its uniqueness. theorem 3.5. the above method is correct and gives a secret sharing scheme whose access structure is a. proof. an authorized coalition a can solve the system and, from the solution x, obtain x (mod lcm(a)) ∈ f(lcm(a)). since n(s) < m+/4 ≤ n(lcm (a))/4, proposition 3.2 guarantees that s ∈ fo(lcm(a)), and by proposition 3.3 this holds for any choice of lcm(a). then, from proposition 3.4, we have x (mod lcm(a)) = s, and a successfully recovers the secret. an unauthorized coalition b may find the solution x of (sb) modulo lcm(b). however, since n(s) ≥ m− ≥ n(lcm(b)) > n(x), it holds that s 6= x. b can also compute x (mod lcm(b)) ∈ f(lcm(b)). but since n(s) > m−/2 > n(lcm (b))/2, again according to proposition 3.2, we have s 6∈ f(lcm(b)). hence x (mod lcm(b)) 6= s. we focus on this on section 3.3. example 3.6. we will now see how example 1 would work under this new method. let n = 3, m : 7+4i,−3−13i,11+8i and take t = 2. the set of secrets will now be s = {s ∈ z[i] : 185 ≤ n(s) < 2892}. let s = 70 − 70i. note that n(s) = 9800 so s is not a valid secret, and thus cannot be shared. this explains why the method proposed in [10] does not work on this case. 3.3. some properties once the method is described we may study some of its properties. computing the information rate of this scheme leads to the so called gauss circle problem, which asks about the number of gaussian integers inside a circle of radius r > 0 centered at the origin. that is to say, the number of gaussian integers z such that n(z) ≤ n(r) = r2, [6]. we denote this number by n(r). since, on average, each unit square contains one gaussian integer, n(r) is approximately equal to the area of a circle of radius r, n(r) ∼ πr2 181 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 it is also known a explicit expression for this number: n(r) = 1 + 4 ∞∑ j=0 (b r2 4j + 1 c−b r2 4j + 3 c) then the number of possible secrets to be shared is |s| = n( √ m+ −1 2 )−n( √ m− −1) = 4 ∞∑ j=0 (b m+ −1 16j + 16 c−b m+ −1 16j + 12 c)−{4 ∞∑ j=0 (b m− −1 4j + 1 c−b m− −1 4j + 3 c)} ∼ π ( m+ −1 4 −m− + 1 ) ∼ π ( m+ 4 −m− ) . on the other hand, the set si of possible shares for participant i is precisely the set of congruence classes of gaussian integers modulo mi. it is well known that the number of such congruence classes is the norm n(mi), [4]. unfortunately, the scheme is not perfect. an unauthorized coalition b can solve (sb) and thus it may compute x = s (mod lcm(b)). so it may deduce that the secret is of the form x + λ lcm(b) for some λ ∈ z[i]. thus it can discard all secrets of s not satisfying this condition. since n(lcm(b)) < m−, this fact reduces for b the set of secrets to a set of cardinality equal to  4 ∞∑ j=0 (b m+ −1 16j + 16 c−b m+ −1 16j + 12 c)−{4 ∞∑ j=0 (b m− −1 4j + 1 c−b m− −1 4j + 3 c)}  /n(lcm(b)) ∼ π(m+ −4m−)/4 n(lcm(b)) > π(m+ −4m−) 4m− . so this number should be large enough in order to guarantee the security of the scheme. note that we have imposed that number to be larger than 2, so that the coalition b never compute the correct secret. example 3.7. let m be the sequence of pairwise coprime gaussian integers m : 15+14i,10−18i,13+16i. take m− = 425,m+ = 178504. this choice leads to a (2,3) threshold access structure. the set of secrets is s = {s ∈ z[i] : 425 ≤ n(s) ≤ 44625}. the number of possible secrets is then |s| = n( √ 44625)−n( √ 424) ∼ 138858 while the set of possible shares has cardinality at most n(13 + 16i) = 425. of course the scheme is not perfect. as noted before, an unauthorized coalition b can reduce the whole set of secrets to a set of size at least (approximately) π ( (m+/4)2 − (m− −1)2 ) m− ≈ 327. 182 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 3.4. any access structure can be realized by a mignotte sss over z[i] an interesting question is to characterize those access structures that can be realized by a mignotte construction. the next proposition is analogous to theorem 2.2 of [5]. even the proof is an adaptation of it, we will include it for the convenience of the reader. theorem 3.8. for any access structure a over n participants and any integer s, there is sequence m : m1, . . . ,mn of gaussian integers such that the mignotte sss over z[i] arising from m realizes the structure a with a set of secrets s of cardinality |s|≥ s. proof. let a be an access structure over a set p of n participants and let b1, . . . ,bt be the maximal unauthorized coalitions fora. choose t pairwise coprime gaussian integers µ(1), . . . ,µ(t) with n(µ(j)) > 8 for all j = 1, . . . , t, and let µ = µ(1) · · ·µ(t). now for i = 1, . . . ,n and j = 1, . . . , t, define µ (j) i = { 1 if i ∈ bj µ(j) if i 6∈ bj and mi = µ (1) i · · ·µ (t) i . then the mignotte scheme associated to the sequence m : m1, . . . ,mn realizes the structure a. to see that let m+ = n(µ) and ` = m+/min{n(µ(1)), . . . ,n(µ(t))}. note that 4` < m+. let a ∈a be an authorized coalition. for each maximal unauthorized coalition bj there is a participant i (depending on j) such that i ∈ a\bj. thus µ (j) i = µ (j) and hence µ(j)|lcm(a). since this reasoning holds for any j = 1, . . . , t, we conclude that lcm(a) = µ, so n(lcm(a)) = m+. conversely, let b be an unauthorized coalition. then there exists j such that b ⊆ bj. it follows that µ (j) i = 1 for all i ∈ b, and hence n(lcm(b)) ≤ m +/n(µ(j)) ≤ `. we have proved that a = {a ⊆ p | n(lcm(a)) ≥ m+}, so the sequence m : m1, . . . ,mn realizes the structure a with set of secrets s = {s ∈ z[i] : m− ≤ n(s) < m + 4 }, where m− = max{n(lcm(b1)), . . . ,n(lcm(bt))} ≤ `. since m+ 4 −m− ≥ `, by choosing the gaussian integers µ(1), . . . ,µ(t) having large enough norm, it is clear that we can always get |s|≥ s. the simplest case of mignotte’s construction arises when all the numbers in the sequence m are pairwise coprime. let us recall that an access structure a is weighted threshold if there is an n-tuple w = (w1, . . . ,wn) of positive weights and a threshold t such that a = {a ⊆p : ∑ i∈a wi ≥ t}. the proof that a weighted access structure defined by real weights can be also written by using integer weights is straightforward. proposition 3.9. if the gaussian integers in the sequence m : m1,m2, · · · ,mn are pairwise coprime, then for any m+ the access structure a(m,m+) is a weighted threshold access structure. proof. a coalition c is authorized if and only if n(lcm(c)) = ∏ i∈c n(mi) ≥ m +, that is if and only if ∑ i∈c log(n(mi)) ≥ log(m +), so a(m,m+) is weighted threshold structure with weights log(n(mi)), j = 1 . . . ,n, and threshold log(m+). example 3.10. let m be the sequence of pairwise coprime gaussian integers m : 6 + 5i,1 − 9i,13 + 16i, and take m− = 5002,m+ = 25925. this sequence leads to the access structure a whose minimal authorized coalitions are {1,3} and {2,3}, which certainly is a weighted threshold access structure with weights 1,1,2 and threshold t = 3. 183 d. munera-merayo / j. algebra comb. discrete appl. 9(3) (2022) 175–184 references [1] c. a. asmuth, j. bloom, a modular approach to key safeguarding, ieee trans. inf. theory 29(2) (1983) 208–210. 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[10] i. ozbek, f. temiz, i. siap, a generalization of the mignotte’s scheme over euclidean domains and applications to secret image sharing, j. algebra comb. discrete appl. 6(3) (2019) 147–161. [11] a. shamir, how to share a secret?, communications of acm 22(11) (1979) 612–613. [12] d. stinson, cryptography: theory and practice, crc press (2005). [13] g. ulutas, m. ulutas, v. nabiyev, secret sharing scheme based on mignotte’s scheme, ieee 19th signal processing and communications applications conference (siu), antalya (2011) 291–294. 184 https://doi.org/10.1109/tit.1983.1056651 https://doi.org/10.1109/tit.1983.1056651 https://link.springer.com/book/10.1007/978-1-4612-2826-4 https://doi.org/10.1142/3254 https://doi.org/10.1142/3254 https://www.worldcat.org/title/first-course-in-abstract-algebra/oclc/991699474?referer=br&ht=edition https://doi.org/10.1016/j.entcs.2006.12.044 https://doi.org/10.1016/j.entcs.2006.12.044 https://link.springer.com/book/10.1007%2f978-0-387-26677-0 https://doi.org/10.1016/j.entcs.2007.01.065 https://doi.org/10.1016/j.entcs.2007.01.065 https://doi.org/10.2307/2315810 https://doi.org/10.1007/3-540-39466-4_27 https://doi.org/10.1007/3-540-39466-4_27 https://mathscinet.ams.org/mathscinet-getitem?mr=4010414 https://mathscinet.ams.org/mathscinet-getitem?mr=4010414 https://doi.org/10.1145/359168.359176 https://www.worldcat.org/title/cryptography-theory-and-practice/oclc/1158700191&referer=brief_results https://doi.org/10.1109/siu.2011.5929644 https://doi.org/10.1109/siu.2011.5929644 introduction some background on sss's and gaussian integers a mignotte sss over gaussian integers references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056547 j. algebra comb. discrete appl. 9(1) • 29–46 received: 24 january 2021 accepted: 25 november 2021 journal of algebra combinatorics discrete structures and applications decomposition of cartesian product of complete graphs into sunlet graphs of order eight∗ research article kaliappan sowndhariya, appu muthusamy abstract: for any integer k ≥ 3, we define the sunlet graph of order 2k, denoted by l2k, as the graph consisting of a cycle of length k together with k pendant vertices such that, each pendant vertex adjacent to exactly one vertex of the cycle so that the degree of each vertex in the cycle is 3. in this paper, we establish necessary and sufficient conditions for the existence of decomposition of the cartesian product of complete graphs into sunlet graphs of order eight. 2010 msc: 05c51 keywords: graph decomposition, cartesian product, corona graph, sunlet graph 1. introduction all graphs considered here are finite, simple and undirected. a cycle of length k is called k-cycle and it is denoted by ck. km denotes the complete graph on m vertices and km,n denotes the complete bipartite graph with m and n vertices in the parts. we denote the complete m-partite graph with n1,n2, . . . ,nm vertices in the parts by kn1,n2,...,nm. for any integer λ > 0, λg denotes the graph consisting of λ edge-disjoint copies of g. let g and h be two graphs of orders m and n, respectively. the corona product g � h is the graph obtained by taking one copy of g and m copies of h such that the ith vertex of g is connected to every vertex in the ith copy of h. we define the sunlet graph l2k with v (l2k) = {x1,x2, . . . ,xk,xk+1,xk+2, . . . ,x2k} and e(l2k) = {xixi+1 ∪xixk+i | i = 1,2, ...,k and subscripts of the first term is taken addition modulo k}. we denote it by l2k = ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) . clearly, l2k ∼= ck �k1. ∗ this work was supported by department of science and technology, university grant commission, government of india. kaliappan sowndhariya, appu muthusamy (corresponding author); department of mathematics, periyar university, salem, tamil nadu, india (email: sowndhariyak@gmail.com, ambdu@yahoo.com). 29 https://orcid.org/0000-0001-9014-6916 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 the cartesian product of two graphs, g and h, denoted by g�h, has the vertex set v (g)×v (h) and two vertices (g,h) and (g′,h′) are adjacent if and only if either g = g′ and h is adjacent to h′ in h or h = h′ and g is adjacent to g′ in g. it is well known that cartesian product is commutative, associative and distributive over edge-disjoint union of graphs. we shall use the following notation throughout the paper. let g and h be simple graphs with vertex sets v (g) = {x1,x2, . . . ,xn} and v (h) = {y1,y2, . . . ,ym}. then for our convenience, we write v (g) × v (h) = ⋃n i=1 xi, where xi stands for xi × v (h). further, in the sequel, we shall denote the vertices of xi as { x j i|1 ≤ j ≤ m } , where xji stands for the vertex (xi,yj) ∈ v (g)×v (h). by a decomposition of a graph g, we mean a list of edge-disjoint subgraphs whose union is g. for a graph g, if e(g) can be partitioned into e1,e2, ...,ek such that the subgraph induced by ei is hi, for all i, 1 ≤ i ≤ k, then we say that h1,h2, ...,hk decompose g and we write g = h1 ⊕ h2 ⊕ ... ⊕ hk, since h1,h2, ...,hk are edge-disjoint subgraphs of g. for 1 ≤ i ≤ k, if hi = h, we say that g has a hdecomposition. study of h-decomposition of graphs is not new. many authors around the world are working in the field of cycle decomposition [4, 7, 8, 18, 19], path decomposition [22, 23], star decompositon [17, 21, 24, 25] and hamilton cycle decomposition [2, 3, 13, 14] problems in graphs. here we consider the sunlet decomposition of product graphs. anitha and lekshmi [5, 6] proved that n-sun decomposition of complete graph, complete bipartite graph and the harary graphs. liang and guo [15, 16] gave the existence spectrum of a k-sun system of order v as k = 2,4,5,6,8. fu et. al. [10, 11] obtained that 3-sun decompositions of kp,p,r, knand embed a cyclic steiner triple system of order n into a 3-sun system of order 2n − 1, for n ≡ 1 (mod 6). further they obtained k-sun system when k = 6,10,14,2t, for t > 1. fu et. al. [9] obtained the existence of a 5-sun system of order v. gionfriddo et.al. [12] obtained the spectrum for uniformly resolvable decompositions of kv into 1-factor and h-suns. akwu and ajayi [1] obtained the necessary and sufficient conditions for the existence of decomposition of kn ⊗ km and (kn − i) ⊗ km, where i denote the 1-factor of a complete graph into sunlet graph of order 2p, p is a prime. sowndhariya and muthusamy [20] obtained necessary and sufficient conditions for the existence of decomposition of km ×kn and km ⊗kn into sunlet graph of order eight. in this paper, we prove the existence of an l8-decomposition of km�kn. in fact, we establish necessary and sufficient conditions for the existence of an l8-decomposition of km�kn. to prove our results, we state the following: theorem 1.1. [11] let t ≥ 2 be an integer. an l2.2t -decomposition of kn exists if and only if n ≡ 0 (or) 1 (mod 2t+2). theorem 1.2. [20] for any m,n ≥ 4, km,n has an l8decomposition if and only if mn ≡ 0 (mod 8) except (m,n) = (4,2 (mod 4)) & (8,5). 2. decomposition of km�kn into sunlet graph of order 8 necessary conditions: lemma 2.1. if km�kn has an l8-decomposition, then either 1. m,n ≡ 0 (mod 4) 2. m ≡ 0 (mod 8), n ≡ 0 (mod 2) 3. m ≡ 4 (mod 8), n ≡ 2 (mod 4) 4. m ≡ 0 (mod 16) 5. m ≡ 1 (mod 16), n ≡ 1 (mod 16) 6. m ≡ 15 (mod 16), n ≡ 3 (mod 16) 30 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 7. m ≡ 13 (mod 16), n ≡ 5 (mod 16) 8. m ≡ 11 (mod 16), n ≡ 7 (mod 16) 9. m ≡ 9 (mod 16), n ≡ 9 (mod 16) proof. the graph km�kn has mn vertices, each having degree m + n−2 and hence has mn(m+n−2) 2 edges. assume that km�kn admits an l8decomposition. then the number of edges in the graph must be divisible by 8. i.e., 16|mn(m + n−2). hence these conditions are met in each of the above nine cases and only in these cases. figure 1. l8-decomposition of k4�k2. sufficient conditions: we now prove the above necessary conditions are also sufficient by proving the following lemmas: lemma 2.2. if m ≡ 0 (mod 4) and n ≡ 0 (mod 4), then the graph km�kn has an l8-decomposition. proof. let m = 4s and n = 4t for some s,t > 0. we can divide the graph km�kn into st(k4�k4), the l8-decomposition of k4�k4 is shown in fig. 2 and the remaining edges are viewed in the following manner; for each row we have a ’t’ set of four vertices and each set is adjacent to other. therefore we get t(t − 1)/2 complete bipartite graph k4,4. then by theorem 1.2, k4,4 has an l8-decomposititon. similarly we can use the same procedure to column vertices. hence the graph km�kn has the desired decomposition. lemma 2.3. if m ≡ 0 (mod 8) and n ≡ 0 (mod 2), then the graph km�kn has an l8-decomposition. proof. for m = 8 and n = 2, the graph k8�k2 has an l8decomposition by fig. 1 and by theorem 1.2. let m ≡ 0 (mod 8) and n ≡ 0 (or) 4 (mod 8), then the proof follows from lemma 2.2. set m ≡ 0 (mod 8) and consider two cases for n. case (1) n ≡ 2 (mod 8). let m = 8s and n = 8t + 2 for some s,t > 0. the graph km�kn can be viewed as s(t−1) (k8�k8) ⊕ s(k8�k10) and the remaining edges viewed as follows; in each row, we have (t−1) set of eight vertices and one set of ten vertices which are form the complete bipartite graph k8,8 and k8,10. similarly each column can be viewed as ’s’ set of eight vertices and each set is adjacent with each other (i.e. k8,8). the l8-decomposition of k8�k10 is given in appendix 3.1.1 and the l8-decomposition of the graphs k8�k8, k8,8 and k8,10 follows from lemma 2.2 and theorem 1.2. case (2) n ≡ 6 (mod 8). let m = 8s and n = 8t+6 for some s,t > 0. then we can view km�kn as st(k8�k8) ⊕ s(k8�k6) and the remaining edges form the complete bipartite graph k8,8 and k8,6 which are obtained by using the above procedure. the l8-decomposition of k8�k6 is given in appendix 3.1.2 and the l8-decomposition of the remaining graphs follows from lemma 2.2 and theorem 1.2. hence the graph km�kn has the desired decomposition. lemma 2.4. if m ≡ 4 (mod 8) and n ≡ 2 (mod 4), then the graph km�kn has an l8-decomposition. 31 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 figure 2. l8-decomposition of k4�k4. proof. let m = 8s + 4 and n = 4t + 2 for some s,t > 0. now, we divide the proof into three cases. case (1) m = 4 and n = 4t + 2 for some t > 0. if t = 1, then the graph k4�k6 has an l8-decomposition, see appendix 3.2.1. if t = 2, then the graph k4�k10 has an l8-decomposition, see appendix 3.2.2. further, for t > 2, the graph k4�k4t+2 can be viewed as k4�k6 ⊕ k4�k4(t−1) ⊕ 4k6,4(t−1). then by appendix 3.2.1, lemma 2.2 and theorem 1.2, we get the desired decomposition. case (2) m = 12 and n = 4t + 2 for some t > 0. if t = 1, then the graph k12�k6 has an l8-decomposition see appendix 3.2.3. if t = 2, then the graph k12�k10 has an l8-decomposition see appendix 3.2.4. further, for t > 2, the graph k12�k4t+2 can be viewed as k12�k6 ⊕ k12�k4(t−1) ⊕ 12k6,4(t−1). then by appendix 3.2.3, lemma 2.2 and theorem 1.2, we get the desired decomposition. case (3) m > 12 and n = 4t + 2 for some t > 0. the graph km�kn can be viewed as k8(s−1)�k4t+2 ⊕ k12�k4t+2 ⊕ (4t + 2)k12,8(s−1). then by lemma 2.3, theorem 1.2 and the above case (2), we get the desired decomposition. lemma 2.5. if m ≡ 0 (mod 16), then the graph km�kn has an l8-decomposition. proof. if n = 1, the graph km has an l8-decomposition by theorem 1.1. let m ≡ 0 (mod 16) and n ≡ 0,2,4,6 (mod 8), then the proof follows from lemma 2.3. set m = 0 (mod 16) and consider four cases for odd n. case (1) n ≡ 1 (mod 8). let m = 16s and n = 8t + 1 for some s,t > 0. the graph km�kn can be viewed as s(t− 1) (k16�k8) ⊕ s(k16�k9). then the remaining edges are viewed as follows; each row contains (t − 1) set of eight vertices and one set of nine vertices. then each set is adjacent to each other and these forms the complete bipartite graphs k8,8, k8,9. similarly each column can be viewed as ’s’ set of sixteen vertices and each set is adjacent with each other (i.e. we have k16,16). finally thel8-decomposition of k16�k9 is shown in appendix 3.3.1 and the l8-decomposition of the remaining graphs follows from lemma 2.2 and theorem 1.2. case (2) n ≡ 3 (mod 8). if n = 3, then the graph km�k3 can be viewed as copies of k16�k3 and k16,16 which has an l8decomposition, see appendix 3.3.2 and theorem 1.2. let m = 16s and n = 8t + 3 for some s,t > 0. 32 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 apply the same procedure as in case (1) and we write km�kn as s(t−1) (k16�k8) ⊕ s(k16�k11) ⊕ 8s(t−1)(t−2)k8,8 ⊕ s(8t+3)(s−1) 2 k16,16 ⊕ 16s(t−1)k8,11. an l8-decomposition of k16�k11 is shown in appendix 3.3.3 and the l8-decomposition of the remaining graphs follows from lemma 2.2 and theorem 1.2. case (3) n ≡ 5 (mod 8). if n = 5, then the graph km�k5 can be viewed as k16�k5 ⊕ k16,16 which has an l8-decomposition, see appendix 3.3.4 and theorem 1.2. let m = 16s and n = 8t + 5 for some s,t > 0. then we can write km�kn as s(t − 1) (k16�k8) ⊕ s(k16�k13) ⊕ 8s(t − 1)(t − 2)k8,8 ⊕ s(8t+5)(s−1) 2 k16,16 ⊕ 16s(t − 1)k8,13. an l8-decomposition of k16�k13 is presented in appendix 3.3.5 and the l8-decomposition of the remaining graphs follows from lemma 2.2 and theorem 1.2. case (4) n ≡ 7 (mod 8). let m = 16s and n = 8t+7 for some s,t > 0. then we can write km�kn as st(k16�k8) ⊕ s(k16�k7) ⊕ 8st(t − 1)k8,8 ⊕ s(8t+7)(s−1) 2 k16,16 ⊕ 16stk8,7. an l8-decomposition of k16�k7 is presented in appendix 3.3.6 and the l8-decomposition of the remaining graphs follows from lemma 2.2 and theorem 1.2. hence the graph km�kn has the desired decomposition. lemma 2.6. if m ≡ 1 (mod 16) and n ≡ 1 (mod 16), then the graph km�kn has an l8-decomposition. proof. let m = 16s + 1 and n = 16t + 1 for some s,t > 0. then we can write km�kn = nkm ⊕ mkn. i.e, (16t + 1)k16s+1 ⊕ (16s + 1)k16t+1. by theorem 1.1, the graph km�kn has the desired decomposition. lemma 2.7. if m ≡ 15 (mod 16) and n ≡ 3 (mod 16), then the graph km�kn has an l8decomposition. proof. let m = 16s+ 15 and n = 16t+ 3 for some s,t > 0. we can write km�kn as (16t+ 3)k16s+15 ⊕ (16s + 15)k16t+3. now the first 16t columns can be viewed as k16s ⊕ k15 ⊕ sk16,15 and the first 16s rows can be viewed as k16(t−1) ⊕ k19 ⊕ (t − 1)k16,19. then k16s(= sk16 ⊕ s(s−1) 2 k16,16), k16(t−1)(= (t− 1)k16 ⊕ (t−1)(t−2) 2 k16,16), k16,15 and k16,19 have l8-decompositions by theorems 1.1, 1.2. the graph k19 can be viewed as k19\k3 ⊕ k3. the l8-decomposition of k19\k3 follows from appendix 3.4.1. then 16s(k19\k3) has an l8-decomposition. the remaining graph can be viewed as s(k16�k3) ⊕ t(k15�k16) ⊕ k15�k3. hence the desired decomposition follows from appendixes 3.3.2, 3.4.2 and lemma 2.5 lemma 2.8. if m ≡ 13 (mod 16) and n ≡ 5 (mod 16), then the graph km�kn has an l8decomposition. proof. let m = 16s+13 and n = 16t+5 for some s,t > 0. then we can write km�kn as st(k16�k16) ⊕ t(k16�k13) ⊕ s(k16�k5) ⊕ k13�k5 ⊕ t(t−1)(16s+13)+s(s−1)(16t+5) 2 k16,16 ⊕ s(16t + 5)k16,13 ⊕ t(16s+13)k16,5. an l8-decomposition of k13�k5 is given in appendix 3.5.1. and the l8-decomposition of the remaining graphs follows from lemma 2.5 and theorem 1.2. hence the graph km�kn has the desired decomposition. lemma 2.9. if m ≡ 11 (mod 16) and n ≡ 7 (mod 16), then the graph km�kn has an l8decomposition. proof. let m = 16s+11 and n = 16t+7 for some s,t > 0. then we can write km�kn as st(k16�k16) ⊕ t(k16�k11) ⊕ s(k16�k7) ⊕ k11�k7 ⊕ s(s−1)(16t+7)+t(t−1)(16s+11) 2 k16,16 ⊕ (16t + 7)sk16,11 ⊕ (16s+ 11)tk16,7. an l8-decomposition of k11�k7 is given in appendix 3.6.1 and the l8-decomposition of the remaining graphs follows from lemma 2.5 and theorem 1.2. hence the graph km�kn has the desired decomposition. lemma 2.10. if m ≡ 9 (mod 16) and n ≡ 9 (mod 16), then the graph km�kn has an l8decomposition. 33 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 proof. let m = 16s+9 and n = 16t+9 for some s,t > 0. then we can write km�kn as st(k16�k16) ⊕ (s + t) (k16�k9) ⊕ k9�k9 ⊕ s(s−1)(16t+9)+t(t−1)(16s+9) 2 k16,16 ⊕ [s(16t + 9) + t(16s + 9)]k16,9. an l8-decomposition of k11�k7 is given in appendix 3.7.1 and the l8-decomposition of the remaining graphs follows from lemma 2.5 and theorem 1.2. hence the graph km�kn has the desired decomposition. 2.1. main theorem combining the results from lemma 2.1 to lemma 2.10, we get the following main result. theorem 2.11. the graph km�kn admits an l8decomposition if and only if one of the following holds: 1. m,n ≡ 0 (mod 4) 2. m ≡ 0 (mod 8), n ≡ 0 (mod 2) 3. m ≡ 4 (mod 8), n ≡ 2 (mod 4) 4. m ≡ 0 (mod 16) 5. m ≡ 1 (mod 16), n ≡ 1 (mod 16) 6. m ≡ 15 (mod 16), n ≡ 3 (mod 16) 7. m ≡ 13 (mod 16), n ≡ 5 (mod 16) 8. m ≡ 11 (mod 16), n ≡ 7 (mod 16) 9. m ≡ 9 (mod 16), n ≡ 9 (mod 16) acknowledgment: the first author thank the department of science and technology, government of india, new delhi for its financial support through the grant no.dst/inspire 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[25] s. yamamoto, h. ikeda, s. shige-eda, k. ushio, n. hamada, on claw decomposition of complete graphs and complete bipartite graphs, hiroshima math. j. 5(1) (1975) 33-42. 3. appendix 3.1. l8-decomposition required for lemma 2.3 3.1.1. an l8decomposition of k8�k10( x1i x 6 i x 2 i x 7 i x3i x 8 i x 4 i x 9 i ) , ( x3i x 8 i x 4 i x 9 i x10i x 5 i x 7 i x 2 i ) , ( x2i x 3 i x 4 i x 5 i x1i x 7 i x 6 i x 9 i ) for i = 1, 2, ..., 8;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 7 x j 4 x j 8 ) , ( x j 3 x j 7 x j 4 x j 8 x j 5 x j 1 x j 6 x j 2 ) for j = 1, 2, ..., 10;( x j 1 x j 2 x j 3 x j 4 x j 8 x j 7 x j 6 x j 5 ) for j = 1, 3, 5, 7, 8, 9; ( x j 5 x j 6 x j 7 x j 8 x j 4 x j 3 x j 2 x j 1 ) for j = 2, 4, 6, 10;( x1i x 5 i x 6 i x 10 i x8i x k1 i x k2 i x k3 i ) for i = 1, 2, 3, 4, (k1, k2, k3) = (3, 9, 4) & i = 5, 6, 7, 8, (k1, k2, k3) = (7, 3, 2);( x7i x 8 i x 9 i x 10 i x6i x 2 i x 1 i x k i ) for i = 1, 2, 3, 4, k = 5 & i = 5, 6, 7, 8, k = 4; 35 https://doi.org/10.1201/9781420010541 https://doi.org/10.1201/9781420010541 https://doi.org/10.1016/j.disc.2013.09.007 https://doi.org/10.1016/j.disc.2013.09.007 https://doi.org/10.11650/twjm/1500406600 https://doi.org/10.11650/twjm/1500406600 https://doi.org/10.1016/j.disc.2012.03.007 https://doi.org/10.1016/j.disc.2012.03.007 https://mathscinet.ams.org/mathscinet-getitem?mr=3495890 https://mathscinet.ams.org/mathscinet-getitem?mr=3495890 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0012-365x(86)90186-x https://doi.org/10.1016/0012-365x(86)90186-x https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1002/(sici)1097-0118(199612)23:4<361::aid-jgt5>3.0.co;2-p https://doi.org/10.1002/(sici)1097-0118(199612)23:4<361::aid-jgt5>3.0.co;2-p https://doi.org/10.1002/jcd.1027 https://doi.org/10.1002/jcd.1027 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.13069/jacodesmath.867617 https://doi.org/10.13069/jacodesmath.867617 https://doi.org/10.1016/0012-365x(79)90034-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/s0012-365x(85)80023-6 https://doi.org/10.1016/s0012-365x(85)80023-6 https://doi.org/10.32917/hmj/1206135570 https://doi.org/10.32917/hmj/1206135570 https://doi.org/10.32917/hmj/1206136782 https://doi.org/10.32917/hmj/1206136782 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (2, 10), (4, 1), (6, 3), (10, 8);( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 8 ) for (j, k) = (1, 4), (3, 5), (5, 10), (7, 5), (8, 10), (9, 6). 3.1.2. an l8decomposition of k8�k6( x j 1 x j 5 x j 2 x j 6 xk1 x j 7 x k 2 x j 8 ) , ( x j 3 x j 7 x j 4 x j 8 xk3 x j 1 x k 4 x j 2 ) for (j, k) = (1, 3), (5, 2), (6, 4);( x j 1 x j 5 x j 2 x j 6 x j 3 x k 5 x j 4 x k 6 ) ; ( x j 3 x j 7 x j 4 x j 8 x j 5 x k 7 x j 6 x k 8 ) for (j, k) = (2, 5), (3, 1), (4, 6);( x1i x 2 i x 3 i x 4 i x1k x 2 9−i x 3 9−i x 4 9−i ) for (i, k) = (1, 3), (2, 4), (3, 6);( x1i x 5 i x 3 i x 6 i x1i+1 x 5 k x 3 i+1 x 6 k ) for (i, k) = (1, 8), (2, 4), (3, 5);( x1i x 2 i x 3 i x 4 i x1i+1 x 2 i+1 x 3 i+1 x 4 i+1 ) for i = 4, 5, 6, 7; ( x2i x 4 i x 5 i x 6 i x2i+1 x 4 i+1 x 5 i+1 x 6 i+1 ) for i = 1, 2, 3;( x18 x 2 8 x 3 8 x 4 8 x15 x 2 5 x 3 5 x 4 5 ) , ( x14 x 5 4 x 3 4 x 6 4 x11 x 5 6 x 3 1 x 6 6 ) , ( x15 x 5 5 x 3 5 x 6 5 x13 x 5 6 x 3 7 x 6 6 ) , ( x16 x 5 6 x 3 6 x 6 6 x14 x 5 7 x 3 8 x 6 7 ) ,( x17 x 5 7 x 3 7 x 6 7 x12 x 5 8 x 3 1 x 6 8 ) , ( x18 x 5 8 x 3 8 x 6 8 x11 x 5 5 x 3 2 x 6 5 ) , ( x24 x 4 4 x 5 4 x 6 4 x21 x 4 1 x 5 1 x 6 1 ) , ( x25 x 4 5 x 5 5 x 6 5 x27 x 4 7 x 5 4 x 6 4 ) ,( x26 x 4 6 x 5 6 x 6 6 x28 x 4 8 x 5 3 x 6 3 ) , ( x27 x 4 7 x 5 7 x 6 7 x21 x 4 1 x 5 2 x 6 2 ) , ( x28 x 4 8 x 5 8 x 6 8 x22 x 4 2 x 5 1 x 6 1 ) . 3.2. l8-decomposition required for lemma 2.4 3.2.1. an l8decomposition of k4�k6( x2i x 3 i x 6 i x 4 i x2k x 3 k x 1 i x 5 i ) for (i, k) = (1, 3), (2, 4); ( x2i x 3 i x 6 i x 4 i x5i x 1 i x 6 k x 4 k ) for (i, k) = (3, 1), (4, 2);( x1i x 5 i x 5 k x 1 k x2i x 6 i x 6 k x 2 k ) for (i, k) = (1, 3), (2, 4); ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 4), (4, 3);( x j 1 x j 2 x j 3 x j 4 x k1 1 x k1 2 x k2 3 x k2 4 ) for (j, k1, k2) = (2, 5, 6), (3, 1, 5), (5, 3, 4), (6, 2, 1). 3.2.2. an l8decomposition of k4�k10( x1i x 9 i x 2 i x 10 i x1k x 3 i x 2 k x 4 i ) , ( x7i x 9 i x 8 i x 10 i x4i x 5 i x 6 i x 10 k ) , ( x3i x 5 i x 5 k x 3 k x1i x 10 i x 2 k x 1 k ) for (i, k) = (1, 3), (2, 4);( x7i x 9 i x 8 i x 10 i x7k x 9 k x 8 k x 3 i ) for (i, k) = (3, 1), (4, 2); ( x4i x 6 i x 6 k x 4 k x5i x 1 i x 8 k x 5 k ) for (i, k) = (1, 3), (2, 4);( x1i x 2 i x 3 i x 4 i x7i x 6 i x 8 i x 9 i ) for i = 1, 2, 3, 4; ( x5i x 6 i x 7 i x 8 i x2i x 10 i x 3 i x 4 i ) for i = 1, 2;( x5i x 6 i x 7 i x 8 i x9i x 10 i x 3 i x 4 i ) , ( x1i x 9 i x 2 i x 10 i x6i x 3 i x 4 i x 5 i ) for i = 3, 4;( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 5), (2, 8), (3, 6), (5, 7), (6, 9), (7, 2), (8, 1), (9, 10);( x41 x 4 2 x 4 3 x 4 4 x21 x 2 2 x 7 3 x 7 4 ) , ( x101 x 10 2 x 10 3 x 10 4 x31 x 3 2 x 4 3 x 4 4 ) . 3.2.3. an l8decomposition of k12�k6( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x j 6 x j 5 ) , ( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x j 10 x j 9 ) , ( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x j 4 x j 3 ) for (j, k) = (1, 3), (2, 4); 36 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) , ( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 48 ) , ( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x k 11 x k 12 ) for (j, k) = (3, 5), (4, 6), (5, 4), (6, 3); ( x j 5 x j 9 x j 6 x j 10 xk12 x k 2 x j 11 x j 1 ) for j = 1, ..., 6;( x j 1 x j 5 x j 2 x j 6 x j 7 x j 11 x j 8 x j 12 ) , ( x j 3 x j 9 x j 4 x j 10 x j 5 x j 1 x j 6 x j 2 ) , ( x j 7 x j 11 x j 8 x j 12 x j 9 x j 3 x j 10 x j 4 ) for j = 1, ..., 6;( x j 1 x j 11 x j 2 x j 12 x j 8 x j 9 x j 7 x j 10 ) , ( x j 3 x j 7 x j 4 x j 8 x j 1 x j 5 x j 2 x j 6 ) for j = 1, 2;( x j 1 x j 11 x j 2 x j 12 x j 8 x j 4 x j 7 x j 3 ) , ( x j 3 x j 7 x j 4 x j 8 x j 6 x j 10 x j 5 x j 9 ) for j = 5, 6;( x j 1 x j 11 x j 2 x j 12 x j 8 x k 11 x j 7 x k 12 ) , ( x j 3 x j 7 x j 4 x j 8 xk3 x k 7 x k 4 x k 8 ) for (j, k) = (3, 1), (4, 2);( x5i x 6 i x 6 k x 5 k x2i x 1 i x 1 k x 2 k ) for (i, k) = (1, 3), (2, 4), (5, 7), (6, 8), (9, 11), (10, 12);( x1i x 2 i x 3 i x 4 i x5i x 6 i x 3 k x 4 k ) for (i, k) = (1, 3), (2, 4), (3, 6), (4, 5), (5, 7), (6, 8), (7, 10), (8, 9), (9, 11), (10, 12), (11, 4), (12, 3). 3.2.4. an l8decomposition of k12�k10( x1i x 2 i x 3 i x 4 i x1k x 2 k x 3 k x 4 k ) , ( x5i x 6 i x 7 i x 8 i x5k x 6 k x 7 k x 8 k ) , ( x1i x 9 i x 2 i x 10 i x6i x 9 k x 5 i x 10 k ) for (i, k) = (1, 7), (2, 8), (3, 6), (4, 5), (5, 11), (6, 12), (7, 10), (8, 9), (9, 2), (10, 1), (11, 3), (12, 4);( x7i x 9 i x 8 i x 10 i x7k x 9 k x 8 k x 10 k ) for (i, k) = (1, 8), (2, 7), (3, 12), (4, 11), (5, 12), (6, 11); c ( x7i x 9 i x 8 i x 10 i x3i x 5 i x 6 i x 4 i ) for i = 7, ..., 12; ( x3i x 5 i x 4 i x 6 i x7i x 10 i x 2 i x 8 i ) for i = 1, ..., 6;( x3i x 5 i x 4 i x 6 i x3k x 5 k x 4 k x 6 k ) for (i, k) = (7, 2), (8, 1), (9, 11), (10, 12), (11, 6), (12, 5);( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 3), (3, 10), (4, 7), (5, 7), (6, 10), (7, 1)(8, 2), (9, 3), (10, 4);( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 8 ) for (j, k) = (1, 3), (3, 7), (4, 7), (5, 7), (6, 8), (7, 1)(8, 2), (9, 3);( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x k 11 x k 12 ) for (j, k) = (1, 3), (2, 4), (3, 7), (4, 7), (5, 7), (6, 8), (7, 1)(8, 2), (9, 3), (10, 5);( x j 1 x j 5 x j 2 x j 6 xi3 x k 5 x i 4 x k 6 ) for (j, k) = (3, 8), (4, 8), (5, 9), (6, 9);( x j 1 x j 5 x j 2 x j 6 xk1 x k 5 x k 2 x k 6 ) for (j, k) = (1, 5), (2, 6), (7, 2), (8, 1), (9, 4), (10, 9);( x j 3 x j 9 x j 4 x j 10 xk3 x k 9 x k 4 x k 10 ) for (j, k) = (2, 6), (3, 8), (4, 8), (6, 9), (7, 2), (8, 1), (9, 4), (10, 9);( x j 7 x j 11 x j 8 x j 12 xk7 x k 11 x k 8 x k 12 ) for (j, k) = (2, 6), (3, 8), (4, 8), (5, 1), (6, 9), (7, 2), (8, 1), (9, 4), (10, 9);( x j 1 x j 11 x j 2 x j 12 x j 3 x j 9 x j 4 x j 10 ) for i = 2, 7, 8, 9, 10;( x j 1 x j 11 x j 2 x j 12 xk1 x j 4 x k 2 x j 3 ) for (j, k) = (3, 9), (4, 8), (5, 9), (6, 9);( x j 5 x j 9 x j 6 x j 10 x j 7 x j 1 x j 8 x j 2 ) , ( x j 3 x j 7 x j 4 x j 8 x j 5 x j 9 x j 6 x j 10 ) , for j = 1, ..., 10;( x21 x 2 2 x 2 3 x 2 4 x28 x 2 7 x 2 12 x 2 11 ) , ( x25 x 2 6 x 2 7 x 2 8 x212 x 2 11 x 4 7 x 4 8 ) , ( x105 x 10 6 x 10 7 x 10 8 x45 x 4 6 x 4 7 x 4 8 ) , ( x13 x 1 9 x 1 4 x 1 10 x53 x 1 11 x 5 4 x 1 12 ) ,( x53 x 5 9 x 5 4 x 5 10 x93 x 1 9 x 9 4 x 1 10 ) , ( x17 x 1 11 x 1 8 x 1 12 x12 x 1 4 x 1 1 x 1 3 ) , ( x11 x 1 11 x 1 2 x 1 12 x13 x 1 6 x 1 4 x 1 5 ) . 37 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 3.3. l8-decomposition required for lemma 2.5 3.3.1. an l8decomposition of k16�k9( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 1, 4, 5, 7, 8;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, 4, 5, 7, 8;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 1, 3, 4, 5, 6, 7, 8;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 1, 3, 4, 5, 6, 7, 8, 9;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) , ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 3, 4, 5, 7, 8;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 3, 4, 5, 6, 7, 8; ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 3, 4, 5, 7, 8;( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 2, 3, 4, 5, 6, 7, 8; ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 16 x j 12 x j 14 ) for j = 6, 9;( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, 2, ..., 8, 9; ( x3i x 7 i x 4 i x 9 i x1i x 5 i x 6 i x 8 i ) for i = 1, 2, ..., 15, 16;( x9i x 9 i+4 x 9 i+1 x 9 i+5 x9i+2 x 5 i+4 x 9 i+3 x 5 i+5 ) for i = 5, 7; ( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 7 i ) for i = 1, 2, ..., 7, 8;( x j i x j 12+i x j i+1 x j 13+i x j 11 x 5 12+i x j 10 x 5 13+i ) for j = 2, 9 & i = 1, 3;( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j k ) for j = 1, 3, 4, 5, 7, 8, k = 14 & j = 2, 6, 9, k = 16;( x5i x 6 i x 7 i x 8 i x4i x 1 i x k i x 3 i ) for i = 1, 2, ..., 7, 8, k = 2, & i = 9, ..., 16, k = 9;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x k 11 x j 16 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x k 12 x j 8 ) for (j, k) = (6, 3), (9, 6);( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x k 10 ) , ( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x k 13 x j 15 ) for (j, k) = (2, 7), (6, 3), (9, 6);( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x k 15 x k 16 ) for (j, k) = (2, 7), (3, 5), (6, 3), (9, 6);( x3i x 3 i+1 x 3 16−i x 3 17−i x3k1 x 3 k2 x516−i x 5 17−i ) ; for (i, k1, k2) = (3, 12, 11)(5, 15, 7), (7, 13, 15);( x j 9 x j 13 x j 10 x j 14 xk9 x j 16 x j 12 x k 14 ) for (j, k) = (2, 7), (6, 3), (9, 6);( x1i x 2 i x 3 i x 4 i x7i x 2 k x 6 i x 8 i ) for (i, k) = (1, 10), (2, 11), (3, 9), (4, 12), (5, 13), (6, 15), (7, 14), (8, 5);( x1i x 2 i x 3 i x 4 i x7i x 6 i x 3 k x 8 i ) for (i, k) = (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (14, 15), (15, 6), (16, 5);( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 9 k ) for (i, k) = (9, 12), (10, 11), (11, 15), (12, 6), (13, 6), (14, 5), (15, 1), (16, 2);( x2i x 5 i x 9 i x 6 i x2k1 x 3 i x 9 k2 x6k2 ) for (i, k1, k2) = (1, 15, 10), (2, 16, 11), (3, 10, 9), (4, 1, 12), (5, 14, 13), (6, 13, 15), (7, 6, 14), (8, 14, 5);( x27 x 2 11 x 2 8 x 2 12 x29 x 2 15 x 2 10 x 5 12 ) , ( x21 x 2 7 x 2 2 x 2 8 x29 x 2 16 x 2 12 x 2 13 ) , ( x29 x 2 15 x 2 10 x 2 16 x59 x 2 12 x 5 10 x 2 11 ) , ( x211 x 2 13 x 2 12 x 2 14 x511 x 2 15 x 7 12 x 2 16 ) ,( x21 x 2 5 x 2 2 x 2 6 x23 x 2 16 x 2 9 x 2 12 ) , ( x25 x 2 6 x 2 11 x 2 12 x215 x 2 14 x 7 11 x 2 16 ) . 38 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 3.3.2. an l8decomposition of k16�k3( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 2, 3;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) for j = 1, 3;( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x 1 10 ) for j = 2, 3;( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x k2 k1 x j 1 ) for (j, k1, k2) = (1, 9, 3), (2, 9, 3), (3, 11, 3);( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x k2 k1 x j 6 ) for (j, k1, k2) = (1, 10, 1), (2, 10, 2), (3, 8, 1);( x j 3 x j 5 x j 4 x j 6 x j 2 x k2 k1 x j 9 x j 14 ) for (j, k1, k2) = (1, 11, 1), (2, 11, 2), (3, 5, 2);( x j 9 x j 15 x j 10 x j 16 x j 1 x k2 k1 x j 3 x j 14 ) for (j, k1, k2) = (1, 12, 1), (2, 15, 3), (3, 12, 3);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x k2 k1 x j 16 ) (j, k1, k2) = (1, 4, 3), (2, 4, 1), (3, 14, 3);( x j 9 x j 13 x j 10 x j 14 x j 3 x k2 k1 x j 12 x j 7 ) for (j, k1, k2) = (1, 16, 1), (2, 13, 1), (3, 13, 1);( x13 x 1 4 x 1 13 x 1 14 x33 x 1 11 x 1 5 x 3 14 ) , ( x23 x 2 4 x 2 13 x 2 14 x33 x 3 4 x 2 5 x 3 14 ) , ( x33 x 3 4 x 3 13 x 3 14 x312 x 3 11 x 3 5 x 3 15 ) , ( x25 x 2 6 x 2 11 x 2 12 x15 x 1 6 x 2 2 x 3 12 ) ,( x21 x 2 13 x 2 2 x 2 14 x11 x 2 6 x 1 2 x 2 5 ) , ( x15 x 1 9 x 1 6 x 1 10 x35 x 1 12 x 1 8 x 1 11 ) , ( x11 x 1 7 x 1 2 x 1 8 x14 x 3 7 x 3 2 x 2 8 ) , ( x21 x 2 7 x 2 2 x 2 8 x24 x 3 7 x 3 2 x 3 8 ) ,( x31 x 3 7 x 3 2 x 3 8 x34 x 3 16 x 3 9 x 3 13 ) , ( x11 x 1 5 x 1 2 x 1 6 x31 x 1 8 x 1 12 x 1 16 ) , ( x21 x 2 5 x 2 2 x 2 6 x31 x 2 8 x 2 12 x 3 6 ) , ( x31 x 3 5 x 3 2 x 3 6 x33 x 3 8 x 3 12 x 1 6 ) ,( x13 x 1 12 x 2 12 x 2 3 x11 x 3 12 x 2 15 x 2 1 ) , ( x114 x 1 15 x 2 15 x 2 14 x14 x 3 15 x 2 5 x 2 4 ) , ( x19 x 1 11 x 2 11 x 2 9 x12 x 3 11 x 2 1 x 2 2 ) , ( x210 x 2 11 x 3 11 x 3 10 x22 x 2 4 x 3 5 x 3 8 ) ,( x17 x 1 16 x 2 16 x 2 7 x15 x 3 16 x 2 12 x 2 6 ) , ( x213 x 2 16 x 3 16 x 3 13 x28 x 2 6 x 3 6 x 3 8 ) . 3.3.3. an l8decomposition of k16�k11( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, 2, 4, 5, 7, 8, 10;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 2, 5, 6, 7, 8, 9;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, 2, 4, 5, ..., 10, 11;( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) , ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 2, 3, 5, ..., 11;( x3i x 7 i x 4 i x 10 i x11i x 6 i x 8 i x 9 i ) , ( x2i x 5 i x 7 i x 11 i x3i x 9 i x 8 i x 10 i ) for i = 1, 2, 5, 6, 7, 8, 10, 11, 12, 14;( x2i x 5 i x 7 i x 11 i x1i x 9 i x 8 i x 10 i ) , ( x1i x 6 i x 2 i x 7 i x3i x 5 i x 4 i x 10 i ) for i = 15, 16;( x3i x 4 i x 9 i x 8 i x5i x 6 i x 11 i x 10 i ) , ( x3i x 7 i x 4 i x 10 i x11i x 6 i x 8 i x 9 k ) for i = 15, 16;( x j 1 x j 2 x j 15 x j 16 xk1 x k 2 x j 6 x j 5 ) , ( x j 3 x j 4 x j 13 x j 14 xk3 x k 4 x k 13 x k 14 ) for (j, k) = (6, 3), (9, 3), (11, 8);( x j 5 x j 6 x j 11 x j 12 xk5 x k 6 x k 11 x k 12 ) , ( x j 7 x j 8 x j 9 x j 10 xk7 x k 8 x k 9 x k 10 ) for (j, k) = (6, 3), (9, 3), (11, 8);( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) for j = 2, 5, 7, 8, 10; ( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) for j = 1, 2, 4, 5, ..., 10, 11; 39 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 2, 3, ..., 10, 11; ( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 6, 7, 8, 9;( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 1, 2, 3, 4, 5, 7, 8, 10;( x1i x 8 i x 2 i x 9 i x4i x 6 i x 10 i x 7 i ) for i = 1, 2, 5, 6, 12, 14, 15, 16;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 2, 4, 5, ..., 11; ( x5i x 10 i x 6 i x 11 i x5k x 1 i x 9 i x 8 i ) for (i.k) = (15, 12), (16, 14);( x j 9 x j 15 x j 10 x j 16 xk9 x j 12 x j 3 x j 14 ) for (j, k) = (1, 10), (4, 1), (10, 11), (11, 3);( x j 3 x j 15 x j 4 x j 16 xk3 x j 1 x k 4 x j 2 ) for (j, k) = (10, 11), (11, 3); ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 2, 3, 5, ..., 11;( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x k 15 x k 16 ) for (j, k) = (1, 11), (3, 9), (4, 5);( x j 1 x j 13 x j 2 x j 14 x j 11 x k 13 x j 10 x j 5 ) for (j, k) = (4, 1), (10, 11), (11, 3);( x j 3 x j 15 x j 4 x j 16 x k1 3 x k2 15 x k1 4 x k2 16 ) for (j, k1, k2) = (1, 10, 5), (4, 1, 11);( x1i x 6 i x 2 i x 7 i x3i x 6 k x 4 i x 10 i ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (6, 7), (7, 13), (8, 15), (9, 11), (10, 1), (11, 2), (13, 5);( x1i x 6 i x 2 i x 7 i x3i x 6 k1 x2k2 x 10 i ) for (i, k1, k2) = (12, 16, 15), (14, 15, 16);( x3i x 4 i x 9 i x 8 i x5i x 6 i x 9 k x 10 i ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (6, 7), (7, 13), (8, 15), (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (14, 15);( x5i x 10 i x 6 i x 11 i x8i x 1 i x 9 i x 11 k ) for (i, k) = (1, 12), (2, 4), (5, 15), (6, 7), (7, 13), (8, 15), (10, 1), (11, 2), (12, 16), (14, 15);( x5i x 10 i x 6 i x 11 i x8i x 10 k1 x9i x 11 k2 ) for (i, k1, k2) = (3, 11, 12), (4, 10, 11), (9, 1, 11), (13, 6, 5);( x1i x 8 i x 2 i x 9 i x1k x 6 i x 10 i x 7 i ) for (i, k) = (3, 11), (4, 10), (7, 5), (8, 6), (9, 1), (10, 2), (11, 1), (13, 6);( x3i x 7 i x 4 i x 10 i x3k x 6 i x 8 i x 9 i ) for (i, k) = (3, 10), (4, 1), (9, 2), (13, 7);( x2i x 5 i x 7 i x 11 i x3i x 9 i x 8 i x 11 k ) for (i, k) = (3, 11), (4, 10), (9, 1), (13, 6);( x1i x 5 i x 4 i x 11 i x2i x 6 i x 4 k x 9 i ) for (i, k) = (1, 15), (2, 16), (3, 11), (4, 10), (5, 16), (6, 15), (7, 15), (8, 16), (9, 1), (10, 12), (11, 16), (12, 4), (13, 6), (14, 8);( x35 x 3 6 x 3 11 x 3 12 x315 x 3 14 x 3 2 x 3 16 ) , ( x37 x 3 8 x 3 9 x 3 10 x36 x 3 15 x 3 11 x 3 1 ) , ( x11 x 1 13 x 1 2 x 1 14 x115 x 10 13 x 1 16 x 1 5 ) , ( x31 x 3 13 x 3 2 x 3 14 x311 x 3 6 x 3 16 x 3 5 ) ,( x33 x 3 15 x 3 4 x 3 16 x311 x 6 15 x 3 10 x 6 16 ) , ( x37 x 3 11 x 3 8 x 3 12 x39 x 3 15 x 3 5 x 3 6 ) , ( x15 x 1 9 x 1 6 x 1 10 x116 x 1 12 x 1 15 x 1 11 ) , ( x33 x 3 5 x 3 4 x 3 6 x32 x 3 11 x 3 9 x 3 16 ) ,( x31 x 3 7 x 3 2 x 3 8 x315 x 3 16 x 3 10 x 3 13 ) , ( x29 x 2 15 x 2 10 x 2 16 x21 x 3 15 x 2 3 x 3 16 ) , ( x39 x 3 15 x 3 10 x 3 16 x31 x 3 12 x 3 8 x 3 14 ) , ( x411 x 4 13 x 4 12 x 4 14 x111 x 4 15 x 2 12 x 2 14 ) ,( x43 x 4 7 x 4 4 x 4 8 x413 x 1 7 x 4 14 x 1 8 ) , ( x31 x 3 5 x 3 2 x 3 6 x33 x 3 16 x 3 12 x 3 15 ) , ( x49 x 4 13 x 4 10 x 4 14 x43 x 4 16 x 1 10 x 4 7 ) , ( x59 x 5 15 x 5 10 x 5 16 x51 x 8 15 x 5 3 x 8 16 ) . 3.3.4. an l8decomposition of k16�k5( x j 1 x j 2 x j 15 x j 16 x51 x 5 2 x 5 15 x 5 16 ) , ( x j 3 x j 4 x j 13 x j 14 x53 x 5 4 x 5 13 x 5 14 ) for j = 1, 2, 3, 4;( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) , ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 2, 5; 40 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x j 5 x j 6 x j 11 x j 12 x55 x 5 6 x 5 11 x 5 12 ) , ( x j 7 x j 8 x j 9 x j 10 x57 x 5 8 x 5 9 x 5 10 ) for j = 1, 2, 3, 4;( x j 12 x j 15 x k 15 x k 12 x j 2 x j 8 x k 14 x k 9 ) , ( x j 11 x j 16 x k 16 x k 11 x j 4 x j 6 x k 13 x k 10 ) for (j, k) = (1, 3), (2, 4);( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) , ( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) for j = 1, 2, 3, 4, 5;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 3, 4, 5; ( x j 3 x j 7 x j 4 x j 8 x j 13 x k 7 x j 14 x j 16 ) for (j, k) = (3, 1), (4, 2);( x j 1 x j 13 x j 2 x j 14 x k2 k1 x j 6 x j 10 x j 5 ) for j = k2 = 3, 4, 5, k1 = 11&(j, k1, k2) = (1, 1, 3), (2, 1, 4);( x j 3 x j 15 x j 4 x j 16 x k2 k1 x j 1 x j 10 x j 2 ) for j = k2 = 1, 2, 5, k1 = 11&(j, k1, k2) = (3, 3, 1), (4, 3, 2);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x k2 k1 x j 14 ) for j = k2 = 3, 4, 5, k1 = 9&(j, k1, k2) = (1, 4, 3), (2, 4, 4);( x j 9 x j 15 x j 10 x j 16 x j 1 x j k x j 3 x j 14 ) for (j, k) = (1, 5), (2, 5), (3, 7), (4, 7);( x j 11 x j 13 x j 12 x j 14 x j k x j 15 x j 4 x j 8 ) for (j, k) = (1, 1), (2, 1), (3, 3), (4, 3);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x k 8 ) for (j, k) = (1, 3), (2, 4);( x1i x 2 i x 3 i x 4 i x1k x 2 k x 3 k x 4 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (6, 7)(7, 13), (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (15, 6), (16, 5);( x1i x 2 i x 3 i x 4 i x1k1 x 2 k1 x3k2 x 4 k2 ) for (i, k1, k2) = (4, 9, 11), (5, 8, 15), (8, 16, 15), (14, 15, 7);( x51 x 5 2 x 5 15 x 5 16 x512 x 5 4 x 5 6 x 5 5 ) , ( x53 x 5 4 x 5 13 x 5 14 x512 x 5 11 x 5 5 x 5 15 ) , ( x55 x 5 6 x 5 11 x 5 12 x515 x 5 7 x 5 2 x 5 16 ) , ( x57 x 5 8 x 5 9 x 5 10 x513 x 5 15 x 5 11 x 5 1 ) ,( x35 x 3 9 x 3 6 x 3 10 x37 x 1 9 x 3 8 x 1 10 ) , ( x45 x 4 9 x 4 6 x 4 10 x47 x 2 9 x 4 8 x 2 10 ) , ( x59 x 5 15 x 5 10 x 5 16 x51 x 5 12 x 5 3 x 5 14 ) , ( x511 x 5 13 x 5 12 x 5 14 x516 x 5 15 x 5 4 x 5 8 ) ,( x53 x 5 7 x 5 4 x 5 8 x513 x 5 15 x 5 14 x 5 16 ) , ( x11 x 1 5 x 1 2 x 1 6 x13 x 3 5 x 3 2 x 3 6 ) , ( x21 x 2 5 x 2 2 x 2 6 x23 x 4 5 x 4 2 x 4 6 ) , ( x39 x 3 13 x 3 10 x 3 14 x33 x 1 13 x 3 12 x 1 14 ) ,( x49 x 4 13 x 4 10 x 4 14 x43 x 2 13 x 4 12 x 2 14 ) . 3.3.5. an l8decomposition of k16�k13( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) , for j = 2, . . . , 8, 10, . . . , 13;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, . . . , 13;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, . . . , 13;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 2, . . . , 6, 8, 10, 12, 13;( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 2, . . . 6, 8, 10, 12, 13;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) , ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, . . . , 6, 8, 9, 10, 12, 13; , ( x1i x 7 i x 2 i x 8 i x5i x 4 i x 6 i x 3 i ) , ( x4i x 5 i x 7 i x 6 i x2i x 3 i x 10 i x 1 i ) for i = 1, 2, ..., 16;( x1i x 2 i x 10 i x 9 i x4i x 12 i x 6 i x 5 i ) , ( x5i x 6 i x 13 i x 10 i x2i x 3 i x 11 i x 12 i ) , for i = 1, 2, ..., 16;( x j 7 x j 11 x j 8 x j 12 x j 14 x j 15 x j 10 x j 6 ) , ( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 16 x j 5 ) , for j = 7, 11; 41 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x3i x 4 i x 8 i x 7 i x13i x 11 i x 5 i x 9 i ) , ( x2i x 3 i x 12 i x 11 i x13i x 1 i x 8 i x 10 i ) , for i = 1, 2, 5, 6, 7, 8, 11, . . . , 16;( x j 3 x j 4 x j 13 x j 14 x k1 3 x k1 4 x k2 13 x j 15 ) , ( x j 9 x j 13 x j 10 x j 14 x k1 9 x j 16 x k1 10 x k2 14 ) for (j, k1, k2) = (1, 10, 12), (9, 7, 13);( x j 3 x j 15 x j 4 x j 16 x j 11 x j 13 x j 10 x k 16 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x k 13 x j 15 x j 8 ) for (j, k) = (7, 13), (11, 8);( x3i x 4 i x 8 i x 7 i x13i x 11 i x 5 i x 7 k ) , ( x2i x 3 i x 12 i x 11 i x13i x 1 i x 8 i x 11 k ) for (i, k) = (3, 1), (4, 12), (9, 7), (10, 2);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 13 x j 9 x j 14 ) for j = 1, 9; ( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x k 16 ) for (j, k) = (1, 12), (9, 11);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, . . . , 13; ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x k 14 ) for (j, k) = (1, 11), (9, 11);( x j 1 x j 5 x j 2 x j 6 x j 15 x j 8 x j 12 x j 16 ) for j = 7, 11; ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 2, . . . , 6, 8, 12, 13;( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 2, . . . , 8, 11, 12, 13;( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x k 14 ) for (j, k) = (7, 13), (11, 8);( x j 9 x j 15 x j 10 x j 16 x j 1 x k 15 x j 3 x j 14 ) for (j, k) = (1, 12), (7, 13), (9, 13), (11, 8);( x3i x 9 i x 4 i x 10 i x11i x 2 i x k i x 1 i ) for i = 1, 2, 5, 6, 7, 8, 11, 12, k = 12 & i = 13, 14, 15, 16 k = 13;( x3i x 9 i x 4 i x 10 i x11i x 2 i x 12 i x 10 k ) for (i, k) = (3, 12), (4, 11), (9, 3), (10, 12);( x5i x 11 i x 6 i x 12 i x13i x 7 i x 8 i x k i ) for i = 1, 2, ..., 12, k = 1 & i = 13, 14, 15, 16 k = 4;( x8i x 9 i x 12 i x 13 i x10i x 6 i x 7 i x k i ) for i = 1, 2, ..., 12, k = 4 & i = 13, 14, 15, 16 k = 1;( x1i x 11 i x 9 i x 13 i x1k x 8 i x 9 k x 7 i ) for (i, k) = (1, 15), (2, 16), (3, 12), (4, 11), (5, 16), (6, 13), (7, 14), (8, 14), (9, 3), (10, 12), (11, 5), (12, 15);( x103 x 10 4 x 10 13 x 10 14 x113 x 11 4 x 10 5 x 10 15 ) , ( x11 x 1 13 x 1 2 x 1 14 x111 x 11 13 x 1 10 x 1 5 ) , ( x91 x 9 13 x 9 2 x 9 14 x911 x 11 13 x 9 10 x 9 5 ) , ( x13 x 1 15 x 1 4 x 1 16 x111 x 11 15 x 1 10 x 11 16 ) ,( x93 x 9 15 x 9 4 x 9 16 x911 x 11 15 x 9 10 x 13 16 ) , ( x109 x 10 13 x 10 10 x 10 14 x119 x 10 16 x 11 10 x 10 7 ) . 3.3.6. an l8decomposition of k16�k7( x j 1 x j 2 x j 15 x j 16 xk1 x k 2 x k 15 x k 16 ) , ( x j 3 x j 4 x j 13 x j 14 xk3 x k 4 x k 13 x k 14 ) for (j, k) = (1, 4), (2, 3), (3, 6), (4, 3), (5, 6), (6, 2), (7, 6);( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) for j = 1, 3, 6, 7;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 11 x j 13 ) for j = 4, 5; ( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 3, 4, 6;( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 1, 6; ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 4, 6, 7;( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 6; ( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, 2, 4, 5, 6;( x j 3 x j 7 x j 4 x j 8 x j 10 x j 15 x j 14 x j 16 ) for j = 3, 7; ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, 3, 4, 5, 6, 7;( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 1, 6, 7; ( x j 1 x j 13 x j 2 x j 14 x j 11 x k 13 x j 10 x j 5 ) for (j, k) = (2, 5), (4, 7), (5, 4);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) for j = 1, ..., 7; ( x j 3 x j 15 x j 4 x j 16 x j 11 x k 15 x j 10 x k 16 ) (j, k) = (2, 5), (4, 7), (5, 4); 42 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 3, 4, 5, 6, 7;( x j 7 x j 8 x j 9 x j 10 xk7 x k 8 x k 9 x k 10 ) for (j, k) = (1, 4), (5, 6), (6, 2), (7, 6);( x j 7 x j 8 x j 9 x j 10 x k1 7 x k2 8 x k2 9 x k2 10 ) for (j, k1, k2) = (2, 4, 3), (3, 4, 6), (4, 6, 3);( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x k 10 x j 14 ) for (j, k) = (3, 7), (4, 7), (5, 2), (7, 2);( x j 5 x j 6 x j 11 x j 12 xk5 x k 6 x k 11 x k 12 ) for (j, k) = (1, 4), (4, 3), (5, 6), (7, 6);( x1i x 3 i x 5 i x 7 i x1k x 3 k x 5 k x 7 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (8, 15), (9, 11), (10, 1), (12, 16), (13, 5);( x1i x 2 i x 4 i x 6 i x5i x 2 k x 4 k x 6 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (8, 15), (12, 16), (14, 15);( x2i x 5 i x 4 i x 7 i x2k x 5 k x 4 k x 3 i ) for (i, k) = (1, 15), (2, 16), (4, 12), (5, 16), (6, 13), (8, 14), (9, 2);( x25 x 2 6 x 2 11 x 2 12 x27 x 3 6 x 3 11 x 3 12 ) , ( x35 x 3 6 x 3 11 x 3 12 x65 x 6 6 x 6 11 x 7 12 ) , ( x65 x 6 6 x 6 11 x 6 12 x25 x 2 6 x 2 11 x 3 12 ) , ( x33 x 3 15 x 3 4 x 3 16 x311 x 3 1 x 3 10 x 7 16 ) ,( x27 x 2 11 x 2 8 x 2 12 x216 x 7 11 x 2 6 x 7 12 ) , ( x57 x 5 11 x 5 8 x 5 12 x59 x 5 15 x 5 10 x 2 12 ) , ( x77 x 7 11 x 7 8 x 7 12 x79 x 3 11 x 7 10 x 7 6 ) , ( x25 x 2 9 x 2 6 x 2 10 x213 x 2 11 x 2 12 x 2 1 ) ,( x35 x 3 9 x 3 6 x 3 10 x316 x 3 12 x 3 8 x 3 11 ) , ( x55 x 5 9 x 5 6 x 5 10 x57 x 5 12 x 5 8 x 4 10 ) , ( x21 x 2 7 x 2 2 x 2 8 x24 x 2 6 x 2 11 x 2 10 ) , ( x29 x 2 15 x 2 10 x 2 16 x27 x 2 12 x 2 3 x 2 14 ) ,( x211 x 2 13 x 2 12 x 2 14 x216 x 2 15 x 6 12 x 5 14 ) , ( x311 x 3 13 x 3 12 x 3 14 x316 x 3 7 x 3 4 x 7 14 ) , ( x411 x 4 13 x 4 12 x 4 14 x511 x 4 15 x 7 12 x 7 14 ) , ( x511 x 5 13 x 5 12 x 5 14 x510 x 5 15 x 4 12 x 4 14 ) ,( x711 x 7 13 x 7 12 x 7 14 x716 x 2 13 x 7 4 x 7 8 ) , ( x21 x 2 5 x 2 2 x 2 6 x29 x 2 8 x 2 12 x 2 16 ) , ( x29 x 2 13 x 2 10 x 2 14 x23 x 2 16 x 2 12 x 7 14 ) , ( x15 x 3 5 x 5 5 x 7 5 x115 x 2 5 x 5 15 x 7 15 ) ,( x16 x 3 6 x 5 6 x 7 6 x17 x 3 15 x 5 12 x 7 7 ) , ( x17 x 3 7 x 5 7 x 7 7 x113 x 3 5 x 5 13 x 7 13 ) , ( x111 x 3 11 x 5 11 x 7 11 x12 x 3 2 x 2 11 x 7 2 ) , ( x114 x 3 14 x 5 14 x 7 14 x115 x 3 8 x 5 15 x 7 15 ) ,( x115 x 3 15 x 5 15 x 7 15 x16 x 3 5 x 5 6 x 7 6 ) , ( x116 x 3 16 x 5 16 x 7 16 x15 x 3 2 x 5 11 x 7 5 ) , ( x16 x 2 6 x 4 6 x 6 6 x56 x 2 15 x 4 7 x 6 7 ) , ( x19 x 2 9 x 4 9 x 6 9 x59 x 2 12 x 4 11 x 6 11 ) ,( x110 x 2 10 x 4 10 x 6 10 x510 x 2 11 x 4 1 x 6 1 ) , ( x111 x 2 11 x 4 11 x 6 11 x511 x 2 15 x 7 11 x 6 2 ) , ( x113 x 2 13 x 4 13 x 6 13 x513 x 2 8 x 4 5 x 6 5 ) , ( x115 x 2 15 x 4 15 x 6 15 x515 x 7 15 x 4 6 x 6 6 ) ,( x116 x 2 16 x 4 16 x 6 16 x516 x 7 16 x 4 11 x 6 5 ) , ( x23 x 5 3 x 4 3 x 7 3 x21 x 5 10 x 4 10 x 3 3 ) , ( x27 x 5 7 x 4 7 x 7 7 x214 x 5 6 x 4 13 x 3 7 ) , ( x17 x 2 7 x 3 7 x 6 7 x57 x 2 13 x 3 6 x 6 13 ) ,( x313 x 7 13 x 7 15 x 3 15 x33 x 7 3 x 7 11 x 3 14 ) . 3.4. l8-decomposition required for lemma 2.7 3.4.1. an l8decomposition of k19\k3 let v (k19) = {x1,x2, ...,x19} and the k3 be (x13x15x17).( x1 x2 x18 x19 x9 x12 x5 x6 ) , ( x3 x4 x16 x17 x9 x1 x18 x19 ) , ( x5 x6 x14 x15 x3 x4 x16 x1 ) , ( x7 x8 x12 x13 x9 x2 x16 x14 ) ,( x9 x10 x11 x12 x4 x16 x7 x17 ) , ( x1 x16 x2 x17 x12 x3 x9 x4 ) , ( x3 x18 x4 x19 x10 x1 x2 x12 ) , ( x5 x12 x6 x13 x11 x18 x3 x16 ) ,( x7 x14 x8 x15 x2 x17 x1 x3 ) , ( x1 x10 x2 x11 x14 x13 x19 x3 ) , ( x3 x12 x4 x13 x1 x7 x11 x2 ) , ( x1 x5 x2 x6 x13 x14 x15 x11 ) ,( x7 x16 x8 x17 x6 x19 x5 x11 ) , ( x9 x18 x10 x19 x11 x6 x4 x5 ) , ( x11 x14 x12 x15 x13 x2 x10 x4 ) , ( x3 x7 x4 x8 x2 x1 x14 x9 ) ,( x5 x16 x6 x17 x7 x9 x8 x18 ) , ( x9 x14 x10 x15 x13 x3 x17 x16 ) , ( x11 x18 x13 x19 x16 x14 x8 x15 ) , ( x5 x9 x6 x10 x4 x17 x15 x8 ) , 43 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x7 x18 x8 x19 x10 x15 x11 x14 ) . 3.4.2. an l8decomposition of k15�k3( x j 7 x j 8 x j 9 x j 10 x j 15 x j 14 x j 1 x j 5 ) for j = 2, 3; ( x j 5 x j 13 x j 6 x j 15 x j 14 x j 4 x j 7 x j 2 ) for j = 1, 2, 3;( x j 3 x j 11 x j 9 x j 12 x j 8 x j 6 x j 15 x j 10 ) for j = 1, 3; ( x j 1 x j 12 x j 2 x j 13 x j 4 x j k x j 7 x j 14 ) for (j, k) = (1, 15), (2, 5), (3, 15);( x j 10 x j 13 x j 11 x j 14 x j 15 x j 8 x j k x j 1 ) for (j, k) = (1, 4), (2, 12), (3, 4);( x j 3 x j 7 x j 4 x j 9 x j 10 x j 13 x j k x j 14 ) for (j, k) = (1, 5), (2, 8), (3, 8);( x11 x 1 2 x 1 14 x 1 15 x17 x 1 8 x 1 12 x 3 15 ) , ( x21 x 2 2 x 2 14 x 2 15 x31 x 2 8 x 3 14 x 3 15 ) , ( x31 x 3 2 x 3 14 x 3 15 x37 x 3 8 x 1 14 x 3 13 ) , ( x13 x 1 4 x 1 12 x 1 13 x33 x 3 4 x 1 5 x 1 9 ) ,( x23 x 2 4 x 2 12 x 2 13 x33 x 3 4 x 2 14 x 3 13 ) , ( x33 x 3 4 x 3 12 x 3 13 x31 x 3 2 x 3 5 x 3 9 ) , ( x15 x 1 6 x 1 10 x 1 11 x13 x 2 6 x 1 2 x 1 15 ) , ( x25 x 2 6 x 2 10 x 2 11 x23 x 2 4 x 3 10 x 3 11 ) ,( x35 x 3 6 x 3 10 x 3 11 x33 x 3 4 x 3 2 x 3 15 ) , ( x17 x 1 8 x 1 9 x 1 10 x37 x 1 14 x 1 1 x 1 5 ) , ( x13 x 1 14 x 1 4 x 1 15 x12 x 2 14 x 1 10 x 1 8 ) , ( x23 x 2 14 x 2 4 x 2 15 x22 x 2 7 x 2 10 x 2 12 ) ,( x33 x 3 14 x 3 4 x 3 15 x32 x 3 7 x 3 10 x 3 8 ) , ( x15 x 1 8 x 1 6 x 1 9 x25 x 1 1 x 3 6 x 1 2 ) , ( x25 x 2 8 x 2 6 x 2 9 x24 x 2 1 x 3 6 x 3 9 ) , ( x35 x 3 8 x 3 6 x 3 9 x34 x 3 1 x 3 3 x 3 2 ) ,( x17 x 1 11 x 1 8 x 1 12 x114 x 1 1 x 3 8 x 1 6 ) , ( x27 x 2 11 x 2 8 x 2 12 x37 x 2 1 x 3 8 x 2 6 ) , ( x37 x 3 11 x 3 8 x 3 12 x39 x 3 1 x 3 10 x 3 6 ) , ( x11 x 1 5 x 1 2 x 1 6 x110 x 3 5 x 1 11 x 1 14 ) ,( x21 x 2 5 x 2 2 x 2 6 x210 x 3 5 x 2 11 x 2 14 ) , ( x31 x 3 5 x 3 2 x 3 6 x310 x 3 7 x 2 2 x 3 14 ) , ( x23 x 2 11 x 2 9 x 2 12 x28 x 2 6 x 2 15 x 3 12 ) ,( x111 x 1 12 x 3 12 x 3 11 x211 x 2 12 x 3 14 x 3 2 ) , ( x11 x 1 3 x 2 3 x 2 1 x31 x 1 6 x 2 6 x 2 7 ) , ( x12 x 1 4 x 2 4 x 2 2 x32 x 1 6 x 2 11 x 2 10 ) ,( x17 x 1 9 x 2 9 x 2 7 x15 x 3 9 x 2 2 x 2 5 ) , ( x18 x 1 10 x 2 10 x 2 8 x14 x 3 10 x 2 12 x 2 15 ) , ( x113 x 1 15 x 2 15 x 2 13 x313 x 1 7 x 2 11 x 2 9 ) . 3.5. l8-decomposition required for lemma 2.8 3.5.1. an l8decomposition of k13�k5( x j 5 x j 9 x j 6 x j 10 x j 3 x 3 9 x j 4 x 3 10 ) , ( x j 3 x j 12 x j 4 x j 13 x j 11 x j 2 x j 9 x j 7 ) for j = 2, 4;( x j 7 x j 8 x j 10 x j 11 x j 9 x j 2 x j 1 x j 13 ) for j = 1, 5; ( x j 1 x j 13 x j 8 x j 12 x j 7 x 3 13 x j 9 x 3 12 ) for j = 2, 5;( x j 9 x j 12 x j 10 x j 13 x j 8 x j 6 x j 4 x j 5 ) for j = 3, 4; ( x1i x 2 i x 4 i x 3 i x1k x 2 k x 4 k x 3 k ) for (i, k) = (5, 7), (8, 11);( x j 1 x j 5 x j 2 x j 6 x j k1 x k2 5 x k2 2 x k2 6 ) for (j, k1, k2) = (1, 9, 4), (2, 8, 5), (3, 9, 2);( x1i x 2 i x 5 i x 4 i x1k1 x 2 k2 x5k1 x 4 k2 ) for (i, k1, k2) = (9, 10, 11), (10, 2, 1), (13, 6, 6);( x41 x 4 5 x 4 2 x 4 6 x51 x 5 5 x 4 11 x 5 6 ) , ( x51 x 5 5 x 5 2 x 5 6 x31 x 3 5 x 3 2 x 3 6 ) , ( x13 x 1 7 x 1 4 x 1 8 x33 x 5 7 x 3 4 x 4 8 ) , ( x23 x 2 7 x 2 4 x 2 8 x21 x 4 7 x 3 4 x 5 8 ) ,( x33 x 3 7 x 3 4 x 3 8 x39 x 3 1 x 3 2 x 2 8 ) , ( x43 x 4 7 x 4 4 x 4 8 x23 x 3 7 x 3 4 x 5 8 ) , ( x53 x 5 7 x 5 4 x 5 8 x33 x 5 5 x 5 11 x 3 8 ) , ( x15 x 1 9 x 1 6 x 1 10 x55 x 3 9 x 1 12 x 3 10 ) ,( x35 x 3 9 x 3 6 x 3 10 x33 x 3 7 x 3 4 x 3 1 ) , ( x55 x 5 9 x 5 6 x 5 10 x513 x 3 9 x 1 6 x 3 10 ) , ( x11 x 1 4 x 1 5 x 1 11 x12 x 1 6 x 1 8 x 3 11 ) , ( x21 x 2 4 x 2 5 x 2 11 x22 x 4 4 x 2 8 x 5 11 ) ,( x31 x 3 4 x 3 5 x 3 11 x41 x 3 13 x 3 8 x 3 6 ) , ( x41 x 4 4 x 4 5 x 4 11 x42 x 4 3 x 4 8 x 1 11 ) , ( x51 x 5 4 x 5 5 x 5 11 x52 x 5 6 x 5 8 x 5 3 ) , ( x12 x 1 3 x 1 6 x 1 7 x111 x 1 4 x 1 5 x 1 13 ) ,( x22 x 2 3 x 2 6 x 2 7 x211 x 2 4 x 2 5 x 2 10 ) , ( x32 x 3 3 x 3 6 x 3 7 x311 x 2 3 x 3 5 x 3 10 ) , ( x42 x 4 3 x 4 6 x 4 7 x49 x 4 1 x 4 5 x 4 10 ) , ( x52 x 5 3 x 5 6 x 5 7 x511 x 5 4 x 5 5 x 3 7 ) , 44 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x27 x 2 8 x 2 10 x 2 11 x29 x 2 2 x 2 3 x 2 4 ) , ( x37 x 3 8 x 3 10 x 3 11 x27 x 3 1 x 3 3 x 4 11 ) , ( x47 x 4 8 x 4 10 x 4 11 x49 x 4 1 x 4 3 x 4 4 ) , ( x11 x 1 13 x 1 8 x 1 12 x17 x 3 13 x 1 9 x 1 5 ) ,( x31 x 3 13 x 3 8 x 3 12 x33 x 3 6 x 3 2 x 3 7 ) , ( x41 x 4 13 x 4 8 x 4 12 x47 x 3 13 x 4 2 x 3 12 ) , ( x13 x 1 12 x 1 4 x 1 13 x111 x 1 2 x 5 4 x 5 13 ) , ( x33 x 3 4 x 3 12 x 3 13 x311 x 3 9 x 3 5 x 3 7 ) ,( x53 x 5 12 x 5 4 x 5 13 x51 x 5 2 x 3 4 x 5 7 ) , ( x19 x 1 12 x 1 10 x 1 13 x111 x 3 12 x 1 4 x 1 5 ) , ( x29 x 2 12 x 2 10 x 2 13 x22 x 2 6 x 2 4 x 2 5 ) , ( x59 x 5 12 x 5 10 x 5 13 x511 x 5 6 x 5 4 x 5 2 ) ,( x11 x 3 1 x 2 1 x 4 1 x13 x 3 2 x 5 1 x 4 9 ) , ( x12 x 2 2 x 4 2 x 3 2 x14 x 2 13 x 4 13 x 3 10 ) , ( x13 x 2 3 x 5 3 x 4 3 x19 x 2 9 x 5 9 x 3 3 ) , ( x14 x 2 4 x 5 4 x 4 4 x111 x 2 2 x 5 2 x 4 2 ) ,( x16 x 2 6 x 4 6 x 3 6 x111 x 2 8 x 4 8 x 3 8 ) , ( x17 x 2 7 x 5 7 x 4 7 x37 x 2 12 x 5 12 x 4 12 ) , ( x211 x 3 11 x 5 11 x 4 11 x26 x 3 13 x 5 6 x 4 6 ) , ( x112 x 2 12 x 5 12 x 4 12 x17 x 2 11 x 5 5 x 4 11 ) ,( x12 x 5 2 x 5 9 x 1 9 x113 x 4 2 x 5 4 x 1 4 ) , ( x29 x 4 9 x 4 10 x 2 10 x21 x 4 3 x 4 2 x 2 2 ) , ( x111 x 5 11 x 5 12 x 1 12 x211 x 5 8 x 5 13 x 1 13 ) , ( x212 x 4 12 x 4 13 x 2 13 x25 x 4 5 x 4 11 x 2 11 ) ,( x32 x 3 9 x 3 11 x 3 12 x313 x 3 10 x 3 4 x 3 3 ) , ( x13 x 5 3 x 5 10 x 1 10 x15 x 5 5 x 5 7 x 1 7 ) , ( x11 x 5 1 x 5 8 x 1 8 x21 x 5 9 x 5 6 x 1 6 ) . 3.6. l8-decomposition required for lemma 2.9 3.6.1. an l8decomposition of k11�k7( x j 1 x j 5 x j 2 x j 6 x j 9 x j 7 x j 4 x j 8 ) for j = 2, 5; ( x j 1 x j 4 x j 5 x j 11 x j 3 x j 9 x j 8 x k 11 ) for (j, k) = (2, 3), (3, 4);( x j 5 x j 9 x j 6 x j 10 x j 3 x j 11 x j 4 x j 1 ) for j = 5, 6; ( x j 7 x j 8 x j 10 x j 11 x j 9 x j 1 x j 2 x j 4 ) for j = 1, 2, 5;( x j 2 x j 3 x j 6 x j 7 x j 1 x j 11 x j 5 x k 7 ) for (j, k) = (5, 2), (6, 2);( x j 7 x j 8 x j 10 x j 11 x j 9 x j 1 x k 10 x j 4 ) for (j, k) = (3, 6), (6, 7);( x1i x 3 i x 5 i x 7 i x4i x 2 i x 6 i x 7 k ) for (i, k) = (5, 9), (6, 8), (7, 1), (8, 9);( x1i x 3 i x 5 i x 7 i x1k1 x 2 i x 6 i x 7 k2 ) for (i, k1, k2) = (1, 9, 9), (4, 3, 3), (9, 8, 2);( x1i x 2 i x 4 i x 6 i x5i x 2 k x 3 i x 7 i ) for (i, k) = (4, 3), (8, 9), (9, 2);( x2i x 5 i x 4 i x 7 i x6i x 5 k x 4 k x 3 i ) for (i, k) = (3, 10), (8, 9);( x11 x 1 5 x 1 2 x 1 6 x110 x 1 3 x 1 9 x 1 8 ) , ( x31 x 3 5 x 3 2 x 3 6 x39 x 3 7 x 3 10 x 3 11 ) , ( x41 x 4 5 x 4 2 x 4 6 x11 x 4 10 x 1 2 x 4 8 ) , ( x61 x 6 5 x 6 2 x 6 6 x31 x 3 5 x 2 2 x 3 6 ) ,( x71 x 7 5 x 7 2 x 7 6 x61 x 6 5 x 6 2 x 7 11 ) , ( x13 x 1 7 x 1 4 x 1 8 x111 x 1 1 x 1 10 x 1 2 ) , ( x23 x 2 7 x 2 4 x 2 8 x29 x 2 1 x 2 10 x 2 2 ) , ( x33 x 3 7 x 3 4 x 3 8 x35 x 3 1 x 3 10 x 3 2 ) ,( x43 x 4 7 x 4 4 x 4 8 x33 x 4 1 x 4 9 x 4 2 ) , ( x53 x 5 7 x 5 4 x 5 8 x59 x 5 1 x 5 10 x 5 5 ) , ( x63 x 6 7 x 6 4 x 6 8 x69 x 3 7 x 6 10 x 6 5 ) , ( x73 x 7 7 x 7 4 x 7 8 x75 x 6 7 x 7 6 x 7 2 ) ,( x15 x 1 9 x 1 6 x 1 10 x18 x 1 11 x 1 4 x 4 10 ) , ( x25 x 2 9 x 2 6 x 2 10 x23 x 2 11 x 2 4 x 3 10 ) , ( x35 x 3 9 x 3 6 x 3 10 x75 x 6 9 x 3 4 x 3 1 ) , ( x77 x 7 8 x 7 10 x 7 11 x79 x 7 1 x 7 2 x 7 3 ) ,( x11 x 1 4 x 1 5 x 1 11 x13 x 1 9 x 1 7 x 1 8 ) , ( x41 x 4 4 x 4 5 x 4 11 x43 x 4 10 x 4 8 x 4 9 ) , ( x51 x 5 4 x 5 5 x 5 11 x53 x 5 9 x 2 5 x 5 6 ) , ( x61 x 6 4 x 6 5 x 6 11 x13 x 6 9 x 2 5 x 2 11 ) ,( x71 x 7 4 x 7 5 x 7 11 x73 x 7 2 x 7 8 x 6 11 ) , ( x12 x 1 3 x 1 6 x 1 7 x111 x 1 9 x 1 5 x 1 10 ) , ( x22 x 2 3 x 2 6 x 2 7 x21 x 3 3 x 2 5 x 2 10 ) , ( x32 x 3 3 x 3 6 x 3 7 x31 x 3 9 x 3 5 x 3 10 ) ,( x42 x 4 3 x 4 6 x 4 7 x41 x 4 11 x 4 9 x 4 10 ) , ( x72 x 7 3 x 7 6 x 7 7 x32 x 7 9 x 7 5 x 7 10 ) , ( x12 x 3 2 x 5 2 x 7 2 x14 x 4 2 x 5 8 x 7 11 ) , ( x13 x 3 3 x 5 3 x 7 3 x110 x 3 4 x 6 3 x 7 10 ) ,( x110 x 3 10 x 5 10 x 7 10 x19 x 3 3 x 5 7 x 7 9 ) , ( x111 x 3 11 x 5 11 x 7 11 x411 x 3 9 x 6 11 x 7 8 ) , ( x11 x 2 1 x 4 1 x 6 1 x12 x 2 10 x 3 1 x 6 9 ) , ( x12 x 2 2 x 4 2 x 6 2 x52 x 2 11 x 4 9 x 6 8 ) ,( x13 x 2 3 x 4 3 x 6 3 x53 x 2 10 x 4 9 x 7 3 ) , ( x15 x 2 5 x 4 5 x 6 5 x55 x 7 5 x 3 5 x 6 7 ) , ( x16 x 2 6 x 4 6 x 6 6 x111 x 2 11 x 3 6 x 6 8 ) , ( x17 x 2 7 x 4 7 x 6 7 x57 x 7 7 x 3 7 x 6 1 ) ,( x110 x 2 10 x 4 10 x 6 10 x510 x 2 9 x 3 10 x 6 7 ) , ( x111 x 2 11 x 4 11 x 6 11 x511 x 2 8 x 4 10 x 6 6 ) , ( x21 x 5 1 x 4 1 x 7 1 x61 x 1 1 x 4 9 x 3 1 ) , ( x22 x 5 2 x 4 2 x 7 2 x32 x 5 11 x 4 11 x 7 1 ) , 45 k. sowndhariya, a. muthusamy / j. algebra comb. discrete appl. 9(1) (2022) 29–46 ( x24 x 5 4 x 4 4 x 7 4 x64 x 5 3 x 1 4 x 3 4 ) , ( x62 x 6 4 x 6 3 x 6 10 x52 x 3 4 x 3 3 x 5 10 ) , ( x26 x 5 6 x 4 6 x 7 6 x66 x 1 6 x 4 11 x 3 6 ) , ( x29 x 5 9 x 4 9 x 7 9 x69 x 5 2 x 1 9 x 7 11 ) ,( x210 x 5 10 x 4 10 x 7 10 x610 x 5 9 x 4 1 x 3 10 ) , ( x211 x 5 11 x 4 11 x 7 11 x23 x 5 8 x 4 8 x 3 11 ) , ( x32 x 3 9 x 3 8 x 3 11 x34 x 3 10 x 3 6 x 3 3 ) , ( x62 x 6 9 x 6 8 x 6 11 x32 x 6 10 x 3 8 x 3 11 ) ( x43 x 4 4 x 4 6 x 4 5 x13 x 1 4 x 4 10 x 4 9 ) , ( x47 x 4 8 x 4 10 x 4 9 x411 x 4 1 x 4 2 x 4 4 ) , ( x74 x 7 9 x 7 6 x 7 10 x711 x 3 9 x 6 6 x 7 1 ) , ( x45 x 4 7 x 7 7 x 7 5 x55 x 5 7 x 3 7 x 7 10 ) . 3.7. l8-decomposition required for lemma 2.10 3.7.1. an l8decomposition of k9�k9( x j 1 x j 2 x j 3 x j 4 x j 7 x j 5 x j 6 x j 8 ) for j = 4, 6, 7, 8; ( x j 5 x j 6 x j 7 x j 8 x j 4 x j 1 x j 2 x j 3 ) for j = 1, 2, 4, 5, 6, 7, 8;( x1i x 2 i x 3 i x 4 i x7i x 5 i x 6 i x 8 i ) for i = 2, 4, 6; ( x j 1 x j 8 x j 2 x j 9 x j 5 x j 6 x j 4 x j 7 ) for j = 1, 2, 3, 4, 7, 8, 9;( x1i x 2 i x 3 i x 4 i x7i x 6 i x 3 k x 8 i ) for (i, k) = (1, 7), (7, 2); ( x1i x 2 i x 3 i x 4 i x7i x 2 k x 6 i x 8 i ) for (i, k) = (3, 5), (8, 4);( x1i x 2 i x 3 i x 4 i x1k x 5 i x 6 i x 4 k ) for (i, k) = (5, 3), (9, 6); ( x5i x 6 i x 7 i x 8 i x4i x 1 i x 2 i x 3 i ) for i = 1, 3, 4, 6, 7, 8;( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 7 i ) for i = 1, 3, . . . , 9; ( x3i x 7 i x 4 i x 9 i x1i x 5 i x 6 i x 8 i ) for i = 1, 3, 4, 5, 7, 8, 9;( x j 3 x j 7 x j 4 x j 9 x j 1 x j 5 x j 6 x j 8 ) for j = 1, 2, 3, 4, 6, 7, 8, 9;( x11 x 1 2 x 1 3 x 1 4 x17 x 5 2 x 1 6 x 1 8 ) , ( x21 x 2 2 x 2 3 x 2 4 x27 x 2 5 x 2 6 x 6 4 ) , ( x31 x 3 2 x 3 3 x 3 4 x61 x 8 2 x 3 6 x 3 8 ) , ( x51 x 5 2 x 5 3 x 5 4 x31 x 7 2 x 5 6 x 3 4 ) ,( x91 x 9 2 x 9 3 x 9 4 x61 x 9 5 x 9 6 x 6 4 ) , ( x35 x 3 6 x 3 7 x 3 8 x34 x 3 1 x 6 7 x 5 8 ) , ( x95 x 9 6 x 9 7 x 9 8 x55 x 9 9 x 9 2 x 9 3 ) , ( x51 x 5 8 x 5 2 x 5 9 x55 x 5 6 x 3 2 x 5 7 ) ,( x61 x 6 8 x 6 2 x 6 9 x65 x 2 8 x 6 4 x 6 7 ) , ( x53 x 5 7 x 5 4 x 5 9 x51 x 2 7 x 5 6 x 5 8 ) , ( x52 x 6 2 x 7 2 x 8 2 x42 x 1 2 x 2 2 x 8 6 ) , ( x55 x 6 5 x 7 5 x 8 5 x57 x 1 5 x 2 5 x 3 5 ) ,( x59 x 6 9 x 7 9 x 8 9 x55 x 9 9 x 2 9 x 3 9 ) , ( x12 x 8 2 x 2 2 x 9 2 x16 x 6 2 x 4 2 x 7 2 ) , ( x32 x 7 2 x 4 2 x 9 2 x12 x 7 6 x 6 2 x 8 2 ) , ( x36 x 7 6 x 4 6 x 9 6 x16 x 5 6 x 4 2 x 8 6 ) ,( x26 x 6 6 x 6 9 x 2 9 x22 x 6 8 x 6 5 x 2 5 ) , ( x36 x 5 6 x 5 9 x 3 9 x32 x 5 2 x 9 9 x 3 5 ) , ( x45 x 8 5 x 8 9 x 4 9 x55 x 8 3 x 8 6 x 5 9 ) , ( x15 x 7 5 x 7 9 x 1 9 x12 x 7 3 x 7 6 x 6 9 ) ,( x63 x 9 3 x 9 5 x 6 5 x23 x 5 3 x 9 9 x 2 5 ) , ( x33 x 5 3 x 5 5 x 3 5 x38 x 2 3 x 5 2 x 3 2 ) , ( x62 x 9 2 x 9 6 x 6 6 x22 x 5 2 x 5 6 x 4 6 ) , ( x51 x 9 1 x 9 7 x 5 7 x21 x 9 6 x 6 7 x 3 7 ) ,( x54 x 9 4 x 9 8 x 5 8 x52 x 9 5 x 6 8 x 2 8 ) . 46 introduction decomposition of km kn into sunlet graph of order 8 references appendix issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.13099 j. algebra comb. discrete appl. 3(1) • 37–44 received: 19 may 2015 accepted: 2 december 2015 journal of algebra combinatorics discrete structures and applications generalized hypercube graph qn(s), graph products and self-orthogonal codes research article pani seneviratne abstract: a generalized hypercube graph qn(s) has fn2 = {0, 1}n as the vertex set and two vertices being adjacent whenever their mutual hamming distance belongs to s, where n ≥ 1 and s ⊆{1, 2, . . . , n}. the graph qn({1}) is the n-cube, usually denoted by qn. we study graph boolean products g1 = qn(s) ×q1, g2 = qn(s) ∧q1, g3 = qn(s)[q1] and show that binary codes from neighborhood designs of g1, g2 and g3 are self-orthogonal for all choices of n and s. more over, we show that the class of codes c1 are self-dual. further we find subgroups of the automorphism group of these graphs and use these subgroups to obtain pd-sets for permutation decoding. as an example we find a full error-correcting pd set for the binary [32, 16, 8] extremal self-dual code. 2010 msc: 05, 51, 94 keywords: graphs, designs, codes, permutation decoding 1. introduction the generalized hypercube graphs qn(s) were introduced in berrachedi and mollard [1], where the authors mainly investigated the graph embeddings especially when the underlying graph is a hypercube. their connections to (0, 2)-graphs were studied in laborde and madani [6]. binary codes from the row span of an adjacency matrix for the n-cube were first examined in key and seneviratne [5] and the codes in the case of n even were found to be self-dual with minimum weight n. further 3-pd-sets were found for partial permutation decoding. in [2], fish, key and mwambene extended the results in [5] to graphs γkn = qn({k}), when k = 1, 2, 3. in this paper we study generalized hypercube graphs and binary codes from the neighborhood designs of their boolean products. similar to the n-cube, we prove that the graphs qn(s) are cayley graphs and hence are vertex transitive. in particular we study the codes from graph boolean products and show that they are self-orthogonal and if the boolean product is the graph cartesian product, then the codes pani seneviratne; texas a&m university-commerce, usa (email: padmapani.seneviratne@tamuc.edu). 37 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 are self-dual. this construction leads to many optimal codes and we use properties of these graphs to determine the properties of the codes. sections 2 gives the necessary background material and definitions. in section 3 properties of the generalized hypercube graph are studied. the binary codes from the graph boolean products are studied in section 4. in section 5 we find pd-sets for permutation decoding. 2. background and terminology 2.1. codes all the codes discussed in this paper are linear codes, i.e. subspaces of the vector space fn where f is the finite field. the support of a vector u in fn is the set of non-zero coordinates positions of u, and the weight of u, denoted by wt(u), is the cardinality of its support. the notation [n,k,d]q will be used for a q-ary code of length n, dimension k, and minimum weight d. the dual code c⊥ of c is the orthogonal complement of c under the standard inner product <,>, i.e. c⊥ = {v ∈ fn |< v,c >= 0 for all c ∈ c}. the dual code c⊥ is linear over the field f. a generator matrix of c is a matrix whose rows are vectors of a basis for c. two linear codes of the same length and over the same field are isomorphic if they can be obtained from one another by permuting the coordinate positions. an isomorphism from a code c into itself is called an automorphism of c, and the group of all automorphisms of c will be denoted by aut(c). any code is isomorphic to a code with generator matrix in so-called standard form, i.e. the form [ik | a]. in this case, a check matrix of c, i.e. a generator matrix of c⊥, is then given by [−at | in−k]. an information set for a code is the set of the first k coordinates in the standard form and the corresponding check set is the set of the last n−k coordinates. 2.2. graphs the graphs γ = (v,e) with vertex set v and edge set e, discussed here are simple graphs. if two distinct vertices x and y in v are adjacent, then we write x ∼ y, and denote [x,y] for the edge they define. the set of vertices in γ that are adjacent to a vertex x is the neighbour set of x and is denoted by n(x). the cardinality of n(x) is the valency of x. a graph is regular if all the vertices have the same valency. an adjacency matrix a of a graph of order n is an n × n matrix with entries aij such that aij = 1 if vertices vi and vj are adjacent, and aij = 0 otherwise. the neighborhood design of a regular graph is the design formed by taking the points to be the vertices of the graph and the blocks to be the neighbor sets of the vertices. the code of a graph γ over a finite field fq is the row span of an adjacency matrix a over the field fq, denoted by cq(γ) or c(γ) if the underlying field is obvious. let j = jp be the p×p matrix with all entries 1 and let i = ip be the identity matrix of order p. let a = [aij] and b = [bij] be matrices of size p1 ×p1 and p2 ×p2 respectively. their tensor product, also known as the kronecker product a∗b is defined as the partitioned matrix [aijb] : a∗b =   a11b a12b · · · a1p1b a21b a22b · · · a2p1b · · · · · · · · · · · · ap11b ap12b · · · ap1p1b   . a boolean operation on an ordered pair of disjoint graphs g1 = (v1,e1) and g2 = (v2,e2) results in a graph g = g1 ◦ g2 which has the cartesian product v = v1 × v2 as its vertex set and the edge set e is expressed in terms of e1 and e2, differently for each boolean operation. in [3], harary and wilcox gave a detailed explanation of the follwoing boolean operations. the cartesian product is the boolean operation g = g1 ×g2 in which for any two points u = (u1,u2) and v = (v1,v2) ∈ v = v1 ×v2, the edge [u,v] is in e(g) whenever u1 = v1 andu2 ∼ v2 or u1 ∼ v1 and,u2 = v2. we can express the adjacency matrix , a(g1 × g2) = (a1 ∗ ip2 ) + (ip1 ∗ a2). the conjunction or the kronecker product 38 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 g = g1 ∧ g2: for any two points u = (u1,u2) and v = (v1,v2) ∈ v = v1 × v2, the edge [u,v] is in e(g) if [u1,v1] ∈ e(g1) and [u2,v2] ∈ e(g2). the adjacency matrix of the conjunction g1 ∧ g2 is the tensor product a(g1 ∧ g2) = a1 ∗ a2 of the adjacency matrices a1 and a2. the composition or the lexicographical product g = g1[g2] is the graph with u = (u1,u2) and,v = (v1,v2) are adjacent whenever u1 ∼ v1 or u1 = v1 and u2 ∼ v2. the adjacency matrix of the composition is given by a(g1[g2]) = (a1 ∗ jp2 ) + (ip1 ∗ a2). similarly we can define the composition [g1]g2 by its adjacency matrix a([g1]g2) = (a1 ∗ ip2 ) + (jp1 ∗a2). 2.3. permutation decoding permutation decoding is described fully in macwilliams and sloane [7, chapter 16] and huffman [4, section 8]. a pd-set defined here will fully use the error-correction potential of the code which follows easily and is proved in [4]. definition 2.1. let c be a t-error-correcting code with information set i and check set c. a pd-set for c is a set s of automorphisms of c which is such that every t-set of coordinate positions is moved by at least one member of s into the check positions c. permutation decoding employs the following theorem in [4, theorem 8.1] to ensure that all the errors in a received vector are moved out of the information symbols. theorem 2.2. let c be a t-error-correcting [n,k,d]q code with check matrix h that has the identity matrix in−k in the redundancy positions. suppose y = c + e is a vector where c ∈ c and e has weight s ≤ t. then the information symbols in y are correct if and only if the weight of the syndrome hyt of y is ≤ s. the algorithm for permutation decoding can then be stated as follows: we have a t-error-correcting [n,k,d]q code c with generator matrix g and check matrix h in standard form, i.e. g = [ik|a] and h = [−at |in−k], where a is a k× (n−k) matrix, so that the first k coordinate positions correspond to the information symbols. any message v of length k is then encoded as vg. suppose x is a sent codeword and y is a received vector with at most t errors. let s = {g1, . . . ,gm} be a pd-set for c. compute the syndromes h(ygi)t for i = 1, . . . ,m until an i is found such that the weight of this vector is t or less. compute the codeword c that has the same information symbols as ygi and decode y as cg −1 i . 3. generalized hypercube graph qn(s) for a positive integer n, let s ⊆ [n] = {1, 2, . . . ,n} and let ⊕ denote the addition in fn2 = {0, 1}n. the hamming distance of vectors u = (u1,u2, . . . ,un) and v = (v1,v2, . . . ,vn) ∈ fn2 is d(u,v) = |{i ∈ s | ui 6= vi}|. definition 3.1. the generalized hypercube graph qn(s) = (v,e) is an undirected graph with the vertex set v (qn(s)) = fn2 and the edge set e(qn(s) = {uv | d(u,v) ∈ s}. the cardinality of the vertex set is independent of the choice of s and is equal to 2n and is regular with valency ∑ i∈s ( n i ) . we will use the following notation: for r ∈ z and 0 ≤ r ≤ 2n − 1, if r = ∑n i=1 ri2 i−1 is the binary representation of r, let r = (r1,r2, . . . ,rn) be the correspondng vector in fn2 . standard basis of the vector space vn will be denoted by e1,e2, . . . ,en. an automorphism σ of a graph γ = (v,e) is a bijection σ : v 7→ v such that [u,v] ∈ e if and only if [σ(u),σ(v)] ∈ e. the set of all automorphisms of γ is a group and is denoted by aut(γ). a group g acts transitively on a set v , if for every u,v ∈ v there is a σ ∈ g such that σ(u) = v. a graph γ = (v,e) is vertex transitive if aut(γ) acts transitively on v . 39 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 definition 3.2. for n ≥ 1, u = (u1,u2, . . . ,un) ∈ fn2 and σ ∈ sn, where sn is the symmetric group on the set [n]. • a translation by u is the map τu : w 7→ w ⊕u, for all w ∈ fn2 . • a rotation by σ is the map rσ : w 7→ wσ, where wσ = (wσ(1),wσ(2), . . . ,wσ(n)) for w = (w1,w2, . . . ,wn). lemma 3.3. the group of translations tn = {τu | u ∈ fn2} and the group of rotations rn are subgroups of aut(qn(s)). proof. clearly, τx · τy = τx⊕y, τ−1x = τx and τ0 = ı. further dh(u⊕w,v ⊕w) = dh(u,v). therefore the set tn = {τu | u ∈ fn2} is a subgroup of aut(qn(s). for rotations, we have rσ ·rρ = rσ·ρ, r−1σ = rσ−1 and rid = ı. hence, the set of all rotations rn is a sugbroup of aut(qn(s)) and in fact rn ∼= sn. theorem 3.4. the generalized hypercube graph qn(s) is vertex transitive. proof. every cayley graph γ = cay(g,s) is vertex transitive. we will show that the graph qn(s) is a cayley graph. it is well known that the the hypercube graph qn can be defined as the cayley graph qn = cay(tn,{e1,e2, . . . ,en}). similarly we can extend this result to the generalized hypercube graph qn(s). let e1 denote the set of weight 1 vectors {e1,e2, . . . ,en} in fn2 , e2 denote the weight 2 vectors { ∑n i,j ei + ej|i 6= j} and so on. then it it easy to see that qn(s) = cay(tn,{e1,e2, . . . ,en}). 4. self-orthogonal codes from qn(s) in this section we determine the binary codes c1,c2 and c3 from the neighborhood designs of graph products g1 = qn(s) ×q1, g2 = qn(s) ∧q1 and g3 = qn(s)[q1] respectively. lemma 4.1. let a be the adjacency matrix of the graph qn(s), then a2 = { 0 mod 2 : if ∑ i∈s ( n i ) is even. i2n mod 2 : otherwise. proof. let vi,vj be vertices of qn(s) such that i 6= j and let n(vi) and n(vj) be the neighborhoods of vi and vj respectively. since the qn(s) is regular |n(vi)| = |n(vj)| and further |n(vi) ∪ n(vj)| is even. therefore |n(vi)∩n(vj)| is even. but, |n(vi)∩n(vj)| is equal to the number of walks of length 2 between vertices vi and vj. also, the (i,j)th entry of a2 counts the number of walks of length 2 between the vertices vi and vj. hence (i,j)th entry = 0 mod 2 for i 6= j. next, suppose if i = j then the (i, i)th entry of a counts the number of walks of length 2 from a vertex vi to itself. since |n(vi)| is equal to the valency of qn(s), (i, i)th entry of a is equal to 0 if valency is even and 1 if odd. hence the result. remark 4.2. if c is the binary code from the neighborhood design of a graph g with the adjacency matrix a then we will use c to denote the corresponding binary code from the matrix a = a + i. the matrix a is the adjacency matrix of the reflexive graph g, which is obtained from g by adding a loop to every vertex. theorem 4.3. let c1,c2 and c3 be the binary codes from the neighborhood designs of the graph products g1 = qn(s) ×q1, g2 = qn(s) ∧q1 and g3 = qn(s)[q1]. then the codes c1,c2 and c3 are selforthogonal if the valency of qn(s) is odd and c1,c2 and c3 are self-orthogonal if the valency of qn(s) is even. proof. let a1,a2 and a3 denote the adjacency matrices of the graph products g1,g2 and g3 respectively. let a denote the adjacency matrix of qn(s) and b denote the adjacency matrix of q1. the identity matrix of size r is denoted by ir and n = 2n. we will use the fact that a binary code with the 40 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 generator matrix g is self-orthogonal if ggt = 0. case i: a1a t 1 = (a⊗ i2 + in ⊗b)(a⊗ i2 + in ⊗b) t = (a⊗ i2 + in ⊗b)(at ⊗ it2 + i t n ⊗b t ) = (a⊗ i2 + in ⊗b)(a⊗ i2 + in ⊗b) = (a⊗ i2)2 + (in ⊗b)(a⊗ i2) + (a⊗ i2)(in ⊗b) + (in ⊗b)2 = (a2 ⊗ i22 ) + 2(a⊗b) + (i 2 n ⊗b 2) = a2 ⊗ i2 + in ⊗ i2 = (a2 + in ) ⊗ i2. if the valency of qn(s) is odd, then a2 = in by lemma 4.1, and hence a1at1 = 0 mod 2. if valency is even then a2 = 0. in this case a1 ·a1 t = a21 + i2n+1 = (a 2 + in ) ⊗ i2 + i2n+1 = 0. case ii: a2a t 2 = (a⊗b)(a⊗b) t = (a⊗b)(a⊗b) = a2 ⊗b2 = a2 ⊗ i2. by lemma 4.1, a2 = 0 if valency of qn(s) is even and hence a2at2 = 0 ⊗ i2 = 0. if the valency of qn(s) is odd, consider a2 ·a2 t = a22 + i2n+1 = a 2 ⊗ i2 + i2n+1 = in ⊗ i2 + i2n+1 = 0. case iii: a3a t 3 = (a⊗j2 + in ⊗b + i2n+1 )(a⊗j2 + in ⊗b + i2n+1 ) t = a2 ⊗j22 + a⊗bj2 + a⊗j2 + a⊗j2b + in ⊗b 2 + in ⊗b + a⊗j2 + in ⊗b + i2n+1 = in ⊗ i2 + i2n+1 = 0. theorem 4.4. the binary code c1 from the neighborhood design of the graph product g1 = qn(s)×q1 is self-dual when the valency of qn(s) is odd and the code c1 is self-dual when the valency is even. further the set of points 0,1,2, . . . ,2n −1 form an information set for c1 and c1. proof. we will change the ordering of points in the adjacency a1 of the graph g1. use the natural ordering of points: 0,1,2, . . . ,2n −1,2n, . . . ,2n+1 −1 to index the columns of a1 and use the ordering 2n,2n + 1, . . . ,2n+1 −1,0,1, . . . ,2n−1 to index the rows. then the (i, i)th entry aii = 1 for 1 ≤ i ≤ 2n−1 and aii = 0 for 2n ≤ i ≤ 2n+1−1. by row reduction it is easy to see that the incidence vectors v0,v1, . . . ,v2n−1 are linear independent. hence dimension of c1 is 2n and c1 is self-dual. remark 4.5. instead of using separate notations c1 and c1 to denote codes from the graphs qn(s)×q1 and qn(s) ×q1 we will only use c1 to denote codes from qn(s)×q1 or qn(s) ×q1 as it is understood when the valency is even c1 refers to c1. 41 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 example 4.6. let n = 3 and s = {1, 3}. then the valency of qn(s) is 4 with the adjacency matrix: a =   0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0   . then c1 = [16, 8, 4] self-dual, c2 = [16, 4, 4] self-orthogonal and c3 = [16, 8, 2] self-dual codes. 5. permutation decoding in this section we will find particular information sets for permutation decoding and use these sets to find partial permutation decoding sets for the codes c1. the vertex set of the graph product g1 = qn(s) ×q1 can be viewed as vectors of the space fn+12 . theorem 5.1. for all n and s and for g1 = qn(s) ×q1: • the translation group t = {τu|u ∈ fn+12 } is a subgroup of aut(g1). • the group of rotations rn is a subgroup of aut(g1). • transpositions of the form ti = (i,n + 1), where 1 ≤ i < n are in aut(g1). proof. since the translation group t and the group of rotations rn are subgroups of the graph qn(s), they are also subgroups of the graph cartesian product g1 = qn(s)×q1. let u = (u1,u2, . . . ,un+1),v = (v1,v2, . . . ,vn+1) ∈ v (g1) such that u ∼ v. that is, d(u,v) ∈ s. now tiu = (u1,u2, . . . ,un+1, . . . ,ui) and tiv = (v1,v2, . . . ,vn+1, . . . ,vi), but d(tiu,tiv) = d(u,v) ∈ s. hence ti ∈ aut(g1). the following result shows that any information set for c1 from the graph qn({1}) can be extended to a code from the graphs qn(s), where {1}⊆ s. lemma 5.2. if i is an information set for c1 with s = {1}, then i is an information set for c1 for all s such that {1}⊆ s. proof. since i is an information set for c1 when s = {1} and since the dimension of c1 is 2n the first 2n incidence vectors are linearly independent. if we take any super set s that contains {1} these first 2n vectors will still be linearly independent and since the dimension of the code c1 is 2n is independent of the choice of s, the set i will be an information set for c1 for all {1}⊆ s. permutation decoding method depends on the information set and hence different information sets will yield different pd-sets and results. the information set obtained in theorem 4.4 is only useful for finding one error-correcting pd sets for c1. we will re-order the vertices so that the resulting information set can be used for correcting more than one error. lemma 5.3. an information set can be obtained for the binary code c1 from the graph g1 = qn(s)×q1 for all n and {1} ⊆ s by making the following interchange between the information and check sets from the natural ordering of the vectors: move 2n −1 = (0, 1, 1, . . . , 1) into check positions and 2n+1 −2 = (1, 1, . . . , 1, 0) into information positions. define pn = {ti|1 ≤ i ≤ n}∪{ı} and tn = tpn. 42 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 proposition 5.4. with i = {0,1, . . . ,2n −2}∪{2n+1 −2} as information set for c1, tn is a 3-pd set of size (n + 1)2n+1 for c1 for all n and {1}⊆ s. proof. let t = {a,b,c} be a set of three points in vn+1. we need to show that there is an automorphism σ ∈ tn that maps t into c, i.e. t σ ⊆c. we consider all the possibilities for the points in t . if t ⊆c then all the errors are in check positions and hence we can use the identity map, ı as σ. thus, assume at least one of the points is in the information positions i, and by using a translation, suppose one of the points, say c, is 0. if t ⊆ i. first suppose both a,b ∈ i1, then σ = t(1, 0, 0, . . . , 0) will map t to c unless a,b 6= (0, 1, 1, . . . , 1, 0). in this case σ = t(1, 1, 0, . . . , 0) will work. next, suppose one of the points, say b ∈i2 and a ∈i1. then b = (1, 1, 1, . . . , 1, 0) and σ = t(1, 0, . . . , 0, 1) will map t into c. the other cases for t involve one or more points in c. case(i): a ∈i1 and b ∈c1. then a = (0,a2, . . . ,an+1) and b = (1,b2, . . . ,bn+1). (1).suppose a = bc and let σ = t(1,a2, . . . ,an+1) then cσ = (1,a2, . . . ,an+1), aσ = (1, 0, . . . , 0) and yσ = (0, 1, . . . , 1). this σ will work unless a 6= (0, 1, . . . , 1, 0). in this case b = ac = (1, 0, . . . , 0, 1) and σ = t(1, 1, . . . , 1, 0) will work. (2). suppose ai = bi for 2 ≤ i ≤ n + 1. then a = (0,a2, . . . ,an+1) and b = (1,a2, . . . ,an+1). if σ = t(ac), we have cσ = ac,aσ = (1, 1, . . . , 1),yσ = (0, 1, . . . , 1) ∈c. (3). suppose there exists an i such that ai = bi = 0 and xj 6= yj for some j. the map σ = t(1, 1, . . . , 1)ti will work unless an+1 = bn+1 = 0 is the only common zero. in this case σ = t(0, . . . , 0, 1)ti will work. case(ii): a ∈i2 and b ∈c1. then a = (1, 1, . . . , 1, 0) and b = (1,b2, . . . ,bn+1). the map σ = t(0, 1, . . . , 1) will work as cσ = (0, 1, . . . , 1) ∈ c2,aσ = (1, 0, . . . , 0, 1) ∈ c2 and bσ = (1,b2c, . . . ,bn+1c) ∈ c1 unless b = (1, 0, . . . , 0, 1), in which case the map σ = t(0, . . . , 0, 1)tn+1 will work. case(iii): a ∈i2 and b ∈c2. then a = (1, . . . , 1, 0) and b = (1, . . . , 1) or b = (0, 1, . . . , 1). if b = (1, . . . , 1) then σ = t(1, 0, . . . , 0)tn+1 will work and otherwise if b = (0, 1, . . . , 1), σ = t(1, 0, 1, . . . , 1)t2 will work. case(iv): a ∈ i1 and b ∈ c∈. then a = (0,a2, . . . ,an+1) and b = (1, 1, . . . , 1) or (0, 1, . . . , 1). if a 6= (0, 1, . . . , 1, 0) then σ = t(1, 0, . . . , 0) will work. if a = (0, 1, . . . , 1, 0) and b = (1, 1, . . . , 1) then σ = t(1, 0, 1, . . . , 1)t2 and if a = (0, 1, . . . , 1, 0),b = (0, 1, . . . , 1) then σ = t(1, 0, . . . , 0, 1tn+1 will work. case(v): both a and b in c1. then a = (1,a2, . . . ,an+1) and b = (1,b2, . . . ,bn+1). then σ = t(0, 1, . . . , 1) will work except when a or b equals (1, 0, . . . , 0, 1). in this case at(1, . . . , 1) and bt(1, . . . , 1) contain at least one common i such that ai = bi = 1. then the map σ = t(1, . . . , 1)ti will work. case(vi): a ∈c1 and b ∈c2. then a = (1,a2, . . . ,an+1) and b = (1, . . . , 1) or (0, 1, . . . , 1). if b = (1, . . . , 1) then σ = t(0, 1, . . . , 1) will work unless a = (1, 0, . . . , 0, 1). in that case then σ = t(1, 0, . . . , 0, 1)t2 will work. if b = (0, 1 . . . , 1) case (vii): both a,b ∈c2. in this case the map σ = t(1, 0, . . . , 0) will work. this completes all the cases. example 5.5. let n = 4 and s = {1, 2}, then qn(s) has valency 10 with the adjacency matrix: a =   0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0   43 p. seneviratne / j. algebra comb. discrete appl. 3(1) (2016) 37–44 and c1 = [32, 16, 8] is the binary extremal doubly even self-dual code. the total error correcting capability of this code is t = 3. then |t | = 32 and |pn| = 5 and hence |tpn| = 160. by proposition 5.4 the set tpn is a full error-correcting pd-set for this code. 6. conclusion in this work we have considered the generalized hypercube graphs and their boolean products. we obtained self-orthogonal codes from the neighborhood designs of these graphs and used subgroups of the automorphism group of the graph to find partial permutation decoding sets for permutation decoding. acknowledgment: the author would like to thank the anonymous referees for their careful reading of the paper and for their insightful comments and suggestions. references [1] a. berrachedi, m. mollard, on two problems about (0, 2)-graphs and interval-regular graphs, ars combin. 49 (1998) 303–309. [2] w. fish, j. d. key, e. mwambene, graphs, designs and codes related to the n-cube, discrete math. 309(10) (2009) 3255–3269. [3] f. harary, g. w. wilcox, boolean operations on graphs, math. scand. 20 (1967) 41–51. [4] w. c. huffman. codes and groups. in v. pless and w. c. huffman, eds., handbook of coding theory, vol. 2, pp. 1345–1440, elsevier science publishers, amsterdam, the netherlands, 1998. [5] j. d. key, p. seneviratne, permutation decoding for binary self-dual codes from the graph qn, where n is even. in t. shaska, w. c. huffman, d. joyner, and v. ustimenko, eds., advances in coding theory and cryptography, series on coding theory and cryptography, vol. 3, pp. 152–159, world scientific publishing co. pte. ltd., hackensack, nj, 2007. [6] j. m. laborde, r. m. madani, generalized hypercubes and (0, 2)-graphs, discrete math. 165/166 (1997) 447–459. [7] f. j. macwilliams, n. j. a. sloane, the theory of error-correcting codes, amsterdam: north-holland, 1998. [8] m. mulder, (0,λ)-graphs and n-cubes, discrete math. 28(2) (1979) 179–188. 44 http://www.ams.org/mathscinet-getitem?mr=1633072 http://www.ams.org/mathscinet-getitem?mr=1633072 http://dx.doi.org/10.1016/j.disc.2008.09.024 http://dx.doi.org/10.1016/j.disc.2008.09.024 http://www.ams.org/mathscinet-getitem?mr=211900 http://dx.doi.org/10.1142%2f9789812772022_0010 http://dx.doi.org/10.1142%2f9789812772022_0010 http://dx.doi.org/10.1142%2f9789812772022_0010 http://dx.doi.org/10.1142%2f9789812772022_0010 http://dx.doi.org/10.1016/s0012-365x(96)00343-3 http://dx.doi.org/10.1016/s0012-365x(96)00343-3 http://dx.doi.org/10.1016/0012-365x(79)90095-5 introduction background and terminology generalized hypercube graph qn(s) self-orthogonal codes from qn(s) permutation decoding conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.12813 j. algebra comb. discrete appl. 4(1) • 13–21 received: 30 september 2015 accepted: 10 april 2016 journal of algebra combinatorics discrete structures and applications on the norms of r−circulant matrices with generalized fibonacci numbers research article amara chandoul abstract: in this paper, we obtain a generalization of [6, 8]. firstly, we consider the so-called r−circulant matrices with generalized fibonacci numbers and then found lower and upper bounds for the euclidean and spectral norms of these matrices. afterwards, we present some bounds for the spectral norms of hadamard and kronecker product of these matrices. 2010 msc: 15b05, 15a60, 65f35 keywords: matrix norm, r-circulant matrices, generalized fibonacci numbers 1. introduction the fibonacci numbers are the elements of integer sequence {fn}∞n=0 defined by the linear recurrence equation   f0 = 0, f1 = 1 fn+2 = fn+1 + fn, for n = 0,1,2, .... the sequence is s given by 0,1,1,2,3,5,8,13, . . . . you cannot go very far in the lore of fibonacci numbers without encountering the companion sequence of lucas numbers {ln}∞n=0, which follows the same recursive pattern as the fibonacci numbers, but begins with other initial values. it is defined by the linear recurrence equation  l0 = 2, l1 = 1 ln+2 = ln+1 + ln, for n = 0,1,2, .... amara chandoul; departamento de matemática, universidade de brasília, campus universitário darcy ribeiro brasília df 70910-900, brazil (email: amarachandoul@yahoo.fr). 13 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 the sequence is s given by 2,1,3,4,7,11,18,29, . . . . the fibonacci sequence has been studied extensively and generalized in many ways. such generalizations are possible in four directions, namely, by changing the initial values, by mixing two lucas sequences, by not demanding that the numbers in the sequences be integers, or by having more than two parameters. one of generalization of fibonacci i sequence is the integer sequence {gn}∞n=0. it has the same recurrence relation as fibonacci and lucas, namely it is defined by the linear recurrence equation  g0 = a, g1 = b gn+2 = gn+1 + gn, for n = 0,1,2, .... the starting values of g0 = a and g1 = b can be specified. it therefore includes both sequences them both as special cases. this generalized fibonacci sequence is giving by a,b,(a + b),(a + 2b),(2a + 3b),(3a + 5b),(5a + 8b),(8a + 13b), ... {gn}∞n=0 is a generalization of {fn}∞n=0 and {ln}∞n=0. furthermore, there is a relationship such that gn = afn−1 + bfn, a,b ∈ r between fibonacci and generalized fibonacci numbers. fibonacci, lucas numbers and their generalization have many intersting properties and applications to almost every field of science and art. for the beauty and rich applications of these numbers and their relatives one may see vajda’s and koshy’s books [7, 12]. solak [10] defined the circulant matrices with fibonacci and lucas numbers, and he obtained lower and upper bounds with the fibonacci number for the euclidean and spectral norms of these matrices. civciv and türkmen [2] constructed the circulant matrix with the lucas number and then presented lower and upper bounds for the euclidean and spectral norms of this matrix as a function of n and ln which is n th lucas number. in [2], they are studied the norm bounds for the hadamard inverse of this matrix. in [1], authors computed some norms of r-circulant matrices associated with a number sequence. in this study, we first construct the so-called r-circulant matrix with the generalized fibonacci numbers and then present some lower and upper bounds for the euclidean and spectral norms of this matrix. we begin with some preliminaries related to our study. a matrix c = [cij] ∈ mn(c) is called a r-circulant matrix if it is of the form cij = { cj−i j ≥ i rcn+j−i j < i. obviously, the r-circulant matrix c is determined by parameter r and its first row elements c0,c1, · · · ,cn−1. especially, for r = 1, the matrix c is called a circulant matrix. for any a = [aij] ∈ mm,n(c), the well known frobenius (or euclidean) norm of matrix a is ‖a‖e = √√√√ n∑ i=1 n∑ j=1 |aij|2. the spectral norm of the matrix a ‖a‖2 = √ max 1≤i≤n λi(a ha) 14 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 where λi is an eigenvalue of aha and ah is conjugate transpose of the matrix a. it is well known that 1 √ n ‖a‖e ≤‖a‖2 ≤‖a‖e. if ‖.‖ is any norm on m×n complex matrices, then [5] ‖a◦b‖≤‖a‖‖b‖, (1) where a◦b is the hadamard product of a and b. define the maximum column lenght norm c1(a) and the maximum row lenght norm r1(a) of any matrix a respectively by c1(a) = max 1≤j≤n √√√√ n∑ i=1 |aij|2 and r1(a) = max 1≤i≤n √√√√ n∑ j=1 |aij|2. let a, b and c be m×n complex matrices. if a = b ◦c, then [4] ‖a‖2 ≤ r1(b)c1(c). (2) 2. results definition 2.1. the r−circulant matrix a with generalized fibonacci number is the matrix of the form a =   g0 g1 . . . . . . gn−1 rgn−1 g0 . . . . . . gn−2 rgn−2 rgn−1 ... . . . gn−3 ... ... ... g0 ... rg1 rg2 . . . rgn−1 g0   . (3) theorem 2.2. let a be the matrix defined in (3). then (i) if |r| ≥ 1, then ‖a‖e ≥ √ n(gn−1gn −g1g0 + g20), ‖a‖2 ≥ √ gn−1gn −g1g0 + g20 and ‖a‖2 ≤ min   √ ((n−1)|r|2 + 1)(gn−1gn −g1g0 + g20),√ ((n−1)|r|2 + g20)(gn−1gn −g1g0 + 1),√ (gn−1gn −g1g0 + g20)(gn−1gn −g1g0 + 1). moreover, if g0 = 0, then √ gngn−1 ≤‖a‖2 ≤ |r|gn−1gn. 15 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 (ii) if |r| < 1, then ‖a‖e ≥ |r| √ n(gn−1gn −g1g0 + g20), ‖a‖2 ≥ |r| √ gn−1gn −g1g0 + g20 and ‖a‖2 ≤ min   √ n(gn−1gn −g1g0 + g20),√ ((n−1) + g20)(gn−1gn −g1g0 + 1),√ ((n−1) + g20)(gn−1gn −g1g0 + g20). moreover, if g0 = 0, then |r| √ gngn−1 ≤‖a‖2 ≤ √ (n−1)gn−1gn. proof. from (3), we have ‖a‖e = √√√√n−1∑ k=0 (n−k)g2k + n−1∑ k=1 k|r|2g2k, let matrices be defined as b1 = (b1,ij) = { b1,ij = r, i > j b1,ij = 1, i ≤ j b2 = (b2,ij) = { b2,ij = g(mod(j−i,n)), i ≥ j b2,ij = 1, i < j b3 = (b3,ij) =   b3,ij = r, i > j b3,ij = g0, i = j b3,ij = 1, i < j c1 = (c1,ij) = (g(mod(j−i,n))). c2 = (c2,ij) =   c2,ij = r, i > j c2,ij = 1, i = j c2,ij = g(mod(j−i,n)), i < j and c3 = (c3,ij) = { c3,ij = g(mod(j−i,n)), i 6= j c3,ij = 1, i = j such that a = bk ◦ck, 1 ≤ k ≤ 3. then we have r1(b1) = max 1≤i≤n √√√√ n∑ j=1 |b1,ij|2 =   √ (n−1)|r|2 + 1, |r| ≥ 1 √ n, |r| < 1. r1(b2) = max 1≤i≤n √√√√ n∑ j=1 |b2,ij|2 = √ gn−1gn −g1g0 + g20. r1(b3) = max 1≤i≤n √√√√ n∑ j=1 |b3,ij|2 =   √ g20 + (n−1)|r|2, |r| ≥ 1√ g20 + (n−1), |r| < 1. (4) 16 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 c1(c1) = max 1≤j≤n √√√√ n∑ i=1 |c1,ij|2 = √ gn−1gn −g1g0 + g20. c1(c2) = max 1≤j≤n √√√√ n∑ i=1 |c2,ij|2 = √ gn−1gn −g1g0 + 1. c1(c3) = max 1≤j≤n √√√√ n∑ i=1 |c3,ij|2 = √ gn−1gn −g1g0 + 1. (5) if g0 = 0, we consider the matrices b4 = (b4,ij) = { b4,ij = rg(mod(j−i,n)), i ≥ j b4,ij = 1, i < j. c4 = (c4,ij) = { c4,ij = g(mod(j−i,n)), i ≤ j c4,ij = 1, i > j. then, we have r1(b4) = max 1≤i≤n √√√√ n∑ j=1 |b4,ij|2 = |r| √ gn−1gn, c1(c4) = max 1≤j≤n √√√√ n∑ i=1 |c4,ij|2 = √ gn−1gn. (6) (i) if |r| ≥ 1, then we have ‖a‖e ≥ √ n(gn−1gn −g1g0 + g20) thus, we obtain from 1 √ n ‖a‖e ≤‖a‖2, ‖a‖2 ≥ √ gn−1gn −g1g0 + g20 on the other hand, using (4), (5) and (2), we have ‖a‖2 ≤ min{r1(b1)c1(c1),r1(b2)c1(c2),r1(b3)c1(c3)}. thus, we have ‖a‖2 ≤ min   √ ((n−1)|r|2 + 1)(gn−1gn −g1g0 + g20),√ ((n−1)|r|2 + g20)(gn−1gn −g1g0 + 1),√ (gn−1gn −g1g0 + g20)(gn−1gn −g1g0 + 1). moreover, if g0 = 0, using (6), then√ gngn−1 ≤‖a‖2 ≤ |r|gn−1gn. (ii) if |r| < 1, then we have ‖a‖e ≥ √ n|r|2(gn−1gn −g1g0 + g20). 17 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 thus, we have from 1 √ n ‖a‖e ≤‖a‖2, ‖a‖2 ≥ |r| √ (gn−1gn −g1g0 + g20). on the other hand, using (4), (5) and (2), we have ‖a‖2 ≤ min{r1(b1)c1(c1),r1(b2)c1(c2),r1(b3)c1(c3)}. thus, we have ‖a‖2 ≤ min   √ n(gn−1gn −g1g0 + g20),√ ((n−1) + g20)(gn−1gn −g1g0 + 1),√ ((n−1) + g20)(gn−1gn −g1g0 + g20). moreover, if g0 = 0, then |r| √ gngn−1 ≤‖a‖2 ≤ √ (n−1)gn−1gn. corollary 2.3. let g0 = a = 0 and g1 = b = 1 be in the theorem. then (i) if |r| ≥ 1, then √ fnfn−1 ≤‖a‖2 ≤ |r|fnfn−1. (ii) if |r| < 1, then |r| √ fnfn−1 ≤‖a‖2 ≤ √ (n−1)fnfn−1. corollary 2.4. let g0 = a = 2 and g1 = b = 1 be in the theorem. then (i) if |r| ≥ 1, then √ lnln−1 + 2 ≤‖a‖2 ≤ min   √ ((n−1)|r|2 + 1)(lnln−1 + 2),√ ((n−1)|r|2 + 4)(lnln−1 −1),√ (lnln−1 + 2)(lnln−1 −1). (ii) if |r| < 1, then |r| √ lnln−1 + 2 ≤‖a‖2 ≤ √ (n + 3)(lnln−1 −1) theorem 2.5. let a be the matrix defined in (3). then (i) if |r| ≥ 1, then ‖a‖2 ≥   √ a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1, n odd √ a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1, n even (ii) if |r| < 1, then ‖a‖2 ≥   |r| √ a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1, n odd |r| √ a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1, n even 18 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 proof. we have, n−1∑ k=0 g2k = gn−1gn −g1g0 + g20 = n−1∑ k=0 (afk−1 + bfk) 2 = a2 n−1∑ k=0 f2k−1 + 2ab n−1∑ k=0 fk−1fk + b 2 n−1∑ k=0 f2k . as n−1∑ k=0 f2k−1 = fn−1fn−2 + 1, n−1∑ k=0 f2k = fnfn−1 and n−1∑ k=0 fk−1fk =   f2n−1, n odd f2n−1 −1, n even the proof of theorem 2.5 become trivial. theorem 2.6. let a be the matrix defined in (3). then (i) if |r| ≥ 1, then ‖a‖2 ≤   min   √ a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1.√ (n−1)|r|2 + 1,√ a2fn−2fn−1 + 2abf 2 n−1 + b 2fnfn−1 + 1.√ (n−1)|r|2 + a2,√ a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1.√ a2fn−2fn−1 + 2abf 2 n−1 + b 2fnfn−1 + 1. n odd min   √ a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1.√ (n−1)|r|2 + 1,√ a2fn−2fn−1 + 2ab(f 2 n−1 −1) + b2fnfn−1 + 1.√ (n−1)|r|2 + a2,√ a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1.√ a2fn−2fn−1 + 2ab(f 2 n−1 −1) + b2fnfn−1 + 1. n even 19 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 (ii) if |r| < 1, then ‖a‖2 ≤   min   √ n(a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1), √ a2fn−2fn−1 + 2abf 2 n−1 + b 2fnfn−1 + 1.√ (n−1) + a2,√ a2(1 + fn−2fn−1) + 2abf 2 n−1 + b 2fnfn−1.√ (n−1) + a2, n odd min   √ n(a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1),√ a2fn−2fn−1 + 2ab(f 2 n−1 −1) + b2fnfn−1 + 1.√ (n−1) + a2,√ a2(1 + fn−2fn−1) + 2ab(f 2 n−1 −1) + b2fnfn−1.√ (n−1) + a2. n even proof. as n−1∑ k=1 f2k−1 = fn−1fn−2, n−1∑ k=1 f2k = fnfn−1 and n−1∑ k=1 fk−1fk =   f2n−1, n odd f2n−1 −1, n even the proof of theorem 2.6 become trivial. corollary 2.7. let g0 = a = 2 and g1 = b = 1 be in the theorem. as fnfn−1 ≥ n−1, ∀n ≥ 0 and as we have fnfn−1 = f2n−1 + fn−2fn−1, then, we obtain (i) if |r| ≥ 1, then  √ 5fnfn−1 + 4 ≤‖a‖2 ≤ √ ((n−1)|r|2 + 4)(5fnfn−1 + 1), n odd√ 5fnfn−1 ≤‖a‖2 ≤ √ ((n−1)|r|2 + 4)(5fnfn−1 −3), n even (ii) if |r| < 1, then  |r| √ 5fnfn−1 + 4 ≤‖a‖2 ≤ √ n(5fnfn−1 + 4), n odd |r| √ 5fnfn−1 ≤‖a‖2 ≤ √ n(5fnfn−1), n even acknowledgment: we would like to thank saäd chandoul and massöuda loörayed for helpful discussions and many remarks. 20 a. chandoul / j. algebra comb. discrete appl. 4(1) (2017) 13–21 references [1] d. bozkurt, t. y. tam, determinants and inverses of r−circulant matrices associated with a number sequences, linear and multilinear algebra 63(10) (2015) 2079–2088. 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3(3) • 201–208 received: 9 january 2016 accepted: 17 february 2016 journal of algebra combinatorics discrete structures and applications on the resolutions of cyclic steiner triple systems with small parameters∗ research article svetlana topalova abstract: the paper presents useful invariants of resolutions of cyclic sts(v) with v ≤ 39, namely of all resolutions of cyclic sts(15), sts(21) and sts(27), of the resolutions with nontrivial automorphisms of cyclic sts(33) and of resolutions with automorphisms of order 13 of cyclic sts(39). 2010 msc: 05b07, 05e18 keywords: resolutions, cyclic designs, point-cyclically resolvable, cyclic sts(v), kts(v) 1. introduction 1.1. basic definitions and notations for the basic concepts and notations concerning combinatorial designs and their resolvability refer, for instance, to [2], [3], [13], [22]. let v = {pi} v i=1 be a finite set of points, and b = {bj} b j=1 a finite collection of k-element subsets of v called blocks. if any 2-subset of v is contained in exactly λ blocks of b, then d = (v,b) is a 2-(v,k,λ) design, or balanced incomplete block design (bibd). each point of d is incident with r blocks. two designs are isomorphic if there exists a one-to-one correspondence between the point and block sets of the first design and respectively, the point and block sets of the second design, and if this one-toone correspondence does not change the incidence. an automorphism is an isomorphism of the design to itself, i.e. a permutation of the points that maps each block to a block of the same design. a 2-(v,k,λ) design is cyclic if it has an automorphism permuting its points in one cycle of length v. ∗ this work was partially supported by the bulgarian national science fund under contract no. i01/0003. svetlana topalova; institute of mathematics and informatics, bulgarian academy of sciences, bulgaria(email: svetlana@math.bas.bg). 201 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 a resolution of the design is a partition of the collection of blocks into parallel classes, such that each point is in exactly one block of each parallel class. a design is resolvable if it has at least one resolution. two resolutions are isomorphic if there exists an automorphism of the underlying design which maps each parallel class of the first resolution to a parallel class of the second one. an automorphism of a resolution is an automorphism of the underlying design which maps each parallel class to a parallel class of the same resolution. a resolution is point-cyclic if it has an automorphism permuting the points in one cycle. only designs which are cyclic can have point-cyclic resolutions. a design is point-cyclically resolvable if it has at least one point-cyclic resolution. a 2-(v,3,1) design is called a steiner triple system of order v (sts(v)) and its resolutions are called kirkman triple systems of order v (kts(v)). 1.2. motivation and known results design resolutions have various important applications (see, for instance, [4], [9], [11], [19], [20], [21], [23]). resolutions with rich automorphism groups might be very useful for some of them ( for instance, point-cyclic resolutions of steiner systems are used in some constructions of regular low-density parity-check codes providing a fast decoding algorithm [11] and of systematic repeat-accumulate codes [8]). that is why resolutions of cyclic designs are of particular interest and have been the subject of many papers. general constructions of resolutions of cyclic designs are presented in [7], [14], [17], [18]. cyclic sts(v) are resolvable iff v ≡ 3 (mod 6) and v ≥ 15. there are several computer-aided classifications of kts(v) from cyclic designs. all resolutions of sts(15) [12] and of cyclic sts(21) [16] are known. the point-cyclic kts(21) and kts(39) are subject of [15], kts(33) with automorphisms of order 11 and cyclic underlying designs are constructed in [24] and the point-cyclic kts(27) are part of the constructed in [5] transitive kts(27). there exist many other works on kts(v) with v ≤ 39, but the author does not know classification results for the rest of the cyclic sts(v) cases considered below. subject of the present work is the classification of the resolutions of cyclic sts(v) with small v. as it can be seen from the previous paragraph, not all of the results are new. the aim of the paper is on the one hand to obtain new kts(v) and on the other to present the properties of all known resolutions of cyclic sts(v) with v ≤ 39 and to make all of them available at a web-page for possible future applications. 2. classification method cyclic 2-(v,k,1) designs with small parameters were first classified in [6]. this classification was extended for some of the next parameters in [1]. since all necessary designs are available, the focus of the present paper is only the construction and study of their resolutions. the resolutions of each cyclic sts(v) are found in the following two ways. 2.1. construction of all nonisomorphic resolutions the blocks of the design are first sorted in lexicographic order and then backtrack search is applied on them. the parallel classes are constructed block by block. if n blocks have been added to a class, the n + 1-st one is chosen among the blocks which • contain the smallest point that is in none of the already added blocks and • contain no points which are in any of the added blocks. 202 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 the resolutions are thus constructed in lexicographic order. to make the classification feasible the algorithm is speeded up by performing a minimality test after adding each whole parallel class. this test checks if the current partial solution can be mapped to a lexicographically smaller one by some of the automorphisms of the design. if it can, an equivalent partial solution has already been considered and therefore the current one is rejected, namely a next possibility for the latest added block is looked for. as a result all non isomorphic resolutions of the design are obtained. 2.2. construction of all nonisomorphic resolutions with a given automorphism α of prime order p there are two possibilities for the length of the parallel class orbits under α: length p and length 1 (fixed parallel classes). since the number of parallel classes is (v −1)/2, there must be at least cf = v −1 2 mod p parallel classes which are fixed by α. the number of blocks in a parallel class is v/3 and therefore a fixed parallel class must contain at least bf = v 3 mod p fixed blocks. that is why with respect to α there must be • at least bfcf fixed blocks and • at least cf ( ⌊ v 3p ⌋ + bf ) block orbits such that any point is contained in at most one block of the orbit. therefore the algorithm starts with checking if these two conditions hold and stops if not. the construction algorithm itself is a modification of the general one described in the previous subsection. this modification adds more requirements, namely: • any two blocks of a non fixed parallel class should be from different orbits under α • if a block is contained in a fixed parallel class, then all blocks of its orbit under α should be in the same parallel class these requirements lead to a differently defined lexicographic order and respectively to a more complex minimality test which must take in account the orbit length ( 1 or p ) under α of the added parallel classes. 3. classification results 3.1. resolution invariants the blocks of all cyclic sts(v) with v ≡ 3 (mod 6) are partitioned by the automorphism of order v in (v−3)/6 orbits of length v and one short block orbit of length v/3. three types of point-cyclic resolutions of cyclic steiner triple systems are defined in [17] with respect to the role of the short block orbit in the resolution. in a similar way we can define three types of resolutions (which may not be point-cyclic) as follows. a resolution of a cyclic steiner triple system is of type 203 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 • t1 if the short orbit is a parallel class for all resolutions of the isomorphism class. • t2 if the short orbit is not a parallel class for any resolution of the isomorphism class. • t3 if the short orbit is a parallel class for some resolutions of the isomorphism class and is not a parallel class for the rest. in the present paper the resolution type, the order of the full automorphism group and the orbits of this group on the parallel classes are found for each constructed resolution. the resolution properties are arranged in tables containing 8 columns. the first four of them contain the invariants which are computed for each resolution. namely: • autd is the order of the full automorphism group of the design from which the resolution was obtained • autr is the order of the full automorphism group of the resolution • t is the type of the resolution • orbr is the number of orbits of the parallel classes under the full automorphism group of the resolution the number of solutions with such invariants is presented in the last four columns. namely: • d is the number of designs from which resolutions with these invariants are obtained • dcyc is the number of those of the d designs which admit point-cyclic resolutions • r is the number of resolutions with these invariants • rcyc is the number of those of the r resolutions which are point-cyclic some comments on the properties of the constructed resolutions follow. 3.2. kts(15) all the 7 kts(15) are well known [12]. there are two cyclic sts(15) with full automorphism groups of orders 60 and 20160. one of them is resolvable (table 1) and this is the sts(15) which can be obtained from the point-line incidence in pg(3,2). there are two nonisomorphic resolutions which correspond to the two well-known doubly transitive parallelisms of pg(3,2) [10]. the transitivity is, however, on the parallel classes (orbr = 1), while these resolutions are not point-cyclic. table 1. resolutions of cyclic sts(15) autd autr t orbr d dcyc r rcyc 20160 168 3 1 1 0 2 0 3.3. kts(21) there are 7 cyclic sts(21) with full automorphism groups of orders 21, 42, 126(2), 504, 882 and 1008. three of them are resolvable with 26 nonisomorphic resolutions constructed in [16] (table 2). two of them are point-cyclic and were obtained in [15] too. there are two other resolutions which have an automorphism group of order 21 but are not point-cyclic. 204 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 table 2. resolutions of cyclic sts(21) autd autr t orbr d dcyc r rcyc 126 9 2 4 1 0 3 0 126 63 2 2 1 1 1 1 882 3 2 4 1 0 1 0 882 9 3 4 1 0 1 0 882 21 2 2 1 0 1 0 882 63 3 2 1 1 1 1 1008 1 1 10 1 0 10 0 1008 3 1 6 1 0 7 0 1008 21 1 4 1 0 1 0 – 1 – – – – 10 0 – 3 – – – – 8 0 – 9 – – – – 4 0 – 21 – – – – 2 0 – 63 – – – – 2 2 126 – – – 1 1 4 1 882 – – – 1 1 4 1 1008 – – – 1 0 18 0 – – – – 3 2 26 2 3.4. kts(27) there are 8 cyclic sts(27) with full automorphism groups of order 27. four of them are resolvable yielding 19336 nonisomorphic resolutions (table 3). none of them has a trivial automorphism group. transitive kts(27) are constructed in [5] which intersect the constructed here kts(27) in the 4 point-cyclic ones. table 3. resolutions of cyclic sts(27) autd autr t orbr d dcyc r rcyc 27 3 1 9 2 0 8 0 27 3 1 11 2 0 64 0 27 3 1 13 2 0 112 0 27 3 2 9 3 0 390 0 27 3 2 11 2 0 5088 0 27 3 2 13 2 0 13648 0 27 9 1 7 2 0 4 0 27 9 2 3 1 0 2 0 27 9 2 5 2 0 16 0 27 27 1 3 2 2 4 4 27 3 – – – – 19310 0 27 9 – – – – 22 0 27 27 – – – – 4 4 27 – – – 4 2 19336 4 205 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 3.5. kts(33) there are 84 cyclic sts(33) with full automorphism groups of orders 33(78), 66(3), 165(2) and 330(1). seventy-nine of them possess altogether 28 resolutions with automorphisms of order 11 which were constructed in [24], 141022 resolutions with automorphisms of order 3 and 703296 resolutions with automorphisms of order 2. all the designs which have resolutions with automorphisms of order 11 possess resolutions with automorphisms of order 3 too, but they do not possess point-cyclic resolutions. the present investigation shows that all resolutions of cyclic sts(33) have full automorphism groups of prime orders 11, 3 or 2 (table 4). there are no resolutions with automorphisms of order 5. table 4. resolutions of cyclic sts(33) with nontrivial automorphisms autd autr t orbr d dcyc r rcyc 33 11 1 6 13 0 18 0 33 11 2 6 8 0 10 0 33 3 2 8 76 0 127621 0 33 3 1 6 11 0 13 0 66 3 2 8 3 0 13388 0 66 2 1 11 3 0 7646 0 66 2 2 11 3 0 695650 0 – 11 – – – – 28 0 – 3 – – – – 141022 0 – 2 – – – – 703296 0 33 – – – 76 0 127662 0 66 – – – 3 0 716684 0 – – – – 79 0 844346 0 3.6. kts(39) there are 798 cyclic sts(39) with full automorphism groups of orders 39(730), 78(4), 117(55), 156(2), 234(4), 468(2) and 3042(1). there are 375 of them which have resolutions with automorphisms of order 13 with no fixed points (table 5). note that an automorphism of order 13 with fixed points (one of the 798 cyclic designs possesses such a one) cannot fix any parallel class and thus cannot be an automorphism of a resolution. the number of resolutions with automorphisms of order 13 is 2827. there are 528 point-cyclic ones among them (constructed in [15]). some of the other resolutions have a full automorphism group of order 39, but are not point-cyclic. their underlying designs possess automorphism groups of orders 117, 234 and 3042. only the resolutions of the design with an automorphism group of order 3042 are of type t3. 4. conclusion the classified kts(v) with cyclic underlying designs can be of use both directly in relevant applications, and as parts of constructions of new infinite families. the results obtained in this paper coincide with those of other authors who have constructed part of the presented resolutions. all computer results are obtained by c++ programs written by the author. files with these resolutions can be downloaded from http://www.moi.math.bas.bg/∼ svetlana. they are available online to everybody who is interested and further investigations on their properties are possible. 206 s. topalova / j. algebra comb. discrete appl. 3(3) (2016) 201–208 table 5. resolutions of cyclic sts(39) with automorphisms of order 13 autd autr t orbr d dcyc r rcyc 39 13 1 7 187 0 769 0 39 13 2 7 44 0 601 0 39 39 2 3 252 252 462 462 117 13 1 7 21 0 327 0 117 13 2 7 13 0 354 0 117 39 1 7 20 0 33 0 117 39 1 5 28 0 50 0 117 39 2 7 9 0 14 0 117 39 2 5 19 0 23 0 117 39 2 3 22 8 42 20 117 117 2 3 27 27 41 41 234 13 1 7 4 0 39 0 234 13 2 7 4 0 55 0 234 39 1 7 3 0 3 0 234 39 1 5 2 0 2 0 234 39 2 5 2 0 3 0 234 39 2 3 2 2 2 2 234 117 2 3 1 1 2 2 3042 13 3 7 1 0 3 0 3042 39 3 7 1 0 1 0 3042 117 3 3 1 1 1 1 – 13 – – – – 2148 0 – 39 – – – – 635 484 – 117 – – – – 44 44 39 – – – 322 252 1832 462 117 – – – 48 36 884 61 234 – – – 4 3 106 4 3042 – – – 1 1 5 1 – – – – 375 292 2827 528 references [1] t. baicheva, s. topalova, 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[24] v. d. tonchev, s. a. vanstone, on kirkman triple systems of order 33, discrete math. 106/107 (1992) 493–496. 208 http://www.ams.org/mathscinet-getitem?mr=0584415 http://www.ams.org/mathscinet-getitem?mr=0584415 http://dx.doi.org/10.1002/(sici)1520-6610(1997)5:3<177::aid-jcd2>3.0.co;2-c http://dx.doi.org/10.1002/(sici)1520-6610(1997)5:3<177::aid-jcd2>3.0.co;2-c http://dx.doi.org/10.1109/tcomm.2012.070912.110164 http://dx.doi.org/10.1109/tcomm.2012.070912.110164 http://dx.doi.org/10.1561/0100000044 http://dx.doi.org/10.1561/0100000044 http://dx.doi.org/10.1023/a:1008321206709 http://dx.doi.org/10.1109/tcomm.2003.816946 http://dx.doi.org/10.1109/tcomm.2003.816946 http://dx.doi.org/10.2307/1402466 http://dx.doi.org/10.2307/1402466 http://dx.doi.org/10.1006/jcta.1998.2924 http://dx.doi.org/10.1006/jcta.1998.2924 http://dx.doi.org/10.1016/s0012-365x(99)00374-x http://dx.doi.org/10.1016/s0012-365x(99)00374-x http://dx.doi.org/10.2307/2007514 http://dx.doi.org/10.2307/2007514 http://dx.doi.org/10./ http://dx.doi.org/10./ http://dx.doi.org/10.1016/s0012-365x(99)00141-7 http://dx.doi.org/10.1016/s0012-365x(99)00141-7 http://www.ams.org/mathscinet-getitem?mr=0347445 http://www.ams.org/mathscinet-getitem?mr=0347445 http://dx.doi.org/10.1109/tit.2012.2220119 http://dx.doi.org/10.1109/tit.2012.2220119 http://dx.doi.org/10.1007/s10878-007-9079-z http://dx.doi.org/10.1016/0012-365x(92)90581-y http://dx.doi.org/10.1016/0012-365x(92)90581-y introduction classification method classification results conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.26764 j. algebra comb. discrete appl. 3(1) • 7–11 received: 2 march 2015 accepted: 10 november 2015 journal of algebra combinatorics discrete structures and applications on the rank functions of h-matroids research article yoshio sano abstract: the notion of h-matroids was introduced by u. faigle and s. fujishige in 2009 as a general model for matroids and the greedy algorithm. they gave a characterization of h-matroids by the greedy algorithm. in this note, we give a characterization of some h-matroids by rank functions. 2010 msc: 05b35, 90c27 keywords: matroid, h-matroid, simplicial complex, rank function 1. introduction and main result the notion of matroids was introduced by h. whitney [10] in 1935 as an abstraction of the notion of linear independence in a vector space. many researchers have studied and extended the theory of matroids (cf. [2, 4, 5, 8, 9]). in 2009, u. faigle and s. fujishige [1] introduced the notion of h-matroids as a general model for matroids and the greedy algorithm. they gave a characterization of h-matroids by the greedy algorithm. in this note, we give a characterization of the rank functions of h-matroids that are simplicial complexes, for any family h. our main result is as follows. theorem 1.1. let e be a finite set and let ρ : 2e → z≥0 be a set function on e. let h be a family of subsets of e with ∅,e ∈ h. then, ρ is the rank function of an h-matroid (e,i) if and only if ρ is a normalized unit-increasing function satisfying the h-extension property. (e) (h-extension property) for x ∈ 2e and h ∈h with x ⊆ h, if ρ(x) = |x| < ρ(h), then there exists e ∈ h \x such that ρ(x ∪{e}) = ρ(x) + 1. moreover, if ρ is a normalized unit-increasing set function on e satisfying the h-extension property and i := {x ∈ 2e | ρ(x) = |x|}, then (e,i) is an h-matroid with rank function ρ and i is a simplicial complex. this work was supported by jsps kakenhi grant number 15k20885. yoshio sano; division of information engineering, faculty of engineering, information and systems, university of tsukuba, japan (email: sano@cs.tsukuba.ac.jp). 7 y. sano / j. algebra comb. discrete appl. 3(1) (2016) 7–11 this note is organized as follows. section 2 gives some definitions and preliminaries on h-matroids. in section 3, we give a proof of theorem 1.1 and an example which shows h-matroids that are not simplicial complexes are not characterized only by their rank functions. 2. preliminaries let e be a nonempty finite set and let 2e denote the family of all subsets of e. for any family i of subsets of e, the extreme-point operator exi : i → 2e and the co-extreme-point operator ex∗i : i → 2 e associated with i are defined as follows: exi(i) := {e ∈ i | i \{e}∈i} (i ∈i), ex∗i(i) := {e ∈ e \ i | i ∪{e}∈i} (i ∈i). for any family i ⊆ 2e, we denote the set of maximal elements of i with respect to set inclusion by max(i). let i be a nonempty family of subsets of a finite set e. the family i is called constructible if it satisfies (c) exi(i) 6= ∅ for all i ∈i \{∅}. note that (c) implies ∅ ∈ i. we call i ∈ i a base of i if ex∗i(i) = ∅. we denote by b(i) the family of bases of i, i.e., b(i) := {i ∈i | ex∗i(i) = ∅}. by definition, it holds that b(i) ⊇ max(i). a constructible family i induces a (base) rank function ρ : 2e → z≥0 via ρ(x) = maxb∈b(i)|x ∩b| = maxi∈i|x ∩ i| = maxi∈max(i)|x ∩ i|. the following is easily verified by definitions. lemma 2.1. the rank function ρ of a constructible family is normalized (i.e. ρ(∅) = 0) and satisfies the unit-increase property (ui) ρ(x) ≤ ρ(y ) ≤ ρ(x) + |y \x| for all x ⊆ y ⊆ e. remark that, by putting x = ∅, we obtain (ui)′ 0 ≤ ρ(y ) ≤ |y | for all y ⊆ e. the restriction of i to a subset a ∈ 2e is the family i(a) := {i ∈ i | i ⊆ a}. note that every restriction of a constructible family is constructible. a simplicial complex is a family i ⊆ 2e such that x ⊆ i ∈ i implies x ∈ i. we can easily check the following lemmas on simplicial complexes. lemma 2.2. a family i ⊆ 2e is a simplicial complex if and only if exi(i) = i holds for any i ∈i. proof. the lemma follows from the definitions of a simplicial complex and exi(·). lemma 2.3. let i ⊆ 2e be a simplicial complex and let x ∈ 2e. then, (a) b(i) = max(i). (b) for x ∈ 2e, x ∈i if and only if ρ(x) = |x|. (c) for h ∈ 2e, the family i(h) ⊆ 2h is a simplicial complex. 8 y. sano / j. algebra comb. discrete appl. 3(1) (2016) 7–11 proof. (a): suppose that there exists an element b ∈b(i) \ max(i). then, since b is not maximal in i, there exists i ∈i such that b ( i. for any e ∈ i \b, we have b ∪{e}∈ i since b ∪{e}⊆ i and i is a simplicial complex. therefore e ∈ ex∗i(b). but this is a contradiction to b ∈b(i). (b): if x ∈ i, then ρ(x) = maxi∈i |x ∩ i| = |x|. take x ∈ 2e with ρ(x) = |x|. then there exists i ∈i such that |x ∩i| = ρ(x) = |x|. therefore, x ⊆ i. since i is a simplicial complex, we have x ∈i. (c): take any x ∈ 2h and i ∈i(h) := {i ∈i | i ⊆ h} with x ⊆ i. since i is a simplicial complex, x ∈i. since x ⊆ h, we have x ∈i(h). we now recall the definitions of an h-independence system and an h-matroid, which were introduced by faigle and fujishige [1]. let e be a finite set and let h be a family of subsets of e with ∅,e ∈h. a constructible family i ⊆ 2e is called an h-independence system if (i) for all h ∈h, there exists i ∈i(h) such that |i| = ρ(h). an h-matroid is a pair (e,i) of the set e and an h-independence system i satisfying the following property: (m) for all h ∈h, all the bases b of i(h) have the same cardinality |b| = ρ(h). 3. proof of theorem 1.1 first, we see an example which shows that h-matroids that are not simplicial complexes are not characterized by their rank functions. example 3.1. let e = {1,2,3} and h = {∅,e}. let i1 = {∅,{2},{1,2},{2,3}}, i2 = {∅,{1},{3},{1,2},{2,3}}, i3 = {∅,{1},{2},{3},{1,2},{2,3}}. then (e,i1), (e,i2), and (e,i3) are h-matroids with the same rank function ρ : 2e → z≥0 such that ρ(∅) = 0, ρ({1}) = ρ({2}) = ρ({3}) = ρ({1,3}) = 1, and ρ({1,2}) = ρ({2,3}) = ρ({1,2,3}) = 2. therefore, we cannot distinguish h-matroids in general by their rank functions. more generally, the following holds. proposition 3.2. for any constructible families i and i′ with max(i) = max(i′), the rank function ρ′ associated with i′ coincides with the rank function ρ associated with i. proof. for any x ∈ 2e, it holds that ρ(x) = maxi∈max(i)|x ∩ i| = maxi∈max(i′)|x ∩ i| = ρ′(x) since max(i) = max(i′). in the following, we give a proof of theorem 1.1. lemma 3.3. for any constructible family, there exists a simplicial complex such that their rank functions are the same. 9 y. sano / j. algebra comb. discrete appl. 3(1) (2016) 7–11 proof. let i ⊆ 2e be a constructible family. define i′ := {x ∈ 2e | x ⊆ i for some i ∈ i}. then it is clear that i′ is a simplicial complex. obviously each y ∈ max(i) is maximal in i′, and i′ does not have new maximal members. therefore max(i) = max(i′). note that any simplicial complex is a constructible family. by proposition 3.2, the rank functions of i and i′ are the same. lemma 3.4. let ρ : 2e → z≥0 be the rank function of an h-matroid (e,i), where i is a simplicial complex. then ρ satisfies the h-extension property. proof. take x ∈ 2e and h ∈ h with x ⊆ h, and suppose that ρ(x) = |x| < ρ(h). by lemma 2.3 (c), i(h) is a simplicial complex since i is a simplicial complex. note that b(i(h)) = max(i(h)) by lemma 2.3 (a). by lemma 2.3 (b), x ∈ i. therefore x ∈ i(h), and x is not a base of i(h) by (i) and (m) since ρ(x) < ρ(h). thus there exists b ∈i such that x ( b ⊆ h and |b| = ρ(h). take any element e ∈ b \x ⊆ h \x. then x ∪{e} ∈ i since x ∪{e} ⊆ b ∈ i and i is a simplicial complex. hence it follows that ρ(x ∪{e}) = |x ∪{e}| = |x|+ 1 = ρ(x) + 1. lemma 3.5. let ρ : 2e → z≥0 be a normalized unit-increasing function satisfying the h-extension property for some family h⊆ 2e with ∅,e ∈h. put iρ := {x ∈ 2e | ρ(x) = |x|}. then (e,iρ) is an h-matroid and iρ is a simplicial complex. proof. first we show that iρ is a simplicial complex. take any i ∈ iρ \ {∅} and any e ∈ i. then we have ρ(i) = |i|. since ρ is unit-increasing, we have ρ(i) ≤ ρ(i \ {e}) + 1 and thus ρ(i \ {e}) ≥ ρ(i) − 1 = |i|− 1 = |i \ {e}|. by (ui) and ρ(∅) = 0, we also have ρ(i \ {e}) ≤ 0 + |i \ {e}| and thus ρ(i \{e}) ≤ |i \{e}|. therefore we have ρ(i \{e}) = |i \{e}| and thus i \{e}∈iρ. by lemma 2.2, iρ is a simplicial complex. hence it follows from definitions that iρ satisfies (c) and (i). now we show that iρ satisfies (m). take any h ∈ h. suppose that there exist b1,b2 ∈ b(i (h) ρ ) such that |b1| 6= |b2|. without loss of generality, we may assume that |b1| < |b2| ≤ ρ(h). note that ρ(b1) = |b1| and ρ(b2) = |b2|. then, by (e), there exists e ∈ h\b1 such that ρ(b1∪{e}) = ρ(b1)+1 = |b1|+ 1 = |b1 ∪{e}|. thus we have b1 ∪{e}∈iρ with b1 ∪{e}⊆ h. but this is a contradiction to the assumption that b1 is a base of i (h) ρ . thus iρ satisfies (m). hence (e,iρ) is an h-matroid. proof of theorem 1.1. it follows from lemmas 2.1, 3.3, 3.4, and 3.5. remark 3.6. strict cg-matroids which were introduced by s. fujishige, g. a. koshevoy, and y. sano [3] in 2007 can be considered as h-matroids (e,i) where h is an abstract convex geometry and i ⊆h. the rank functions ρ : h→ z≥0 of strict cg-matroids (e,h;i) are characterized in [6]. for more study on cg-matroids, see [7]. remark 3.7. faigle and fujishige gave a characterization of the rank functions h-matroids when h is a closure space (see [1, theorem 5.1]). acknowledgment: the author is grateful to the anonymous referees for careful reading and valuable comments. references [1] u. faigle, s. fujishige, a general model for matroids and the greedy algorithm, math. program. ser. a 119(2) (2009) 353–369. 10 http://dx.doi.org/10.1007%2fs10107-008-0213-1 http://dx.doi.org/10.1007%2fs10107-008-0213-1 y. sano / j. algebra comb. discrete appl. 3(1) (2016) 7–11 [2] s. fujishige, submodular functions and optimization, annals of discrete mathematics, elsevier, amsterdam, 2005. [3] s. fujishige, g. a. koshevoy, y. sano, matroids on convex geometries (cg-matroids), discrete math. 307(15) (2007) 1936–1950. [4] b. korte, l. lovász, r. schrader, greedoids, algorithms and combinatorics, vol. 4, springer-verlag, berlin, 1991. [5] j. oxley, matroid theory, oxford university press, oxford, 1992. [6] y. sano, rank functions of strict cg-matroids, discrete math. 38(20) (2008) 4734–4744. [7] y. sano, matroids on convex geometries: subclasses, operations, and optimization, publ. res. inst. math. sci. 47(3) (2011) 671–703. [8] a. schrijver, combinatorial optimization. polyhedra and efficiency, algorithms and combinatorics, vol. 24, springer-verlag, berlin, 2003. [9] d. j. a. welsh, matroid theory, academic press, london, 1976. [10] h. whitney, on the abstract properties of linear dependence, amer. j. math. 57(3) (1935) 509–533. 11 http://dx.doi.org/10.1016/j.disc.2006.09.037 http://dx.doi.org/10.1016/j.disc.2006.09.037 http://dx.doi.org/10.1016/j.disc.2007.08.095 http://dx.doi.org/10.2977/prims/48 http://dx.doi.org/10.2977/prims/48 http://dx.doi.org/10.2307/2371182 introduction and main result preliminaries proof of theorem 1.1 references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056581 j. algebra comb. discrete appl. 9(1) • 47–55 received: 16 may 2021 accepted: 19 august 2021 journal of algebra combinatorics discrete structures and applications protection of a network by complete secure domination research article girish v. rajasekharaiah, usha p. murthy, umesh subramanya abstract: a complete secure dominating set of a graph g is a dominating set d ⊆ v (g) with the property that for each v ∈ d, there exists f = {vj|vj ∈ n(v) ∩ (v (g) − d)}, such that for each vj ∈ f , (d −{v}) ∪{vj} is a dominating set. the minimum cardinality of any complete secure dominating set is called the complete secure domination number of g and is denoted by γcsd(g). in this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of g are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of g. 2010 msc: 05c69 keywords: domination, secure domination, complete secure domination 1. introduction the graphs considered here are undirected, finite, connected, without multiple edges or loops and without isolated vertices. as usual n and q denote the number of vertices and edges of a graph g. for basic graph theoretic notation and terminology we refer to [4]. a set of vertices d is said to dominate the graph g if for each vertex v ∈ v (g)−d , there is a vertex u ∈ d with v is adjacent to u. the minimum cardinality of any dominating set is called the domination number of g and it is denoted by γ(g). a secure dominating set x of a graph g is a dominating set with the property that each vertex u ∈ v (g)−x is adjacent to a vertex v ∈ x such that(x −{v})∪{u} is dominating set. the minimum cardinality of such a set is called the secure domination number, denoted by γs(g). the cartesian graph product g1×g2 called graph product of graphs g1 = (v1,e1) and g2 = (v2,e2) with disjoint vertex sets is the graph with the vertex set v1×v2 and u = (u1,u2) adjacent with v = (v1,v2) whenever [u1 = v1 and u2 adj v2] or [u2 = v2 and u1 adj v1]. girish v. rajasekharaiah, (corresponding author), umesh subramanya; department of science and humanities, pes university (ec campus), electronic city, bengaluru, karnataka, india (email: girishvr1@pes.edu, giridsi63@gmail.com, umeshsubbu@gmail.com). usha p. murthy; department of mathematics, siddaganga institute of technology, b.h.road, tumakuru karnataka, india (email: ushapmurhty@yahoo.com). 47 https://orcid.org/0000-0002-0036-6542 https://orcid.org/0000-0001-9855-1887 https://orcid.org/0000-0001-9672-8603 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 a friendship graph fn is the graph obtained by joining n copies of c3 with a common vertex. a vertex of degree one is called an end vertex and a vertex adjacent to an end vertex is called non-end vertex. a two-dimensional grid graph gm,n is the cartesian product pm ×pn of path graphs on m and n vertices. a stacked prism graph is the cartesian product of cm ×pn. the protection of a (simple) graph g = (v,e) involves placing a set (possibly empty) of guards at each vertex, and it is assumed that a guard can deal with a problem (called an attack) at any vertex in its closed neighborhood. various strategies(i.e., rules for guard placements) have been devised, under each of which the entire graph g may be considered protected. the minimum number of guards required for protection under each strategy is clearly of interest. the concept of secure domination was introduced by cockayne et.al. [3]. later this concept was studied extensively in [1, 2]. in social network theory, we can assume the graph nodes as message centers and its edges as transmission lines. the message is transferred through the transmission line from one message center to another. the intent of this process is to transfer the messages to all message centers with minimum number of message centers. to find all the minimum message centers, we can make use of domination and these minimum message centers are called as domination message centers. and if any of the domination message centers has an issue in sending the message, then it’s neighbor message center should act as a domination message center to complete the process. this solution is illustrated in secure domination concept, the drawback in this secure domination concept is that, it is just mentioned to select their neighbors as domination message centers, but not mentioned about which particular neighbor has to be chosen as a domination message center to achieve the objective or to complete the process. so, to overcome these problems we introduce a new concept called as complete secure domination with which one can select a neighbor as domination message centers. so that, we would be able to provide an analogy to retrieve a more efficient and robust domination message centers to solve the above mentioned issue and to transfer messages in a more faster and constructive manner. v1 e1 v2 v7 e7 e26 e27 e3 e4 v5 e8 v6 e6 v8 v14 v3 e5 v4 e28 e11 e13 e14 v15 v16 e9 e9 v9 v13 e15 e19 e20 e10 e12 e16 v17 v18 v10 e24 e17 e18 e21 v11 e25 v12 v20 v19 e22 e23 v21 figure 1. example for complete secure domination minimum dominating set d = {v5,v4,v11,v13,v18,v17,v21}. minimum secure dominating set c = {v6,v2,v3,v9,v13,v18,v17,v21}. minimum complete secure dominating set f = {v6,v2,v3,v9,v13,v18,v17,v21,v15}. |d| = 7, |c| = 8, |f| = 9 48 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 a complete secure dominating set of a graph g is a dominating set d ⊆ v (g) with the property that for each v ∈ d, there exists f = {vj|vj ∈ n(v)∩(v (g)−d)}, such that for each vj ∈ f, (d−{v})∪{vj} is a dominating set. the minimum cardinality of any complete secure dominating set is called the complete secure domination number of g and is denoted by γcsd(g). 2. complete secure domination for standard graphs theorem 2.1. for any path pn, γcsd(pn) = dn2e. theorem 2.2. for any wheel graph wn, γcsd(wn) = dn3e. theorem 2.3. for any cycle cn, γcsd(cn) = dn2e. theorem 2.4. for any friendship graph fn with n vertices, γcsd(fn) = dn3e. 3. main results theorem 3.1. for any graph g = p2 ×pn, γcsd(g) = n,n ≥ 2. proof. let v (p2×pn) = {(ui,vs),s = 1,2,3, . . . ,n}i=2i=1 be the vertices of first and second row, respectively. we consider the following cases. case 1. suppose n is even. let a = {(u1,vs),s = 2p,1 ≤ p ≤ n2} and b = {(u2,vs),s = 2p − 1,1 ≤ p ≤ n 2 } with |a| = n 2 and |b| = n 2 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n2 + n 2 = n. case 2. suppose n is odd. let a = {(u1,vs),s = 2p,1 ≤ p ≤ n−12 } and b = {(u2,vs),s = 2p − 1,1 ≤ p ≤ n+1 2 } with |a| = n−1 2 and |b| = n+1 2 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n−12 + n+1 2 = n. the proof is complete. theorem 3.2. for any graph g = p3 ×pn,n ≥ 2, γcsd(g) =   3n 2 n is even 3n−1 2 n is odd. proof. let v (p3 ×pn) = {(ui,vs),s = 1,2,3, . . . ,n}i=3i=1 be the vertices of first, second and thrid row, respectively. we consider the following cases. case 1. suppose n is even. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n2 ,r = 1,3} and b = {(ur,vs),s = 2p− 1,1 ≤ p ≤ n 2 ,r = 2} with |a| = 2 ∗ n 2 and |b| = n 2 . let d = a ∪ b, every neighborhood vertex of v (g) − d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n + n2 = 3n 2 . 49 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 case 2. suppose n is odd. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n−12 ,r = 1,3} and b = {(ur,vs),s = 2p− 1,1 ≤ p ≤ n+1 2 ,r = 2} with |a| = 2(n−1 2 ) and |b| = n+1 2 . let d = a∪b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d−{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n−1 + n+12 = 3n−1 2 . the proof is complete. theorem 3.3. for any graph g = p4 ×pn, γcsd(g) = 2n,n ≥ 2. proof. let v (p4×pn) = {(ui,vs),s = 1,2,3, . . . ,n}i=4i=1 be the vertices of first, second, thrid and fourth row, respectively. we consider the following cases. case 1. suppose n is even. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n2 ,r = 1,3} and b = {(ur,vs),s = 2p−1,1 ≤ p ≤ n 2 ,r = 2,4} with |a| = 2∗ n 2 and |b| = 2∗ n 2 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n + n = 2n. case 2. suppose n is odd. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n−12 ,r = 1,3} and b = {(ur,vs),s = 2p− 1,1 ≤ p ≤ n+1 2 ,r = 2,4} with |a| = 2(n−1 2 ) and |b| = 2(n+1 2 ). let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d−{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = n−1 + n + 1 = 2n. the proof is complete. theorem 3.4. for any graph g = p5 ×pn,n ≥ 2, γcsd(g) =   5n 2 n is even 5n−1 2 n is odd. proof. let v (p5 ×pn) = {(ui,vs),s = 1,2,3, . . . ,n}i=5i=1 be the vertices of first, second, third, fourth and fifth row, respectively. we consider the following cases. case 1. suppose n is even. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n2 ,r = 1,3,5} and b = {(ur,vs),s = 2p− 1,1 ≤ p ≤ n 2 ,r = 2,4} with |a| = 3∗n 2 and |b| = 2∗n 2 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d−{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+|b| = 3n2 +n = 5n 2 . case 2. suppose n is odd. let a = {(ur,vs),s = 2p,1 ≤ p ≤ n−12 ,r = 1,3,5} and b = {(ur,vs),s = 2p − 1,1 ≤ p ≤ n+1 2 ,r = 2,4} with |a| = 3(n−1 2 ) and |b| = 2(n+1 2 ). the set d = a ∪ b is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = 3(n−12 ) + n + 1 = 5n−1 2 . the proof is complete. 50 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 theorem 3.5. for any graph g = pm ×pn,m,n ≥ 2, γcsd(g) =   mn 2 m is even mn 2 m is odd, n is even mn−1 2 m is odd, n is odd. proof. let v (pm × pn) = {(ur,vs),s = 1,2,3, . . . ,n}r=mr=1 be the vertices of first, second, third, . . . , mth row, respectively. we consider the following cases. case 1. suppose m is even and n is even. let a = {(ur,vs),r = 2p− 1,1 ≤ p ≤ m2 ,s = 2q,1 ≤ q ≤ n 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m 2 ,s = 2q − 1,1 ≤ q ≤ n 2 } with |a| = mn 4 and |b| = mn 4 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a| + |b| = mn 4 + mn 4 = mn 2 . case 2. suppose m is even and n is odd. let a = {(ur,vs),r = 2p − 1,1 ≤ p ≤ m2 ,s = 2q,1 ≤ q ≤ n−1 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m 2 ,s = 2q − 1,1 ≤ q ≤ n+1 2 } with |a| = m(n−1) 4 and |b| = m(n+1) 4 . let d = a ∪ b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = m(n−1) 4 + m(n+1) 4 = mn 2 . case 3. suppose m is odd and n is even. let a = {(ur,vs),r = 2p − 1,1 ≤ p ≤ m+12 ,s = 2q,1 ≤ q ≤ n 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m−1 2 ,s = 2q − 1,1 ≤ q ≤ n 2 } with |a| = (m+1)n 4 and |b| = (m−1)n 4 . let d = a ∪ b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = (m+1)n 4 + (m−1)n 4 = mn 2 . case 4. suppose m is odd and n is odd. let a = {(ur,vs),r = 2p− 1,1 ≤ p ≤ m+12 ,s = 2q,1 ≤ q ≤ n−1 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m−1 2 ,s = 2q−1,1 ≤ q ≤ n+1 2 } with |a| = (m+1)(n−1) 4 and |b| = (m−1)(n+1) 4 . let d = a∪b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = (m+1)(n−1) 4 + (m−1)(n+1) 4 = mn−1 2 . the proof is complete. theorem 3.6. for any graph, g = c3 ×pn, γcsd(g) = n,n ≥ 3. proof. let v (c3 ×cn) = {(ui,vs),s = 1,2,3, . . . ,n}i=3i=1 be the vertices of first, second and third row, respectively. the set a = {(u1,vs),s = 1,2,3, . . . ,n} with |a| = n. since a is the γ-set of g and therefore a is the γcsd-set of g. hence γcsd(g) = n. theorem 3.7. for any graph, g = c4 ×cn, γcsd(g) = 2n,n ≥ 3. 51 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 proof. let v (c3 × cn) = {(ui,vs),s = 1,2,3, . . . ,n}i=4r=1 be the vertices of first, second, third and fourth row, respectively. the set d = {(u1,vs),(u3,vs),s = 1,2,3, . . . ,n} with |d| = 2n. since every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d−{vi})∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = 2n. theorem 3.8. for any graph g = cm ×pn,m 6= 3,n ≥ 3, γcsd(g) =   mn 2 m is even or odd and n is even mn 2 m is even, n is odd. mn−1 2 m is odd, n is odd. proof. let v (cm × cn) = {(ur,us),s = 1,2,3, . . . ,n}r=mr=1 be the vertices of first, second, third, . . . mth row, respectively. we consider the following cases. case 1. suppose m is even and n is even. let a = {(ur,vs),r = 2p− 1,1 ≤ p ≤ m2 ,s = 2q,1 ≤ q ≤ n 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m 2 ,s = 2q − 1,1 ≤ q ≤ n 2 } with |a| = mn 4 and |b| = mn 4 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪{vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a| + |b| = mn 4 + mn 4 = mn 2 . case 2. suppose m is even and n is odd. let a = {(ur,vs),r = 2p − 1,1 ≤ p ≤ m2 ,s = 2q,1 ≤ q ≤ n−1 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m 2 ,s = 2q − 1,1 ≤ q ≤ n+1 2 } with |a| = m(n−1) 4 and |b| = m(n+1) 4 . let d = a∪b, every neighborhood vertex of v (g)−d is in d, d is the complete secure dominating set of pm ×pn. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = m(n−1) 4 + m(n+1) 4 = mn 2 . case 3. suppose m is odd and n is even. let a = {(ur,vs),r = 2p − 1,1 ≤ p ≤ m+12 ,s = 2q,1 ≤ q ≤ n 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m−1 2 ,s = 2q − 1,1 ≤ q ≤ n 2 } with |a| = (m+1)n 4 and |b| = (m−1)n 4 . let d = a ∪ b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (g), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = (m+1)n 4 + (m−1)n 4 = mn 2 . case 4. suppose m is odd and n is odd. let a = {(ur,vs),r = 2p− 1,1 ≤ p ≤ m+12 ,s = 2q,1 ≤ q ≤ n−1 2 } and b = {(ur,vs),r = 2p,1 ≤ p ≤ m−1 2 ,s = 2q−1,1 ≤ q ≤ n+1 2 } with |a| = (m+1)(n−1) 4 and |b| = (m−1)(n+1) 4 . let d = a∪b, every neighborhood vertex of v (g) −d is in d, d is the complete secure dominating set of g. hence γcsd(g) ≤ |d|. if γcsd(g) < |d|, then there exits at least one vertex say vj ∈ v (pm ×pn), (d −{vi}) ∪ {vj},vi ∈ d is not a dominating set. therefore d is the γcsd-set of g. hence γcsd(g) = |d| = |a|+ |b| = (m+1)(n−1) 4 + (m−1)(n+1) 4 = mn−1 2 . the proof is complete. theorem 3.9. for any graph g, if every non-end vertex is adjacent with an end-vertex then, γcsd(g) = p, where p is the number of end-vertices of g. 52 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 proof. let a = {vi ∈ v (g)|d(vi) = 1}, b = {vj ∈ v (g) − a|vj ∈ n(vi)} and c = (a −{vr}) ∪ {vj},vr ∈ a,vj ∈ b ∩ n(vi). let d = a or c. now for each v ∈ d, there exists f = {vj|vj ∈ n(v) ∩ v (g) − d}, such that for each vj ∈ f, (d −{v}) ∪{vj} is a dominating set. hence d is a complete secure dominating set of g. now we show that there exists no other minimum complete secure dominating set other than d. if suppose f 6= d is a minimum complete secure domination set of g with |f| < |d|, then either f is not a dominating set or for atleast one vertex v ∈ f there exists vj ∈ n(v) such that (f −{v})∪{vj} is not a dominating set of g. hence γcsd(g) = |d| = p. corollary 3.10. for any graph g, if every non-end vertex is adjacent with an end-vertex then, γcsd(g) = γns(g), where γns(g) is the nonsplit domination number of g. proof. if every non-end vertex is adjacent with an end-vertex then, γns(g) = p, p is the number of end vertices of g, then by theorem 3.9, the result follows. theorem 3.11. for any tree t with n vertices, γcsd(t) ≤ p+dn−2p2 e,p is the number of end vertices of t. proof. let a = {vi ∈ v (t)|d(vi) = 1} and b = {vj ∈ v (t) − a|vj ∈ n(vi)}. we consider the following cases. case 1. if every non-end vertex of t is adjacent to an end-vertex then by theorem 3.9, γcsd(t) = p. case 2. if atleast one vertex say vj ∈ v (g), vj is not adjacent to an end-vertex say vi ∈ a, then the graph t − (a∪b) will be a tree with n− 2p vertices and partition the vertex set of t − (a∪b) into two disjoint sets say r and s such that vi ∈ r and vj ∈ s,vi ∈ n(vj) and |r| ≥ |s|. now consider the set d = a ∪ r, then for each v ∈ d, there exists f = {vj|vj ∈ n(v) ∩ v (t) − d}, such that for each vj ∈ f, (d −{v}) ∪{vj} is a dominating set. therefore f is the complete secure dominating set of g. hence γcsd(t) ≤ f = p +dn−2p2 e. theorem 3.12. for any graph g = k2 ×k1,n−1, γcsd(g) = n,n ≥ 3. proof. let v (k2) = {u1,u2} and v (k1,n−1) = {v1,v2,v3, . . . ,vn} with d(v1) = n − 1. let v (g) = v (k2 × k1,n−1) = a ∪ b, a = {(ui,vj) ∈ v (g)/d((ui,vj)) = n}, b = {(ur,vs) ∈ v (g) − a} with |a| = 2, |b| = 2(n−1). partition the vertex of b into disjoint vertex sets, b = h∪r such that (ui,vj) ∈ n((ur,vs)),(ui,vj) ∈ h,(ur,vs) ∈ r with |h| = |r| = n−1. any minimum dominating set d of g has to contain two vertices say f = {(u1,v1),(u2,v1)} where {(u1,v1),(u2,v2)} ∈ a. suppose d is a complete secure dominating set, then the induced graph m = 〈v (g)−({(u1,v1)} or {(u2,v1)})〉 will contains n−1 support vertices and by theorem 3.9, γcsd(m) = n−1 and if γcsd(g) = n−1, then there exists at least one vertex say (ui,vj) ∈ v (g) which is not dominated by (d−{(um,vr)}∪(ui,vj),(um,vr) ∈ d∪n((ui,vj)). therefore γcsd > n − 1. since each vertex in b is of degree 2 and belongs to neighborhood of d, therefore the dominating set has to contain the vertices of h or r together with any vertex of a. now consider the set k = h ∪{(u1,v1)}, for each vertex (ui,vj) ∈ k, there exists f = {(ur,vs)/(ur,vs) ∈ n((ui,vj)) ∩ (v (g) −d)}, such that for each (ur,vs) ∈ f, (d −{(ui,vj)}) ∪{(ur,vs)} is a dominating set. hence k is a complete secure dominating set, γcsd = |k| = n. definition 3.13. planar honeycomb graphs are the graphs obtained by connecting some equal regular hexagons such that any two adjacent hexagons have one edge in common. the planar honeycomb lattice is also called benzoid and it is denoted by bn. theorem 3.14. for any planar honeycomb graph bn with n vertices, n ≥ 6, γcsd(bn) = dn2e. proof. let v (bn) = {u1i,u2i,u3i, . . . ,upi, i = 1,2,3, . . . ,q} denotes the first, second, third, fourth,. . . ,pthrow, respectively. we consider the following cases. case 1. when n = 6. in this case the graph b6 = c6. the result follows from theorem 2(iii). 53 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 figure 2. example for honeycomb graph case 2. when n = 10. then the graph b10 contains two cycles with one edge in common, which is isomorphic to the graph h, where v (h) = v (c6) ∪v (p4) and e(h) = e(c6) ∪e(p4) ∪e1 ∪e2, where e1 and e2 joins (v1,u1) and (v2,u2), where v1,v2 ∈ v (c6),v1 ∈ n(v2) and u1 and u2 are the end vetices of p4. hence γcsd = γcsd(c6) + γcsd(p4) = 3 + 2 = dn2e. case 3. when n ≥ 10. in this case. the planar honey comb graph can be obtained by adding an edge that connects the end vertices of m copies of either p1 or p2 or p3 or p4 with an adjacent vertices of b10. by using case 2 and theorem 3.9, γcsd = γcsd(b10) + mγcsd(p1orp2orp3orp4) =5 + dn−10 2 e. =5 + dn 2 e−5. =dn 2 e. 4. application whenever we transfer a message from one mobile device which is in different signal range, to another mobile device which is in some other different signal range, then sometimes can be a loss of data or the message may be delivered after a long time. these are mainly due to the unstructured or unorganized way of locating message service systems and unsecured network. these problems can be solved using our complete secure domination. through secure domination, we are providing the least or minimum number of message centers with which the entire block or chain of message centers can be covered and also secured. in this way, we propose the complete secure domination with minimum number of message centers to overcome the above mentioned issues. 54 g. v. rajasekharaiah et.al. / j. algebra comb. discrete appl. 9(1) (2022) 47–55 references [1] m. anderson, c. barrientos, r. brigham, j. carrington, r. vitray, j. yellen, maximum demand graphs for eternal security, j. combin. math. combin. comput. 61 (2007) 111–128. [2] s. benecke, higher order domination of graphs, master thesis, university of stellenbosch (2004). [3] s. benecke, e. j. cockayne, c. m. mynhardt, secure total domination in graphs, utilitas math. 74 (2007) 247–259. [4] f. harary, graph theory, addison-wesely, reading mass, 1st edition (1969). 55 https://mathscinet.ams.org/mathscinet-getitem?mr=2322206 https://mathscinet.ams.org/mathscinet-getitem?mr=2322206 https://mathscinet.ams.org/mathscinet-getitem?mr=2364004 https://mathscinet.ams.org/mathscinet-getitem?mr=2364004 https://doi.org/10.1201/9780429493768 introduction complete secure domination for standard graphs main results application references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.67265 j. algebra comb. discrete appl. 3(3) • 209–216 received: 30 october 2015 accepted: 7 april 2016 journal of algebra combinatorics discrete structures and applications weak isometries of hamming spaces research article ryan bruner, stefaan de winter abstract: consider any permutation of the elements of a (finite) metric space that preserves a specific distance p. when is such a permutation automatically an isometry of the metric space? in this note we study this problem for the hamming spaces h(n, q) both from a linear algebraic and combinatorial point of view. we obtain some sufficient conditions for the question to have an affirmative answer, as well as pose some interesting open problems. 2010 msc: 05c12, 05c50 keywords: hamming space, weak isometry, eigenvalue collapsing 1. introduction distance preserving permutations of metric spaces, better known as isometries, play a prominent role throughout mathematics. their weaker counterparts, not surprisingly called weak isometries, are permutations of a metric space that preserve only a prescribed subset of distances. in certain cases it is known that a weak isometry necessarily has to be an isometry. a famous example is the beckman-quarles theorem that states that every mapping on the real euclidean space rn, n > 1, that preserves distance 1 has to be an isometry [2]. beckman-quarles type results for certain finite metric spaces related to geometries and buildings were obtained by govaert and van maldeghem in [6, 7], and by abramenko and van maldeghem in [1]. in most cases the conclusion is that weak isometries are automatically isometries, however some interesting counter examples do exist. in particular, there exist permutations of the point set of the so-called split cayley hexagon that preserve distance 6 between points, but do not arise from automorphisms of the hexagon, and hence do not give rise to isometries of the related metric space (see section 3 of [7]). an important class of finite metric spaces are provided by the distance regular graphs, and an important subclass of these are the hamming spaces h(n,q). these are the spaces {0, 1, 2, . . . ,q − 1}n equipped with the hamming distance. in [8] necessary and sufficient conditions were derived for a weak isometry of h(n, 2) to be an isometry. in [5] all weak isometries of h(n, 2) were classified. in this note we will focus on weak isometries of h(n,q), q > 2. ryan bruner, stefaan de winter (corresponding author); department of mathematical sciences, michigan technological university, usa (email: rwbruner@mtu.edu, sgdewint@mtu.edu). 209 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 2. definitions and preliminary observations let γ be any finite connected simple graph without loops. then γ can in an obvious way be viewed as a metric space in which the maximal distance equals the diameter, say d, of the graph. now let φ be any permutation of the vertex set of γ that maps vertices at distance p to vertices at distance p, for some specific 1 ≤ p ≤ d. we will call such permutation a p-isometry of γ. one can easily generalize this definition to that of a p-isometry, where p ⊂{1, 2, . . . ,d}. these are exactly the permutations of γ that preserve all distances p ∈ p . we are interested in the following question: “under which conditions is a p-isometry automatically an isometry (= automorphism) of γ?”. in general this question will be very hard to answer, and if one does not assume some extra conditions on γ, there might exist many p-isometries that are not isometries. assume for example that the distance p-graph of γ (this is the graph in which two vertices are adjacent iff they are at distance p in γ) is not connected, and that one of the connected components, say c1, has a non-trivial automorphism, say σ. then one could consider the map τ on γ defined by τ(v) = σ(v) for v ∈ c1, and τ(v) = v otherwise. in many cases τ will then not induce an isometry of γ. an example of this idea is the construction of so-called even isometries of the boolean cube (see theorem 3 in [5]). it is however not true that all proper p-isometries arise in this way (see for example theorem 6 and 8 in [5]). nevertheless, the following restrictions on γ seem to be quite natural for our problem: • the distance p-graph of γ is connected; • γ admits a primitive automorphism group. even under these conditions it seems to be hopeless to find a general answer to our problem, and restricting to specific interesting classes of graphs seems to be the natural thing to do (and is what has been done so far, see for example the references mentioned in sections 1). a very interesting class of graphs in this respect is the class of primitive distance regular graphs, as these graphs naturally fulfill both conditions mentioned above (for all p). however, even this class seems to be too general to allow a uniform answer to our question as there are both examples where a p-isometry is automatically an isometry, and cases where the group of p-isometries is larger than the automorphism group of the original graph. a small example in the latter class is given by the 3-isometries of the odd graph o4 on 35 vertices. in this case the group of 3-isometries is 8 times larger than the automorphism group of the original graph (this example was provided to the authors by j. koolen and j. williford, and can easily be confirmed with sage or gap). in this paper we will focus on the class of hamming graphs h(n,q) with q > 2. 3. linear algebraic approach when trying to show that every p-isometry of h(n,q), q > 2, is an isometry, a first natural approach is to make use of the bose-mesner algebra. the advantage of this approach is that it can be applied to any distance regular graph. for details on distance regular graphs and the bose-mesner algebra we refer to [3]. so for now, let γ be a distance regular graph of diameter n with intersection numbers ai,bi and ci, i = 0, 1, . . . ,n. furthermore, let γi denote the distance i-graph of γ, that is, the graph on the same vertices as γ in which two vertices are adjacent if they are at distance i in γ. let ai denote the adjacency matrix of γi. now assume φ is a p-isometry of γ. this is equivalent with saying that φ is an automorphism of γp, and hence, if we let p denote the permutation matrix corresponding to φ, with papp −1 = ap. theorem 3.1. every p-isometry of the distance regular graph γ of diameter n is a graph automorphism (of γ) if the degree of the minimal polynomial of ap equals n + 1. proof. let φ be a p-isometry of γ. as above, let ai denote the adjacency matrix of γi, set a = a1, and let p denote the permutation matrix corresponding to φ. we want to show that pap−1 = a under 210 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 the conditions of the theorem. a natural way to do this will be to prove that a can be written as a polynomial in ap. it is well known, see for example [3], that a0 = i,a1 = a, aai = ci+1ai+1 + aiai + bi−1ai−1 (i = 0, . . . ,n) a0 + a1 + . . . + an = j, where j is the all one matrix, a−1 = an+1 = 0, and b−1 and cn+1 are unspecified. from this it follows right away that ap is a polynomial of degree p in a. however, we want to obtain the converse. recall that a = {α0i + α1a + . . . + αnan : αi ∈ c}, the so-called bose-mesner algebra, is a commutative algebra. obviously ap and all its powers belong to a. we can write ap = α01i + α11a + α21a 2 + . . . + αn1a n a2p = α02i + α12a + α22a 2 + . . . + αn2a n ... anp = α0ni + α1na + α2na 2 + . . . + αnna n as we want to write a as a polynomial in ap, we may see the above as a system of n linear equations in the “unknowns” a,a2, . . . ,an. it now is obvious we will be able to solve for a (in terms of ap,a 2 p, . . . ,a n p ) if and only if ap,a 2 p, . . . ,a n p are linearly independent, that is, if and only if the degree of the minimal polynomial of ap is at least n + 1. on the other hand, as ap ∈a the degree of this minimal polynomial is at most n + 1. the most obvious way to guarantee that the minimal degree of ap will be n + 1 is that ap has n + 1 distinct eigenvalues. we call the phenomenon of ap having fewer than n + 1 distinct eigenvalues eigenvalue collapsing. from here on let γ be the hamming graph h(n,q). it is well known that in this case the intersection numbers are given by ai = i(q − 2), bi = (n− i)(q − 1), and ci = i, for i = 0, 1, 2, . . . ,n. let fi denote the polynomial of degree i such that ai = fi(a). from the above it follows that the fi are recursively defined by f−1(x) = 0, f0(x) = 1, f1(x) = x, ci+1fi+1(x) = (x−ai)fi(x) − bi−1fi−1(x) (i = 0, . . . ,n). we know that the n + 1 eigenvalues of a are λi = (q − 1)n − qi for i = 0, 1, . . . ,n. hence the eigenvalues of ap are γi = fp(λi) for i = 0, 1, . . . ,n. if all γi are distinct, then indeed by the above every p-isometry would be an isometry. a straightforward check in mathematica yielded the results presented in tables 1 through 4 for q = 3, 4, 5 and 6. in these tables a value of “true” indicates that ap has n + 1 distinct eigenvalues, 211 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 whereas a value of “false” indicates that eigenvalue collapsing occured. we choose to present the tables for q = 3, 4, 5 and 6 as these represent two primes, a prime power and a composite number. our values of n and p run up to 12, although one could easily compute many more. however, when we did so we did not uncover any interesting patterns. these results seem to indicate that in “most” cases ap has indeed n + 1 distinct eigenvalues, but that eigenvalue collapsing occurs for seemingly random triples (q,n,p). at this point we were unable to derive any meaningful theoretical conclusions on when eigenvalue collapsing for h(n,q) will occur. the only exception is in the case p = n where the recent results of brouwer and fiol [4] imply that no eigenvalue collapsing will occur. in their paper brouwer and fiol study eigenvalue collapsing in the case where p equals the diameter for general distance regular graphs. in the next section we will discuss the p = n case for h(n,q) in a combinatorial way. at this point one may wonder whether eigenvalue collapsing is equivalent with the existence of p-isometries that are not isometries. however, the next section will show this is not the case. table 1. q = 3 q=3 p = 2 3 4 5 6 7 8 9 10 11 12 n = 3 true true 4 false false true 5 true false true true 6 true true true true true 7 false false true false false true 8 true true true true true true true 9 true true true true true true true true 10 false false false false false false false false true 11 true false true true false true false false true true 12 true true true true true true true true true true true table 2. q = 4 q=4 p = 2 3 4 5 6 7 8 9 10 11 12 n = 3 false true 4 true true true 5 false false false true 6 true true true true true 7 false true false true false true 8 true true true true true true true 9 false false true false true false false true 10 true true true true true true true true true 11 false true false true false true false true false true 12 true true true true true true true true true true true table 3. q = 5 q=5 p = 2 3 4 5 6 7 8 9 10 11 12 n = 3 true true 4 true true true 5 true true true true 6 false false true false true 7 true true true true true true 8 true true true true true true true 9 true true true true true true true true 10 true true true true true true true true true 11 false false true false true true true false false true 12 true true true true true true true true true true true 212 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 table 4. q = 6 q=6 p = 2 3 4 5 6 7 8 9 10 11 12 n = 3 true true 4 false true true 5 true false true true 6 true true true true true 7 false false true true false true 8 true true true true true true true 9 true true true true true true true true 10 false false false false true false false true true 11 true false false true false true true true true true 12 true true true true true true true true true true true 4. combinatorial approach in this section we will present a proof that shows that in at least half of all possible cases every p-isometry of h(n,q), q > 2 is an isometry. our main result is the following. theorem 4.1. let φ be a p-isometry of h(n,q), q > 2. if 2p < n or p = n then φ is an isometry of h(n,q). we will proceed through a series of lemmas, but first need a few definitions. we will call the elements of h(n,q) words. the weight of a word is simply its number of non-zero positions, and the support of a word is the set of nonzero positions in that word. the layer lk of weight k of h(n,q) is the set of all words of weight k in h(n,q). for a < b, the cloud c(a,b) of h(n,q) is the union of layers ∪bi=ali. we will denote the hamming distance between two words a and b by d(a,b). we will let 0 denote the unique word of weight zero. finally, addition and subtraction in h(n,q) are defined component wise modulo q. throughout this section we always assume q > 2. lemma 4.2. let φ be a p-isometry of h(n,q) fixing 0 and let 2p < n. then, for positive integers k, the layers lkp, kp ≤ n, and the clouds c((k − 1)p + 1, kp− 1), kp ≤ n, are preserved set wise. proof. we start by noticing that only words of weight p are at distance p from 0. let w be in lp. then p = d(0,w) = d(φ(0),φ(w)) = d(0,φ(w)). thus φ(w) ∈ lp, and lp is preserved as a set under φ. then c(1, p−1)∪c(p + 1, 2p−1)∪l2p are the only remaining words of h(n,q) that are at distance p from some word in lp. so c(1, p− 1) ∪c(p + 1, 2p− 1) ∪l2p is preserved as a set under φ. we know c(2p + 1, n) is not empty since n > 2p. we also know c(2p + 1, n) must be preserved set wise under φ as its complement is preserved. because words in c(p + 1, 2p− 1) and l2p are at distance p from some words in c(2p + 1, n) while words in c(1, p− 1) are not, we see that c(1,p− 1) is preserved set wise as well as c(p + 1, 2p− 1) ∪l2p. next notice that words in l2p can never be at distance p from words in c(1, p− 1) while every word in c(p + 1, 2p− 1) is. therefore both c(p + 1, 2p− 1) and l2p are preserved as sets. the result now easily follows by induction. lemma 4.3. let 2p < n. two words of weight p have disjoint support if and only if there exists a unique word of weight 2p at distance p from both words. proof. this is straightforward. lemma 4.4. let φ be a p-isometry of h(n,q) fixing 0 and let 2p < n. then φ maps words of weight p with disjoint support to words of weight p with disjoint support. 213 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 proof. let α and β be the two words of weight p with disjoint support. let γ be the unique word of weight 2p at distance p from both α and β. by lemma 4.2 φ maps α and β to words of weight p and γ to a word of weight 2p. furthermore, as φ is a p-isometry φ(γ) must be the unique word of weight 2p at distance p from φ(α) and φ(β). hence, by the previous lemma, φ(α) and φ(β) have disjoint support. lemma 4.5. assume 2p < n. a word w in c(1,p− 1) ∪c(p + 1, 2p− 1) has even weight if and only if there are two words x1 and x2 of weight p with disjoint support such that both x1 and x2 are at distance p from w. proof. we first show that for a word w of even weight in the given clouds there are two words x1 and x2 satisfying the conditions of the lemma. let w be a word of weight 2k in c(1,p−1)∪c(p + 1, 2p−1). then we can construct two words x1 and x2 as desired using the simple construction shown below: 2k w : ︷ ︸︸ ︷ ∗·· ·∗ ∗· · ·∗ 0 · · · · · · · 0 0 · · · · · · · 0 x1 : ∗·· ·∗ 0 · · ·0 ∗·· · · · · · ∗ 0 · · · · · · · 0 x2 : 0 · · ·0︸ ︷︷ ︸ ∗·· ·∗︸ ︷︷ ︸ k k 0 · · · · · · · 0︸ ︷︷ ︸ ∗·· · · · · · ∗︸ ︷︷ ︸ p−k p−k next let w be a word of weight k in c(1,p− 1) ∪c(p + 1, 2p− 1) admitting two words x1 and x2 satisfying the conditions of the lemma. let a (respectively a′) be the number of positions in the support of w that are not in the support of x1 (respectively x2). let b (respectively b′) be the number of positions where the support of w and x1 (respectively x2) overlap but have different entries. let c (respectively c′) be the number of positions where the support of w and x1 (respectively x2) overlap and have equal entries. finally let d (respectively d′) be the number of positions in the support of x1 (respectively x2) that are not in the support of w. the following pictographic representation of this situation allows to deduce a few key relationships: k w : ︷ ︸︸ ︷ ∗ · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ∗ 0 · · · · · · · · · · · · · · · · · · · 0 x1 : 0 · · · · · · · · · · · · · ·0︸ ︷︷ ︸ + · · ·+︸ ︷︷ ︸ ∗·· ·∗︸ ︷︷ ︸ a b c ∗·· · · · · · ∗︸ ︷︷ ︸ 0 · · · · · · · 0 d x2 : + · · ·+︸ ︷︷ ︸ ∗·· ·∗︸ ︷︷ ︸ 0 · · · · · · · · · · · · · ·0︸ ︷︷ ︸ b′ c′ a′ 0 · · · · · · · 0 ∗·· · · · · · ∗︸ ︷︷ ︸ d′ a + b + c = k, a′ + b′ + c′ = k, b + c + d = p, b′ + c′ + d′ = p, a + b + d = p, a′ + b′ + d′ = p, b′ + c′ ≤ a, b + c ≤ a′. from b + c + d = p = a + b + d we deduce that a = c, and similarly a′ = c′. then a + b = b + c ≤ a′ ≤ a′ + b′ = b′ + c′ ≤ a. it follows that b = 0. hence k = a + b + c = 2a, and w has even weight. 214 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 lemma 4.6. let 2p < n, and let φ be a p-isometry of h(n,q) fixing 0. then words of even (respectively odd) weight in c(1,p− 1) and c(p + 1, 2p− 1) are mapped to words of even (respectively odd) weight in c(1,p− 1) and c(p + 1, 2p− 1). proof. this follows immediately by combining lemmas 4.2, 4.4 and 4.5. lemma 4.7. let 2p < n, and let φ be a p-isometry of h(n,q) fixing 0. then φ preserves the layer l1. proof. first assume p is odd. set p = 2k + 1. notice that the only words at distance p from a word of weight 1 in c(p + 1, 2p − 1) are words of weight p + 1 = 2k + 2. so all words at distance p from a word of weight 1 in c(p + 1, 2p−1) have even weight. now let us look at words in c(2,p−1). all these words are at distance p from both words of even weight and words of odd weight in c(p + 1, 2p− 1). by lemma 4.6 the parity of the words under consideration must be preserved by φ. it follows that layer l1 is stabilized by φ. in the same way, interchanging the roles of even and odd, we see that when p is even, layer l1 is preserved. lemma 4.8. let p = n, and let φ be a p-isometry of h(n,q) fixing 0. then φ preserves the layer l1. proof. let w be a word of weight x. then the number of words of weight n at distance n from w equals f(x) = (q−2)x(q−1)n−x. so the number of words of weight n at distance n from a word of weight 1 equals (q − 2)1(q − 1)n−1. as f(x) is a strictly decreasing function it follows that words of weight 1 are the only words that have (q − 2)1(q − 1)n−1 words of weight n at distance n. now we know that n-isometries that fix 0 preserve ln. hence, if there are k words of weight n at distance n from a given word w, there should also be k words of weight n at distance n from φ(w). it follows that φ stabilizes layer l1. we are now ready to prove theorem 4.1. proof. let a and c be words such that d(a,c) = 1. let φ(a) = b. then we construct the p-isometry ψ := τ−b ◦φ◦ τa where τ−b and τa are isometries defined by τa(w) = w + a and τ−b(w) = w − b for all words w. we note ψ(0) = τ−b ◦φ◦ τa(0) = τ−b ◦φ(a) = τ−b(b) = 0. so ψ is a p-isometry that fixes 0. we obtain d(φ(a),φ(c)) =d(τ−b ◦φ(a),τ−b ◦φ(c)) =d(τ−b ◦φ◦ τa(0),τ−b ◦φ◦ τa(c−a)) =d(ψ(0),ψ(c−a)) = d(0,ψ(c−a)) now d(c,a) = 1 implies that d(0,c−a) = 1, and by lemmas 4.7 and 4.8, ψ preserves layer l1. hence d(0,ψ(c−a)) = 1, implying d(φ(a),φ(c)) = 1. this proves that φ preserves distance 1, and hence that φ is an isometry. 215 r. bruner and s. de winter / j. algebra comb. discrete appl. 3(3) (2016) 209–216 5. concluding remarks the results from sections 3 and 4 show that the concepts of p-isometries that are not isometries and eigenvalue collapsing are closely related but not equivalent. the existence of a p-isometry that is not an isometry implies eigenvalue collapsing but not vice versa. it is unfortunate that the restriction 2p < n or p = n is necessary in the proofs of section 4. although we strongly believe the following conjecture is true, we did not succeed in proving it. (however, by computing orders of automorphism groups in sage we did not succeed in finding any p-isometry that is not an isometry of h(n,q), q > 2.) conjecture 5.1. every p-isometry of h(n,q), q > 2, is an isometry. we would like to end with several open problems. • prove conjecture 5.1. • determine all triples (q,n,p) for which the adjacency matrix of the distance p-graph of h(n,q) has fewer than n + 1 distinct eigenvalues. • more generally, find necessary and sufficient conditions for eigenvalue collapsing to occur in a distance regular graph. • generalize our results on p-isometries to other (primitive) distance regular graphs. references [1] p. abramenko, h. van maldeghem, maps between buildings that preserve a given weyl distance, indag. math. 15(3) (2004) 305–319. [2] f. s. beckman, d. a. jr. quarles, on isometries of euclidean spaces, proc. amer. math. soc. 4 (1953) 810–815. [3] a. brouwer, a. cohen, a. neumaier, distance-regular graphs, springer-verlag, berlin, heidelberg, 1989. [4] a. e. brouwer, m. a. fiol, distance-regular graphs where the distance d-graph has fewer distinct eigenvalues, linear algebra appl. 480 (2015) 115–126. [5] s. de winter, m. korb, weak isometries of the boolean cube, discrete math. 339(2) (2016) 877–885. [6] e. govaert, h. van maldeghem, distance-preserving maps in generalized polygons. i. maps on flags, beitrage algebra. geom. 43(1) (2002) 89–110. [7] e. govaert, h. van maldeghem, distance-preserving maps in generalized polygons. ii. maps on points and/or lines, beitrage algebra geom. 43(2) (2002) 303–324. [8] v. yu. krasin, on the weak isometries of the boolean cube, diskretn. anal. issled. oper. ser. 1 13(4) (2006) 26–32; translation in j. appl. ind. math. 1(4) (2007) 463–467. 216 http://dx.doi.org/10.1016/s0019-3577(04)80001-6 http://dx.doi.org/10.1016/s0019-3577(04)80001-6 http://dx.doi.org/10.1090/s0002-9939-1953-0058193-5#sthash.3ixszh9n.dpuf http://dx.doi.org/10.1090/s0002-9939-1953-0058193-5#sthash.3ixszh9n.dpuf http://dx.doi.org/10.1016/j.laa.2015.04.020 http://dx.doi.org/10.1016/j.laa.2015.04.020 http://dx.doi.org/10.1016/j.disc.2015.10.006 http://www.ams.org/mathscinet-getitem?mr=1913772 http://www.ams.org/mathscinet-getitem?mr=1913772 http://www.ams.org/mathscinet-getitem?mr=1957740 http://www.ams.org/mathscinet-getitem?mr=1957740 http://dx.doi.org/10.1134/s1990478907040096 http://dx.doi.org/10.1134/s1990478907040096 introduction definitions and preliminary observations linear algebraic approach combinatorial approach concluding remarks references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.70490 j. algebra comb. discrete appl. 3(3) • 187–194 received: 06 november 2015 accepted: 03 march 2016 journal of algebra combinatorics discrete structures and applications group divisible designs of four groups and block size five with configuration (1,1,1,2) research article ronald mwesigwa, dinesh g. sarvate, li zhang abstract: we present constructions and results about gdds with four groups and block size five in which each block has configuration (1,1,1,2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. we provide the necessary conditions of the existence of a gdd(n,4,5;λ1,λ2) with configuration (1,1,1,2), and show that the necessary conditions are sufficient for a gdd(n,4,5;λ1, λ2) with configuration (1,1,1,2) if n 6≡ 0(mod 6), respectively. we also show that a gdd(n,4,5; 2n,6(n − 1)) with configuration (1,1,1,2) exists, and provide constructions for a gdd(n = 2t,4,5;n,3(n− 1)) with configuration (1,1,1,2) where n 6= 12, and a gdd(n = 6t,4,5; 4t,2(6t−1)) with configuration (1,1,1,2) where n 6= 6 and 18, respectively. 2010 msc: 05b05, 05b30 keywords: group divisible designs (gdds), latin squares, block configurations, 1-factors, rgdds, rbibds 1. introduction group divisible designs (gdds) have been studied for their usefulness in statistics and for their universal application to constructions of new designs [13, 17, 18]. certain difficulties are present especially when the number of groups is smaller than the block size. in [3, 4], the question of existence of gdds d. g. sarvate thanks the college of charleston for granting sabbatical. r. mwesigwa thanks mbarara university of science and technology for its support. d. g. sarvate and r. mwesigwa also thank council for international exchange of scholars and the u. s. department of state’s bureau of educational and cultural affairs for granting d. sarvate a fulbright core fellowship which made this collaboration possible. l. zhang thanks the citadel foundation for its support. ronald mwesigwa; mbarara university of science and technology, mbarara, uganda (email: ronmwesigwa@yahoo.com). dinesh g. sarvate (corresponding author); college of charleston, department of mathematics, charleston, sc, 29424 (email: sarvated@cofc.edu). li zhang; the citadel, department of mathematics and computer science, charleston, sc, 29409 (email: li.zhang@citadel.edu). 187 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 for block size three was settled. there is a more technical proof given in the book “triple system" [2]. similar results were established for gdds with block size four in [6, 8, 9, 14, 19]. in [7, 16], results about gdds with two groups and block size five with fixed block configuration were presented. in [10], results about gdds with block size six with fixed block configuration were established. a group divisible design gdd(n,m,k;λ1,λ2) is a collection of k-element subsets of a v-set v called blocks which satisfies the following properties: each point of v appears in r (called replication number) of the b blocks; the v = nm elements of v are partitioned into m subsets (called groups) of size n each; points within the same group are called first associates of each other and appear together in λ1 blocks; any two points not in the same group are called second associates of each other and appear together in λ2 blocks. we note that in [13], the term gdd always refer to the case where λ1 = 0. when λ1 is not zero, the designs here are called group divisible pbibds [18]. in [6, 19], the necessary conditions are proved to be sufficient for the existence of a gdd(n,3,4;λ1,λ2) with configuration (1,1,2) where each block has exactly one point from two of the three groups and two points from the third group. the purpose of this paper is to establish results for gdds with block size five and four groups (i.e. gdd(n,4,5;λ1,λ2)) in which each block has configuration (1,1,1,2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. unless otherwise stated, gdds addressed in this paper all have the configuration (1,1,1,2). first we find the relationship between λ2 and λ1. theorem 1.1. the necessary conditions for the existence of a gdd(n,4,5; λ1,λ2) are n ≥ 2 and λ2 = 3(n−1)λ1 n . proof. suppose a gdd(n,4,5;λ1,λ2) exists, then the replication number r for an arbitrary point is λ1(n−1)+λ2(3n) 4 . also, since vr = bk, we have b = n×[λ1(n−1)+λ2(3n)] 5 . on the other hand, since every block must contain exactly one first associate pair (with configuration (1,1,1,2)), the group size n should be greater than or equal to 2, and the number of the first associates pairs 4n(n−1) 2 times λ1 must be equal to the number of blocks b. we have 2n(n−1)λ1 = n×[λ1(n−1)+λ2(3n)] 5 , that is, λ2 = 3(n−1)λ1 n . corollary 1.2. a necessary condition for the existence of a gdd(3,4,5; λ1,λ2) is λ2 = 2λ1 and a necessary condition for the existence of a gdd(n,4,5;λ1,λ2) reduces to λ2 = (n − 1)t (for t ≥ 1) if n 6= 3. proof. by theorem 1.1, λ2 = 2λ1 if n = 3. if n 6= 3, then 3λ1 ≡ 0(mod n) = nt (t ≥ 1), thus λ1 = nt3 , and λ2 = (n−1)t for t ≥ 1. corollary 1.3. for n 6≡ 0(mod 3), the minimum λ1 for the existence of a gdd(n,4,5;λ1,λ2) is n. for n ≡ 0(mod 3), the minimum λ1 is n3 . proof. by theorem 1.1, if n 6≡ 0(mod 3), then λ1 ≡ 0(mod n), thus the minimum λ1 for the existence of a gdd(n,4,5;λ1,λ2) is n. if n ≡ 0(mod 3), then λ1 ≡ 0(mod n3 ), thus the minimum λ1 is n 3 . notice that if a gdd(n,4,5;λ1,λ2) exists, then a gdd(n,4,5;tλ1, tλ2) exists by taking t multiples of gdd(n,4,5;λ1,λ2). therefore, we can reduce the problem to find a gdd(n,4,5;λ1,λ2) for the minimum value of λ1 (which are given in corollary 1.3). remark 1.4. if a gdd(n,4,5;λ1,λ2) for the minimum value of λ1 exists (it’s n for n 6≡ 0(mod 3) and n 3 for n ≡ 0(mod 3)), then a gdd(n,4,5;tλ1, tλ2) exists for t ≥ 1. 2. gdd(n,4,5;λ1,λ2) for n = 2,3,4 and n ≡ 1,5(mod 6) theorem 2.1. necessary conditions given in theorem 1.1 are sufficient for the gdds with n = 2,3 and 4. 188 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 proof. by theorem 1.1, the necessary condition for the existence of a gdd(2,4,5;λ1,λ2) is 2λ2 = 3λ1, that is, λ1 ≡ 0(mod 2) and λ2 ≡ 0(mod 3). the minimum values of λ1 and λ2 are 2 and 3, respectively. a gdd(2,4,5; 2,3) on the four groups {1,2}, {3,4}, {5,6} and {7,8} is as follows: {1,3,5,7,8}, {2,4,6,7,8}, {3,6,7,1,2}, {4,5,8,1,2}, {5,7,3,2,4}, {6,8,1,3,4}, {7,1,4,5, 6}, and {8,2,3,5,6}. by remark 1.4, we have a gdd(2,4,5;λ1,λ2). by corollary 1.2, the necessary condition for the existence of a gdd(3,4,5; λ1,λ2) is λ2 = 2λ1. the minimum values of λ1 and λ2 are 1 and 2, respectively. a construction for a gdd(3,4,5; 1,2) on the four groups {1,2,3}, {4,5,6}, {7,8,9} and {a,b,c} is as follows: {1,2,6,7,b}, {1,3,4,9,a}, {2,3,5,8,c}, {4,5,7,3,b}, {5,6,9,2,a}, {6,4,8,1, c}, {7,8,a,1,5}, {8,9,b, 2,4}, {9,7,c,3,6}, {c,b,1,5,9}, {b,a,3,6,8}, {c,a, 2,4,7}. note that this construction is also listed in clatworthy’s table (number 513 on page 902 in [1]). by remark 1.4, we have a gdd(3, 4,5;λ1,λ2 = 2λ1). the necessary condition for the existence of a gdd(4,4,5; λ1,λ2) is λ1 ≡ 0(mod 4). the minimum values of λ1 and λ2 are 4 and 9, respectively. a construction for a gdd(4,4,5; 4,9) on the four groups {1,2,3,4}, {5,6,7,8}, {9,10,11,12} and {13,14,15,16} is as follows in figure 1 (where each column represents a block). by remark 1.4, we have a gdd(4,4,5;λ1,λ2). figure 1. a gdd(4,4,5; 4,9) we use a different construction below for a gdd(n,4,5;λ1 = 2n,λ2 = 6(n − 1)) where λ1 is not of its minimum value (but twice of its minimum value if n 6≡ 0(mod 3) or six times of its minimum value if n ≡ 0(mod 3)). it’s an interesting construction as it uses a special kind of group divisible design gdd(n,k,k; 0,1). such a gdd is called a transversal design, td(k,n). the construction also uses a resolvable gdd (rgdd). a design is resolvable if the blocks of the design can be partitioned into parallel classes p1, . . . ,ps, where every point of v occurs exactly once in each pi. similarly, one can define a resolvable transversal design, rtd(k,n). the following several theorems from the handbook of combinatorial designs (2nd edition) [1], also see rees [15] and ge and ling [5], are well-known theorems of rgdds that we will use in our proof. theorem 2.2. [1] (theorem 5.35 on page 264) the necessary condition for the existence of a rgdd(n, m,k; 0,λ)) are (1) m ≥ k, (2) nm ≡ 0(mod k), and (3) λn(m−1) ≡ 0(mod (k −1)). theorem 2.3. [1] (theorem 5.43 on page 265) a rgdd(n,m,3; 0,λ) exists if and only if m ≥ 3, λn(m−1) is even, nm ≡ 0(mod 3), and (λ,n,m) 6∈ {(1,2,6),(1,6,3)}∪{(2j+1,2,3),(4j+2,1,6) : j ≥ 0}. theorem 2.4. [1] (theorem 5.44 on page 265) the necessary conditions for the existence of a rgdd(n, m,4; 0,1), namely, m ≥ 4, nm ≡ 0(mod 4) and n(m − 1) ≡ 0(mod 3), are also sufficient except for (n,m) ∈ {(2,4),(2,10),(3,4),(6,4)} and possibly excepting: n = 2 and m ∈ {34,46, 52,70,82,94,100,118,130,142,178,184, 202,214,238,250,334,346}; n = 10 and m ∈ {4,34,52,94}; n ∈ [14,454] ∪ {478,502,514,526,614,626,686} and m ∈ {10,70,82}; n = 6 and m ∈ {6,54,68}; n = 18 and m ∈ {18,38,62}; n = 9 and m = 44; n = 12 and m = 27; n = 24 and m = 23; and n = 36 and m ∈{11,14,15,18,23}. 189 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 a latin square l of side (or order) n is an n×n array in which each cell contains a single symbol from an n-set s, such that each symbol occurs exactly once in each row and exactly once in each column. two latin squares l1 and l2 of the same order are orthogonal if l1(a,b) = l1(c,d) and l2(a,b) = l2(c,d), implies a = c and b = d. a set of latin squares l1, . . . ,lm is mutually orthogonal, or a set of mols, if for every 1 ≤ i ≤ j ≤ m, li and lj are orthogonal. theorem 2.5. [1] (theorem 3.18 on page 161) the existence of k mols (n), the existence of a td(k+ 2,n) and the existence of a rtd(k + 1,n) are equivalent where k ≥ 1. theorem 2.4 implies the existence of a rgdd(n,4,4; 0,1) = rtd(4,n) except for n = 2,3,6 and 10. a construction of a td(4,3) (it is also a rtd(4,3)) and a set of 2 mols(10) (which implies the existence of a td(4,10) by theorem 2.5) can be found in examples 6.5.1 and 6.5.10 in [11], respectively. therefore, we have the following lemma 2.6. lemma 2.6. a td(4,n) and a rtd(3,n) exist except for n = 2 and 6. theorem 2.7. if a rgdd(n,3,3; 0,n− 1) exists (i.e. n 6= 2 by theorem 2.3), then a gdd(n,4,5; 2n, 6(n−1)) also exists. hence a gdd(n,4,5; 2n,6(n−1)) exists for all n > 2. proof. a rgdd(n,3,3; 0,n−1) has n2(n−1) blocks and these are partitioned into n(n−1) parallel classes. first, we construct a rgdd(n,3,3; 0, n−1) on three groups g1, g2, and g3. there are n(n−1) parallel classes. attach each pair of distinct points from g4 with blocks of two parallel classes to make blocks of size 5. in the same way, we construct rgdds on g1, g2, and g4 and attach a pair from g3, and then construct rgdds on g1, g3, and g4 and attach a pair from g2, and then construct rgdds on g2, g3, and g4 and attach a pair from g1, on two parallel classes each. now a parallel class has n triples and each pair from a gi is attached to these triples of two parallel classes, λ1 is 2n. now we show λ2 = 6(n−1). let i ∈ gi and j ∈ gj. when we attach a pair from gi containing i to two parallel classes from the rgdd that misses gi, the pair {i,j} occurs in 2(n − 1) blocks. likewise, when a pair from gj containing j is attached to two parallel classes from the rgdd that misses gj, the pair {i,j} occurs 2(n−1) times. the other two rgdds intersect both gi and gj. thus the pair {i,j} occurs n−1 times in each of these rgdds. hence the pair {i,j} occurs a total of 6(n−1) times. by using the same proof as in theorem 2.7, we have the following corollary. corollary 2.8. if a rgdd(n,3,3; 0, n−1 2 ) exists, then a gdd(n,4,5; n,3(n− 1)) also exists. hence a gdd(n,4,5;n,3(n−1)) exists for all n ≡ 1(mod 2), i.e. odd numbers. combine corollary 1.3, remark 1.4 and corollary 2.8, we have the following result. corollary 2.9. necessary conditions are sufficient for a gdd(n,4,5;λ1, λ2) if n ≡ 1,5(mod 6), i.e. n ≡ 1(mod 2) and n 6≡ 0(mod 3). the following is a different proof of theorem 2.7. theorem 2.10. a gdd(n,4,5;λ1 = 2n,λ2 = 6(n−1)) exists for all n ≥ 2. proof. in theorem 2.1, we have proved that the necessary conditions are sufficient for n = 2. a td(4,n) exists for every n except for n = 2 and 6 by lemma 2.6. for any n > 2 and n 6= 6, a td(4,n) has n2 blocks each of size 4 and replication number r = n. any two elements from group gi occur together 0 times and pairs from different gi’s occur once. take any block {a1,a2,a3,a4}, where aj ∈ gj. and replace it by {x,a1,a2,a3,a4}, where x ∈ gi −{ai}. we have 4n2(n − 1) blocks. it is easy to check the parameters λ1 = 2n, (because replication for td(4,n) is n) and λ2 = 6(n− 1). therefore, a gdd(n,4,5;λ1 = 2n,λ2 = 6(n−1)) always exists. the reason for λ2 = 6(n−1) is that suppose we have two elements a ∈ gi, b ∈ gj, and i 6= j. there are only three types of blocks which will involve a and/or b, that is, when both appear in one block, when a appears and b does not appear, and when a does not appear while b appears, in a block. in case 1, the number of pairs are 4(n− 1), while in cases 2 and 3, there are n−1 pairs in each, and this gives a total of 6(n−1). 190 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 for n = 6, an rgdd(6,3,3; 0,5) exists by theorem 2.3 since m ≥ 3 and (λ,n,m) = (5,6,3) does not belong to that set of exceptions in theorem 2.3. hence we get a gdd(6,4,5; 12,30) by theorem 2.7, that is, a gdd(n,4,5; 2n,6(n−1)) with n = 6 exists. thus, a gdd(n,4,5;λ1 = 2n,λ2 = 6(n−1)) exists for all n ≥ 2. 3. gdd(n,4,5;λ1,λ2) for n ≡ 2,4(mod 6) a balanced incomplete block design bibd(v,k,λ) (λ ≥ 1) is a pair (v,b) where b is a collection of binary blocks of v such that every block contains exactly k < v points and every pair of distinct elements is contained in exactly λ blocks. a resolvable bibd(v,k,λ) is denoted as rbibd(v,k,λ). a 1-factor of a graph g is a set of pairwise disjoint edges which partition the vertex set. a 1factorization of a graph g is the set of 1-factors which partition the edge set of the graph. a 1-factorization of a k2n (also a rbibd(2n,2,1)) exists, and for all n ≥ 1 contains 2n−1 1-factors [12]. theorem 3.1. necessary conditions are sufficient for a gdd(n,4,5;λ1, λ2) if n ≡ 2,4(mod 6), i.e. a gdd(6t + 2,4,5; 6t + 2,3(6t + 1)) and a gdd(6t + 4,4,5; 6t + 4,3(6t + 3)) exist. proof. the construction provided in this proof uses a td(4, t), and it works for all n = 2t except for t = 2 and t = 6 (since a td(4, t) does not exist for t = 2 or 6 from lemma 2.6). let n = 2t where t 6= 2 and 6. a 1-factorization of a ki2t on 2t elements of gi has (2t− 1) 1-factors. each 1-factor has t edges. let fij be the jth 1-factor. we partition 2t elements of gi according to the edges of f i j , that is, fij1,f i j2, . . . ,f i jt. construct a td(4, t) on four groups hi = {f i j1,f i j2, . . . ,f i jt}. from each block of the td(4, t), which gives naturally four groups each of size 2, we construct a gdd(2,4,5; 2,3) with 8 blocks. we repeat this for each 1-factor of the 1-factorization. a detailed counting gives the required values for λ1 and λ2 (see example 3.2 below for an illustration of the construction). for t = 2 (i.e., n = 4), a gdd(4,4,5;λ1,λ2) exists from theorem 2.1. for t = 6 (i.e., n = 12), it is considered in the case of a gdd(n,4,5;λ1,λ2) for n ≡ 0(mod 6), a gdd(6t + 2,4,5; 6t + 2,3(6t + 1)) and a gdd(6t + 4,4,5; 6t + 4,3(6t + 3)) exist. example 3.2. a gdd(6,4,5; 6,15) based on the construction procedure in theorem 3.1 is as follows. here t = 3 and we want to construct a gdd(6,4,5; 6,15). the number of blocks for the gdd is 360. we start with a td(4,3). if we use the groups {a1,a2,a3}, {b1,b2,b3}, {c1,c2,c3}, and {d1,d2,d3} then the blocks of the td are {{a1,b1,c1,d1}, {a1,b2,c2,d2}, {a1,b3,c3,d3}, {a2,b1,c2,d3}, {a2,b2,c3,d1}, {a2,b3,c1,d2}, {a3,b1,c3,d2}, {a3, b2,c1,d3}, {a3,b3,c2,d1}}. take a rbibd(6,2,1) on gi and call it βi. essentially this is a 1-factorization on ki6, a complete graph on six vertices where the vertices are the elements of group gi and we have 6 − 1 = 5 1-factors. the sets gi are given as, say, g1 = {x1,x2,x3,x4,x5,x6}, g2 = {y1,y2,y3,y4,y5,y6}, g3 = {z1,z2,z3,z4,z5,z6}, and g4 = {w1,w2,w3,w4,w5,w6}. taking, for example, the set gi, with i = 1, the five 1-factors will appear as fi1 = {(x1,x2),(x3,x4),(x5,x6)}, fi2 = {(x1,x3),(x2,x5),(x4,x6)}, fi3 = {(x1,x4),(x2,x6),(x3,x5)}, fi4 = {(x1,x5),(x2,x4),(x3,x6)}, and fi5 = {(x1,x6),(x2,x3),(x4,x5)}. in general, we write fij , where i,j are the group number and one-factor position, respectively. take for example, j = 3 and i = 1,2,3,4. this gives f13 = {(x1,x4),(x2,x6),(x3,x5)}, f23 = {(y1,y4),(y2,y6),(y3,y5)}, f33 = {(z1,z4),(z2,z6),(z3,z5)}, and f43 = {(w1,w4),(w2,w6),(w3,w5)}. construct a td(4,3) where the groups are {f1j1,f 1 j2,f 1 j3}, {f 2 j1,f 2 j2, f 2 j3}, {f 3 j1,f 3 j2,f 3 j3}, and {f4j1,f 4 j2,f 4 j3}. take each block of the transversal design, for example, the first block, {{x1,x4},{y1,y4},{z1,z4},{w1,w4}}. the elements of this block give the groups for a gdd(2,4,5; 2,3). the second block will have the groups {{x1,x4},{y2,y6},{z2,z6},{w2,w6}}, and so on, up to the ninth block with groups {{x3,x5},{y3,y5},{z2,z6},{w1,w4}}. from each of these nine blocks from a 1-factor we get 9 × 8 = 72 blocks. from 5 1-factors, we have constructed 360 required blocks for a gdd(6,4,5; 6,15). now we show that λ1 = 6 and λ2 = 15. for example, observe that the pair {x1,x4} appears in 191 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 one 1-factor. through that 1-factor, the element {x1,x4} appears in three blocks of the td(4,3). in each block of the td(4,3), the pair {x1,x4} appears two times. thus, λ1 = 6. on the other hand, the pair {x1,y1} appears in 3 blocks of the gdd(2,4,5; 2,3). since there are five 1-factors, we get λ2 = 15. remark 3.3. theorem 3.1 provides constructions for a gdd(n,4,5;n, 3(n−1)) for n = 2t (n 6= 4 and 12). 4. gdd(n,4,5;λ1,λ2) for n ≡ 0,3(mod 6) if n ≡ 0,3(mod 6), then n ≡ 0(mod 3) = 3s. the minimum value of λ1 is n3 = s by corollary 1.3. thus, if a gdd(n,4,5;s,3s− 1) exists, then a gdd(n,4,5;λ1,λ2) for n ≡ 0(mod 3) exists by theorem 1.1 and remark 1.4. theorem 4.1. necessary conditions are sufficient for a gdd(n,4,5;λ1, λ2) if n ≡ 3(mod 6), i.e., a gdd(6t + 3,4,5; 2t + 1,6t + 2) exists for t ≥ 0. proof. we know that a td(4,2t+1) exists (lemma 2.6) and has a replication number 2t+1. we also know that a rbibd(6t+3,3,1) exists and has 3t+1 parallel classes [1]. we also have a gdd(3,4,5; 1,2). we wish to construct a gdd(6t + 3,4,5; 2t + 1,6t + 2). let the groups be g1 = {a1,a2, . . . ,a6t+3}, g2 = {b1,b2, . . . ,b6t+3}, g3 = {c1,c2, . . . ,c6t+3}, and g4 = {d1,d2, . . . ,d6t+3}. let π1, π2, . . . , π3t+1 be parallel classes of a rbibd(6t + 3,3,1) on {1,2, . . . ,6t + 3}. use each πi to partition each of the four groups by relabelling the elements, i.e., if {j1,j2,j3} is the jth block of πi, then the jth partition set g1j of g1 is {aj1,aj2,aj3}. similarly for other gi, for i = 2,3,4. use a td(4,2t + 1) on groups {gi1,gi2, . . . ,gi,2t+1}, i = 1,2,3,4. if a block of the td(4,2t + 1) is {g1r,g2s,g3t,g4u}, construct a gdd(3,4,5; 1,2) on groups g1r, g2s, g3t, and g4u. the union of all the blocks of the gdds thus constructed using all the πi’s is a gdd(6t + 3,4,5; 2t + 1,6t + 2). clearly λ1 = 2t + 1 because in a td(4,2t + 1) each element occurs 2t + 1 times. it means that gij will be in 2t + 1 blocks of the td and hence, when gdd(3,4,5; 1,2) is formed with gij as one of the groups, pairs of elements within gij will occur 2t + 1 times. also, λ2 is 6t + 2 because pairs of elements between gi and gj (i 6= j) occur twice for each parallel class πi and there are 3t + 1 parallel classes. for n ≡ 0(mod 6) = 6t, we provide constructions for a gdd(6t,4,5; 4t, 2(6t− 1)) where λ1 = 4t is not of its minimum value (but twice of its minimum value which is 2t). example 4.2. a construction of a gdd(12,4,5; 8,22) is as follows. first note that a td(4,4) exists (by lemma 2.6), and it has 16 blocks of size 4. let g1, g2, g3, and g4 be the groups, each of size four for a gdd(12,4,5; 8,22) which we wish to construct. let hi = {gi1,gi2,gi3, gi4}, i = 1,2,3,4, where each gij has size 3. we construct td(4,4)s on groups h = {h1,h2,h3,h4}. any gij is in four blocks. but a block of h gives four subsets each of size 3. so these groups can be used to get a gdd(3,4,5; 1,2). do this for each block of h. the pair of elements within gij occur four times and the pairs from gij,gst, i 6= s occur two times. now, construct a rbibd(12,3,2) with 11 parallel classes. use each of the parallel classes and apply the construction. as pairs in bibd appear twice we have any two elements (a,b) from gi in eight blocks and (c,d) where c ∈ gi and d ∈ gj, i 6= j occur 22 times. remark 4.3. a gdd(6t,4,5; 4t,2(6t−1)) where t 6= 1 and 3 can be constructed using a td(4,2t) and a rbibd(6t,3,2). by lemma 2.6, a td(4,n) exist except for n = 2 and 6. use similar ideas as in example 4.2, we construct a gdd(6t,4,5; 4t,2(6t−1)) except for t = 1 and 3 using a td(4,n = 2t) and a gdd(3,4,5; 1,2). we use a partition of 6t elements according to the parallel classes πij of a rbibd(6t,3,2) on gi, i = 1,2,3,4 and j = 1,2, · · · ,6t − 1. note for a rbibd(6t,3,2), there are 6t − 1 parallel classes and the 192 r. mwesigwa et. al. / j. algebra comb. discrete appl. 3(3) (2016) 187–194 number of blocks is b = 2t(6t − 1). we use a partition πs, to get groups, say gij, i = 1,2,3,4 and j = 1,2, · · · ,2t. the pair of elements within gij occur 2t times and pairs from gij,gst, i 6= s occur two times. this construction is repeated 6t− 1 times once for each of the 6t− 1 parallel classes. therefore elements from the same group will occur 4t times and pairs of elements from two different groups will occur 2(6t−1) times and we have gdd(6t,4,5; 4t,2(6t−1)) where t 6= 1 and 3. to completely solve the case for n ≡ 0(mod 6) = 6t, one should construct a gdd(6t,4,5; 2t,6t−1). 5. summary in this paper we studied constructions and results about gdd(n,4,5;λ1, λ2) with configuration (1,1,1,2). we provide the necessary conditions of the existence of a gdd(n,4,5;λ1,λ2) with configuration (1,1,1,2), and show that the necessary conditions are sufficient for a gdd(n,4,5;λ1,λ2) with configuration (1,1,1,2) if n 6≡ 0(mod 6), respectively. we also show that a gdd(n,4,5; 2n,6(n−1)) with configuration (1,1,1,2) exists, and provide constructions for a gdd(n = 2t,4,5;n,3(n− 1)) with configuration (1,1,1,2) where n 6= 12, and a gdd(n = 6t,4,5; 4t,2(6t − 1)) with configuration (1,1,1,2) where n 6= 6 and 18, respectively. the remaining case of the problem is to show that the necessary conditions are sufficient for n ≡ 0(mod 6), i.e., to show the existence of a gdd(6t,4,5; 2t,6t − 1) with configuration (1,1,1,2). acknowledgment: we are thankful to both the referees for their useful comments. our special thanks to one of the referees as we have used his/her wordings to count the value of λ1 in theorem 2.7 and the values for λ1 and λ2 in example 3.2 verbatim. references [1] c. j. colbourn, d. h. dinitz (eds.), handbook of combinatorial designs, second edition, chapman and hall, crc press, boca raton, fl, 2007. 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[19] m. zhu, g. ge, mixed group divisible designs with three groups and block size four, discrete math. 310(17-18) (2010) 2323–2326. 194 http://www.ams.org/mathscinet-getitem?mr=2408798 http://www.ams.org/mathscinet-getitem?mr=2408798 http://dx.doi.org/10.1002/jcd.3180010104 http://dx.doi.org/10.1002/jcd.3180010104 http://dx.doi.org/10.1016/j.disc.2010.05.014 http://dx.doi.org/10.1016/j.disc.2010.05.014 introduction gdd(n, 4, 5; 1, 2) for n = 2, 3, 4 and n 1, 5 (mod 6) gdd(n, 4, 5; 1, 2) for n 2, 4 (mod 6) gdd(n, 4, 5; 1, 2) for n 0, 3 (mod 6) summary references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(2) • 117-149 received: 1 december 2014; accepted: 22 april 2015 doi 10.13069/jacodesmath.28239 journal of algebra combinatorics discrete structures and applications identifying long cycles in finite alternating and symmetric groups acting on subsets research article steve linton1∗, alice c. niemeyer2∗∗, cheryl e. praeger3§ 1. school of computer science, university of st. andrews, north haugh, st. andrews, fife, ky16 9sx, scotland 2. department of mathematics and statistics, maynooth university, co. kildare, ireland 3. centre for the mathematics of symmetry and computation, the university of western australia, 35 stirling hwy, crawley, wa 6009, australia abstract: let h be a permutation group on a set λ, which is permutationally isomorphic to a finite alternating or symmetric group an or sn acting on the k-element subsets of points from {1, . . . ,n}, for some arbitrary but fixed k. suppose moreover that no isomorphism with this action is known. we show that key elements of h needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n−1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in λ. 2010 msc: 20b30, 60c05, 20p05, 05a05 keywords: symmetric and alternating groups in subset actions, large base permutation groups, finding long cycles 1. introduction the second and third authors predicted in [9] that, for a permutation group h on a set λ, which is permutationally isomorphic to a symmetric group sn acting on the k-element subsets of points from {1, . . . ,n} (that is in its k-set action), for some arbitrary but fixed k, it should be possible to recognise an element in h corresponding to an n-cycle in sn by the lengths of just four of its cycles in λ. the purpose of this paper is to prove this result. theorem 1.1. let h be a permutation group on a set λ, which is permutationally isomorphic, via an unknown isomorphism ϕ, to a finite symmetric group sn in its k-set action, for some k. let h be ∗ e-mail: sal@cs.st-andrews.ac.uk ∗∗ e-mail: alice.niemeyer@nuim.ie § e-mail: cheryl.praeger@uwa.edu.au 117 identifying long cycles a uniformly distributed random element of h and let λ1, . . . ,λ4 be independent, uniformly distributed random points of λ. then there exist positive constants n0 and c such that, for n ≥ n0, prob ( ϕ(h) is an n-cycle ∣∣∣∣ the h-cycle containing λi haslength n, for i = 1, . . . , 4 ) > 1 − c n 1 6 . subset actions of sn and the alternating group an play a crucial role in algorithms for permutation groups. they are examples of ‘large-base’ actions. most primitive permutation groups are ‘small-base’ and very efficient algorithms are available to compute with them (for a detailed definition see [12, p. 51]). however these algorithms become prohibitively expensive when applied to large-base groups and, therefore, alternative means of handling large-base groups are essential (see [12, chapter 10] for a discussion on currently available algorithms for this case). the large-base primitive permutation groups all contain in their socles alternating groups with associated subset actions. hence finding efficient algorithms for these actions is important. the probabilistic algorithm described in [6] recognises alternating and symmetric groups in their actions on k-sets constructively. it takes as input a group h and an integer n. under the assumption that h is isomorphic to sn, the algorithm examines a number of random elements of h seeking to find key-elements, namely elements for which good estimates for their proportions in sn are known and which have particular properties. should this search fail, the algorithm concludes that the assumption that h is isomorphic to sn is incorrect and reports that h is not isomorphic to sn. otherwise, it attempts to construct an isomorphism from h to sn. since this algorithm is randomised, there is a possibility that the search appears to succeed but in fact has not found suitable elements. in most settings this will be detected as part of the larger algorithm, see [6]. the theoretical underpinning of the algorithm described above is to determine very good bounds for the probability of finding the key-elements in h under the assumption that h is isomorphic to sn. this is the purpose of the current paper. for the connection between estimation results and probabilistic algorithms in the context of recognition algorithms for groups see [10]. the groups the algorithm can take as input need not be permutation groups. rather, they may belong to a more general class of groups called groups of black box permutations (see [6]). this allows the algorithm to be employed in different computational models, for example, to deal with groups given by a set of generating matrices, which are permutation isomorphic to sn acting on the underlying vector space, without converting such a group into a permutation group. the crucial requirement is the ability to compute the image of a point under the action of a generator of the group. suppose that t is an upper bound for the time taken to perform this action, which might be viewed as a black box procedure. then the time required to determine that a word of length r in the group generators permutes a given point in a cycle of length m is at most mrt. in some contexts this time can even be very much less than the time required to find the product of two group generators. for suitable n and k the algorithms in [6] have running time (excluding any time needed to read the input) growing significantly more slowly than ( n k ) , implying that they can use the input black box permutations only through computing their action on a selection of the ( n k ) points, and not through examining any other aspect of their structure, or computing any other elements of the group they generate. for example, checking that the n-th power of an element is the identity may already be more expensive than our entire algorithm. the key elements sought by the algorithm to recognise an or sn in its k-set action are elements containing an m-cycle for large m, as described in table 1. our theorem 1.1 follows from a more general theorem, theorem 3.1, which determines the probability of finding elements of each of these types among a certain number of random elements of h. once these key elements have been found, a permutational isomorphism from h on λ to an or sn on k-sets can be implemented using the methods described in [2, section 4], especially method b, or those described in [1, sections 4 and 5], especially lemmas 4.1, 4.3 and 5.5. alternative methods, focussing in particular on the k-set action, are developed in [6]. 118 s. linton, a.c. niemeyer c.e. praeger 1.1. context of our results in a seminal collection of papers, erdös and turán initiated the study of asymptotic behaviour of the proportions of various kinds of elements in permutation groups. for example, they showed [4, 5] that for n large enough, most elements in the symmetric group sn of degree n have order n( 1 2 +o(1)) log(n). in the same vein warlimont [13] proved that the conditional probability that a random element g in sn is an n-cycle, given that gn = 1, is 1 −o(n−1). applied algorithmically, warlimont’s result is used to conclude, from the fact that the nth power of a ‘hidden’ permutation g ∈ sn is the identity, that g is almost certainly an n-cycle. finding an n-cycle is a key step in many algorithms that ‘constructively recognise’ sn, so this is valuable. however, testing whether gn = 1 requires log(n) multiplications of black box permutations. in computational models where a single multiplication costs about ( n k ) operations, employing this approach would not yield an algorithm whose running time grows significantly more slowly than ( n k ) . the results of [9] provide a basis for extending this to a situation where we know only that 〈g〉 has an orbit of length n in some action. an extension of this nature to k-set actions of sn and an is the subject of this paper. we refine and improve significantly the main result of [9]. for example, we employ a similar division of the elements of sn into several families according to properties of points which lie in cycles of lengths dividing n. however, examining this subdivision alone is not sufficient to achieve the results in this paper. we need to study the probability that several k-element subsets of {1, . . . ,n} have exactly n distinct images under 〈g〉 for g an element in one of the families. moreover, in our algorithmic applications we also required analogous results for elements of sn and an containing m-cycles, for m ≥ n− 6. line g n m r key-elements ρ(g,n,m) 1 sn n 1 n-cycle 1 2 odd n− 2 2 2-cycle 1 3 even n− 3 2 2-cycle 2/3 4 an odd n 1 n-cycle 1 5 even n− 1 1 (n−1)-cycle 1 6 2 or 4 (mod 6) n− 3 3 3-cycle 1 7 3 or 5 (mod 6) n− 4 3 3-cycle 3/4 8 0 (mod 6) n− 5 3 3-cycle 7/20 9 1 (mod 6) n− 6 3 3-cycle 9/40 table 1. groups and types of elements in section 2 we briefly describe the algorithmic application, and in particular we explain the meaning of the parameters r and ρ in table 1. in section 3 we introduce the notation which we shall use throughout the paper and give the precise statement of the main result (theorem 3.1). the proof of theorem 3.1 (and hence of theorem 1.1) is given in section 4. in particular we exhibit an explicit value for the constant c of theorem 3.1(a) and theorem 1.1. we present some background material in sections 5 and 6. sections 7 11 contain the various parts which are pulled together in section 4 for the proof of theorem 3.1. 2. algorithmic application the results in this paper are motivated by algorithmic applications in [6] and [7]. in these applications, h is a permutation group acting on a set λ of ( n k ) points. we wish to test whether h is permutation isomorphic to g = an or g = sn acting on the set ( ω k ) of k-element subsets of ω = {1, . . . ,n}. that is 119 identifying long cycles to say, whether there is a group isomorphism ϕ : h → g and a bijection f : λ → ( ω k ) such that, for each h ∈ h and λ ∈ λ, (λh)f = (λ)fhϕ. these isomorphisms will be constructed in the form of a computer program rather than listing the image of each element. we say that an element h ∈ h corresponds to an element g ∈ g if the permutation isomorphism ϕ maps h to g. the algorithms construct a ‘nice generating’ set for h of size 2. in the case where h is permutation isomorphic to sn in its action on ( ω k ) , this generating set consists of elements that, in the natural representation of sn on n points, correspond to an n-cycle and a 2-cycle interchanging two consecutive points of the n-cycle. in the case where h is permutation isomorphic to an in its action on( ω k ) the nice generating set consists of elements that in an correspond to an n-cycle or (n−1)-cycle, and to a 3-cycle. we wish to find these elements by selecting independent, uniformly distributed random elements from the group h. however, the proportion of 2-cycles in sn or 3-cycles in an is too small to allow us to find such elements directly by random selection. therefore, we seek elements in h which correspond to permutations containing a 2-cycle or a 3-cycle together with one long cycle of length m, say, where m is at least n− 6 and m is coprime to 2 or 3, respectively. the algorithms in [6] and [7] seek elements h ∈ h which correspond to the kinds of elements g listed in table 1, where h is permutation isomorphic to g = sn or g = an, with g,n as in the second and third columns. the fourth column, labelled m, lists the length of the m-cycle which the element g contains. the fifth column, labelled r, lists an integer between 1 and 3. ultimately we wish to find an element h in h which corresponds to an element in g with cycle type as recorded in the sixth column. this element is constructed as a power of the element h. the first element in the nice generating set for h corresponds to an element satisfying the conditions of line 1, 4, or 5, namely it corresponds to an n-cycle or an (n − 1)-cycle. the second nice generator corresponds to a 2-cycle if g = sn and is constructed from an element h ∈ h which corresponds to g as in line 2 or 3. if g = an, the second nice generator corresponds to a 3-cycle and is constructed from h ∈ h corresponding to an element g as in line 6, 7, 8 or 9. the last column, labelled ρ(g,n,m), records a rational number such that the proportion of elements h of h which correspond to elements of g containing an m-cycle and with order dividing rm is ρ(g,n,m) m (see (1)). the group h acts on a set λ of size |λ| = ( n k ) , and in the context of the algorithm m, n and k are so large that it is ‘too expensive’ to compute the full cycle structure of elements of h in their action on λ. instead we compute the cycle lengths of elements h ∈ h on a handful of randomly chosen points of λ, that is to say, we ‘trace’ these points under the action of 〈h〉. in computer experiments in gap [3], we discovered that if h is permutation isomorphic to g = sn or an on ( ω k ) then, for m,r as in one of the lines of table 1, most elements of h which produced cycles of lengths a multiple of m and dividing rm, when we traced each of four or five independent random points of λ, corresponded to elements of g containing an m-cycle. this computer experiment is formalised in procedures findmcycle and tracecycle. our experimental observation turns out to be true in general, and is proved in theorem 3.1 for sufficiently large n. our experiments also suggest that the results hold for smaller values of n. for clarity of exposition the proofs of theorem 3.1 are written in terms of the action of g on ( ω k ) . for n,m and r as in one of the lines of table 1, define n(n,m) to be the set of all g ∈ sn that contain an m-cycle and ngood(g,n,m) to be the set of all g ∈ n(n,m) ∩g for which o(g) divides rm. note that, for given g,n,m, only one of the lines of table 1 is satisfied, and hence r is determined by g,n,m. we define ρ(g,n,m) to be the rational number satisfying |ngood(g,n,m)| |g| = ρ(g,n,m) m . (1) as an example of how to interpret this information, consider line 3 of table 1. the proportion of elements g of sn containing an (n−3)-cycle is 1n−3, and 2/3 of these elements contain also a 2-cycle or three 1-cycles on the remaining 3 points. thus the proportion of elements of sn containing an (n−3)-cycle and having order dividing 2(n− 3) is 2/3 n−3 = ρ(sn,n,n−3) n−3 . in order to construct a 2-cycle (the entry in column 6 for 120 s. linton, a.c. niemeyer c.e. praeger this line), we raise the element g to the (n − 3)rd power producing x = gn−3. since n − 3 is odd, the element x is the identity if g has three fixed points, a 2-cycle if g contains a 2-cycle, or possibly a 3-cycle if g contains a 3-cycle and 3 does not divide n. thus three quarters of the elements of ngood(sn,n,n−3) yield a 2-cycle by powering. the algorithm findmcycle can therefore easily be incorporated into a monte carlo algorithm to construct a transposition in this case: by repeating findmcycle a number of times we will with high probability construct a transposition by powering the output of findmcycle. the other lines have a similar interpretation for ρ(g,n,m). we now describe the two algorithms. algorithm 1 assumes that we have a function randomgrpelt which takes as input a generating set y for a group h and returns independent, uniformly distributed random elements of h. algorithm 2 assumes that we have a function randompoint which takes as input a finite set λ and returns independent, uniformly distributed random points of λ. note that algorithm 1 calls algorithm 2 and that we assume that algorithm 2 has access to the variables of algorithm 1. algorithm 1: findmcycle(n,m,r,h, λ,ε,m) data: let (n,m,r) be as in one of the lines of table 1. let h be a permutation group with a generating set y acting on a finite set λ. let ε be a real number with 0 < ε < 1 and let m be an integer with m ≥ 4. result: an element h ∈ h or fail; this algorithm inspects up to o(n log(ε−1)) uniformly distributed independent random elements from h to find one which has orbits of length a multiple of m and dividing rm on each of m randomly selected points from λ. if such an h ∈ h is found it returns h, otherwise it returns fail. set n := ⌈ 5n log( 2 ε ) ⌉ ; for i = 1, . . . ,n do hi := randomgrpelt(y ); if tracecycle(hi) = true then return hi; return fail; algorithm 2: tracecycle(h) data: a permutation h ∈ h; result: a boolean ‘true’ or ‘false’ this algorithm tests whether the permutation h ∈ h has orbits of length a multiple of m and dividing rm on m randomly selected points from λ. if this is the case it returns true, otherwise it returns false. for i = 1, ..,m do λi := randompoint(λ); put γ = {λj}mj=1; for λ ∈ γ do if |λ〈g〉| 6= r0m for some r0 | r then return false; return true; remark 2.1. (a) the number m of random points of λ tested in the algorithm tracecycle is often a bounded constant (as, for example, in theorem 3.1), but in our analysis we allow it to be as large as o(n), see (2). (b) the algorithm tracecycle performs o(n) image computations to check whether |λ〈g〉| = r0m, for each random point λ. thus if ξrp, ξrge, νim, are upper bounds for the costs of producing a random point using randompoint, producing a random group element using randomgrpelt, and computing the image of a point of λ under an element of h, respectively, then the cost of findmcycle is o(n log(ε−1)(ξrge + mξrp + mnνim)). 121 identifying long cycles this cost is modest when compared with the cost ( n k ) νim of computing the product of two permutations of λ (especially when k = o(n)) or the cost of directly computing the order of any element. our main result theorem 3.1 shows that these simple and inexpensive procedures provide an effective way to find and identify elements of sn and an containing m-cycles from their actions on k-element subsets. 3. statement of the main theorem and notation in order to state our main theorem we introduce several parameters that are used throughout the paper. suppose that the triple (g,n,m) satisfies one of the lines of table 1, and note that r is determined by g,n,m. the integer m used in the algorithm findmcycle is assumed to satisfy 4 ≤ m ≤ log ( 9 8 ) n− 2 2 . (2) let d(x) be the number of positive divisors of an integer x. by [11, pp. 395-396], d(x) = xo(1). in fact, for every δ > 0, there is a positive constant cδ such that d(x) ≤ cδxδ (3) for all x. choose real numbers δ and s satisfying 0 < δ < min{1 −s, s 3 ,s− 1 2 } and 1 2 < s < m − 1 m . (4) further let ` = min{m(1 −s), 3 − 2s− 2δ, 1 + s− 3δ, 2s− 2δ} . (5) by (4), all of m(1 −s) > 1, 3 − 2s− 2δ > 1, 1 + s− 3δ > 1 and 2s− 2δ > 1 hold. hence ` > 1. next we define the constant aδ by aδ := 5 4 ( 1 + 3 cδ 150s−δ + ( cδ 150s−δ )2) , (6) with cδ as in (3), and the constant bm,δ,s, which we usually abbreviate to bm, by bm = ( 33 8 )m + 72 aδc 2 δr 2s+2δ + 6.24 aδc 3 δr 3δ + c2δ r2s−2δ + ( 31 r1−s )m . (7) the theorem involves an ‘error probability’ ε, that is, a real number satisfying 0 < ε < 1. we assume that the integer n satisfies the following inequalities: n ≥   12(rn)s + 6 (rn)s log n( 10bm ε )1/(`−1) . (8) theorem 3.1. let (g,n,m) be as in one of the lines of table 1, and let k be a positive integer satisfying 2 ≤ k ≤ n/2. suppose that h is a permutation group permutation isomorphic to g acting on k-element subsets of {1, . . . ,n} (via the unknown isomorphism ϕ : h → g). then the following hold (a) let h be a uniformly distributed random element of h and let λ1, . . . ,λ4 be independent, uniformly distributed random points of λ. then there exist positive constants n0 and c such that, for n ≥ n0, prob   ϕ(h) contains an m-cycle ∣∣∣∣∣∣∣ for i = 1, . . . , 4 the h-cycle on λi has length rim for some ri | r   > 1 − c n 1 6 . 122 s. linton, a.c. niemeyer c.e. praeger (b) let m be an integer satisfying (2), and let s,δ be real numbers satisfying (4), and ` as in (5). then findmcycle is a monte carlo algorithm which, given as input the permutation group h, an error probability ε > 0 and the integer m, returns an output h such that, provided n satisfies (8), (i) the probability that h ∈ h and ϕ(h) contains an m-cycle is at least 1 −ε, (ii) the probability that h ∈ h and ϕ(h) does not contain an m-cycle is at most ε/2, and (iii) the probability that h = fail is at most ε/2. notation 3.2. for the rest of the paper we assume that n,m,r and g are as in one of the lines of table 1, noting that r is determined by g,n,m. let m be an integer satisfying (2), let s,δ be real numbers satisfying (4), and let `,cδ,aδ and bm be as in (5), (3), (6) and (7) respectively. let sn act naturally on ω = {1, 2, . . . ,n}. let k and k0 be positive integers satisfying 2 ≤ k ≤ n/2, and 1 ≤ k0 ≤ k. a k0-element subset of ω is called a k0-subset. we use the notation in table 2 to describe an element g ∈ sn, where γ0 is a k0-subset of ω. here we identify a cycle of g with the subset of ω it permutes. ck0 (γ0,g) length of the g-cycle containing γ0 on k0-subsets s-small g-cycle g-cycle in ω of length less than (rn)s s-large g-cycle g-cycle in ω of length at least (rn)s ∆(g) union of g-cycles in ω whose lengths divide rm σ(g) ω \ ∆(g) v cardinality of ∆(g) u cardinality of σ(g) table 2. table for notation 3.2 we define in table 3 several classes of elements in g. we usually omit mentioning n and m in our notation. for example, we refer to n(n,m) (defined in section 2) simply as n and to ngood(g,n,m) simply as ngood. n set of all g ∈ sn that contain an m-cycle ngood set of all g ∈n ∩g for which o(g) divides rm f set of all g ∈ g\n such that m | o(g) r set of all g ∈f such that |∆(g)| ≤ 4(rn)s s0 set of all g ∈f such that |∆(g)| > 4(rn)s and all g-cycles in ∆(g) are s-small s+1 set of all g ∈f such that |∆(g)| > 4(rn) s, exactly one g-cycle c in ∆(g) is s-large, and |∆(g) \c| > 3(rn)s s−1 set of all g ∈f such that |∆(g)| > 4(rn) s, exactly one g-cycle c in ∆(g) is s-large, and |∆(g) \c| ≤ 3(rn)s s≥2 set of all g ∈f such that |∆(g)| > 4(rn)s and at least two g-cycles in ∆(g) are s-large table 3. families of elements 123 identifying long cycles remark 3.3. (a) the definition of aδ is not too critical. we simply need aδ to be greater than or equal to the right hand side of (6) for the values of rm we are considering, see remark 8.2 and lemma 8.3. for example, if rm ≥ c1/(s−δ)δ then we may take aδ = 25/4. (b) currently equation (8) limits the practical applicability of theorem 3.1 severely, but we note that in our analysis we allow k to be as large as n/2. the first two inequalities of (8) imposed on n are due to the subdivision of the set of permutations of order divisible by m into disjoint subsets which depend on s. we give a uniform proof that holds for all values of k in the range 2 ≤ k 6= n/2. if, for example, k were bounded as n increases, then several of the arguments would be simpler and the constraints on n correspondingly less severe. (c) the main constraint forcing n to be very large is the third inequality in (8). for example, for our parameter choice in theorem 3.1, namely m = 4,s = 17 24 and δ = 1 6 , we have cδ ≤ 138.32 and, for n large enough, aδ = 254 . in this case we find bm > 2·10 8 and the last inequality of (8) dictates n > 3.3·10112/ε12. moreover, even though a larger value of m allows us to choose a smaller value for cδ, the choice might result in a smaller value for `, which in turn has undesired consequences, making bm larger, and hence requiring n to be larger. 4. proof of the main theorem the proof of the main theorem, theorem 3.1, relies on many supporting results. in this section we subdivide the proof into various parts and show how these parts are then brought together to give a complete proof. the individual parts of the proof are proved in later sections. the main idea of the proof is to divide the elements of sn that could possibly be returned by findmcycle into disjoint families, and to compute the probability that tracecycle returns true for an element of each of these families. the families of elements in this subdivision are defined in table 3, namely n ,r,s0,s+1 ,s − 1 ,s≥2, and we use the notation introduced in this table throughout the paper. proof of theorem 3.1(b). we prove this theorem by analysing the algorithm findmcycle. let n = ⌈ 5n log( 2 ε ) ⌉ . a call to algorithm findmcycle can terminate in one of three possible ways: (g) for some i with 1 ≤ i ≤ n the i-th iteration of the for-loop returns an element in n . we call this a good outcome. (b) for some i with 1 ≤ i ≤ n the i-th iteration of the for-loop returns an element which is not in n . we call this a bad outcome. (u) the for-loop is executed n times and tracecycle returns false for each of the selected random elements. in this case the algorithm returns fail. we call this an ugly outcome. thus to prove the three parts of theorem 3.1 we must prove prob(g) ≥ 1 −ε, prob(b) ≤ ε/2, prob(u) ≤ ε/2. clearly any two of these inequalities implies the third. we shall therefore prove only prob(b) ≤ ε/2 and prob(u) ≤ ε/2. to study these outcomes more closely we define the following events. ei the i-th iteration of the for-loop is executed. let gi denote the random element selected in the i-th iteration. gi event ei occurs, gi ∈n and tracecycle(gi) = true bi event ei occurs, gi /∈n and tracecycle(gi) = true ui event ei occurs and tracecycle(gi) = false 124 s. linton, a.c. niemeyer c.e. praeger note that ei = gi∪̇bi∪̇ui and that prob(e1) = 1. further, for i > 1 we have that ei = u1 ∩ . . .∩ui−1 = ui−1. (9) thus g = g1 ∨g2 ∨ . . .∨gn b = b1 ∨b2 ∨ . . .∨bn u = u1 ∧u2 ∧ . . .∧un = un. (10) proof that prob(u) ≤ ε/2: for a uniformly distributed random element g ∈ g, let p1 = prob(tracecycle(g) = false | g ∈ngood) p2 = prob(tracecycle(g) = false | g /∈ngood) and let p = ρ m p1 + m−ρ m p2, where ρ := ρ(g,n,m) (see table 1), the proportion of elements of g containing an m-cycle that have order dividing rm. note that, since the proportion of elements containing an m-cycle in sn is 1/m, we have prob(g ∈ngood) = ρm. given ei, the event ui is the disjoint union of the events ui1, that gi ∈ngood and tracecycle(gi) = false, and ui2, that gi 6∈ngood and tracecycle(gi) = false. thus prob(ui | ei) = ρ m prob(tracecycle(gi) = false | gi ∈ngood) + m−ρ m prob(tracecycle(gi) = false | gi /∈ngood) = ρ m p1 + m−ρ m p2 = p. note, in particular, that this probability is independent of i. by (9) we have ei = ui−1, and hence prob(ui) = prob(ei)prob(ui | ei) = prob(ui−1) ·p. as this is true for all i with 1 ≤ i ≤ n, we have prob(ui) = pi, (11) and in particular, prob(u) = prob(un ) = pn. the required inequality prob(u) ≤ ε/2 holds whenever pn ≤ ε/2. we now prove the latter inequality. by proposition 7.5 we have 1 −p1 ≥ ( n−2 n )m . therefore, p ≤ ρ m p1 + m−ρ m = 1 − ρ m (1 −p1) ≤ 1 − ρ n ( n− 2 n )m . (12) now n = ⌈ 5n log( 2 ε ) ⌉ = ⌈ log((ε/2)−1) (5n)−1 ⌉ , and so by lemma 5.2, (1 − 1 5n )n ≤ ε/2. thus pn ≤ ε/2 holds if 1− ρ n ( n−2 n )m ≤ 1− 1 5n , or equivalently, if ( n n−2 )m ≤ 5ρ. since ρ ≥ 9/40 (see table 1), it is sufficient to prove that ( n n−2 )m ≤ 9 8 . by our assumption, m ≤ log( 9 8 ) n−2 2 , and hence m log ( n n− 2 ) = m log ( 1 + 2 n− 2 ) ≤ m 2 n− 2 ≤ log ( 9 8 ) and exponentiating both sides gives the required inequality. thus pn ≤ ε/2 and hence prob(u) ≤ ε/2 is proved. 125 identifying long cycles proof that prob(b) ≤ ε/2: recall the definition of b in (10). note that, if tracecycle(g) = true, then o(g) is divisible by m. thus, by the definition of f, for a uniformly distributed, random element g ∈ g, q := prob(g ∈f and tracecycle(g) = true) (13) = prob(g 6∈n and tracecycle(g) = true). now, for all i with 1 ≤ i ≤ n, we have that prob(bi | ei) = prob(gi 6∈n and tracecycle(gi) = true) = q. hence prob(bi) = prob(ei)prob(bi | ei) = prob(ei) q. if i ≥ 2 then ei = ui−1 by (9), and so by (11), prob(bi) = pi−1q. therefore, prob(b) = n∑ i=1 prob(bi) = q n∑ i=1 pi−1 = q 1 −pn 1 −p < q 1 −p . (14) the most substantial part of the paper is devoted to finding an upper bound for q. it follows from table 3 that f = r∪̇s0 ∪̇s+1 ∪̇s − 1 ∪̇s≥2. hence q = q(r) + q(s0) + q(s+1 ) + q(s − 1 ) + q(s≥2), where we estimate these proportions in sections 8 11. recall the definition of ` in (5), and that ` > 1. q(r) = prob(g ∈r and tracecycle(g) = true) q(s0) = prob(g ∈s0 and tracecycle(g) = true) q(s+1 ) = prob(g ∈s + 1 and tracecycle(g) = true) q(s−1 ) = prob(g ∈s − 1 and tracecycle(g) = true) q(s≥2) = prob(g ∈s≥2 and tracecycle(g) = true). table 4. subdivision of the probability q of (13). define bm (r) = ( 33 8 )m and note that q(r) = prob(g ∈ r) · prob(tracecycle(g) = true | g ∈ r) ≤ prob(tracecycle(g) = true | g ∈r). then proposition 7.3 gives q(r) ≤ bm (r) nm(1−s) ≤ bm (r) n` . define bm (s0) = aδc2δr 2s+2δ72. then proposition 8.4 and (3) give q(s0) ≤ bm (s0) n3−2s−2δ ≤ bm (s0) n` . define bm (s+1 ) = aδc 3 δr 3δ6.24. then proposition 9.1 and (3) give q(s+1 ) ≤ bm (s+1 ) n1+s−3δ ≤ bm (s+1 ) n` . 126 s. linton, a.c. niemeyer c.e. praeger define bm (s≥2) = c2δr 2δ−2s. then proposition 10.1 gives q(s≥2) ≤ bm (s≥2) n2s−2δ ≤ bm (s≥2) n` . define bm (s−1 ) = ( 31 r1−s )m . then proposition 11.1(b) yields q(s−1 ) ≤ bm (s−1 ) nm(1−s) ≤ bm (s−1 ) n` . thus by (7), bm (r) + bm (s0) + bm (s+1 ) + bm (s≥2) + bm (s − 1 ) ≤ ( 33 8 )m + aδc 2 δr 2s+2δ72 + aδc 3 δr 3δ6.24 + c2δ r2s−2δ + ( 31 r1−s )m = bm and q ≤ bm n` . (15) remark 4.1. we make a critical observation that the argument up to this point relies only on the first two inequalities of (8), and does not depend on the third inequality of (8). by (15) and the inequalities (14) and (12), we have that prob(b) < q 1 −p ≤ bm n` ρ n ( n−2 n )m = bm ρ ( n n− 2 )m 1 n`−1 . we showed above that ( n n−2 )m ≤ 9 8 ≤ 5ρ. thus prob(b) < 5bm n`−1 . by assumption n ≥ ( 10bm ε )1/(`−1) and so this is at most ε/2. hence prob(b) < ε/2. the proof of theorem 3.1(a) requires a short argument applying theorem 3.1(b). proof of theorem 3.1(a). we use the algorithm tracecycle with m = 4. note first that the probability that a random element h ∈ h corresponds to an element g ∈ g containing an m-cycle, given that the h-cycles containing four random k-subsets λ1, . . . ,λ4 all have lengths of the form rim with ri | r, is prob(g ∈n | tracecycle(g) = true). recall the definition of q in (13). then prob(g ∈n | tracecycle(g) = true) = prob(g ∈n and tracecycle(g) = true) prob(tracecycle(g) = true) = prob(tracecycle(g) = true) −q prob(tracecycle(g) = true) = 1 − q prob(tracecycle(g) = true) ≥ 1 − q prob(g ∈ngood and tracecycle(g) = true) = 1 − q prob(tracecycle(g) = true | g ∈ngood) ·prob(g ∈ngood) . 127 identifying long cycles set s = 5 8 , δ = 1 24 and let ` = 1 + 1 6 . note that ` = min{m(1 −s), 3 − 2s− 2δ, 1 + s− 3δ, 2s− 2δ}, so in particular the inequalities (4) and (5) all hold. we choose n0 to be the least natural number for which inequality (8) holds. hence the inequality (2) holds and in particular also 12(rn)s + 6 ≤ n and (rn)s log(n) ≤ n. inequality (15) holds by remark 4.1, so we have q ≤ b4 n` , where, since m = 4, the constant b4 given by (7), satisfies b4 ≤ ( 33 8 )4 + 72 aδc 2 δr 5/3 + 6.24 aδc 3 δr 3/8 + c2δ r7/6 + ( 31 r7/24 )4 . by proposition 7.5 we have that prob(tracecycle(g) = true | g ∈ ngood) ≥ ( n−2 n )4 . also, by equation (1), prob(g ∈ngood) = ρ(g,n,m) m . hence, using n ≥ n0, and the displayed inequality above, we have prob(g ∈n | tracecycle(g) = true) ≥ 1 − b4 n1+ 1 6 ( n n− 2 )4 m ρ(g,n,m) ≥ 1 − ( n0 n0 − 2 )4 b4 ρ(g,n,m) · 1 n 1 6 = 1 − c n 1 6 , where c = ( n0 n0−2 )4 b4 ρ(g,n,m) . 5. preliminaries it is useful to collect together some of the arithmetic facts we use in the rather delicate estimations in the remaining sections. lemma 5.1. let n,m,r be as in one of the lines of table 1, and let d be a divisor of rm with d ≤ n. then either d = m, or d ≤ 2m/7, or r,d are as in table 5. in particular, either d ≤ 2m/7 or d is one of r d 1 m 3 m 2 2 m 3 2m 5 2m 3 3 3m 5 3m 7 table 5. possibilities for r and d at most 3 different divisors of rm greater than 2m/7 and in the latter case d ≤ 2m/3 ≤ 2n/3. proof. we have d = r0 mj , where r0 divides r and j divides m. if j = 1 then d = m since 2m ≥ 2(n− 6) > n. so assume j ≥ 2. assume also that d > 2m/7, or equivalently 7r0 > 2j. if m is even, then (see table 1) r = 1. hence r0 = 1 and j ≤ 2. thus d = m/2 or m/3 as in table 5. so assume now that m is odd, so j ≥ 3. if j = 3 then we have the examples (r,d) = (1, m 3 ), (2, m 3 ), (1, 2m 3 ) in table 5 and no others since if r = 3 then (see table 1) gcd(m, 6) = 1. now assume that j ≥ 5. then r0 > 1 and we find (r,d) = (2, 2m 5 ), (3, 3m 5 ), (3, 3m 7 ) in table 5 and no others (since gcd(m, 6) = 1 when r = 3.). the next result follows from the fact that log(1 −p) > −p for 0 < p < 1. 128 s. linton, a.c. niemeyer c.e. praeger lemma 5.2. let ε,p be real numbers such that 0 < ε < 1 and 0 < p < 1. set n(ε,p) := ⌈ log(ε−1) p ⌉ . if m ≥ n(ε,p) then (1 −p)m ≤ ε. lemma 5.3. let s be a real number with 1 2 < s < 1 and n,r,t positive integers such that 12(rn)s + 6 ≤ n. then (i) ms/n < ns/n < (rn)s/n < 1/12. (ii) n ≥ 156. (iii) 2(rn)s − t > 24−t 12 (rn)s. (iv) if s = 2/3 then n ≥ 1746. proof. (i) this follows directly from 12(rm)s < 12(rn)s < 12(rn)s + 6 ≤ n. (ii) as s > 1/2 and r ≥ 1 we have 12 √ n + 6 ≤ 12 √ rn + 6 < 12(rn) s + 6 ≤ n. an easy calculation shows that this implies n ≥ 156. (iii) note that n ≥ 156 implies ns > n1/2 ≥ √ 156 > 12 and so 2(rn)s−t = (2rs− t ns )ns > (2rs− t 12 )ns = 24rs−t 12 ns ≥ 24−t 12 rsns. (iv) by calculator. the next inequalities are easily verified. lemma 5.4. let x ∈ r with x > 12. then (a) x ( 1 2 )x < 1 4x , and (b) ( 11 12 )x < 5 x . for the estimates in our last arithmetic result lemma 5.6, we first restate how to estimate sums via integrals. lemma 5.5. let a,b ∈ z with a < b, and let f(x) be a function defined on the interval [a − 1,b + 1], satisfying one of the lines of table 6. then b∑ x=a f(x) ≤ ∫ b+ε a−δ f(t)dt. conditions on f δ ε increasing in [a,b + 1] 0 1 decreasing in [a− 1,b] 1 0 non-negative in [a− 1,b + 1] and for some c ∈ (a,b) decreasing in [a− 1,c] and increasing in [c,b + 1] 1 1 table 6. conditions on f 129 identifying long cycles lemma 5.6. let a,c ∈ r+ and n ∈ z+ with n > a > c + 2 ≥ 3, and let t,` ∈ z+ with t ≥ 2 and t ≥ `. then, summing over integers x in the interval (a,n], ∑ a<x≤n xt (x− c)` < `−2∑ i=0 ( t i ) ct−i(a− 1 − c)i+1−` `− i− 1 + ( t `− 1 ) ct+1−` log(n) + t∑ i=` ( t i ) ct−i(n + 1 − c)i+1−` i + 1 − ` . proof. note first that if t > ` the function f(x) = x t (x−c)` is decreasing on (a, tc t−`] and increasing on [ tc t−`,n], while if t = ` then f(x) is decreasing on (a,n]. in either case, by lemma 5.5 we have∑ a<x≤n f(x) < ∫n+1 a−1 f(x)dx. now∫ n+1 a−1 xt (x− c)` dx = ∫ n+1−c a−1−c (y + c)t y` dy = ∫ n+1−c a−1−c y−` t∑ i=0 ( t i ) yict−idy = t∑ i=0 ( t i ) ct−i ∫ n+1−c a−1−c yi−`dy =   ∑ 0≤i≤t,i6=`−1 ( t i ) ct−i yi+1−` i + 1 − ` + ( t `− 1 ) ct+1−` log y  n+1−c y=a−1−c = ∑ 0≤i≤t,i 6=`−1 ( t i ) ct−i (n + 1 − c)i+1−` − (a− 1 − c)i+1−` i + 1 − ` + ( t `− 1 ) ct+1−`(log(n + 1 − c) − log(a− 1 − c)) < `−2∑ i=0 ( t i ) ct−i (a− 1 − c)i+1−` `− i− 1 + ( t `− 1 ) ct+1−` log(n) + t∑ i=` ( t i ) ct−i (n + 1 − c)i+1−` i + 1 − ` . 6. binomial inequalities and partitions in this section we prove a result about partitions that will be needed in sections 11 and 7. as preparation, we prove an inequality about certain binomial coefficients. lemma 6.1. let a be an integer such that a > 1, and let c,` be integers such that 1 ≤ ` < c. then( ca− 1 a− 1 )( c ` ) ≤ ( ca `a ) . proof. the proof is by induction on `, for fixed c,a. since ( c ` ) = ( c c−` ) and ( ca `a ) = ( ca (c−`)a ) , it is sufficient to prove this for 1 ≤ ` ≤ bc/2c. suppose first that ` = 1. here it is straightforward to check that ( ca− 1 a− 1 )( c 1 ) = ( ca a ) . 130 s. linton, a.c. niemeyer c.e. praeger now suppose that 1 ≤ ` < bc/2c and that the inequality holds for `. then, using induction we have( ca− 1 a− 1 )( c ` + 1 ) = ( ca− 1 a− 1 )( c ` ) c− ` ` + 1 ≤ ( ca `a ) c− ` ` + 1 . this latter quantity is at most ( ca (`+1)a ) if and only if c− ` ` + 1 · 1 (`a)!(ca− `a)! ≤ 1 (`a + a)!(ca− `a−a)! (16) and this is equivalent to c− ` ` + 1 ≤ ca− `a `a + a · ca− `a− 1 `a + a− 1 . . . ca− `a−a + 1 `a + 1 . now the first factor on the right hand side is equal to (c− `)/(` + 1), and each of the other factors is at least 1 since c ≥ 2` + 1. thus the inequality (16) holds, and so the induction proof is complete. lemma 6.2. (a) for 2 ≤ k ≤ d < n we have( d k ) ≤ ( d n )k ( n k ) and moreover, if d ≤ αn for some α < 1 then( d k )( n k ) ≤ αk−1 d−k + 1 n−k + 1 ≤ αk. (b) for 2 ≤ k ≤ 2n/3 we have ( bn/2c bk/2c ) < 2 ( n k )( 3k 4n )dk/2e . proof. every part of the proof depends on the following observation: fact 1: for 0 ≤ i ≤ t ≤ n with i < n we have t−i n−i ≤ t n with strict inequality if t < n. for (a) observe that ( v k )( n k ) = k−1∏ i=0 v − i n− i ≤ k−1∏ i=0 v n = (v n )k . if d ≤ αn for some α < 1 then( d k )( n k ) = ( k−2∏ i=0 d− i n− i ) · d−k + 1 n−k + 1 ≤ ( k−2∏ i=0 αn− i n− i ) · d−k + 1 n−k + 1 . now, again by fact 1, (αn− i)/(n− i) ≤ α and d−k+1 n−k+1 ≤ d n ≤ α, and therefore( d k )( n k ) ≤ αk−1 d−k + 1 n−k − 1 ≤ αk. 131 identifying long cycles for (b) let n0 = bn/2c and k0 = bk/2c. note then that( n0 k0 )( n k ) = n0(n0 − 1) · · ·(n0 −k0 + 1) n(n− 1) · · ·(n−k + 1) k(k − 1) · · ·(k0 + 1) = k0−1∏ i=0 n0 − i n− i · k−1∏ j=k0 k + k0 − j n− j . now k + k0 ≤ 2n/3 + n/3 ≤ n. applying fact 1 with t = n0 to the first product and with t = k + k0 to the second, we obtain ( n0 k0 )( n k ) ≤ k0−1∏ i=0 n0 n · k−1∏ j=k0 k + k0 n = (n0 n )k0 · ( k + k0 n )k−k0 ≤ ( 1 2 )k0 · ( 3k 2n )k−k0 = 3dk/2e 2k · ( k n )dk/2e ≤ 2 · 3dk/2e 4dk/2e · ( k n )dk/2e . note that the first inequality is strict if either k0 ≥ 2 or k − 1 > k0, that is, if k ≥ 3. if k = 2 then( n0 k0 ) = bn/2c, while 2 ( n k )( 3k 4n )dk/2e = 3 2 (n− 1) > bn/2c. thus (b) is proved for all k. lemma 6.3. let d,k,t be positive integers and a > 0 such that k ≤ d and t d−k+1 ≤ a. then (d + t)(d + t− 1) . . . (d + t−k + 1) < d(d− 1) . . . (d−k + 1)(1 + (1 + a)kt a(d−k + 1) ). proof. note first that (d + t)(d + t− 1) . . . (d + t−k + 1) = d(1 + t d )(d− 1)(1 + t (d− 1) ) . . . (d−k + 1)(1 + t (d−k + 1) ) ≤ d(d− 1) . . . (d−k + 1)(1 + t (d−k + 1) )k. set x = t d−k+1, so 0 < x ≤ a. then (1 + x)k = k∑ j=0 ( k j ) xj = 1 + x k∑ j=1 ( k j ) xj−1 ≤ 1 + x k∑ j=1 ( k j ) aj−1 132 s. linton, a.c. niemeyer c.e. praeger ≤ 1 + x a k∑ j=0 ( k j ) aj = 1 + x a (1 + a)k. now we state and prove the result on partitions. proposition 6.4. let u be a finite set of size u > 1, and let p be a partition of u in which all parts have size at least 2. for 2 ≤ k0 ≤ u, let np(k0) denote the number of k0-subsets of u that are unions of parts of p. then np(k0) ≤ (bu/2c bk0/2c ) , and moreover, if k0 is odd and u is even, then u ≥ 4 and np(k0) ≤ ( (u−2)/2 (k0−1)/2 ) . in particular, np(k0) = 1 if k0 = u and np(k0) ≤ 1u−1 ( u k0 ) otherwise. proof. first we construct a partition p′ of u having at most two parts of size 1, and all parts of size at most 2. start with p′ = ∅ and run through the parts of p. for each part p ∈p of even size, choose any partition of p with all parts of size 2, and add the parts of this partition to p′. if all parts of p have even size, then the construction of p′ is completed in this way. so suppose that p has at least one part of odd size. in this case p′ will have 1 or 2 parts of size 1, and its construction is completed as follows. for each part p ∈ p of odd size p := |p |, add (p− 1)/2 parts of size 2 to p′ formed from p− 1 of the points of p. let p1, . . . ,pr be the odd length parts of p. pair up the remaining r points into parts of size 2 and add them to p′, leaving exactly 1 or 2 of these points to form singleton parts of p′. next we define, for each k0-subset η of u that is a union of parts of p, a k0-subset η′ that is a union of parts of p′. note that if k0 is odd then η must contain a part of p of odd size, and in this case p′ has one or two singleton parts. if k0 is odd and p′ has two singleton parts, then we choose one of them, and we always place this chosen singleton part in η′. to define η′ for a given η, we start with η′ = ∅ and build it up by considering in turn each of the parts p of p contained in η. if |p | is even, then p is a union of parts of p′ of size 2, and we add all of these parts to η′. if |p | is odd, then we add to η′ all the parts of size 2 of p′ contained in p . at this stage |η′| = k0 −`, where ` is the number of odd sized parts of p contained in η. next we add to η′ up to b`/2c parts of p′ of size 2 that contain points from two different parts of p. if η′ cannot be completed in this way then either (i) ` is odd, or (ii) ` is even and is equal to the number of odd sized parts of p. case (i) occurs if and only if k0 is odd, and here we add to η′ the designated singleton part of p′. in case (ii) there are two singleton parts of p′, and we add to η′ these two singleton parts. note that, if ` ≥ 2, then we may have had some freedom in choosing the b`/2c parts of p′ of size 2 that contain points from two different parts of p, so η′ may not be determined uniquely by η. on the other hand, η′ always determines η uniquely, since η is the union of the parts of p that have at least two points in η′. thus distinct sets η correspond to distinct sets η′. it follows that np(k0) ≤ n′ where n′ is the number of k0-subsets γ ⊆ u such that γ is a union of parts of p′ and in addition, if k0 is odd and p′ has two singleton parts, then γ contains a designated one of these singleton parts. suppose that γ is such a k0-subset. if p′ has at most one part of size 1, then γ contains bk0/2c of the parts of p′ of size 2 (and also a singleton part if k0 is odd). thus n′ ≤ (bu/2c bk0/2c ) . note that in this case, if k0 were odd, then p would have at least one odd part, and so p′ would have exactly one odd part, whence u would be odd. thus the first assertion is proved in this case. so suppose that p′ has two singleton parts, in which case u is even. if k0 is odd then k0 ≥ 3 and γ consists of bk0/2c of the parts of p′ of size 2 and the designated singleton part, whence u ≥ 4 and n′ ≤ ( (u−2)/2 (k0−1)/2 ) < (bu/2c bk0/2c ) . on the other hand, if k0 is even then γ consists of k0/2 of the two-point parts (or k0/2 − 1 parts of size two and the two singleton parts). again n′ ≤ (bu/2c bk0/2c ) . this proves the first assertion in all cases. note that bu/2c = bk0/2c if and only if either k0 = u, or k0 = u−1 with u odd. if k0 = u obviously np(k0) = n ′ = 1. if k0 = u− 1 with u odd then p′ has a unique part of size 1 and its complement is 133 identifying long cycles the unique k0-subset of u that is a union of parts of p′ it may or may not be a union of parts of p. thus np(k0) ≤ n′ = 1 ≤ 1u−1 ( u k0 ) . so suppose from now on that bk0/2c < bu/2c, and set u1 = bu/2c and k1 = bk0/2c. then (bu/2c bk0/2c ) =( u1 k1 ) , and by lemma 6.1, this is at most 1 2u1−1 ( 2u1 2k1 ) . if k0 and u are even, then k0 < u and this quantity is at most 1 u−1 ( u k0 ) . if k0 is even and u is odd, then 2 ≤ k0 and 12u1−1 ( 2u1 2k1 ) = 1 u−2 ( u−1 k0 ) . this in turn is at most 1 u−1 ( u k0 ) . now suppose k0 and u are odd. then bk0/2c < bu/2c implies k0 ≤ u − 2, and 1 2u1−1 ( 2u1 2k1 ) = 1 u−2 ( u−1 k0−1 ) which is at most 1 u−1 ( u k0 ) . finally consider k0 odd and u is even. as shown above u ≥ 4 and n′ ≤ ( (u−2)/2 (k0−1)/2 ) . by lemma 6.1, this is at most 1 u−3 ( u−2 k0−1 ) , which in turn is at most 1 u−1 ( u k0 ) . for a prime p and an integer n, let np denote the p-part of n, that is the highest power of p dividing n. recall that, for a positive integer k0 ≤ n, a k0-subset γ′ of ω, and an element g ∈ sn, we denote by ck0 (γ ′,g) the length of the g-cycle containing γ′ in the action of g on k0-sets. lemma 6.5. let g ∈ sn, let c be a g-cycle of length t, let k0 be a positive integer such that k0 ≤ t and let p be a prime dividing t. (a) suppose that γ′ is a k0-subset of c such that the p-part tp does not divide ck0 (γ ′,g). then γ′ is a union of z(c,p)-orbits, where z(c,p) is the subgroup of order p of the cyclic group 〈gc〉 ∼= zt induced by g on c. in particular p divides gcd(k0, t). (b) the number σ(k0,c) of k0-subsets γ′ of c such that tp does not divide ck0 (γ ′,g) is at most ( bt/2c bk0/2c ) , and in particular, is 1 if k0 = t, and at most 1t−1 ( t k0 ) if k0 < t. proof. (a) since tp does not divide ck0 (γ ′,g) and 〈gc〉∼= zt, it follows that the setwise stabiliser h of γ′ in 〈gc〉 contains the unique subgroup z(c,p) of 〈gc〉 of order p. as γ′ is h-invariant, γ′ is a union of h-orbits in c, and hence γ′ is a union of z(c,p)-orbits in c. in particular, p divides k0 as well as t. (b) if k0 = t then c is its unique k0-subset and σ(k0,c) = 1. if k0 < t then, by proposition 6.4, σ(k0,c) ≤ ( bt/2c bk0/2c ) and also σ(k0,c) ≤ 1t−1 ( t k0 ) . corollary 6.6. let g,n,m,r be as in one of the lines of table 1, and let g ∈ g. let σ(g) be as in table 2 with u = |σ(g)|, and let k0 be a positive integer such that k0 ≤ u. then the number σ(k0, σ(g)) of k0-subsets γ′ of σ(g) such that ck0 (γ ′,g) divides rm satisfies σ(k0, σ(g)) ≤   0 if k0 = 1 1 if k0 = u 1 u−1 ( u k0 ) if 1 < k0 < u. proof. for each g-cycle c in σ(g), by the definition of σ(g), |c| does not divide rm, and hence there exists a prime p(c) such that |c|p(c) does not divide rm. let z(c,p(c)) denote the subgroup of order p(c) of the cyclic group 〈gc〉 induced by g on c, let p(c) denote the set of z(c,p(c))-orbits in c (all of length p(c)), and let p = ∪cp(c) denote the corresponding partition of σ(g). suppose that γ′ is a k0-subset of σ(g), and for each g-cycle c in σ(g), let k(c) = |γ′ ∩ c|. then ck0 (γ ′,g) is the least common multiple of ck(c)(γ′ ∩c,g), over all g-cycles c such that k(c) 6= 0. note that ck(c)(γ′ ∩c,g) divides |c|. suppose now that ck0 (γ ′,g) divides rm. then for each c such that k(c) 6= 0, also ck(c)(γ′ ∩c,g) divides rm, and hence |c|p(c) does not divide ck(c)(γ′∩c,g). by lemma 6.5, γ′∩c is a union of parts of p(c). thus γ′ is a union of parts of p. since all parts of p have size at least 2, this implies that σ(k0, σ(g)) = 0 if k0 = 1, and the inequality for 1 < k0 ≤ u follows from proposition 6.4. 134 s. linton, a.c. niemeyer c.e. praeger 7. tracing k-subsets for the remainder of this paper we assume that k is an integer with 2 ≤ k ≤ n/2. we use ∆(g), σ(g) and other notation introduced in tables 2 and 3. further, we use without further reference the number m of independent uniformly distributed random k-subsets in algorithm 2 tracecycle, where m satisfies (2), in particular m ≥ 4. proposition 7.1. let g,n,m,r be as in one of the lines of table 1, and suppose that g ∈ g does not contain an m-cycle. set v = |∆(g)| and suppose that v ≤ n−k − 1. then the proportion of k-subsets γ of ω such that ck(γ,g) = r0m, for some r0 dividing r, is at most v k nk + 1 n−v−1. proof. set u = n−v = |σ(g)|. suppose that γ is a k-subset of ω such that ck(γ,g) = r0m for some r0 dividing r, and set k0 := |γ ∩ σ(g)|. then k0 ≤ min{k,u}. by assumption, v ≤ n − k − 1 and so u = n−v ≥ k + 1 and k0 ≤ min{k,u} = k. also ck0 (γ ∩ σ(g),g) divides ck(γ,g), and hence divides rm. by corollary 6.6, the number σ(k0, σ(g)) of k0-subsets γ′ of σ(g) such that ck0 (γ ′,g) divides rm is 0 if k0 = 1, 1 if k0 = u, and at most 1u−1 ( u k0 ) , otherwise. if k0 = 0 then γ is one of the ( v k ) k-subsets of ∆(g). thus the number of possibilities for γ is at most( v k ) + k∑ k0=2 σ(k0, σ(g)) ( n−u k −k0 ) ≤ ( v k ) + 1 u− 1 k∑ k0=2 ( u k0 )( n−u k −k0 ) < ( v k ) + 1 u− 1 ( n k ) . now u− 1 = n−v− 1, hence the above is ( v k ) + 1 n−v−1 ( n k ) . by lemma 6.2(a), ( v k ) is at most (v/n)k ( n k ) , which completes the proof. lemma 7.2. let g,n,m,r be as in one of the lines of table 1. let g be a uniformly distributed random element of g, and suppose that g does not contain an m-cycle, and that v = |∆(g)| ≤ n−k − 1. then the following both hold. (a) prob(tracecycle(g) = true) ≤ 2m (( vk nk )m + ( 1 n−v−1 )m) , (b) prob(tracecycle(g) = true) ≤ 16 max {( v n )4 , ( 1 n−v−1 )4} . moreover, if 3 ≤ v ≤ n− 3 then prob(tracecycle(g) = true) ≤ 16 ( v n )4 . proof. now tracecycle(g) = true if and only if ck(γ,g) = r0m, for some r0 dividing r, for each of the m independent uniformly distributed random k-sets γ tested during the algorithm. thus if g does not contain an m-cycle, the probability that tracecycle(g) = true is pm, where p is the proportion of ksubsets γ such that ck(γ,g) = r0m for some r0 dividing r. by proposition 7.1, p ≤ v k nk + 1 n−v−1 . note that pm ≤ p4 since p ≤ 1 and m ≥ 4. set x = v k nk and y = 1 n−v−1 . if x ≤ y then (x + y) m ≤ (2y)m = 2mym, and similarly if x ≥ y then (x + y)m ≤ 2mxm. it follows that pm ≤ 2m (xm + ym ), proving part (a). for (b), we observe that pm ≤ p4 ≤ (x + y)4 ≤ (2 max{x,y})4 = 16 · max{x,y}4. part (b) follows on noting that x ≤ v/n (since v ≤ n). finally suppose that 3 ≤ v ≤ n − 3. then n ≥ v + 3 ≥ v + 2 + 2 v−1 so n(v−1) ≥ v 2 + v and hence (n−v−1)v ≥ n, that is, v n ≥ 1 (n−v−1) . the last assertion now follows from part (b). 135 identifying long cycles now we analyse the effect of tracecycle applied to elements of r. proposition 7.3. let g,n,m,r be as in one of the lines of table 1 and suppose that 12(rn)s + 6 ≤ n. then, for a uniformly distributed random element g ∈ g, prob(tracecycle(g) = true | g ∈r) ≤ ( 33 8n1−s )m . proof. by definition, for g ∈ r, v = |∆(g)| ≤ 4(rn)s and g does not contain an m-cycle. by our assumptions on n and k and the hypothesis, we have n−k − 1 ≥ n/2 − 1 > 4(rn)s ≥ v. thus by proposition 7.1, the proportion of k-subsets γ such that ck(γ,g) = r0m, for some r0 dividing r, is at most v k nk + 1 n−v−1 ≤ (4rs)k nk(1−s) + 1 n−4(rn)s−1 . now tracecycle(g) = true if and only if ck(γ,g) = r0m, for some r0 dividing r, for each of m independent uniformly distributed random k-sets γ tested during the algorithm. thus, given g ∈r, the probability of this occurring is at most( (4rs) k nk(1−s) + 1 n− 4(rn)s − 1 )m . now 12(rn)s < n, that is to say, 4r s n1−s < 1 3 . also k ≥ 2, r ≤ 3 and s < 1. therefore ( 4r s n1−s )k ≤ ( 4r s n1−s )2 < 4rs 3n1−s < 4 n1−s . also, by assumption, n − 4(rn)s − 1 ≥ 8(rn)s + 5 > 8rsns > 8rsn1−s. therefore, the probability that tracecycle(g) = true is at most( 4 n1−s + 1 8rsn1−s )m ≤ ( 33 8n1−s )m . next we analyse the effect of tracecycle applied to elements of ngood (defined in table 3). lemma 7.4. let g,n,m,r be as in one of the lines of table 1, and let k0 be an integer satisfying 0 ≤ k0 ≤ k. let g ∈n and let c be the m-cycle contained in g. then the number of k0-subsets of c that can occur as γ ∩c, for a k-subset γ of ω such that ck(γ,g) is not divisible by m, is at most σk0 =   1 if k0 = 0 and k ≤ n−m 0 if gcd(m,k0) = 1 or if k0 < k −n + m ω(gcd(m,k0)) (bm/2c bk0/2c ) if gcd(m,k0) > 1 and k0 ≥ max{1,k −n + m} where ω(d) is the number of distinct prime divisors of an integer d. proof. let σ′ be the number of k0-subsets of c that can occur as γ ∩ c, for a k-subset γ of ω such that ck(γ,g) is not divisible by m. note that, if γ is such a k-subset, then γ \ c is contained in the complement c of c and hence k = |γ| ≤ k0 + |c| = k0 + n−m. thus if k0 < k −n + m then σ′ = 0. also if k0 = 0 ≥ k −n + m, then γ ∩c = ∅ so σ′ ≤ 1. suppose now that k0 > 0 and k0 ≥ k −n + m, that is, k0 ≥ max{1,k −n + m}. let γ be such that ck(γ,g) is not divisible by m. then ck0 (γ ∩c,g) properly divides m, and hence there exists a prime p dividing m such that the p-part mp does not divide ck0 (γ∩c,g). by lemma 6.5(a), p divides gcd(m,k0) (and in particular if gcd(m,k0) = 1 then σ′ = 0). if such a prime p exists then, by lemma 6.5(b), the number of k0-subsets γ ∩c such that mp does not divide ck0 (γ ∩c,g) is at most(bm/2c bk0/2c ) . finally there are at most ω(gcd(m,k0)) primes p to consider, and the proof is complete. 136 s. linton, a.c. niemeyer c.e. praeger proposition 7.5. let g,n,m,r be as in one of the lines of table 1 and suppose that g ∈ ngood, and 12(rn) s + 6 ≤ n. then the proportion of k-subsets γ of ω such that ck(γ,g) 6= mr0, for any r0 dividing r, is at most k∑ k0=max{k−(n−m),0} σk0 ( n−m k −k0 ) / ( n k ) ≤ √ 8k ( 3k 4m )dk/2e , where σk0 is as in lemma 7.4. moreover, for a uniformly distributed random element g ∈ g, prob(tracecycle(g) = true | g ∈ngood) ≥ ( n− 2 n )m . proof. let c denote the m-cycle in g and let γ be a k-subset of ω such that ck(γ,g) 6= mr0 for any r0 dividing r. by the definition of ngood, this implies that ck0 (γ ∩ c,g) is not divisible by m, where k0 = |γ ∩c|. now 0 ≤ k0 ≤ min{k,m} = k, and moreover k0 ≥ k− (n−m) since γ ⊆ (γ ∩c) ∪ (ω \c). given γ ∩ c, there are at most ( n−m k−k0 ) choices for γ \ c. hence, by lemma 7.4, the number of such k-subsets γ is at most x := k∑ k0=max{k−n+m,0} σk0 ( n−m k −k0 ) (17) where σ0 = 1, and σk0 = ω((gcd(m,k0)) (bm/2c bk0/2c ) for k0 > 0. now ω(gcd(m,k0)) ≤ ω(k0) ≤ √ 2k0 ≤ √ 2k (see for example, [11, p. 395]). hence, x ≤ √ 2k ∑k k0=max{k−n+m,0} (bm/2c bk0/2c )( n−m k−k0 ) and by lemma 6.2(b), we have, x ≤ √ 2k k∑ k0=max{k−n+m,0} 2 ( m k0 )( 3k0 4m )dk0/2e(n−m k −k0 ) ≤ √ 8k ( 3k 4m )dk/2e k∑ k0=max{k−n+m,0} ( m k0 )( n−m k −k0 ) ≤ √ 8k ( 3k 4m )dk/2e( n k ) and hence the proportion x/ ( n k ) ≤ p where p := √ 8k ( 3k 4m )dk/2e . now we consider the final assertion. note that tracecycle(g) = true if and only if, for each of the m independent uniformly distributed random k-subsets γ tested, we have ck(γ,g) = r0m for some r0 dividing r. the class ngood is, for some lines of table 1, a union of several conjugacy classes of elements of sn, say ngood = ∪cn(c). for g ∈n(c), the proportion p(c) of k-subsets γ of ω, such that ck(γ,g) 6= r0m for any r0 dividing r, may depend on the class c, although, as we have shown above, p(c) ≤ p for all c. thus, given g ∈ n(c), the probability that tracecycle(g) = true is (1 − p(c))m ≥ (1 − p)m. this implies that prob(tracecycle(g) = true | g ∈ngood) ≥ (1 −p) m . thus to complete the proof it is sufficient to prove that p ≤ 2 n for some upper bound p of x/ ( n k ) . note that, by lemma 5.3(ii), m ≥ n − 6 ≥ 150. suppose first that 4 ≤ k ≤ n 2 . we consider the function f(x) = ( 3x 4m )x 2 = e x 2 log 3x 4m on the interval [4, n 2 ]. note that 3x 4m ≤ 3n 8m < 1 and k 2 ≤ dk 2 e, so 137 identifying long cycles f(k) ≥ ( 3k 4m )dk/2e , and hence p ≤ √ 8kf(k). differentiating we have f ′(x) = f(x) 1 2 ( log( 3x 4m ) + 1 ) , and since f(x) > 0 for x > 0, it follows that f(x) has a unique minimum at log 3x 4m = −1, that is, when x = 4m 3e (which may or may not lie in the interval [4, n 2 ]). thus the maximum of f(x) on the interval [4, n 2 ] occurs at one of the endpoints. we claim that max{f(4),f(n 2 )} < 1 n3/2 . it follows from a proof of this claim that p ≤ √ 8kf(k) ≤ √ 8k 1 n3/2 ≤ 2 n , since k ≤ n 2 . since m ≥ n − 6 ≥ 150, we have m2 > 9n3/2, which implies that f(4) = ( 3 m )2 < 1 n3/2 . also 3n 8m ≤ 3 8 + 6 8m < 1 2 , and n3/2 < 2n/4. then, applying lemma 5.4(a), we find f( n 2 ) = ( 3n 8m )n/4 < ( 1 2 )n/4 < 1 n3/2 proving the claim for k ≥ 4. for the remaining cases where k = 2 or 3, note that ω(gcd(m,k0)) ≤ 1 for 1 ≤ k0 ≤ 3, σk0 = 0 when k0 = 1, n ≥ 156, and n−m ≤ 6. if k = 2 then by (17), x( n 2 ) ≤ (62)(n 2 ) + (bm/2c1 ) ·(60)(n 2 ) ≤ 15 · 2 155 · 1 n + m n− 1 · 1 n < 2 n . if k = 3 then, again by (17), x( n 3 ) ≤ (63)(n 3 ) + (bm/2c1 )(61)(n 3 ) + (bm/2c1 )(60)(n 3 ) ≤ 20 · 6 154 · 155 · 1 n + 3 ·m(6 + 1) 154(n− 1) · 1 n < 2 n . 8. bounding s0 let g,m,n,r be as in one of the lines of table 1, so g is an or sn. to estimate the probability of a uniformly distributed random element g ∈ g being in s0 or s+1 , and tracecycle(g) = true we use the following result from [8]. recall the definitions of an s-small and an s-large cycle and of v from notation 3.2. let i ∈{1, 2, 3}. in the next two sections we use the following notation: notation 8.1. 1. for v ≥ 1 let p(v,rm) denote the proportion of elements of sv of order dividing rm, and let p(0,rm) = 1. 2. for v ≥ 1 let p0(v,rm) denote the proportion of elements of sv of order dividing rm, all of whose cycles are s-small, and let p0(0,rm) = 1. 3. let p +1 (v,rm) denote the proportion of elements g ∈ sv of order dividing rm, and such that g has exactly one s-large cycle of length d, say, where in addition, d satisfies (rn)s ≤ d < v − 3(rn)s. 4. let d denote the set of all divisors of rm which are at most n. 5. let d+1 (v) denote the set of all divisors d of rm satisfying (rn) s ≤ d < v − 3(rn)s. note that r = 1 or r is a prime. hence the number d(rm) of positive divisors of rm is at most 2d(m), as d|rm if and only if either d|m or d = rd0 and d0|m. remark 8.2. the following result is essentially [8, lemma 2.4]. suppose that s, δ and cδ are as in notation 3.2. in particular, s > δ. in lemma 8.3 we may use as a′δ any constant such that a ′ δ ≥ 138 s. linton, a.c. niemeyer c.e. praeger a′δ m0 25/4 c 1/(s−δ) δ aδ as in (6) 150 table 7. possible values of a′δ for lemma 8.3 5 4 (1 + 3 cδ (rm)s−δ + ( cδ (rm)s−δ )2 ) for all sufficiently large values of rm, say rm ≥ m0. these conditions hold in particular for a′δ,m0 in one of the lines of table 7. note that, for the proof of theorem 3.1, we have n ≥ 156 by lemma 5.3(ii), so rm ≥ 150, as in line 2 of table 7. lemma 8.3. let m,n,r be as in one of the lines of table 1. further, let v ≥ 16 and s, δ, cδ and aδ be as in notation 3.2. let a′δ and m0 be as in one of the lines of table 7 (or more generally as in remark 8.2) and suppose that rm ≥ m0. then (a) p0(v,rm) < a′δd(rm)r 2sn2s v3 . (b) if 3(rn)s < v then p0(v,rm) ≤ a′δd(rm) 2r2sn2s v(v − (rn)s)3 . (c) p +1 (v,rm) = ∑ d∈d+1 (v) 1 d p0(v −d,rm). proof. this result follows from [8, lemma 2.4] and its proof. a direct application of [8, lemma 2.4] would require that rm ≥ v, which we cannot guarantee to hold. however, the proof of that lemma shows, without the assumption that rm ≥ v, that p0(v,rm) ≤ d(rm)(rm) 2s (1 + 3cδ(rm) δ−s + (cδ(rm) δ−s )2) v(v − 1)(v − 2) whenever v ≥ 3. statement (a) follows from this, since m ≤ n, δ < s and, for v ≥ 16, v(v−1)(v−2) > 4 5 v3. to prove (b) we let ds denote the set of all divisors d of rm such that d < min{v, (rn) s}. by [8, lemma 2.3(a)] we have that p0(v,rm) = 1v ∑ d∈ds p0(v−d,rm), where p0(j,m) = 0 for j ≤ 0. since, using lemma 5.3(ii), v−d > 3(rn)s−(rn)s > 24 for d ∈ ds, we have by (a) that p0(v−d,rm) ≤ a′δd(rm)r 2sn2s (v−d)3 . thus p0(v,rm) ≤ 1v ∑ d∈ds a′δd(rm)r 2sn2s (v−d)3 . since v −d > v − r sns > 0 for d ∈ ds, and |ds| ≤ d(rm), we have p0(v,rm) ≤ a′δd(rm) 2r2sn2s v(v−(rn)s)3 . finally, we prove (c). the number of permutations in sv of order dividing rm with exactly one s-large cycle of a given length, d say, where d divides rm and (rn)s ≤ d < v−3(rn)s is ( v d ) (d−1)!p0(v− d,rm)(v − d)!. hence the proportion in sv of such permutations is 1dp0(v − d,rm). summing over all d ∈ d+1 (v) yields the desired result. proposition 8.4. let g,m,n,r be as in one of the lines of table 1. if 12(rn)s+6 ≤ n and (rn)s log(n) ≤ n then, for a uniformly distributed random element g ∈ g, prob(g ∈s0 ∩g and tracecycle(g) = true) ≤ aδd(rm)2r2s 72 n3−2s where aδ is as in (6). 139 identifying long cycles proof. the set s0 = ∪̇s0(v), where s0(v) is the set of all g ∈s0 with |∆(g)| = v, where v ranges over all integers satisfying 4(rn)s < v ≤ n. for g ∈ s0(v), the restriction g∆(g) of g to ∆(g) is a permutation in sym(∆(g)) of order dividing rm with all cycles of length less than (rn)s. consider a fixed v-set ∆. if g = sn, then all elements of sym(∆) are induced by permutations in g. on the other hand if g = an, then one of the lines 4-9 of table 1 holds and hence rm is odd; thus all elements of sym(∆) of order dividing rm actually lie in alt(∆) and are therefore induced by elements of g. therefore in all cases the number of possibilities for the restriction g∆ of elements g ∈ g, for a given v-subset ∆ = ∆(g), is v!p0(v,rm) and the restriction gς where σ = ω\∆ lies in sym(σ) or alt(σ) according as g = sn or an, respectively. hence the number of permutations in s0 ∩g corresponding to this value of v satisfies |s0(v) ∩g| ≤ ( n v ) v!p0(v,rm) (n−v)! |sn : g| = n! p0(v,rm) |sn : g| = |g| ·p0(v,rm). as 3(rn)s < 4(rn)s < v, we have n ≥ 156 by lemma 5.3(ii) so rm ≥ 150, and hence we can apply lemma 8.3(b) with a′δ = aδ. thus, for a random g ∈ g, prob(g ∈s0(v) ∩g) ≤ p0(v,rm) ≤ aδd(rm) 2r2sn2s v(v − (rn)s)3 . for any g ∈ sn with |∆(g)| = v and v ≤ n − k − 1, we have in particular 3 ≤ v ≤ n − 3. hence by lemma 7.2(b), given that g ∈s0(v) ∩g with v ≤ n−k − 1, prob(tracecycle(g) = true) ≤ 16 (v n )4 . hence, if v ≤ n − k − 1, then the probability that g ∈ s0(v) and tracecycle(g) = true is at most aδd(rm)2r2sn2s 16v(v−(rn)s)3 ( v n )4 ; and if n − k − 1 < v ≤ n, this probability is at most aδd(rm) 2r2sn2s 1 v(v−(rn)s)3 . summing over the values of v, we find prob(g ∈s0 ∩g and tracecycle(g) = true) ≤ σ1 + σ2 where σ1 = 16aδd(rm) 2 r 2sn2s n4 ∑ 4(rn)s<v≤n−k−1 v3 (v − (rn)s)3 , σ2 = aδd(rm) 2r2sn2s ∑ n−k≤v≤n 1 (v − (rn)s)4 . we first consider σ1 and apply lemma 5.6 with a = 4(rn) s, c = (rn)s, t = ` = 3, and n−k − 1 in place of n. we also use a− 1 − c = 3(rn)s − 1 > 2(rn)s, and find σ1 = 16aδd(rm) 2 r 2sn2s n4 ∑ 4(rn)s<v≤n−k−1 v3 (v − (rn)s)3 < 16aδd(rm) 2 r 2sn2s n4 ( (rn) 3s 8(rn) 2s + 3(rn) 2s 2(rn) s + ( 3 2 ) (rn) s log(n−k − 1) + ( 3 3 ) ((rn)s) 0 (n−k − (rn)s)1 1 ) < 16 aδd(rm) 2r2s n3−2s ( (rn) s 8n + 3(rn) s 2n + 3 log(n)(rn) s n + n− (rn)s n ) . 140 s. linton, a.c. niemeyer c.e. praeger the assumption 12(rn)s + 6 ≤ n implies by lemma 5.3(i) that (rn)s/n < 1/12. also, by our hypothesis, (rn)s log(n) ≤ n and, therefore, σ1 < 16aδd(rm) 2r2s n3−2s ( 1 96 + 3 24 + 3 + 1) < 66.2aδd(rm) 2r2s n3−2s . finally, we estimate σ2. σ2 = aδd(rm) 2r2sn2s ∑ n−k≤v≤n 1 (v − (rn)s)4 . since k ≤ n/2, and since 1 v−(rn)s is decreasing for v in the interval [n−k− 2,n], we have by lemma 5.5 and lemma 5.3 that σ2 < aδd(rm) 2r2sn2s ∫ n n/2−1 1 (v − (rn)s)4 dv = aδd(rm) 2r2sn2s [ −1 3(v − (rn)s)3 ]n n/2−1 < aδd(rm) 2r2sn2s 1 3(n/2 − 1 − (rn)s)3 = aδd(rm) 2r2sn2s 8 3n3(1 − 2/n− 2(rn)s/n)3 < aδd(rm) 2r2sn2s 8 3n3(1 − 2/156 − 2/12)3 < 4.83 aδd(rm) 2r2s 1 n3−2s . adding the upper bounds for σ1 and σ2 we find that prob(g ∈s0 ∩g and tracecycle(g) = true) < aδd(rm)2r2s 72 n3−2s . 9. bounding s+1 let g,m,n,r be as in one of the lines of table 1, so g is an or sn. recall the definitions of an s-small and an s-large cycle and of v from notation 3.2 and the notation set out in notation 8.1. proposition 9.1. let g,n,m,r be as in one of the lines of table 1. if n is such that 12(rn)s + 6 ≤ n and (rn)s log(n) ≤ n , then for a uniformly distributed random element g ∈ g, prob(g ∈s+1 ∩g and tracecycle(g) = true) ≤ aδd(rm) 3 6.24 n1+s . proof. the set s+1 = ∪̇s + 1 (v), where s + 1 (v) is the set of all g ∈s + 1 with |∆(g)| = v and v ranges over integers satisfying 4(rn)s < v ≤ n. for a given v, an analogous argument to that given in the second paragraph of the proof of proposition 8.4 shows that |s+1 (v) ∩g| ≤ ( n v ) ·v!p +1 (v,rm) · (n−v)! |sn : g| = n! p +1 (v,rm) |sn : g| = p +1 (v,rm) · |g|. 141 identifying long cycles thus applying lemma 8.3(c) we have, for a random g ∈ g, prob(g ∈s+1 (v) ∩g) ≤ p + 1 (v,rm) = ∑ d∈d+1 (v) 1 d p0(v −d,rm). if |∆(g)| = v and v ≤ n−k − 1, then in particular 3 ≤ v ≤ n− 3. hence by lemma 7.2(b), given that g ∈s+1 (v) ∩g with |∆(g)| = v with v ≤ n−k − 1, prob(tracecycle(g) = true) ≤ 16 (v n )4 . thus, if v ≤ n−k − 1, the probability that g ∈s+1 (v) ∩g and tracecycle(g) = true is at most  ∑ d∈d+1 (v) 1 d p0(v −d,rm)   16 (v n )4 , and if n−k ≤ v this probability is at most∑ d∈d+1 (v) 1 d p0(v −d,rm). summing over v we find prob(g ∈s+1 ∩g and tracecycle(g) = true) ≤ σ1 + σ2 where σ1 = 16 ∑ 4(rn)s<v≤n−k−1   ∑ d∈d+1 (v) 1 d p0(v −d,rm)  (v n )4 , σ2 = ∑ n−k≤v≤n   ∑ d∈d+1 (v) 1 d p0(v −d,rm)   . first we consider σ1. interchanging the two summations and taking the sum up to n, we obtain the following upper bound, where d` denotes the set of all divisors d of rm satisfying d ≥ (rn) s . note that v > d + 3(rn) s (see notation 3.2). σ1 < 16 ∑ d∈d` 1 d   ∑ 3(rn)s+d<v≤n p0(v −d,rm) · v4 n4   . since rm ≥ 150 by lemma 5.3(ii), we may apply lemma 8.3(b) with a′δ = aδ, and find that this expression is at most 16 ∑ d∈d` 1 d   ∑ 3(rn)s+d<v≤n aδd(rm) 2r2sn2s (v −d)(v −d− (rn)s)3 · v4 n4   < 16 aδd(rm) 2r2sn2s n4 ∑ d∈d` 1 d   ∑ 3(rn)s+d<v≤n v4 (v −d− (rn)s)4   . 142 s. linton, a.c. niemeyer c.e. praeger now we apply lemma 5.6 with t = ` = 4, a = 3(rn)s + d and c = d + (rn)s. noting that a − c − 1 = 2(rn) s − 1, we obtain that this expression is at most 16 aδd(rm) 2r2sn2s n4 ∑ d∈d` 1 d (( 4 0 ) (d + (rn) s )4 3(2(rn) s − 1)3 + ( 4 1 ) (d + (rn) s )3 2(2(rn) s − 1)2 + ( 4 2 ) (d + (rn) s )2 (2(rn) s − 1) + ( 4 3 ) (d + (rn) s )1 log(n) + ( 4 4 ) (d + (rn) s )0(n + 1 −d− (rn)s)1 ) . note that 2(rn)s − 1 > 23 12 rsns by lemma 5.3(iii) and, since d ≥ (rn)s, also d+(rn) s d ≤ 2. note also that d + (rn)s < n and n + 1 −d− (rn)s < n. hence σ1 ≤ 16 aδd(rm) 3r2sn2s n4 ( 2 · 123 ·n3 3 · 233 ·r3sn3s + 4 · 122n2 232r2sn2s + 12 · 12n1 23rsns + 8 · log(n) + n1 (rn) s ) = 16aδd(rm) 3 n1+s ( 3456 36501rs + 576 529n1−s + 144rs 23n2−2s + 8r2s log(n) n3−3s + rs n2−2s ) . since, by hypothesis (rn)s log(n) ≤ n and by lemma 5.3(i) ns/n ≤ rsns/n ≤ 1/12 and r ≥ 1, the last expression is at most 16aδd(rm) 3 n1+s ( 3456 36501 + 576 529 · 12 + 144 23 · 122 + 8 122 + 1 122 ) ≤ 4.7aδ d(rm)3 n1+s . we now consider σ2 = ∑ n−k≤v≤n (∑ d∈d+1 (v) 1 d p0(v −d,rm) ) . as v−d > 3(rn)s and n−k ≥ n/2 we have by lemma 8.3(b) (with a′δ = aδ) that σ2 aδd(rm)2(rn) 2s ≤ ∑ n/2≤v≤n   ∑ d∈d+1 (v) 1 d(v −d− (rn)s)4   = ∑ d∈d+1 (v) 1 d   ∑ v(d)≤v≤n 1 (v −d− (rn)s)4   where v(d) = max{n 2 ,d + 3(rn)s} since, by notation 8.1, each d ∈ d+1 (v) is less than v − 3(rn) s. by 143 identifying long cycles lemma 5.5, this quantity is at most ∑ d∈d+1 (v) 1 d (∫ n v(d)−1 1 (v −d− (rn)s)4 dv ) = ∑ d∈d+1 (v) 1 d [ − 1 3 1 (v −d− (rn)s)3 ]n v(d)−1 < ∑ d∈d+1 (v) 1 3d 1 (v(d) − 1 −d− (rn)s)3 . in particular each d ∈ d+1 (v) is less than m. by lemma 5.1, there are at most three divisors of rm which are less than m and greater than 2m/7, and the sum of the reciprocals 1 d of these divisors is at most 7 m , which is less than 7.3 n since n ≥ 156 (by lemma 5.3(ii)). using v(d) ≥ d + 3(rn)s and lemma 5.3(iii), the contribution from these exceptional divisors is therefore at most 1 (2(rn)s − 1)3 ∑ d∈d+1 (v),d>2m/7 1 3d < ( 12 23(rn)s )3 7.3 3n < 0.35 (rn)3sn . finally we estimate the contribution of the remaining elements d of d+1 (v). we note that each such d is at most 2n 7 and at least (rn)s, and that (rn)s < n−6 12 by our hypothesis. thus, using v(d) ≥ n 2 , the remaining contribution is at most d(rm) 3(rn) s 1 (n 2 − 1 − 2n 7 − n−6 12 )3 . observe that n 2 −1− 2n 7 − n−6 12 = 11n−42 84 and since n > 84 by lemma 5.3(a) we have 11n−42 84 > n 8 . hence, using also that (rn) s n < 1 12 (by lemma 5.3(i)), the above expression is less than d(rm) (rn)s 83 3n3 < d(rm) 83 122 · 3 (rn)3sn < 1.19 d(rm) (rn)3sn . thus σ2 aδd(rm)2(rn)2s < 0.35 (rn)3sn + 1.19 d(rm) (rn)3sn ≤ 1.54 d(rm) (rn)2sn1+s and hence prob(g ∈s+1 ∩g and tracecycle(g) = true) < 6.24 aδ d(rm)3 n1+s . 10. bounding s≥2 proposition 10.1. let g,m,n,r be as in one of the lines of table 1. then |s≥2 ∩g| |g| ≤ d(rm)2 (rn) 2s . 144 s. linton, a.c. niemeyer c.e. praeger proof. if g is an element of s≥2∩g then it has two cycles of lengths d1,d2, where di|rm, and di ≥ (rn) s. there are at most d(rm) choices for each di. thus, there are at most d(rm)2 choices for the two divisors d1 and d2. for a given d1,d2, the proportion of elements in g having cycles of lengths d1 and d2 is at most (d1d2) −1 ≤ (rn)−2s. thus altogether we get a proportion of at most d(rm)2(rn)−2s. 11. bounding s−1 proposition 11.1. let g,m,n,r be as in one of the lines of table 1. suppose that n is such that 12(rn) s + 6 ≤ n. let k be a fixed integer with 2 ≤ k ≤ n/2. then (a) the proportion of k-subsets γ such that ck(γ,g) = r0m, for some r0 dividing r, for g ∈ s−1 ∩g, is less than 31/((rn)1−s). (b) if tracecycle is algorithm 2 and m is as defined there, then for a uniformly distributed random element g ∈ g, prob(tracecycle(g) = true | g ∈s−1 ∩g) < ( 31 (rn) 1−s )m , and so prob(g ∈s−1 ∩g and tracecycle(g) = true) < ( 31 (rn) 1−s )m . proof. we start by recording some important facts used throughout the proof. let g ∈ s−1 ∩g and put v = |∆(g)| and u = |σ(g)|, such that u + v = n. the definition of s−1 implies that g has a unique s-large cycle c in ∆(g) of length d and we have (i) d ≤ n and d 6= m since g ∈f; (ii) v > 4(rn)s and v −d ≤ 3(rn)s. by lemma 5.1 and the hypothesis n ≥ 12(rn)s + 6, it follows that d ≤ 2m/3 ≤ 2n/3. hence u = n−v ≥ n−d−3(rn)s ≥ n 3 −3(rn)s ≥ 4(rn)s + 2−3(rn)s = (rn)s + 2. also, v ≤ d + 3(rn)s ≤ 2n 3 + 3(rn) s . this implies that v = n−u ≤ n− 2 − (rn)s and hence in particular v ≤ n− 3 (18) and 1 u− 1 < 1 (rn) s < 1 (rn) 1−s . (19) set t = v −d so that t = v −d ≤ 3(rn)s. then v = d + t ≤ 2n/3 + 3(rn)s. (20) suppose that γ is a k-subset for which ck(γ,g) = r0m, for some r0 dividing r, and set k0 := |γ∩σ(g)|. then ck0 (γ ∩ σ(g),g) divides rm, and hence the number of possibilities for the k0-subset γ ∩ σ(g) is at 145 identifying long cycles most the number σ(k0, σ(g)) of corollary 6.6. in particular σ(k0, σ(g)) = 0 if k0 = 1. thus k0 = 0 or 2 ≤ k0 ≤ min{u,k}, and the case k0 = 0 is only possible if v ≥ k. first we prove the following upper bound for the number k¬0 = k¬0(g) of k-subsets γ such that k0 = |γ ∩ σ(g)| ≥ 2. k¬0( n k ) < 97 96(rn) 1−s . (21) by the remarks above k¬0 ≤ min{k,u}∑ k0=2 σ(k0, σ(g)) ( n−u k −k0 ) . if k0 ≤ u−1 then, by corollary 6.6 and our considerations above, σ(k0, σ(g)) ≤ 1u−1 ( u k0 ) ≤ 1 (rn)(1−s) ( u k0 ) , while if k0 = u then σ(k0, σ(g)) = 1. thus k¬0( n k ) ≤ 1(n k ) (rn)1−s min{u−1,k}∑ k0=2 ( u k0 )( n−u k −k0 ) + r¬0, where r¬0 =  0 if k ≤ u− 1,(n−uk−u) (nk) if k ≥ u. hence k¬0( n k ) ≤ 1(n k ) (rn)1−s min{u,k}∑ k0=0 ( u k0 )( n−u k −k0 ) + r¬0 = 1 (rn) 1−s + r¬0. (22) thus (21) is proved if k ≤ u− 1, so suppose that k ≥ u. recall that u > (rn)1−s + 1 by (19). hence r¬0 ≤ ( n−u k−u )( n k ) = u∏ i=1 (k −u + i) (n−u + i) ≤ ( k n )u ≤ ( 1 2 )u ≤ 1 2 ( 1 2 )(rn)(1−s) . using n1−s > 12 > 8 (see lemma 5.3(i)) and lemma 5.4(a), we have r¬0 ≤ 12 ( 1 2 )(rn)(1−s) < 1 2 1 4(rn)2(1−s) < 1 96 1 (rn)1−s , and now the inequality (21) follows from inequality (22). to complete the proof of part (a) it remains to estimate the number k=0 = k=0(g) of k-subsets γ ⊆ ∆(g) such that ck(γ,g) = r0m for some r0 dividing r. since this number is zero if v < k, we assume that v ≥ k. recall that c is the unique s-large cycle of g contained in ∆(g) and d = |c|. by lemma 5.1, d ≤ 2m/3 < 2n/3. since m divides ck(γ,g) it follows that γ 6⊆ c. we prove k=0( n k ) ≤ 30.6 (rn) 1−s . (23) the number k=0 of such k-subsets is at most ( v k ) − ( d k ) . 146 s. linton, a.c. niemeyer c.e. praeger set t = v −d so that t = v −d ≤ 3(rn)s. then we have( v k ) = 1 k! (d + t)(d + t− 1) . . . (d + t−k + 1). we consider separately the cases (i) (rn)s < k, (ii) k ≤ min{(rn)s,d− t + 1}, and (iii) d− t + 1 < k ≤ (rn)s. recall that (rn)s ≤ d. consider first case (ii), so k ≤ (rn)s and d − t + 1 ≥ k. if d ≤ m/2 define a = 1 and observe that t d−k+1 ≤ a. if d > m/2 then, by lemma 5.1, it follows that d ≥ 3m/5. in this case t d−k+1 ≤ 3(rn)s 3m/5−(rn)s = 3(rn)s m(3/5−(rn)s/m). by the hypothesis (rn) s ≤ (n − 6)/12 ≤ m/12 and by lemma 5.3(i) we have then t d−k+1 ≤ 3 12 1 (3/5−1/12) = 15 31 . in this case define a = 15 31 . then again t d−k+1 ≤ a. setting d(k) := d(d− 1) . . . (d−k + 1), by lemma 6.3 we obtain( v k ) = 1 k! (d + t)(d + t− 1) . . . (d + t−k + 1) < 1 k! ( d(k) ( 1 + (1 + a)kt a(d−k + 1) )) = ( d k )( 1 + (1 + a)kt a(d−k + 1) ) . (24) if d ≤ m/2 we have a = 1 and so ( v k ) ≤ ( d k ) + ( d k ) 2kt d−k + 1 . applying lemma 6.2(a) with α = 1 2 , ( d k ) ≤ 1 2k−1 ( n k ) d−k + 1 n−k + 1 . hence k=0 ≤ ( v k ) − ( d k ) < ( d k ) 2kt d−k + 1 ≤ ( n k ) 2t n−k + 1 < ( n k ) 2t n−k . on the other hand, if m/2 < d ≤ 2n/3, then a = 15 31 , and (24) becomes( v k ) ≤ ( d k ) + ( d k )( 46 31 )k 31t 15(d−k + 1) . by lemma 6.2(a) with α = 2 3 , ( d k ) ≤ 2k−1 3k−1 ( n k ) d−k + 1 n−k + 1 and hence k=0 = ( v k ) − ( d k ) < ( d k ) (46/31) k 31t 15(d−k + 1) < ( n k ) 2k−1 3k−1 (46/31) k 31t 15(n−k + 1) < ( n k ) 92k 93k 31t 10(n−k) < ( n k ) 31t 10(n−k) . 147 identifying long cycles note that by lemma 5.3, since k ≤ (rn)s and by our assumptions, t n−k ≤ 3(rn)s n(1−k/n) ≤ 3(rn)s n(1−(rn)s/n) ≤ 3(rn)s n(11/12) = 36(rn)s 11n . thus for all d we have k=0( n k ) < 31 · 36 10 · 11 · (rn) s n < 10.2 r (rn) 1−s ≤ 30.6 (rn) 1−s and (23) is proved for case (ii). now consider cases (i) and (iii). recall from (20) that v = d + t ≤ 2n/3 + 3(rn)s. by lemma 5.3(i), (rn) s ≤ 1 12 n. therefore v ≤ 2n/3 + 3 12 n = 11 12 n. this shows, using lemma 6.2(a), that k=0( n k ) ≤ (vk)−(dk)(n k ) < (vk)(n k ) ≤ (v n )k ≤ ( 11 12 )k . in case (i) we have k > (rn)s and hence, observing that (rn)s > n1/2 > 12 by lemma 5.3(i), and using lemma 5.4(b), we have ( 11 12 )k < ( 11 12 )(rn)s < 5 (rn)s < 5 (rn)1−s . thus k=0 (nk) < 5 (rn)1−s and (23) holds for case (i). in case (iii) we have (rn)s ≥ k > d− t + 1 and so d < 4(rn)s as t ≤ 3(rn)s. therefore, v = d + t < 7(rn) s, and using lemmas 5.3(i) and 6.2(a), k=0( n k ) ≤ (v n )k < ( 7(rn) s n )k ≤ ( 7(rn) s n )2 < 49(rn) s 12n ≤ 49 4(rn) 1−s . thus (23) holds for case (iii) and hence in all cases. combining (23) with (21), we conclude that the proportion of k-subsets γ such that ck(γ,g) = r0m, for some r0 dividing r, is less than 31/((rn) 1−s ) for all values of k and v. this proves (a). now tracecycle(g) = true if and only if ck(γ,g) = r0m, for some r0 dividing r, for each of the m independent uniformly distributed random k-sets γ tested in the algorithm. thus, given g ∈s−1 ∩g, the probability that tracecycle(g) = true is at most ( 31/((rn) 1−s ) )m . the last assertion follows on noting that for events a and b we have prob(a∩b) = prob(a)prob(b | a) ≤ prob(b | a). acknowledgment: the first author acknowledges the support of epsrc grant ep/c523229 and the second and third authors acknowledge the support of arc discovery grants dp0879134 and dp140100416. we thank yohei negi and sven reichard for some discussions on an early draft of this paper. we thank an anonymous referee for valuable suggestions. 148 s. linton, a.c. niemeyer c.e. praeger references [1] r. m. beals, c. r. leedham-green, a. c. niemeyer, c. e. praeger, and á. seress, a black-box algorithm for recognizing finite symmetric and alternating groups, i, trans. amer. math. soc., 355, 2097-2113, 2003. [2] s. bratus and i. pak, fast constructive recognition of a black box group isomorphic to sn or an using goldbach’s conjecture, j. symbolic comput., 29(1), 33-57, 2000. [3] the gap group, gap groups, algorithms, and programming, version 4.7.7, 2015, http://www.gap-system.org. [4] p. erdős, and p. turán, on some problems of a statistical group-theory. i, wahrscheinlichkeitstheorie verw. gebiete, 4, 175-186, 1965. [5] p. erdős, and p. turán, on some problems of a statistical group-theory. iii, acta math. acad. sci. hungar., 18 , 309-320, 1967. [6] s. linton, a. c. niemeyer and c. e. praeger, constructive recognition of sn in its action on k-sets, in preparation. [7] y. negi, recognising large base actions of finite alternating groups, honours thesis, school of mathematics and statistifcs, the university of western australia, 2006. [8] a. c. niemeyer and c. e. praeger, on permutations of order dividing a given integer, j. algebraic combinatorics, 26, 125-142, 2007. [9] a. c. niemeyer and c. e. praeger, on the proportion of permutations of order a multiple of the degree, j. london math. soc., 76, 622-632, 2007. [10] a. c. niemeyer, c. e. praeger and á. seress, estimation problems and randomised group algorithms, probabilistic group theory, combinatorics, and computing, lecture notes in math., 2070, springer, london, 35-82, 2013. [11] i. niven, h. s. zuckerman and h. l. montgomery, an introduction to the theory of numbers, john wiley & sons inc., new york, fifth edition, 1991. [12] á. seress, permutation group algorithms, cambridge tracts in mathematics, 152, cambridge university press, cambridge, 2003. [13] r. warlimont, über die anzahl der lösungen von xn = 1 in der symmetrischen gruppe sn, arch. math., 30, 591-594, 1978. 149 introduction algorithmic application statement of the main theorem and notation proof of the main theorem preliminaries binomial inequalities and partitions tracing k-subsets bounding s0 bounding s1+ bounding s2 bounding s1 references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1056511 j. algebra comb. discrete appl. 9(1) • 17–27 received: 19 october 2020 accepted: 8 october 2021 journal of algebra combinatorics discrete structures and applications recent results on choi’s orthogonal latin squares research article jon-lark kim∗, dong eun ohk, doo young park, jae woo park abstract: choi seok-jeong studied latin squares at least 60 years earlier than euler although this was less known. he introduced a pair of orthogonal latin squares of order 9 in his book. interestingly, his two orthogonal non-double-diagonal latin squares produce a magic square of order 9, whose theoretical reason was not studied. there have been a few studies on choi’s latin squares of order 9. the most recent one is ko-wei lih’s construction of choi’s latin squares of order 9 based on the two 3 × 3 orthogonal latin squares. in this paper, we give a new generalization of choi’s orthogonal latin squares of order 9 to orthogonal latin squares of size n2 using the kronecker product including lih’s construction. we find a geometric description of choi’s orthogonal latin squares of order 9 using the dihedral group d8. we also give a new way to construct magic squares from two orthogonal non-double-diagonal latin squares, which explains why choi’s latin squares produce a magic square of order 9. 2010 msc: 05b15, 05b20 keywords: choi seok-jeong, koo-soo-ryak, latin squares, magic squares 1. introduction a latin square of order n is an n×n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. this latin square is one of the most interesting mathematical objects. it can be applied to a lot of branches of discrete mathematics including finite geometry, coding theory and cryptography [8], [9]. in particular, orthogonal latin squares have been one of the main topics in latin squares. the superimposed pair of two orthogonal latin squares is also called a graeco-latin sqaure by leonhard euler (1707-1783) in 1776 [4]. it is known that the study of latin squares was researched by euler in the 18th century. however the korean mathematician, choi seok-jeong [choi is a family name] (1646-1715) already studied latin squares at least 60 years before euler’s work [11]. a ∗ this author is supported by basic research program through the national research foundation of korea (nrf) funded by the ministry of education (nrf-2019r1a2c1088676). jon-lark kim (corresponding author), dong eun ohk, doo young park, jae woo park; department of mathematics, sogang university, seoul, 04107, south korea (email: jlkim@sogang.ac.kr, tony_to@naver.com, dy9723@naver.com, 67670711@naver.com). 17 https://orcid.org/0000-0002-0517-9359 https://orcid.org/0000-0002-7737-5199 https://orcid.org/0000-0001-7404-0492 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 pair of two orthogonal latin squares of order 9 was introduced in koo-soo-ryak (or gusuryak) written by choi seok-jeong [2]. the koo-soo-ryak was listed as the first literature on latin squares in the handbook of combinatorial designs [3]. let k be the matrix form of the superimposed latin square of order 9 from koo-soo-ryak: k = (5,1) (6,3) (4,2) (8,7) (9,9) (7,8) (2,4) (3,6) (1,5) (4,3) (5,2) (6,1) (7,9) (8,8) (9,7) (1,6) (2,5) (3,4) (6,2) (4,1) (5,3) (9,8) (7,7) (8,9) (3,5) (1,4) (2,6) (2,7) (3,9) (1,8) (5,4) (6,6) (4,5) (8,1) (9,3) (7,2) (1,9) (2,8) (3,7) (4,6) (5,5) (6,4) (7,3) (8,2) (9,1) (3,8) (1,7) (2,9) (6,5) (4,4) (5,6) (9,2) (7,1) (8,3) (8,4) (9,6) (7,5) (2,1) (3,3) (1,2) (5,7) (6,9) (4,8) (7,6) (8,5) (9,4) (1,3) (2,2) (3,1) (4,9) (5,8) (6,7) (9,5) (7,4) (8,6) (3,2) (1,1) (2,3) (6,8) (4,7) (5,9) then we can separate k into two latin squares l and n. to get a visible effect, let us color in each square. figure 1. a colored two latin squares l and n, respectively we paint colors for each numbers, 1,2, · · · ,9. in details, 1,2,3 are colored in red, 4,5,6 are colored in green, and 7,8,9 are colored in blue. then we observe that the latin squares have self-repeating patterns. this simple structure of choi’s latin squares motivates some generalization of his idea. we generalize choi’s latin squares in three directions: the kronecker product approach, the dihedral group approach, and magic squares from choi’s latin squares. in this paper, we give a new generalization of choi’s orthogonal latin squares of order 9 to orthogonal latin squares of size n2 using the kronecker product including lih’s construction [9]. there has been some attempt that the dihedral group d8 acts on the latin squares [5]. we find a geometric description of choi’s orthogonal latin squares of order 9 using d8. we also give a new way to construct magic squares from two orthogonal non-double-diagonal latin squares, which explains why choi’s latin squares produce a magic square of order 9. 2. a generalization of choi’s orthogonal latin squares definition 2.1. ([9]) let a = (aij) be a latin square of order n(i,j ∈ {1,2, · · · ,n}) and b = (bst) be a latin square of order m(s,t ∈ {1,2, · · · ,m}). then the kronecker product of a and b, which is an 18 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 mn×mn square a⊗b given by a⊗b = (a11,b) (a12,b) · · · (a1n,b) (a21,b) (a22,b) · · · (a2n,b) ... ... ... ... (an1,b) (an2,b) · · · (ann,b) where (aij,b) is the m×m square (aij,b) = (aij,b11) (aij,b12) · · · (aij,b1m) (aij,b21) (aij,b22) · · · (aij,b2m) ... ... ... ... (aij,bm1) (aij,bm2) · · · (aij,bmm) lemma 2.2. ([9]) a⊗b is a latin square if a and b are both latin squares. theorem 2.3. ([9]) if two latin squares a1 and a2 of order n are orthogonal and two latin squares b1 and b2 of order m are orthogonal, then a1 ⊗b1 and a2 ⊗b2 of order mn are orthogonal. now it is natural to substitute m(aij −1) + bkl for the entry (aij,bkl) in a⊗b. thus we define the substituted kronecker product ⊗s of two latin squares a and b by the following block matrix a⊗s b =  (m(a11 −1)×nm + b) · · · (m(a1n −1)×nm + b)... ... ... (m(an1 −1)×nm + b) · · · (m(ann −1)×nm + b)   where a = (aij) is a matrix of order n, b is a matrix of order m, and nm is the m×m all-ones matrix. let us return to latin squares. judging from figure 1, we can expect that l is closely related to a latin square of order 3. let a3 = (aij) = 2 3 1 1 2 3 3 1 2 then the following block matrix (3(a11 −1)×n3 + a3) (3(a12 −1)×n3 + a3) (3(a13 −1)×n3 + a3)(3(a21 −1)×n3 + a3) (3(a22 −1)×n3 + a3) (3(a23 −1)×n3 + a3) (3(a31 −1)×n3 + a3) (3(a32 −1)×n3 + a3) (3(a33 −1)×n3 + a3)   produces l. in other words, l = a3 ⊗s a3. similarly, let b3 = 1 3 2 3 2 1 2 1 3 then n = b3⊗s b3. these two latin squares a3 and b3 are elements of mols(3) which is the mutually orthogonal latin squares of order 3. we recall that lih [10] also found this relation. however he did not explain why l = a3 ⊗s a3 and n = b3 ⊗s b3 are orthogonal from the kronecker product point of view. corollary 2.4. choi’s two latin squares of order 9 are orthogonal. proof. by the above notation, we can put choi’s two latin squares of order 9 by l = a3 ⊗s a3 and n = b3 ⊗s b3. note that a3 and b3 are orthogonal. therefore, by taking a1 = b1 = a3 and a2 = b2 = b3 in theorem 2.3 we see that l = a3 ⊗s a3 and n = b3 ⊗s b3 are also orthogonal. 19 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 hence it appears that choi might know how to get the orthogonal latin squares of order 9 by expanding orthogonal latin squares of order 3. it is natural to generalize choi’s approach to obtain orthogonal latin squares by copying a smaller latin square several times. if a is a latin square of order n, we call a⊗s a choi type latin square of order n2. since there exists a pair of orthogonal latin squares of order n ≥ 3 and n 6= 6, the following is immediate. corollary 2.5. there exists a pair of choi’s type latin squares of order n2 which are orthogonal whenever n ≥ 3 and n 6= 6. we remark that lih’s construction [10] gives only the case when n = 3. corollary 2.5 extends this result to any n ≥ 3 and n 6= 6. 3. latin squares acted by the dihedral group d8 we have noticed that l is symmetric to n with respect to the 5th column of l. in other word, if we let l = (lij), then n = (li(n+1−j)). so we define some operation. definition 3.1. let a = (aij) be an n×n matrix (or square or array). define the n×n matrix s2(a) by s2(a) = (ai(n+1−j)). we can consider more symmetries. the dihedral group of degree n denoted by d2n is a well-known group of order 2n consisting of symmetries on a regular n-polygon consisting rotations and reflections. in this case, we concentrate on a square, so the dihedral group of order 8, denoted by d8, is needed. in d8, there are eight elements, s0,s1,s2,s3,r1,r2,r3,r4. note that si for i = 0,1,2,3 denotes a reflection. more precisely, s0 is a horizontal reflection, s1 is a main diagonal reflection, s2 is a vertical reflection, and s3 is an antidiagonal reflection. note that r0 denotes the rigid motion and ri’s (i = 1,2,3) denote counterclockwise rotations by 90, 180, 270 degrees respectively so that r2 = r21 and r3 = r 3 1. we can define a set d8(a) = {a,r1(a),r2(a),r3(a),s0(a),s1(a),s2(a),s3(a)} for a given latin square a. definition 3.2. let ln be the set of all latin squares of order n. then σ ∈ d8 is a function with σ : ln → ln defined by r0(a) = (aij), r1(a) = (aj(n+1−i)), r2(a) = (a(n+1−i)(n+1−j)), r3(a) = (a(n+1−j)i), s0(a) = (a(n+1−i)j), s1(a) = (aji) , s2(a) = (ai(n+1−j)), s3(a) = (a(n+1−j)(n+1−i)) where a ∈ ln and a = (aij). then we can regard an element in d8 as a function acting on ln. in fact, the dihedral group d8 acts on ln (or ln is a d8-set) as follows. lemma 3.3. ln is a d8-set. proof. let σ ∈ d8 and a ∈ ln. since a = (aij) is a latin square of order n, {a1j,a2j, · · · ,anj} = {ai1,ai2, · · · ,ain} = {1,2, · · · ,n} for all i,j = 1,2, · · · ,n. thus by definition, σ(a) is a latin square. if σ = r0, then r0(a) = a for any a ∈ ln. suppose that σ1,σ2 ∈ d8. let σ3 = σ1 ◦σ2 ∈ d8. it is straightforward to check that σ3(a) = σ1(σ2(a)) by definition 3.2. 20 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 in the choi’s latin squares, n = s2(l) (or l = s2(n)). since l and n are orthogonal, we can say that l and s2(l) are orthogonal. then we can have some questions. is l orthogonal to σ(l) for another σ in d8? and how many mutually orthogonal latin squares are in the set d8(l)? moreover, for any latin square a, what is the maximum number of mutually orthogonal latin squares in the set d8(a)? lemma 3.4. suppose a and b are latin squares of order n and take an arbitrary σ ∈ d8. then a is orthogonal to b if and only if σ(a) is orthogonal to σ(b). by lemma 3.4, we have a criteria when two latin squares in the set d8(a) are orthogonal. if two latin squares a and b are orthogonal, we denote it by a⊥b: r0(a)⊥s0(a) ⇐⇒ r1(a)⊥s1(a) ⇐⇒ r2(a)⊥s2(a) ⇐⇒ r3(a)⊥s3(a) r0(a)⊥s1(a) ⇐⇒ r1(a)⊥s2(a) ⇐⇒ r2(a)⊥s3(a) ⇐⇒ r3(a)⊥s0(a) r0(a)⊥s2(a) ⇐⇒ r1(a)⊥s3(a) ⇐⇒ r2(a)⊥s0(a) ⇐⇒ r3(a)⊥s1(a) r0(a)⊥s3(a) ⇐⇒ r1(a)⊥s0(a) ⇐⇒ r2(a)⊥s1(a) ⇐⇒ r3(a)⊥s2(a) r0(a)⊥r1(a) ⇐⇒ r1(a)⊥r2(a) ⇐⇒ r2(a)⊥r3(a) ⇐⇒ r3(a)⊥r0(a) ⇐⇒ s0(a)⊥s1(a) ⇐⇒ s1(a)⊥s2(a) ⇐⇒ s2(a)⊥s3(a) ⇐⇒ s3(a)⊥s0(a) r0(a)⊥r2(a) ⇐⇒ r1(a)⊥r3(a) ⇐⇒ s0(a)⊥s2(a) ⇐⇒ s1(a)⊥s3(a) thus for finding mutually orthogonal latin squares in d8(a), we should look at the orthogonality of a = r0(a) and σ(a) for σ ∈ d8. lemma 3.5. for any a ∈ ln, a is not orthogonal to r2(a). proof. let a = (aij) and r2(a) = (bij). suppose that a and r2(a) are orthogonal. then we have {(aij,bij) |i,j = 1,2, · · · ,n} = {(x,y) |x,y = 1,2, · · · ,n} . therefore there exist some integers sk, tk such that (asktk,bsktk) = (k,k) for each nonnegative integer k = 1,2, . . .n. let n + 1 − sk = s′k and n + 1 − tk = t ′ k. since bsktk = as′kt′k and bs′kt′k = asktk, so (asktk,bsktk) = (bsktk,asktk) = (as′kt′k,bs′kt′k). it means that the two ordered pairs (asktk,bsktk) and (as′ k t′ k ,bs′ k t′ k ) are the same in the set {(aij,bij)}. since a and r2(a) are orthogonal, we have (sk, tk) = (s′k, t ′ k). that is, sk = s ′ k and tk = t ′ k. this implies that n = 2sk − 1 = 2tk − 1, that is, sk = tk for any k. it contradicts. lemma 3.6. let a ∈ ln and n be even. then a is not orthogonal to either s0(a) or s2(a). proof. suppose that a is orthogonal to s0(a). let s0(a) = (a(n+1−i)j) = bij. by the similar argument of proof of lemma 3.5, there exist integer u and v such that auv = buv = k for some k. so auv = buv = a(n+1−u)v. since a is a latin square, the entries in the v-th column are all distinct. thus auv = a(n+1−u)v implies u = n+1−u and so u = (n+1)/2. however, n is even so that u is not an integer. it contradicts. hence a is not orthogonal to s0(a). we can show that a is not orthogonal to s2(a) in a similar manner. theorem 3.7. let a ∈ ln and n be odd. then the maximum number of mutually orthogonal latin squares of order n in the set d8(a) is less than or equal to 4. and if we assume that n is even, then the maximum number of mutually orthogonal latin squares in the set d8(a) is 2. proof. let m be the set of mutually orthogonal latin squares, which has the maximum number of mutually orthogonal latin squares in the set d8(a). by lemma 3.5, we can get r0(a) 6⊥ r2(a), r1(a) 6⊥ r3(a), s0(a) 6⊥ s2(a) and s1(a) 6⊥ s3(a). if we take three or more elements of m from the set 21 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 {r0(a),r1(a),r2(a),r3(a)}, then there should appear a pair of non-orthogonal latin squares. similarly, we cannot take three or more elements from the set {s0(a),s1(a),s2(a),s3(a)}. it means that the set m can be m = {ri1(a),ri2(a),sj1(a),sj2(a)}. therefore we have that the maximum number of mutually orthogonal latin squares in the set d8(a) is less than or equal to four. suppose n is even and m = {ri1(a),ri2(a),sj1(a),sj2(a)}. it is possible that m does not contain r0(a), however, we can get the set of 4 mutually orthogonal latin squares containing r0(a) by the group action. so without loss of generality, assume that ri1 = r0. by lemma 3.6, sj1,sj2 should be 1 and 3. however s1(a) 6⊥ s3(a) by lemma 3.5, so |m | 6= 4. now suppose that m = {r0(a),ri2(a),sj1(a)}. note that i2 = 1,3 and j1 = 1,3. however, lemma 3.6 also implies that r1(a) 6⊥ s1(a), r1(a) 6⊥ s3(a), r3(a) 6⊥ s1(a) and r3(a) 6⊥ s3(a). thus |m | 6= 3. therefore |m | = 2 if n is even. corollary 3.8. let l be one of choi’s latin squares of order 9. then the maximum number of mutually orthogonal latin squares in d8(l) is two. proof. by theorem 3.5 there are at most 4 mutually orthogonal latin squares in d8(l). without loss of generality, we may assume that l is one of them. we first show that there are only two mutually orthogonal latin squares among l,r1(l),r2(l),r3(l). by lemma 3.3, l is not orthogonal to r2(l). this also implies that r1(l) is not orthogonal to r3(l). on the other hand, we have checked by enumerating all ordered pairs that l is orthogonal to both r1(l) and r3(l). therefore we have only two cases {l,r1(l)} and {l,r3(l)} among rotations. we can easily check that l is orthogonal to both s0(l) and s2(l) while l is neither orthogonal to s1(l) nor to s3(l) because the two diagonal reflections do not change the value of 5 in the main diagonal. however s0(l) cannot be orthogonal to s2(l) because they reduce to l and r2(l) which are not orthogonal by lemma 3.3. therefore we have the following four possibilities. 1. {l,r1(l),s0(l)} 2. {l,r1(l),s2(l)} 3. {l,r3(l),s0(l)} 4. {l,r3(l),s2(l)} r1(l) = 132798465 321987654 213879546 798465132 987654321 879546213 465132798 654321987 546213879 s0(l) = 978312645 789123456 897231564 312645978 123456789 231564897 645978312 456789123 564897231 s2(l) = 132798465 321987654 213879546 798465132 987654321 879546213 465132798 654321987 546213879 r3(l) = 978312645 789123456 897231564 312645978 123456789 231564897 645978312 456789123 564897231 we have checked that r1(l) is not orthogonal to s0(l) because (4,6) is repeated and r1(l) is not orthogonal to s2(l) because (4,4) is repeated. similarly, r3(l) is not orthogonal to s0(l) because (6,6) is repeated and r3(l) is not orthogonal to s2(l) because (6,4) is repeated. these are visualized by pairing the bold face numbers in r1(l),s0(l),s2(l),r3(l). therefore, we have {l,r1(l)}, {l,s0(l)}, {l,s2(l)}, or {l,r3(l)} as a maximal mutually orthogonal latin square subset of d8(l). hence the maximum number of mutually orthogonal latin squares in d8(l) is two. 22 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 if a latin square a of order n is orthogonal to σ(a) for some σ ∈ d8(a), we call such a a dihedral latin square. we recall that a latin square a is self-orthogonal if it is orthogonal to its transpose [12]. since the transpose of a can be represented as s1(a) (s1 is a main diagonal reflection), the concept of a dihedral latin square includes the concept of a self-orthogonal latin square. for example, choi’s two latin squares l,n of order 9 are dihedral since n = s2(l). let us take another example as follows. a = 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 r1(a) = 4 2 1 3 3 1 2 4 2 4 3 1 1 3 4 2 s1(a) = 1 3 4 2 2 4 3 1 3 1 2 4 4 2 1 3 then a and r1(a) are a pair of orthogonal latin squares. so a is a dihedral latin square. similarly, a and s1(a) are orthogonal. so a is self-orthogonal too. however r1(a) is not orthogonal to s1(a) since (1,4) is repeated. by the previous theorem, the maximum number of mutually orthogonal latin squares in the set d8(a) is 2. consider choi’s type latin squares a⊕s a, r1(a) ⊕s r1(a), and s1(a) ⊕s s1(a). then a⊕s a is orthogonal to both r1(a)⊕s r1(a) and s1(a)⊕s s1(a). 4. magic squares from latin squares definition 4.1. a magic square of order n is an n×n array (or matrix) of the n2 consecutive integers with the sums of each row, each column, each main diagonal, and each antidiagonal are the same. for example, 4 9 2 3 5 7 8 1 6 is a magic square of order 3 since the sums of each row, column, main diagonal and antidiagonal are the same. similarly, for order n latin square, we assume the symbols are {1,2, · · · ,n2}. then the question is what the relation between latin squares and magic squares is. we need the following definition. definition 4.2. let a be a latin square of order n. then, a is called a double-diagonal latin square [6], [7] if the n entries in main diagonal are all distinct and the n entries in antidiagonal are also all distinct. a construction of orthogonal double-diagonal latin squares has been actively studied [8], [1], [12]. theorem 4.3. ([9]) suppose a pair of orthogonal double-diagonal latin squares of order n exist. then a magic square of order n can be constructed from them. definition 4.4. suppose a = (aij) and b = (bij) are orthogonal latin squares of order n. then define an n×n square a +s b by a +s b = (n(aij −1) + bij). this a +s b is not necessarily a magic square since its sums of two main diagonals is not the same as its sums of columns or rows. theorem 4.3 states that if the two latin squares a and b are orthogonal and double-diagonal, then a +s b is a magic square. 23 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 and the another noticeable point is that the pair of choi’s orthogonal latin squares is not doublediagonal. however, choi’s squares also can produce a magic square even though they are not doublediagonal. theorem 4.5. if there is a pair of orthogonal latin squares a and b of order n such that the sum of main diagonal of each of a and b is n(n + 1)/2 and the sum of antidiagonal of each of a and b is n(n + 1)/2, then a +s b is a magic square of order n. proof. suppose a = (aij) and b = (bij),(i,j ∈{1,2, · · · ,n}) are orthogonal latin squares such that n∑ i=1 aii = n∑ i=1 bii = n(n + 1) 2 and n∑ i=1 ai(n+1−i) = n∑ i=1 bi(n+1−i) = n(n + 1) 2 . now define m = (mij) by m = (mij) = (n(aij − 1) + bij). we want to show that m is a magic square. since 1 ≤ aij, bij ≤ n for all (i,j), 1 ≤ n(aij − 1) + bij ≤ n2. we first show that each mij is distinct. suppose n(auv −1) + buv = n(ast −1) + bst. then n(auv −ast) = bst −buv. so n | (bst −buv). however, 1 ≤ bij ≤ n for all (i,j), so 1 − n ≤ bst − buv ≤ n − 1. thus n | (bst − buv) implies bst − buv = 0 and so auv = ast. since a and b are orthogonal latin squares, (u,v) = (s,t). hence if (u,v) 6= (s,t) then n(auv −ast) 6= bst − buv so all mij are distinct. now calculate the sums. n∑ i=1 {n(aij −1) + bij} = n n∑ i=1 aij −n2 + n∑ i=1 bij = n(n2 + 1) 2 , n∑ j=1 {n(aij −1) + bij} = n n∑ j=1 aij −n2 + n∑ j=1 bij = n(n2 + 1) 2 , n∑ i=1 {n(aii −1) + bii} = n n∑ i=1 aii −n2 + n∑ i=1 bii = n(n2 + 1) 2 , and similarly, n∑ i=1 {n(ai(n+1−i) −1) + bi(n+1−i)} = n(n2 + 1) 2 . thus the sums are the same. hence m is a magic square. we have an existence theorem satisfying theorem 4.5. theorem 4.6. for any odd number n ≥ 3, there exists a pair of orthogonal latin squares each of whose sum of main diagonal (and antidiagonal respectively) is n(n + 1)/2. 24 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 proof. suppose n = 2k − 1 where k ≥ 2. let an = (aij) be a matrix where each descending diagonal from left to right is constant like following matrix: an = k n k −1 n−1 ... k + 2 2 k + 1 1 1 k ... ... ... ... k + 2 2 k + 1 k + 1 ... k ... ... n−1 ... k + 2 2 2 ... ... ... n k −1 n−1 ... k + 2 ... ... ... 1 k n ... ... ... k −2 ... 2 k + 1 1 ... ... ... n−1 n−1 k −2 ... 2 ... ... k ... k −1 k −1 n−1 k −2 ... ... ... ... k n n k −1 n−1 k −2 ... 2 k + 1 1 k in particular, if n = 3 and k = 2, we get latin square a3 in section 2. since latin square b3 in section 2 is obtained by reflecting a3 along the 2nd column of a3, it is natural to reflect an along the kth column of an as follows. the sum of main diagonal and the sum of antidiagonal of an are n(n+1)/2 since ∑n i=1 aii = n×k = n(n + 1)/2 and ∑n i=1 ai(n+1−i) = ∑n i=1 i = n(n + 1)/2. recall that s2(a) is the latin square obtained by reflecting along the middle vertical line of an. then s2(an) has the same sum of the main diagonal (and antidiagonal respectively) of an since the trace of an, tr(an) is the sum of antidiagonal (and main diagonal respectively) of s2(an). now it remains to show that a and s2(an) are orthogonal. there is a one-to-one correspondence between pandiagonals of an and line equations; let y = x+α be a line with α ∈ zn. then each constant pandiagonal corresponds to each equation of line. for example, y = x corresponds to the diagonal constant k in an since k = aij ⇔ i = j in zn. (i,e. (i,j) is a root of y = x in zn). similarly, x = y − 2 corresponds to the constant k−1, · · · , x = y−(n−1) corresponds to the constant 1. and x = y+(n−1) corresponds to the constant n, x = y+(n−3) corresponds to the constant n−1, · · · , x = y+2 corresponds to the constant k + 1. then we can do this to s1(a); similarly, x = −y corresponds to the constant k, · · · , x = −y− (n−1) corresponds to the constant 1. and x = −y + (n−1) corresponds to the constant n, · · · , x = −y + 2 corresponds to the constant k + 1. any two lines x = y + α and x = −y + β have exactly one unique root. it means that an entry (aij,ai(n+1−j)) appears only once. by the above theorem, we get a magic square constructed from a pair of orthogonal latin squares which are not double-diagonal. although there are many other ways to construct magic squares, our method is the way choi obtained magic squares from two orthogonal non-double-diagonal latin squares. however, we can ask a question ”what does happen if n is even?” it is well known that a pair of orthogonal latin square does not exist when n = 2 and n = 6, and so it is more difficult to get an even order magic square consisting of a pair of latin squares. so we construct magic squares of some even order cases in a different way. lemma 4.7. suppose that a latin square a1 of order n has main diagonal and antidiagonal sums n(n + 1)/2 respectively and that a latin square b1 of order m has main diagonal and antidiagonal sums m(m+1)/2 respectively. then a1⊗b1 is a latin square of order mn with main diagonal and antidiagonal sums nm(nm + 1)/2 respectively. proof. the fact that a1 ⊗s b1 is a latin square of order mn follows from theorem 2.2. it remains to show that the two sums give nm(nm + 1)/2. 25 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 first we consider the sum of main diagonal of a1⊗s b1. by definition of a1⊗s b1, its main diagonal sum is equal to m{m n∑ i=1 aii −mn}+ ( m∑ i=1 bii)n = m 2 { n(n + 1) 2 −n } + mn(m + 1) 2 = mn(mn + 1) 2 . similarly its antidiagonal sum is equal to m{m n∑ i=1 ai(n+1−i) −mn}+ ( m∑ i=1 bi(n+1−i))n = m 2 { n(n + 1) 2 −n } + mn(m + 1) 2 = mn(mn + 1) 2 . this completes the proof. theorem 4.8. for every n with n ≡ 2 (mod 4), there exists a pair of orthogonal latin squares each of whose sum of main diagonal (and antidiagonal, respectively) is n(n + 1)/2. proof. define four latin squares by a1 = 1 4 3 2 4 1 2 3 2 3 4 1 3 2 1 4 a2 = 3 1 2 4 4 2 1 3 2 4 3 1 1 3 4 2 b1 = 1 5 8 4 2 6 7 3 3 8 5 2 4 7 6 1 8 3 2 5 7 4 1 6 6 2 3 7 5 1 4 8 2 6 7 3 1 5 8 4 4 7 6 1 3 8 5 2 7 4 1 6 8 3 2 5 5 1 4 8 6 2 3 7 b2 = 1 4 5 8 2 6 3 7 4 1 8 5 6 2 7 3 3 8 1 6 5 7 2 4 8 3 6 1 7 5 4 2 7 5 4 2 8 3 6 1 5 7 2 4 3 8 1 6 6 2 7 3 4 1 8 5 2 6 3 7 1 4 5 8 then a1 and a2 are orthogonal. b1 and b2 are also orthogonal. so we can construct two orthogonal latin squares of order 4k (where k is an integer) using the following way. if we want to construct of orthogonal latin squares of order 4t with t odd, then we can make two latin squares a1 ⊗s c1 and a2 ⊗s c2 where c1 and c2 are orthogonal latin squares of order t and the sums of diagonal and antidiagonal are t(t + 1)/2 (by theorem 4.6, we can get such pair of latin squares). then a1 ⊗s c1 and a2 ⊗s c2 are orthogonal by theorem 2.2 and each sum of their diagonal and antidiagonal is 4t(4t + 1)/2 by lemma 4.4. or if we want to construct orthogonal latin squares of order 2p with p ≥ 3, we recursively use the substituted kronecker products of a1,b1,a2, and b2. so we can construct orthogonal latin squares of an even order n which is not of the form of 2r (r is odd) each of whose sum of diagonal (and antidiagonal, respectively) is n(n + 1)/2. corollary 4.9. for any integer n with n 6= 2r where r is odd, there exists a pair of non-double-diagonal orthogonal latin squares of order n such that the pair of latin squares can produce a magic square of order n. proof. by theorems 4.2, 4.3, and 4.5, we can construct a magic square of order n where n 6= 2r (r is odd). therefore, choi’s orthogonal latin squares of various orders give a new way to construct magic squares based on non-double-diagonal orthogonal latin squares. 26 j. l. kim et.al. / j. algebra comb. discrete appl. 9(1) (2022) 17–27 references [1] j. w. brown, f. cherry, l. most, m. most, e. t. parker, w. d. wallis, completion of the spectrum of orthogonal diagonal latin squares, graphs, matrices and desings, dekker (1993) 43–49. [2] s. j. choi, gusuryak, seoul national university kyujanggak institute for korean studies. [3] c. j. colbourn, j. h. dinitz, handbook of combinatorial designs, crc press, second edition (2007). [4] l. euler, de quadratis magicis, commentationes arithmeticae collectae 2 (1849) 593-602 and opera omnia 7 (1911) 441–457. [5] m. a. francel , d. j. john, the dihedral group as the array stabilizer of an augmented set of mutually orthogonal latin squares, ars combin. 97 (2010) 235–252. [6] a. j. w. hilton, some simple constructions for double diagonal latin squares, sankhya: the indian journal of statistics 36(3) (1974) 215–229. [7] a. j. w. hilton, s. h. scott, a further construction of double diagonal orthogonal latin squares, discrete mathematics 7 (1974) 111–127. [8] a. d. keedwell, j. dãľnes, latin squares and their applications, academic press, second edition (2015). [9] c. f. laywine, g. l. mullen, discrete mathematics using latin squares, john wiley & sons, new york (1998). [10] k. w. lih, a remarkable euler square before euler, mathematics magazine 83(3) (2010) 163–167. [11] h. y. song, choi’s orthogonal latin squares is at least 67 years earlier than euler’s, global kms conference, jeju, korea (2008). [12] y. zhang, k. chen, n. cao, h. zhang, strongly symmetric self-orthogonal diagonal latin squares and yang hui type magic squares, discrete mathematics 328 (2014) 79–87. 27 https://doi.org/10.1201/9780203719916-4 https://doi.org/10.1201/9780203719916-4 https://doi.org/10.1201/9781420010541 https://scholarlycommons.pacific.edu/euler-works/795/ https://scholarlycommons.pacific.edu/euler-works/795/ https://mathscinet.ams.org/mathscinet-getitem?mr=2721801 https://mathscinet.ams.org/mathscinet-getitem?mr=2721801 https://www.jstor.org/stable/25051907 https://www.jstor.org/stable/25051907 https://doi.org/10.1016/s0012-365x(74)80023-3 https://doi.org/10.1016/s0012-365x(74)80023-3 https://doi.org/10.1016/c2014-0-03412-0 https://doi.org/10.1016/c2014-0-03412-0 https://books.google.ru/books?id=vwqn86g68sic https://books.google.ru/books?id=vwqn86g68sic https://doi.org/10.4169/002557010x494805 https://doi.org/10.1016/j.disc.2014.04.002 https://doi.org/10.1016/j.disc.2014.04.002 introduction a generalization of choi's orthogonal latin squares latin squares acted by the dihedral group d8 magic squares from latin squares references issn 2148-838x j. algebra comb. discrete appl. -(-) • 1–15 received: 29 march 2022 accepted: 29 august 2022 ar ti cl e in pr es s journal of algebra combinatorics discrete structures and applications walled klein-4 brauer algebras research article annamalai tamilselvi, dhilshath shajahan abstract: the new class of diagram algebras known as walled klein-4 brauer algebras, denoted by −→−→ dr,s(l), where r, s ∈ n and l ∈ k, is an indeterminate, is studied in this paper. the walled klein-4 brauer algebras are explained in terms of generators and relations. the indexing set of the simple modules of the walled klein-4 brauer algebras was described. we established that walled klein-4 brauer algebras are iterated inflations of the group algebra of the group (z2 ×z2) osr ×(z2 ×z2) oss, and we concluded that −→−→ dr,s(l) is cellular as a consequence. 2010 msc: 16g30, 05e10, 20c30 keywords: cellularity, walled brauer algebras, walled signed brauer algebras 1. introduction richard brauer introduced the brauer algebra bn(δ) in the representation theory of orthogonal groups. undirected diagrams are the basis elements of brauer algebra. the centralizers of the orthogonal groups are these algebras. g-brauer algebras were introduced by parvathi and savithri, with signed diagram as basis and studied in [7] its structure over k(x), where x is an indeterminate. the symmetric group sr and the general linear group gln(c) representation theories over c are linked through schur-weyl duality. the walled brauer algebras br,s(δ) is the result of a third version of schur-weyl duality. this algebra was examined by turaev, koike, and benkart et al. kethesan introduced a new class of diagram algebras called walled signed brauer algebras denoted by −→ dr,s(x), where x is an indeterminate. the fr,s map gives a vector space isomorphism between the group algebra of the hyperoctahedral group z2 o sr+s and the walled singed brauer algebras −→ dr,s(x). the structure of walled signed brauer algebras has been studied in [4]. in [11], tamilselvi et al., studied annamalai tamilselvi (corresponding author); ramanujan institute for advanced study in mathematics, university of madras, chennai, india (email: tamilselvi.riasm@gmail.com). dhilshath shajahan; ramanujan institute for advanced study in mathematics, university of madras, chennai, india (email: dhilshajahan@gmail.com). 1 https://orcid.org/0000-0002-0682-2705 https://orcid.org/0000-0002-1448-0792 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 the robinson-schensted correspondence for the walled brauer algebras and the walled signed brauer algebras. cellularity and representations of walled cyclic g-brauer algebras were studied by tamilselvi [10]. in [8], parvathi and sivakumar studied about a new class of diagram algebras called klein-4 diagarm algebras denoted by rk(n). these algebras are the centralizer algebras of the group (z2 ×z2) osn acting on v ⊗k, where v is the signed permutation module for (z2 ×z2) o sn. in that paper [8], they give a combinatorial rule for the decomposition of the tensor powers of the signed permutation representation of (z2 ×z2) osn, by explicitly constructing the basis for the irreducible modules. we were motivated by this work to create a new class of algebras over k, known as walled klein-4 brauer algebras and denoted by −→−→ dr,s(l), where l ∈ k is an indeterminate. in section 3, we define walled klein-4 brauer algebras and show how they are represented in terms of generators and relations. we define the map flipr,s, which is a vector space isomorphism between the group algebra of the group (z2 ×z2) o sr+s and the walled klein-4 brauer algebras, in that section. using the representations of the group algebra of the group [8, 9], we described the indexing set of the simple modules of the walled klein-4 brauer algebras in section 4. in section 5, we proved that the walled klein-4 brauer algebras are the iterated inflations of the group algebra of the group (z2 ×z2) osr×(z2 ×z2) oss, and we concluded that walled klein-4 diagram algebras are cellular. 2. preliminaries now, we will present some definitions that we will use to proceed on. definition 2.1. [3] a partition β = (β1,β2, . . . ,βt) of (r + s) is said to be p-regular if either p > 0 and there is no 1 ≤ i ≤ t such that βi = βi+1 = . . . = βi+p or p = 0. definition 2.2. ([9]) a 4-partition ((β1), (β2), (β3), (β4)) of size (r+s), is an ordered 4-tuple of partitions (β1), (β2), (β3) and (β4) such that |(β1), (β2), (β3), (β4)| = |(β1)| + |(β2)| + |(β3)| + |(β4)| = (r + s). , , , the 4-partition ((β1), (β2), (β3), (β4)) = ([2, 2][3, 2][2, 1][12]) is represented by the above 4-tableau and the sum |(β1)| + |(β2)| + |(β3)| + |(β4)| = 14. the set of all 4-partitions of (r + s) is denoted by b4(r+s). definition 2.3. if the partitions (β1), (β2), (β3) and (β4) are p-regular then the 4-partition ((β1), (β2), (β3), (β4)) of (r + s) is said to be p-regular. theorem 2.4. ([2], theorem (10.33)) let k be a field which is a splitting field for the finite groups g1 and g2. then: • for each simple kg1 module m and each simple kg2 module n, the outer tensor product m �n is a simple k(g1 ×g2)-module. • every simple k(g1 × g2)-module is of the above form m � n, with m and n are uniquely determined (up to isomorphism). remark 2.5. [8] a finite dimensional associative algebra a with unit over c, the field of complex numbers is said to be semi-simple if a is isomorphic to a direct sum of full matrix algebras a∼= ⊕ λ∈â mdλ(c) for â a finite index set, and dλ positive integers. corresponding to each λ ∈ â, there is a single irreducible a module call it v λ which has dimension dλ. if â is a singleton set then a is said to be simple. maschke’s theorem says that for g finite, c[g] is semi-simple. 2 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 remark 2.6. [5] an algebra is cellular if an only if it can be written as the iterated inflations of copies of full matrix algebras. moreover iterated inflation of cellular algebras is always cellular again ([5], proposition 3.4). 3. walled klein-4 brauer algebras in this section, we define the walled klein-4 brauer algebras which are subalgebras of the g-brauer algebras [7], where g is the klein-4 group and give a presentation of walled klein-4 brauer algebras in terms of generators and relations. throughout this paper we are considering 1 as the identity. fix an algebraically closed field k of characteristic p ≥ 0 and l an indeterminate. for r,s ∈ n the walled klein-4 brauer algebras −→−→ dr,s(l) can be defined in the following manner. a diagram is said to be signed diagram if every edge is labeled by a sign. in the walled klein-4 brauer algebras we are considering four types of signs (e,e), (e,b), (a,e) and (a,b). the klein-4 brauer algebra −→−→ dr+s (diagrams without wall) which is the g-brauer algebra defined in [7], is the set of all signed diagrams −→−→ d with n signed edges and 2n vertices arranged in two rows of n vertices each. each signed edge links to exactly two vertices in these signed diagrams, and each vertex links to exactly one signed edge. edges connecting the top row vertex and the bottom row vertex are called vertical edges, and the remaining edges are called upper (connecting two vertices on the top row) or lower (connecting two veritices on the bottom row) horizontal curves (edges connecting two vertices in the same row). 3.1. multiplication of walled klein-4 brauer diagrams to find the product of two signed diagrams −→−→ d 1 and −→−→ d 2, draw −→−→ d 2 below −→−→ d 1 and connect the ith upper vertex of −→−→ d 2 with the ith lower vertex of −→−→ d 1. a new edge obtained in the product −→−→ d 1. −→−→ d 2 is labeled by a sign (e,e) or (e,b) or (a,e) or (a,b) according as the product of the signs (coordinate wise) of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this edge, where the product of the coordinates of the signs defined to be e = e.e = a.a = b.b; a = e.a = a.e; b = e.b = b.e. the product of the diagrams produces a new diagram with closed curves called loops. a loop α in −→−→ d 1. −→−→ d 2 is replaced by the variable l2, if the product of signs of the edges obtained from−→−→ d 1 and −→−→ d 2 to form this loop is (e,e) or (a,b). similary the loop α in −→−→ d 1. −→−→ d 2 is replaced by the variable l, if the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this loop is (a,e) or (e,b). now we have −→−→c , which is a signed diagram with each edge is marked as above and −→−→ d 1. −→−→ d 2 = l 2e1+e2. −→−→c , where e1 is the number of loops with the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form those loops is (e,e) or (a,b), e2 is the number of loops with the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form those loops is (e,b) or (a,e). it is usual to represent basis elements by diagrams with (r + s) vertices on the top row numbered 1 to (r + s) from left to right and (r + s) vertices on the bottom row numbered 1 to (r + s) from left to right, where each vertex is connected to precisely one other by a signed edge. we now define the walled klein-4 brauer algebra when n = r + s. partition the basis diagrams of−→−→ dr+s with the wall separating the first r vertices in the top row and the first r vertices in the bottom row from the remaining s vertices in the top row and s vertices in the bottom row, obtained the walled 3 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 klein-4 brauer algebras −→−→ dr,s(l) which is the diagram algebra with basis of those signed diagrams with no vertical edges crosses the wall and every upper or lower horizontal curves does cross the wall. the example of the basis elements −→−→ d 1 and −→−→ d 2 in −→−→ d3,2(l), and the multiplication of −→−→ d 1 and −→−→ d 2 are given below, −→−→ d 1 = 1 2 3 4 5 1 2 3 4 5 (a, e) (e, b) (e, e) (e, e) (e, e) −→−→ d 2 = 1 2 3 4 5 1 2 3 4 5 (a, e) (a, b) (e, b) (e, e) (e, e) 1 1 2 3 4 5 1 2 3 4 5 (a, e) (e, b) (e, e) (e, e) (e, e) 1 2 3 4 5 1 2 3 4 5 (a, e) (a, b) (e, b) (e, e) (e, e) −→−→ d 1. −→−→ d 2 = = l 2 1 2 3 4 5 1 2 3 4 5 (e, b)(a, e) (e, e) (e, e) (e, e) 1 the edge (1, 5) in the resulting diagram is labeled by the sign (e,b), since the coordinate wise product of the signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this edge is (a,e).(a,b).(e,e) = (a.a.e,e.b.e) = (e,b). similarly the other edges are labeled by the corresponding signs in the resulting diagram. also the resulting diagram is multiplied with l2 since the product of the signs of the edges from −→−→ d 1 and −→−→ d 2 to form the loop in the product −→−→ d 1. −→−→ d 2 is (e,b).(e,b) = (e.e,b.b) = (e,e). it is useful to compare −→−→ dr,s(l) with the group algebra k((z2 × z2) o sr+s) of the group g = (z2 ×z2) osr+s; where sr+s is a symmetric group of (r + s) symbols and (z2 ×z2) is the klein-4 group consisting of four elements. the group algebra can be viewed diagrammatically if we identify the element g of g with its signed diagram with no horizontal edge. we define a map, flipr,s : k((z2 ×z2) o sr+s) → −→−→ dr,s(l), mapping a klein-4 signed diagram with no horizontal curves to the walled klein-4 signed diagram obtained by adding a wall between the rthand (r + 1)th vertices, then without disconnecting any edges or changing the sign flip the portion of the diagram that is already on the right side of the wall. for example, 1 2 3 4 5 1 2 3 4 5 (e, e) (e, e) −→−→ f = (e, a) (e, b) (a, b) 1 2 3 4 5 1 2 3 4 5 (e, e) (e, e) (e, a) (e, b) (a, b) −→−→ h = 1 the walled klein-4 brauer diagram −→−→ h above arise by applying flip3,2 to the klein-4 signed diagram 4 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 −→−→ f . once add the wall in −→−→ f between the vertices 3 and 4 flip it on the right side of the wall, so that the edges (3, 4) and (1, 4) in −→−→ f changes to (3, 4) and (1, 4) in −→−→ h respectively. it follows in particular that the map flipr,s is a vector space isomorphism. since dim −→−→ d r,s (l) = dim k((z2 ×z2) osr+s) = 4r+s.(r + s)! 3.2. generators and relations of the group algebra k((z2 ×z2) osr+s) [9] the algebra g(r+s) = kg of the group g = (z2 × z2) o sr+s is generated by t1, t2 and the transpositions si := (i, i + 1) for i = 1, 2, ..., (r + s− 1) subject to the following relations • t21 = 1 • t22 = 1 • si2 = 1 for 1 ≤ i ≤ r + s− 1 • t1t2 = t2t1 • t1s1t1s1 = s1t1s1t1 • t2s1t2s1 = s1t2s1t2 • t2s1t1s1 = s1t1s1t2 • sisj = sjsi if |i− j| ≥ 2 • sisi+1si= si+1sisi+1 for 1 ≤ i ≤ r + s− 1 • sit1 = t1si for i ≥ 2 • sit2 = t2si for i ≥ 2 let flipr,s(t1) = q1, flipr,s(t2) = q2, and for i = 1, 2, . . . , (r−1), (r + 1), . . . , (r + s−1), flipr,s(si) = si(with the addition of the wall). when i = r then flipr,s(sr) = er is the diagram, 3.3. generators and relations of the walled klein-4 brauer algebras theorem 3.1. the walled klein-4 brauer algebras −→−→ dr,s(l) is generated by the elements −→−→q 1, −→−→q 1′, −→−→q r+1,−→−→q (r+1)′, s1, s2, ..., sr−1, er, sr+1, ..., sr+s−1 and satisfying the following relations: 1. s2i = 1 for i = 1, 2, ...(r − 1), (r + 1)...(r + s− 1). 2. sisj = sjsi if |i− j| > 1. 3. si+1sisi+1 = sisi+1si for i 6= r,r − 1. 5 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 4. ersi = sier for 1 ≤ i ≤ r − 2 or (r + 2) ≤ i ≤ (r + s− 1). 5. (er)2 = l2er. 6. ersr−1er = er. 7. ersr+1er = er. 8. sr−1sr+1ersr−1sr+1er = ersr−1sr+1er. 9. ersr−1sr+1ersr−1sr+1 = ersr−1sr+1er. 10. −→−→q 2i = 1 for i = 1, 1 ′ , 2, 2 ′ , ...(r + s), (r + s) ′ . 11. −→−→q i sj = sj −→−→q i for i = 1, 1 ′ and j 6= i. 12. −→−→q i s1 −→−→q i s1 = s1 −→−→q i s1 −→−→q i for i = 1, 1′. 13. −→−→q isj = sj −→−→q i for i = (r + 1), (r + 1) ′ and j 6= r, (r + 1). 14. −→−→q isr+1 −→−→q isr+1 = sr+1 −→−→q isr+1 −→−→q i for i = (r + 1), (r + 1) ′ . 15. er −→−→q ier = ler for i = r,r ′ , (r + 1), (r + 1) ′ 16. si −→−→q i+1 = −→−→q isi for i 6= r,r ′ . 17. er −→−→q j = er −→−→q j−1 for j = (r + 1), (r + 1) ′ . 18. −→−→q jer = −→−→q j−1er for j = (r + 1), (r + 1) ′ . 19. er −→−→q i = −→−→q ier for i 6= r,r ′ , (r + 1), (r + 1)′. 20. er −→−→q isr+1er = er −→−→q i+2 for i = r,r′. 21. er −→−→q isr+1er = er −→−→q i+1 for i = r + 1, (r + 1)′. where r′ + i = (r + i)′ 6 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 for 1 ≤ i ≤ r − 1, for i = r, for r + 1 ≤ i ≤ r + s− 1 proof. the walled brauer algebra dr,s(l2) for the symmetric group sr+s is generated by s1, s2, . . . , sr−1, er, sr+1, . . . , sr+s−1 with the relations from (1) to (9). by the universal property of free algebra any homomorphism from {s1, s2, . . . , sr−1, er, sr+1, . . ., sr+s−1} to b, where b is the associative algebra over k and is generated by p1, p1′, pr+1, p(r+1)′, t1, t2, . . . , tr−1, xr, tr+1, . . . , tr+s−1 satisfies the relations from (1) to (21) can be extended to a homomorphism from dr,s(l2) to b. so there exist a unique algebra homomorphism, ψ ′ : dr,s(l 2) → b such that ψ′(s1), ψ ′ (s2), . . . , ψ ′ (sr−1), ψ ′ (er), ψ ′ (sr+1), . . . , ψ ′ sr+s−1 satisfies the relation from (1) to (9), where ψ′(si) = ti, for i = 1, 2, . . . , r− 1, r + 1, . . . , r + s− 1 and ψ′(er) = xr. similarly the generators −→−→q 1, −→−→q 1′, −→−→q r+1, −→−→q (r+1)′, s1, s2, . . . , sr−1, sr+1, . . . , sr+s−1 satisfies the relations of the groups (z2 ×z2) o sr and (z2 ×z2) o ss that is the relations (1), (2), (3), (10), (11), (12), (13), (14) and (16). again by the universal property of free algebra any homomorphism from { −→−→q 1, −→−→q 1′, −→−→q r+1, −→−→q (r+1)′, s1, s2, . . . , sr−1, sr+1, . . . , sr+s−1 } to b can be extended to a homomorphism from k((z2 ×z2) o sr × (z2 ×z2) oss) to b. so there exist a unique algebra homomorphism, ψ′′ : k((z2 ×z2) osr × (z2 ×z2) oss) → b such that ψ′′( −→−→q 1), ψ′′( −→−→q 1′), ψ′′( −→−→q r+1), ψ′′( −→−→q (r+1)′), ψ′′(s1), ψ ′′(s2), . . . , ψ ′′(sr−1), ψ ′′(sr+1), . . . , ψ ′′(sr+s−1) satisfies the relations of the groups (z2×z2)osr and (z2×z2)oss that is the relations (1), (2), (3), (10), (11), (12), (13), (14) and (16). also ψ′|k(sr×ss) = ψ′′|k(sr×ss) and ψ′′( −→−→q 1) = p1, ψ′′( −→−→q 1′) = p1′, ψ′′( −→−→q r+1) = pr+1, ψ′′( −→−→q (r+1)′) = p(r+1)′ and ψ′′(si) = ti. first, for the walled brauer diagram w ∈ dr,s(l2), we prove the following 7 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 (i) if w −→−→q i −→−→q j = w, then ψ′(w)ψ′′( −→−→q i)ψ′′( −→−→q j) = ψ′(w) for 1, 1′ ≤ i ≤ r,r′ and (r + 1), (r + 1)′ ≤ j ≤ (r + s), (r + s)′. (ii) if −→−→q i −→−→q jw = w, then ψ′′( −→−→q i)ψ′′( −→−→q j)ψ′(w) = ψ′(w) for 1, 1′ ≤ i ≤ r,r′ and (r + 1), (r + 1)′ ≤ j ≤ (r + s), (r + s)′. (iii) if −→−→q sw −→−→q t = w, then ψ′′( −→−→q s)ψ′(w)ψ′′( −→−→q t) = ψ′(w) for 1, 1′ ≤ s,t ≤ r,r′ and (r + 1), (r + 1)′ ≤ s,t ≤ (r + s), (r + s)′. by extending the calculations for 1′ ≤ i ≤ r′, (r + 1)′ ≤ j ≤ (r + s)′ and considering ψ′ is an algebra homomorphism on dr,s(l2), ψ′′ is an algebra homomorphism on k((z2 ×z2) osr × (z2 ×z2) oss) and ψ′|k(sr×ss) = ψ′′|k(sr×ss)), the proofs of (i), (ii) and (iii) are very similar to the proofs (a), (b) and (c) of the theorem 3.2, [4]. next, using the results (i), (ii) and (iii), we define the homomorphism from −→−→ dr,s(l) to b by the following way. if −→−→w ∈ −→−→ dr,s(l), then there exist −→−→q , −→−→ q′ ∈ h so that −→−→w = −→−→q w −→−→ q′ , with w being the underlying walled klein-4 brauer diagram of −→−→w and h being the subgroup of (z2 ×z2) osr × (z2 ×z2) oss generated by−→−→q i, where 1, 1′ ≤ i ≤ (r + s), (r + s)′. if there are other elements −→−→ q′′, −→−→ q′′′ ∈ h and make −→−→w = −→−→ q′′w −→−→ q′′′ then −→−→q w −→−→ q′ = −→−→ q′′w −→−→ q′′′ implies that w = −→−→q −→−→ q′′w −→−→ q′′′ −→−→ q′ . so we get, ψ′′( −→−→q −→−→ q′′)ψ′(w)ψ′′( −→−→ q′′′ −→−→ q′ ) = ψ′(w) ψ′′( −→−→q )ψ′′( −→−→ q′′)ψ′(w)ψ′′( −→−→ q′′′)ψ′′( −→−→ q′ ) = ψ′(w) ψ′′( −→−→ q′′)ψ′(w)(ψ′′( −→−→ q′′′) = ψ′′( −→−→q )ψ′(w)ψ′′( −→−→ q′ ) now define ψ : −→−→ dr,s(l) → b by ψ( −→−→w ) = ψ′′( −→−→q ) ψ′(w) ψ′′( −→−→ q′ ). ψ = ψ′ on dr,s(l2), ψ = ψ′′ on k((z2 ×z2) o sr × (z2 ×z2) o ss) are obvious. we can use the linearity condition to extend it to the entire space. again by extending the calculations for 1′ ≤ i ≤ r′, (r + 1)′ ≤ j ≤ (r + s)′, the proof of the following identity, ψ(eĭ,j̆ −→−→q eŭ,v̆) = ψ′(eĭ,j̆)ψ ′′( −→−→q )ψ′(eŭ,v̆) (1) where, eĭ,j̆ = ∏ p eip,jp, eŭ,v̆ = ∏ q euq,vq, −→−→q = ∏ n −→−→q kn, where 1 ≤ ip,uq ≤ r, (r + 1) ≤ jp,vq ≤ (r + s), 1, 1′ ≤ kn ≤ (r + s), (r + s)′, is similar to the proof of the identity (3.1.1) in theorem 3.2, [4]. using the identity (1) and the proof of theorem 3.2, [4], we get that ψ is an isomorphism. 4. indexing set for the simple modules of the walled klein-4 brauer algebras this section will cover the indexing set for the simple modules of the walled klein-4 brauer algebra−→−→ dr,s(l), using the idempotent of that algebra and the simple modules of the group algebra of the groups (z2 ×z2) o sr and (z2 ×z2) o ss. the walled klein-4 brauer algebra −→−→ dr,s(l) is also used to build the recollment towers discussed in [1]. for k an algebraically closed field, r,s > 0 and l 6= 0 the following is a description of an idempotent element of the walled klein-4 brauer algebra. let −→−→e r,s be the walled klein-4 brauer diagram multiplied 8 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 by l−2, with one horizontal curve linking r and (r + 1) at the top marked by the sign (e,e), one horizontal curve linking r and (r + 1) at the bottom marked by the sign (e,e). the remaining edges for i = 1, 2, . . .r−1,r + 2, . . . ,s, are vertical edges that link i and i, and they are marked by the sign (e,e). the diagram below demonstrates this. −→−→e r,s is certainly an idempotent in −→−→ dr,s(l). because ( −→−→e r,s)2 equals −→−→e r,s. lemma 4.1. the algebra −→−→ dr−1,s−1(l) is isomorphic to the algebra −→−→e r,s −→−→ dr,s(l) −→−→e r,s, for r,s > 0. proof. for every diagram −→−→ d that belongs to −→−→e r,s −→−→ dr,s(l) −→−→e r,s, −→−→ d can be constructed from −→−→ dr−1,s−1(l) by inserting two signed vertical edges (vertical edges marked by the sign (e,e) or (a,e) or (e,b) or (a,b)) before and after the wall in some −→−→ f that belongs to −→−→ dr−1,s−1(l), so that r is linked with r and (r + 1) is linked with (r + 1). by this way we can define an injective homomorphism between the algebras −→−→ dr−1,s−1(l) and −→−→e r,s −→−→ dr,s(l) −→−→e r,s. for any image −→−→ d belongs to −→−→e r,s −→−→ dr,s(l) −→−→e r,s, we can get a pre-image −→−→ f belongs to −→−→ dr−1,s−1(l) by removing the vertical edges with the sign that we inserted to get −→−→ d . as a result, the algebras −→−→ dr−1,s−1(l) and −→−→e r,s −→−→ dr,s(l) −→−→e r,s are isomorphic. between the algebras, the preceding lemma aids in defining the necessary exact localization and the right exact globalisation functor [3]. next we will collect some details about the representations and the simple modules of the group (z2 ×z2) osn from [9]. the collection of all 4-partitions of (r + s), that is b4(r+s) is the indexing set for the inequivalent irreducible representations of the group (z2×z2)osr+s. for each λ ∈ b4(r+s) the set, {πλ: (z2×z2)osr+s → end(vλ)} is a complete set of inequivalent irreducible representation of the group (z2 ×z2) o sr+s. the set {vλ: λ ∈ b4(r+s)} is a complete set of simple modules of (z2 ×z2) osr+s. by extending the multiplication turn the representations of the group (z2×z2)osr+s into the group algebra k((z2 ×z2) osr+s) the simple modules of k((z2 ×z2) osr×(z2 ×z2) oss) are the outer tensor product of the simple modules of the form vλl � vλr, where vλl is the simple module of the group algebra k((z2 ×z2) o sr), vλr is the simple module of the group algebra k(z2 ×z2) o ss) and λl ∈ b4r, λr ∈ b4s, λl and λr are the p-regular 4-partition of r and p-regular 4-partition of s respectively [[2], theorem (10.33)]. consider the indexing set of the simple modules of the group algebra k((z2×z2)osr×(z2×z2)oss) as −→−→ in r,s 4−reg. so we get, −→−→ in r,s 4−reg= {(λl, λr) : λl is a p-regular 4-partition of r and λr is a p-regular 4-partition of s}. if the characteristic p of the field k is zero or p > max(r,s) then the group algebra k((z2 ×z2) o sr × (z2 ×z2) oss) is semisimple [2]. next, we will give the definition and lemma that we will need to construct the indexing set simple modules of the algebra −→−→ dr,s(l). 9 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 definition 4.2. the vertical pair of a diagram −→−→ d that belongs to −→−→ dr,s(l) is an ordered pair (α,β), where α represents the number of signed vertical edges on the left side of the wall and β represents the number of singed vertical edges on the right side of the wall, and the remaining top and bottom vertices are linked in pairs with signed horizontal curves. the edges marked by the signs (e,e) or (a,e) or (e,b) or (a,b) are referred to as signed edges. remark 4.3. whenever we multiply the two walled klein-4 brauer diagrams −→−→ d and −→−→e with the vertical pairs (α,β) and (γ,δ), we get a walled klein-4 brauer diagram with the vertical pair (σ,τ), with σ ≤ min(α,γ) and τ ≤ min(β,δ). lemma 4.4. (a) the set −→−→ dr,s(l) −→−→e r,s −→−→ dr,s(l) has a basis of all diagrams with vertical pair (a,b) for some a ≤ (r − 1) and b ≤ (s− 1). (b) −→−→ dr,s(l) −→−→e r,s −→−→ dr,s(l) is an ideal of −→−→ dr,s(l). (c) the quotient −→−→ dr,s(l) / −→−→ dr,s(l) −→−→e r,s −→−→ dr,s(l) has a basis of all diagrams with vertical pair (r,s). (d) the quotient −→−→ dr,s(l) / −→−→ dr,s(l) −→−→e r,s −→−→ dr,s(l) is isomorphic to the algebra k((z2 ×z2) osr ×(z2 × z2) oss). proof. the proofs of (a), (b) and (c), follows from the multiplication of the walled klein-4 brauer diagrams that we defined in section(3.1), and the definition(4.2), of the vertical pair of the diagram. again the proof (d), follows from (c). subsequently, for l = 0, we define an alternate idempotent −̃→−→e r,s in a different way as we cannot define the idempotent element of the walled klein-4 algebra as we did before. if r or s is greater than or equal to 2, then −̃→−→e r,s is the walled klein-4 brauer diagram with one upper horizontal curve marked by the sign (e,e) linking r and (r + 1), one lower horizontal curve marked by the sign (e,e) connecting r and (r + 2), nodes (r + 2) and (r + 1) are linked by a vertical edge marked by the sign (e,e), and all other edges being vertical lines from i to i marked by the sign (e,e). this is depicted in the diagram below, −̃→−→e r,s is definitely an idempotent. the lemma(4.1) and lemma(4.4) also applicable to −̃→−→e r,s. theorem 4.5. the indexing set −→−→ inr,s of the simple modules of the walled klein-4 brauer algebra −→−→ dr,s(l) equals to the disjoint union of −→−→ in r−i,s−i 4−reg , where 0 ≤ i ≤ min(r,s), for l 6= 0 or r 6= s. proof. the isomorphism defined in lemma(4.1) allows us to define an exact localization functor [3], f : −→−→ dr,smod → −→−→ dr−1,s−1mod by lr,s(m)= −→−→e r,s(m); m ∈ −→−→ dr,smod, and a corresponding right exact globalization functor, gr−1,s−1 : −→−→ dr−1,s−1mod → −→−→ dr,s mod by, 10 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 gr−1,s−1(n) = −→−→ dr,s −→−→e r,s ⊗−→−→e r,s −→−→ dr,s −→−→e r,s (n); n ∈ −→−→ dr−1,s−1 mod. by theorem 1 in [1] and the isomorphism (d) in lemma(4.4), we have that for r,s > 0, −→−→ inr,s = −→−→ inr−1,s−1 t −→−→ in r,s 4−reg = −→−→ inr−2,s−2 t −→−→ inr−1,s−1 t −→−→ in r,s 4−reg and so on, we have −→−→ inr,s = min(r,s)⊔ i=0 −→−→ in r−i,s−i 4−reg as −→−→ dr,0 ∼= −→−→ d0,r ∼= k((z2 ×z2) osr) next we define the sequence of idempotents −→−→e r,s,i in the walled klein-4 brauer algebra −→−→ dr,s (l) to produce the chain of two sided ideals in that algebra. consider, −→−→e r,s,0 = 1 and −→−→e r,s,i is the isomorphic image of ( −→−→e r−1,s−1,i−1), which we defined in the lemma(4.1), for 1 ≤ i ≤ min(r,s). the element −→−→e r,s,i is undefined when l = 0, i = r, and r = s. to these sequence of idempotents we define associate quotients, −→−→ dr,s,i = −→−→ dr,s / −→−→ dr,s −→−→e r,s,i −→−→ dr,s. when l 6= 0, we can use our explicit explanation of the isomorphism defined in the lemma 4.1 to provide an equivalent description of −→−→e r,s,i as l−2i times the walled klein-4 brauer diagram with i upper horizontal curves marked by the sign (e,e) linking r− t to r + 1 + t, j lower horizontal curves marked by the sign (e,e) linking r − t to r + 1 + t for 0 ≤ t ≤ i−1, and the remaining edges are all vertical marked by the sign (e,e) and linking i to i. in the case of l = 0, a similar justification can be provided. when i = 2 and l 6= 0, the idempotent −→−→e r,s,2 is shown in the diagram below. let −→−→ ai = −→−→ dr,s −→−→e r,s,i −→−→ dr,s then, −→−→ a0 = −→−→ dr,s −→−→e r,s,0 −→−→ dr,s = −→−→ dr,s , −→−→ a1 = −→−→ dr,s −→−→e r,s,1 −→−→ dr,s ⊂ −→−→ dr,s and so on. so we get the sequence of ideals −→−→ ai such that, . . . ⊂ −→−→ ai+1 ⊂ −→−→ ai ⊂ −→−→ ai−1 ⊂ . . . ⊂ −→−→ a1 ⊂ −→−→ a0 = −→−→ dr,s (2) remark 4.6. the ideal −→−→ aj, for j = 1, 2, 3, . . . , i−1, i, i + 1, . . . , has a basis of all diagrams with veritcal pair −−−→−−−→ (α,β) for some α ≤ r − j and β ≤ s− j. in particular the quotient −→−→ aj / −→−→ aj+1 for j = 1, 2, 3, . . . , i−1, i, i + 1, . . . , has a basis of all diagrams with vertical pair −−−−−−−−−→−−−−−−−−−→ (r − j,s− j). 11 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 5. walled klein-4 brauer algebras are iterated inflations we show that walled klein-4 brauer algebras are iterated inflations [5], of group algebra of the group ((z2×z2)osr×(z2×z2)oss) in this section. as a result, we will see that walled klein-4 brauer algebras are cellular, and we can also see that the simple modules of the walled klein-4 brauer algebra over a field k are described. we will start by proving some results for constructing inflated algebras. 5.1. one row representation of walled klein-4 brauer diagram each of the walled klein-4 brauer diagram −→−→ d ∈ −→−→ dr,s with f top signed horizontal curves, f bottom signed horizontal curves and the remaining edges are being vertical can be represented uniquely by a partial one row diagram such that −→−→ d = y−→−→µ, −→−→ν ,ρ(µν) . here −→−→µ stands for the top signed horizontal curves configuration, −→−→ν for the bottom signed horizontal curves configuration and ρ(µν) for the remaining signed vertical edge configuration obtained by renumbering the top vertices of the signed vertical edges from left to right as 1, 2, . . . ,r−f,r−f + 1,r−f + 2, . . . ,r + s−2f and their bottom vertices from left to right as 1̄, 2̄, . . . ,r −f,r −f + 1,r −f + 2, . . . ,r + s− 2f. also ρ(µν) = (ρ,ζ) ∈ (z2×z2)osr−f×(z2×z2)oss−f such that ρ(i) = j̄ if the ith top vertex of the the vertical edge is linked to the bottom vertex j̄, where ρ ∈ sr−f ×ss−f and ζ is a map from {1, 2, . . . ,r−f,r−f + 1,r−f + 2, . . . ,r + s−2f}−→ z2 ×z2 with, ζ(i) =   (e,e), if the sign (e,e) is marked on the vertical edge from i to some j̄; (a,e), if the sign (a,e) is marked on the vertical edge from i to some j̄; (e,b), if the sign (e,b) is marked on the vertical edge from i to some j̄; (a,b), if the sign (a,b) is marked on the vertical edge from i to some j̄. the set of elements −→−→µ that arise in this way is denoted by vfr,s, and the same notation is used to represent the set of elements −→−→ν that arise. this is considered to as the set of partial one-row (r,s,f) diagrams. lemma 5.1. an involution i from −→−→ dr,s(l) to −→−→ dr,s(l) is defined by transferring a diagram −→−→ d ∈ −→−→ dr,s(l) to a diagram i( −→−→ d ) ∈ −→−→ dr,s(l), which is the reflection in the horizontal axis of that diagram −→−→ d . proof. it is obvious that i2 = i since the diagram −→−→ d ∈ −→−→ dr,s(l) will be the same if we reflect it two times around its horizontal axis. we now prove the algebra vm ⊗vm ⊗k((z2 ×z2) osr−m × (z2 ×z2) oss−m) is an inflation of the group algebra of the group (z2 ×z2) osr−m ×(z2 ×z2) oss−m along a free k-module vm of rank |vmr,s| with respect to some bilinear form φm from vm⊗vm → k((z2×z2)osr−m×(z2×z2)oss−m) which we will define in the following lemma. also the multiplication is defined on the basis elements of the algebra vm⊗vm⊗k((z2 ×z2) osr−m×(z2 ×z2) oss−m) as follows. for −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 , −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ∈ vm ⊗vm ⊗k((z2 ×z2) osr−m × (z2 ×z2) oss−m) we have, ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 ).( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) = −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1φm( −→−→ν 1, −→−→µ 2)ρ(µν)2. (3) lemma 5.2. fix an index m, then the k-algebra −→−→ am/ −→−→ am+1 is isomorphic to the algebra vm ⊗vm ⊗ k((z2 ×z2) osr−m × (z2 ×z2) oss−m) with respect to some bilinear form which we will describe later in the proof. proof. by applying the partial one row diagram representation of a walled klein-4 brauer diagram that we mentioned previously we can define a k-isomorphism ψ : vm⊗vm⊗k((z2×z2)osr−m×(z2× z2) oss−m) → −→−→ am/ −→−→ am+1 as follows, 12 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 ψ( −→−→µ ⊗ −→−→ν ⊗ρ(µν)) = y−→−→µ,−→−→ν ,ρ(µν) we now define the bilinear form φm, which is required for multiplication of elements from the algebra vm ⊗ vm ⊗ k((z2 × z2) o sr−m × (z2 × z2) o ss−m). for −→−→µ 1 ⊗ −→−→ν 1 ⊗ ρ(µν)1, −→−→µ 2 ⊗ −→−→ν 2 ⊗ ρ(µν)2 ∈ vm ⊗vm ⊗k((z2 ×z2) osr−m × (z2 ×z2) oss−m) we have, ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1) = y−→−→µ 1,−→−→ν 1,ρ(µν)1 ∈ −→−→ dr,s(l) and ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2) = y−→−→µ 2,−→−→ν 2,ρ(µν)2 ∈ −→−→ dr,s(l) using the multiplication of the elements of the walled klein-4 diagram algebra −→−→ dr,s(l), that we described in the section 3.1 we have, y−→−→µ 1, −→−→ν 1,ρ(µν)1 .y−→−→µ 2, −→−→ν 2,ρ(µν)2 = lt. −−→−−→ (µν) (4) where t is the number of closed loops in the product (4), and −−→−−→ (µν) ∈ −→−→ dr,s(l) having 2m or more signed horizontal edges. if the product (4), does not have vertical pair −−−−−−−−−−→−−−−−−−−−−→ (r −m,s−m) then, set φm( −→−→ν 1, −→−→µ 2) = 0, otherwise φm( −→−→ν 1, −→−→µ 2) = lt.ρµν; where ρ(µν) ∈ k((z2 ×z2) osr−m × (z2 ×z2) oss−m) such that, y−→−→µ 1, −→−→ν 1,ρ(µν)1 ·y−→−→µ 2,−→−→ν 2,ρ(µν)2 = lt · −−→−−→ (µν) = lt ·y−→−→µ 1,−→−→ν 2,ρ(µν)1ρ(µν)ρ(µν)2 so we have, ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 · −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) = ψ( −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1φm( −→−→µ 1, −→−→ν 2)ρ(µν)2 ) = ψ( −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1 (ltρ(µν))ρ(µν)2 ) = lt ·y−→−→µ +1 ⊗ −→−→ν −1 ⊗ρ(µν)1ρ(µν)ρ(µν)2 = y−→−→µ 1, −→−→ν 1,ρ(µν)1 ·y−→−→µ 2,−→−→ν −2 ,ρ(µν)2 = ψ( −→−→µ 1 ⊗ −→−→ν −1 ⊗ρ(µν)1 ) · ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) ψ is an algebra homomorphism. suppose that, ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 ) = ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) then, y−→−→µ 1, −→−→ν 1,ρ(µν)1 = y−→−→µ +2 , −→−→ν 2,ρ(µν)2 so we have −→−→µ 1 = −→−→µ 2, −→−→ν 1 = −→−→ν 2, ρ(µν)1 = ρ(µν)2. now let −−→−−→ (µν) ∈ am/am+1 then −−→−−→ (µν)= y−→−→µ +, −→−→ν −,ρ(µν) and ψ( −→−→µ ⊗ −→−→ν ⊗ ρ(µν)) = y−→−→µ,−→−→ν ,ρ(µν) = −−→−−→ (µν), here −→−→µ ⊗ −→−→ν ⊗ρ(µν) ∈ vm⊗vm ⊗ k((z2×z2)osr−m×(z2×z2)oss−m). hence ψ is an isomorphism. lemma 5.3. the involution i from −→−→ am/ −→−→ am+1 to −→−→ am/ −→−→ am+1 under the map ψ corresponds to the standard involution on vm ⊗vm ⊗ k((z2 ×z2) osr−m ×(z2 ×z2) oss−m), which sends −→−→µ ⊗ −→−→ν ⊗ρ(µν) to −→−→ν ⊗ −→−→µ ⊗ρ(µν)−. 13 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 proof. the involution on −→−→ am/ −→−→ am+1 is derived from the involution on −→−→ dr,s(l) stated in lemma 5.1. as a result, the proof follows from the previous lemma 5.2. lemma 5.4. let −→−→ d 1 ∈ −→−→ am/ −→−→ am+1 and −→−→ d 2 ∈ −→−→ an/ −→−→ an+1 be two walled klein-4 brauer diagrams from−→−→ dr,s(l) and let their respective ψ-pre images be −→−→µ 1 ⊗ −→−→ν 1 ⊗ ρ(µν)1 and −→−→µ 2 ⊗ −→−→ν 2 ⊗ ρ(µν)2. we assume that n ≥ m. then the product −→−→ d 1. −→−→ d 2 is either an element of −→−→ an/ −→−→ an+1, or is an element of −→−→ an+1. under ψ it corresponds to the scalar multiple of an element −→−→ µ′ ⊗ −→−→µ 2 ⊗ηρ(µν)2; where −→−→ µ′ ∈ vn and η ∈ k((z2 ×z2) osr−n × (z2 ×z2) oss−n). in the case of n ≤ m, a similar assumption holds true. proof. both statements have a proof that is extremely similar to lemma 5.2. as a result of the lemma 5.4, we can now define the bilinear mappings α,β,γ described in [5], which are required for iterated inflation. we have proven the following theorem in total. theorem 5.5. the walled klein-4 brauer algebra −→−→ dr,s(l) is an iterated inflation of group algebras of groups ((z2 ×z2) osr−m ×(z2 ×z2) oss−m), where 0 ≤ m ≤ min(r,s). specifically as a free k-module−→−→ dr,s is equal to k((z2 ×z2) osr × (z2 ×z2) oss) ⊕ (v1 ⊗v1 ⊗k((z2 ×z2) osr−1 × (z2 ×z2) oss−1)) ⊕ (v2 ⊗v2 ⊗ k((z2×z2)osr−2×(z2×z2)oss−2)) ⊕ ··· , and inflation begins with k((z2×z2)osr×(z2×z2)oss) inflates it along v1 ⊗v1 ⊗k((z2 ×z2) osr−1 ×(z2 ×z2) oss−1) and so on, concluding with an inflation of k((z2 ×z2) os1) or k((z2 ×z2) os0) as bottom layer depending on whether (r + s) is even or odd. proof. the proof follows from the above lemma 5.2, lemma 5.3 and lemma 5.4. corollary 5.6. the walled klein-4 brauer algebra −→−→ dr,s(l) over any field of characteristic p = 0 or p > max(r,s) is cellular. proof. from the theorem 5.5, the walled klein-4 brauer algebra −→−→ dr,s is an iterated inflation of group algebras of the group ((z2 ×z2) o sr−m × (z2 ×z2) o ss−m) for 0 ≤ m ≤ min(r,s) along vm. so the proof follows from the remarks 2.5 and 2.6. corollary 5.7. if l 6= 0 or r 6= s then the simple modules of the walled klein-4 brauer algebras −→−→ dr,s(l) are indexed by all pair (m,λl,λr), where 0 ≤ m ≤ min(r,s) and (λl,λr) ∈ ∧r−m,s−m4−reg . if l = 0 and r = s we get the same indexing set for the simple modules as above but with the single simple corresponding to l = min(r,s). proof. by theorem 4.5, for l 6= 0 or r 6= s, the simple modules of the walled klein-4 brauer algebras−→−→ dr,s(l) are indexed by all pair (m,λl,λr), where 0 ≤ m ≤ min(r,s) and (λl,λr) ∈ ∧r−m,s−m4−reg . in the case of l = 0, the above assertion is also valid except that the case m = 0 (which occurs only for r even) does not contribute a simple module [6]. the exception in this situation arises from the fact that in the iterated inflation, there is a piece with φm equal to zero [6]. here φm is the bilinear form defined in the lemma 5.2. 14 ar ti cl e in pr es s a. tamilselvi, s. dhilshath / j. algebra comb. discrete appl. -(-) (2023) 1–15 references [1] a. g. cox, p. p. martin, a. e. parker, c. xi, representation theory of towers of recollement: theory, notes, and examples, j. algebra 302 (2006) 340–360. [2] c. w. curtis, i. reiner, methods of representation theory, vol. 1, wiley (1981). [3] j. a. green, polynomial representations of gln, lecture notes in mathematics 830, springer, berlin (1980). [4] b. kethesan, the structure of walled signed brauer algebras, kyungpook math. j 56 (2016) 1047– 1067. [5] s. konig, c. xi, cellular algebras: inflations and morita equivalences, j. london math. soc. 60(3) (1999) 700–722. [6] s. konig, c. xi, a characteristic free approach to brauer algebras, trans. am. math. soc. 353(4) (2001) 1489–1505. [7] m. parvathi, d. savithri, representation of g-brauer algebras, southeast asian bulletin of mathematics 26(3) (2002) 453–468. [8] m. parvathi, b. sivakumar, the klein-4 diagram algebras, j. algebra appl. 07(02) (2008) 231–262. [9] b. sivakumar, matrix units for the group algebra kgf = k((z2 ×z2) osf ), asian-eur. j. math. 2(2) (2009) 255–277. [10] a. tamilselvi, cellularity and representations of walled cyclic g-brauer algebras, malaya j. mat. 9(1) (2021) 402–408. [11] a. tamilselvi, a. vidhya, b. kethesan, robinson-schensted correspondence for the walled brauer algebras and the walled signed brauer algebras, algebra and its applications, springer proc. math. stat., springer, singapore, 174 (2016) 195-223. 15 https://doi.org/10.1016/j.jalgebra.2006.01.009 https://doi.org/10.1016/j.jalgebra.2006.01.009 http://dx.doi.org/10.5666/kmj.2016.56.4.1047 http://dx.doi.org/10.5666/kmj.2016.56.4.1047 https://doi.org/10.1112/s0024610799008212 https://doi.org/10.1112/s0024610799008212 https://www.ams.org/journals/tran/2001-353-04/s0002-9947-00-02724-0/s0002-9947-00-02724-0.pdf https://www.ams.org/journals/tran/2001-353-04/s0002-9947-00-02724-0/s0002-9947-00-02724-0.pdf https://doi.org/10.1007/s10012-002-0453-6 https://doi.org/10.1007/s10012-002-0453-6 https://doi.org/10.1142/s0219498808002795 https://doi.org/10.1142/s1793557109000212 https://doi.org/10.1142/s1793557109000212 https://doi.org/10.26637/mjm0804/0095 https://doi.org/10.26637/mjm0804/0095 https://doi.org/10.1007/978-981-10-1651-6_11 https://doi.org/10.1007/978-981-10-1651-6_11 https://doi.org/10.1007/978-981-10-1651-6_11 introduction preliminaries walled klein-4 brauer algebras indexing set for the simple modules of the walled klein-4 brauer algebras walled klein-4 brauer algebras are iterated inflations references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.00924 j. algebra comb. discrete appl. 3(2) • 105–123 received: 30 october 2015 accepted: 25 january 2016 journal of algebra combinatorics discrete structures and applications erratum to “a further study for the upper bound of the cardinality of farey vertices and applications in discrete geometry” [j. algebra comb. discrete appl. 2(3) (2015) 169-190] erratum daniel khoshnoudirad abstract: the equation (4) on the page 178 of the paper previously published has to be corrected. we had only handled the case of the farey vertices for which min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ n∗. in fact we had to distinguish two cases: min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ n∗ and min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) = 0. however, we highlight the correct results of the original paper and its applications. we underline that in this work, we still brought several contributions. these contributions are: applying the fundamental formulas of graph theory to the farey diagram of order (m, n), finding a good upper bound for the degree of a farey vertex and the relations between the farey diagrams and the linear diophantine equations. 2010 msc: 05a15, 05a16, 05a19, 05c30, 68r01, 68r05, 68r10 keywords: combinatorial number theory, farey diagrams, theoretical computer sciences, discrete planes, diophantine equations, arithmetical geometry, combinatorial geometry, discrete geometry, graph theory in computer sciences 1. introduction in [9], one of the strategies for the enumeration of pieces of discrete planes, was to estimate the number of vertices in a farey diagram. this work, combined with a basic property of graph theory, yields an upper bound. this upper bound is an homogeneous polynomial of degree 8: m3n3(m + n)2. in [17], i found that the number of straight farey lines is asymptotically mn(m + n) ζ(3) when m and n go to infinity. daniel khoshnoudirad (email: daniel.khoshnoudirad@hotmail.com). 105 http://eds.yildiz.edu.tr/ajaxtool/getarticlebypublishedarticleid?publishedarticleid=2169 http://eds.yildiz.edu.tr/ajaxtool/getarticlebypublishedarticleid?publishedarticleid=2169 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 henceforth, the strategy consisting in focusing on farey lines to study farey vertices combinatorics is not sufficient if we want to have a deeper understanding of the combinatorics of the (m,n)-cubes, and we can directly focus on the farey vertices [17] with some tools of number theory. the work which has been done for the case where min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ n∗, remains correct. but the case where min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) = 0 has to be handled. the goal of this study is to understand better how to bound |fv (m,n)| in an optimal way. 2. preliminaries let j−m,mk denote the set {−m,.. . ,−1,0,1, . . . ,m} of consecutive integers between −m and m. definition 2.1. [17](farey lines of order (m,n)) a farey line of order (m,n) is a line whose equation is uα + vβ + w = 0 with (u,v,w) ∈ j−m,mk × j−n,nk × z, and which has at least 2 intersection points with the frontier of [0,1]2. (u,v,w) are the coefficients. (α,β) are the variables. let denote the set of farey lines of order (m,n) by fl(m,n). definition 2.2. [14](farey sequences of order n) the farey sequence of order n is the set fn = { 0 }⋃{p q , ∣∣∣1 ≤ p ≤ q ≤ n,p∧q = 1} we mention [14] as a forthcoming modern reference work on the farey sequences. several standard variants of the notion of farey diagram are mentioned there. definition 2.3. (farey vertex) a farey vertex of order (m,n) is the intersection of two farey lines in [0,1]2. we will denote the set of farey vertices of order (m,n), obtained as intersection points of farey lines of order (m,n), by fv (m,n). definition 2.4. (farey diagrams for the pieces of discrete planes of order (m,n) (or (m,n)-cubes)) the farey diagram for the (m,n)-cubes of order (m,n) is the diagram defined by the passage of farey lines in [0,1]2. we recall that b c denotes the integer part, d e denotes the upper integer part, and 〈 〉 denotes the fractional part. if a and b are two integers, a ∧ b denotes the greatest common divisor of a and b, and a∨b denotes the least common multiple. ϕ denotes the euler’s totient function. card(a) or |a| denotes the cardinality of the set a. definition 2.5. (farey edge) a farey edge of order (m,n) is an edge of the farey diagram of order (m,n). we denote the set of farey edges by fe(m,n). definition 2.6. (farey graph) the farey graph of order (m,n) is the graph fg(m,n) = (fv (m,n),fe(m,n)). definition 2.7. (farey facet) a farey facet of order (m,n) is a facet of the farey graph of order (m,n). we will denote the set of farey facets of order (m,n) by ff(m,n). let m and n be two positive integers. we let fm,n denote the set = j0,m− 1k × j0,n− 1k. um,n denotes the set of all (m,n)-cubes. furthermore, the proposition 3 of [9] shows that the set of (m,n)-cubes of the discrete planes pα,β,γ only depends of (α,β), and is denoted by cm,n,α,β. definition 2.8. [9]((m,n)-pattern) let m and n be two positive integers. a (m,n)-pattern is a map w:fm,n −→ z. m × n is called the size of the (m,n)-pattern w. the set of the (m,n)-patterns will be denoted by mm,n. 106 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 figure 1. farey lines of order (3, 3) definition 2.9. [9]((m,n)-cube, see figure 2) the (m,n)-cube wi,j(α,β,γ) at the position (i,j) of a discrete plane pα,β,γ is the (m,n)-pattern w defined by: w(i′,j′) = pα,β,γ(i + i ′,j + j′)−pα,β,γ(i,j) for all (i′,j′) ∈fm,n = bα(i + i′) + β(j + j′) + γc−bαi + βj + γc for all (i′,j′) ∈fm,n where pα,β,γ(i,j) = bαi + βj + γc and { (i,j,pα,β,γ(i,j)), ∣∣∣(i,j) ∈ z2} defines the discrete plane pα,β,γ. now, we recall some results obtained in [9], and some direct consequences of this result. proposition 2.10. (recall [9]) 1. the (k,l)-th point of the (m,n)-cube at the position (i,j) of the discrete plane pα,β,γ can be computed by the formula : wi,j (α,β,γ) (k,l) = { bαk + βlc if 〈αi + βj + γ〉 < cα,βk,l bαk + βlc+ 1 else 107 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 figure 2. example of two (4, 3)-cubes (red and green) where cα,βk,l = 1−〈αk + βl〉 2. the (m,n)-cube wi,j(α,β,γ) only depends on the interval [b α,β h ,b α,β h+1[ containing 〈αi + βj + γ〉 where the bα,βh are the number c α,β k,l ordered by ascending order. 3. for all h ∈ j0,mn−1k, if [bα,βh ,b α,β h+1[ is non-empty, then there exists i,j such that 〈αi + βj + γ〉 ∈ [b α,β h ,b α,β h+1[. such a way, the number of (m,n)-cubes in the discrete plane pα,β,γ is equal to card ({ c α,β k,l ∣∣∣(k,l) ∈fm,n}) ≤ mn. corollary 2.11. [9] 1. ∀(α,β,γ) ∈ [0,1]2 ×r,w0,0(α,β,γ) = w0,0(α,β,〈γ〉) 108 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 2. ∀(α,β,γ) ∈ [0,1]2 ×r,∀(i,j) ∈ z2, wi,j(α,β,γ) = w0,0(α,β,αi + βj + γ) = w0,0(α,β,〈αi + βj + γ〉) 3. by the proposition 2.10, the set of (m,n)-cubes of the discrete planes pα,β,γ only depends of (α,β) and is denoted by cm,n,α,β. corollary 2.12. [9] let o be a farey connected component, then o is a convex polygon and if p1,p2,p3 are distinct vertices of the polygon o, then : • for any point p ∈o, cm,n,p = cm,n,p1 ∪cm,n,p2 ∪cm,n,p3 • for any point p ∈o in the interior of the segment of vertices p1 and p2, cm,n,p = cm,n,p1 ∪cm,n,p2 by this corollary, all the (m,n)-cubes are associated to farey vertices. and according to the proposition 2.10, there are at most mn (m,n)-cubes associated to a farey vertex, therefore∣∣∣um,n∣∣∣ ≤ mn∣∣∣fv (m,n)∣∣∣. 3. fundamental properties and lemmas lemma 3.1. (reminder of graph theory) let us consider n straight lines. the number of vertices constructed from these n lines is at most n(n−1) 2 . we know by [17], that the number of farey lines, is equivalent to a polynomial of degree 3 in m and n, when m and n go to infinity. according to lemma 3.1, these lines form a number of vertices, given at most by a polynomial of order 6 ([9]). but this method is far from giving an optimal upper bound for the cardinality of the farey vertices. in order to obtain a new and more powerful result of combinatorics on this set of vertices, we are going to study the properties of the farey lines passing through a farey vertex. our idea is to use the theorem: proposition 3.2. (reminder of graph theory) in a simple graph g = (v,e), we have:∑ x∈v deg(x) = 2 |e| where v is the set of vertices, and e is the set of edges, and deg(x) is the degree of the vertex x, that is the number of edges which are adjacent to the vertex x. moreover, we remind the euler’s formula: theorem 3.3. (euler’s formula for the connex planar graphs) in a connex planar multi-graph, having v vertices, e edges, and f facets, we have: v −e + f = 2 109 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 4. bound for the degree of a farey vertex 4.1. modeling corollary 4.1. in fg(m,n), |fe(m,n)| ≤ ∑ x∈fv (m,n) nl(x,m,n) where nl(x,m,n) denotes the number of farey lines of order (m,n) passing through the vertex x. proof. in fg(m,n), because a farey line generates at most 2 edges passing through the farey vertex p , we have: deg(p) ≤ 2×card ({ farey lines passing through p }) (1) so, by the handshaking proposition 3.2, 2 |fe(m,n)| ≤ ∑ x∈fv (m,n) 2nl(x,m,n) we simplify by 2, and we obtain the result. theorem 4.2. (gauss theorem) if (a,b,c) ∈ z3, such that a | bc, and a∧ b = 1. then, a | c. theorem 4.3. [2](asymptotic development of the harmonic series) if x ≥ 1, then∑ n≤x 1 n = log x + c +o( 1 x ). where c is euler’s constant, and τ the divisor function. we can apply this theorem and we are able to say in particular: corollary 4.4. there exists k > 0 such that, ∀n ∈ n\{0,1}, we have n∑ i=1 1 i ≤ k log n. lemma 4.5. let (a,b) ∈ n∗ ×n. let x ∈ r.⌊ bbxc a ⌋ = ⌊ bx a ⌋ proof. there is a classical equality which already exists, where a = b. here, we generalize it : bbxc a −1 < ⌊ bbxc a ⌋ ≤ bx a we multiply by a all the members : bbxc−a < a ⌊ bbxc a ⌋ ≤ bx ⇔bbxc−a + 1 ≤ a ⌊ bbxc a ⌋ ≤ bx 110 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 so, using the definition of the integer part of bx, we have bx−a < a ⌊ bbxc a ⌋ ≤ bx ⇔ bx a −1 < ⌊ bbxc a ⌋ ≤ bx a so, by the definition of the integer part of bx a , we obtain the claim. proposition 4.6. (upper bound for the number of farey lines of order (m,n) passing through a farey vertex of order (m,n)) let p = ( p q , p′ q′ ) be a farey vertex of order (m,n). let us define r,r′,s,s′,d and d′ as follows:   p = (p∧p′)r, q = (q ∧q′)s p′ = (p∧p′)r′, q′ = (q ∧q′)s′ d = p∧p′, d′ = q ∧q′ (2) let us define nlmax(p,m,n) as following: nlmax(p,m,n) = ( min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) + 1 ) × ( 2 ⌊ 1 d ( m p q + n p′ q′ )⌋) + 2 ⌊ m sd′ ⌋ + min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) . • if (p,p′) ∈ n∗2. then, we have nl(p,m,n) ≤ nlmax(p,m,n) • if p = 0 then we have nl ( 0, p′ q′ ) ≤ ( 1 + ⌊ n q′ ⌋) (2m + 1). the vertices such that p = 0, are the vertices of the set{( 0, p′ q′ ) with p′ q′ ∈ fn } • if p′ = 0, then we have nl ( p q ,0 ) ≤ ( 1 + 2 ⌊ m q ⌋) (n + 1). the vertices such that p′ = 0 are the vertices of the set{( p q ,0 ) with p q ∈ fm } proof. we can always suppose that in the equation of a farey line, (of the type: uα + vβ + w = 0, with (u,v,w) ∈ j−m,mk × j−n,nk ×z), we have v ≥ 0. because if v < 0, it is sufficient to multiply the equation by −1. and we obtain the same line, but (−u,−v,−w) ∈ j−m,mk × j0,nk ×z. first, we handle the case where p = 0 or p′ = 0. p = 0 ⇒ p′v + q′w = 0 ⇒ { v = q′k w = −p′k ⇒ 0 ≤ k ≤ n q′ (because of the preliminary.) 111 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 there are at most 1 + ⌊ n q′ ⌋ such integers. and there are 2m + 1 integers in the interval j−m,mk. the vertices such that p = 0, are the vertices of the set{( 0, p′ q′ ) with p′ q′ ∈ fn } . p′ = 0 ⇒ pu + qw = 0 ⇒ { u = qk w = −pk ⇒ 0 ≤ |k| ≤ m q there are at most 1 + 2 ⌊ m q ⌋ such integers. the vertices such that p′ = 0, are the vertices of the set {( p q ,0 ) with p q ∈ fm } . then, it remains to handle the general case: (p,p′) ∈ n∗2 so, we are looking for an optimal bound for the cardinality of (u,v,w) ∈ j−m,mk × j0,nk ×z such that u p q + v p′ q′ = −w ⇔ upq′ + vp′q qq′ = −w (with the condition u∧v ∧w = 1), that is u(p∧p′)r(q ∧q′)s′ + v(p∧p′)r′(q ∧q′)s qq′ = −w. (p∧p′)(q ∧q′) urs′ + vr′s (q ∧q′)2ss′ = −w. after simplification: (p∧p′) urs′ + vr′s (q ∧q′)ss′ = −w. (p∧p′)(urs′ + vr′s) = −w(q ∧q′)ss′ (p∧p′)urs′ = −w(q ∧q′)ss′ − (p∧p′)vr′s ⇒ s | (p∧p′)urs′ as s∧ [(p∧p′)rs′] = 1, the gauss theorem 4.2 implies that s | u. so, { ∃u′ ∈ z such that u = su′ ∃v′ ∈ z such that v = s′v′ (3) if v = 0, then we have: u = qk ⇒ 1 ≤ |k| ≤ ⌊ m sd′ ⌋ w = −pk ⇒ 1 ≤ |k| ≤ ⌊ 1 p ( m p q + n p′ q′ )⌋ 112 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 so, in the case where v = 0, 1 ≤ |k| ≤ min (⌊ m sd′ ⌋ , ⌊ 1 p ( m p q + n p′ q′ )⌋) . we come back to the general equation (with v′ ≥ 1): in particular, 0 ≤ |u′| ≤ ⌊m s ⌋ and 1 ≤ v′ ≤ ⌊n s′ ⌋ . (p∧p′) su′rs′ + s′v′r′s (q ∧q′)ss′ = −w ⇒ (p∧p′) u′r + v′r′ (q ∧q′) = −w (theorem of gauss 4.2) ⇒ (p∧p′) | w and (q ∧q′) | u′r + v′r′. and because of the non-redundancy hypothesis, we have: u′ ∧v′ | q ∧q′ the diophantine equation becomes: u′r + v′r′ = − w d d′ (4) when w is fixed, the consequence of the hypothesis of primality enables to solve this diophantine equation: let us fix w,   u′ = u0 ( − w p∧p′ (q ∧q′) ) + r′k v′ = v0 ( − w p∧p′ (q ∧q′) ) −rk for k ∈ z (5) where (u0,v0) is a particular solution of the diophantine equation in (x,y): rx + r′y = 1. in particular,   − ⌊m s ⌋ + u0 ( w p∧p′ ) q ∧q′ ≤ r′k ≤ ⌊m s ⌋ + u0 ( w p∧p′ ) q ∧q′ − ⌊n s′ ⌋ + v0 ( − w p∧p′ ) q ∧q′ ≤ rk ≤−1−v0 ( w p∧p′ ) q ∧q′ (6) the determinant of this system in ( w p∧p′ ,k ) is: u0q ∧q′r + v0(q ∧q′)r′ = (q ∧q′)[u0r + v0r′] = q ∧q′ moreover, we have seen that as we have: |w| ≤ m p q + n p′ q′ , and as p∧p′ | w, we can deduce that there exists w′ such that w = w′(p∧p′). so, 0 ≤ |w′| ≤ ⌊ 1 p∧p′ ( m p q + n p′ q′ )⌋ now, we distinguish 2 cases: 113 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 • if w = 0, by the lemma 4.5, the number of suitable integers k is bounded by min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) • w 6= 0. we can always choose u0 < 0 and v0 > 0. in these conditions, the number of suitable integers k is bounded by: min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) and the number of ( k, w d ) is bounded by: ( 1 + min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋)) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋ and finally, the total number of couples ( k, w d ) is at most: ( min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) + 1 ) × ( 2 ⌊ 1 d ( m p q + n p′ q′ )⌋) + 2 min (⌊ m sd′ ⌋ , ⌊ 1 p ( m p q + n p′ q′ )⌋) + min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) that is, ( min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) + 1 ) × ( 2 ⌊ 1 d ( m p q + n p′ q′ )⌋) + 2 ⌊ m sd′ ⌋ + min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) so, nl(p,m,n) ≤ nlmax(p,m,n) lemma 4.7. if we consider a farey vertex v = ( p q , p′ q′ ) of order (m,n), then q ∨q′ ≤ 2mn. proof. ( p q , p′ q′ ) ∈ fv (m,n) ⇒   ∃((u,u′),(v,v′),(w,w′)) ∈ j−m,mk2 × j−n,nk2 ×z2 u′v −uv′ 6= 0 such that: ( p q , p′ q′ ) = ( |w′v −wv′| |uv′ −u′v| , |wu′ −w′u| |uv′ −u′v| ) . 114 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 so, q ∨q′ | |uv′ −u′v|. in particular, q ∨q′ ≤ 2mn. in particular, ss′d′ ≤ 2mn. proposition 4.8. let ( p q , p′ q′ ) ∈ fv (m,n).   a(m,n,p,q,p′,q′) = min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) b(m,n,p,q,p′,q′) = 2 ⌊ 1 d ( m p q + n p′ q′ )⌋ c(m,n,p,q,p′,q′) = a(m,n,p,q,p′,q′)×b(m,n,p,q,p′,q′) a′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ a(m,n,p,q,p′,q′) b′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ b(m,n,p,q,p′,q′) c′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ c(m,n,p,q,p′,q′) d′(m,n) = ∑ x∈fv (m,n) min(b2m sr′c,b ns′rc)=0 nl(x,m,n) we have: |fe(m,n)| ≤ a′(m,n) + 2b′(m,n) + c′(m,n) + d′(m,n) (7) + ∑ p q ∈fm nl ( p q ,0 ) + ∑ p′ q′∈fn nl ( 0, p′ q′ ) (8) proof. p(α,β) ∈ fv (m,n) ⇒∃ ( p q , p′ q′ ) such that:   α = p q with p∧q = 1,p ≤ q ≤ 2mn β = p′ q′ with p′ ∧q′ = 1,p′ ≤ q′ ≤ 2mn by the corollary 4.1, we have: 115 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 |fe(m,n)| ≤ ∑ x∈fv (m,n) nl(x,m,n) ≤ ∑ x∈fv (m,n) min(b2m sr′c,b ns′rc)∈n∗ nl(x,m,n) + ∑ x∈fv (m,n) min(b2m sr′c,b ns′rc)=0 nl(x,m,n) + ∑ p q ∈fm nl ( p q ,0 ) + ∑ p′ q′∈fn nl ( 0, p′ q′ ) to conclude, we use the result of the proposition 2. 4.2. case of the vertices for which min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) ∈ n∗ proposition 4.9. ∃k > 0,∀(m,n) ∈ (n\{0,1})2, a′(m,n) ≤ km2n2(m + n) ln2(mn). proof. a′(m,n) ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ a(m,n,p,q,p′,q′) ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) i point out that i choosed ⌊ n s′r ⌋ , and after ⌊ 2m sr′ ⌋ , in order to obtain a symmetric upper bound. in the following, we use the boundaries for r,r′,s,s′ given by: let us permute the sums and let us change the variables by using, as before,   r = p p∧p′ ,s = q q ∧q′ r′ = p′ p∧p′ ,s′ = q′ q ∧q′ d = p∧p′,d′ = q ∧q′ ⇒   1 ≤ s ≤ 2mn d′ 1 ≤ s′ ≤ 2mn d′ 1 ≤ r ≤ d′s d 1 ≤ r′ ≤ d′s′ d in particular,   r ≤ d′s d r′ ≤ d′s′ d ⇒ d ≤ ⌊ min ( d′s r , d′s′ r′ )⌋ 116 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 a′(m,n) ≤ m+n∑ d=1 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 ⌊ d′s d ⌋∑ r=1 ⌊ d′s′ d ⌋∑ r′=1 ⌊ n s′r ⌋ ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ n s′r ⌋⌊min( d′sr , d′s′r′ )⌋∑ d=1 1 now, we have to distinguish the case where d′s = 1 and the case d′s > 1 in the sums in order to use the corollary 4.4 (and the same for d′s′). ∃k > 0,∀(m,n) ∈ (n\{0,1})2, a′(m,n) ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s>1 d′s′∑ r′=1 d′s′>1 d′ n rr′ + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 2mn d′s + km2n3 ln2(mn) a′(m,n) ≤ k′n ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 2mn s + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ s=1 b2mns c∑ d′=1 2mn s + km2n3 ln2(mn) ≤ k′m2n3 ln2(mn) 2mn∑ s=1 1 s2 + km2n3 ln2(mn) ≤ k′′m2n3 ln2(mn) ∃k > 0,∀(m,n) ∈ (n\{0,1})2, a′(m,n) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 a(m,n,p,q,p′,q′) ≤ m+n∑ d=1 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s d∑ r=1 d′s′ d∑ r′=1 ⌊ 2m sr′ ⌋ ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ m sr′ ⌋⌊min( d′sr , d′s′r′ )⌋∑ d=1 1 117 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 m sr′ d′s r ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s>1 d′s′∑ r′=1 d′s′>1 d′ m rr′ + kn2m3 ln2(mn) ≤ k′m ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 + kn2m3 ln2(mn) ≤ k′m ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 2mn d′s + kn2m3 ln2(mn) ≤ k′m2n ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 1 s + kn2m3 ln2(mn) ≤ k′′n2m3 ln2(mn) proposition 4.10. ∃k > 0,∀(m,n) ∈ (n\{0,1})2, b′(m,n) ≤ km2n2(m + n). proof. b′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ b(m,n,p,q,p′,q′) ≤ [b ′ 1(m,n) + b ′ 2(m,n)] b ′ 1(m,n) ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 r q b ′ 2(m,n) ≤ n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 r′ q′ let us study further b ′ 1(m,n), then the results for b ′ 2(m,n) are computed in a similar manner. 118 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 b ′ 1(m,n) ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 r q ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ 1 ≤ m2n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 ss′ ≤ k′m3n2 2mn∑ s=1 2mn∑ s′=1 1 s2s′2 ≤ k′′m2n2(m + n) the computation is exactly the same for b ′ 2(m,n). proposition 4.11. ∃k > 0,∀(m,n) ∈ (n\{0,1})2, c′(m,n) ≤ km2n2(m + n) ln(mn). proof. c′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ c(m,n,p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 min(b2m sr′c,b ns′rc)∈n∗ a(m,n,p,q,p′,q′)×b(m,n,p,q,p′,q′) ≤ k[c ′ 1(m,n) + c ′ 2(m,n)] c ′ 1(m,n) = mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 s′r 1 d p q c ′ 2(m,n) = mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 sr′ 1 d p′ q′ let us study further c ′ 1(m,n), then the results for c ′ 2(m,n) are computed in a similar manner. 119 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 ∃k > 0,∀(m,n) ∈ (n\{0,1})2, c ′ 1(m,n) ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 ss′d′ ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ d′s′ r′ 1 ss′d′ ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′c≥r d′s′∑ r′=1 b2ms c≥r′ d′s′>1 1 r′s + km2n3 ln(mn) ≤ k′mn2 ln(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 ss′ + km2n3 ln(mn) ≤ k′m2n3 ln(mn) b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 s2s′2 + km2n3 ln(mn) ≤ k′′m2n3 ln(mn) the computation is exactly the same for c ′ 2(m,n). 4.3. cases of the vertices for which p = 0 or p′ = 0 now, we treat the two simple cases where p = 0 or p′ = 0 of the proposition 4.6: proposition 4.12. ∃k > 0,∀(m,n) ∈ n∗2, ∑ p′ q′∈fn nl ( 0, p′ q′ ) + ∑ p q ∈fm nl ( p q ,0 ) ≤ kmn(m2 + n2). proof. • [( 1 + ⌊ n q′ ⌋) (2m + 1) ] ≤ 2m + 1 + 2nm + n ≤ 5mn + 1 ∑ p′ q′∈fn nl ( 0, p′ q′ ) ≤ ∑ p′ q′∈fn [( 1 + ⌊ n q′ ⌋) (2m + 1) ] ≤ ∑ p′ q′∈fn (5mn + 1) ≤ (5mn + 1) |fn| • [( 1 + 2 ⌊ m q ⌋) (n + 1) ] ≤ n + 1 + 2mn + 2m ≤ 5mn + 1 ∑ p q ∈fm nl ( p q ,0 ) ≤ ∑ p q ∈fm [( 1 + 2 ⌊ m q ⌋) (n + 1) ] ≤ ∑ p q ∈fm (5mn + 1) ≤ (5mn + 1) |fm| 120 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 we know [12] that |fn| = 1 + n∑ k=1 ϕ(k) ∼ +∞ n2 2ζ(2) so there exists k > 0 such that∑ p′ q′∈fn nl ( 0, p′ q′ ) + ∑ p q ∈fm nl ( p q ,0 ) ≤ kmn(m2 + n2). 4.4. case of the vertices for which min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) = 0 it remains to handle the case of the vertices for which min (⌊ 2m sr′ ⌋ , ⌊ n s′r ⌋) = 0. at this stage of our research, we are not yet able to bound this term d′(m,n). 5. conclusion of this strategy by the strategy of the farey vertices, we obtained some interesting results: • we applied the fundamental formulas of graph theory to the farey diagram of order (m,n). • we found a good upper bound for the degree of a farey vertex. • we made relations between the farey diagrams and the linear diophantine equations by solving explicit systems of linear diophantine equations. however, at the moment, this method does not help to improve the known upper bound for the cardinality of the farey vertices. we suggest two possible ways of future research for bounding this term d′(m,n). • either ∃k > 0,∀(m,n) ∈ (n\{0,1})2, d′(m,n) ≤ km2n2(m + n) ln2(mn). in that case, we could conclude that : ∃k > 0,∀(m,n) ∈ (n\{0,1})2, ∣∣∣fv (m,n)∣∣∣ ≤ km2n2(m + n) ln2(mn) • otherwise we have to search a bound whose order is between 5 and 6. if the optimal order is 6, that would strenghten the importance of our work [17], as it would probably mean that the order of the cardinality of farey vertices is a homogeneous polynomial of order 6. acknowledgment: many thanks to grégory apou for his help. i am greatly thankful to the referee for his remarks, helps, propositions of improvements. and many thanks to the members of the team of j. algebra comb. discrete appl., for their patience. 121 d. khoshnoudirad / j. algebra comb. discrete appl. 3(2) (2016) 105–123 references [1] d. m. acketa, j. d. žunić, on the number of linear partitions of the (m,n)−grid, inform. process. lett. 38(3) (1991) 163–168. 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[32] v. trifonov, l. pasqualucci, r. dalla-favera, r. rabadan, fractal-like distributions over the rational numbers in high-throughput biological and clinical data, sci. rep. 1(591) (2011). 123 http://dx.doi.org/10.1007/bf02579194 http://dx.doi.org/10.1007/bf02579194 https://hal.inria.fr/file/index/docid/46382/filename/tel-00005113.pdf https://hal.inria.fr/file/index/docid/46382/filename/tel-00005113.pdf http://dx.doi.org/10.1109/icpr.1992.201971 http://dx.doi.org/10.1109/icpr.1992.201971 http://dx.doi.org/10.1109/icpr.1992.201971 http://dx.doi.org/10.1103/physrevstab.17.014001 http://dx.doi.org/10.1103/physrevstab.17.014001 http://arxiv.org/abs/1406.6991 http://arxiv.org/abs/1406.6991 http://dx.doi.org/doi:10.1038/srep00191 http://dx.doi.org/doi:10.1038/srep00191 introduction preliminaries fundamental properties and lemmas bound for the degree of a farey vertex conclusion of this strategy references issn 2148-838x j. algebra comb. discrete appl. -(-) • 1–23 received: 18 august 2021 accepted: 20 january 2022 ar ti cl e in pr es s journal of algebra combinatorics discrete structures and applications toric code distances from terraced polytopes research article andrew wilfong abstract: we call a polytope terraced if upon projecting onto a one-dimensional coordinate space, each fiber of the projection is contained in the fiber below it. we present a technique to compute the minimum distance for toric codes arising from terraced polytopes. this technique is demonstrated by determining the minimum distance for all toric codes that correspond to smooth n–polytopes with n + 2 facets. we also find the minimum distance for all toric codes coming from smooth polygons with five edges and from smooth 3–polytopes with six facets. 2010 msc: 94b27, 52b20, 14m25 keywords: toric code, lattice polytope 1. introduction a toric code, first introduced by hansen in [4], is an algebraic geometry code whose underlying variety is a toric variety. it is typically very difficult to determine basic properties of algebraic geometry codes, but this is often more easily accomplished for toric codes. there is a bijective correspondence between toric varieties and convex polytopes, and we can use combinatorial and geometric properties of polytopes to reveal information about toric codes. these codes provide a nice balance of computability and diversity. they allow us to create algebraic geometry codes that are interesting enough to merit exploration while still being simple enough to reveal their essential characteristics. as one example application, for certain block lengths and dimensions, toric codes have been used to obtain the maximum possible minimum distance among all linear codes [2, 8]. when working over a sufficiently large field, the dimension of a toric code equals the number of lattice points in the corresponding polytope. but even with the extra combinatorial structure from the polytope, the minimum distance of a toric code can still be quite difficult to compute. several techniques have been developed to provide minimum distance formulas for special cases. hansen first determined the minimum distance of codes coming from hirzebruch surfaces by studying their cohomology and intersection theory [5]. decomposing a polytope into a minkowski sum is another useful strategy for bounding the minimum distance [11, 18]. a lower bound for minimum distance was found by little and andrew wilfong; eastern michigan university, usa (email: awilfon2@emich.edu). 1 http://orcid.org/0000-0002-5338-9289 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 schwarz using multivariate vandermonde matrices [12]. soprunov and soprunova determined natural relationships for the distances of toric codes arising from products of polytopes and from pyramids over polytopes [17, 19]. these techniques have been used to calculate minimum distance for certain classes of toric codes. for example, the minimum distances for toric surface codes (i.e., those arising from polygons) have been computed up to dimension seven (i.e., for all lattice polygons that contain up to seven lattice points) [7, 12, 14]. little and schenck found the minimum distance for most of the toric codes corresponding to smooth polygons with five edges [11]. kimball derived minimum distance results for certain smooth 3–polytopes with five facets [9]. in [17], soprunov derived a lower bound for minimum distance based on the fiber structure of a polytope. in this paper, we build on these results. if a polytope has some extra geometric structure, then soprunov’s lower bound becomes equality. the necessary extra structure involves fibers in the direction of one of the coordinate axes. if each successive fiber is contained in the previous fiber, then we say that the corresponding polytope is terraced, and the minimum distance of the corresponding toric code can be computed exactly. terraced polytopes account for a large and varied class of polytopes, so this should provide a powerful tool for computing the minimum distance for toric codes. to demonstrate the utility of the technique, we focus on smooth polytopes with few facets. all smooth n–polytopes with n + 2 facets are terraced, as are all smooth polygons with five edges and all smooth 3–polytopes with six facets. we will complete and extend the results in [11] and [9] by calculating the minimum distance for all of the corresponding toric codes. while it is still possible to derive the minimum distance for these codes using the techniques from [18] and [11], these codes illustrate the benefits of working with the terraced polytope structure. with our new technique, the computations are often simpler and we do not need to make extra assumptions about the size of the underlying field. the techniques also extend beyond the special class of smooth polytopes, allowing the potential for generalization to other terraced polytopes. in section 2, we provide the necessary background and notation for toric codes. in section 3, we derive the main results about the minimum distance of toric codes coming from terraced polytopes. we then demonstrate the technique for several examples that will be helpful in the later sections. in section 4, we compute the minimum distance for all toric codes coming from smooth n–polytopes with n + 2 facets. in section 5, we explore smooth polytopes with three more facets than the dimension of the polytope. we find the minimum distance of the corresponding toric codes for all such polytopes in dimensions two and three. section 6 concludes with some questions to motivate further exploration. 2. toric codes for background on error-correcting codes, and on algebraic geometry codes in particular, see [6, 13, 20]. refer to [3] for background on toric varieties. we will bypass the algebraic geometric structure of toric varieties and approach toric codes from a purely combinatorial and geometric direction. to construct a toric code, we start with a polytope p ⊆ rn and a prime power q. (see [21] for background on polytopes.) we construct a vector space of laurent polynomials whose exponents correspond to the lattice points in p : l(p) = spanfq { xλ11 · · ·x λn n : (λ1, . . . ,λn) ∈ p ∩z n } . let ( f∗q )n = { t1, t2 . . . , t(q−1)n } , so we are establishing a linear order for the points in this algebraic torus. define ev : l(p) → f(q−1) n q by ev (f) = ( f (t1) ,f (t2) , . . . ,f ( t(q−1)n )) . the toric code cp is the image ev (l(p)). it is a linear code with block length (q − 1)n. example 2.1. let p = [0,a] ⊆ r, where a is an integer and 0 ≤ a ≤ q − 2 for some prime power q. 2 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 then l(p) is generated by 1,x1,x21, . . . ,xa1 and cp is generated by the rows of the following matrix:  1 1 . . . 1 t1 t2 . . . tq−1 t21 t 2 2 . . . t 2 q−1 ... ... ... ta1 t a 2 . . . t a q−1   . in this case, cp is a reed–solomon code with dimension a + 1 and minimum distance q − 1 −a (cf. [6, chapter 5]). in the previous example, the dimension of cp equals the number of lattice points in p . this is true in general, as long as q is large enough. theorem 2.2. [16, theorem 3.3] let p ⊆ [0,q − 2]n be a polytope, where q is a prime power. then the dimension of cp equals the number of lattice points contained in p. for the remainder of this paper, we will assume that q is a sufficiently large prime power so that all polytopes under consideration are contained in [0,q − 2]n. for a polytope p ⊆ [0,q − 2]n, let d (p) represent the minimum distance of the corresponding toric code cp . for brevity, we will often call this the distance of the code. theorem 2.3. [19, theorem 2.1] let p ⊆ [0,q − 2]m and q ⊆ [0,q − 2]n be lattice polytopes, and consider p ×q ⊆ rm+n. then we have d (p ×q) = d (p) d (q). example 2.4. consider the interval [0,a]×{0} as a subset of [0,q − 2]2, where a ≥ 0 is an integer. then d ([0,a] ×{0}) = d ([0,a]) d ({0}) = (q − 1 −a) (q − 1) by example 2.1. in general, if [0,a] is considered as a subset of rn for some n ∈ n, then d ([0,a]) = (q − 1)n−1 (q − 1 −a). in calculating distances, we will often need to apply certain affine transformations to polytopes to produce more convenient geometric realizations. in many situations, such a transformation preserves the distance of the corresponding toric codes. definition 2.5. [12] two polytopes p,q ⊆ rn are lattice equivalent if there is an invertible integer affine transformation t : rn → rn for which t (p) = q. theorem 2.6. [12, theorem 4] if p and q are lattice equivalent, then cp and cq are monomially equivalent. in particular, we get d (p) = d (q). 3. minimum distance for terraced polytopes one technique that allows us to compute the minimum distance of a toric code is to project the corresponding polytope onto a lower-dimensional space. the distances of the fibers under this projection can reveal information about the distance of the original polytope. theorem 3.1. [17, corollary 4.3] let p ⊆ [0,q − 2]n be a polytope. let πk : rn → r be the projection onto the kth coordinate, whose corresponding indeterminate is xk. suppose πk (p) = {0, 1, . . . , l}. let pi be the projection of the ith fiber p ∩π−1k (i) to all but the k th coordinate. if d (p0) ≤ d (p1) ≤ ···≤ d (pl), then d (p) ≥ min 0≤i≤l {(q − 1 − i) d (pi)}. with certain extra restrictions on p , the inequality above becomes an equality. definition 3.2. let p ⊆ [0,q − 2]n be a polytope. let πk : rn → r be the projection onto the kth coordinate, whose corresponding indeterminate is xk. for each i = 0, . . . ,q − 2, let pi be the projection of the ith fiber p ∩π−1k (i) to all but the k th coordinate. we say that p is terraced in the xk–direction if p0 ⊇ p1 ⊇ ···⊇ pq−2. we call pi the ith–level of p. 3 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 theorem 3.3. suppose p ⊆ [0,q − 2]n is terraced in the xk–direction, with πk (p) = {0, 1, . . . , l}. then d (p) = min 0≤i≤l {(q − 1 − i) d (pi)}. proof. suppose p0 ⊇ p1 ⊇ ···⊇ pl. decreasing the number of lattice points also decreases the number of codewords. this can never decrease the minimum distance of the corresponding codes. thus we have d (p0) ≤ d (p1) ≤ ···≤ d (pl), and d (p) ≥ min 0≤i≤l {(q − 1 − i) d (pi)} by theorem 3.1. it remains to verify that min 0≤i≤l (q − 1 − i) d (pi) ≥ d (p). note that (q − 1 − i) d (pi) = d ([0, i]) d (pi) = d ([0, i] ×pi) by example 2.1 and theorem 2.3. since p is terraced, [0, i]×pi ⊆ p for each i. thus d ([0, i] ×pi) ≥ d (p) for each i, so min 0≤i≤l (q − 1 − i) d (pi) ≥ d (p). let’s explore some examples. first, we use the method of terraced polytopes to find the distance of the toric code corresponding to an n–dimensional simplex ∆na with vertices (0, . . . , 0) , (a, 0, . . . , 0) , (0,a, 0, . . . , 0) , . . . , (0, . . . , 0,a) where a ∈ n. this has been derived in the past using various other techniques [12, 19]. lemma 3.4. let a ∈ n. then d (∆na) = (q − 1) n−1 (q − 1 −a). proof. induct on n. if n = 1, then ∆1a = [0,a] is an interval and d ( ∆1a ) = q − 1 − a by example 2.1. for n > 1, the simplex ∆na is terraced in the xn–direction. its i th level is itself a simplex ∆n−1a−i . by induction, we can assume d ( ∆n−1a−i ) = (q − 1)n−2 (q − 1 − (a− i)). then theorem 3.3 yields d (∆na) = min 0≤i≤a (q − 1 − i) d ( ∆n−1a−i ) = (q − 1)n−2 · min 0≤i≤a (q − 1 − i) (q − 1 − (a− i)) . the expression being minimized is quadratic in i, and it attains a minimum at one of the endpoints i = 0 or i = a. but the value is the same at both endpoints, and we get d (∆na) = (q − 1) n−2 (q − 1) (q − 1 −a) = (q − 1)n−1 (q − 1 −a) . we can also apply theorem 3.3 to much more complicated terraced polytopes. for example, we can combine that theorem with the previous lemma to study truncated simplices stacked on top of prisms. the following result appears to be new in dimensions three and higher. proposition 3.5. choose positive integers a, b, and c. stack a simplex ∆nb+c on top of (in the xn– direction) a prism ∆n−1b+c × [0,a]. truncate the simplex with xn ≤ a + b to obtain a polytope p ⊆ [q − 2] n. (see figure 1 for an example of such a polytope in three dimensions.) then d (p) = { (q − 1)n−2 (q − 1 −a) (q − 1 − (b + c)) if c ≥ a (q − 1)n−2 (q − 1 − (a + b)) (q − 1 − c) if c ≤ a . proof. the polytope p is terraced in the xn–direction. for 0 ≤ i ≤ a, the ith–level is pi = ∆n−1b+c and d (pi) = (q − 1) n−2 (q − 1 − (b + c)). then min 0≤i≤a (q − 1 − i) d (pi) = (q − 1 −a) (q − 1) n−2 (q − 1 − (b + c)) 4 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 1. a truncated simplex stacked on top of a triangular prism. comes from the i = a level. now we must determine if a higher level gives a smaller value for (q − 1 − i) d (pi). for a ≤ i ≤ a + b, we have pi = ∆n−1b+c−(i−a) and d (pi) = (q − 1) n−2 (q − 1 − (b + c− (i−a))) . then d (p) = min a≤i≤a+b (q − 1 − i) (q − 1)n−2 (q − 1 − (b + c− (i−a))) . = (q − 1)n−2 · min a≤i≤a+b (q − 1 − i) (q − 1 − (a + b + c) + i) . the expression (q − 1 − i) (q − 1 − (a + b + c) + i) is quadratic with respect to i, and it is minimized at one of the endpoints i = a or i = a + b. thus d (p) = (q − 1)n−2 · min{(q − 1 −a) (q − 1 − (b + c)) , (q − 1 − (a + b)) (q − 1 − c)} = { (q − 1)n−2 (q − 1 −a) (q − 1 − (b + c)) if c ≥ a (q − 1)n−2 (q − 1 − (a + b)) (q − 1 − c) if c ≤ a . in some common situations, the minimum of (q − 1 − i) d (qi) occurs at the base level of a terraced polytope q. theorem 3.3 can then provide the distance corresponding to certain polytopes contained in q. this gives a powerful tool for computing a toric code’s distance by comparing the corresponding polytope to a polytope with known distance. corollary 3.6. let p ⊆ q ⊆ [0,q − 2]n be polytopes that are terraced in the xk–direction. if d (p0) = d (q0) and d (q) = (q − 1) d (q0), then d (p) = d (q) = (q − 1) d (p0). proof. decreasing the number of lattice points can never decrease distance. thus p ⊆ q implies d (qi) ≤ d (pi) for each i. suppose d (p0) = d (q0) and d (q) = (q − 1) d (q0). then d (q) is determined by the i = 0 level of q. then for all i, we have (q − 1) d (p0) = (q − 1) d (q0) ≤ (q − 1 − i) d (qi) ≤ (q − 1 − i) d (pi) . thus the i = 0 level of p minimizes (q − 1 − i) d (pi), and the result follows from theorem 3.3. 5 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 example 3.7. let p ⊆ [q − 2]2 be the trapezoid with vertices (0, 0), (0,a), (b,a), and (b + ca, 0), where a,b,c ∈ n (see figure 2). then p is terraced in the x2–direction, p ⊆ ∆2b+ca, and d (p0) = d (( ∆2b+ca ) 0 ) = q − 1 − (b + ca). by corollary 3.6, d (p) = d ( ∆2b+ca ) = (q − 1) (q − 1 − (b + ca)). figure 2. a trapezoid contained in a simplex. the distances of the corresponding toric codes are equal. the toric variety corresponding to such a trapezoid is a hirzebruch surface. hansen first determined the distance of the corresponding toric code using topological methods [4]. little and schenck proved the same result using minkowski sum decompositions [11, example 3.3]. the method of terraced polytopes offers a third approach. but this method also offers much more general results. a trapezoid is just one of many polytopes that shares the same distance as the simplex in which it is contained. in general, if p ⊆ ∆na ⊆ [q − 2] n is terraced in the xn–direction and d (p0) = d ( ∆n−1a ) = (q − 1)n−2 (q − 1 −a), then d (p) = d (∆na) = (q − 1) n−1 (q − 1 −a). the distance of any such terraced polytope p is completely determined by its base-level distance and the simplex in which it is contained. we can also derive the above distance immediately by noting that [0,b + ca] ⊆ p ⊆ ∆2b+ca. then d ([0,b + ca]) ≥ d (p) ≥ d ( ∆2b+ca ) . using example 2.4 and lemma 3.4, we get d (p) = d ([0,b + ca]) = d ( ∆2b+ca ) = (q − 1) (q − 1 − (b + ca)) . this observation will be useful in later sections, so we state it here in more generality. the converse of the following statement is also true (see [15, proposition 6.6], for example), but we will not need it for our results. proposition 3.8. suppose a polytope p is contained in the simplex ∆na. if [0,a] ⊆ p, then d (p) = (q − 1)n−1 (q − 1 −a). we can now determine the distance of the toric code corresponding to a more general simplex ∆na1,...,an with vertices (0, . . . , 0) , (a1, 0, . . . , 0) , (0,a2, 0, . . . , 0) , . . . , (0, . . . , 0,an). for a1, . . . ,an ∈ n, this distance formula was first determined by little and schwarz [12, corollary 2]. here, we generalize to simplices ∆na1,...,an for which a1, . . . ,an ≥ 1 are not necessarily integers. the corresponding code comes from the lattice polytope that is the convex hull of all lattice points in the simplex. these polytopes can be quite complicated. for example, consider the hexagon h = conv{(0, 0) , (9, 0) , (8, 1) , (4, 4) , (1, 6) , (0, 6)} shown in figure 3. it has exactly the same set of lattice points as the simplex ∆29.9,6.8. then the following proposition gives us d (h) = d ( ∆29.9,6.8 ) = (q − 1) (q − 10). 6 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 3. a hexagon with integer vertices whose lattice points coincide with those of a 2–simplex with non-integer vertices. proposition 3.9. consider the simplex ∆na1,...,an ∈ [q − 2] n, where a1, . . . ,an ≥ 1 are real numbers. let a = max{ba1c , . . . ,banc}. then d ( ∆na1,...,an ) = (q − 1)n−1 (q − 1 −a). proof. let p be the convex hull of the lattice points of ∆na1,...,an, so d ( ∆na1,...,an ) = d (p). since [0,a] ⊆ p ⊆ ∆na, the result follows immediately from proposition 3.8. combining this result with theorem 2.3 yields the following. corollary 3.10. let q = ∆mt1,...,tm × ∆ n tm+1,...,tm+n ⊆ [q − 2]m+n, where t1, . . . , tm+n ≥ 1 are real numbers and m,n ∈ n. let t = max{bt1c , . . . ,btmc} and u = max{btm+1c , . . . ,btm+nc}. 1. then d (q) = (q − 1)m+n−2 (q − 1 − t) (q − 1 −u). 2. suppose n ≥ 2 and assume u = btm+1c (using theorem 2.6 if needed). then q is terraced in the xm+n–direction. the base level is q0 = ∆mt1,...,tm × ∆ n tm+1,...,tm+n−1 and the distance of this level is d (q0) = (q − 1) m+n−3 (q − 1 − t) (q − 1 −u). thus d (q) = (q − 1) d (q0). the second part of this corollary allows us to apply corollary 3.6 using a product of simplices. in the next section, this will allow us to determine the distance for certain polytopes contained in such a product that share the same base-level distance. 4. distances for smooth n–polytopes with n + 2 facets an n–polytope is simple if exactly n–many edges meet at each vertex. a smooth n–polytope is a simple lattice polytope for which the primitive generators of the edges emanating from each vertex form a basis of zn. there is a bijective correspondence between smooth n–polytopes and complete nonsingular n–dimensional toric varieties (see [10], for example). to describe all smooth n–polytopes with n + 2 facets, we start by considering all corresponding fans whose generating rays are the inner normal vectors of the facets of such a polytope. (see [21, chapter 7], for example, for background on fans.) 7 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 let n ≥ 2 and 1 ≤ r ≤ n− 1 be integers. let 0 ≤ an−r+1 ≤ an−r+2 ≤ ··· ≤ an be integers. let ei be the ith standard basis vector in rn. define the following vectors in rn: u = − n∑ i=n−r+1 ei v = − n−r∑ i=1 ei − n∑ i=n−r+1 aiei. define fn (an−r+1, . . . ,an) to be the fan in rn with generating rays e1, . . . ,en,u,v, whose maximal cones are given by any subset of all but one element of {e1, . . . ,en−r,v}, combined with all but one element of {en−r+1, . . . ,en,u}. theorem 4.1. [10] every complete nonsingular n–dimensional toric variety whose fan has n + 2 generating rays is isomorphic to a variety with normal fan fn (an−r+1, . . . ,an). thus any smooth n–polytope comes from one of the normal fans fn (an−r+1, . . . ,an). to realize such a polytope, we must choose a position in space for each of its facets. since we are ultimately interested in the distances of the corresponding toric codes, we can use lattice equivalence (definition 2.5 and theorem 2.6) to limit the smooth polytopes under consideration. in particular, we can assume that the facet with inner normal ray ei is xi ≥ 0. just two rays remain to consider. let b1,b2 > 0 be integers. assume u is normal to the boundary of the half space xn−r+1 + . . . + xn ≤ b1 and v is normal to the boundary of the half space x1 + · · · + xn−r + an−r+1xn−r+1 + · · · + anxn ≤ b2. denote the corresponding polytope as p . for dimension n = 2, such a polytope p is a rectangle or a trapezoid. for a rectangle, the distance of the corresponding toric code can be found by applying example 2.1 and theorem 2.3. a trapezoid corresponds to a hirzebruch surface, and the distance of the corresponding code was calculated in example 3.7. hansen first computed the distance of this hirzebruch surface code [5], and kimball later described the distances of some toric codes corresponding to smooth 3–polytopes with five facets [9]. viewing these as terraced polytopes allows us to generalize their results to any dimension. theorem 4.2. let p be a smooth n–polytope with n + 2 facets. assume p is defined by the following inequalities, where r,an−r+1, . . . ,an,b1,b2 are integers for which 1 ≤ r ≤ n− 1, 0 ≤ an−r+1 ≤ ··· ≤ an, and b1,b2 ≥ 1. x1, . . . ,xn ≥ 0 xn−r+1 + · · · + xn ≤ b1 x1 + · · · + xn−r + an−r+1xn−r+1 + · · · + anxn ≤ b2 let k be the smallest index for which ak > 0. (if an−r+1 = · · · = an = 0 then set k equal to n + 1.) then d (p) = { (q − 1)n−1 (q − 1 − b2) if k = n−r + 1 (q − 1)n−2 (q − 1 − b1) (q − 1 − b2) if k > n−r + 1 . proof. we use corollary 3.6 to derive a formula for d (p). thus we need another polytope q that contains p . define q to be the product of simplices given by the following inequalities: x1, . . . ,xn ≥ 0 xn−r+1 + · · · + xk−1 ≤ b1 x1 + · · · + xn−r + akxk + · · · + anxn ≤ b2 if k = n−r + 1, then we omit the second inequality, and q is a simplex. let pj and qj be the projections of p and q, respectively, onto the first j coordinates. for any j, the inequalities tell us that both pj and qj are terraced in the xj–direction. this will allow us to use corollary 3.6 iteratively, starting in a dimension where pj = qj, and building up to dimension n. 8 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 first, note that pj0 = p j−1 and qj0 = q j−1 for any j. by proposition 3.9 or corollary 3.10, d ( qj ) = (q − 1) d ( q j 0 ) whenever j ≥ k. finally, note that pj ⊆ qj for all j. corollary 3.6 tells us that for any j, if d ( p j 0 ) = d ( q j 0 ) , then d ( pj ) = d ( qj ) . by projecting onto the first k − 1 coordinates, we get pk−1 = qk−1. this common polytope is a simplex or a product of simplices. then d ( pk0 ) = d ( pk−1 ) = d ( qk−1 ) = d ( qk0 ) , so d ( pk ) = d ( qk ) . then d ( pk+10 ) = d ( qk+10 ) , so d ( pk+1 ) = d ( qk+1 ) . we can continue in this fashion until we obtain d (p) = d (q). since q is a simplex if k = n−r + 1 and a product of simplices if k > n−r + 1, the result follows from proposition 3.9 and corollary 3.10. as in example 3.7, theorem 4.2 could be generalized considerably to describe the distance of any toric code corresponding to a polytope that is contained in q and that shares the same base-level distance. in particular, d (p) is completely described by the placement of the facets, determined by b1 and b2. the parameters ai essentially indicate the angle at which one of the facets is tilted, and the amount of tilting does not affect d (p). only the presence or absence of tilting influences this distance. if ai = 0 for some i, then there is no tilting in that direction and p has the same distance as a product of simplices. but if ai 6= 0 for all i, then each direction is tilted inward and p has the same distance as the simplex ∆nb2. 5. distances for smooth n–polytopes with n + 3 facets little and schenck used minkowski sums to find the toric code distances corresponding to smooth polygons with five edges [11]. here, we reprove their results using the method of terraced polytopes. then, we extend the results to dimension three. there is much more variation in the possible structure of a smooth n–polytope with n + 3 facets compared to those with n + 2 facets. there may not be enough commonality to find the distance for such polytopes in every dimension using the technique of terraced polytopes. 5.1. smooth polygons with five edges little and schenck provided four cases to consider [11]. possibly using lattice equivalence (theorem 2.6), each polygon from [11, section 4] matches one of the polygons in figure 4. polygons i (if r > 0), ii (if r > 1), and iii correspond to the left polygon in the figure. polygon iv (if r > 0) from [11, section 4] corresponds to the center polygon in figure 4. polygon ii (if r = 1) or any of polygons i, ii, or iv with r = 0 (which were not explicitly covered in [11]) correspond to the right polygon in figure 4. while all smooth polygons with five edges fit in these cases, there are many other polygons described by figure 5.1 that are not smooth. the same distance results hold for those polygons. figure 4. every smooth polygon with five edges is lattice equivalent to one of these three, where a, b, c > 0 are integers. the locations of the unlabeled vertices do not affect the distance computations. both the left and center polygons in figure 4 are contained in ∆2a and contain [0,a]. the distance of the corresponding toric codes follows immediately from proposition 3.8. 9 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 proposition 5.1. for the left and center polygons in figure 4, the distance of the corresponding toric code equals d ( ∆2a ) = (q − 1) (q − 1 −a). the right polygon in figure 4 is a truncated simplex (a triangle here) stacked on top of a triangular prism (a rectangle here). the distance for such a polytope was determined in proposition 3.5. proposition 5.2. let p represent the right polygon in figure 4. then d (p) = { (q − 1 −a) (q − 1 − (b + c)) if c ≥ a (q − 1 − (a + b)) (q − 1 − c) if c ≤ a . 5.2. smooth wedges over pentagons batyrev first classified the inner normal fans corresponding to smooth n–polytopes with n+ 3 facets. theorem 5.3. [1] let p be a smooth n–polytope with n + 3 facets, and assume p is not combinatorially equivalent to a product of polytopes, each of positive dimension. then the inner normal fan for p has generating rays that belong to the nonempty sets x0 = {v1, . . . ,vp0} x1 = {y1, . . . ,yp1} x2 = {z1, . . . ,zp2} x3 = {t1, . . . , tp3} x4 = {u1, . . . ,up4} where p0 +p1 +p2 +p3 +p4 = n+3. a collection of generating rays forms a cone in the fan if the collection does not contain all rays from two cyclically adjacent sets xi. in particular, to form a maximal cone, we start by using all generating rays from two of the sets xi that are not cyclically adjacent. we then include all but one generating ray from each of the remaining sets. to ensure smoothness, the generating rays must satisfy the following for some nonnegative integers c2, . . . ,cp2,b1, . . . ,bp3. v1 + · · · + vp0 + y1 + · · · + yp1 − c2z2 −···− cp2zp2 − (b1 + 1) t1 −···− (bp3 + 1) tp3 = 0 y1 + · · · + yp1 + z1 + · · · + zp2 −u1 −···−up4 = 0 z1 + · · · + zp2 + t1 + · · · + tp3 = 0 t1 + · · · + tp3 + u1 + · · · + up4 −y1 −···−yp1 = 0 u1 + · · · + up4 + v1 + · · · + vp0 − c2z2 −···− cp2zp2 − b1t1 −···− bp3tp3 = 0 we will focus on three-dimensional polytopes with six facets. in dimension three, a smooth polytope with six facets is combinatorially equivalent either to a wedge over an edge of a pentagon or to a cube. we consider the wedges in this section and cubes in the next section. let e1, . . . ,e5 be the edges of a pentagon p . we can construct a wedge over the edge ei of p as follows. embed p ×{0} in r3. consider the plane x3 = 0 along with a second plane that contains the edge ei. these planes determine exactly one bounded region in p × r. this region is a 3–polytope, which we will call the wedge over the edge ei of p , denoted pi. the facets fj of pi for i 6= j can be associated with the corresponding edges ej of the pentagon p . let fi represent the base of pi (i.e., the original pentagon p), and let w represent the new facet produced by wedging (see figure 5). the pi are all combinatorially equivalent, but they must be realized in very specific ways to yield smooth polytopes. let v1, . . . ,v5,w represent the inner normal vectors for the corresponding facets f1, . . . ,f5,w of pi. now we can apply theorem 5.3. we can simplify the notation in this dimension so that v1, v2, v3, v4, and v5 correspond to t1, u1, v1, y1, and z1, respectively, from the theorem. the generating ray w corresponds to the final generating ray, choosing just one ray with subscript 2 from the theorem. for all possible i, we will need at most two symbols to represent the coefficients, so we use s and t instead of c2, . . . ,cp2,b1, . . . ,bp3. 10 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 5. p i, a wedge over a pentagon. case 1 first, we consider p1. by theorem 5.3, for p1 to be smooth, the generating rays of the inner normal fan must satisfy the following, for some integers s,t ≥ 0. v3 + v4 − (s + 1) v1 − (t + 1) w = 0 v4 + v5 −v2 = 0 v5 + v1 + w = 0 v1 + w + v2 −v4 = 0 v2 + v3 −sv1 − tw = 0 without loss of generality (using theorem 2.6), let v1 = e1, v3 = e2, and v5 = e3, where the ei are the standard basis vectors. then the equations above yield the following: v2 = −(t−s) e1 −e2 − te3 v4 = −(t−s) e1 −e2 − (t + 1) e3 w = −e1 −e3 without loss of generality (using theorem 2.6), assume s ≤ t. to obtain the polytope p1, we choose facets corresponding to each generating ray of the fan. let x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0 correspond to v1, v3, and v5, respectively. the facets (i.e., the corresponding half-spaces) for v2, v4, and w are given below, where a,b,c > 0 are integers. v2 : (t−s) x1 + x2 + tx3 ≤ b v4 : (t−s) x1 + x2 + (t + 1) x3 ≤ c w : x1 + x3 ≤ a possibilities for p1 are shown in figure 6. theorem 5.4. the minimum distance of the toric code corresponding to p1 is d ( p1 ) = { (q − 1)2 (q − 1 − b) if s < t (q − 1) (q − 1 −a) (q − 1 − b) if s = t . proof. first, suppose s < t. it is straightforward to verify that p1 ⊆ ∆3b (by comparing the facet for v2 to the facet x1 + x2 + x3 ≤ b of ∆2b). since [0,b] ⊆ p 1, the distance formula follows from proposition 3.8. 11 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 6. p 1, with s < t on the left and s = t on the right. the unlabeled vertices are not needed in the distance computations. next, suppose s = t. the polytope p1 is terraced in the x3–direction. we have p10 = [0,a] × [0,b] and d ( p10 ) = (q − 1 −a) (q − 1 − b). since p1 ⊆ ∆2a × [0,b] and d ( ∆2a × [0,b] ) = (q − 1) (q − 1 −a) (q − 1 − b) = (q − 1) d (( ∆2a × [0,b] ) 0 ) the result follows from corollary 3.6. case 2 as in case 1, we can use theorem 5.3 to obtain a convenient geometric realization for the polytope p2. if we let x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0 correspond to v2, v5, and v3, respectively, then we can use the procedure from case 1 to describe the remaining facets v1, v4, and w. these facets are given below, where a,b,c > 0 are integers. v1 : x2 ≤ b v4 : (s + 1) x2 + x3 ≤ c w : x1 + sx2 + x3 ≤ a possibilities for p2 are shown in figure 7. theorem 5.5. the distance of the toric code corresponding to p2 is d ( p2 ) = { (q − 1)2 (q − 1 −a) if s > 0 (q − 1) (q − 1 −a) (q − 1 − b) if s = 0 . the proof is nearly identical to that of theorem 5.4. 12 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 7. p 2, with s > 0 on the left and s = 0 on the right. case 3 we can use theorem 5.3 again to realize the polytope p3. let x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0 correspond to v3, w, and v5, respectively. the facets for v1, v2, and v4 are given below, where a,b,c > 0 are integers. v1 : x3 ≤ c v2 : x1 + x2 + sx3 ≤ a v4 : x1 + x2 + (s + 1) x3 ≤ b possibilities for p3 are shown in figure 8. theorem 5.6. the distance of the toric code corresponding to p3 is d ( p3 ) =   (q − 1)2 (q − 1 −a) if s > 0 (q − 1) (q − 1 − (b−a)) (q − 1 −a) if s = 0 and a ≥ c (q − 1) (q − 1 − c) (q − 1 − (b− c)) if s = 0 and a ≤ c . proof. first, suppose s > 0. then [0,a] ⊆ p3 ⊆ ∆3a, and the formula follows from proposition 3.8. next, suppose s = 0. then p3 is a truncated simplex ∆3a stacked on top of a triangular prism ∆2a × [0,b−a], and d ( p3 ) can be computed using proposition 3.5. 13 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 8. p 3, with s > 0 on the left and s = 0 on the right. case 4 we use theorem 5.3 to realize p4. let x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0 correspond to v3, v4, and v5, respectively. the facets for v1, v2, and w are given below, where a,b,c > 0 are integers. v1 : x3 ≤ c v2 : x1 + sx3 ≤ a w : x1 + x2 + (s + 1) x3 ≤ b a possibility for p4 is shown in figure 9. it is straightforward to verify that [0,b] ⊆ p4 ⊆ ∆3b. proposition 3.8 gives the following result. figure 9. p 4. theorem 5.7. the distance of the toric code corresponding to p4 is d ( p4 ) = (q − 1)2 (q − 1 − b). case 5 once more, we apply theorem 5.3 to describe p5. let x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0 correspond to v5, w, and v3, respectively. the facets for v1, v2, and v4 are given below, where a,b,c > 0 are integers. v1 : x1 + x2 ≤ a v2 : sx1 + (s− t) x2 + x3 ≤ c v4 : (s + 1) x1 + (s− t + 1) x2 + x3 ≤ b 14 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 this case is significantly more complicated. depending on the values of s and t, we must examine p5 from different perspectives to find a useful terracing that leads to a distance computation. we will consider five possibilities for p5. see figure 10. theorem 5.8. the distance of the toric code corresponding to p5 depends on the parameters s and t as follows. 1. if t < s, then d ( p5 ) = (q − 1)2 (q − 1 − c). 2. if s = t = 0, then d ( p5 ) = { (q − 1) (q − 1 − (b−a)) (q − 1 −a) if a ≥ c (q − 1) (q − 1 − c) (q − 1 − (b− c)) if a ≤ c . 3. if s = t ≥ 1, then d ( p5 ) = (q − 1) (q − 1 − (b− c)) (q − 1 − c). 4. if t = s + 1, then d ( p5 ) = (q − 1) (q − 1 − (a− b + c)) (q − 1 − b). 5. if t ≥ s + 2, then d ( p5 ) = (q − 1)2 (q − 1 − (b− (s− t + 1) a)). proof. 1. note that [0,c] ⊆ p5 ⊆ ∆3c. the result follows from proposition 3.8. 2. this follows immediately from proposition 3.5. 3. consider p5 as a terraced polytope in the x2–direction. if 0 ≤ i < b − c, then p5i is a pentagon like that shown in the center of figure 4, whose longest edge has length c. by proposition 5.1, d ( p5i ) = (q − 1) (q − 1 − c) for each such i, and (q − 1 − i) d ( p5i ) is minimized at i = b− c− 1 in this interval. if b−c ≤ i < a, then p5i is a trapezoid. example 3.7 yields d ( p5i ) = (q − 1) (q − 1 − (b− i)) in this interval. note that d ( p5b−c ) = (q − 1) (q − 1 − c) = d ( p5i ) for 0 ≤ i < b−c. thus the minimum of (q − 1 − i) d ( p5i ) is achieved for some i ≥ b− c. if i = a, then p5a is an interval and d ( p5a ) = (q − 1) (q − 1 − (b−a)). this fits the same formula as the distances for b− c ≤ i < a. by theorem 3.3, d ( p5 ) = (q − 1) · min b−c≤i≤a (q − 1 − i) (q − 1 − (b− i)) . since (q − 1 − i) (q − 1 − (b− i)) is quadratic in i, the minimum happens at one of the endpoints i = b− c or i = a. thus d ( p5 ) = (q − 1) · min{(q − 1 − (b− c)) (q − 1 − c) , (q − 1 −a) (q − 1 − (b−a))} = (q − 1) · min { (q − 1)2 − b (q − 1) + c (b− c) , (q − 1)2 − b (q − 1) + a (b−a) } = (q − 1) · [ (q − 1)2 − b (q − 1) + min{c (b− c) ,a (b−a)} ] . for p5 to be a valid polytope, we need b− c < a and b− (s + 1) a > 0. (see the vertices in figure 10.) then b−a < c and b−sa > a. since we assumed s ≥ 1, we have a < b−sa ≤ b−a < c. we get b < a + c and c−a > 0. then b < a + c =⇒ b (c−a) < (a + c) (c−a) =⇒ c (b− c) < a (b−a) . therefore, d ( p5 ) = (q − 1) (q − 1 − (b− c)) (q − 1 − c). 15 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 10. possibilities for p 5. 16 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 4. consider p5 as being terraced in the x1–direction. if 0 ≤ i < b−c, then p5i is a pentagon, as shown in figure 11. by proposition 5.2, d ( p5i ) = { (q − 1 − (a− b + c)) (q − 1 − (b− (s + 1) i)) if si ≤ b−a (q − 1 − (a− i)) (q − 1 − (c−si)) if si ≥ b−a . but we need b− (s + 1) a > 0 for the polytope in figure 10 to have the desired structure. then b− (s + 1) i > (s + 1) a− (s + 1) i = (s + 1) (a− i) ≥ a− i since s ≥ 0. then b−a > si and we get d ( p5i ) = (q − 1 − (a− b + c)) (q − 1 − (b− (s + 1) i)). figure 11. p 5i , if t = s + 1 and 0 ≤ i < b − c. if b − c ≤ i < a, then p5i is a rectangle with dimensions (a− i) × (b− (s + 1) i). then d ( p5i ) = (q − 1 − (a− i)) (q − 1 − (b− (s + 1) i)). when i = b−c, d ( p5i ) agrees with the formula for d ( p5i ) for 0 ≤ i < b − c. if i = a, then p5a is an interval and d ( p5a ) = (q − 1) (q − 1 − (b− (s + 1) a)). this agrees with the formula for d ( p5i ) in b− c ≤ i < a. to summarize, we have d ( p5i ) = { (q − 1 − (a− b + c)) (q − 1 − (b− (s + 1) i)) if 0 ≤ i ≤ b− c (q − 1 − (a− i)) (q − 1 − (b− (s + 1) i)) if b− c ≤ i ≤ a . let’s first compute the minimum of (q − 1 − i) d ( p5i ) for 0 ≤ i ≤ b− c. we have min 0≤i≤b−c (q − 1 − i) d ( p5i ) = min 0≤i≤b−c (q − 1 − i) (q − 1 − (a− b + c)) (q − 1 − (b− (s + 1) i)) = (q − 1 − (a− b + c)) · min 0≤i≤b−c (q − 1 − i) (q − 1 − (b− (s + 1) i)) . the expression (q − 1 − i) (q − 1 − (b− (s + 1) i)) is quadratic in i, so it is minimized at one of the endpoints i = 0 or i = b− c. since 0 < b− c < q − 1, c > 0, and s ≥ 0, we have 0 < (b− c) (s (q − 1 − (b− c)) + c) . this is equivalent to (q − 1) (q − 1 − b) < (q − 1 − (b− c)) (q − 1 − (b− (s + 1) (b− c))) so d ( p5i ) is minimized at the i = 0 level over this interval. using theorem 3.3, we must now determine if (q − 1 − i) d ( p5i ) can attain a lesser value over b− c ≤ i ≤ a. consider min b−c≤i≤a (q − 1 − i) d ( p5i ) = min b−c≤i≤a (q − 1 − i) (q − 1 − (a− i)) (q − 1 − (b− (s + 1) i)) . the product being minimized is now cubic with respect to i. its graph has two negative roots and one positive root at i = q − 1. thus the graph is concave down on b − c ≤ i ≤ a and the 17 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 minimum value on this interval must occur at one of the endpoints. but we already know that (q − 1 − (b− c)) d ( p5b−c ) > (q − 1) d ( p50 ) , so we only get a potentially lesser value in this range at i = a. according to the construction of p5, we have −s (q − 1 −a) ≤ 0 < b−a. then −as (q − 1 −a) < a (b−a), which is equivalent to (q − 1) (q − 1 − b) < (q − 1 −a) (q − 1 − (b− (s + 1) a)). multiplying both sides by the positive integer q−1−(a− b + c) reveals that (q − 1 − i) d ( p5i ) is minimized at the i = 0 level, so by theorem 3.3, d ( p5 ) = (q − 1) (q − 1 − (a− b + c)) (q − 1 − b) . 5. apply the transformation t     x1x2 x3     =   0 −1 01 0 0 0 0 1   ·     x1x2 x3  −   0a 0     to p5. by theorem 2.6, the resulting polytope yields the same distance. the transformed polytope t ( p5 ) satisfies [0,b− (s− t + 1) a] ⊆ t ( p5 ) ⊆ ∆3 b−(s−t+1)a, and the distance formula follows from proposition 3.8. 5.3. smooth combinatorial cubes the remaining smooth 3–polytopes with six facets are combinatorially equivalent to a cube. to describe these polytopes, we start with their inner normal fans. let u1,u2,u3,v1,v2,v3 be the generating rays of a fan in r3. if the cones of this fan are given by any subset of generating rays not containing both ui and vi for some i, then the fan is the normal fan to a combinatorial cube. it remains to determine the restrictions on the generating rays that will make the corresponding polytope smooth. without loss of generality (using theorem 2.6), we can assume that ui = ei for each i. let the columns of the following matrix represent the coordinates of the generating rays, where the last three columns give v1, v2, and v3, respectively.  1 0 0 b1 c1 d10 1 0 b2 c2 d2 0 0 1 b3 c3 d3   to be smooth, the minors corresponding to maximal cones must equal ±1. to make the fan complete (i.e., so that it spans all of r3), adjacent maximal cones must have opposite determinants. any minor comprising an odd number of ui’s must equal +1, and any minor with an even number of ui’s must equal −1. to denote a minor, we enclose the corresponding columns in vertical bars. for example, |u1,u2,u3| = 1 gives the determinant of the first three columns. the minor |u1,u2,v3| = −1 gives us d3 = −1. similarly, we need b1 = c2 = −1. the minor |u1,v2,v3| = 1 gives us c3d2 = 0. without loss of generality (using theorem 2.6 if needed), assume d2 = 0. similarly, we can assume c1 = d1 = 0 using |u3,v1,v2| = 1 and |u2,v1,v3| = 1. we get the following matrix for the generating rays.  1 0 0 −1 0 00 1 0 b2 −1 0 0 0 1 b3 c3 −1   using theorem 2.6, we can assume c3 ≤ 0. (if c3 > 0, then we reflect across a plane normal to e3, switch u3 and v3, and then translate as needed.) similarly, we can assume b2 ≤ 0. by choosing facets normal to the generating rays, we can now describe all smooth combinatorial cubes up to lattice equivalence. 18 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 proposition 5.9. let c be a smooth combinatorial cube with integer vertices. then c is lattice equivalent to a 3–polytope whose facets correspond to the following inequalities, where a1,a2,a3 ∈ n, b2,b3,c3 ∈ z, and b2,c3 ≤ 0. x1,x2,x3 ≥ 0 x1 − b2x2 − b3x3 ≤ a1 x2 − c3x3 ≤ a2 x3 ≤ a3 the way in which we calculate the distance of the toric code corresponding to c again depends on the values of b2, b3, and c3. there are numerous cases to consider, but there is significant overlap in the results and in their proofs. theorem 5.10. let c be a smooth combinatorial cube as described in proposition 5.9 (see figure 12). 1. if b3 < 0 and b2 < 0, then d (c) = (q − 1) 2 (q − 1 −a1). 2. if b3 < 0 and c3 = b2 = 0, then d (c) = (q − 1) (q − 1 −a1) (q − 1 −a2). 3. if b3 < 0 and c3 < b2 = 0, then d (c) = (q − 1) (q − 1 −a1) (q − 1 −a2). 4. if b3 = 0 and b2 < 0, then d (c) = (q − 1) (q − 1 −a1) (q − 1 −a3). 5. if b3 = 0 and c3 = b2 = 0, then d (c) = (q − 1 −a1) (q − 1 −a2) (q − 1 −a3). 6. if b3 = 0 and c3 < b2 = 0, then d (c) = (q − 1) (q − 1 −a1) (q − 1 −a2). 7. if b3 > 0 and b2 < 0, then d (c) = (q − 1) 2 (q − 1 − (a1 + b3a3)). 8. if b3 > 0 and c3 = b2 = 0, then d (c) = (q − 1) (q − 1 −a2) (q − 1 − (a1 + b3a3)). 9. if b3 > 0 and c3 < b2 = 0, then d (c) = (q − 1) · min{(q − 1 −a1) (q − 1 −a2) , (q − 1 − (a1 + b3a3)) (q − 1 − (a2 + c3a3))} . proof. (5) in this case, c is a product of three intervals, and the result follows from theorem 2.3. (2,6,8) in each of these cases, c is a product of a trapezoid and an interval. the results follow from theorem 2.3 and example 3.7. (1) here, we can apply proposition 3.8, noting that [0,a1] ⊆ c ⊆ ∆3a1. (3,4) in each of these cases, c is contained in the product of a trapezoid and an interval. the results follow immediately from corollary 3.6. (7) first, apply the transformation   1 0 00 1 0 0 0 −1   ·   x1x2 x3   +   00 a3   to c. it is straightforward to verify that the transformed polytope contains [0,a1 + b3a3] and is contained in ∆3a1+b3a3. the result follows from proposition 3.8. 19 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 figure 12. smooth combinatorial cubes. the remaining case b2 = b3 = c3 = 0 is a rectangular prism and is not shown. 20 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 (9) the polytope c is terraced in the x1–direction. for 0 ≤ i ≤ a1, the level ci is a trapezoid with d (ci) = (q − 1) (q − 1 −a2). for a1 ≤ i < a1 + b3a3, the polygon ci is still a trapezoid, but it no longer necessarily has integer vertices. the vertices of ci are now ( i, 0, i−a1 b3 ) ,( i,a2 + c3(i−a1) b3 , i−a1 b3 ) , (i,a2 + c3a3,a3), and (i, 0,a3). then ci is contained in a 2–simplex with the same (possibly non-integer) base length. by proposition 3.9 and 3.6, we get d (ci) = (q − 1) ( q − 1 − ⌊ a2 + c3 b3 (i−a1) ⌋) for a1 ≤ i < a1 + b3a3. for i = a1 + b3a3, the polygon ca1+b3a3 is an interval and d (ca1+b3a3 ) = (q − 1) (q − 1 − (a2 + c3a3)) in r2. this agrees with the distance formula for a1 ≤ i < a1 + b3a3. thus we have d (ci) = { (q − 1) (q − 1 −a2) if 0 ≤ i ≤ a1 (q − 1) ( q − 1 − ⌊ a2 + c3 b3 (i−a1) ⌋) if a1 ≤ i ≤ a1 + b3a3 . since d (ci) does not depend on i for 0 ≤ i ≤ a1, we get min 0≤i≤a1 (q − 1 − i) d (ci) = (q − 1 −a1) d (ca1 ) . then d (c) = min a1≤i≤a1+b3a3 (q − 1 − i) d (ci) = (q − 1) min a1≤i≤a1+b3a3 (q − 1 − i) ( q − 1 − ⌊ a2 + c3 b3 (i−a1) ⌋) . for all i, we have q − 1 − ( a2 + c3 b3 (i−a1) ) ≤ q − 1 − ⌊ a2 + c3 b3 (i−a1) ⌋ . since the expression (q − 1 − i) ( q − 1 − ( a2 + c3 b3 (i−a1) )) is quadratic in i and c3 b3 < 0, it is minimized at one of the endpoints i = a1 or i = a1 +b3a3 of the given interval. for these values of i, the floor function is not needed since the result is already an integer. since one of these values gives the minimum of all the (q − 1 − i) ( q − 1 − ( a2 + c3 b3 (i−a) )) , and all of these are less than or equal to the corresponding expressions with the floor function, d (c) itself must be minimized at either i = a1 or i = a1 + b3a3. the result follows from theorem 3.3. 6. questions it would be interesting to complete the classification of minimum distance for all toric codes coming from smooth n–polytopes with n + 3 facets. problem 6.1. are all smooth n–polytopes with n + 3 facets terraced? if so, what happens as dimension increases? are there few enough cases to allow us to explicitly describe the distances of the corresponding toric codes, or does the number of cases continue to grow as dimension increases? in most of our proofs, we used very little of the smooth structure of the polytopes. for example, corollary 3.6 appeared frequently in our distance computations. in these cases, the same techniques and results would apply to many other similar polytopes, as long as the base level and containment requirements persisted. this suggests that the method of terraced polytopes could apply far beyond smooth polytopes. 21 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 problem 6.2. what other classes of polytopes are terraced? can we use theorem 3.3 to find the minimum distance of the corresponding toric codes? in particular, can we compute distances for sequences of terraced polytopes with increasing dimension? then we could study asymptotic parameters for the corresponding sequences of toric codes. definition 6.3. (see [6], for example) the information rate of a linear code with dimension k and codeword length n is k n . the relative minimum distance of a linear code with minimum distance d and codeword length n is d n . ideally, we want a sequence of codes that balances information rate and relative distance, as this would give sufficient error-correcting capabilities while still providing a large amount of codewords. consider a sequence of toric codes whose corresponding polytopes are smooth n–polytopes with n+ 2 facets. by theorem 4.2, the relative minimum distance of a toric code in this sequence will be d (p) (q − 1)n ∈ { q − 1 − b2 q − 1 , (q − 1 − b1) (q − 1 − b2) (q − 1)2 } and d(p) (q−1)n tends toward positive values as n →∞. but since each such polytope p lies within a simplex or a product of simplices, the information rate tends toward zero as the polytope dimension increases (cf. [19, section 4]). perhaps terraced polytopes provide enough diversity to allow us to find a sequence of codes for which both the information rate and the relative distance tend toward positive values. problem 6.4. does there exist a sequence of terraced polytopes for which the corresponding toric codes have relative distances and information rates that both tend toward positive numbers as codeword length increases? references [1] v. v. batyrev, on the classification of smooth projective toric varieties, tohoku math. j. (2) 43(4) (1991) 569–585. [2] g. brown, a. m. kasprzyk, seven new champion linear codes, lms j. comput. math. 16 (2013) 109–117. [3] d. a. cox, j. b. little, h. k. schenck, toric varieties, ams grad. stud. math. 124 (2011). [4] j. p. hansen, toric surfaces and error-correcting codes, in : coding theory, cryptography and related areas, proceedings of an international conference, springer, berlin, heidelberg (2000) 132–142. 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[13] j. b. little, algebraic geometry codes from higher dimensional varieties, in: advances in algebraic geometry codes, world scientific, series on coding theory and cryptology (2008) 257–293. 22 https://doi.org/10.2748/tmj/1178227429 https://doi.org/10.2748/tmj/1178227429 https://doi.org/10.1112/s1461157013000041 https://doi.org/10.1112/s1461157013000041 http://dx.doi.org/10.1090/gsm/124 https://doi.org/10.1007/978-3-642-57189-3_12 https://doi.org/10.1007/978-3-642-57189-3_12 https://doi.org/10.1007/s00200-002-0106-0 https://doi.org/10.1007/s00200-002-0106-0 https://doi.org/10.1017/cbo9780511807077 https://dx.doi.org/10.4310/cag.2020.v28.n2.a3 https://dx.doi.org/10.4310/cag.2020.v28.n2.a3 https://doi.org/10.1007/s00200-004-0152-x https://hdl.handle.net/1969.1/etd-tamu-2992 https://hdl.handle.net/1969.1/etd-tamu-2992 https://doi.org/10.1007/bf01830946 https://doi.org/10.1137/050637054 https://doi.org/10.1137/050637054 https://doi.org/10.1007/s00200-007-0041-1 https://doi.org/10.1007/s00200-007-0041-1 https://doi.org/10.1142/9789812794017_0007 https://doi.org/10.1142/9789812794017_0007 ar ti cl e in pr es s a. wilfong / j. algebra comb. discrete appl. -(-) (2023) 1–23 [14] x. luo, s. s.-t. yau, m. zhang, h. zuo, on classification of toric surface codes of low dimension, finite fields appl. 33 (2015) 90–102. [15] k. meyer, i. soprunov, j. soprunova, fq-zeros of sparse trivariate polynomials and toric 3-fold codes, arxiv:2105.10071. [16] d. ruano, on the parameters of r-dimensional toric codes, finite fields appl. 13(4) (2007) 962–976. [17] i. soprunov, lattice polytopes in coding theory, j. algebra comb. discrete struct. appl. 2(2) (2015) 85–94. [18] i. soprunov, j. soprunova, toric surface codes and minkowski length of polygons, siam j. discrete math. 23(1) (2009) 384–400. [19] i. soprunov, j. soprunova, bringing toric codes to the next dimension, siam j. discrete math. 24(2) (2010) 655–665. [20] j. l. walker, codes and curves, ams stud. math. libr. 7 (2000). [21] g. m. ziegler, lectures on polytopes, springer-verlag, grad. texts math. 152 (1995). 23 https://doi.org/10.1016/j.ffa.2014.11.007 https://doi.org/10.1016/j.ffa.2014.11.007 https://arxiv.org/abs/2105.10071 https://arxiv.org/abs/2105.10071 https://doi.org/10.1016/j.ffa.2007.02.002 https://jacodesmath.com/index.php/jacodesmath/article/view/14 https://jacodesmath.com/index.php/jacodesmath/article/view/14 https://doi.org/10.1137/080716554 https://doi.org/10.1137/080716554 https://doi.org/10.1137/090762592 https://doi.org/10.1137/090762592 http://dx.doi.org/10.1090/stml/007 https://doi.org/10.1007/978-1-4613-8431-1 introduction toric codes minimum distance for terraced polytopes distances for smooth n–polytopes with n+2 facets distances for smooth n–polytopes with n+3 facets questions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.52790 j. algebra comb. discrete appl. 3(3) • 135–144 received: 23 october 2015 accepted: 17 february 2016 journal of algebra combinatorics discrete structures and applications the 3-gdds of type g3u2 research article charles j. colbourn, melissa s. keranen, donald l. kreher abstract: a 3-gdd of type g3u2 exists if and only if g and u have the same parity, 3 divides u and u ≤ 3g. such a 3-gdd of type g3u2 is equivalent to an edge decomposition of kg,g,g,u,u into triangles. 2010 msc: 05b05, 05b07, 05c70 keywords: group divisible designs, partial triple systems, graph decomposition 1. introduction a group divisible design (gdd) is a decomposition of the complete multipartite graph into complete subgraphs. the complete subgraphs used are the blocks of the gdd and are presented by giving the subset of the vertices they span. the partite sets are groups. formally a k -gdd is a triple (v, b, g ) where 1. v is a finite set of points; 2. b is a collection of subsets of v , where |b| ∈ k , for all b ∈ b; 3. g is a partition of v into groups. 4. every pair of points is in exactly one block or group. the type of a gdd is the multiset of its group sizes. thus a decomposition of kg0,g1,g2,...,gt−1 into complete subgraphs is a gdd of type {g0,g1,g2, . . . ,gt−1}. if the gdd has ti groups of size gi it is our custom to specify the type with the notation: gt00 g t1 1 g t2 2 · · ·g t` ` . also if k = {k} we write k-gdd instead of {k}-gdd. the blocks of a 3-gdd are usually called triples or triangles. for example a 3-gdd of type 4321 is a decomposition of k4,4,4,2 into triangles. charles j. colbourn; school of cidse, arizona state university, tempe, arizona 85287-8809, u.s.a. (email: charles.colbourn@asu.edu). melissa s. keranen, donald l. kreher (corresponding author); department of mathematical sciences, michigan technological university, houghton, michigan 49931-1295, u.s.a. (email: msjukuri@mtu.edu, kreher@mtu.edu). 135 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 theorem 1.1. for a 3-gdd of type g1g2 · · ·gs with g1 ≥ ··· ≥ gs ≥ 1, s ≥ 2, and v = ∑s i=1 gi to exist, necessary conditions include (colbourn [2]): 1. ( v 2 ) ≡ ∑s i=1 ( gi 2 ) (mod 3); 2. gi ≡ v (mod 2) for 1 ≤ i ≤ s; 3. g1 ≤ ∑s i=3 gi; 4. whenever αi ∈{0,1} for 1 ≤ i ≤ s and v0 = ∑s i=1 αigi, v0(v −v0) ≤ 2 [( v0 2 ) + ( v −v0 2 ) − s∑ i=1 ( gi 2 )] 5. 2g2g3 ≥ g1[g2 + g3 − ∑s i=4 gi]; and 6. if g1 = ∑s i=3 gi then 2g3g4 ≥ (g1 −g2)[g3 + g4 − ∑s i=5 gi]. these conditions are known to be sufficient when 1. (wilson [8, 9]) g1 = · · · = gs; 2. (colbourn, hoffman, and rees [5]) g1 = · · · = gs−1 or g2 = · · · = gs; 3. (colbourn, cusack, and kreher [3]) 1 ≤ t ≤ s, g1 = · · · = gt, and gt+1 = · · · = gs = 1; 4. (bryant and horsley [1]) g3 = · · · = gs = 1; and 5. (colbourn [2]) ∑s i=1 gi ≤ 60. surprisingly, in no other cases are necessary and sufficient conditions known for any other class of 3-gdds (of index 1). partial results are known when g3 = · · · = gs = 2 [6]. theorem 1.1 establishes that no 3-gdd with two groups exists; every 3-gdd with three groups has g1 = g2 = g3; and every 3-gdd with four groups has type g4 or g3u1; moreover, the first and second sufficient conditions ensure that all such 3-gdds exist. turning to five groups, the situation is much less satisfactory. while theorem 1.1 handles all types g5, g4u1, and g1 · · ·g5 with ∑5 i=1 gi ≤ 60, many more cases are possible. indeed it may happen that a 3-gdd with five groups has all groups of different sizes; for example, a 3-gdd of type 171111917151 exists [2]. hence the general existence problem for five groups appears to be substantially more complicated than cases with fewer groups. we address one part of this problem, when there are only two group sizes. the focus of this article is to prove theorem 1.2 (main theorem). a 3-gdd of type g3u2 exists if and only if g ≡ u (mod 2), u ≡ 0 (mod 3), and u ≤ 3g. if a 3-gdd of type g3u2 exist, then v = 3g + 2u ≡ 2u (mod 3) and v ≡ g (mod 3). thus it follows from theorem 1.1 conditions (1) and (2) that g ≡ u (mod 2) and u ≡ 0 (mod 3). condition 3 of theorem 1.1 is exactly the necessary conditions for the existence of a 3-gdd of type g3u2 are established by theorem 1.1. sufficiency is proved in the sections that follow. 2. 3-gdds of type g3u2 let γij` be the number of triples that contain points of groups gi, gj, and g`. elementary counting establishes that when |g1| = |g2| = |g3| = g and |g4| = |g5| = u, we have γ123 = g2 − 13(u(3g − u)), γ124 = γ125 = γ134 = γ135 = γ234 = γ235 = 1 6 (u(3g − u)), and γ145 = γ245 = γ345 = 13u 2. an easy case arises when u = 3g: 136 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 lemma 2.1. there exists a 3-gdd of type g3(3g)2, for all g. proof. a 3-gdd of type (3g)3 exists. partition one of the groups into three groups of size g on these groups place the triples of a 3-gdd of type g3. a one-factor on a set s is a set of |s|/2 vertex-disjoint edges. a holey one-factor on a set s with hole h is a set of (|s|− |h|)/2 vertex-disjoint edges in which no edge is incident to a vertex in h. we use the following result. lemma 2.2. (rees [7]) let h ≥ 1 and 0 ≤ r ≤ 2h, (h,r) 6∈ {(1,2),(3,6)}. there exists a {2,3}-gdd of type (2h)3 which is resolvable into r parallel classes of blocks of size 3 and 4h− 2r parallel classes of blocks of size 2. consequently whenever 0 ≤ x ≤ r, the edges of k2h,2h,2h can be partitioned into 4h−2r one-factors, 3x holey one-factors (x for each group), and r −x parallel classes of triples. theorem 2.3. if there exists a 3-gdd of type x3u2 with g ≡ x (mod 2) and g ≥ 2x+u, then there exists a 3-gdd of type g3u2. proof. write h = g−x 2 . without loss of generality, u 6= 0 so u ≥ 3. because g ≥ 2x + u, then h ≥ x+u 2 ≥ 2. when h = 3, we have (g,x,u) ∈ {(7,1,3),(9,3,3)}, and the required gdds are from theorem 1.1. henceforth h 6∈ {1,3}. choose groups {gi : 1 ≤ i ≤ 5} with |g1| = |g2| = |g3| = g and |g4| = |g5| = u. for i ∈ {1,2,3} partition gi into parts gi,1 and gi,2 where |gi,1| = x and |gi,2| = g − x = 2h. place a 3-gdd of type x3u2 aligning the groups on g1,1, g2,1, g3,1, g4, and g5. now r = 2h−u = g −x−u ≥ 2x + u−x−u = x. so use lemma 2.2 with groups g1,2, g2,2, g3,2 to construct a partition of k2h,2h,2h into 4h− 2r one-factors {fy : y ∈ g4 ∪g5}; for i ∈ {1,2,3}, x holey one-factors {hi,x : x ∈ gi,1} missing gi,2; and r − x parallel classes of triples. include all (2h)(r − x) triples in the r − x parallel classes. then for each y ∈ g4 ∪ g5, adjoin y to each edge in fy, forming 2u(3(2h)) triples. finally, for i ∈{1,2,3} and x ∈ gi,1, adjoin point x to each edge in hi,x to form 6xh additional triples. corollary 2.4. there exists a 3-gdd of type g3u2, whenever g ≥ 5 3 u, u ≡ 0 (mod 3), and g ≡ u (mod 2). proof. apply theorem 2.3 with x = u/3. in the remainder, the expression give weight w to the point x means to replace x with a set of w new points x1,x2, . . . ,xw; and if s = {s1,s2, . . . ,sk} is a set of points given weights (w(si) : 1 ≤ i ≤ k), then to place a 3-gdd of type {w(s1),w(s2), . . . ,w(sk)} on s means to include all triples in a 3-gdd of type {w(s1),w(s2), . . . ,w(sk)} with groups {{si,1,si,2,si,3, . . . ,si,w(si)} : 1 ≤ i ≤ k}. a synonymous expression is to fill the inflated block with a 3-gdd of type {w(s1),w(s2), . . . ,w(sk)}. this is illustrated in the following. lemma 2.5. if a 3-gdd of type (g/w)3(u/w)2 exists, then a 3-gdd of type g3u2 also exists. proof. starting with a 3-gdd of type (g/w)3(u/w)2, give weight w to the points using a 3-gdd of type w3, which always exists. recall that a 5-gdd of type k5 is equivalent to 3 mutually orthogonal latin squares of order k, which are known to exist when k 6∈ {2,3,6,10} [4]. (when k = 10 existence remains uncertain, but they do not exist for k ∈{2,3,6}.) lemma 2.6. if there exists a 5-gdd of type k5, and integers g,u with g ≡ u ≡ k (mod 2), 3k ≤ g,u ≤ 9k, and u ≡ 0 (mod 3), then there exists a 3-gdd of type g3u2. 137 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 proof. form a 5-gdd of type k5 with groups g1,g2,g3,g4,g5. let the points of gi be {xi1, . . . ,xik}, so that {x1k,x2k,x3k,x4k,x5k} is a block. write g = 3a + 9(k−1−a) + b with 0 ≤ a ≤ k−1 and b ∈{3,5,7,9}. write u = 3c + 9(k−c) with 0 ≤ c ≤ k. set w(xij) =   b if 1 ≤ i ≤ 3 and j = k, 3 if 1 ≤ i ≤ 3 and 1 ≤ j ≤ a, 3 if 4 ≤ i ≤ 5 and 1 ≤ j ≤ c, 9 if 1 ≤ i ≤ 3 and a + 1 ≤ j < k, 9 if 4 ≤ i ≤ 5 and c + 1 ≤ j ≤ k. according to [2], there exist 3-gdds of types 35, 5134, 7134, 9134, 915133, 5332, 917133, 7332, 9233, 925132, 927132, 9332, 9253, 935131, 937131, 9273, 9431, 9451, 9471, and 95. each block of the 5-gdd of type k5 has weights forming one of these types, so we place a 3-gdd on the points arising from each. theorem 2.7. suppose u ≡ 0 (mod 3), u ≡ g (mod 2) and 3 ≤ u ≤ 3g. then a 3-gdd of type g3u2 exists, except possibly when 3g + 2u > 60 and g ∈{9,10,11,13,18,20,22,30,32,34} and 3 5 g < u < 3g; or (1) g ≡ 1 (mod 3) and u = 3g −6; or (2) u ∈{18,30} and u < g < 5 3 u. (3) proof. using lemma 2.1 and corollary 2.4, assume that 3 5 g < u < 3g. write u = 3` and g = 3m + r, where r ∈ {0,1,2}; then ` ≡ m (mod 2) if and only if r ∈ {0,2}. to handle cases with u ≥ g and g 6∈ s = {2,4,6,8,9,10,11,13,18,20,22,30,32,34}, apply lemma 2.6 with k = m when r ∈ {0,2} and k = m− 1 when r = 1. when applied with k = m− 1, u ≤ 9m− 9, leading to the possible exceptions in (2). when g ∈ {2,4,6,8} and u < 3g, all required gdds are from [2]. thus when u ≥ g, the possible exceptions are listed in (1) and (2). to handle cases when u ≤ g and u 6∈ t = {6,9,18,30}, apply lemma 2.6 with k = `. when u ∈{6,9} and u ≤ g ≤ 5 3 u, all required gdds are from [2]. thus when u ≤ g, the possible exceptions are listed in (3). lemma 2.8. there exists a 3-gdd of type 133152. proof. begin with a 5-gdd of type 55. fix a block b and give weights 1,1,1,3,3 to it. on the remaining points give weight 3. fill the inflated blocks with 3-gdds of type 1331, 1134, 35 from [2]. theorem 2.9. a 3-gdd of type g3u2 exists if and only if g ≡ u (mod 2) and u ≡ 0 (mod 3) except possibly when g ≡ 1 (mod 3), g ≥ 16, and u = 3g −6; or g3u2 ∈   93212, 103242, 113152, 113212, 113272, 133212, 133272, 133332, 183422, 183482, 203422, 203482, 203542, 223422, 223482, 223542, 223602, 303842, 323782, 323902, 343842, 343962.   proof. apply lemma 2.1, lemma 2.5, theorem 2.3, and theorem 2.7. then apply lemma 2.6 with k = 4 to handle types 183u1 for u ∈ {12,24} and 223u1 for u ∈ {24,30,36}; and with k = 8 to handle 303u1 for u ∈ {24,48,72}, 323u1 for u ∈ {30,42,54,66}, and 343u1 for u ∈ {24,36,48,60,72}. apply lemma 2.8 to handle 133152. 138 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 3. incomplete group divisible designs let k be a set of positive integers, each at least 2. an incomplete group divisible design (k-igdd) of type (g1 : h1)u1 · · ·(gs : hs)us is a quadruple (v, b, g ,h) where 1. v is a set of ∑s i=1 uigi elements; 2. h ⊂ v , the hole, contains ∑s i=1 uihi elements; 3. g = {g1, . . . ,gm} is a partition of v into m = ∑s i=1 ui groups g1, . . .gm so that ui of the groups have size gi and contain hi points of h, for 1 ≤ i ≤ s; 4. b is a set of blocks with |b| ∈ k whenever b ∈ b, so that every pair of elements that are in the hole or in a group do not appear in a block, and every other pair occurs in exactly one block. when k = {k}, we write k-igdd. lemma 3.1. suppose that k is a set of odd positive integers. if a k-igdd of type (g1 : h1)u1 · · ·(gs : hs)us exists and w ≥ 2, then a 3-igdd of type (wg1 : wh1)u1 · · ·(wgs : whs)us exists. proof. give weight w to each point and fill with a 3-gdd of type wk for k ∈ k. corollary 3.2. a 3-igdd of type (12 : 3)1(6 : 3)4 exists. proof. a {3,5}-igdd of type (4 : 1)1(2 : 1)4 exists with groups {{di,xi} : 0 ≤ i ≤ 3}∪{y,z1,z2,z3}, hole {d0,d1,d2,d3,y}, and blocks {{di,x(i+j) mod 4,zj} : 0 ≤ i ≤ 3,1 ≤ j ≤ 3} and {x0,x1,x2,x3,y}. apply lemma 3.1 with w = 3. lemma 3.3. if a 3-igdd of type (3g : 3h)3 and a 3-igdd of type (g : h)3 exist, then a 3-igdd of type (3g : 3h)2(g : h)3 exists. proof. fill one group of the 3-igdd of type (3g : 3h)3 with the 3-igdd of type (g : h)3. corollary 3.4. when 1 ≤ h ≤ 1 2 g, a 3-igdd of type (3g : 3h)2(g : h)3 exists. in particular, a 3-igdd of type (6 : 3)2(2 : 1)3 and a 3-igdd of type (12 : 3)2(4 : 1)3 exist. proof. a 3-igdd of type (g : h)3 is equivalent to a latin square of side g with a subsquare of side h, which exist whenever 1 ≤ h ≤ 1 2 g, [4]. lemma 3.5. a 3-igdd of type (4 : 1)i(2 : 1)5−i exists when i ∈ {0,2}. hence a 3-igdd of type (6 : 3)5 and a 3-igdd of type (12 : 3)2(6 : 3)3 exist. proof. when i = 0, form blocks {{ai, i + 1, i + 4},{ai, i + 2, i + 3} : i ∈ z5} with groups {ai, i} and hole {ai : i ∈ z5}. when i = 2, a solution follows: blocks: {5,7,10}, {5,6,11}, {4,9,10}, {4,8,12}, {4,7,13}, {3,8,10}, {3,6,12}, {3,4,11}, {2,9,12}, {2,7,11}, {2,6,13}, {2,5,8}, {1,8,11}, {1,7,12}, {1,4,6}, {1,2,10}, {0,9,11}, {0,8,13}, {0,6,10}, {0,5,12}, {0,3,7}, {0,2,4}. groups: {0,1}, {2,3}, {4,5}, {6,7,8,9}, {10,11,12,13}. hole: {1,3,5,9,13}. use lemma 3.1 with weight 3 to obtain the specific igdds. 139 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 lemma 3.6. there exist 3-igdds of type (4 : 1)3(6 : 3)i(12 : 3)2−i for i ∈{0,1,2}. proof. when i = 0, apply corollary 3.4. when i = 2, start with points {xj : x ∈ z5, j ∈ z3}; elements with the same x-coordinate are in the same group of the igdd. place orbits of triples {00,10,20}, {00,30,40}, {10,30,41}, and {20,30,42}, developing the subscript modulo 3. then the remaining pairs {xi,yj} with x 6= y can be partitioned into a holey 1-factor missing {x0,x1,x2} for x ∈{0,1,2} and three holey 1-factors missing {x0,x1,x2} for x ∈{3,4}. extending these 9 holey 1-factors gives the 9 points in the hole of the igdd. to construct a 3-igdd of type (4 : 1)3(6 : 3)1(12 : 3)1 first form seven sets of size 3: {ai = {aij : j ∈ z3} : i ∈ z3}, b = {bj : j ∈ z3}, and {ci = {cij : j ∈ z3} : i ∈ z3}. let h = {αi,βi,γi : i ∈ z3} be 9 additional points. we construct the 3-igdd with groups:( a0 ∪{α0} ) , ( a1 ∪{α1} ) , ( a2 ∪{α2} ) , ( b ∪{β0,β1,β2} ) , ( c0 ∪c1 ∪c2 ∪{γ0,γ1,γ2} ) and hole h. now form 1. the triples of a 3-gdd of type 33 on groups {β0,β1,β2}, ai, and ci for i ∈ z3, 2. {{γi,bj,aij},{γi,a i+1 j ,a i+2 j } : i,j ∈ z3}, 3. {{αi,ai+1j ,c i+2 j },{αi,a i+2 j ,c i+1 j },{αi,bj,c i j} : i,j ∈ z3}, 4. {{bj,aij+1,c i+1 j+2},{bj,a i+1 j+2,c i j+1} : i,j ∈ z3}, 5. {{aij,c i+1 j+2,a i+2 j+1} : i,j ∈ z3}, 6. {{a0j,a 1 j+1,a 2 j+2} : j ∈ z3}. it is an easy but tedious exercise to verify that these triples provide the desired igdd. lemma 3.7. a 3-igdd of type (5 : 1)3(9 : 3)2 exists. proof. form a set x = z3 × z4 × z2 of points. let gi = {i} × {0,1} × z2 for i ∈ z3, and let gj+1 = z3×{j}×z2 for j ∈{2,3}. on x with groups {gi : 0 ≤ i ≤ 4} we construct a partition of pairs not in a group into one holey parallel class of ten pairs missing gi for each i ∈ {0,1,2}; three parallel clases of nine pairs missing gi for each i ∈ {3,4}; and 48 triples. once constructed, extending holey parallel classes produces the desired igdd. first we make the triples. form a 3-gdd of type 43 on z3 ×z4 having a parallel class on {z3 ×{j} : j ∈ z4} and groups on {{i}×z4 : i ∈ z3}. this has 12 triples; give weight 2 to form 48 triples. for i ∈ z3 let fi =   {(i,2,0),(i,3,0)}, {(i,2,1),(i,3,1)}, {(i + 1,2,0),(i + 1,3,1)},{(i + 1,0,0),(i + 1,2,1)},{(i + 1,1,1),(i + 1,3,0)}, {(i + 2,2,1),(i + 2,3,0)},{(i + 2,0,1),(i + 2,2,0)},{(i + 2,1,0),(i + 2,3,1)}, {(i + 1,0,1),(i + 2,0,0)},{(i + 1,1,0),(i + 2,1,1)}.   then fi is a holey parallel class for gi for i ∈ z3. for i ∈ z3 and σ ∈ z2 let σ = 1−σ and let hiσ =   {(i + 1,σ,σ),(i + 2,σ,σ)},{(i,σ,σ),(i + 1,σ,σ)}, {(i,σ,σ),(i + 2,σ,σ)}, {(i,σ,σ),(i,2 + σ,σ)}, {(i + 1,σ,σ),(i + 1,2 + σ,σ)}, {(i + 2,σ,σ),(i + 2,2 + σ,σ)}, {(i,σ,σ),(i,2 + σ,σ)}, {(i + 1,σ,σ),(i + 1,2 + σ,σ)},{(i + 2,σ,σ),(i + 2,2 + σ,σ)}.   then {hiσ : i ∈ z3} contains three holey parallel classes for g3+σ for σ ∈ z2. 140 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 4. using incomplete group divisible designs theorem 4.1. let m,k be integers. if 5 ≤ m ≤ k ≤ 3m, m ≡ k (mod 2), and m 6∈ {6,10}, then there exist 3-gdds of type (3m + 1)3(3k + 3)2 and (3m + 3)3(3k + 3)2. proof. there exists a 5-gdd of type m5 that has a parallel class p of blocks (this is equivalent to three idempotent mols of side m, see [4]). let g1,g2,g3,g4,g5 be its groups. give weight 3 to all points in g1 ∪g2 ∪g3. in each of 3m−k2 of the blocks of p give weight 3 to the two points of the block in g4 or g5; for the remaining k−m2 of the blocks of p , give weight 9. add a set h of 15 or 9 points distributed 3,3,3,3,3 or 1,1,1,3,3 to the groups to obtain group types (3m + 3)3(3k + 3)2 and (3m + 1)3(3k + 3)2 respectively. fill blocks not in the parallel class p with a 3-gdd of type 9i35−i, i = 0,1 or 2 from [2]. fill blocks that are in the parallel class p with a 3-igdd of type (6 : 3)5 and a 3-igdd of type (6 : 3)3(12 : 3)2, or a 3-igdd of type (4 : 1)3(6 : 3)2 and a 3-igdd of type (4 : 1)3(12 : 3)2, all having hole h. fill h with a 3-gdd of type 35 or a 3-gdd of type 1332 respectively. corollary 4.2. there exist 3-gdds with g ≡ 1 (mod 3), g ≥ 16, g 6∈ {19,31}, and u = 3g−6; and when g3u2 ∈ { 183422,183482,223422,223482,223542,223602,303842,343842,343962 } . proof. apply the first statement of theorem 4.1 with (m,k) = (g−1 3 ,g − 3) when g ≡ 1 (mod 3), g ≥ 16, g 6∈ {19,31} and u = 3g − 6. apply the first statement with m = 7 and k ∈ {13,15,17,19} to treat the cases with g = 22; and with m = 11 and k ∈{27,31} to treat the cases with g = 34. apply the second statement with m = 5 and k ∈{13,15} to handle the cases with g = 18, and with (m,k) = (9,27) to handle 303842. it remains to treat g3u2 ∈ { 93212, 103242, 113152, 113212, 113272, 133212, 133272, 133332, 193512, 203422, 203482, 203542, 313872, 323782, 323902. } next we extend theorem 4.1: theorem 4.3. let m ≥ 5 be an integer with m 6∈ {6,10,14,18,22}. let k ≡ m (mod 2) be an integer, where m ≤ k ≤ 3m. let α be an integer with 1 ≤ α ≤ m, and let a be an integer for which α ≤ a ≤ 3α and a ≡ α (mod 2). then a 3-igdd of type ((3m + a : a)3(3k + 3α : 3α)2 exists. if in addition a 3-gdd of type a3(3α)2 exists, then a 3-gdd of type (3m + a)3(3k + 3α)2 exists. proof. there are 4 mols of order m [4] and hence there exists a 5-gdd of type m5 with α disjoint parallel classes {pi : 1 ≤ i ≤ α}. let g1,g2,g3,g4,g5 be its groups. give weight 3 to all the points in g1 ∪g2 ∪g3. in each of g4 and g5 give weight 3 to 3m−k2 of the points and weight 9 to the remaining k−m 2 points. now for 1 ≤ i ≤ α, let hi contain three new points in each of g4 and g5; and either three or one new points in each of g1,g2,g3 according to whether i ≤ a−α2 or not. fill blocks not in ⋃α i=1 pα using a 3-gdd of type 9 i35−i with i ∈{0,1,2}. for each parallel class pi, fill each block with an igdd of type (6 : 3)3(12 : 3)2, (6 : 3)4(12 : 3)1, or (6 : 3)5, that has hole hi; these are from corollary 3.2 and lemma 3.5. this produces the 3-igdd of type ((3m + a : a)3(3k + 3α : 3α)2. if a 3-gdd of type a3(3α)2 exists, use it to fill the hole. corollary 4.4. there exist 3-gdds of types 193512, 203u2 for u ∈ {42,48,54}, 313872, and 323u2 for u ∈{78,90}. proof. theorem 4.3 handles 193512 using (m,α,a) = (5,2,4) and k = 15; 203u2 for u ∈ {42,48,54} using (m,α,a) = (5,3,5)and k ∈{11,13,15}; 313872 using (m,α,a) = (9,2,4) and k = 27; and 323u2 for u ∈{78,90} using (m,α,a) = (9,3,5) and k ∈{23,27}. 141 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 it remains only to treat a few cases with g ≤ 13. a variant of theorem 4.3 uses a different weighting: theorem 4.5. let m ≥ 5 be an integer with m 6∈ {6,10,14,18,22}. let α be an integer with 1 ≤ α ≤ m, and let a be an integer for which α ≤ a ≤ 3α and a ≡ α (mod 2). then a 3-igdd of type ((m + α : α)3(3m + a : a)2 exists. if in addition a 3-gdd of type α3a2 exists, then a 3-gdd of type ((m + α)3(3m + a)2 exists. proof. let g1,g2,g3,g4,g5 be the groups of a 5-gdd of type m5 with α disjoint parallel classes {pi : 1 ≤ i ≤ α}. give weight 1 to all points in g1 ∪g2 ∪g3, and weight 3 to all points in g4 ∪g5. fill blocks not in ⋃α i=1 pα using a 3-gdd of type 1 332. for 1 ≤ i ≤ α, let hi contain three new points in each of g1,g2,g3; and either three or one new points in each of g4 and g5 according to whether i ≤ a−α2 or not. for 1 ≤ i ≤ a−α 2 , fill each block of pi with an 3-igdd of type (2 : 1)3(6 : 3)2 that has hole hi. for a−α2 < i ≤ α, fill each block of pi with an 3-igdd of type (2 : 1)3(4 : 1)2 that has hole hi. this produces the 3-igdd of type ((m + α : α)3(3k + a : a)2. corollary 4.6. there exist 3-gdds of types 93212, 103242, 113272, 133272, and 133332. proof. apply theorem 4.5 with (m,α,a) = (5,4,6) to handle 93212; (m,α,a) = (7,3,3) to handle 103242; (m,α,a) = (7,4,6) to handle 113272; (m,α,a) = (7,6,6) to handle 133272; and (m,α,a) = (9,4,6) to handle 133332. theorem 4.7. if there exist 3-gdds of types g3u2 and a3b2 and a 3-igdd of type (g+a : a)3(u+b : b)2, then for all w ≥ 3 there also exists a 3-gdd of type (wg + a)3(wu + b)2. proof. let {gi : 1 ≤ i ≤ 5} be groups of size g,g,g,u,u respectively and set g = ⋃5 i=1 gi. let {hi : 1 ≤ i ≤ 5} be groups of size a,a,a,b,b respectively and set h = ⋃5 i=1 hi. if x ∈ g , then we denote by x, the w-element set x = x×zw and set gi = ⋃ x∈gi x. we construct the 3-gdd of type (wg +a) 3(wu+b)2 on groups {(gi ∪hi) : 1 ≤ i ≤ 5}. if x,y,z ∈ g , let p(x,y,z) = { {x×{i},y ×{i},z ×{i}} : i ∈ zw } ; this is a parallel class of triples. because w ≥ 3 there is an idempotent latin square of side w; consequently a 3-gdd of type w3 can be constructed with groups x,y,z that contains the parallel class p(x,y,z). let d(x,y,z) be the triples in this gdd that are not in p(x,y,z). let a be a 3-igdd of type (g + a : a)3(u + b : b)2 on the groups (gi ∪hi), i = 1,2,3,4,5 and hole h . let b be a 3-gdd of type g3u2 on the groups gi, i = 1,2,3,4,5. to construct the 3-gdd of type (wg + a)3(wu + b)2 we take the triples in: {{h,x ×{i},y ×{i}} : i ∈ zw} whenever { h,x,y } is a triple in a intersecting the hole h in the point h; p(x,y,z) whenever{ x,y,z } is a triple in a disjoint from the hole h ; d(x,y,z) whenever { x,y,z } is a triple in b; and a 3-gdd of type a3b2 on the hole h . corollary 4.8. there exists a 3-gdd of type 133212. proof. there exist 3-gdds of types 4362 and 1332 and a 3-igdd of type (5 : 1)3(9 : 3)2 from lemma 3.7. apply theorem 4.7 with w = 3. theorem 4.9. if there exist 3-gdds of types g3u2 and a3b2 and a 3-igdd of type (g +a : a)3(u+b : b)2, then for all w ≥ 3 there also exists a 3-gdd of type (wg + (w −1)a)3(wu + (w −1)b)2. 142 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 ap bs cr ev fw hm io ku gg hl io jq kd ln mj nt pq rx bu co dv er fq ht js lm gm ho in ji kw lk ng px qp ra am bw cn dx gv hq it kr gu hm ie jf kn lo op pl qs rj au cm dq gn hr jw ko ls gv ht iq je kp lm nx of pb ri as bv cw eq ix jn gm hd il jo kf lt mk np or pg qh ru at ep hs iu kx lo ag bc cd db ew fj mq nf ov pm qn rr aw bx em fn gq ir kp as bh co du ed fp mt nl oc qj rv ao bm dw fr gp hx jv lu ae br cq dq ec ft mi nn os pk cv dt fm hu iw jr lq ap bp cx dn ea fo ms nk oe qb rg dn et fu gm jp kw lx aq bv cm di eb fs nh oa po qr rc ax bq do fv is km lw ag be cg dp er fc hn iu jt kh lj an bt cp ds ew fo gu iv jx kq al bm cl dg ei fk hr jh ar bo cu dm fs gt hv in lp ai bk cw dj ex fl gq he jk bp ct dr eu gs hn jo kv ax bw ci dh eq fa gi jm kl lf dp en fx go hw jq kt lv ai bh ca dc ej fu gb im kr ls ab bj ck dp el fd gk hi ia jm le nc oh qg rf ao bl cp di ei fh ga hg jb kk lc me nj qf rd an bf ce dr ek fg ha ib jg kc lh ml oj pi qd af bb cc dm eg fl gj hr ik jn ki la od ph qe ad bl cf dl ee fj gn hb ic ka mg ok pj qi rh al bo cj dh eg fi hf id ja kg mc nb pe qk rl am bg cl dk en fe gf hh ii jj ko ld pa qc rb ac ba cb df eh fk gd hj im jo kp lg ni ql re aj bi ck de eo fg gc hq ih jl kb lr ma nd pf av bq cm dn el fw gr hp ix jp kn mo ou qt rs ap bt co do em fr gq hw is jr kq lx mn nu pv ar bg cs dx ev fh ir jp kt lm mu nw oq pn qo au bx cr dk eo fp gw hs ir jv ln mm on pt qq at br cu ds en fi go ho jq km lv nm pw qx rp am bo cq dt eq fo gp hv in jn ks lu mx pr rw an bj ct do es fq gx hu iv ko lq mw nr pp rm ak bn cv dm em fr gp hx iq ju lp ns oo qw rt ar bn cn dq ep fx gt hm io js ku lp mv ow rq aho ahk ajw aqa bks bip bmd bqu cip cgh chn cjr dav dgr djd dlw eft eju ehp ekr fbn fmf fnv fqm ges gln gro hcq ifp igw ijt jcx kex kjm kqv liq llr lob mbr mhp naq neo ogx oim olt pcs pdu rkn figure 1. a partition of k6,6,6,12,12 proof. let {gi : 1 ≤ i ≤ 5} be groups of size g,g,g,u,u respectively and set g = ⋃5 i=1 gi. let {hi : 1 ≤ i ≤ 5} be groups of size a,a,a,b,b respectively and set h = ⋃5 i=1 hi. if x ∈ g , then we denote by x, the w-element set x = x× zw and set gi = ⋃ x∈gi x. if h ∈ h , then we denote by h, the (w − 1)-element set x = x × (zw \{0}) and set hi = ⋃ h∈hi h. we construct the 3-gdd of type (wg + (w − 1)a)3(wu + (w − 1)b)2 on groups {(gi ∪ hi) : 1 ≤ i ≤ 5}. p(x,y,z) and d(x,y,z) are as in the proof of theorem 4.7. let a be a 3-igdd of type (g + a : a)3(u + b : b)2 on the groups (gi ∪hi), i = 1,2,3,4,5 and hole h . let b be a 3-gdd of type g3u2 on the groups gi, i = 1,2,3,4,5. to construct the 3-gdd of type (wg+(w−1)a)3(wu+(w−1)b)2 we take the triples in: {{h×{j},x× {i},y ×{i + j mod w}} : i,j ∈ zw,j 6= 0} whenever { h,x,y } is a triple in a intersecting the hole h in 143 c. j. colbourn et al. / j. algebra comb. discrete appl. 3(3) (2016) 135–144 the point h; d(x,y,z) whenever { x,y,z } is a triple in a disjoint from the hole h ; p(x,y,z) whenever{ x,y,z } is a triple in b; and a 3-gdd of type ((w −1)a)3((w −1)b)2 on the hole ⋃5 i=1 hi. corollary 4.10. there exists a 3-gdd of type 113152. proof. there exist 3-gdds of types 3332 and 1332 and a 3-igdd of type (4 : 1)3(6 : 3)2. apply theorem 4.9 with w = 3. lemma 4.11. there exists a 3-gdd of type 113212. proof. figure 1 provides a partition of k6,6,6,12,12 that was found using a hill-climbing algorithm on points a-r and a-x with groups g1 = a-f, g2 = g-l, g3 = m-r, g4 = a-l, and g5 = m-x into five holey parallel classes of pairs for each of g1, g2, and g3; nine holey parallel classes of pairs for each of g4 and g5; and 48 triples. to form a 3-igdd of type (11 : 5)3(21 : 9)2, extend the holey parallel classes. then fill the hole with a 3-gdd of type 5392. this completes the proof of the main theorem. references [1] d. bryant, d. horsley, steiner triple systems with two disjoint subsystems, j. combin. des. 14(1) (2006) 14–24. [2] c. j. colbourn, small group divisible designs with block size three, j. combin. math. combin. comput. 14 (1993) 153–171. [3] c. j. colbourn, c. a. cusack, d. l. kreher, partial steiner triple systems with equal-sized holes, j. combin. theory ser. a 70(1) (1995) 56–65. [4] c. j. colbourn, j. h. dinitz (eds.), handbook of combinatorial designs, second edition, crc/chapman and hall, boca raton, fl, 2007. [5] c. j. colbourn, d. hoffman, r. rees, a new class of group divisible designs with block size three, j. combin. theory ser. a 59(1) (1992) 73–89. [6] c. j. colbourn, m. a. oravas, r. s. rees, steiner triple systems with disjoint or intersecting subsystems, j. combin. des. 8(1) (2000) 58–77. [7] r. rees, uniformly resolvable pairwise balanced designs with blocksizes two and three, j. combin. theory ser. a 45(2) (1987) 207-225. [8] r. m. wilson, an existence theory for pairwise balanced designs. i. composition theorems and morphisms, j. combinatorial theory ser. a 13 (1972) 220–245. [9] r. m. wilson, an existence theory for pairwise balanced designs. ii. the structure of pbd-closed sets and the existence conjectures, j. combinatorial theory ser. a 13 (1972) 246–273. 144 http://dx.doi.org/10.1002/jcd.20071 http://dx.doi.org/10.1002/jcd.20071 http://www.ams.org/mathscinet-getitem?mr=1238867 http://www.ams.org/mathscinet-getitem?mr=1238867 http://dx.doi.org/10.1016/0097-3165(95)90080-2 http://dx.doi.org/10.1016/0097-3165(95)90080-2 http://dx.doi.org/10.1016/0097-3165(92)90099-g http://dx.doi.org/10.1016/0097-3165(92)90099-g http://dx.doi.org/10.1002/(sici)1520-6610(2000)8:1<58::aid-jcd8>3.0.co;2-2 http://dx.doi.org/10.1002/(sici)1520-6610(2000)8:1<58::aid-jcd8>3.0.co;2-2 http://dx.doi.org/10.1016/0097-3165(87)90015-x http://dx.doi.org/10.1016/0097-3165(87)90015-x http://www.ams.org/mathscinet-getitem?mr=0304203 http://www.ams.org/mathscinet-getitem?mr=0304203 http://www.ams.org/mathscinet-getitem?mr=0304204 http://www.ams.org/mathscinet-getitem?mr=0304204 introduction 3-gdds of type g3u2 incomplete group divisible designs using incomplete group divisible designs references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.40139 j. algebra comb. discrete appl. 3(3) • 165–176 received: 28 october 2015 accepted: 29 november 2015 journal of algebra combinatorics discrete structures and applications some new large sets of geometric designs of type ls[3][2, 3, 28] research article michael r. hurley, bal k. khadka, spyros s. magliveras abstract: let v be an n-dimensional vector space over fq. by a geometric t-[qn,k,λ] design we mean a collection d of k-dimensional subspaces of v , called blocks, such that every t-dimensional subspace t of v appears in exactly λ blocks in d. a large set, ls[n][t,k,qn], of geometric designs, is a collection of n t-[qn,k,λ] designs which partitions the collection [ v k ] of all k-dimensional subspaces of v . prior to recent article [4] only large sets of geometric 1-designs were known to exist. however in [4] m. braun, a. kohnert, p. östergard, and a. wasserman constructed the world’s first large set of geometric 2-designs, namely an ls[3][2,3,28], invariant under a singer subgroup in gl8(2). in this work we construct an additional 9 distinct, large sets ls[3][2,3,28], with the help of lattice basis-reduction. 2010 msc: 05b25, 05b40, 05e18 keywords: geometric t-designs, large sets of geometric t-designs, t-designs over gf(q), parallelisms, lattice basis reduction, lll algorithm. 1. introduction in this article we deal with large sets of geometric t-designs. by a geometric t-design we mean what earlier authors have called t-designs over a finite field, or designs on vector spaces. geometric t-designs are the fq-analogs of ordinary t-(v,k,λ) designs. the earliest mention of t-[qn,k,λ] designs, although not using our terminology or notation, was by p.j. cameron in 1974 [5, 6] and p. delsarte in 1976 [7]. in 1987, s. thomas [20] exhibited the first simple geometric 2-design, and in the 1990’s h. suzuki [19], m. miyakawa et al. [17] , and t. itoh [10] constructed new geometric 2-designs and families of such designs. in 1994, d.k. ray-chaudhuri and e.j. schram [18] studied and constructed geometric t-designs from quadratic forms, allowing repeated blocks. for the first time, the latter authors also studied large sets of geometric t-designs. michael r. hurley, bal k. khadka, spyros s. magliveras (corresponding author); department of mathematical sciences, florida atlantic university, boca raton, florida 33472, usa (email: mhurley6@my.fau.edu, bkhadka@my.fau.edu, spyros@fau.edu). 165 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 m. braun, a. kerber and r. laue [3] constructed in 2005 the first simple geometric 3-design. in 2013, braun et al. [2] constructed the first example of a q-steiner system, that is a simple, geometric t-design with λ = 1, namely a 2-[213, 3, 1] design. in a short recent arxiv preprint [8], and based on a probabilistic existence theorem of g. kuperberg, s. lovett and r. peled in preprint [14], a. fazeli, s. lovett, and a. vardy, appear to have proved the remarkable theorem that simple geometric t-designs exist for all values of t. this would be a q-analog of the famous theorem of l. tierlinck for ordinary t-designs. it should be noted however, that the result in [8] is purely existential and there is no known efficient algorithm which can produce t-[qn,k,λ] designs for t > 3. the authors present the following challenge: problem 1.1. design an efficient algorithm to produce simple, non-trivial t-[qn,k,λ] designs for large t, (say t ≥ 4). of course, finding large sets of geometric t-designs is even harder than just finding geometric tdesigns. prior to recent article [4] only large sets of geometric 1-designs were known to exist. however in [4] m. braun, a. kohnert, p. östergard, and a. wasserman constructed the world’s first large set of geometric 2-designs, namely an ls[3][2,3,28], invariant under a singer subgroup in gl8(2). in this paper we construct 9 distinct large sets ls[3][2, 3, 28], all different from the large set constructed in [4]. the computation involved our apl package knuth for group theoretic matters, and various lll variants in the ntl library, augmented by certain optimization techniques for parallel lattice basis reduction. it should be noted that some of the recent work on geometric t-designs has been motivated by present day coding theoretic applications as discussed in [9] and [11]. 2. preliminaries let v be an n-dimensional vector space over the field fq. if u is a j-dimensional subspace of v , we say that u is a j-subspace of v . if x is a set and 0 ≤ s ≤ |x|, ( x s ) denotes the collection of all subsets of cardinality s of x. a geometric t-[qn,k,λ] design is a pair (v,b) where b is a multiset of k-subspaces of v , called blocks, such that any t-subspace t of v is contained in exactly λ blocks. (v,b) is said to be simple if b is a set, i.e. if there are no repeated blocks. in this paper we deal only with simple geometric designs, and the square brackets of the symbol t-[qn,k,λ] will imply “geometric” in contrast to the round parentheses for an ordinary t-(v,k,λ) design. we denote the collection of all k-subspaces of v by [ v k ] and note that | [ v k ] | = [ n k ] q , where [ n k ] q is the well known gaussian binomial coefficient, given by: [ n k ] q = [n]q! [k]q![n−k]q! (1) where for positive integer r, [r]q! := [1]q[2]q · · · [r]q, and [j]q := (1 + q + · · · + qj−1). (2) analogously to the case of ordinary t-(v,k,λ) designs, a geometric t-[qn,k,λ] design (v,b) is also an s-[qn,k,λi] design for every 0 ≤ s ≤ t with: λs = λ [ n−s t−s ] q / [ k −s t−s ] q , (3) 166 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 thus, a necessary condition for the existence of a t-[qn,k,λ] design is that the λs given by the equations (3) must be integral for all 0 ≤ s ≤ t. by a large set ls[n][t,k,qn] we mean a collection l = {(v,bi)}ni=1 of simple t-[q n,k,λ] designs where {bi}ni=1 is a partition of [ v k ] . we can immediately see that for a given large set ls[n][t,k,qn], n can be expressed in terms of the other parameters as : n = [ n− t k − t ] q /λ (4) two t-[qn,k,λ] designs d = (v,b) and d′ = (v,b′) are said to be isomorphic if there exists α ∈ gln(q) such that bα = b′, that is, bα ∈b′ for all b ∈b, in which case we also write dα = d′. if dα = d, then α is said to be an automorphism of d. the group of all automorphisms of d is denoted by aut(d). let b = {bi}ni=1 be the collection of designs in a large set l. a group g ≤ gln(q) is said to be an automorphism group of l if bg=b for all g ∈ g, that is, if bgi ∈ b for all bi ∈ b and g ∈ g. equivalently, we say that a large set with this property is g-invariant. the group of all automorphisms of l is denoted by aut(l). if the stronger condition holds, that bgi =bi for all bi ∈ b and g ∈ g, we say that the large set l is [g]-invariant. in 1976, e.s. kramer and d.m. mesner [12] presented a theorem which provides necessary and sufficient conditions for the existence of an ordinary g-invariant t-(v,k,λ) design. beginning with a given group action g|x, the authors define certain integer matrices, presently known as the kramer-mesner (km) matrices. roughly speaking such a matrix at,k is the result of fusing under g the incidence matrix between ( x t ) and ( x k ) where incidence is set inclusion (fused r.wilson matrix). these matrices extend naturally to the case of a group g ≤ gln(q) acting on agn(q) or pgn−1(q), and provide necessary and sufficient conditions for the existence of geometric, g-invariant t-[qn,k,λ] designs. we proceed to define these matrices in the context of geometric t-designs, and state the analog of the kramer-mesner theorem. let v be an n-dimensional vector space over fq, and g ≤ gln(q). suppose that t and k are integers, 0 ≤ t < k ≤ n, and consider the actions of g on [ v t ] and [ v k ] respectively, with corresponding g-orbit decompositions: [ v t ] = ∆1 + ∆2 + · · · + ∆ρ(t), (5) and [ v k ] = γ1 + γ2 + · · · + γρ(k). (6) where ρ(s) denotes the number of g-orbits on [ x s ] . just as in [12], it can be shown that for any fixed t-subspaces t,t ′ ∈ ∆i, we have that |{k ∈ γj : t ≤ k}| = |{k ∈ γj : t ′ ≤ k}|, (7) that is, the number at,k(i,j) = |{k ∈ γj : t ≤ k}| is independent of the choice of a fixed t ∈ ∆i. the kramer-mesner matrix at,k is then defined as the ρ(t) ×ρ(k) matrix : at,k = (at,k(i,j)) (8) dually, for k fixed in γj, let bt,k(i,j) := |{t ∈ ∆i : t ≤ k}|, and define the dual km matrix bt,k by: bt,k = (bt,k(i,j)) (9) in the following lemma we state without proof geometric analogs of some properties of the at,k and bt,k as included for the ordinary t-design context in [13]. lemma 2.1. let at,k and bt,k, ∆i, γj be as defined above. 167 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 (i) if t ≤ s ≤ k ≤ n, then [ k−t k−s ] q at,k = at,s ·as,k (ii) at,k has constant row sums [ v−t k−t ] q (iii) |∆i| ·at,k(i,j) = |γj| ·bt,k(i,j) keeping in mind that we are only interested in simple geometric t-designs, we now state, without proof, the kramer-mesner theorem for geometric t-designs : theorem 2.2. if g ≤ gln(q), there is a g-invariant (simple) t-[qn,k,λ] design if and only if there is a ρ(k) × 1 0-1 vector u which is solution of the matrix equation at,ku = λj (10) where j is the ρ(t) × 1 vector of all 1’s. here, the 1’s in a solution u select the g-orbits of [ v k ] whose union will constitute the design. the following corollary follows immediately: corollary 2.3. there is a [g]-invariant large set ls[n][t,k,qn] of geometric designs if and only if there exist n distinct solutions, u1, . . . ,un, to the matrix equation (10), whose sum is the ρ(k) × 1 all 1’s vector. 3. main result it is well known that gln(q) has a cyclic subgroup of order qn−1 , called a singer subgroup, acting regularly on the non-zero vectors of v = fnq . it is also known that all singer subgroups are conjugate in gln(q). a singer subgroup g of γ = gl8(2) is the centralizer of a sylow-17 subgroup of γ and its normalizer n in γ is a split extension of g by its frobenius group φ8, thus |n| = 2040. in particular, for the rest of the paper we adopt the notation v = f82, γ = gl8(2), and g = 〈α〉, where α is the same singer cycle as the one used in [4], that is : α =   0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0   . we will presently construct 9 distinct large sets of geometric 2 − [28, 3, 21] designs which are [g]invariant under singer subgroup g of γ. we have used the exact same singer subgroup g = 〈α〉 as in [4] so that it will be easy to check that our large sets are different from the one constructed in [4]. 3.1. computing and presenting a2,3 members of [ v 2 ] are klein 4-groups, and those of [ v 3 ] elementary abelian groups of order 8. viewed projectively, the 2and 3-spaces can be seen as collinear triples and fano planes respectively. there are in all 10795 2-spaces, and 97155 3-spaces. we begin by computing the g-orbits on [ v 2 ] and [ v 3 ] , where g = 〈α〉. there are exactly 43 g-orbits on [ v 2 ] , all of which have length 255, except for one which has length 85. the short orbit is explained by 168 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 the fact that the cyclic subgroup of order 3 in g fixes a collinear triple. there are 381 g-orbits on [ v 3 ] all of length 255. the vectors of v = f82 are represented by the radix-2 representation of integers in z256. orbits of 2and 3-spaces are represented by the lexically smallest basis among all members of the orbit, but since g is transitive on the non-zero vectors, each such basis will consist of the vector 1 ↔ 00000001, and one (or two) elements of z256 −{1}. hence, to represent 〈α〉-orbits of 2-spaces, it suffices to specify the second vector in the lexically minimal basis over all 2-spaces for that orbit. thus, the 〈α〉-orbits of 2-spaces are represented by the following 43 integers : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 34, 36, 38, 40, 42, 44, 50, 54, 56, 58 60, 62, 70, 74, 76, 78, 80, 86, 88, 96, 100, 106, 114, 128, 136, 146, 164, 210, 218. similarly, orbit representatives of the 381 orbits of 3-spaces are given by the pair of integers in z256 which together with 1, form the lexically minimal basis among the members of the orbit of 3-spaces. the pairs x,y ∈ z256 representing the g-orbits on 3-spaces will appear in our display of the km-matrix a2,3 below. to compute a2,3, we found it easier to first compute matrix b2,3 and then compute the a2,3(i,j) entries, using lemma 2.1, equation (iii) : a2,3(i,j) = |γj| |∆i| b2,3(i,j). almost all ratios |γj||∆i| are 1, that is all, except for those involving the short orbit ∆43 of length 85, in which case the ratio is 3. for any particular fano plane f in orbit γj, it is easy to determine how the 7 lines of f are distributed among the orbits {∆i}, thus computation of b2,3 is straightforward. in an effort to overcome the difficulty of presenting in this article the 43 × 381 matrix a2,3, the next two pages display a coded version of a2,3 from which, with a little effort, a user-friendly version of a2,3 can be recovered. each column of a2,3 is a vector consisting of 43 elements from {0, 1, 3}. we adjoin two extra 0’s at the top of the column and transpose, transforming the column to a row vector v ∈ z454 . we then use the following alphabet of 64 characters, as digits with values from 0 to 63: 0123456789abcdefghijklmnopqrstuvwxyzabcdegfhijklmnopqrstuvwxyz+− . the vector v ∈ z454 is separated into 15 triples, and each triple, belonging to z34, is encoded as a symbol in the alphabet using radix-4 notation. for example, 1000110101000101000000000000000000000000000 =⇒ 001 000 110 101 000 101 000 000 000 000 000 000 000 000 000 =⇒ 1 0 k h 0 h 0 0 0 0 0 0 0 0 0 =⇒ 10kh0h000000000 the next page displays the first 192 columns, and the subsequent page the remaining columns of a2,3. the 381 columns of a2,3 correspond to 381 short rows in the display. for example the short row 5 2 20 10kh0h000000000 means the following: i) 5 is the column index for a2,3, corresponding to the 5th orbit of 3-spaces under g. (ii) augmenting the pair 2 20 with 1, yields the basis of 3 vectors {1, 2, 20} in f82 from which a fano plane is constructed, and from which the complete 5th g-orbit can be generated. (iii) by reversing the encoding process discussed earlier, the code “10kh0h000000000” yields the 5th column of a2,3, as the transpose of 1000110101000101000000000000000000000000000 . remark 3.1. in passing, we present some properties of a2,3 which may be used to establish still unknown features of designs and large sets related to a2,3. we say that a vector with integer entries has type x λ1 1 x λ2 2 · · ·x λm m if the value xi appears λi times in the vector, for 1 ≤ i ≤ m. (i) the row sums of a2,3 are all 63, as expected, (ii) the vector of column sums of a2,3 is of type 7360921, (iii) the row vectors of the long orbits of 2-spaces are all of type 032016031, (iv) the row vector for the short orbit of 2-spaces is of type 0360321, 169 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 (v) there are 4 column types for a2,3 as follows: 1 2 4 3k0000000040004 65 4 26 k00l0040001000 129 6 80 400g000001hg0g 2 2 8 1lg000100000100 66 4 32 h000h040g00003 130 6 82 404400401004g0 3 2 12 1k5000000011000 67 4 34 gg004g10000140 131 6 88 4000000400s004 4 2 16 111k0000g000040 68 4 40 hg0004g00gg000 132 6 96 40g40100001500 5 2 20 10kh0h000000000 69 4 42 h040g500400000 133 6 98 40g00g0000kk00 6 2 24 1140m0400000000 70 4 48 gc00000h010000 134 6 106 405000g0000k10 7 2 28 10h050000000013 71 4 50 g10ggg50000000 135 6 112 40g014g00g0g00 8 2 32 100g1k00000040g 72 4 56 g540000h000010 136 6 114 41g040g0001100 9 2 36 100541g000000g0 73 4 58 g1001004g01010 137 6 120 40544010040000 10 2 40 1101g05000000g0 74 4 64 ggh00001gg0000 138 6 122 5000g000150g00 11 2 44 110gg04g4000000 75 4 66 g100001g410400 139 6 130 40010014g0g040 12 2 48 104gg004000g400 76 4 72 g04100g00k4000 140 6 136 4001004140004g 13 2 52 1045g0010004000 77 4 74 ggg000001400k0 141 6 138 4k00k004000100 14 2 56 100g400gk00000g 78 4 80 g0h10040410000 142 6 144 44000g0g0h0010 15 2 60 100110001g10040 79 4 82 h04000000gm000 143 6 146 4000g0140g00h0 16 2 64 10144g001g00000 80 4 90 g040441000g010 144 6 154 40400004h400g0 17 2 68 100004001l00100 81 4 96 g00g0000g11g0g 145 6 160 44000104004440 18 2 72 10001h00gh00000 82 4 98 g0g101000044g0 146 6 162 4000000140k04g 19 2 76 1g0000g040k1000 83 4 104 k00l00g0400000 147 6 168 4g00044g001400 20 2 80 104400k01010000 84 4 106 g0500000504400 148 6 170 50100050g000g0 21 2 84 1000001c000g010 85 4 112 g1040045000g00 149 6 176 400000g140005g 22 2 88 1000000j000400g 86 4 114 g00g1g0400010g 150 6 178 400gg01104g000 23 2 92 1h40000000gg004 87 4 120 k10l0000040000 151 6 192 4001g0g001g100 24 2 96 10400g0000g5040 88 4 122 g000400000ggk4 152 6 194 401400g00h00g0 25 2 100 10g004140000g10 89 4 128 g000g004040gg4 153 6 200 404000001100hg 26 2 104 14k004g00000400 90 4 130 k00l0100010000 154 6 202 50104004100400 27 2 108 1000010hg500000 91 4 136 g000c400000050 155 6 216 40001c0g0g0000 28 2 112 11000000g400hg0 92 4 138 gg000hg0500000 156 6 218 4000104gg00007 29 2 116 101g110g4000000 93 4 144 g0041g10401000 157 6 224 4000g0014g004g 30 2 120 100040k05000g00 94 4 146 h0000000ggg050 158 6 232 4040k0g0g0000g 31 2 124 140004100g40004 95 4 152 g004ggg0g00g00 159 6 234 40g0041000h100 32 2 128 100014k010000g0 96 4 160 g100g40000140g 160 6 240 4g0014110g0000 33 2 132 151000004040004 97 4 162 g0100g001g4040 161 6 242 40000m0g000410 34 2 136 10gg40000g00440 98 4 168 h00011g0000410 162 6 248 40000g0g404014 35 2 140 100004000c00gg0 99 4 170 g00045000000hg 163 6 250 41g01100004004 36 2 144 100010g000g0k10 100 4 176 g00g10g0040g40 164 8 20 10144010040004 37 2 148 1000000101g4110 101 4 178 g000g01g0004h0 165 8 22 10g10g400000gg 38 2 152 10004100g1g0040 102 4 192 g44000g001gg00 166 8 34 10045100014000 39 2 156 1gh000000g4000g 103 4 194 g01000g0g10440 167 8 38 30010g0100000g 40 2 160 10000g000015440 104 4 200 g01000411040g0 168 8 50 11g00054010000 41 2 164 10040g00000504g 105 4 202 gg00000400030g 169 8 52 14000gg10010g0 42 2 168 1000001400005h0 106 4 216 g000141g000440 170 8 66 11004040500g00 43 2 172 1104g040000gg00 107 4 218 g000h00g400007 171 8 70 11100400440100 44 2 176 10000g001105040 108 4 224 g400000h000gg4 172 8 80 1g00l000014000 45 2 180 1000g01l0000010 109 4 226 g0001040000ggk 173 8 82 1g44001000g040 46 2 188 10000010004h40g 110 4 232 g004g10400010g 174 8 98 10005000014k00 47 2 192 100100k0100h000 111 4 234 g041001g001400 175 8 100 140g00g0100k00 48 2 196 1010000g0014013 112 4 240 g400040h0g1000 176 8 102 1g00ghg0100000 49 2 200 1000010044g0gg0 113 4 242 gg004405000010 177 8 112 1104g00g000h00 50 2 204 10010011gg0000g 114 4 248 g0g00g40g44000 178 8 114 1040000g044h00 51 2 212 100000000gh0504 115 4 250 l0000040400007 179 8 116 11041000040050 52 2 216 1000504g0010003 116 6 16 40m00000001h00 180 8 118 11005000015000 53 2 220 10000100g300100 117 6 18 c05000001000g0 181 8 128 10g0001gg000k0 54 2 224 10000004140ggg0 118 6 32 4g00gg0g000g10 182 8 130 1g4100000g5000 55 2 228 1g000400400010k 119 6 34 4l005100000000 183 8 144 10k40h01000000 56 2 232 10110g000005040 120 6 40 5400040g000h00 184 8 146 100000041k1010 57 2 236 10g40101000010g 121 6 42 410g11g0000040 185 8 148 10000014k010g0 58 2 240 1000400000hg0h0 122 6 48 5k004010001000 186 8 160 101000g4000c00 59 2 244 10g51000g000g00 123 6 50 440014400g0040 187 8 162 100101010041g0 60 2 248 140000g0005g004 124 6 56 5000g00gk00003 188 8 166 10g0040g0k0040 61 2 252 100g040k4400000 125 6 58 4h000005004010 189 8 182 10001000g01030 62 4 16 m0h00000104000 126 6 64 400g0001k0004g 190 8 192 140g0010g0g0g0 63 4 18 g0k00101110000 127 6 66 41011101g00000 191 8 194 101g0004g00410 64 4 24 k00l0010000100 128 6 74 401000401g0g0g 192 8 196 10100104410400 170 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 193 8 198 10g000g10g000k 256 12 192 4h1h00000g000 319 18 226 400040gg0504 194 8 212 10005040054000 257 12 194 41400400g000k 320 18 230 4000hg00401g 195 8 224 1004g04100400g 258 12 196 500000g01g1g0 321 18 234 401g040010g4 196 8 226 10010041101004 259 12 198 44040g4g00400 322 18 236 4h00004g4100 197 8 228 1k04000000010k 260 12 214 4000000040hk3 323 20 38 101g500g000g 198 8 230 1041gg0000010g 261 12 224 40000g1010114 324 20 44 1100gkg0g000 199 8 240 14010g001g000g 262 12 226 44g0444100000 325 20 78 1k4110040000 200 8 242 111000g0001014 263 12 228 4h0l000000004 326 20 102 10010004h110 201 8 244 10g4000g0000c0 264 12 230 5004000h0100g 327 20 132 1g4g05000003 202 8 246 1g000010510004 265 12 240 4h0h0004g0000 328 20 140 14501004g000 203 10 16 gggg0000k00g0 266 12 242 4g10g00440010 329 20 162 1g0001404110 204 10 18 g5g1000400004 267 14 20 11g0g00040044 330 20 164 100040gh004g 205 10 50 g01004g003000 268 14 22 1gg400h0g0000 331 20 170 1000g10011k0 206 10 52 l040000h000g0 269 14 38 1100k000041g0 332 20 192 14010g01g00g 207 10 64 g00g001h50000 270 14 66 30000500g0g00 333 20 202 10010q00000g 208 10 66 h0104g0004003 271 14 86 1000g0400k044 334 20 206 100g0410g00j 209 10 68 mg00000g400g0 272 14 100 100g0110g0k00 335 20 230 10010g000g1k 210 10 70 g0000h04g400g 273 14 128 100400gk0g0g0 336 20 234 10g500401004 211 10 84 g00014040g440 274 14 130 10h41000040g0 337 24 38 g0g10k04100 212 10 100 g000hg0100g40 275 14 148 1000g141g0g00 338 24 106 g0001gg0504 213 10 102 h000hg010g000 276 14 150 10k00401g0400 339 24 110 g10g0100150 214 10 112 g110000000m10 277 14 160 100040004040m 340 24 128 k400g0h00g0 215 10 118 h0g1001040400 278 14 164 10104001004gg 341 24 130 h400g1g1000 216 10 130 g110040000g44 279 14 166 104000g00g4k0 342 24 162 g4g0g0k4000 217 10 132 g400444410000 280 14 178 140gh00000044 343 24 164 g4100011g0g 218 10 144 k000hh0100000 281 14 194 1300400001100 344 24 166 g0000111g4g 219 10 146 g040400014013 282 14 198 1000gg00100k4 345 24 168 g10g4400440 220 10 160 kg000000404gg 283 14 214 110g4g000g100 346 24 170 gm0100000h0 221 10 162 g04100040k400 284 14 224 1s00101000000 347 24 234 g1100g01014 222 10 164 h00040040g40g 285 14 226 1004g400h0100 348 24 236 k000g504100 223 10 180 g4g004g001004 286 14 240 1g000104g0h00 349 28 32 cg005000g00 224 10 192 g00000k00k00k 287 14 242 1k00000410014 350 28 74 41000h0h010 225 10 194 g1g0000g100k0 288 14 244 10gg00g00g0g4 351 28 96 40h0044h000 226 10 196 g0410000h00gg 289 16 38 g00h10g01g00 352 28 102 4010l004g00 227 10 198 g11g0g00g0g00 290 16 42 g1011k010000 353 28 226 40440500404 228 10 210 g050410001004 291 16 70 g0500040hg00 354 28 236 40000gg0514 229 10 212 g10000l010010 292 16 74 g00414110010 355 30 36 31004g0g000 230 10 224 g00404h005000 293 16 78 g0040014g01g 356 30 38 1kg0g010400 231 10 226 g0gg045000010 294 16 98 g40400g04h00 357 30 40 10411010404 232 10 228 g4g0004000144 295 16 108 gh00kg040000 358 30 64 1101110g100 233 10 240 g4g00000hg004 296 16 110 h0400110g003 359 30 78 10g40050140 234 10 244 g0g0g0g40g400 297 16 140 g04000gg0504 360 30 110 1k000100k04 235 12 16 4g4000000401k 298 16 164 gg0g0100g00j 361 30 162 105000040m0 236 12 18 4510000440010 299 16 166 k40400100050 362 30 174 14510000044 237 12 32 400g1k000g100 300 16 196 g00014430000 363 30 228 1041010000c 238 12 50 40000l0g0400g 301 16 200 gg04001040h0 364 32 78 g10015h000 239 12 54 4000154000g40 302 16 202 gg400410000j 365 32 94 k0gg10g100 240 12 66 501000044001g 303 16 226 g0g05000g0h0 366 32 156 g500k4000g 241 12 68 5010004h0g000 304 16 228 k0010hg00004 367 32 196 g400150044 242 12 70 5400100l00000 305 16 232 gggg00000414 368 32 216 gk40g10100 243 12 84 4h0h040004000 306 16 234 g40gg0401004 369 34 192 404100g150 244 12 98 4040040105400 307 18 32 44g4k4000000 370 34 200 40g14g10g0 245 12 112 4400010040k03 308 18 40 404k44000040 371 36 64 140104gk00 246 12 118 400100400kg04 309 18 44 4040kgg40000 372 36 78 1013040400 247 12 128 40440g00000gk 310 18 68 4440g0ggg000 373 36 196 1101050500 248 12 130 40100004k0050 311 18 96 400010001h43 374 36 198 14kg040100 249 12 132 4011104000140 312 18 100 4g1000011g0g 375 38 68 k00m04g00 250 12 144 4040k00k01000 313 18 132 40g0044k1000 376 40 68 4g0g54g00 251 12 148 4144g10040000 314 18 160 41k0000g4400 377 42 76 3000m0003 252 12 162 40g0500g04g00 315 18 164 414000000g5g 378 42 78 111140g04 253 12 166 40000100014k3 316 18 196 401g400101g0 379 44 64 m10010g4 254 12 176 400g504400040 317 18 200 4000000415g3 380 44 78 kg0c4000 255 12 182 4050000440110 318 18 206 403g000000gg 381 58 128 c0k10g0 a.) 320 columns of type 03617 b.) 40 columns of type 0381431 c.) 20 columns of type 0361631 d.) a single column of type 04033 (vi) since all g-orbits on 3-spaces have length 255, each of the three constituent designs of any [g]−invariant large set ls[3][2, 3, 28] will be comprised of 127 g-orbits of 3-spaces. in particular, properties (v) d.) and (vi) imply that a large set ls[3][2, 3, 28] whose automorphism group contains a singer subgroup as a normal subgroup, can not have a group of automorphisms transitive on the 3 2-[28, 3, 21] designs. 171 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 3.2. constructing and presenting the designs and large sets as the number of columns of a2,3 is rather large, a backtrack, depth-first search or similar algorithm would be hopeless in finding solutions to equation (10). instead, we use lattice basis reduction to seek solutions. this technique is nicely described in [15], pages 277-300. for each of the 9 large sets of 2-[28, 3, 21] designs we proceed using the following non-deterministic procedure, which, in general, is not guaranteed to terminate. procedure 3.2. (i) determine a 0-1 solution u1 to equation a2,3u1 = 21j, thus extracting a 2-[28, 3, 21] design d1 as the union of 127 g-orbits of 3-spaces. if this step succeeds, proceed to step (ii), otherwise stop. (ii) remove from a2,3 the 127 columns corresponding to design d1 to obtain a 43×254 matrix c2,3, and find a 0-1 solution u2 to equation c2,3u2 = 21j, thus extracting a second design d2 consisting of 127 g-orbits among the orbits corresponding to the columns of c2,3. if this step succeeds, proceed to step (iii), otherwise stop. (iii) remove the 127 columns constituting d2 from c2,3. the remaining 127 columns of c2,3 correspond to orbits whose union is a third design d3, and l = {d1,d2,d3} is a ls[3][2, 3, 28] large set. if steps (i) and (ii) are successful, so is (iii) and we have a successful termination with output large set l. thus, the procedure of finding u1 and u2 becomes a matter of solving systems of integer equations through lattice basis reduction [15]. the following procedure describes briefly how the problems are set up so that lattice basis reduction can be used. procedure 3.3. first we construct a matrix that will constitute a basis for an integral lattice λ1 by adjoining the identity matrix of order 381 above km matrix a2,3. to the right of the 424×381 matrix just formed we adjoin a 424×1 column vector which has zeros in the first 381 positions and −21’s in the remaining 43 positions. let m1 denote the 424 × 382 matrix just formed. m1 = [ i 0 a2,3 −21j ] , m2 = [ i 0 c2,3 −21j ] if basis reduction produces a short enough basis m′1 for λ1 which contains a short vector v1 with 0’s and 1’s (or 0’s and −1’s) in the first 381 positions and all 0’s below, then the projection u1 of v1 (or −v1) to the first 381 coordinates is likely to be a solution to a2,3u1 = 21j (see [15].) the weight of u1 will be 127, and the union of orbits of 3-spaces corresponding to the 1’s in u1 will form a 2-[28, 3, 21] design d1. if a solution u1 is found, then replacing a2,3 by c2,3 yields a 297×255 matrix m2 which spans a lattice λ2, and by the same process as above, m2 can yield a solution to c2,3u2 = 21j, that is, a design d2 disjoint from d1. it is now clear that when the 127 columns corresponding to the orbits forming d2 are removed from c2,3, the remaining 127 orbits will form a 2-[28, 3, 21] design d3, and that {d1,d2,d3} will be a large set. however, the above procedure is not guaranteed to find a solution at first try, so if the basis reduction algorithm was unable to find a column in reduced basis m′1 that met the conditions in procedure 3.2.2 , we would repeat the process, twiking the order of the columns of m1, and the same later for m2. the above procedure was repeated a number of times and we successfully constructed 9 distinct large sets {l1, . . . ,l9} which we exhibit below. 3.3. reconstruction of the large sets we briefly describe the display, to enable the reader to reconstruct the large sets and related designs. the first column is the index of the g-orbits on 3-spaces. there are 9 additional columns, each corresponding to one of the large sets. each column has 127 1’s, 127 2’s and 127 3’s in it, which select the orbits contained in d1, d2 and d3 respectively, for each large set. since the orbits can be computed from the representative bases in the presentation of a2,3, the reader can readily reconstruct the 9 large sets and the designs involved. direct computation shows that indeed the 9 large sets are different from each other and different from the large set l0 constructed in [4]. however, a peculiar visual symmetry is observed in the structure of our 9 large sets 172 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 figure 1. the 10 large sets {l0, . . . ,l9} which is perhaps only related to our search method. the 9 large sets can be divided into 3 clusters of 3 large sets per cluster. the large sets of each cluster share a common 2-[28, 3, 21] design forming a triad of large sets, with a central design and three peripheral pairs of designs as illustrated in figure 1. the three centers are different from each other, and the 18 peripheral designs are also different from each other and the 3 centers. actually, there are no elements of γ permuting non-trivially the 3 clusters of large sets, nor elements of order 3 permuting the 3 designs of any one of the large sets. checking the list of maximal subgroups of γ = gl8(2) shows that n = nγ(g) is not maximal in γ. let φ8 = 〈ζ〉≤ n , ζ : α → α2 be the frobenius subgroup normalizing g. we have checked that φ8 does not fix any of the 9 large sets, and does not move any one of the 9 large sets to any other. let l0 be the ls[3][2, 3, 28] discovered by the authors of [4], and let s = {li : 0 ≤ i ≤ 9}. we already know that g ≤ aut(li) for each i ∈ {0, . . . , 9}. it is conceivable that the automorphism groups of the 10 large sets in s are not all identical, but we think this is very unlikely and we conjecture that in fact aut(li) are all identical, and equal to the singer subgroup g. ζ =   1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1   we presently rephrase, in the context of our notation, a very useful theorem of a. betten, r. laue, and a. wassermann in [1]. this will immediately yield a corollary concerning the question of isomorphism between the 10 large sets in s = {l0,l1, . . . ,l9}. theorem 3.4. (theorem 3.1 in [1]) let g be a finite group acting on a set x. suppose that x1,x2 ∈ x and g ∈g such that xg1 = x2. moreover, suppose that a sylow subgroup p of g is contained in the stabilizers gx1 and gx2. then, x n 1 = x2 for some n ∈ ng(p). 173 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 1 222332233 65 311113121 129 122221313 193 233223323 257 111111111 321 123111112 2 111111111 66 211232221 130 333112123 194 233333323 258 132321213 322 311333321 3 211232221 67 233333333 131 223333232 195 122111113 259 111221311 323 211222321 4 133111112 68 111111111 132 211323231 196 223332232 260 132111112 324 223112133 5 132111112 69 232112123 133 132111112 197 133321213 261 233112122 325 123221213 6 311323331 70 322332323 134 323113123 198 311222331 262 232112133 326 233112133 7 333233333 71 323113133 135 323223233 199 311112131 263 111321311 327 233232223 8 332112122 72 322323333 136 123111113 200 322112133 264 211232231 328 211333321 9 133111112 73 111221311 137 311222231 201 322322323 265 311112131 329 232332322 10 211222331 74 122111113 138 133111112 202 211223321 266 211323221 330 223333232 11 311333221 75 311223321 139 211322221 203 332233323 267 223113132 331 111331311 12 133321313 76 233113132 140 111111111 204 123111112 268 223112123 332 211333221 13 311223331 77 322233223 141 332322232 205 322323333 269 211113131 333 233223232 14 122331213 78 223333332 142 232113132 206 133111112 270 322323232 334 122221212 15 211223221 79 232322322 143 332113123 207 333113123 271 322233233 335 111111111 16 333333223 80 111331211 144 111331211 208 211322331 272 111331211 336 323333323 17 222222332 81 111331311 145 323232223 209 223322332 273 123221213 337 111111111 18 232322323 82 211112131 146 323112123 210 332322233 274 323332222 338 232233233 19 111111111 83 222112122 147 111231311 211 211113131 275 122331312 339 211113121 20 223332232 84 222333232 148 123231313 212 123221213 276 222222232 340 223333322 21 233223332 85 211232321 149 133321212 213 111111111 277 232232323 341 133111113 22 222332323 86 332113133 150 132231212 214 233332233 278 223333332 342 133221212 23 322232223 87 111111111 151 323233223 215 311112121 279 332113132 343 233113132 24 223233322 88 133321313 152 322322323 216 332232222 280 233323322 344 311232221 25 322333322 89 111111111 153 211223331 217 132231313 281 332333223 345 233333232 26 211333331 90 332323232 154 223113132 218 332333223 282 111231311 346 223333332 27 132231212 91 332333332 155 223222222 219 132111112 283 332113122 347 332233222 28 323113132 92 233332222 156 333113123 220 223323232 284 223112123 348 111221311 29 111111111 93 123111113 157 111221311 221 211232331 285 211222221 349 222232333 30 311323321 94 122331312 158 322333233 222 323332322 286 333112123 350 323323323 31 222112133 95 222222232 159 223332322 223 222222222 287 111321211 351 323232232 32 132111113 96 132221213 160 133231212 224 332113123 288 223332233 352 111111111 33 311113121 97 332222323 161 222332223 225 311112121 289 132111112 353 132331212 34 233113133 98 322232233 162 311333331 226 111111111 290 111221211 354 311112131 35 322223223 99 133111112 163 211232331 227 211332321 291 132221213 355 222233223 36 133221313 100 322112133 164 332232233 228 111111111 292 223113133 356 322223333 37 133111112 101 233112122 165 133111112 229 322112132 293 311112121 357 122111113 38 232113132 102 211113131 166 122111112 230 332333333 294 211113121 358 222112132 39 222233222 103 133221312 167 333333332 231 122221212 295 132111112 359 311332231 40 333112132 104 211322221 168 332112132 232 111321211 296 111111111 360 311323321 41 111221211 105 323223233 169 223113122 233 133231212 297 322322222 361 322222223 42 111231211 106 111321211 170 232323223 234 311112121 298 322323333 362 223112122 43 322233232 107 223112132 171 123111112 235 123331212 299 333232233 363 332223333 44 111111111 108 111221211 172 323223233 236 132231312 300 332332333 364 111111111 45 233322233 109 311323331 173 232233333 237 111321211 301 122221313 365 311112131 46 133321312 110 133331312 174 111231211 238 333113122 302 211332221 366 311113131 47 111111111 111 133231212 175 123111113 239 323112122 303 222222232 367 122321213 48 111221311 112 333233233 176 111221311 240 222223332 304 311332321 368 122111112 49 211333231 113 223113133 177 233332322 241 333233333 305 211232321 369 311233321 50 311112121 114 323322223 178 222222333 242 132231313 306 333323322 370 311222221 51 123111113 115 111111111 179 211322221 243 123221213 307 311223321 371 111321311 52 323322222 116 232233332 180 133111112 244 322113123 308 133111113 372 333323232 53 323332322 117 332323332 181 211233331 245 323113122 309 111111111 373 222223322 54 333112133 118 211232221 182 211322221 246 233112122 310 223223332 374 311332331 55 222222322 119 311112121 183 322223222 247 232332232 311 322323232 375 332333332 56 232322233 120 211233321 184 211323231 248 123111113 312 223112123 376 233222232 57 323223223 121 132231313 185 311112121 249 122221312 313 232332332 377 322333333 58 311113121 122 222112122 186 322322223 250 211222231 314 111111111 378 211223331 59 122321312 123 232322332 187 132231213 251 111221311 315 311112121 379 232233332 60 333222323 124 133321213 188 111231211 252 332322323 316 222332223 380 322333323 61 111231311 125 111221311 189 322322223 253 211233331 317 232232323 381 333322322 62 333223322 126 222112132 190 111321211 254 223113123 318 332333333 63 311332231 127 133321312 191 133231312 255 133231213 319 111111111 64 133321313 128 123231213 192 232323333 256 311233231 320 111111111 174 m.r. hurley et al. / j. algebra comb. discrete appl. 3(3) (2016) 165–176 corollary 3.5. the ten large sets in s = {l0,l1, . . . ,l9} are pairwise non-isomorphic. proof. let l be the collection all large sets of type ls[3][2, 3, 28], and g = 〈α〉 be the singer subgroup as defined earlier. then, γ acts on l, and for any λ ∈ l the stabilizer γλ is the full automorphism group of λ. in particular, for each λ ∈ s we have that p < g ≤ γλ where p is the sylow-17 subgroup of g. let β,γ ∈ s, β 6= γ, and suppose there is g ∈ γ such that βg = γ. then, by theorem 3.4, there would exist an element n ∈ nγ(p) = nγ(g) such that βn = γ. but we know, by direct checking, that no element of the frobenius group φ8, and therefore no element of nγ(g) = g · φ8 sends β to γ, a contradiction. 3.4. conclusions until 2014, the only large sets of geometric t-[qn,k,λ] designs known were for t = 1. in finite geometry, ls[n]-[t,k,qn] large sets with t = 1, are known as (k-1)-parallelisms of (k-1)-spreads in pg(n− 1,q). the first large set l0 of geometric 2-designs, a ls[3]-[2, 3, 28], was constructed by the authors of [4]. in this paper we construct an additional nine pairwise different large sets l1, . . . ,l9 which are also different from l0. all these large sets are [g]-invariant, under the same singer subgroup g of order 255. in fact, the large sets {l0, . . . ,l9} are pairwise non-isomorphic. 3.5. possible future work the necessary conditions for the existence of a ls[3]-[3, 4, 29] are satisfied and we are close to settling the question of existence of a ls[3]-[3, 4, 29]. acknowledgment: the authors would like to express their thanks to dr. igor kliakhandler whose generous support made possible a most significant conference on algebraic combinatorics and applications at michigan technical university, in august, 2015. the authors also wish to thank prof. vladimir tonchev for putting together a superbly organized conference. references [1] a. betten, r. laue, a. wassermann, simple 7−designs with small parameters, j. combin. des. 7(2) (1999) 79–94. 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[20] s. thomas, designs over finite fields, geom. dedicata 24(2) (1987) 237–242. 176 http://dx.doi.org/10.1109/tit.2008.926449 http://dx.doi.org/10.1109/tit.2008.926449 http://dx.doi.org/10.1016/0012-365x(76)90030-3 http://dx.doi.org/10.1016/s0304-0208(08)72985-2 http://dx.doi.org/10.1016/s0304-0208(08)72985-2 http://arxiv.org/abs/1302.4295v2 http://arxiv.org/abs/1302.4295v2 http://dx.doi.org/10.1002/1520-6610(2001)9:1<40::aid-jcd4>3.0.co;2-0 http://dx.doi.org/10.1002/1520-6610(2001)9:1<40::aid-jcd4>3.0.co;2-0 http://dx.doi.org/10.1002/jcd.3180030108 http://dx.doi.org/10.1002/jcd.3180030108 http://dx.doi.org/10.1006/jnth.1994.1036 http://dx.doi.org/10.1006/jnth.1994.1036 http://dx.doi.org/10.1007/bf02351594 http://dx.doi.org/10.1007/bf00150939 introduction preliminaries main result references issn 2148-838x j. algebra comb. discrete appl. -(-) • 1–6 received: 4 january 2022 accepted: 31 march 2022 ar ti cl e in pr es s journal of algebra combinatorics discrete structures and applications new good quasi-cyclic codes and codes with improved minimum distance research article eric zhi chen, fredrik jönsson abstract: one of the most important and challenging problems in coding theory is to construct codes with optimal parameters. as a generalization of cyclic codes, quasi-cyclic (qc) codes as well as quasitwisted (qt) codes have been shown to contain record-breaking codes. in this paper, various computer algorithms have been used to search for good qc codes. a lot of good new qc codes have been found and they have been used to construct new linear codes. a total 11 new codes that improve the bound on the minimum distance are presented. 2010 msc: 94b05, 94b65 keywords: finite fields, linear codes, quasi-cyclic codes, algorithms 1. introduction a linear [n,k,d]q code over finite field gf(q) is a k-dimensional subspace of gf(q)n, where n is the block length, k is the dimension of the code, and d is the minimum distance between any two different codewords. the minimum distance determines the error-correcting or error-detecting capability. a central and fundamental problem in coding theory is to find the optimal values of the parameters of a linear code and construct codes with these parameters. grassl [16] maintains online code tables of linear codes for small block length and code dimension over small finite fields. the code tables contain both the lower bounds and upper bounds on the minimum distance. a code with a minimum distance meeting the upper bound is said to be optimal, while a code with a minimum distance meeting the lower bound is called best-known (since no other code with the same block length n, code dimension k, and with larger minimum distance is known). to construct codes with the best possible minimum distances is shown to be very difficult and challenging. for small code dimension and block length, it is possible to do exhaustive computer search. the problem becomes intractable when both the code dimension and block length become large. it has been shown that subclasses of linear codes with rich mathematical structures can be used to reduce the search time complexity. during the last decades, the classes of eric zhi chen (corresponding author), fredrik jönsson; department of computer science, kristianstad university, 291 88 kristianstad, sweden (email: eric.chen@hkr.se, fredrik.jonsson@hkr.se). 1 https://orcid.org/0000-0002-2492-7754 https://orcid.org/0000-0001-7589-1723 ar ti cl e in pr es s e. z. chen, f. jãűnsson / j. algebra comb. discrete appl. -(-) (2023) 1–6 quasi-cyclic (qc) codes and quasi-twisted (qt) codes have been shown to contain many good codes, and many record-breaking qc/qt codes have been constructed [1–3, 5–9, 11–15, 17–23] a lot of codes that reach the lower bound on the minimum distance are qc/qt codes [16]. an online database of good qc/qt codes is available [10]. in this paper, various algorithms to search for good qc/qt codes have been applied, and lot of good new qc/qt codes have been obtained. by applying construction x with new constructed qc codes, 5 new linear codes have been constructed. a total of 11 new linear codes that improve the lower bounds on the minimum distance have been presented in this paper. 2. computer search for quasi-cyclic codes a linear [n,k,d]q code c is called cyclic if a codeword (a0,a1, . . . ,an−1) is in c, then so is (an−1,a0,a1, . . . ,an−2). a code is said to be quasi-cyclic (qc) if a cyclic shift of any codeword by p positions is also a codeword. therefore, a cyclic code is a qc code with p = 1. the length n of a qc code is a multiple of p, i.e., n = pm. a cyclic matrix is also called a circulant matrix. an m × m cyclic matrix is defined as a =   a0 a1 a2 . . . am−1 am−1 a0 a1 . . . am−2 am−2 am−1 a0 . . . am−3 ... ... ... ... ... a1 a2 a3 . . . a0   , (1) and the algebra of m × m cyclic matrices over gf(q) is isomorphic to the algebra in the ring gf(q)[x]/(xm − 1), if a is mapped onto the polynomial formed by the elements of its first row, a(x) = a0 + a1x + . . . + am−1x m−1, with the least significant coefficient on the left. the polynomial a(x) is also called the defining polynomial of the matrix a. the polynomials bxja(x), where b is a non-zero element in gf(q), and j = 0,1,2, . . . ,m − 1, form an equivalent class, and they generate the equivalent cyclic codes. therefore, it is enough to take one representative from each equivalent class. the number of nonzero representatives (used as defining polynomials) for m × m circulant matrices over gf(q) is given below [23]: b(m,q) = 1 (q − 1)m ∑ d|m φ(d)(q m d − 1) gcd(d,q − 1), (2) where φ(d) is euler’s totient function. the generator matrix of a qc code can be transformed into rows of m × m circulant matrices by suitable permutation of columns. an h-generator qc code has a generator matrix of the following form: g =   g1,1 g1,2 g1,3 . . . g1,p g2,1 g2,2 g2,3 . . . g2,p g3,1 g3,2 g3,3 . . . g3,p ... ... ... ... ... gh,1 gh,2 gh,3 . . . gh,p   , (3) where gi,j are m×m circulant matrices, for i = 1,2, . . . ,h, and j = 1,2, . . . ,p. let gij(x) be the defining polynomial of the matrix gi,j. then the defining polynomials for the h-generator qc code with generator matrix given in (3) can be written as (g11(x),g12(x),g13(x), . . . ,g1p(x), . . . ,gh1(x),gh2(x),gh3(x), . . . ,ghp(x)). in magma [4], the parameter h is called the height. 2 ar ti cl e in pr es s e. z. chen, f. jãűnsson / j. algebra comb. discrete appl. -(-) (2023) 1–6 in the computer search algorithms presented in [8, 17–19], a weight matrix w is used in the computation of the minimum distance of a 1-generator qc code. the general r × s weight matrix has the following form: w =   w0,0 w0,1 . . . h0,s−1 w1,0 w1,1 . . . h1,s−1 ... ... ... ... wr−1,0 wr−1,1 . . . wr−1,s−1   , (4) where the entry wi,j is the hamming weight of ii(x)gj(x) mod xm − 1, ii(x) is the i-th distinct information polynomial after the equivalent reduction, and gj(x) is the j-th defining polynomial [17–20, 23]. with this weight matrix, to construct a best qc [pm,k] code, it is sufficient to find p columns that give the maximum of minimum row sums. in practical implementation of search algorithms, the computer storage is limited. when the code dimension becomes large, the number of defining polynomials would be too large, which makes the weight matrix too large for a general computer to complete the search in reasonable time. in [6], defining polynomials of specific weights were selected, while in [8, 9], a specified number of randomly chosen defining polynomials were selected. for example, the number of defining polynomials for m = 88, k = 18 and q = 3 is 2204293. it is too large to store a weight matrix of 2204293 × 2204293 inside the computer memory. if 100 randomly chosen defining polynomials are used, then the weight matrix is reduced to the size of 2204293 × 100, which is possible in most laptop or desktop computers. of course, the selection of the number of defining polynomials during the search would limit how good a code can be found. but the experience shows that even 100 randomly chosen defining polynomials are used, many good qc codes can be found. for example, by applying the limited search algorithms, a new qc [176,18,88]3 code is found, which improves the bound on minimum distance. 3. the new good and improved codes for a given size m of the circulant matrix and code dimension k, first the non-equivalent defining polynomials and distinct information polynomials were calculated [17, 19]. then 100 defining polynomials are selected randomly to compute the weight matrix. for small p (p = 2 or 3), an exhaustive search among these polynomials is taken, otherwise the iterative search algorithm is applied [8]. via the computer search, more than 300 good qc codes have been obtained. these codes are included in online database of quasi-twisted codes [10]. for example, for m = 11, all best-known qc [pm,10]7 codes with p = 2, . . . ,9 have been found, as shown in table 1. the details of the codes can be found in the online database [10]. in the rest of this paper, the codes that improve the minimum distances in [16] are presented. table 1. best-known [pm, 10, d]7 codes with m = 11. p n k d reference 2 22 10 10 [10] 3 33 10 18 [16] 4 44 10 26 [10] 5 55 10 34 [16] 6 66 10 42 [16] 7 77 10 50 [10] 8 88 10 59 [10] 9 99 10 68 [10] 3 ar ti cl e in pr es s e. z. chen, f. jãűnsson / j. algebra comb. discrete appl. -(-) (2023) 1–6 theorem 3.1. there exist qc [93,9,62]5, [75,11,45]5, and [176,18,88]3 codes. proof. the qc [93,9,62]5 code is constructed with m = 31, and its defining polynomials are g1(x) = x27 + x26 + 2x25 + x24 + 2x23 + 4x22 + 2x21 + 4x20 + 3x19 + 4x18 + 3x16 + x13 + x11 + x10 + 4x8 + 3x7 + 4x6 + 4x5 + 3x4 + 3x + 4,g2(x) = x 27 + x24 + 3x23 + 2x21 + 2x20 + 2x19 + 2x18 + x17 + 3x15 + 3x14 + x13 + 2x12 + x11 + 2x10 + 2x9 + 3x8 + 4x7 + 3x6 + 2x5 + 2x4 + x3 + 2x2 + 3x + 2, and g3(x) = x29 + 3x28 + 4x26 +x25 +x24 + 3x22 + 4x21 + 3x20 +x19 + 4x18 + 2x17 +x16 + 3x15 + 3x14 + 3x13 + 2x12 + x11 + x10 + 3x9 + 3x8 + x7 + 2x6 + x5 + 2x4 + x3 + 3x2 + x + 2. the qc [75,11,45]5 code is constructed with m = 15, and its defining polynomials are g1(x) = x14 + x13 + 3x12 + 4x11 + x10 + x9 + 4x8 + x7 + 3x6 + 2x5 + x4 + 3x3 + x2 + 3x + 2, g2(x) = x12 + 3x11 + 4x10 + 3x9 + 4x8 + 4x7 + 2x5 + 3x4 + 4x3 + 3x2 + x + 4, g3(x) = x13 + 2x12 + 3x10 + x9 + 4x8 + 4x6 + 3x5 + x3 + 4x2 + 2x + 3, g4(x) = x13 + x12 + 3x10 + 4x9 + 3x8 + 3x7 + x6 + x5 + 3x3 + 2x2 + 4x + 2, and g5(x) = x 11 + 4x10 + x9 + 2x8 + 2x7 + x5 + 2x4 + x2 + 2x + 4. the qc [176,18,88]3 code is constructed with m = 88 and its defining polynomials are g1(x) = x83 + 2x82 + 2x81 + x80 + 2x79 + 2x77 + x76 + x75 + x74 + 2x73 + x72 + x70 + 2x69 + 2x68 + 2x63 + 2x62 + 2x60+2x59+x58+2x56+x55+x54+x53+x52+x51+x50+x47+x46+2x45+2x43+2x42+2x41+2x40+2x37+ 2x36 +x35 +x34 +x33 +2x32 +2x30 +2x29 +x28 +x27 +x26 +2x24 +x22 +2x21 +2x20 +x17 +2x15 +2x14 + 2x12+2x11+2x10+2x8+x6+x5+2x4+x3+x+1, and g2(x) = x83+x81+x79+2x78+2x77+2x76+2x75+ x73 +x72 +2x71 +2x70 +2x69 +x68 +x67 +x64 +2x63 +2x62 +2x61 +2x59 +x58 +2x57 +x55 +2x54 +x51 + 2x49 +2x47 +2x46 +2x45 +x44 +2x42 +2x41 +x39 +2x38 +x37 +2x35 +x34 +x32 +2x30 +2x29 +x28 +x25 + 2x24+2x23+2x22+x20+x18+2x16+2x14+x13+x11+x10+2x9+2x8+2x7+2x6+2x4+x3+x2+x+1. all the codes have been checked in magma algebraic system [4] [4] and their weight distributions of these codes can be found in online database of quasi-twisted codes [10]. it should be noted, that a new [176,18,89]3 code was found after our codes were reported [16], and it was based on our reported [176,17,90]3 code as given in theorem 3.2 below. construction x is a method to construct new codes by combining 3 existing codes. let c1 = [n,k1,d1]q and c2 = [n,k2,d2]q be a pair of nested codes, where c1 ⊂ c2. let c3 = [n3,k2 − k1,d3]q be an auxiliary code. then there exists a c = [n + n3,k2,d]q code with d ≥ min(d1,d2 + d3). theorem 3.2. there exists [39,10,21]5, [73,9,47]5, [68,11,40]5, [178,18,90]3, and [119,30,34]2 codes. proof. let c1 be the qc [38,10,20]5 code with m = 19. its defining polynomials are g1(x) = x15 + x14+4x13+x12+x11+3x10+3x9+x7+2x6+x5+4x4+2x3+4x2+4, and g2(x) = x16+2x15+4x14+x13+ 4x12 +x11 +x9 +3x8 +x6 +2x5 +3x4 +3x3 +4x2 +4. let c2 be the qc [38,9,21]5 code with m = 19. its defining polynomials are g1(x) = x16+3x14+2x13+2x11+2x9+x8+x7+4x6+3x5+3x4+2x3+x2+4x+1, and g2(x) = x17 +x16 +2x15 +2x14 +3x13 +2x12 +4x11 +x10 +2x9 +2x8 +x7 +x6 +x5 +x3 +x2 +4x+1. let c3 be an [1,1,1]5 code. by applying construction x, the new [39,10,21]5 code can be constructed. let c1 be the qc [72,9,46]5 code with m = 24. its defining polynomials are g1(x) = x22 + 4x20 + 3x19+2x18+x17+x16+2x15+3x14+4x13+2x12+2x10+3x8+3x7+4x6+2x4+2x3+4x2+3x+4, g2(x) = x20+3x19+x18+2x17+x16+3x15+x14+2x13+x12+x11+4x9+2x8+4x7+3x6+2x5+3x4+3x3+3x2+3x+2, and g3(x) = x21 + 4x20 + 4x18 + 4x17 + 3x16 + 4x15 + 4x14 + x13 + 4x12 + x11 + 3x10 + 4x9 + 4x8 + 2x7 + x6 +3x5 +x4 +3x2 +2x+2. let c2 be the qc [72,8,47]5 code with m = 24. its defining polynomials are g1(x) = x 23+3x22+4x21+x19+2x18+4x17+4x15+3x14+4x13+x12+2x11+x10+3x9+2x8+3x7+2x6+2x5+ 3x4+3x+2, g2(x) = x 21+x20+2x17+x16+2x13+4x12+3x11+4x10+4x9+x6+4x5+2x4+2x3+2x2+x+1, and g3(x) = x22 + 2x21 + 2x20 + 4x19 + x18 + 3x16 + x15 + 3x14 + 2x13 + 3x12 + x11 + 3x10 + x9 + 4x8 + 2x7 + x6 + 3x4 + 3x3 + x2 + 3x + 1. let c3 be an [1,1,1]5 code. by applying construction x, the new [73,9,47]5 code can be constructed. let c1 be the qc [66,11,38]5 code with m = 22. its defining polynomials are g1(x) = x19 + x18 + 4x17 + 4x16 + x15 + 3x14 + 3x13 + 2x11 + 2x9 + 4x8 + 4x7 + x6 + 2x5 + 2x4 + x2 + 4x + 2, g2(x) = x 19 +x18 +4x17 +x15 +4x14 +3x13 +2x12 +4x11 +x10 +3x9 +4x7 +2x6 +4x5 +3x4 +x3 +4x+1, and g3(x) = x18 + 4x17 + 2x16 + 3x15 + 3x14 + x13 + 2x11 + 4x10 + 3x9 + 2x5 + 4x2 + 1. let c2 be the qc [66,10,40]5 code with m = 22. its defining polynomials are g1(x) = x20 + 3x18 + 2x16 + 2x15 + 4 ar ti cl e in pr es s e. z. chen, f. jãűnsson / j. algebra comb. discrete appl. -(-) (2023) 1–6 2x13 + 2x12 + 3x11 + 2x10 + 2x9 + 2x7 + x6 + 3x4 + x3 + 3x2 + 3x + 3, g2(x) = x20 + 3x18 + x17 + x16 + 3x15 + 4x14 + 4x13 + 2x12 + 2x11 + 2x10 + 2x9 + 4x8 + 3x7 + 2x6 + 4x5 + 3x4 + 4x3 + 4x2 + 2x + 4, and g3(x) = x 19 + 3x18 + 3x17 + x16 + 3x14 + 4x13 + 2x12 + 2x11 + 4x10 + 2x9 + 2x6 + 3x5 + 4x3 + x2 + x + 4. let c3 be an [2,1,2]5 code. by applying construction x, a new [68,11,40]5 code can be constructed. let c1 be the qc [176,17,90]3 code with m = 88. its defining polynomials are g1(x) = x84 + x82 + 2x79 + 2x78 + 2x76 + 2x75 + x72 + x71 + x69 + 2x68 + 2x64 + x63 + 2x62 + 2x61 + x60 + x58 + 2x57 + 2x55 + 2x54 + 2x53 + 2x52 + 2x51 + x50 + x48 + 2x47 + 2x45 + 2x44 + x43 + x42 + x41 + 2x40 + 2x38 + x37 + 2x35 + 2x34 + 2x32 + 2x31 + x30 + 2x28 + 2x27 + x26 + 2x25 + 2x24 + x23 + x21 + 2x20 + x18 + x17 + 2x16 + x15 + 2x14 + 2x13 + x12 + x11 + 2x10 + 2x9 + 2x8 + x7 + 2x6 + x3 + x2 + 2x + 1, and g2(x) = x 84 +x83 +x82 +x81 +x80 +x78 +x77 +x76 + 2x75 +x74 + 2x73 +x71 +x70 + 2x68 +x67 +x65 + x63 + x62 + 2x61 + 2x60 + 2x57 + x56 + 2x54 + x52 + x51 + 2x50 + 2x49 + 2x48 + x47 + x46 + x44 + 2x43 + x42 + 2x41 + x40 + x37 + 2x36 + x34 + x33 + x32 + 2x31 + x30 + x28 + x26 + x24 + x23 + 2x22 + x21 + x20 + x19 + x18 + 2x17 + 2x16 + 2x15 + x13 + x12 + 2x11 + x9 + x8 + x7 + 2x6 + 2x5 + 2x3 + 2x2 + 2x + 1. let c2 be the qc [176,18,88]3 code given above. let c3 be the [2,1,2]3 code. by applying construction x, a new [178,18,90]3 can be constructed. let c1 be the 3-generator qc [116,30,32]2 code with m = 29. its defining polynomials are g1(x) = x + 1, g2(x) = x12 + x11 + x9 + x6 + x5 + x3 + x2 + 1, g3(x) = x24 + x23 + x18 + x17 + x16 + x15 + x14 + x12 + x9 + x7 + x5 + x4 + x3 + x2 + x + 1, g4(x) = x20 + x19 + x17 + x15 + x9 + x4 + x + 1, g5(x) = g7(x) = g10(x) = g12(x) = x 28 + x27 + x26 + x25 + x24 + x23 + x22 + x21 + x20 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x12 + x11 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1, and g6(x) = g8(x) = g9(x) = g11(x) = 0. let c2 be the qc [116,28,34]2 code with m = 29. its defining polynomials are g1(x),g2(x),g3(x) and g4(x) as given in the [116,30,32]2 code above. let c3 be an [3,2,2]2 code. by applying construction x, a new [119,30,34]2 code can be constructed. all the codes given above improve the minimum distances in [16]. by applying puncturing method, new improved [92,9,61]5, [67,11,39]5, [118,30,33]2, and [177,18,89]3 codes are obtained. all the codes given in the paper have been checked with the magma algebraic system [4] and included in [16] now. references [1] n. aydin, i. siap, d. k. ray-caudhuri, the structure of 1-generator quasi-twisted codes and new linear codes, des. codes crypt. 24 (2001) 313–326. 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[23] v. ch. venkaiah, t. a. gulliver, quasi-cyclic codes over f13 and enumeration of defining polynomials, j. discrete algorithms 16 (2012) 249–257. 6 https://doi.org/10.1007/s10623-015-0059-5 https://doi.org/10.1007/s10623-015-0059-5 https://doi.org/10.1109/18.623167 https://doi.org/10.1109/18.623167 https://doi.org/10.13069/jacodesmath.66269 https://doi.org/10.13069/jacodesmath.66269 https://doi.org/10.1007/s002000050117 https://doi.org/10.1007/s002000050117 https://doi.org/10.13069/jacodesmath.784999 https://doi.org/10.13069/jacodesmath.784999 http://www.codetables.de http://www.codetables.de https://doi.org/10.1109/18.79911 https://doi.org/10.1109/18.79911 https://doi.org/10.1109/18.144718 https://doi.org/10.1109/18.144718 https://doi.org/10.1109/18.144719 https://doi.org/10.1109/18.144719 https://doi.org/10.13069/jacodesmath.645015 https://doi.org/10.13069/jacodesmath.645015 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/tit.1978.1055929 https://doi.org/10.1109/tit.1978.1055929 https://doi.org/10.1016/j.jda.2012.04.006 https://doi.org/10.1016/j.jda.2012.04.006 introduction computer search for quasi-cyclic codes the new good and improved codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.34390 j. algebra comb. discrete appl. 4(1) • 37–47 received: 25 november 2015 accepted: 16 may 2016 journal of algebra combinatorics discrete structures and applications refined analysis of rghws of code pairs coming from garcia-stichtenoth’s second tower∗ research article olav geil, stefano martin, umberto martínez-peñas, diego ruano abstract: asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. their security is given by the relative generalized hamming weights of the corresponding codes. in this paper we demonstrate how to obtain refined information on the rghws when the codimension of the codes is small. for general codimension, we give an improved estimate for the highest rghw. 2010 msc: 94a62, 94b27, 94b65 keywords: algebraic geometric codes, asymptotically good ramp secret sharing schemes, generalized hamming weights, relative generalized hamming weights, secret sharing 1. introduction relative generalized hamming weights (rghws) of two linear codes are fundamental for evaluating the security of ramp secret sharing schemes and wire-tap channels of type ii [6, 7, 9, 12]. until few years ago only the rghws of mds codes and a few other examples of codes were known [8], but recently new results were discovered for one-point algebraic geometric codes [6], q-ary reed-muller codes [4] and cyclic codes [13]. in [5] it was discussed how to obtain asymptotically good sequences of ramp secret sharing schemes by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. the tools used in [5] were the goppa bound and the feng-rao bounds. in the present paper we focus on secret sharing schemes coming from the garcia-stichtenoth’s second tower [3]. we give a method for obtaining new information on the rghws when the used codes have small codimension. for general codimension we give an improved estimate on the highest rghw. the new results are obtained ∗ supported by the danish council for independent research, grant dff-4002-00367 and the spanish ministry of economy, grant mtm2015-65764-c3-2-p. olav geil, stefano martin, umberto martínez-peñas, diego ruano (corresponding author); department of mathematical sciences, aalborg university, fredrik bajers vej 7g, 9220-aalborg øst, denmark (email: olav@math.aau.dk, stefano.martin87@gmail.com, umberto@math.aau.dk, diego@math.aau.dk). 37 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 by studying in detail the sequence of weierstrass semigroups related to the sequence of rational places [10]. we recall the definition of rghws and briefly mention their use in connection with secret sharing schemes. definition 1.1. let c2 ( c1 ⊆ fnq be two linear codes. for m = 1, . . . ,` = dimc1 − dimc2 the m-th relative generalized hamming weight of c1 with respect to c2 is mm(c1,c2) = min{#suppd | d ⊆ c1 is a linear space, dimd = m,d ∩c2 = {~0}}. here suppd = #{i ∈ n | exists (c1, . . . ,cn) ∈ d with ci 6= 0}. note that for m = 1, . . . ,dimc1, the m-th generalized hamming weight (ghw) of c1 dm(c1) is mm(c1,{~0}). given c2 ( c1 linear codes, by definition, we have that the m-th generalized hamming weight is a lower bound for the m-th relative generalized hamming weight of c1 with respect to c2, i.e. mm(c1,c2) ≥ dm(c1). in [2], a general construction of a linear secret sharing scheme with n participants is defined from two linear codes c2 ( c1 of length n. it was proved in [6, 7] that it has rm = n−m`−m+1(c1,c2) + 1 reconstruction and tm = mm(c⊥2 ,c⊥1 )−1 privacy for m = 1, . . . ,`. here, rm and tm are the unique numbers such that the following holds: it is not possible to recover m q-bits of information about the secret with only tm shares, but it is possible with some tm + 1 shares. with any rm shares it is possible to recover m q-bits of information about the secret, but it is not possible to recover m q-bits of information with some rm −1 shares. we shall focus on one-point algebraic geometric codes cl(d,g) where d = p1 + . . . + pn, g = µq, and p1, . . . ,pn,q are pairwise different rational places over a function field. by writing νq for the valuation at q, the weiestrass semigroup corresponding to q is h(q) = −νq ( ∞⋃ µ=0 l(µq) ) = {µ ∈ n0 | l(µq) 6= l((µ−1)q)}. we denote by g the genus of the function field and by c the conductor of the weierstrass semigroup. we consider c1 = cl(d,µ1q) and c2 = cl(d,µ2q), with −1 ≤ µ2 < µ1. observe that for ` = dim(c1)−dim(c2) and µ = µ1 −µ2 we have that ` ≤ µ, with equality if 2g−1 ≤ µ2 < µ1 ≤ n−1 holds. from [6, theorem 19] we have the following bound: theorem 1.2. for m = 1, . . . ,` we have that: mm(c1,c2) ≥ n−µ1 + z(h(q),µ,m), where z(h(q),µ,m) = min{#{α ∈∪m−1s=1 (is + h(q)) | α /∈ h(q)} | −(µ−1) ≤ i1 < i2 < ... < im−1 ≤−1}. for m > g, one has that dm(c) = n−k + m, that is the singleton bound is reached [11, corollary 4.2]. for other values of m, using theorem 1.2, the following result was found [5, proposition 14]. proposition 1.3. let cl(d,µq) be a one-point algebraic geometric code of length n and dimension k. if −1 ≤ µ < n and 1 ≤ m ≤ min{k,g}, then: dm(cl(d,µq)) ≥ n−k + 2m− c + hc−m ≥ n−k + 2m− c where hc−m = #(h(q)∩ (0,c−m]). 38 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 moreover, in the proof of [5, proposition 14], one has that dm(cl(d,µq)) ≥ n−µ + g −1 + 2m− c + hc−m, which may allow us to improve the bound in proposition 1.3 for µ ≤ 2g−2, since in this case k ≥ µ+1−g. furthermore, we can apply it to bound the rghws of a pair of codes. proposition 1.4. let cl(d,µ2q) ⊆ cl(d,µ1q) be two one-point algebraic geometric codes of length n and dimension k1 and k2, respectively. if −1 ≤ µ2 < µ1 < n and 1 ≤ m ≤ min{k1,g} then dm(cl(d,µ1q)) ≥ n−µ1 + g −1 + 2m− c + hc−m where hc−m = #(h(q)∩ (0,c−m]). moreover, if 1 ≤ m ≤ min{k1 −k2,g} then mm(cl(d,µ1q),cl(d,µ2q)) ≥ n−µ1 + g −1 + 2m− c + hc−m. from garcia-stichtenoth’s second tower [3] one obtains codes over any field fq where q is an even power of a prime. garcia and stichtenoth analyzed the asymptotic behaviour of the number of rational places and the genus, from which one has that the codes beat the gilbert-varshamov bound for q ≥ 49. this allows us to create sequences of asymptotically good codes. the garcia-stichtenoth’s tower (f1,f2,f3, . . .) in [3] over fq, for q an even power of a prime, is given by: • f1 = fq(x1), • for ν > 1, fν = fν−1(xν) with xν satisfying x √ q ν + xν = x √ q ν−1 x √ q−1 ν−1 +1 . the number of rational points of fν is nq(fν) ≥ q ν−1 2 (q− √ q) and its genus is gν = g(fν) = (q 1 2bν+12 c− 1)(q 1 2dν+12 e−1). for every function field fν the following complete description of the weierstrass semigroups corresponding to a sequence of rational places was given in [10]. let qν be the rational point that is the unique pole in x1. the weierstrass semigroups h(qν) at qν in fν are given recursively by: h(q1) = n0, h(qν) = √ q ·h(qν−1)∪{i ∈ n0 : i ≥ cν}, where cν = q ν 2 −q 1 2bν+12 c is the conductor of h(qν). an alternative way to obtain these weiestrass semigroups was described in [1]. definition 1.5. first we define h(q1) = n0. for ν positive integer and j = 2 ⌊ ν 2 ⌋ , we define: cν = q ν 2 −q 1 2 (ν−j 2 ), h(qν) = s 0 ν ∪s 1 ν ∪s 2 ν ∪ . . .∪s j 2 ν ∪s∞ν , where: • s0ν = {xν,1} = {0}, • for 1 ≤ i ≤ j 2 , siν = {x ν,q i−1 2 +1 ,x ν,q i−1 2 +2 ,x ν,q i−1 2 +3 , . . . ,x ν,q i 2 } where for 1 ≤ k ≤ q i 2 − q i−1 2 we have that x ν,q i−1 2 +k = q j 2 −q ν−i+1 2 + kq ν−2i+1 2 , • s∞ν = [cν + 1,∞). 39 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 using the previous description of the weierstrass semigroup h(qν), we can see that it has the following properties: lemma 1.6. with the same notation as before, one has that: 1. for any i1, i2 ∈{0,1,2, . . . , j2 −1, j 2 ,∞}, i1 6= i2, we have that si1ν ∩si2ν = ∅. 2. for i ∈{1, . . . , j 2 } we have that #siν = q i 2 −q i−1 2 and # ( ∪ir=0srν ) = q i 2 . 3. for i ∈ {1, . . . , j 2 } and for any two consecutive elements x,y ∈ siν, with x > y, we have that x−y = q ν−2i+1 2 . 4. for i ∈ {1, . . . , j 2 }. let x be the first element of siν and y the last element of si−1ν , we have that x−y = q ν−2i+1 2 . 5. for i ∈{1, . . . , j 2 }, and for any x,y ∈∪ir=0srν, x > y we have that x−y ≥ q ν−2i+1 2 . proof. 1. by theorem 1 in [1]. 2. let i ∈ {1, . . . , j 2 }, the cardinality of siν follows by its definition. for the second part, by (1), we have that: # ( ∪ir=0s r ν ) = #s0ν + i∑ r=1 #srν = 1 + i∑ r=1 (q r 2 −q r−1 2 ) = 1 + q i 2 −1 = q i 2 . 3. consider two consecutive elements x,y ∈ siν, x > y. there exists a k ∈{1, . . . ,q i 2 −q i−1 2 −1} such that x = q j 2 −q ν−i+1 2 +kq ν−2i+1 2 and y = q j 2 −q ν−i+1 2 +(k+1)q ν−2i+1 2 . it follows that x−y = q ν−2i+1 2 . 4. let y be the last element of si−1ν , i.e. y = q j 2 − q ν−i+1 2 , and x be the first element of siν, i.e. x = q j 2 −q ν−i+1 2 + q ν−2i+1 2 . we have that x−y = q ν−2i+1 2 . 5. for i ∈{1, . . . , j 2 }, consider x,y ∈∪ir=0srν with x > y. this mean that there exists i1, i2 ∈{1, . . . , i}, i1 ≥ i2 such that x ∈ si1ν , y ∈ si2ν . let x2 be the element that precedes x in h(qν), then we have that x−y = (x−x2) + (x2 −y) = q ν−2i1+1 2 + (x2 −y) ≥ q ν−2i1+1 2 ≥ q ν−2i+1 2 . the second inequality follows by (3) and (4), the third one since x2 ≥ y and the last one since i1 ≤ i. applying proposition 1.3 to code pairs coming from garcia-stichtenoth’s second tower [3], an asymptotic result was given in [5, theorem 23], which combined with proposition 1.4 allows us to obtain the following result. corollary 1.7. let (fi)∞i=1 be garcia-stichtenoth’s second tower of function fields over fq, where q is an even power of a prime. let (ci)∞i=1 be a sequence of one-point algebraic geometric codes constructed from (fi)∞i=1. consider r̃,r,ρ with 0 ≤ r ≤ 1 − 1√ q−1, 0 ≤ r̃ < 1 and 0 ≤ ρ ≤ min{r, 1√ q−1}, and assume that dim(ci)/ni → r and µi/ni → r̃. for all sequences of positive integers (mi)∞i=1 with mi/ni → ρ, it holds that δ = lim infi→∞dmi(ci)/ni satisfies δ ≥ 1−r + 2ρ− 1 √ q −1 . (1) δ ≥ 1− r̃ + 2ρ. (2) note that the bound (2) is sharper than (1) for 1√ q−1 ≤ r ≤ 1− 1√ q−1. 40 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 2. small codimension in this section we give a refined bound on the rghws of two nested one-point algebraic geometric codes coming from garcia-stichtenoth’s towers when the codimension is small. before giving such bound, we illustrate the main idea with an example. example 2.1. consider q = 9 and let f6 be the 6-th function field defined by the garcia-stichtenoth’s tower over fq. the weierstrass semigroup h(q6) at q6 in f6 is h(q6) = {0,243,486,513,540,567,594,621,648}∪ ∪{3n | n ∈ n and 3n ∈ [654,702]}∪{n ∈ n | n > 702}. we denote these three sets as a0, b0 and c0 respectively. for computing z(h(q6),µ,m) one should find i1, . . . , im−1 such that −(µ − 1) ≤ i1 < i2 < · · · < im−1 ≤ −1 and minimize #{α ∈ ∪m−1s=1 (is + h(q6)) | α /∈ h(q6)}. in this example we fix i1 = −20, thus: i1 + h(q6) = {−20,223,466,493,520,547,574,601,628}∪ ∪{3n−20 | n ∈ n and 3n ∈ [654,702]}∪{n ∈ n | n > 682} = = (i1 + a 0)f ∪ (i1 + b0)∪ (i1 + c0). note that i1 + a0 and h(q6) are disjoint since −i1 = 20 < 27 and |x−y| ≥ 27 for any x,y ∈ a0 x 6= y. for the same reason for any −20 < i2 < · · · < im−1 ≤−1, we have that im−1 + a0, im−2 + a0, . . . , i2 + a0, i1+a 0 and h(q6) are disjoint. it follows that ∪m−1v=1 (iv+a 0) ⊆{α ∈∪m−1v=1 (iv+h(q6)) | α /∈ h(q6)} and #∪m−1v=1 (iv + a 0) = ∑m−1 v=1 #(iv + a 0) = (m−1)#a0 = 9(m−1). the same argument does not hold for i + b0 (or i + c0) because there exists x,y ∈ b0 (or c0) such that |x−y| = 3 and −i1 > 3 thus it is possible that (iv + b0)∩b0 6= ∅ (or (iv + c0)∩c0 6= ∅) for some v = 1, . . . ,m−1. note that #((i1 +c0)\c0) = i1 = 20, but (i1 +c0)\c0 may intersect a0∪b0. therefore we consider (i1 + c 0)\ √ qn, in this way (i1 + c0)\(c0 ∪ √ qn) and h(q6) are disjoint. it follows that if i1 = −20, then z(h(q6),µ,m) ≥ #∪m−1v=1 (iv + a 0) + #((i1 + c 0)\(c0 ∪ √ qn)) = (m−1) ·9 + ⌊ 202 3 ⌋ . as we can see from previous example, we do not consider the sets b0, i1 +b0, . . ., im−1 +b0 because of their intersections with other sets. in general, we will also consider all the possible values −i1 in the range [m−1,µ−1] to obtain the following bound. theorem 2.2. let ν be an even positive integer and q an even power of a prime. consider two onepoint algebraic geometric codes c2 = cl(d,µ2q) ( c1 = cl(d,µ1q) of length n built on the ν-th garcia-stichtenoth’s function field over fq and µ = µ1 − µ2. for µ < q ν+1 2 , m = 1, . . . ,µ, consider u∗ = 2 3 ( 1 + ν 4 + logq ( m−1 2( √ q−1) )) and β = min{2q− ν+1 4 ( √ q − 1)(µ− 1) 3 2 + 1, 1 4 q ν−5 2 ( √ q − 1)−2 + 1}, we have that: mm(c1,c2) ≥ n−µ1 + g(m) where g(m) =   min { (m−1)q ν 4 −u 2 + qu− 1 2 (1−q− 1 2 )−1 : if m > β, u ∈{logq(m−1)− 1 2 , logq(µ−1) + 3 2 } } (m−1)q ν 4 −u ∗ 2 + qu ∗−1 2 (1−q− 1 2 )−1 if m ≤ β,m 6= 1, 0 if m = 1. 41 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 proof. by theorem 1.2 we have that mm(c1,c2) ≥ n − µ1 + z(h(qν),µ,m) thus we will estimate z(h(qν),µ,m). if m = 1, z(h(qν),µ,1) = 0, otherwise we denote the conductor of h(qν) by c. set −(µ − 1) ≤ i1 < · · · < im−1 ≤ −1, we define u(i1) = ⌊ logq(−i1) + 1 2 ⌋ , then qu(i1)− 1 2 ≤ −i1 < qu(i1)+ 1 2 . for the sake of simplicity we write u instead of u(i1). to estimate z(h(qν),µ,m) we consider the following two sets: a(i1, i2, . . . , iv) = {α ∈ ∪m−1v=1 (iv + a0(u)) | α /∈ h(qν)} where a0(u) = ⋃ν 2 −u i=0 s i ν and c(i1) = (i1 + c 0)\h(qν) where c0 = {α ∈ n | α > c}. again, to simplify the notation we write a = a(i1, i2, . . . , iv), a0 = a0(u) and c = c(i1). by construction a ∪ c ⊆ {α ∈ ∪m−1s=1 (is + h(qν)) | α /∈ h(qν)} and a ∩ c = ∅. thus we have that z(h(qν),µ,m) ≥ #a + #c. we start by computing the cardinality of a. by definition of a0 for any x,y ∈ a0, x 6= y there exist ix, iy ∈ {0, . . . , ν2 − u} such that x ∈ s ix ν and y ∈ s iy ν . we can assume without loss of generality that ix ≥ ij and x > y, then we obtain by (6) in lemma 1.6 that x − y ≥ q ν−2ix+1 2 . since µ < q ν+1 2 , then ix ≤ ν2 −u ≤ 0 and |x−y| ≥ q ν−2ix+1 2 ≥ q ν−2( ν 2 −u)+1 2 = qu+ 1 2 . thus for x,y ∈ a0, x > y, we have that x−y ≥ qu+ 1 2 . since −i1 < qu+ 1 2 , it follows that (j1 + a0)∩(j2 + a0) = ∅ for any j1,j2 ∈ [i1,0]. therefore we have that #a = # ⋃m−1 v=1 (iv+a 0) = (m−1)#a0. by (2) in lemma 1.6, we have #a0 = #( ⋃ν 2 −u i=0 s i ν) = q ν 4 −u 2 . thus #a = (m−1)q ν 4 −u 2 . furthermore, #c = #((i1 + c0)\h(qν)) = #([c + i1,c)\h(qν)) ≥ #([c + i1,c)\ √ qn) =⌊ −i1(1−q− 1 2 ) ⌋ where the inequality follows since h(qν)∩ [0,c) ⊂ √ qn. hence, mm(c1,c2) ≥ n−µ1 + z(h(qν),µ,m) ≥ n−µ1 + min i1∈{−(µ−1),...,−(m−1)} (#a + #c) ≥ n−µ1 + min { (m−1)q ν 4 −u 2 + ⌊ −i1 ( 1−q− 1 2 )⌋ | | i1 ∈{−(µ−1), . . . ,−(m−1)} } . one could try to minimize the previous expression bounding u by logq(−i1) + 1 2 . however, the obtained bound is too loose. hence, we consider the minimum among all possible values of u instead of i1: mm(c1,c2) ≥ n−µ1 + min { (m−1)q ν 4 −u 2 + ⌊ qu− 1 2 ( 1−q− 1 2 )⌋ | | u ∈ {⌊ logq(m−1) + 1 2 ⌋ , ⌊ logq(m) + 1 2 ⌋ , . . . , ⌊ logq(µ−1) + 1 2 ⌋}} ≥ n−µ1 + min { (m−1)q ν 4 −u 2 + qu− 1 2 ( 1−q− 1 2 ) −1 | | u ∈ {⌊ logq(m−1) + 1 2 ⌋ , ⌊ logq(m) + 1 2 ⌋ , . . . , ⌊ logq(µ−1) + 1 2 ⌋}} ≥ n−µ1 + min { (m−1)q ν 4 −u 2 + qu− 1 2 ( 1−q− 1 2 ) −1 | | logq(m−1)− 1 2 ≤ u ≤ logq(µ−1) + 1 2 } , where the second to last inequality is obtained since −i1 ≥ qu− 1 2 . we define f(u) = (m − 1)q ν 4 −u 2 + qu− 1 2 ( 1−q− 1 2 ) −1. in this way our bound becomes: mm(c1,c2) ≥ n−µ1 + min { f(u) | logq(m−1)− 1 2 ≤ u ≤ logq(µ−1) + 1 2 } . 42 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 by looking the derivative of f(u), one can see that f(u) only has a minimum at u∗ = 2 3 ( 1 + ν 4 + logq ( m−1 2( √ q−1) )) . however, it does not always hold that logq(m − 1) − 1/2 ≤ u∗ ≤ logq(µ − 1) + 1/2. this happens when either u∗ < logq(m − 1) − 1 2 or u∗ > logq(µ − 1) + 1 2 . the first case is equivalent to m > 1 4 q ν−5 2 ( √ q−1)−2 + 1, the second one to m > 2q− ν+1 4 ( √ q−1)(µ−1) 1 2 + 1. thus if m > β = min{2q− ν+1 4 ( √ q−1)(µ−1) 3 2 + 1, 1 4 q ν−5 2 ( √ q−1)−2 + 1}, then the minimum is reached in logq(m−1)− 1 2 or logq(µ−1) + 1 2 . the previous result has an asymptotic implication as well. corollary 2.3. let q be an even power of a prime, 0 ≤ r̃2 ≤ r̃1 < 1, and r̃ = r̃1 − r̃2 < 1√q−1. there exists a sequence of pairs of one-point ag codes c2,i = cl(di,µ2,iq) ( c1,i = cl(di,µ1,iq), such that: ni = n(c2,i) = n(c1,i) →∞, µj,i/ni → r̃j when i →∞, for j = 1,2. for a given ρ let mi be such that mi/ni → ρ when i →∞ and let m = lim inf mmi(c1,i,c2,i)/ni. it holds that: m ≥ 1− r̃1 + g(ρ), where g(ρ) =   minw∈{ρ,r̃} { ρ(w(q − √ q))− 1 2 + w q (q − √ q) } if ρ > β,( 2ρ2 q )1 3 + 1√ q ( ρ 2 )2 3 if ρ ≤ β,ρ 6= 0, 0 if ρ = 0, and β = min { 1 4 q− 5 2 ( √ q −1)−3,2q− 1 4 (r √ q −r) 3 2 } . proof. consider the garcia-stichtenoth’s tower (f1,f2, . . .) over fq described at the end of section 1, and 0 ≤ µ2,i < µ1,i ≤ ni − 1 with µj,i/ni → r̃j for j = 1,2. now consider cj,i = cl(di,µj,iq) for j = 1,2, where di is a divisor of degree ni − 1 and with ni − 1 distinct places not containing qi, which is the unique pole of x1 ∈ fi. by taking the limit of the bound obtained in theorem 2.2, the corollary holds. note that if we assume that c2,i is the zero code for all i, then lim inf mmi(c1,i,{~0}) is the asymptotic value of the mi-th general hamming weight of ci,1. for r̃ < 14(q−√q), the bound in corollary 2.3 is sharper than the one obtained in [5, theorem 23]. in figure 1 we compare the bound from corollary 1.7 (the dashed curve) with the bound from corollary 2.3 (the solid curve). the first axis represents ρ = limmi/ni, and the second axis represents δ = lim inf mmi(c1,i,{~0}). 3. the highest rghw as we illustrated at the beginning of section 1, for any n − m`(c1,c2) + 1 obtained shares an eavesdropper may recover at least one q-bit of the secret. in this section, for 2g − 1 ≤ µ2 < µ1 ≤ n− 1, we obtain a refined bound for the highest rghw of two one-point algebraic geometric codes obtained from garcia-stichtenoth’s towers, i.e. m`(c1,c2). proposition 3.1. let ν be an even positive integer and 2g−1 ≤ µ2 < µ1 ≤ n−1. consider two one-point algebraic geometric codes c2 = cl(d,µ2q) ( c1 = cl(d,µ1q) built on the ν-th garcia-stichtenoth’s tower. we have that µ = µ1 −µ2 = dim(c1)−dim(c2) = ` and mµ(c1,c2) ≥ n−dimc2 if µ ≥ q ν−1 2 , 43 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 figure 1. mµ(c1,c2) ≥ n−dimc2 − ( q ν−1 2 bν+1 2 −logq(µ)c−1∑ i=1 (q1− i 2 −q− i 2 )+ +(q ν+1 2 −bν+1 2 −logq(µ)c −µ)q bν+1 2 −logq(µ)c 2 ) if µ < q ν−1 2 . proof. since 2g − 1 ≤ µ2 < µ1 ≤ n − 1, then µ = µ1 − µ2 = ` [5, lemma 12]. by theorem 1.2, we have that mµ(c1,c2) ≥ n−µ1 + z(h(q),µ,µ), where z(h(q),µ,µ) = #{α ∈∪µ−1v=1(−v + h(q)) | α /∈ h(q)}. for any x,y ∈ h(q), we have that |x − y| ≤ q ν−1 2 , thus if µ ≥ q ν−1 2 then (∪µ−1v=0 − v + h(q)) = n0 ∪{−1, . . . ,−(µ−1)}. it follows that z(h(q),µ,µ) = (n0\h(qν))∪{−1, . . . ,−(µ−1)}) = g +µ−1. thus mµ(c1,c2) ≥ n−µ1 + g + µ− 1. moreover since µ2 ≥ 2g − 1, then µ2 −g + 1 = dimc2 and the first part of the proposition holds. for ` ≤ q ν−1 2 we claim that: z(h(q),µ,µ) = µ + g −1− (q ν−1 2 u1(µ)−1∑ i=1 (q1− i 2 + q− i 2 ) + u2(µ)q u1(µ) 2 ), where u1(µ) = ⌊ ν+1 2 − logq(µ) ⌋ and u2(µ) = q ν−2u1(µ)+1 2 −µ. this means that µ = q ν+1 2 −u1(µ) −u2(µ). we prove it by decreasing induction on µ, for q ν−1 2 ≥ µ ≥ 1. for the basis step we have µ = q ν−1 2 , thus u1(µ) = 1 and u2(µ) = 0. according to our claim, z(h(q),µ,µ) is equal to µ + g − 1, which has been already proven in the first part of this proposition. for the inductive step, we now assume that our claim is true for z(h(q),µ,µ) and we want to prove it for z(h(q),µ−1,µ−1). we note that: z(h(q),µ,µ) = #{α ∈∪µ−2v=1(−v + h(q)) | α /∈ h(q)}+ #{α ∈−(µ−1) + h(q)) | α /∈∪µ−2v=0(−v + h(q))} = z(h(q),µ−1,µ−1) + #t(µ), where t(µ) = {α ∈ −(µ − 1) + h(q) | α /∈ ∪µ−2v=0(−v + h(q))}. thus z(h(q),µ − 1,µ − 1) = z(h(q),µ,µ) − #t(µ). we consider two cases: µ such that q ν−2u1(µ−1)−1 2 < µ− 1 < q ν−2u1(µ−1)+1 2 and µ−1 = q ν−2u1(µ−1)+1 2 −1. 44 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 let us consider the first case, µ such that q ν−2u1(µ−1)−1 2 < µ−1 < q ν−2u1(µ−1)+1 2 , then u1(µ−1) = u1(µ) and u2(µ−1) = u2(µ) + 1. by induction we have that z(h(q),µ,µ) = µ + g−1−(q ν−1 2 ∑u1(µ)−1 i=1 (q 1− i 2 −q− i 2 ) + u2(µ)q u1(µ) 2 ). we claim that #t(µ) = q u1(µ) 2 . by (5) in lemma 1.6, for any x,y ∈ ∪u1(µ−1)i=0 s i ν, x > y we have that x−y ≥ q ν−2u1(µ−1)+1 2 , moreover µ−1 < q ν−2u1(µ−1)+1 2 . therefore, one has that (−(µ−1) +∪u1(µ−1)i=0 s i ν)∩ (∪µ−2v=0 − v + h(q)) = ∅. then −(µ − 1) + ∪ u1(µ−1) i=0 s i ν ⊆ t(µ). actually, the previous inclusion is an equality. we shall prove it by contradiction: we assume that there exists an element x ∈ t(µ) but not in −(µ−1) +∪u1(µ−1)i=0 s i ν. by definition of t(µ), we have that x ∈−(µ−1) + (s∞ν ∪ (∪ j/2 i=u1(µ−1)+1 siν)) where j = 2bν/2c. consider y < x to be the previous element of x in −(µ−1) + h(q). by (3) and (4) in lemma 1.6, we have that x−y ≤ q ν−2u1(µ−1)−1 2 < µ−1. thus, −(µ−2) ≤−(µ−1) + (x−y) ≤ 0 and x ∈−(µ−1) + (x−y) + h(q) ⊆∪µ−2v=0(−v + h(q)). this means x /∈ t(µ), which is a contradiction. it follows that #t(µ) = # ( −(µ−1)+∪u1(µ−1)i=0 s i ν ) = # ( ∪u1(µ−1)i=0 s i ν ) = q u1(µ−1) 2 = q u1(µ) 2 . we consider now the second case, µ−1 = q ν−2u1(µ−1)+1 2 −1, then u1(µ−1) = u1(µ)+1 and u2(µ−1) = 0. by using the same argument as in the first case, one may also prove that #t(µ) = q u1(µ) 2 . corollary 3.2. by using the same notation of the previous proposition, for 2g ≤ µ2 < µ1 < n− 1 and ` ≥ q ν−1 2 we have that m`(c1,c2) = n−dimc2. proof. by 3.1, m`(c1,c2) ≥ n− dimc2. and m`(c1,c2) ≤ n− dimc2, by the singleton bound for one-point algebraic geometric codes and the result holds. this means that for ` ≥ q ν−1 2 the singleton bound is reached. note that for ` < q ν−1 2 , the bound in proposition 3.1 allows us to get a refined bound since we could consider hc−m. as before, this result has an asymptotically implication: corollary 3.3. let q be an even power of a prime, 2√ q−1 ≤ r̃2 ≤ r̃1 < 1, and r̃ = r̃1 − r̃2. there exists a sequence of one-point algebraic geometric codes c2,i = cl(di,µ2,iq) ( c1,i = cl(di,µ1,iq), µi = µ1,i − µ2,i, such that: ni = n(c2,i) = n(c1,i) → ∞, µj,i/ni → r̃j when i → ∞, for j = 1,2. let `i = dimc1,i −dimc2,i, m = lim inf m`i(c1,i,c2,i)/ni, rj = lim dim ci,j ni for j = 1,2, and r = r1 −r2, we have that: m = 1−r2 if r ≥ 1 q − √ q and m ≥ 1−r2 − ( 1 q − √ q (−blogq(r(1− 1√q ))c−1∑ i=1 (q1− i 2 −q− i 2 )+ +q 1+ 1 2 blogq(r(1− 1√ q ))c ) + rq −1 2 blogq(r(1− 1√ q ))c ) if r < 1 q − √ q . 45 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 proof. let (f1,f2, . . .) be the tower of function fields defined in section 1, and 0 ≤ µ2,i < µ1,i ≤ ni−1 with µj,i/ni → r̃j for j = 1,2, where ni is the length of rational places of fi. now consider cj,i = cl(di,µj,iq) for j = 1,2, where di is a divisor of degree ni−1, with ni−1 distinct places not containing q, which is the unique pole of x1 ∈fi. since we assume that 2√q−1 ≤ r̃2 ≤ r̃1 < 1 then rj = r̃j − 1√q−1, for j = 1,2 and r = r̃. by taking the limit of the result obtained in proposition 3.1 and corollary 3.2 the result holds. note that if blogq(r(1− 1√ q ))c = logq(r(1− 1√ q )) then the formulas in corollary 3.3 become: m = 1−r1 if r ≥ 1 q − √ q and m ≥ 1−r1 − 1 q − √ q − logq(r(1− 1√ q ))−1∑ i=1 ( q1− i 2 −q− i 2 ) if r < 1 q − √ q . corollary 1.7 can be used for ρ ≤ min{r, 1√ q−1}. if c2,i = {0} for all i, then the value m of corollary 3.3 represents the asymptotic value of the highest ghw of ci,1. note that corollary 3.3 can be used for any value of r, but 1.7 cannot. references [1] m. bras–amorós, m. o’sullivan, the order bound on the minimum distance of the one–point codes associated to the garcia–stichtenoth tower, ieee trans. inform. theory 53(11) (2007) 4241–4245. [2] h. chen, r. cramer, s. goldwasser, r. de haan, v. vaikuntanathan, secure computation from random error correcting codes, in advances in cryptology—eurocrypt 2007, lecture notes in comput. sci. 4515 (2007) 291–310. [3] a. garcia, h. stichtenoth, on the asymptotic behaviour of some towers of function fields over finite fields, j. number theory 61(2) (1996) 248–273. [4] o. geil, s. martin, relative generalized hamming weights of q−ary reed–muller codes, to appear in adv. appl. commun. [5] o. geil, s. martin, u. martínez-peñas, r. matsumoto, d. ruano, asymptotically good ramp secret sharing schemes, arxiv preprint arxiv:1502.05507, 2015. [6] o. geil, s. martin, r. matsumoto, d. ruano, y. luo, relative generalized hamming weights of one–point algebraic geometric codes, ieee trans. inform. theory 60(10) (2014) 5938–5949. 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[12] v. k. wei, generalized hamming weights for linear codes, ieee trans. inform. theory 37(5) (1991) 1412–1418. 46 http://dx.doi.org/10.1109/tit.2007.907522 http://dx.doi.org/10.1109/tit.2007.907522 http://link.springer.com/chapter/10.1007%2f978-3-540-72540-4_17 http://link.springer.com/chapter/10.1007%2f978-3-540-72540-4_17 http://link.springer.com/chapter/10.1007%2f978-3-540-72540-4_17 http://dx.doi.org/10.1006/jnth.1996.0147 http://dx.doi.org/10.1006/jnth.1996.0147 https://arxiv.org/abs/1407.6185 https://arxiv.org/abs/1407.6185 http://dx.doi.org/10.1109/tit.2014.2345375 http://dx.doi.org/10.1109/tit.2014.2345375 http://doi.org/10.1587/transfun.e95.a.2067 http://doi.org/10.1587/transfun.e95.a.2067 http://doi.org/10.1587/transfun.e95.a.2067 http://dx.doi.org/10.1007/s10623-008-9170-1 http://dx.doi.org/10.1007/s10623-008-9170-1 http://dx.doi.org/10.1109/tit.2004.842763 http://dx.doi.org/10.1109/tit.2004.842763 http://dx.doi.org/10.1006/ffta.1998.0217 http://dx.doi.org/10.1006/ffta.1998.0217 http://dx.doi.org/10.1109/18.476213 http://dx.doi.org/10.1109/18.476213 http://dx.doi.org/10.1109/18.133259 http://dx.doi.org/10.1109/18.133259 o. geil et al. / j. algebra comb. discrete appl. 4(1) (2017) 37–47 [13] j. zhang k. feng, relative generalized hamming weights of cyclic codes, arxiv preprint arxiv:1505.07277, 2015. 47 introduction small codimension the highest rghw references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.22530 j. algebra comb. discrete appl. 3(3) • 159–164 received: 07 december 2015 accepted: 19 march 2016 journal of algebra combinatorics discrete structures and applications regular handicap tournaments of high degree research article dalibor froncek, aaron shepanik abstract: a handicap distance antimagic labeling of a graph g = (v, e) with n vertices is a bijection f : v → {1, 2, . . . , n} with the property that f(xi) = i and the sequence of the weights w(x1), w(x2), . . . , w(xn) (where w(xi) = ∑ xj∈n(xi) f(xj)) forms an increasing arithmetic progression with difference one. a graph g is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling. we construct (n−7)-regular handicap distance antimagic graphs for every order n ≡ 2 (mod 4) with a few small exceptions. this result complements results by kovář, kovářová, and krajc [p. kovář, t. kovářová, b. krajc, on handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than n−7. 2010 msc: 05c78 keywords: incomplete tournaments, handicap tournaments, distance magic labeling, handicap labeling 1. motivation the study of handicap distance antimagic graphs has been motivated by incomplete round-robin type tournaments with various properties. a complete round robin tournament of n teams is a tournament in which every team plays the remaining n−1 teams. when the teams are ranked 1, 2, . . . ,n according to their strength, it is apparent that the sum of rankings of all opponents of the i-th ranked team, denoted w(i), is w(i) = n(n + 1)/2−i, and the sequence w(1),w(2), . . . ,w(n) is a decreasing arithmetic progression with difference one. because complete round robin tournaments are generally considered to be fair, a tournament of n teams in which every team plays precisely r opponents, where r < n − 1 and the sequence w(1),w(2), . . . ,w(n) is a decreasing arithmetic progression with difference one is called a fair incomplete round robin tournament. a disadvantage of such a tournament is that the best team plays the weakest opponents, while the weakest team plays the strongest opponents. this disadvantage is eliminated in equalized incomplete round robin tournaments in which the sum of rankings of all opponents of every team is the same. some results on fair incomplete round robin tournaments can be found in [6] and [3]. dalibor froncek (corresponding author), aaron shepanik; department of mathematics and statistics, university of minnesota duluth, usa (email: dfroncek@d.umn.edu, shepa107@d.umn.edu). 159 d. froncek, a. shepanik / j. algebra comb. discrete appl. 3(3) (2016) 159–164 however, if we want to give the weaker teams a better chance of winning, the weakest team should play the weakest opponents, while the strongest one should play the strongest opponents. that is, the sequence w(1),w(2), . . . ,w(n) should be an increasing arithmetic progression. a tournament in which this condition is satisfied, and every team plays r < n− 1 games, is called a handicap incomplete round robin tournament. the existence of such tournaments with n ≡ 0 (mod 4) is studied by the authors in [d. froncek and a. shepanik, handicap incomplete tournaments of order n ≡ 0 (mod 4), manuscript, personal communication, 2016], kovář [p. kovář, on regular handicap graphs, personal communication, june 16, 2016] and kovářová [t. kovářová, on regular handicap graphs, personal communication, june 16, 2016]. kovář, kovářová, and krajc [p. kovář, t. kovářová, b. krajc, on handicap labeling of regular graphs, manuscript, personal communication, 2016] found such tournaments for n ≡ 2 (mod 4) and r ≤ n− 11 and proved that they can exist only when r is odd and at most n − 7. we provide a construction of handicap incomplete round robin tournaments for n ≡ 2 (mod 4) and the missing regularity r = n− 7 with a few small exceptions. 2. basic notions by a graph g = (v,e) we mean a finite undirected graph without loops or multiple edges. for graph theoretic terminology we refer to chartrand and lesniak [2]. motivated by properties of magic squares, vilfred [12] introduced the concept of sigma labelings. the same concept was introduced by miller et al. [10] under the name 1-vertex magic vertex labeling. sugeng et al. [11] introduced the term distance magic labeling, which currently seems to be most commonly used. a survey on distance magic graphs was published recently [1]. many newer results can by found in an extensive survey with much wider focus by gallian [7]. definition 2.1. a distance magic labeling of a graph g of order n is a bijection f : v → {1, 2, . . . ,n} with the property that there is a positive integer µ such that ∑ y∈n(x) f(y) = µ for every x ∈ v. the constant µ is called the magic constant of the labeling f. the sum ∑ y∈n(x) f(y) is called the weight of the vertex x and is denoted by w(x). when we think of the vertices as of teams and identify their labels with their rankings, we can see that a distance magic graph is providing a structure of a fair incomplete tournament described above. in [4] the first author introduced two closely related concepts, namely the distance antimagic and handicap distance antimagic labelings (which was called an ordered distance antimagic labeling in that paper) and showed their relationship to certain types of incomplete round robin tournaments. the term “handicap distance antimagic labeling” was originally coined by kovářová [t. kovářová, on regular handicap graphs, personal communication, june 16, 2016]. definition 2.2. a distance d-antimagic labeling of a graph g = (v,e) with n vertices is a bijection f : v → {1, 2, . . . ,n} with the property that there exists an ordering of the vertices of g such that the sequence of the weights w(x1),w(x2), . . . ,w(xn) forms an arithmetic progression with difference d. when d = 1, then f is called just distance antimagic labeling. a graph g is a distance d-antimagic graph if it allows a distance d-antimagic labeling, and a distance antimagic graph when d = 1. it should be obvious that a graph g is distance magic if and only if its complement g is distance antimagic. in distance antimagic graphs the weight of a vertex is not tied to its own label. all that we require is that the sequence w(x1),w(x2), . . . ,w(xn) forms an arithmetic progression. we now impose an additional condition on the labeling and require that a vertex with a lower label has a lower weight than a vertex with a higher label. 160 d. froncek, a. shepanik / j. algebra comb. discrete appl. 3(3) (2016) 159–164 definition 2.3. a handicap distance d-antimagic labeling of a graph g = (v,e) with n vertices is a bijection f : v → {1, 2, . . . ,n} with the property that f(xi) = i and the sequence of the weights w(x1),w(x2), . . . ,w(xn) forms an increasing arithmetic progression with difference d. when d = 1, the labeling is called just a handicap distance antimagic labeling (or a handicap labeling for short). a graph g is a handicap distance d-antimagic graph if it allows a handicap distance d-antimagic labeling, and a handicap distance antimagic graph or a handicap graph when d = 1. again, if we identify each team in a tournament with its ranking, then an r-regular handicap distance d-antimagic graph is nothing else than a model of a handicap incomplete round robin tournament, since the sum of rankings of opponents of team i is its weight w(i) and the sequence of weights is an increasing arithmetic progression. our constructions will be based on the properties of magic rectangles, which are a generalization of the magic squares mentioned above. definition 2.4. a magic rectangle mr(a,b) is an a × b array whose entries are 1, 2, . . . ,ab, each appearing once, with all row sums equal to a constant ρ and all column sums equal to a constant σ. it is easy to observe that a and b must be either both even or both odd. the following existence result was proved by harmuth [8, 9] more than 130 years ago. theorem 2.5. [8, 9] a magic rectangle mr(a,b) exists if and only if a,b > 1, ab > 4, and a ≡ b (mod 2). 3. known results kovář, kovářová, and krajc [p. kovář, t. kovářová, b. krajc, on handicap labeling of regular graphs, manuscript, personal communication, 2016], kovář [p. kovář, on regular handicap graphs, personal communication, june 16, 2016] and kovářová [t. kovářová, on regular handicap graphs, personal communication, june 16, 2016] proved the following results. theorem 3.1. let g be an r-regular handicap graph on n vertices, where n ≡ 2 (mod 4). then r ≡ 3 (mod 4) and r ≤ n− 7. theorem 3.2. for n ≡ 2 (mod 4), there exists an r-regular handicap graph if 3 ≤ r ≤ n− 11 and r ≡ 3 (mod 4) except when r = 3 and n ≤ 26. their result leaves open the case of n ≡ 2 (mod 4) and r = n−7 for n ≥ 14. in the following section, we prove the existence of such graphs with the exception of n = 14, 18, 22, 26, 34, 38, which remain in doubt. in our constructions, we will also use the following result by kovář [p. kovář, on regular handicap graphs, personal communication, june 16, 2016] and kovářová [t. kovářová, on regular handicap graphs, personal communication, june 16, 2016]. theorem 3.3. for n ≡ 0 (mod 4) there exists an (n− 7)-regular handicap graph whenever n ≥ 16. in [5] the first author made made an observation, which was a special case of the following. observation 3.4. let g be an r-regular distance 2-antimagic graph with vertices x1,x2, . . . ,xn, labeling f and weight function w such that f(xi) = i and w(xi) = k − 2i for some constant k. then g, the complement of g, is an (n− r − 1)-regular handicap graph with labeling f and weight function w such that w(xi) = n(n + 1)/2 −k + i. the converse is obviously also true. proof. label the vertices of the complete graph kn so that vertex xi is labelled i. the sum of labels of all neighbors of xi is then indeed equal to n(n + 1)/2− i. every neighbor of xi contributes its label to 161 d. froncek, a. shepanik / j. algebra comb. discrete appl. 3(3) (2016) 159–164 either w(xi) or w(xi). therefore, we have w(xi) + w(xi) = n(n + 1)/2 − i and w(xi) = n(n + 1)/2 − i−w(xi). because w(xi) = k − 2i, it follows that w(xi) = n(n + 1)/2 − i− (k − 2i) = n(n + 1)/2 −k + i. this completes the proof. we will use the observation in our constructions and instead of constructing directly (n−7)-regular handicap graphs, we will construct 6-regular distance 2-antimagic graphs satisfying assumptions of observation 3.4. we will call such graphs genuine distance 2-antimagic graphs and the labeling will be called a genuine distance 2-antimagic labeling. the following observation was proved in a more general form in [4]. to avoid introduction of a new notion that would be only used in its special form, we state the observation as follows. observation 3.5. [4] the graph g = ka�kb admits a genuine distance 2-antimagic labeling f such that f(x) = p implies w(x) = (a + b)(ab + 1)/2 − 2p for every x ∈ v (g) whenever there exists a magic rectangle mr(a,b). 4. new results lemma 4.1. there exists a 6-regular genuine distance 2-antimagic graph on 12 vertices. proof. let g = k2�k6. by theorem 2.5 there exist a magic rectangle mr(2, 6). the assertion follows directly from observation 3.5. lemma 4.2. there exists a 6-regular genuine distance 2-antimagic graph h on 30 vertices. proof. we denote the vertices of h by yst and zst for 1 ≤ s ≤ 3 and 1 ≤ t ≤ 5. the edge set will consist of edges ystypt, zstzpt, and ystzsq for every for every 1 ≤ s ≤ p ≤ 3 and 1 ≤ t ≤ 5 with t 6= q. let mr(3, 5) be a 3 × 5 magic rectangle with entries mst for 1 ≤ r ≤ 3 and 1 ≤ s ≤ 5. we label the vertices yst by entries mst in the natural way, that is, f(yst) = mst while the vertices zst obtain the labels raised by 15, that is, f(zst) = mst + 15. notice that the vertices yst have the highest weights. we now rename the vertices so that vertex yst or zst becomes xi when w(yst) = 124−2i or w(zst) = 124 − 2i, respectively. one can check that this labeling has the required property and h is a 6-regular genuine distance 2-antimagic graph. now we are ready to present our construction. lemma 4.3. there exists a 6-regular genuine distance 2-antimagic graph g on n vertices for every n ≡ 2 (mod 4) and n ≥ 42. proof. we use two building blocks, graph h on 30 vertices from the previous lemma, and a graph j on m ≡ 0 (mod 4) vertices whose existence is guaranteed by lemma 4.1 and theorem 3.3. our graph g will then have n = m + 30 vertices. let h be the graph on 30 vertices constructed in lemma 4.2 with vertices x1,x2, . . . ,x30, genuine distance 2-antimagic labeling fh and vertex weights wh satisfying fh (xi) = i and wh (xi) = 124 − 2i. 162 d. froncek, a. shepanik / j. algebra comb. discrete appl. 3(3) (2016) 159–164 it follows from theorem 3.3 and observation 3.4 that for any m ≡ 0 (mod 4) there exists a genuine distance 2-antimagic labeling graph j with vertices u1,u2, . . . ,um, labeling fj and vertex weights wj satisfying fj (uj) = j and wj (uj) = 4m− 2j. we use h and j as components of g and rename the vertices so that xi becomes vi for i = 1, 2, . . . , 15, uj becomes vj+15 for j = 1, 2, . . . ,m and xi becomes vi+m for i = 16, 17, . . . , 30. then we label all vertices using labeling function fg as fg(vi) = i to obtain the desired genuine distance 2-antimagic labeling. we can check that for i = 1, 2, . . . , 15 we have wg(vi) = wh (xi) + 4m = 124 − 2i + 4m = 4(m + 30) + 4 − 2i = 4n + 4 − 2i, because vi has two neighbors in the “lower” part of h (that is, among vertices v1,v2, . . . ,v15) where the labels have not changed, and four neighbors in the “upper” part (among vertices vm+16,vm+17, . . . ,vm+30), where each label was increased by m. for i = 16, 17, . . . ,m + 15 we have wg(vi) = wj (ui−15) + 90 = 4m + 4 − 2(i− 15) + 90 = 4(m + 30) + 4 − 2i = 4n + 4 − 2i, because the labels of all six neighbors were increased by 15. finally, for i = m + 16,m + 17, . . . ,m + 30 we have wg(vi) = wh (xi−m) + 2m = 124 − 2(i−m) + 2m = 4(m + 30) + 4 − 2i = 4n + 4 − 2i, because vi has four neighbors in the “lower” part of h where the labels have not changed, and two neighbors in the “upper” part where each label was increased by m. our main result now follows directly from lemma 4.3 and observation 3.4. theorem 4.4. for n ≡ 2 (mod 4), there exists an (n − 7)-regular handicap graph when n = 30 or n ≥ 42. this, together with the result by kovář [p. kovář, on regular handicap graphs, personal communication, june 16, 2016] and kovářová [t. kovářová, on regular handicap graphs, personal communication, june 16, 2016], gives an almost complete characterization of handicap graphs for n ≡ 2 (mod 4). theorem 4.5. for n ≡ 2 (mod 4), there exists an r-regular handicap graph if and only if 3 ≤ r ≤ n− 7 and r ≡ 3 (mod 4) except when r = 3 and n ≤ 26 and possibly when r = n − 7 and n ∈ {14, 18, 22, 26, 34, 38}. since no magic rectangles of orders 14, 22, 26, 34, or 38 exist, one has to hope that a computer aided search would help to settle the existence question for these orders. a construction for n = 18 similar to that for n = 30 may exist, but the authors were unable to find one. references [1] s. arumugam, d. froncek, n. kamatchi, distance magic graphs – a survey, j. indones. math. soc. special edition (2011) 1–9. [2] g. chartrand, l. lesniak, graphs and digraphs, chapman and hall, crc, fourth edition, 2005. [3] d. froncek, fair incomplete tournaments with odd number of teams and large number of games, congr. numer. 187 (2007) 83–89. [4] d. froncek, handicap distance antimagic graphs and incomplete tournaments, akce int. j. graphs comb. 10(2) (2013) 119–127. [5] d. froncek, handicap incomplete tournaments and ordered distance antimagic graphs, congr. numer. 217 (2013) 93–99. 163 http://www.ams.org/mathscinet-getitem?mr=3082443 http://www.ams.org/mathscinet-getitem?mr=3082443 http://www.ams.org/mathscinet-getitem?mr=2408780 http://www.ams.org/mathscinet-getitem?mr=2408780 http://www.ams.org/mathscinet-getitem?mr=3114157 http://www.ams.org/mathscinet-getitem?mr=3114157 http://www.ams.org/mathscinet-getitem?mr=3220296 http://www.ams.org/mathscinet-getitem?mr=3220296 d. froncek, a. shepanik / j. algebra comb. discrete appl. 3(3) (2016) 159–164 [6] d. froncek, p. kovář, t. kovářová, fair incomplete tournaments, bull. inst. combin. appl. 48 (2006) 31–33. [7] j. gallian, a dynamic survey of graph labeling, electron. j. combin. 5 (# ds6) (2015) 43 pp. [8] t. harmuth, ueber magische quadrate und ähnliche zahlenfiguren, arch. math. phys. 66 (1881) 286–313. [9] t. harmuth, ueber magische rechtecke mit ungeraden seitenzahlen, arch. math. phys. 66 (1881) 413–447. [10] m. miller, c. rodger, r. simanjuntak, distance magic labelings of graphs, australas. j. combin. 28 (2003) 305–315. [11] k. a. sugeng, d. froncek, m. miller, j. ryan, j. walker, on distance magic labeling of graphs, j. combin. math. combin. comput. 71 (2009) 39–48. [12] v. vilfred, σ-labelled graph and circulant graphs, ph.d. thesis, university of kerala, trivandrum, india, 1994. 164 http://www.ams.org/mathscinet-getitem?mr=2259699 http://www.ams.org/mathscinet-getitem?mr=2259699 http://www.ams.org/mathscinet-getitem?mr=1668059 http://www.ams.org/mathscinet-getitem?mr=1999203 http://www.ams.org/mathscinet-getitem?mr=1999203 http://www.ams.org/mathscinet-getitem?mr=2568905 http://www.ams.org/mathscinet-getitem?mr=2568905 motivation basic notions known results new results references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(1) • 39-63 received: 20 october 2014; accepted: 6 december 2014 doi 10.13069/jacodesmath.36866 journal of algebra combinatorics discrete structures and applications recent progress on weight distributions of cyclic codes over finite fields∗ research article hai q. dinh1∗∗, chengju li2§, qin yue2∗∗∗ 1. departments of mathematical sciences, kent state university, warren, oh 44484, usa 2. department of mathematics, nanjing university of aeronautics and astronautics, nanjing, 211100, p.r. china abstract: cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. in coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. in this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. the cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions. 2010 msc: 94b15, 11t71, 11t24 keywords: linear codes, weight distribution, cyclic codes, finite fields, gauss periods, gauss sums, exponential sums, quadratic forms 1. introduction the classes of cyclic codes play a very significant role in the theory of error-correcting codes. cyclic codes can be efficiently encoded using shift registers, and they have rich algebraic structures for efficient error detection and correction, which explains their preferred role in engineering. information theory and coding theory have been widely considered to be born in 1948, when claude shannon’s1 landmark paper [75] on the mathematical theory of communication, showed that good codes exist2. cyclic codes were ∗ the second and third authors are supported by nnsf of china (no. 11171150) ∗∗ e-mail: hdinh@kent.edu § e-mail: lichengju1987@163.com ∗∗∗ e-mail: yueqin@nuaa.edu.cn 1 claude elwood shannon (april 30, 1916 february 24, 2001) was an american mathematician, electronic engineer, and cryptographer, who is refered to as "the father of information theory" [43]. shannon is also credited 39 weight distributions of cyclic codes over finite fields introduced as early as 1957, nine years after that, in a series of papers by prange [67]-[71]. since then, cyclic codes have been the most studied of all codes. many well known codes, such as bch, kerdock, golay, reed-muller, preparata, justesen, and binary hamming codes, are either cyclic codes or can be constructed from cyclic codes. in this paper, we survey some results on the weight distributions of cyclic codes over finite fields that have been recently determined by exponential sums. for a prime p, let fq be a finite field of characteristic p with q elements, i.e., q = ps, for some positive integer s. an [n,k,d] linear code c is a k-dimensional subspace of fnq with minimum distance d. hereafter, we always assume that the code length n and the field characteristic p are coprime 3. the code c is called cyclic if (c0,c1, . . . ,cn−1) ∈ c implies (cn−1,c0,c1, . . . ,cn−2) ∈c. by identifying the vector (c0,c1, . . . ,cn−1) ∈ fnq with c0 + c1x + c2x 2 + · · · + cn−1xn−1 ∈ fq[x]/(xn − 1), any code c of length n over fq corresponds to a subset of fq[x]/(xn−1). then c is a cyclic code if and only if the corresponding subset is an ideal of fq[x]/(xn−1). note that every ideal of fq[x]/(xn−1) is principal. hence there is a monic polynomial g(x) with least degree such that c = 〈g(x)〉 and g(x) | (xn−1). then g(x) is called the generator polynomial and h(x) = (xn − 1)/g(x) is called the check polynomial of the cyclic code c. suppose that h(x) has t irreducible factors over fq, we call c the dual of the cyclic code with t zeros. let ai be the number of codewords with hamming weight i in the code c of length n. the weight enumerator of c is defined by 1 + a1x + a2x 2 + · · · + anxn. the sequence (1,a1,a2, . . . ,an) is called the weight distribution of the code c. in coding theory it is often desirable to know the weight distributions of the codes because they can be used to estimate the error correcting capability and the error probability of error detection and correction with respect to some decoding algorithms. this is quite useful in practice. unfortunately, it is a very hard problem in general and remains open for most cyclic codes. let r = qm for a positive integer m and α a generator of f∗r. let h(x) = h1(x)h2(x) · · ·ht(x), where hj(x) (1 ≤ j ≤ t) are distinct monic irreducible polynomials over fq. let gj = α−sj be a root of hj(x) and as the founder of both digital computer and digital circuit design theory, when, in 1937, as a 21-year-old master’s student at mit, he wrote a thesis establishing that electrical application of boolean algebra could construct and resolve any logical, numerical relationship. it has been claimed that this has been the most important master’s thesis of all time. shannon contributed to the field of cryptanalysis during world war ii and afterwards, including basic work on code breaking. 2 shannon’s theorem ensures that our hopes of getting the correct messages to the users will be fulfilled a certain percentage of the time. based on the characteristics of the communication channel, it is possible to build the right encoders and decoders so that this percentage, although not 100%, can be made as high as we desire. however, the proof of shannon’s theorem is probabilistic and only guarantees the exixtence of such good codes. no specific codes were constructed in the proof that provides the desired accuracy for a given channel. the main goal of coding theory is to establish good codes that fulfill the assertions of shannon’s theorem. during the last 50 years, while many good codes have been constructed, but only from 1993, with the introduction of turbo codes [7], the rediscoveries of ldpc codes, and the study of related codes and associated iterative decoding algorithms, researchers started to see codes that approach the expectation of shannon’s theorem in practice. 3 the case when the code length n is divisible by the characteristic p of the field yields the so-called repeated-root codes, which were first studied since 1967 by berman [5], and then in the 1970s and 1980s by several authors such as massey et al. [61], falkner et al. [33], roth and seroussi [72]. however, repeated-root codes were first investigated in the most generality in the 1990s by castagnoli et al. [16], and van lint [77], where they showed that repeated-root cyclic codes have a concatenated construction, and are asymptotically bad. to distinguish the two cases, codes when the code-length is not divisible by the characteristic p of the field are called simple-root codes. 40 h. q. dinh, c. li, q. yue nj the order of gj for 0 ≤ sj ≤ r − 2 (1 ≤ j ≤ t). let mj be the least positive integer such that qmj ≡ 1 (mod nj). in fact, we have deg(hj(x)) = mj for j = 1, 2, . . . , t. denote δ = gcd(r − 1,s1,s2, . . . ,st) and n = r−1 δ . a cyclic code c can be defined by c = {c(a1,a2, . . . ,at) : aj ∈ fqmj}, (1) where c(a1,a2, . . . ,at) = ( t∑ j=1 trqmj/q(aj), t∑ j=1 trqmj/q(ajgj), . . . , t∑ j=1 trqmj/q(ajg n−1 j )) (2) and trqmj/q denotes the trace function from fqmj to fq. it follows from delsarte’s theorem [22] that the code c is an [n,k] cyclic code over fq with the check polynomial h(x), where k = m1 + m2 + · · · + mt. in the rest of this paper, we use gi to denote the corresponding cyclic code. if we only give g1 and g2, we mean that the dual of cyclic code has two zeros and the product of the minimal polynomials of g1 and g2 over fq is the check polynomial of such cyclic code. it is similar for cyclic codes whose duals have more zeros. in most cases, we also only list the cyclic codes whose weight distributions are known because they may have many nonzero weights. the reader can get the details on weight distributions in the corresponding references which are given. for any a1,a2, . . . ,at ∈ fr, the hamming weight of c(a1,a2, . . . ,at) is equal to wh(c(a1,a2, . . . ,at)) = n−z(r,t), where z(r,t) = |{0 ≤ i ≤ n− 1 : t∑ j=1 trqmj/q(ajg i j) = 0}|. let φ be the canonical additive character of fq. then ψj = φ ◦ trqmj/q is the canonical additive character of fr. by the orthogonal property of additive characters we have z(r,t) = n−1∑ i=0 1 q ∑ y∈fq φ(y t∑ j=1 trqmj/q(ajg i j)) = 1 q t∑ j=1 n−1∑ i=0 ∑ y∈fq ψj(yajg i j). (3) hence determining the weight distribution of cyclic code is equivalent to determining the multiset {z(r,t) : aj ∈ fqmj for j = 1, 2, . . . t}. in general, it is very difficult and remains open for most cases. however, the weight distributions of cyclic codes had been determined in a few cases by using mathematical tools, such as gauss periods, gauss sums, quadratic forms, and the numbers of the solutions of equations over finite fields. in view of the trace representation (2) of c, it is natural to study the weight distributions of irreducible cyclic codes (i.e., t = 1) and the duals of cyclic codes with two or three zeros (i.e., t = 2 or 3). there are few results [53, 90] on weight distributions of cyclic codes with arbitrary zeros. moreover, ding and yang [27] used gauss periods to give an excellent survey on weight distributions of irreducible cyclic codes. in this paper, we mainly investigate the weight distributions of reducible cyclic codes which had been determined by exponential sums. the cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. the rest of this paper is organized as follows. in section 2, we study the weight distributions of cyclic codes whose duals have two or three zeros. in section 3, we present the results on weight distributions 41 weight distributions of cyclic codes over finite fields of cyclic codes whose duals have arbitrary zeros. in section 4, we investigate the cyclic codes with niho exponents. in section 5, the cyclic codes with few weights are discussed and their existence conditions are listed. section 6 discusses the more general case of constacyclic codes, we present some methods to study the equivalence classes of constacyclic codes. all constacyclic codes that are in the same equivalence class of cyclic codes share the same weight distributions and all results from previous sections hold for such constacyclic codes. it is impractical to mention all recent work on weight distributions of cyclic codes in this paper. we focus on the weight distributions determined by exponential sums and some results may be omitted. an apparent omission is the weight distributions determined by combinatorial methods. however, we hope that this paper will show that weight distributions of cyclic codes which are determined by exponential sums in general. 2. weights of the duals of cyclic codes with two or three zeros we begin with the weight distributions of cyclic codes whose duals have two or three zeros because ding and yang had given an elegant survey on irreducible cyclic codes. for details we refer the readers to [27] and the references therein. below we consider the cyclic codes whose duals have two zeros. the weight distributions of such codes are settled for a few special cases and is quite complex in general [17]. let g1, g2, and g3 be three zeros of h1(x),h2(x), and h3(x), respectively, and c the cyclic code as (1) with the check polynomial h(x) = h1(x)h2(x)h3(x). now we assume that m1 = m2 = m3 = m if we do not give a special statement and α is a primitive element of fqm. the weights of c were first studied in [14, 18, 81] by using exponential sums and combinatorial methods. yuan et al. [91] used exponential sums to present the weight distributions of cyclic codes from perfect nonlinear functions. we refer the reader to [15] for a survey of highly nonlinear functions. feng and luo [34] presented a unified way to determine the weight distributions of cylic codes defined by perfect nonlinear functions. theorem 2.1. the weight distributions of the following cyclic codes defined by perfect nonlinear functions are known: 1. g1 = α−1,g2 = α−(p l+1), where q = p, l ≥ 0 is an integer, and m/ gcd(m,l) is odd [35, 91]; 2. g1 = α−1,g2 = α− 3l+1 2 , where q = 3, l is odd, and gcd(m,l) = 1 [35]. remark 2.2. let the assumptions and the notations be as above theorem. then f(x) = xp l+1 is called dembowski-ostrom function [23] and f(x) = x 3l+1 2 is called coulter-matthews function [21]. 2.1. quadratic forms and weight distributions quadratic form is an effective tool to determine the weight distributions of cyclic codes. below we recall some results on quadratic forms. we refer the readers to [54] for more details on quadratic forms. let h be an m×m symmetric matrix over fp. by identifying fpm with fmp , a function q(x) from fpm to fp is called a quadratic form over fp if q(x) = xhx⊥, where x = (x1,x2, . . . ,xm) ∈ fmp . suppose that r = rank(h). then there exists m ∈ glm(fp) such that h′ = mhm⊥ is a diagonal matrix and h′ = diag(a1, . . . ,ar, 0, . . . , 0), where ai ∈ fp for 1 ≤ i ≤ r. let ∆ = a1 · · ·ar (∆ = 1 if r = 0). then we have the following proposition. 42 h. q. dinh, c. li, q. yue proposition 2.3. [35, 54] suppose that p is an odd prime. let ( ∆ p ) denote the legendre symbol and ζp = e 2π √ −1 p be a complex p-th root of unity. then we have ∑ x∈fmp ζxhx ⊥ p = { ( ∆ p )pm− r 2 , if p ≡ 1 (mod 4); ( ∆ p )( √ −1)rpm− r 2 , if p ≡ 3 (mod 4). by employing the above proposition, feng and luo [35, 55] presented the weight distributions of several classes of cyclic codes. since then a series of jobs were motivated by their original idea. theorem 2.4. let l be a positive integer. then the weight distributions of cyclic codes over fp (p odd prime) had been determined by using quadratic forms in the following cases: 1. g1 = α−2,g2 = α−(p l+1), where m ≥ 3 and gcd(m,l) = 1 [35]; 2. g1 = α−2,g2 = α−(p l+1), and g3 = α−1, where m ≥ 3 and gcd(m,l) = 1 [35]; 3. g1 = α−2,g2 = α−(p l+1), where m ≥ 2 and 1 ≤ l ≤ m− 1 [55]; 4. g1 = α−2,g2 = α−(p l+1), and g3 = α−1, where m ≥ 2 and 1 ≤ l ≤ m− 1 [55]; 5. g1 = α−1,g2 = α− pl+1 2 , where l/ gcd(m,l) is odd [58]; 6. g1 = α−1,g1 = α−(p l+1), and g3 = α−(p 3l+1), where m/ gcd(m,l) is odd [92]; 7. g1 = −α−1,g2 = α− pl+1 2 , where m/ gcd(m,l) ≥ 3 is odd [96]; 8. g1 = α−1,g2 = α− p2l+1 2 , and g3 = α− p4l+1 2 , where m ≥ 5 is odd and gcd(m,l) = 1 [98]; 9. g1 = α−(p l+1),g2 = α −(p3l+1), where m/ gcd(m,l) is even [95]; 10. g1 = α−1,g1 = α−(p l+1), and g3 = α−(p 3l+1), where m/ gcd(m,l) is even [95]; 11. g1 = α−2,g2 = α−(p 2l+1), and g3 = α−(p 4l+1), where m/ gcd(m,l) is odd [94]; 12. g1 = α−1,g2 = −α−1, and g3 = α− pl+1 2 , where m ≥ 3 is odd and gcd(m,l)=1 [57]; 13. g1 = α−2,g2 = α−4, and g3 = α−10, where p = 3 [56]. there is a parallel result on the exponential sums over quadratic forms for even p. luo et al. [59] investigated these exponential sums and gave the weight distributions of cyclic codes associated with generalized kasami sequences. theorem 2.5. [59] for even m, let l be an integer with 1 ≤ l ≤ m − 1 and l 6= m 2 . then the weight distributions of binary cyclic codes c are known in the following cases: 1. g1 = α−(2 m 2 +1) and g2 = α−(2 l+1); 2. g1 = α−(2 m 2 +1),g2 = α −(2l+1), and g3 = α−1. remark 2.6. in the above theorem, m1 = m2 ,m2 = m and m3 = m. for the binary cyclic codes whose duals have two zeros, the calculations of their weight distributions is more important because it is equivalent to determine the value distribution of the cross-correlation function between two m-sequences and the walsh transforms of monomials over finite fields. in fact, they represent the same mathematical problem (i.e., the calculation of exponential sum) in most cases. for more results of their relationships, cross-correlation function between two m-sequences, and the walsh transforms of monomials, we refer the readers to [13, 36, 38, 40, 41, 44, 51, 65, 93] and references therein. 43 weight distributions of cyclic codes over finite fields 2.2. gauss periods and weight distributions there is another useful tool which is called gauss periods to determine the weight distributions of cyclic codes. now we recall the definition of gauss period. let r − 1 = nn and α be a fixed primitive element of fr, where r = qm = psm. we define c (n,r) i = α i〈αn〉 for i = 0, 1, . . . ,n − 1, where 〈αn〉 denotes the subgroup of f∗r generated by αn. the gauss periods of order n are given by η (n,r) i = ∑ x∈c(n,r) i ψ(x), where ψ is the canonical additive character of fr and η (n,r) i = η (n,r) i (mod n) if i ≥ n. in general, the explicit evaluation of gauss periods is a very difficult problem. however, they can be computed in a few cases: n = 2, 3, 4, semi-primitive case, and index 2 case [27, 64]. by using these known gauss periods, the weight distributions of some classes of cyclic codes were determined. for future use, here we also introduce gauss sums which are closely related to gauss periods. let λ : f∗r → c ∗ be a multiplicative character of f∗r. now we define the gauss sum over fr by g(λ) = ∑ x∈f∗r λ(x)ψ(x). it is easy to see that g(λ0) = −1, where λ0 is the trivial multiplicative character, i.e., λ0(x) = 1 for all x ∈ f∗r. gauss sums can be viewed as the fourier coefficients in the fourier expansion of the restriction of ψ to f∗r in terms of the multiplicative characters of fr, i.e., ψ(x) = 1 r − 1 ∑ λ∈f̂∗r g(λ)λ(x), for x ∈ f∗r. (4) by (4), we can obtain η (n,r) i = 1 n n−1∑ j=0 ζ −ij n g(λ j) = 1 n (−1 + n−1∑ j=1 ζ −ij n g(λ j)), where λ is a primitive multiplicative character of order n over f∗r. generally, the explicit determination of gauss sums is a difficult problem. however, they can be explicitly evaluated in the following cases [6, 89]: quadratic gauss sums, semi-primitive gauss sums, and index 2 gauss sums. ding [24] used gauss periods to determine the weight distributions of irreducible cyclic codes. moreover, a survey on the weight distributions of irreducible cyclic codes determined by gauss periods was given by ding and yang [27]. below we consider the weight distributions of reducible cyclic codes. let m1 = m2 = m, r = qm, and α a generator of f∗r. let h be a positive factor of q − 1 and e > 1 an integer such that e | gcd(q − 1,hm). define g = α q−1 h ,n = h(r − 1) q − 1 ,β = α r−1 e ,n = gcd( r − 1 q − 1 , e(q − 1) h ). we easily see that the order of g is n and (βg)n = 1. it had been proved that the minimal polynomials of g−1 and (βg)−1 over fq are distinct except when q = 3,h = 1,e = m = 2 [87]. hence their product is a factor of xn − 1. let g1 = g−1 and g2 = (βg)−1. in general, we have m1 = m2 = m. thus the corresponding cyclic code c is an [n, 2m] code. if f∗r = 〈g1〉 = 〈g2〉, then the weight distribution of the code c which is 44 h. q. dinh, c. li, q. yue called the dual of primitive cyclic code with two zeros had been studied in [9, 13, 14, 18, 62, 63, 74]. in this subsection, we only consider cyclic codes whose weight distributions are determined by gauss periods, so we do not describe the results on primitive cyclic codes here. in fact, to determine the weight distributions of cyclic codes, more mathematical tools are employed, such as gauss sums, jacobi sums, and elliptic curves. theorem 2.7. let g1 = g−1 and g2 = (βg)−1. then the weight distribution of cyclic code c were determined by gauss periods in the following cases: 1. e > 1 and n = 1 [60]; 2. e = 2 and n = 2 [60]; 3. e = 2 and n = 3 [26]; 4. e = 2 and pj + 1 ≡ 0 (mod n) for some positive integer j [26]; 5. e = 3 and n = 2 [82]; 6. e = 2, n ≡ 3 (mod 4) is a prime, n−1 2 | sm, and p is of index 2 modulo n which means [z∗n : 〈p〉] = 2 and −1 6∈ 〈p〉, where 〈p〉 is a subgroup generated by p in z∗n [37]; 7. e = 4 and n = 2 [86]; 8. e = 3 and n = 3 [87]; 9. e = 2 and pj + 1 ≡ 0 (mod n) for some positive integer j [88]. in [79], vega presented an extended version for the class of cyclic codes studied by ma et al. [60] and gave their weight distributions. moreover, a general description for such reducible cyclic codes which generalizes the code c with e = 2 was given by vega and morales [80]. the weight distributions of these general cyclic codes were determined explicitly and the main tool is also gauss periods. theorem 2.8. [80] suppose that q is odd and sm is even. let d,a1,a2, and δ be four integers such that 2d | sm,a1 − a2 = ±r−12 , and δ = gcd( r−1 q−1,a1). let λ1,λ2 be two divisors of q − 1 such that gcd(q − 1, ai δ ) = q−1 λi for i = 1, 2. fix δ′ = gcd(2, ∆ δ ) and λ = max{λ1,λ2}. let g1 = α−a1 and g2 = α−a2. if δδ ′ | (pd + 1) and 2δ < r−1 pd1 , then 1. the corresponding cyclic code c is an [n, 2m] code with n = λ∆ δ and its weight distribution can be computed explicitly; 2. c is a projective linear code which means the minimum weight of its dual code is at least three if and only if δ′ = 1 and λ = 2. in this subsection, we have investigated the weight distributions of cyclic codes in the case m1 = m2 = m. now we concentrate on the case m1 6= m2. in [46, 47], the authors used gauss periods to express the weight distributions of such cyclic codes. moreover, a more general result on cyclic codes whose duals have two zeros was given in [48]. based on the expression via gauss periods, the weight distributions of several classes of cyclic codes were explicitly presented. let α be a fixed primitive element of fr and f∗qmi = 〈αi〉, where αi = α r−1 qmi−1 for i = 1, 2. denote qmi − 1 = nini, gi = α−nii , d = gcd(n1,n2), and n = n1n2 d . theorem 2.9. [48] if gcd(n1,n2) = d, n1n1 = qm1−1, n2n2 = qm2−1, m1 = q m1−1 q−1 ,m2 = qm2−1 q−1 ,d1 = gcd(m1,n1),d2 = gcd(m2,n2),d3 = gcd( m1dn2 d4 ,dn1), and d4 = gcd(m2,dn2), then the weight distribution of the cyclic code c with g1 = α−n11 and g2 = α −n2 2 is given by table 1. 45 weight distributions of cyclic codes over finite fields table 1. weight distribution of cyclic code c with g1 = α−n11 and g2 = α −n2 2 . weight frequency 0 1 (q−1)n1n2 dq − (q−1)d1n2 qdn1 η (d1,q m1 ) j qm1−1 d1 (0 ≤ j ≤ d1 −1) (q−1)n1n2 dq − (q−1)d2n1 qdn2 η (d2,q m2 ) j qm2−1 d2 (0 ≤ j ≤ d2 −1) (q−1)n1n2 dq − (q−1)d3d4 qd2n1n2 d−1∑ t=0 dn2 d4 −1∑ i=0 η (dn2,q m2 ) n2t+m2i+j η (d3,q m1 ) n1t+m1i+k (qm1−1)(qm2−1) dd3n2 ( 0 ≤ j ≤ dn2 −1 0 ≤ k ≤ d3 −1 ) table 2. weight distribution of c from two distinct finite fields. weight frequency 0 1 qm1−1(qm2 −1) qm1 −1 qm2−1(qm1 −1) qm2 −1 (qm1−1)(qm2−1) q − δ q δ−1∑ v=0 η (δ,qm1 ) v+k η (δ,qm2 ) v+j (qm1−1)(qm2−1) δ2 (0 ≤ k,j ≤ δ −1) as an application of theorem 2.9, the weight distribution of cyclic code from two distinct finite fields (i.e., n1 = n2 = 1) was presented. theorem 2.10. [48] let m1,m2 be two positive divisors of m with gcd(m1,m2) = 1, n1 = qm1 − 1, and n2 = q m2 −1. if gcd(q−1,m1 −m2) = δ, then c is a [ (qm1−1)(qm2−1) q−1 ,m1 +m2] cyclic code and its weight distribution is given by table 2. if gcd(m1,m2) = 1, then the weight distributions of the cyclic codes c can be explicitly determined when the gauss periods of order δ are known. corollary 2.11. [48] let r = qm, m1,m2 be two divisors of m with gcd(m1,m2) = 1. 1. if gcd(q − 1,m1 − m2) = 1, then the corresponding cyclic code c is a [ (qm1−1)(qm2−1) q−1 ,m1 + m2] three-weight cyclic code and its weight distribution can be explicitly determined. 2. if gcd(q − 1,m1 − m2) = 2, then the corresponding cyclic code c is a [ (qm1−1)(qm2−1) q−1 ,m1 + m2] four-weight cyclic code and its weight distribution can be explicitly determined. in particular, if (q − 1) | m1 or (q − 1) | m2, then we have δ = 1 by gcd(m1,m2) = 1. thus c is a three-weight cyclic code. moreover, if q = 2, then the corresponding code c is a three-weight binary cyclic code which is more interesting in communication and storage systems. if n1 = n2 = 2, then we have the following theorem. theorem 2.12. [48] let r = qm with q odd, m1,m2 be two divisors of m with gcd(m1,m2) = 1 and (q − 1) | m1 or (q − 1) | m2, n1 = q m1−1 2 , and n2 = qm2−1 2 . then the corresponding code c is a [ (qm1−1)(qm2−1) 2(q−1) ,m1 +m2] cyclic code with five nonzero weights and its weight distribution can be explicitly determined. more classes of cyclic codes can be presented by theorem 2.9 and it is unnecessary to state them here. we refer the readers to [46–48] for more results. in fact, the weight distributions of most cyclic codes whose duals have few zeros are open. moreover, zeta functions were also employed to determine the weight distributions of the duals of cyclic codes with 46 h. q. dinh, c. li, q. yue two zeros [9]. it is a good research problem to present the weight distributions of cyclic codes by using zeta functions, quadratic forms, gauss periods, or other mathematical tools. 3. weight distributions of cyclic codes with arbitrary zeros in this section, we survey the weight distributions of cyclic codes with arbitrary zeros. it is in general very difficult to compute z(r,t) if the dual of cyclic code has more zeros. hence there are few results on such cyclic codes. 3.1. hermitian forms graphs and weight distributions let g be a finite abelian group and d a subset of g. the cayley graph cay(g,d) on g with connection set d is the directed graph with vertex set g and edge set {(g,h) : g,h ∈ g,gh−1 ∈ d}. let a = (agh) with entries in {0, 1} be a square matrix such that agh = 1 if gh−1 ∈ d and agh = 0 otherwise. we call a the adjacency matrix of cay(g,d). it is known that each character χ of g corresponds to an eigenvector of a with eigenvalue χ(d) = ∑ d∈d χ(d) . furthermore, the spectrum of cay(g,d) is the multiset {χ(d) : χ ∈ ĝ}, where ĝ is the character group of g. in this subsection, we always suppose that m = 2l for some integer l and s = 1, i.e., q = p. a matrix h over fp2 is called hermitian if h = h∗, where h∗ is the conjugate transpose of h. let h denote the abelian group formed by all l×l hermitian matrices over fp2 under the matrix addition. the hermitian forms graph is the cayley graph cay(h,d), where d = {h ∈ h : rank(h) = 1}. let w = fl p2 . then the hermitian forms graph on w is the cayley graph cay(h,d). the eigenvalues of the hermitian forms graph were first computed by stanton [76] and a more accessible formula was given in [10] by using the gaussian binomial coefficients. for details and more information on the spectrum of hermitian forms graph, we refer the readers to [10]. li et al. [53] proposed an elegant method to study this problem by building a connection between the corresponding exponential sums and the spectra of hermitian forms graphs. for odd l, we denote t = l−1 2 . suppose that α is a primitive element of fr. let gj = α−(p 2i−1+1) for j = 1, 3, . . . , t and gt+1 = α−(p l+1). then we have m1 = m2 = · · · = mt = m and mt+1 = m2 and . theorem 3.1. [53] the corresponding code c is a [r− 1, m 2 4 ] cyclic code and its weight distribution can be exactly determined. very recently, zhou et al. [99] generalized this class of p-ary cyclic codes proposed in [53] and the weight distributions of the generalized cyclic codes were settled for both even l and odd l along with the idea of li, hu, feng, and ge. theorem 3.2. [99] let t = bm 2 c. then the weight distributions of the following cyclic codes over fq (q is a prime power here) are known: 1. gj = α−(p 2j−1+1) (j = 1, 2, . . . , t), gt+1 = α−(p l+1) for odd m; 2. gj = α−(p 2j−1+1) (j = 1, 2, . . . , t), gt+1 = α−(p l+1), and gt+2 = α−1 for odd m; 3. gj = α−(p 2j−1+1) (j = 1, 2, . . . , t) for even m; 4. gj = α−(p 2j−1+1) (j = 1, 2, . . . , t), gt+1 = α−1 for even m. 47 weight distributions of cyclic codes over finite fields 3.2. yang-xiong-ding-luo cyclic codes by yang-xiong-ding-luo cyclic codes we mean a class of cyclic codes with arbitrary number of zeros proposed in [90]. now we describe this class of cyclic codes. main assumptions: let r = qm = psm be a prime power for two integers s,m and let e ≥ t ≥ 2. assume that 1. a 6≡ 0 (mod r − 1) and e | (r − 1); 2. aj ≡ a + r−1e ∆j (mod r− 1), 1 ≤ j ≤ t, where ∆u 6= ∆v for any u 6= v and gcd(∆2 − ∆1, . . . , ∆t − ∆1,e) = 1; 3. gj = α−aj for 1 ≤ j ≤ t, their minimal polynomials over fq are pairwise distinct, and m1 = m2 = · · · = mt = m. denote δ = gcd(r − 1,a1,a2, . . . ,at),n = r − 1 δ , and n = gcd( r − 1 q − 1 ,ae). we easily see that eδ | n(q − 1). it follows from delsarte’s theorem [22] that the corresponding code c is an [n,tm] cyclic code over fq. it was proved that condition (3) can be met by the following simple criterion. criterion: [90] suppose that for any proper factor ` of m (i.e. ` | m and ` < m) we have r − 1 q` − 1 n. then condition (3) in the main assumptions holds. in particular, if n ≤ √ r, then condition (3) in the main assumptions is met. if t = 2, let a1 = q−1 h and a2 = q−1 h + r−1 e for positive integers e,h such that e | h and h | (q − 1), the code c had been studied in [26, 37, 60, 82, 86–88]. hence this class of cyclic codes with arbitrary zeros can be viewed as the generalization of cyclic codes whose duals have two zeros. the proper choices of these ai’s is key to compute the weight distribution of the code c. it may be very difficult to find the weight distribution if the integers ai are not chosen in the right way. if t = e ≥ 2, the set {∆j : 1 ≤ j ≤ e} is a complete residue system modulo e, so we may take ∆1 = 0, ∆2 = 1, . . . , ∆e = e− 1. theorem 3.3. [90] under the main assumptions, when n = 1 and t = e ≥ 2, the corresponding code c is a t-weight cyclic code over fq and its weight distribution can be explicitly presented. the gauss periods are known for n = 2, 3, 4, semi-primitive case, and index 2 case. hence the weight distributions of more cyclic codes can be determined. theorem 3.4. [90] suppose that the gaussian periods η(n,r)j of order n have µ distinct values {η1,η2, . . . ,ηµ}, and for each i(1 ≤ i ≤ µ), there are exactly τi distinct js such that η (n,r) j = ηi. (note that τ1 + τ2 + · · · + τµ = n.) then, the corresponding code c is an [n,em] cyclic code over fq with at most ( µ+e e ) −1 nonzero weights and its weight distribution can be explicitly presented when gauss periods are known. 48 h. q. dinh, c. li, q. yue 3.3. cyclic codes from fl conjugates let fq be a finite field with q = lt and γ a primitive element of fq, where l is a prime power and t is a positive integer. let g be an element in the algebraic closure of fq and mg(x) its minimal polynomial over fq. suppose that deg(mg(x)) = m and fqm = fq(g). then g = α−n and n | (qm −1), where α is a primitive element of fqm. let c be a cyclic code over fq with check polynomial h(x) = mg(x)mgl(x) · · ·mglt−1 (x), where mglu (x) is the minimal polynomial of g lu over fq for u = 0, 1, . . . , t− 1. it follows from delsarte’s theorem [22] that the code c is an [n,tm] cyclic code over fq, where n = q m−1 n . it is well known that g(x) = (xn − 1)/h(x) ∈ fq[x] and every codeword of c is c(x) = a(x)g(x), where a(x) ∈ fq[x] and deg(a(x)) ≤ tm − 1. note that the roots of h(x) are all the conjugates of g with respect to fl. then h(x) ∈ fl[x] is the minimal polynomial of g over fl and g(x) ∈ fl[x]. for a(x) ∈ fq[x], deg(a(x)) ≤ tm−1, by fq = fl ⊕γfl ⊕···⊕γt−1fl we have a(x) = s0(x) + γs1(x) + · · · + γt−1st−1(x), where su(x) ∈ fl[x] and deg(su(x)) ≤ tm− 1 for u = 0, 1, . . . , t− 1. then we get c(x) = a(x)g(x) = s0(x)g(x) + γs1(x)g(x) + · · · + γt−1st−1(x)g(x). it is easy to see that each su(x)g(x) is a codeword of the irreducible cyclic code over fl whose check polynomial is h(x) for u = 0, 1, . . . , t − 1. let t := trqm/l denote the trace function from fqm to fl. then by the trace representation of the irreducible cyclic code, the cyclic code c can be expressed by c = {c(a0,a1, . . . ,at−1) : a0,a1, . . . ,at−1 ∈ fqm}, where c(a0,a1, . . . ,at−1) = ( t−1∑ u=0 γut(au), t−1∑ u=0 γut(auαn ), . . . , t−1∑ u=0 γut(au(αn )n−1)). (5) when gcd(q m−1 l−1 ,n) = 1, the zeros of the check polynomial of the cyclic code c are α −nlu for u = 0, 1, . . . , t−1. in [90], yang et al. also dealt with such problem and the zeros of the check polynomials of yang-xiong-ding-luo cyclic codes are α−(a+ qm−1 t u) for u = 0, 1, . . . , t−1, where t | (qm−1) and a 6≡ 0 (mod qm − 1). hence this class of cyclic codes with arbitrary number of zeros are different from yangxiong-ding-luo cyclic codes. theorem 3.5. [49] let the notations be as above. if gcd(q m−1 l−1 ,n) = 1, then the corresponding code c is a t-weight cyclic code and its weight distribution can be explicitly determined. 4. weight distributions of cyclic codes with niho exponents in this section, we always assume that q = p, i.e., s = 1. now we consider the weight distributions of cyclic codes over fp with niho exponents which are due to [65]. a positive integer d is of niho exponent if d ≡ pi (mod pl − 1), where m = 2l for some integer l. without loss of generality, we can assume that d ≡ 1 (mod pl−1). for two niho exponents d = s(pl−1)+1 and d′ = s′(pl − 1) + 1, we call them equivalent if d′ ≡ pid (mod pm − 1) for some integer i. moreover, d ≡ pld (mod pm − 1) if and only if s + s′ ≡ 1 (mod pl + 1). hence, s can be restricted in the range 1 ≤ s ≤ pl−1 + 1. 49 weight distributions of cyclic codes over finite fields let g1 = α−d1 and g2 = α−d2 for two niho exponents d1,d2. suppose that g1 and g2 are not conjugates over fp. to determine the weight distribution of the corresponding cyclic code c, by (3) we have to deal with the exponential sums t(a,b) = ∑ x∈fr ψ1(a1x d1 )ψ2(a2x d2 ) = ζ trpm1/p(a1xd1 )+trpm2/p(a2xd2 ) p for a1,a2 ∈ fr. if one of d1,d2 is equal to 1 and p = 2, this class of cyclic codes with few nonzero weights were studied [18]. li, feng, and ge [52] gave some sufficient conditions for these codes to have few nonzero weights for both p = 2 and odd p. some preliminaries are necessary for determining the exponential sums t(a,b). let fpm be a finite field with m = 2l and r = pm. denote s = {x ∈ fr : xx̄ = 1}, where x̄ = xp l . then s is a cyclic group of order pl + 1 and s = 〈η〉 with η = αp l−1. lemma 4.1. [52, 65] for two niho exponents d1 = s1(pl − 1) + 1 and d2 = s2(pl − 1) + 1, we have t(a,b) = (n(a,b− 1))pl, where n(a,b) is the number of z ∈ s satisfying azs1 + āz1−s1 + bzs2 + b̄z1−s2 = 0. from the properties of trace function, we easily obtain the following moment identities which is very important to determine the value distribution of t(a,b). lemma 4.2. 1. ∑ a,b∈fr t(a,b) = p2m. 2. ∑ a,b∈fr t(a,b)2 = p2mn2(d1,d2), where n2(d1,d2) is the number of solutions of the equations { xd1 + yd1 = 0 xd2 + yd2 = 0 , x,y ∈ fr. 3. ∑ a,b∈fr t(a,b)3 = p2mn3(d1,d2), where n3(d1,d2) is the number of solutions of the equations { xd1 + yd1 + zd1 = 0 xd2 + yd2 + zd2 = 0 , x,y,z ∈ fr. from the above two lemmas we see that determining the weight distributions of cyclic codes with niho exponents is equivalent to count the number of solutions of the equation and the system of equations over finite fields. hence, if cyclic codes with niho exponents have many nonzero weights, it is very difficult to determine their weight distributions. recently, li, feng, and ge [52] presented the weight distributions of three classes of cyclic codes with niho exponents. theorem 4.3. [52] let c be a cyclic code defined by g1 = α−d1 and g2 = α−d2 for two niho exponents d1,d2. then the weight distribution of the p-ary cyclic code c with the following niho exponents are known: 1. p = 2, d1 = 2l + 1, and d2 = s2(2l − 1) + 1, where s2 6≡ 12 (mod 2 l + 1); 2. p = 2, l ≥ 2, d1 = s1(2l−1)+1, and d2 = s2(2l−1)+1, where s1 = 2k−1t− t−12 and s2 = 2 k−1t+ t+1 2 for integers k (1 ≤ k ≤ l) and t (t is odd and 1 ≤ t ≤ 2l +1) satisfying 2k−1t, 2k+1t 6≡ 0 (mod 2l +1) and m ≡−1 (mod k) or gcd(k, 2m) = 1; 3. p is odd, d1 = s1(pl − 1) + 1, and d2 = s2(pl − 1) + 1, where s1 = t+24 and s2 == 3t+2 4 for integer t satisfying t ≡ 2 (mod 4) and t 6≡ 0 (mod pl + 1). 50 h. q. dinh, c. li, q. yue 5. cyclic codes with few weights cyclic codes with few weights are of much interest in coding theory due to their applications in cryptography and combinatorics. in this section, we begin with some definitions. a linear code is called to be projective if the minimum weight of its dual code is at least three. moreover, a linear code is a n-weight code if the number of non-zero weights of this code is n. a cyclic code of length n over fq is irreducible if its check polynomial is irreducible (its polynomial representation is a minimal ideal). it is said to be non-degenerate if its check polynomial is a primitive divisor of xn − 1 over fq (that is, the order of this polynomial is n). 5.1. one-weight cyclic codes let fr be a finite field with r = qm elements. when the length of a cyclic code c is r − 1 and the check polynomial is the minimal polynomial over fq of a primitive root of fr (in fact, c is an irreducible cyclic code), then the code c is called a simplex code or a subfield code. it is easily proved that c is a 1-weight code with (q − 1)qm−1 as its unique non-zero weight. in [84], wolfmann first gave some descriptions of one-weight cyclic codes via pless identities [66]. furthermore, vega and wolfmann [81] presented a better and more simple characterization of one-weight irreducible cyclic codes. theorem 5.1. [81] let c be an [n,k] irreducible cyclic code over fq with n = λr−1q−1 , where λ divides q−1. let ρ be the order of the check polynomial of c, that is, the common order of its roots. the following assertions are equivalent: 1. c is a one-weight cyclic code; 2. c contains a codeword of weight λqm−1; 3. ρ gcd(ρ,q−1) = qm−1 q−1 . 5.2. two-weight cyclic codes two-weight linear codes are closely related to strongly regular graphs, partial difference sets, and finite projective spaces. there is a survey [12] to investigate their relationships. for two-weight irreducible cyclic codes, schmidt and white [73] in 2002 gave a classification by gauss sums. they presented some necessary and sufficient numerical conditions on the parameters of an irreducible cyclic code to have at most two nonzero weights. it is conjectured that an irreducible cyclic code is a two-weight code if and only if it is a semi-primitive code or one of the eleven sporadic examples. moreover, they gave a partial proof of this conjecture via generalized riemann hypothesis. let q = p be a prime, and let u,m be positive integers such that u divides p m−1 p−1 . let c be an irreducible cyclic code over fp defined by g1 = α−u. for a positive integer x, let sp(x) denote the sum of the p-digits of x, that is, if x = l0 + l1p + · · · + lvpv, where 0 ≤ li ≤ p− 1 and lv 6= 0, then sp(x) = l0 + l1 + · · · + lv. denote f =: ordu(p) (i.e., the least positive integer such that pf ≡ 1 (mod u)) and θ = θ(u,p) =: 1 p− 1 min{sp( j(pf − 1) u ) : 1 ≤ j < u}. 51 weight distributions of cyclic codes over finite fields table 3. eleven sporadic examples with u ≤ 100000 u p s f θ κ � 11 3 1 5 2 5 +1 19 5 1 9 4 9 +1 35 3 1 12 5 17 +1 37 7 1 9 4 9 +1 43 11 1 7 3 21 +1 67 17 1 33 16 33 +1 107 3 1 53 25 53 +1 133 5 1 18 8 33 -1 163 41 1 81 40 81 +1 323 3 1 144 70 161 +1 499 5 1 249 123 249 +1 then we have the following theorem. theorem 5.2. [73] let the notations be as above. if m = fl for some integer l, then c is a two-weight code if and only if there exists a positive integer κ satisfying κ | (u− 1), κplθ ≡ � (mod u), κ(u−κ) = (u− 1)pl(f−2θ), where � = ±1. moreover, the two nonzero weights are w1 = (p− 1)plθ−1(pl(f−θ) − �κ)/u, w2 = w1 + �(p− 1)plθ−1. conjecture 5.3. [73] an irreducible cyclic code is a two-weight code if and only if it is a semi-primitive code or one of the eleven sporadic examples in table 3. in [85], wolfmann gave a characterization of projective two-weight linear codes. furthermore, a family of projective 2-weight irreducible cyclic codes were presented. theorem 5.4. [85] let c be a non-degenerate irreducible cyclic code of length n over fq with gcd(n,q) = 1. let fqm be the splitting field of xn − 1 over fq and let nn = qm − 1. if : 1. gcd(n,q − 1) = 1, 2. m = 2fl such that (q − 1)(qf + 1) ≡ 0 (mod n), 3. n 6= q − 1 and d 6= (q − 1)(qfl + 1), then c is a projective two-weight code. similarly, a conjecture on projective two-weight non-degenerate irreducible cyclic codes was proposed in [85]. 52 h. q. dinh, c. li, q. yue conjecture 5.5. [85] any projective two-weight non-degenerate irreducible cyclic code is a code satisfying conditions (1)-(3) of theorem 5.4 except for eleven special cases deduced from table 3. wolfmann [84] also characterized projective two-weight cyclic codes and proved that if a linear code c is a two-weight projective cyclic code of dimension m over fq, then either: (1) c is irreducible, or (2) if q 6= 2, c is the direct sum of two one-weight irreducible cyclic codes of length n = λq m−1 q−1 , where λ divides q − 1 and λ 6= 1 and direct sum means direct sum as vector spaces. it is clear that the code c is reducible in case (2) of wolfmann’s characterization [84]. two-weight reducible cyclic codes had also been presented in [39] and [81]. motivated by these results, in 2008, vega [78] presented a family of two-weight reducible cyclic codes which were constructed as the direct sum of two one-weight cyclic codes and obtained their weight distributions. moreover, this new family gives a unified explanation for all these two-weight cyclic codes that were presented in [39] and [81]. to get vega’s result, gauss sum introduced in section 2 is a necessary tool. theorem 5.6. [78] let p,q, and m be defined as before. denote ∆ = q m−1 q−1 . let a1,a2 and v be integers such that a1qi 6≡ a2 (mod qm − 1), for all i ≥ 0, v = gcd(a1 − a2,q − 1), and a2 ∈ z∗∆. for some integer ` satisfying ` | gcd(a1,a2,q − 1), we set λ = (q−1)` gcd(a1,a2,q−1) , n = λ∆, µ = q−1 ∆ , and ξ = q−1 v . let h1(x),h2(x) ∈ fq[x] be the minimal polynomials of α−a1 and α−a2, respectively. suppose that at least one of the following two conditions holds: 1. p = 2,k = 2,v = 1, and a1 is a unit in the ring z∆, or 2. for some integer j , with 1 ≤ pj < qm , we have (1 + ã2(a1 −a2))pj ≡ 1 (mod ∆v), where ã2 is the inverse of a2 in z∆. then the following four assertions are true: (a) h1(x) and h2(x) are the check polynomials for two different one-weight cyclic codes of length n and dimension m. (b) µ | v and λ > v µ . (c) if c is the cyclic code with check polynomial h1(x)h2(x), then c is an [n, 2m] two-weight cyclic code with weight enumerator polynomial a(x) = 1 + µ v n(q − 1)z(λ− v µ )qm−1 + (q2m − 1 − µ v n(q − 1)zλq m−1 . (d) c is a projective code if and only if µ = v. 5.3. three-weight cyclic codes cyclic codes with three nonzero weights have been applied in association schemes [11] and secret sharing schemes [97]. hence constructing three-weight cyclic codes is a good research problem. recently, perfect nonlinear (or planar) and almost perfect nonlinear functions are employed to find three weight cyclic codes. in [91], yuan, ding, and carlet used planar functions to get two classes of three-weight cyclic codes. feng and luo [34] presented a unified way to investigate the weight distributions of cyclic codes from planar functions. theorem 5.7. let m ≥ 3 be odd and let q be an odd prime. then the corresponding cyclic code c is a three-weight [qm − 1, 2m] cyclic code in the following cases: 53 weight distributions of cyclic codes over finite fields table 4. weight distribution of cyclic code from planar functions. weight frequency 0 1 (q −1)qm−1 −q m−1 2 (q−1)(qm−1)(qm−1+q m−1 2 ) 2 (q −1)qm−1 (qm −1)(qm−1 + 1) (q −1)qm−1 + q m−1 2 (q−1)(qm−1)(qm−1−q m−1 2 ) 2 table 5. weight distribution of cyclic code in [34]. weight frequency 0 1 (q −1)qm−1 − q−1 2 q m−1 2 (qm −1)(qm−1 + q m−1 2 ) (q −1)qm−1 (qm −1)(qm −2qm−1 + 1) (q −1)qm−1 + q−1 2 q m−1 2 (qm −1)(qm−1 −q m−1 2 ) 1. g1 = α−1 and g2 = α−(q l+1) [91]; 2. g1 = α−1 and g2 = α− ql+1 2 , where q = 3, gcd(m,l) = 1, and h is odd [34, 91]. moreover, its weight distribution is presented in table 4. luo and feng [58] extended the second construction in theorem 5.7. theorem 5.8. [58] let m ≥ 3 be odd and let q be an odd prime. then the code c over fq defined by g1 = α −1 and g2 = α−v is a three-weight [qm − 1, 2m] cyclic code with the weight distribution in table 5 if v = q l+1 2 , where l is a positive integer satisfying gcd(2m,l) = 1. we remark that table 4 and table 5 are same when p = 3. additionally, ding, gao, and zhou [25, 96] presented several classes of three-weight cyclic codes over f3 from almost perfect nonlinear functions. theorem 5.9. let q = 3 and c be the ternary cyclic code defined by g1 = α−1 and g2 = α−v. then c is a [qm − 1, 2m] three-weight cyclic code with weight distribution depicted in table 4 or 5 in the following cases: 1. m is odd and v = 3 m+1−1 4 [25]; 2. m is odd and v = 3 m+1 2 − 1 [97]; 3. m ≡ 3 (mod 4) and v = 3 m+1 2 −1 2 [97]; 4. m ≡ 1 (mod 4) and v = 3 m+1 2 −1 2 + 3 m−1 2 [97]; 5. m ≡ 3 (mod 4) and v = 3 m+1−1 8 [97]; 6. m ≡ 1 (mod 4) and v = 3 m+1−1 8 + 3 m−1 2 [97]; 7. m ≡ 3 (mod 4) and v = (3 m+1 4 − 1)(3 m+1 2 + 1) [97]; 8. m ≡ 7 (mod 8) and v = (3 m+1 8 − 1)(3 m+1 4 + 1)(3 m+1 2 + 1) [25]. 54 h. q. dinh, c. li, q. yue table 6. weight distribution of three classes of cyclic code in [25]. weight frequency 0 1 (q −1)(qm−1 −q m−1 2 ) 1 2 (qm −1)(qm−1 + q m−1 2 ) (q −1)qm−1 (qm −1)(qm −qm−1 + 1) (q −1)(qm−1 + q m−1 2 ) 1 2 (qm −1)(qm−1 −q m−1 2 ) table 7. case: v ≡ 1 + q−1 2 (mod q −1). weight frequency 0 1 (q −1)qm−1 − q−1 2 q m+d−2 2 (qm −1)(qm−d + q m−d 2 ) (q −1)qm−1 (qm −1)(qm −2qm−d + 1) (q −1)qm−1 + q−1 2 q m+d−2 2 (qm −1)(qm−d −q m−d 2 ) remark 5.10. for q = 3, xv is an almost perfect nonlinear function over fqm for the following v: 1. v = 3 m+1 2 − 1; 2. v = 3 m+1 2 −1 2 if m ≡ 3 (mod 4); 3. v = 3 m+1 2 −1 2 + 3 m−1 2 if m ≡ 1 (mod 4); 4. v = 3 m+1−1 8 if m ≡ 3 (mod 4); 5. v = 3 m+1−1 8 + 3 m−1 2 if m ≡ 1 (mod 4). there are another three classes of three-weight cyclic codes whose weight distributions are given in table 6 and are different from the one in table 4 or 5. theorem 5.11. [25] let q = 3 and c be the ternary cyclic code defined by g1 = α−1 and g2 = α−v. then c is a [qm − 1, 2m] three-weight cyclic code with weight distribution depicted in table 6 if 1. v = 3 m+1−1 3h+1 + 3 m−1 2 , where m+1 h is even; or 2. v = (3 m+1 8 − 1)(3 m+1 4 + 1)(3 m+1 2 + 1) + 3 m−1 2 , where m ≡ 7 (mod 8); or 3. v = (3 m+1 4 − 1)(3 m+1 2 + 1) + 3 m−1 2 , where m ≡ 3 (mod 4). in 2014, li et al. [50] gave a more general description of three-weight cyclic codes defined by g1 = α−1 and g2 = α−v. theorem 5.12. [50] let m ≥ 3 be odd. let q be any odd prime. if v is an integer satisfying (ql + 1)v ≡ 2 (mod qm−1) for some positive integer v with gcd(m,l) = d, then c is a [qm−1, 2m] cyclic code with the weight distribution in table 7 if v ≡ 1 + q−1 2 (mod q − 1) and table 8 when v ≡ 1 (mod q − 1). there are more classes of three-weight cyclic codes presented in the literature. three-weight cyclic codes were also constructed from niho exponents [52] and two distinct finite fields [48]. it is unnecessary to list all the results on three-weight cyclic codes and we have to omit some results here. in [35, 96], the cyclic codes were proved to be three-weight by using quadratic forms. 55 weight distributions of cyclic codes over finite fields table 8. case: v ≡ 1 (mod q −1). weight frequency 0 1 (q −1)(qm−1 −q m+d−2 2 ) 1 2 (qm −1)(qm−d + q m−d 2 ) (q −1)qm−1 (qm −1)(qm −qm−d + 1) (q −1)(qm−1 + q m+d−2 2 ) 1 2 (qm −1)(qm−d −q m−d 2 ) theorem 5.13. let q be a prime and c a cyclic code over fq defined by g1 = α−v1 and g2 = α−v2. then c is a [qm − 1, 2m] three-weight cyclic codes in the following cases: 1. v1 = 2 and v2 = pl + 1, where m ≥ 3 is odd and gcd(m,l) = 1 [35]; 2. v1 = qm+1 2 and v2 = ql+1 2 , where gcd(m,l) = 1 [96]. 6. generalization to constacyclic codes the concept of cyclic codes was extended naturally to negacyclic codes,4 and then to constacyclic codes. given a nonzero element λ of fq, a linear code c of length n over fq is called λ-constacyclic if (λcn−1,c0, · · · ,cn−2) ∈ c for every (c0,c1, · · · ,cn−1) ∈ c. just like cyclic codes, λ-constacyclic codes of length n over fq are classified as the ideals 〈g(x)〉 of the quotient ring fq[x]/〈xn − λ〉, where the generator polynomial g(x) is the unique monic polynimial of minimum degree in the code, which is a divisor of xn−λ. when λ =1, λ-constacyclic codes are just cyclic codes and when λ = −1, λ-constacyclic codes are negacyclic codes. in general, the dual of a λ-constacyclic code of length n is a λ−1-constacyclic code of length n. there are cases when one code can be mapped onto another by means of a map which preserves the hamming distances. two codes c1,c2 are considered to be of the same quality if there exists a mapping ϕ : fnq −→ fnq with ϕ(c1) = c2 which preserves the hamming distance, i.e. dh ( ϕ(a),ϕ(a′) ) = dh(a,a ′), for any a,a′ ∈ fnq . mappings with the latter property are called isometries, and such codes are naturally called equivalent. there are various ways in which such an equivalence relation can be defined. for example, if c1,c2 are linear codes, then we can naturally assume furthermore that the isometry ϕ is a linear map (e.g., [8]). since each λ-constacyclic code is an ideal of fq[x]/〈xn − λ〉, it is natural to assume that isometries between constacyclic codes preserve the algebraic structures and hamming distances. it turns out that if we can classify all the equivalence classes of constacyclic codes, we then only have to study the represtentative of those equivalence classes. in particular, if we can determine all constacyclic codes that are equivalent to cyclic codes, then all our results about hamming weight and weight distributions of cyclic codes in previous sections hold true for those constacyclic codes. we devote this section to consider two type of equivalences, and for each type, we give the necessary and sufficient conditions for λand µ-constacyclic codes to be equivalent. 4 as mentioned before, cyclic codes were introduced in 1957. just about 11 years after that, negacyclic codes over finite fields fp were initiated by berlekamp in 1968 [3, 4], where he showed that these codes are more useful for correcting errors measured relative to the lee metric. berlekamp also designed a decoding algorithm that can correct errors with lee weight at most p−1 2 . a couple of years after that, in 1971, kelsch and green [45] were sucessful to provide non-binary negacyclic codes exceeding berlekamp’s p−1 2 bound. they constructed 2-errorcorrecting negacyclic codes of length 3 m−1 2 with redundancy 2m over f3, and all negacyclic codes of length p m−1 2 with redundancy mt over fp. 56 h. q. dinh, c. li, q. yue first of all, some special results have been obtained in the literature. lemma 6.1. [42, lemma 3.1] or [1, corollary 2.1] let n be a positive integer, and λ ∈ f∗q. if µnλ = 1 for some µ ∈ f∗q, then fq[x]/〈xn −λ〉 −→ fq[x]/〈xn − 1〉, x 7→ µx is an fq-algebra isomorphism which is hamming distance preserving. in particular, in case n is odd, for λ = −1, it is obvious that µ = −1 satisfies the hypothesis of the above lemma. that means that negacyclic codes of odd length are scalar equivalent to cyclic codes of the same length, which is a well known result that was proven to be true for the more general case when the alphabet is a finite commutative ring. noting this fact, dinh [29] established a one-to-one correspondence between negacyclic and cyclic codes, carrying results on negacyclic codes to cyclic codes accordingly. proposition 6.2. [29, proposition 6.1] let p be an odd prime and q a power of p. then the map ξ : fq[x] 〈xps+1〉 7→ fq[x] 〈xps−1〉, given by f(x) 7→ f(−x), is an fq-algebra isomorphism. in particular, for a ⊆ fq[x]〈xps+1〉, b ⊆ fq[x] 〈xps−1〉 such that ξ(a) = b, then a is an ideal of fq[x] 〈xps+1〉 if and only if b is an ideal of fq[x]〈xps−1〉. equivalently, a is a negacyclic code of length p s over fq if and only if b is a cyclic code of length ps over fq. later on, dinh in [30] showed that all constacyclic codes of length ps over fq are scalar equivalent to negacyclic codes. proposition 6.3. [30, proposition 3.1] let p be an odd prime and q a power of p. let λ ∈ f∗q. then there exists a unique element λ0 in f∗q such that λ ps 0 = −λ −1. let φ be the map φ : fq[x]〈xps+1〉 7→ fq[x] 〈xps−λ〉, given by φ(f(x)) = f(λ0x). then φ is an fq-algebra isomorphism, and it is hamming distance preserving. for the more general alphabets of finite rings, [83] showed that cyclic and negacyclic codes over z4 have the same structure for odd code lengths. dinh and lópez-permouth in [28] generalized that to obtain that this fact holds true for cyclic and negacyclic codes of odd lengths over any finite chain ring. batoul et al. in [2, proposition 3.4] extended this result to a more general setting. generalizing the ideas above, chen et al. in [19] introduced a concept called “isometry” for the nonzero elements of fq to classify constacyclic codes over fq such that the constacyclic codes belonging to the same isometry class have the same distance structures and the same algebraic structures. definition 6.4. [19, definition 3.1] let λ,µ ∈ f∗q. we say that an fq-algebra isomorphism ϕ : fq[x]/〈xn −µ〉 −→ fq[x]/〈xn −λ〉 is an isometry if it preserves the hamming distances on the algebras, i.e. dh ( ϕ(a),ϕ(a′) ) = dh(a,a ′), ∀ a,a′ ∈ fq[x]/〈xn −µ〉. and, if there is an isometry between fq[x]/〈xn−λ〉 and fq[x]/〈xn−µ〉, then we say that λ is n-isometric to µ in fq, written λ ∼=n µ. clearly, the n-isometry “∼=n” is an equivalence relation on f∗q, hence f∗q is partitioned into n-isometry classes. if λ ∼=n µ, then the λ-constacyclic codes of length n are in one to one correspondence with the µconstacyclic codes of length n such that the corresponding constacyclic codes have the same dimension and the same distance distribution, specifically, have the same minimum distance; at that case for convenience, the λ-constacyclic codes of length n are said to be isometric to the µ-constacyclic codes of length n. so, it is enough to study the n-isometry classes of constacyclic codes. we have the following result. 57 weight distributions of cyclic codes over finite fields theorem 6.5. [19, theorem 3.2] for any λ,µ ∈ f∗q, the following three statements are equivalent to each other: (i) λ ∼=n µ. (ii) 〈λ,ξn〉 = 〈µ,ξn〉, where 〈λ,ξn〉 denotes the subgroup of f∗q generated by λ and ξn. (iii) there is a positive integer k < n with gcd(k,n) = 1 and an element a ∈ f∗q such that anλ = µk and the following map ϕa : fq[x]/〈xn −µk〉 −→ fq[x]/〈xn −λ〉, (6) which maps any element f(x) + 〈xn −µk〉 of fq[x]/〈xn −µk〉 to the element f(ax) + 〈xn −λ〉 of fq[x]/〈xn −λ〉, is an isometry. in particular, the number of n-isometry classes of f∗q is equal to the number of positive divisors of gcd(n,q − 1). taking µ = 1, we see that λ ∼=n 1 implies that there is an isometry ϕa : fq[x]/〈xn − 1〉 → fq[x]/〈xn −λ〉 such that ϕ(x) = ax. thus for the constacyclic codes n-isometric to cyclic codes, the following consequence is closely related to [42, lemma 3.1] or [1, corollary 2.1]. corollary 6.6. [19, corollary 3.4] let n be a positive integer, and λ ∈ f∗q. the λ-constacyclic codes of length n are isometric to the cyclic codes of length n if and only if anλ = 1 for an element a ∈ f∗q; further, in that case the map ϕa : fq[x]/〈xn − 1〉 −→ fq[x]/〈xn −λ〉, (7) which maps f(x) to f(ax), is an isometry, and xn −λ = λ ·mr1 (ax) psmr2 (ax) ps · · ·mrρ(ax) ps (8) is an irreducible factorization of xn −λ in fq[x], where n = n′ps with s ≥ 0 and p 6 |n′, mri(x) is the irreducible factor of xn ′ −1 over fq corresponding to the q-cyclotomic coset containing ri. in particular, any λ-constacyclic code c has a generator polynomial as follows: ρ∏ i=1 mηri (ax) ei, 0 ≤ ei ≤ ps, for any i = 1, · · · ,ρ. (9) as an immediate application of corollary 6.6, the next result can be regarded as a generalization of proposition 6.3. corollary 6.7. (cf. [19, corollary 3.5]) if n is a positive integer coprime to q−1, then there is only one n-isometry class in f∗q; in particular, for any λ ∈ f∗q the λ-constacyclic codes of length n are isometric to the cyclic codes of length n, i.e. anλ = 1 for an a ∈ f∗q and all the (7), (8) and (9) hold. although λ ∼=n µ means there exists an isometry φ between the rings fq[x]/〈xn − λ〉 and fq[x]/〈xn − µ〉, it is not easy to connect the generator polynomial of the λ-constacyclic code c with the generator polynomial of φ(c), and as a result, it is not easy to describe the relationship between the duals c⊥ and φ(c)⊥. to overcome this problem, chen, dinh and liu in [20] considered a more specified relationship than the isometry “ ∼=n ”, that enabled us to obtain a much more explicit description of the generator polynomials of all constacyclic codes. this detailed description also allows us to establish the generator polynomials of the dual codes. a new equivalence relationship “ ∼n ” is introduced on the nonzero elements of fq to classify constacyclic codes of length n over fq. some necessary and sufficient conditions for any two nonzero elements of fq to be equivalent to each other are established. it is shown that, if λ ∼n µ then there exists a very explicit fq-algebra isomorphism ϕ between fq[x]/〈xn − λ〉 and fq[x]/〈xn −µ〉. furthermore, the generator polynomial of the λ-constacyclic code c and the generator polynomial of the µ-constacyclic code ϕ(c) are connected in a very simple way. 58 h. q. dinh, c. li, q. yue definition 6.8. [20, definition 3.1] let n be a positive integer. for any elements λ, µ of f∗q we say that λ and µ are n-equivalent in f∗q and denote by λ ∼n µ if the polynomial λxn−µ has a root in fq. in this case, we say λ-constacyclic codes are n-equivalent to µ-constacyclic codes. it is routine to check that ∼n is an equivalence relationship on f∗q. the next result shows that λ and µ are n-equivalent if and only if they are belonging to the same coset of 〈ξn〉 in 〈ξ〉. in other words, the cosets of 〈ξn〉 in 〈ξ〉 give all the n-equivalence classes, thus each n-equivalence class contains the same number of elements. theorem 6.9. [20, theorem 3.2] for any λ,µ ∈ f∗q, the following four statements are equivalent: (i) there exists an a ∈ f∗q such that ψ : fq[x]/〈xn −µ〉 → fq[x]/〈xn −λ〉 f(x) 7→ f(ax), is an fq-algebra isomorphism. (ii) λ and µ are n-equivalent in f∗q. (iii) λ−1µ ∈ 〈ξn〉. (iv) (λ−1µ)d = 1, where d = q−1 gcd(n,q−1) . in particular, the number of the n-equivalence classes in f∗q is gcd(n,q − 1). comparing with the equivalence relation “ ∼=n ” mentioned previously, one can easily find that λ ∼n µ implies λ ∼=n µ. however, the converse of this statement is not true in general. in fact, theorem 6.5 implies that if λ ∼=n µ then there exists a positive integer k coprime to n such that λ ∼n µk. therefore, every isometry class is equal to some unions of n-equivalence classes. we give the following illustrative example. example 6.10. take q = 24 and n = 6 in theorem 6.9. clearly, gcd(6, 24 − 1) = 3 and f∗24 = 〈ξ〉 ⋃ ξ〈ξ〉 ⋃ ξ2〈ξ〉. this implies that ξ and ξ2 are not 6-equivalent. however, it is readily seen that there are just two 6-isometry classes and ξ ∼=n ξ2. 7. concluding remarks in this paper, we investigated the weight distributions of cyclic codes determined by exponential sums. it is clear that gauss periods, gauss sums, and quadratic forms are important tools. the weight distributions of cyclic codes have been studied for many years and are known in some cases. however, it remains open for most cyclic codes. thus there are many challenging problems to be solved. acknowledgment: the authors are very grateful to the reviewers and the editor for their valuable comments and suggestions that improved the quality of this paper. 59 weight distributions of cyclic codes over finite fields references [1] n. aydin, i. siap, d. k. ray-chaudhuri, the structure of 1-generator quasi-twisted codes and new linear codes, designs codes cryptogr. 24, 313-326, 2001. 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[99] z. zhou, a. zhang, c. ding, and m. xiong, the weight enumerator of three families of cyclic codes, ieee trans. inform. theory, 59(9), 6002-6009, sep. 2013. 63 introduction weights of the duals of cyclic codes with two or three zeros weight distributions of cyclic codes with arbitrary zeros weight distributions of cyclic codes with niho exponents cyclic codes with few weights generalization to constacyclic codes concluding remarks references issn 2148-838x j. algebra comb. discrete appl. 9(3) • 185–191 received: 8 august 2021 accepted: 24 november 2021 journal of algebra combinatorics discrete structures and applications a database of quantum codes research article nuh aydin, peihan liu, bryan yoshino abstract: quantum error correcting codes (qecc) is becoming an increasingly important branch of coding theory. for classical block codes, a comprehensive database of best known codes exists which is available online at [17]. the same database contains data on best known quantum codes as well, but only for the binary field. there has been an increased interest in quantum codes over larger fields with many papers reporting such codes in the literature. however, to the best of our knowledge, there is no database of best known quantum codes for most fields. we established a new database of qecc that includes codes over fq2 for q ≤ 29. we also present several methods of constructing quantum codes from classical codes based on the css construction. we have found dozens of new quantum codes that improve the previously known parameters and also hundreds new quantum codes that did not exist in the literature. 2010 msc: 94b05, 94b60, 94b65 keywords: quantum error-correcting codes, css construction, constacyclic codes, polycyclic codes 1. introduction and motivation compared to classical information theory, the field of quantum information theory is relatively young. the idea of quantum error correcting codes was first introduced in [31] and [11]. a method of constructing quantum error correcting codes (qecc) was given in [10]. since then researchers have investigated various methods of using classical error correcting codes to construct new qeccs. there are many publications in the literature with new quantum codes. however, compared to the classical block codes, the database of quantum codes is very limited. while the database ([17]) covers the finite fields of orders 2, 3, 4, 5, 7, 8 and 9, it only presents a database of qecc for q = 2. there are static tables of some quantum codes given at [9], based on the work [8]. in recent years, researchers have conducted searches for good quantum codes, and some results are presented in the papers listed in the bibliography. we notice that the online tables [9] may have been overlooked by some researchers. in many cases qeccs over finite fields of characteristic greater than 2 are presented in the literature. hence nuh aydin (corresponding author), bryan yoshino; department of mathematics, kenyon college, usa (email: aydinn@kenyon.edu, yoshino1@kenyon.edu). peihan liu; department of mathematics, university of michigan, usa (email: paulliu@umich.edu). 185 https://orcid.org/0000-0002-5618-2427 https://orcid.org/0000-0003-2403-1517 codetables.de http://codetables.de/ https://www.mathi.uni-heidelberg.de/~yves/matritzen/qtbch/qtbchindex.html https://www.mathi.uni-heidelberg.de/~yves/matritzen/qtbch/qtbchindex.html n. aydin et. al. / j. algebra comb. discrete appl. 9(3) (2022) 185–191 there is a need to have an interactive database of qeccs that contains best known quantum codes over larger alphabets which is continually updated. we now have compiled such a database. it is available at http://quantumcodes.info, and it is being updated continually. in addition to the creation of the database, we have also devised and implemented some search algorithms to find new quantum codes from classical codes using the css construction. the paper also describes and reports the results of these searches. one of the search methods we consider in this paper is about deriving quantum codes from polycyclic codes associated with trinomials, which is based on the css construction. this methods has been used in a recent work [3]. here, we expand this method to construct more quantum codes. this paper is organized as follows. in section 2, we recall some basic concepts about quantum codes and polycyclic codes. in section 3, we present some search algorithms along with the new quantum codes that we have found. a new code means it either has a better minimum distance than existing best known quantum code with the same length and dimension over the same alphabet or a code with these parameters does not exist in literature. all computations are performed using the magma software. 2. preliminaries let c be the field of complex numbers and consider the hilbert space (cq)⊗n = cq ⊗cq ⊗···⊗cq︸ ︷︷ ︸ n times . a q-ary quantum code of length n and dimension k is a qk-dimensional subspace of (cq)⊗n. if the minimum distance of a quantum code is d then we denote it by [[n, k, d]]q. a connection between classical block codes and quantum codes was given in [10] which has been used extensively in the literature. we include the relevant results here. theorem 2.1. [10] suppose s̄ is an (n−k)-dimensional linear subspace of ē (a 2n-dimensional binary vector space) which is contained in its dual s̄⊥ (with respect to the inner product, and is such that there are no vectors of weight ≤ d − 1 in s̄⊥ \ s̄. then there is a quantum-error-correcting code mapping k qubits to n qubits which can correct [(d− 1)/2] errors. theorem 2.2. [10] given two codes [[n1, k1, d1]] and [[n2, k2, d2]] with k2 ≤ n1 we can construct an [[n1 + n2 −k2, k1, d]] code, where d ≥ min{d1, d1 + d2 −k2}. the method of constructing quantum codes this way is known as css construction. there is a large number of papers in the literature that present new quantum codes obtained by the css construction. recently, we have used polycyclic codes for the same purpose [3]. definition 2.3. a linear code c is said to be polycyclic with respect to v = (v0, v1, ..., vn−1) ∈ fnq if for any codeword (c0, c1, ..., cn−1) ∈ c, its right polycyclic shift, (0, c0, c1, . . . , cn−2) + cn−1(v0, v1, . . . , vn−1) is also a codeword. similarly, c is left polycyclic with respect to v = (v0, v1, ..., vn−1) ∈ fnq if for any codeword (c0, c1, ..., cn−1) ∈ c, its left polycyclic shift (c1, c2, . . . , cn−1, 0) + c0(v0, v1, . . . , vn−1) is also a codeword. if c is both left and right polycyclic, then it is bi-polycyclic. in this work, we only work with right polycyclic codes, which we simply refer to as polycyclic codes. under the usual identification of vectors with polynomials, each polycyclic code c of length n is associated with a vector v of length n (or a polynomial v(x) of degree less than n). we call v (v(x)) an associate vector (polynomial) of c, and we say that c is a polycyclic code associated with xn − v(x). moreover, polycyclic codes of length n associated with f(x) = xn − v(x) are ideals of the factor ring fq[x]/〈f(x)〉. note that an associate polynomial of a polycyclic code may not be unique. polycyclic codes are a generalization of cyclic codes and its several important generalizations. the following are some of the most important special cases of polycyclic codes: • a right polycyclic code with respect to v = (1, 0, 0, ..., 0) is a cyclic code. a left polycyclic code with respect to v = (0, 0, 0..., 1) is a cyclic code. 186 http://quantumcodes.info http://magma.maths.usyd.edu.au/magma/ n. aydin et. al. / j. algebra comb. discrete appl. 9(3) (2022) 185–191 • a right polycyclic code with respect to v = (−1, 0, 0, ..., 0) is a negacyclic code. a left polycyclic code with respect to v = (0, 0, 0...,−1) is a negacyclic code. • a right polycyclic code with respect to v = (a, 0, 0, ..., 0) is a constacyclic code. a left polycyclic code with respect to v = (0, 0, 0, ..., a−1) is a constacyclic code. 3. quantum css codes from polycyclic codes many of the works in the literature with new qeccs use the css construction method given in [10] or something related to it. in this method, self-dual, self-orthogonal and dual-containing linear codes are used to construct quantum codes. the css construction requires two linear codes c1 and c2 such that c⊥2 ⊆ c1. hence, if c1 is a self-dual code, then we can construct a css quantum code using c1 alone since c⊥1 ⊆ c1. if c1 is self-orthogonal, then we can construct a css quantum code with c⊥1 and c1 since c1 ⊆ c⊥1 . similarly in the case when c1 is a dual-containing code. 3.1. css codes by two polycyclic codes associated with trinomials by the definition of css construction, we need two codes such that one is contained in the dual of the other one. let c be a polycylic code generated by g(x). by ideal inclusion we know that 〈g(x)f(x)〉⊆ 〈g(x)〉 for any polynomial f(x)and we consider c⊥2 = 〈g(x)f(x)〉 ⊆ 〈g(x)〉 = c1. here g(x) is a divisor of a trinomial t(x) = xn − axi − b and it generates a polycyclic code associated with t(x). then to find subcodes of c1, we use the factorization of t(x) into irreducibles. for each divisor f(x) of t(x) we get a subcode of c1. the following results from [3] are useful in formulating the search. lemma 3.1. [3] fix n, i ∈ z, and let a1, a2, b ∈ f∗q be nonzero with a1 6= a2. then gcd(xn − a1xi − b, xn −a2xi − b) = 1 lemma 3.2. [3] fix n, i ∈ z, and let a, b1, b2 ∈ fq, with b1 6= b2. then gcd(xn−axi−b1, xn−axi−b2) = 1. theorem 3.3. [3] let n and i be positive integers with i < n. for two distinct trinomials xn − axi − b and xn − cxi −d, we have gcd(xn −axi − b, xn − cxi −d) is either 1 or a binomial of degree gcd(n, i). lemma 3.4. [3] for any xn−axi−b 6= xn−cxi−d ∈ f3[x], we have gcd(xn−axi−b, xn−cxi−d) = 1. we present some examples of quantum codes with new parameters in table 1 where t(x) refers to the trinomial, g1(x) is the generator of c1, and h2(x) is the generator of c⊥2 , i.e. 〈g1(x)〉 = c1 and 〈g1(x)f(x)〉 = 〈h2(x)〉 = c⊥2 . the last column in the table refers to the literature where a comparable code is presented. every code in this table has a higher minimum distance than the comparable code presented in the literature. in the tables below, the coefficients of a polynomial are listed in increasing powers of x, with leading coefficient at the right. also, for alphabet of size greater than 10, a denotes 10, b denotes 11, ..., h denotes 17. hence the entry [79f 1 for g1 on the first row of table 2 represents the polynomial 7 + 9x + 15x2 + x3 table 1: new quantum codes constructed from polycyclic codes associated with trinomials [[n, k, d]]q2 t(x) g1(x) h2(x) ref. [[60, 36, 5]]32 x 60 − 2x6 − 2 [1121011011001] [1222021012001] [7] [[81, 52, 5]]32 x 81 − x30 − 1 [200210101101001] [1021111212022001] [29] [[80, 54, 4]]52 x 80 − x − 1 [4231342331] [101124230104302301] [6] 187 n. aydin et. al. / j. algebra comb. discrete appl. 9(3) (2022) 185–191 [[96, 80, 4]]52 x 96 − x − 1 [310032201] [101110301] [2] [[52, 28, 4]]112 x 52 − 2x2 − 2 [7263851] [84782120107151973361] [23] [[11, 1, 5]]132 x 11 − x − 3 [8a0c31] [352721] [20] [[7, 1, 4]]172 x 7 − x2 − 2 [bfc1] [99101] [20] 3.2. css codes by a polycyclic code associated with trinomials and a random linear code in the search above, c1 and c2 are both polycyclic codes associated with the same trinomial. however, this is not required. hence, we can find some f(x) with desired degrees, and then construct c2 by g(x)f(x), where c1 = 〈g(x)〉 and c2 = 〈g(x)f(x)〉, i.e., c2 is not neccessarily polycyclic. note that this method works particularly well for target search by which we mean, we fix the field q, length n and dimension k, and find the best known minimum distance for this set of parameters. next, we randomly generate a number of polynomials f(x) of degree k and iterate through divisors g(x) of t(x), where t(x) is a trinomial, so that the css code constructed by c1 = 〈g(x)〉 and c⊥2 = 〈g(x)f(x)〉 is in the form [[n, k]]q2. table 2: new quantum codes constructed from a polycyclic code and a random linear code [[n, k, d]]q2 t g1 g2 ref. [[7, 1, 4]]172 x 17 − x − 1 [79f1] [36dd1] [20] [[24, 18, 3]]172 x 24 − x − 1 [eg41] [a6872366ca28406ae407f1] [16] [[36, 30, 3]]172 x 36 − x − 1 [3791] [94f9161a374963b0e663e35aa4a6efdgg1] [16] [[48, 36, 4]]172 x 48 − x − 3 [cd3be1] [60942809g709fe411f4910dcc6a7b5bc1545d270681][16] 3.3. css construction by two polycyclic codes associated with multinomials it is also possible to expand the search space by considering multinomials instead of trinomials. even though data suggests that binomials and trinomials have more divisors than other multinomials, we still found some good codes using this search. some good quantum code have been found using this method in [3]. table 3: new quantum codes constructed from multinomials [[n, k, d]]q2 m(x) g1(x) g2(x) ref. [[36, 6, 5]]32 [2011110210112212022010120211000010112102000101111] [222101212100200021] [222101212100200021][26] [[22, 2, 6]]52 [13343122443414240410122] [4223310221] [442143133401] [9] 4. obtaining new codes from existing codes like classical codes, we can also obtain new quantum codes from existing ones using the standard method of extending (e), shortening (s) or puncturing (p) a code or taking the direct sum (ds) of two given codes. all of these standard constructions are available in the magma software. we applied these constructions on exiting coded and obtained many new codes that have better minimum distances than 188 n. aydin et. al. / j. algebra comb. discrete appl. 9(3) (2022) 185–191 the codes with the same length and dimension presented in the literature. in many cases we started with the codes in the tables [9]. table 4: new quantum codes from existing codes [[n, k, d]]q2 existing code(s) (method) [[n ′, k′, d′]]q2 ref. [[41, 1, 10]]32 [[33, 1, 10]] (e) [[41, 1, 8]]32 [9] [[45, 21, 7]]32 [[41, 21, 7]] (e) [[45, 21, 5]]32 [7] [[65, 29, 6]]32 [[25, 7, 6]], [[40, 22, 6]] (ds) [[65, 29, 5]]32 [9] [[119, 7, 13]]32 [[52, 3, 13]], [[67, 4, 13]] (ds) [[119, 7, 11]]32 [9] [[120, 6, 14]]32 [[53, 3, 14]], [[67, 3, 14]] (ds) [[20, 6, 12]]32 [9] [[120, 96, 6]]32 [[121, 96, 7]] (p) [[120, 96, 4]]32 [7] [[34, 2, 8]]52 [[26, 2, 8]] (e) [[34, 2, 6]]52 [9] [[38, 2, 8]]52 [[26, 2, 8]] (e) [[38, 2, 7]]52 [9] [[80, 48, 5]]52 [[11, 1, 5]], [[69, 47, 6]] (ds) [[80, 48, 2]]52 [19] [[80, 54, 7]]52 [[78, 54, 7]] (e) [[80, 54, 3]]52 [6] [[80, 56, 5]]52 [[16, 4, 5]], [[64, 52, 5]] (ds) [[80, 56, 3]]52 [15] [[88, 8, 12]]52 [[52, 8, 12]] (e) [[88, 8, 5]]52 [19] [[88, 48, 7]]52 [[22, 2, 7]], [[66, 46, 7]] (ds) [[88, 48, 2]]52 [25] [[99, 3, 11]]52 [[29, 1, 11]], [[70, 2, 11]] (ds) [[99, 3, 9]]52 [9] [[27, 17, 5]]72 [[25, 17, 5]] (e) [[27, 17, 3]]72 [25] [[49, 40, 5]]72 [[48, 40, 5]] (e) [[49, 40, 3]]72 [29] [[60, 36, 5]]72 [[18, 8, 5]], [[42, 28, 5]] (ds) [[60, 36, 4]]72 [25] [[55, 35, 6]]72 [[58, 35, 9]] (p) [[55, 35, 4]]72 [9] 5. features of the database we introduce an interactive, online database [4] which stores the parameters of best known [[n, k, d]]q2 qeccs over fields gf (q2) for q ≤ 29 where q is a power of a prime and n ≤ 200. users of the database can search for best known qeccs by fixing q, n and k or by fixing q, n and d. also, users can search for multiple qeccs at the same time by fixing q and searching over ranges for n and d. every qecc displayed in the search results has a reference which is a link to the paper in which the code was introduced or an online table where it is listed. in addition, all mds codes are marked with an asterisk. to construct such a database, we surveyed recent literature on qeccs and kept a record of parameters of codes in the literature. the bibliography gives the complete list of papers that have been examined. references [1] a. alahmadi, h. islam, o. prakash, p. sole, a. alkenani, n. muthana, h. rola, new quantum codes from constacyclic codes over a non-chain ring, quantum inf. process 20(2) (2021). 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60 (2021) 172–184. 191 https://doi.org/10.1007/s10773-020-04673-0 https://doi.org/10.1007/s10773-020-04673-0 introduction and motivation preliminaries quantum css codes from polycyclic codes obtaining new codes from existing codes features of the database references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1000842 j. algebra comb. discrete appl. 8(3) • 197–212 received: 30 september 2020 accepted: 20 may 2021 journal of algebra combinatorics discrete structures and applications on metric dimension of plane graphs jn, kn and ln research article sunny kumar sharma, vijay kumar bhat abstract: let γ = γ(v,e) be a simple (i.e., multiple edges and loops and are not allowed), connected (i.e., there exists a path between every pair of vertices), and an undirected (i.e., all the edges are bidirectional) graph. let dγ(%i,%j ) denotes the geodesic distance between two nodes %i,%j ∈ v. the problem of characterizing the classes of plane graphs with constant metric dimensions is of great interest nowadays. in this article, we characterize three classes of plane graphs (viz., jn, kn, and ln) which are generated by taking n-copies of the complete bipartite graph (or a star) k1,5, and all of these plane graphs are radially symmetrical with the constant metric dimension. we show that three vertices is a minimal requirement for the unique identification of all vertices of these three classes of plane graphs. 2010 msc: 05c10, 05c12 keywords: resolving set, metric basis, independent set, metric dimension, planar graph 1. introduction let γ be a simple connected graph with the vertex set v and the edge set e, and let dγ(%i,%j) denotes the geodesic distance between two vertices %i,%j ∈ v. a subset of vertices r ⊆ v is said to be a resolving set (metric generator or locating set) if for every pair of distinct vertices ς,% ∈ v there exists at least one α ∈ r such that dγ(α,ς) 6= dγ(α,%). in other words, for an ordered subset of vertices r = {%1,%2,%3, ...,%k} of γ, any vertex α ∈ v may be represented uniquely in the form of the vector γ(α|r) = (dγ(α,%1),dγ(α,%2), ...,dγ(α,%k)). then r is the metric generator of γ if γ(δ|r) = γ(β|r) implies that δ = β, ∀ β,δ ∈ v. the metric generator r with the minimum possible cardinality is the metric basis for γ, and this minimum cardinality is known as the metric dimension of γ, denoted by β(γ). a set s consisting of vertices of the graph γ is said to be an independent metric generator for γ, if s is both metric generator and independent. for the given ordered subset of vertices, r = {ρ1,ρ2,ρ3, ...,ρk} of γ, the fth component (or distance coordinate) of the code γ(ρ|r) is zero iff ρ = ρf . subsequently, to see that the set r is the metric generator, it is sufficient to prove that ζ(ρ|r) 6= ζ(%|r) for any pair of distinguishable nodes %,ρ ∈ v(γ) \r. sunny kumar sharma, vijay kumar bhat (corresponding author); school of mathematics, shri mata vaishno devi university, katra-182320, india (email: sunnysrrm94@gmail.com, vijaykumarbhat2000@yahoo.com). 197 https://orcid.org/0000-0001-8423-9067 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 the notions of locating or resolving set and that of the metric dimension go back to 1950s. these were characterized by blumenthal [2] with regard to metric space, and were acquainted with graph networks independently by melter and harary in 1976 [6], and slater in 1975 [11]. graph theory has applications in numerous zones of figuring, social, and normal sciences and is likewise an affable play area for the investigation of the verification procedure in discrete science. utilizations of this invariant to the problem of picture preparing (or image processing) and design acknowledgment (or pattern recognition) are given in [9], to the route of exploring specialist (navigating agent or robots) in systems (or networks) are examined in [8], applications to science are given in [5], application to combinatorial enhancement (or optimization) is yielded in [10], and to problems of network discovery and verification in [1]. let ℵ represent a family of connected graphs. we say that the family ℵ has constant metric dimension if dim(γ) does not depend upon the choice of the graph γ in ℵ and is finite. in other words, if all the graphs in ℵ have an indistinguishable metric dimension, at that point ℵ is known as a family with a constant (steady) metric dimension [12]. chartrand et al. in [5], demonstrated that graphs on n vertices have metric dimension one iff it is a path ℘n. additionally, cycle cn has metric dimension two for each positive integer n; n ≥ 3. with this, cn (n ≥ 3) and ℘n (n ≥ 2) establish a family of graphs with a steady metric dimension. additionally, harary graphs h4,n and generalized petersen graphs p(n, 2), are also the families of graphs with constant metric dimension [7]. by joining of two graphs γ1 = γ1(v1,e1) and γ2 = γ2(v2,e2), denoted by γ = γ1 + γ2, we mean a graph γ = γ(v,e) such that v = v1 ∪v2 and e = e1 ∪e2 ∪{%ς : % ∈ v1 and ς ∈ v2}. then a fan graph fm is characterized as fm = k1 + ℘m for m ≥ 1, a wheel graph wm is characterized as wm = k1 +cm, for m ≥ 3, and the jahangir graph j2m (m ≥ 2) is obtained from the wheel graph w2m by alternately deleting m spokes of the wheel graph (which is otherwise known as the gear graph). in [4], caceres et al. decided the location number of the fan graph fm (m ≥ 1) which is b2m+25 c for m /∈ {1, 2, 3, 6}. tomescu and javaid [13] acquired the location number of the jahangir graph j2m (m ≥ 4) which is b2m 3 c, and in [3] chartrand et al. decided the location number of the wheel graph wm (m ≥ 3) which is b2m+2 5 c for m /∈{3, 6}. it is important to note that the metric dimension of these three graphs depend upon the number of vertices in the graph and thus these three families of graphs do not constitute the families with constant metric dimensions. for the simple connected graphs with the metric dimension 2, khuller et al. [8] proved the following important result: theorem 1.1. [8] let a ⊆ v(γ) be the basis set of the connected graph γ = γ(v,e) of cardinality two i.e., |a| = dim(γ) = β(γ) = 2, and say a = {$,ξ}. then, the following are true: 1. between the vertices $ and ξ, there exists a unique shortest path ℘. 2. the valencies (or degrees) of the nodes $ and ξ can never exceed 3. 3. the valency of any other node on ℘ can never exceed 5. the main motivation in characterizing the families of plane graphs with constant metric dimension (or with non-constant metric dimension) is towards making metric dimension of possibly all plane graphs known. in this article, we determine the metric dimension of three classes of plane graphs (viz., jn, kn, and ln) which are generated by taking n-copies of the complete bipartite graph (or a star) k1,5 (see figure 1). these classes of plane graphs are radially symmetric and possess an independent minimum resolving set with cardinality three i.e., three vertices is a minimal requirement for the unique identification of all vertices of these three classes of plane graphs. throughout this article, all vertex indices are taken to be modulo n. in the accompanying section, we acquire the exact metric dimension of the radially symmetrical plane graph jn (see figure 2), and for each positive integer n: n ≥ 6 we prove that β(jn) = 3. 198 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 figure 1. n-copies of the star k1,5 2. metric dimension of the planar graph jn the construction of the plane graph jn can be done in the following four steps: 1. construct n-copies of the complete bipartite graph (or the star) k1,5. denote the central node of each star by rl and the outer nodes of the star k1,5 by pl, ql, sl, al, and bl (1 ≤ l ≤ n). this results in a disconnected graph on 6n nodes with 5n edges (rlpl, rlql, rlsl, rlal, and rlbl for 1 ≤ l ≤ n). 2. placing new edges between these stars as blal+1 and slql+1 for 1 ≤ l ≤ n. this adds 2n new edges. 3. adding n new edges in each star as slql for 1 ≤ l ≤ n. 4. finally, adding n new nodes {cl : 1 6 l 6 n}, and 2n new edges as plcl and clpl+1 for 1 ≤ l ≤ n. thus, the radially symmetrical plane graph jn comprises of 7n nodes and 10n edges. it has n 7-sided cycles, n 6-sided cycles, n 3-sided cycles, and a pair of 2n-sided faces (see figure 2). figure 2. the radially symmetrical graph jn for our purpose, we name the cycle generated by the set of vertices {pl : 1 6 l 6 n}∪{cl : 1 6 l 6 n} in the graph, jn as the inner cycle, the cycle generated by the set of vertices {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} in the graph, jn as the middle cycle, and the cycle generated by the set of vertices {ql : 1 6 l 6 n}∪{sl : 1 6 l 6 n} in the graph, jn as the outer cycle. in the following theorem, we obtain that the minimum cardinality of resolving set for the plane graph, jn is 3 i.e., three vertices is a minimal requirement for the unique identification of all vertices in the graph jn. theorem 2.1. let jn be the planar graph on 7n vertices as defined above. then for each n ≥ 6, we have β(jn) = 3 i.e., it has metric dimension 3. 199 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 proof. to establish this, we study the following two cases relying upon the positive integer n i.e., when n is even and when it is odd. case(1) when the integer n is even. in this case, the positive integer n can be written as n = 2w, where w ∈ n and w ≥ 3. let r = {q1,qw+1,p1}⊂ v(jn). now, to unveil that r is the resolving set for the graph jn, we consign the metric codes for each vertex of the graph jn r r regarding the set r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{cl : 1 6 l 6 n} are γ(pl|r) = { (2l, 2w − 2l + 3, 2l− 2), 2 ≤ l ≤ w (4w − 2l + 3, 2l− 2w, 4w − 2l + 2), w + 1 ≤ l ≤ 2w. and γ(cl|r) =   (2l + 1, 2w − 2l + 2, 2l− 1), 1 ≤ l ≤ w − 1 (2w + 1, 3, 2w − 1), l = w; (4w − 2l + 2, 2l− 2w + 1, 4w − 2l + 1), w + 1 ≤ l ≤ 2w − 1; (3, 2l− 2w + 1, 4w − 2l + 1), l = 2w. the metric codes for the vertices of the middle cycle {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w, 1), l = 1; (2l− 1, 2w − 2l + 2, 2l− 1), 2 ≤ l ≤ w; (2w, 1, 2w + 1), l = w + 1; (4w − 2l + 2, 2l− 2w − 1, 4w − 2l + 3), w + 2 ≤ l ≤ 2w. γ(al|r) =   (2, 2w + 1, 2), l = 1; (2l− 1, 2w − 2l + 3, 2l− 1), 2 ≤ l ≤ w; (2w + 1, 2, 2w + 1), l = w + 1; (4w − 2l + 3, 2l− 2w − 1, 4w − 2l + 4), w + 2 ≤ l ≤ 2w. and γ(bl|r) =   (2, 2w, 2), l = 1; (2l, 2w − 2l + 2, 2l), 2 ≤ l ≤ w − 1; (2w, 3, 2w), l = w; (2w, 2, 2w + 1), l = w + 1; (4w − 2l + 2, 2l− 2w, 4w − 2l + 3), w + 2 ≤ l ≤ 2w − 1; (3, 2l− 2w, 4w − 2l + 3), l = 2w. at last, the metric codes for the vertices of outer cycle {sl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(sl|r) = { (2l− 1, 2w − 2l + 1, 2l), 1 ≤ l ≤ w; (4w − 2l + 1, 2l− 2w − 1, 4w − 2l + 3), w + 1 ≤ l ≤ 2w. and γ(ql|r) = { (2l− 2, 2w − 2l + 2, 2l− 1), 2 ≤ l ≤ w; (4w − 2l + 2, 2l− 2w − 2, 4w − 2l + 4), w + 2 ≤ l ≤ 2w. 200 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 from above, we find that there do not exist two vertices with the same metric codes, which suggest that β(jn) ≤ 3 i.e., the location number of the plane graph jn is less than or equal to 3. now, to finish the evidence for this case, we show that β(jn) ≥ 3 by working out that there does not exist a resolving set r such that |r| = 2. on contrary, suppose that β(jn) = 2. now, from theorem 1.1, we find that the degree of basis vertices can be at most 3. but except the vertices pl, ql, sl, al, bl, and cl (1 ≤ l ≤ n), all other vertices of the radially symmetrical plane graph jn have a degree 5. then, without loss of generality, we suppose that the first resolving vertex is one of the vertices p1, q1, s1, a1, bl, or c1 and other nodes lie in the inner cycle, the middle cycle, and in the outer cycle. therefore, we have the following possibilities to be discussed. resolving sets contradictions {p1,ph}, ph (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(r1|{p1,ph}) = γ(cn|{p1,ph}); and for h = w + 1, we have γ(c1|{p1,pw+1}) = γ(cn|{p1,pw+1}), a contradiction. {c1,ch}, ch (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(r1|{c1,ch}) = γ(cn|{c1,ch}); and for h = w + 1, we have γ(p1|{c1,cw+1}) = γ(p2|{c1,cw+1}), a contradiction. {a1,ah}, ah (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(p1|{a1,ah}) = γ(q1|{a1,ah}), a contradiction. {b1,bh}, bh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(p1|{b1,bh}) = γ(q1|{b1,bh}); and for h = w + 1, we have γ(q1|{b1,bw+1}) = γ(s1|{b1,bw+1}), a contradiction. {q1,qh}, qh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(bn|{q1,qh}) = γ(cn|{q1,qh}); and for h = w + 1, we have γ(sn|{q1,qw+1}) = γ(s1|{q1,qw+1}), a contradiction. {s1,sh}, sh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(bn|{s1,sh}) = γ(cn|{s1,sh}); and for h = w + 1, we have γ(q2|{s1,sw+1}) = γ(q1|{s1,sw+1}), a contradiction. {p1,ch}, ch (1 ≤ h ≤ n) for 1 ≤ h ≤ w, we have γ(cn|{p1,ch}) = γ(r1|{p1,ch}); and for h = w + 1, we have γ(q2|{p1,cw+1}) = γ(a2|{p1,cw+1}), a contradiction. {p1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(q1|{p1,a1}) = γ(b1|{p1,a1}); when h = 2, we have γ(s2|{p1,a2}) = γ(b2|{p1,a2}); when h = 3, we have γ(s3|{p1,a3}) = γ(b3|{p1,a3}); and for 4 ≤ h ≤ w + 1, we have γ(q1|{p1,ah}) = γ(b1|{p1,ah}), a contradiction. {p1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(q1|{p1,b1}) = γ(s1|{p1,b1}); when h = 2, we have γ(a2|{p1,b2}) = γ(q2|{p1,b2}); when 3 ≤ h ≤ w, we have γ(q1|{p1,bh}) = γ(b1|{p1,bh}); and for h = w + 1, we have γ(r2|{p1,bw+1}) = γ(bn|{p1,bw+1}), a contradiction. {p1,qh}, qh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{p1,q1}) = γ(b1|{p1,q1}); when h = 2, we have γ(p2|{p1,q2}) = γ(q1|{p1,q2}); and for 3 ≤ h ≤ w + 1, we have γ(q1|{p1,qh}) = γ(b1|{p1,qh}), a contradiction. {p1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{p1,s1}) = γ(b1|{p1,s1}); when 2 ≤ h ≤ w, we have γ(q1|{p1,sh}) = γ(b1|{p1,sh}); and for h = w + 1, we have γ(r2|{p1,sw+1}) = γ(bn|{p1,sw+1}), a contradiction. {c1,ah}, ah (1 ≤ h ≤ n) for 1 ≤ h ≤ 2, we have γ(s1|{c1,ah}) = γ(q1|{c1,ah}); when h = 3, we have γ(a2|{c1,a3}) = γ(q2|{c1,a3}); and for 4 ≤ h ≤ w + 1, we have γ(a2|{c1,ah}) = γ(s1|{c1,ah}), a contradiction. {c1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(s1|{c1,b1}) = γ(q1|{c1,b1}); when h = 2, we have γ(a2|{c1,a2}) = γ(q2|{c1,a2}); and for 3 ≤ h ≤ w + 1, we have γ(a2|{c1,bh}) = γ(s1|{c1,bh}), a contradiction. {c1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{c1,s1}) = γ(b1|{c1,s1}); and for 2 ≤ h ≤ w + 1, we have γ(a2|{c1,sh}) = γ(s1|{c1,sh}), a contradiction. {c1, th}, th (1 ≤ h ≤ n) for h = 1, we have γ(a1|{c1, t1}) = γ(b1|{c1, t1}); when h = 2, we have γ(a2|{c1, t2}) = γ(b2|{c1, t2}); and for 3 ≤ h ≤ w + 1, we have γ(a2|{c1, th}) = γ(s1|{c1, th}), a contradiction. {a1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(s1|{a1,b1}) = γ(q1|{a1,b1}); when h = 2, we have γ(a2|{a1,b2}) = γ(q2|{a1,b2}); when h = 3, we have γ(a3|{a1,b3}) = γ(q2|{a1,b3}); when 4 ≤ h ≤ w, we have γ(q1|{a1,bh}) = γ(b1|{a1,bh}); and for h = w + 1, we have γ(an|{a1,bw+1}) = γ(sn|{a1,bw+1}), a contradiction. {a1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(b1|{a1,s1}) = γ(p1|{a1,s1}); when 2 ≤ h ≤ w, we have γ(q1|{a1,sh}) = γ(b1|{a1,sh}); and for h = w + 1, we have γ(an|{a1,sw+1}) = γ(sn|{a1,sw+1}), a contradiction. 201 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 resolving sets contradictions {a1,qh}, qh (1 ≤ h ≤ n) for h = 1, we have γ(b1|{a1,q1}) = γ(p1|{a1,q1}); when h = 2, we have γ(sn|{a1,q2}) = γ(c1|{a1,q2}); and for 3 ≤ h ≤ w + 1, we have γ(q1|{a1,qh}) = γ(b1|{a1,qh}), a contradiction. {b1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{b1,s1}) = γ(p1|{b1,s1}); when h = 2, we have γ(b2|{b1,s2}) = γ(p2|{b1,s2}); and for 3 ≤ h ≤ w + 1, we have γ(q2|{b1,sh}) = γ(b2|{b1,sh}), a contradiction. {b1,qh}, qh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{b1,q1}) = γ(p1|{b1,q1}); when 2 ≤ h ≤ 3, we have γ(b2|{b1,q2}) = γ(p2|{b1,q2}); and for 4 ≤ h ≤ w + 1, we have γ(q2|{b1,qh}) = γ(b2|{b1,qh}), a contradiction. {s1,qh}, qh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{s1,q1}) = γ(b1|{s1,q1}); and for 2 ≤ h ≤ w + 1, we have γ(r1|{s1,qh}) = γ(q1|{s1,qh}), a contradiction. thus, from the above table, we obtain that there does not exist a resolving set consisting of two vertices for v(jn), suggesting that β(jn) = 3 in this case. case(2) when the integer n is odd. in this case, the positive integer n can be written as n = 2w + 1, where w ∈ n and w ≥ 3. let r = {q1,qw+1,p1}⊂ v(jn). now, to unveil that r is the resolving set for the graph jn, we consign the metric codes for each vertex of the graph jn r r regarding the set r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{cl : 1 6 l 6 n} are γ(pl|r) =   (2l, 2w − 2l + 3, 2l− 2), 2 ≤ l ≤ w (2w + 2, 2, 2w), l = w + 1; (4w − 2l + 5, 2l− 2w, 4w − 2l + 4), w + 2 ≤ l ≤ 2w + 1. and γ(cl|r) =   (2l + 1, 2w − 2l + 2, 2l− 1), 1 ≤ l ≤ w − 1 (2w + 1, 3, 2w − 1), l = w; (4w − 2l + 4, 2l− 2w + 1, 4w − 2l + 3), w + 1 ≤ l ≤ 2w; (3, 2l− 2w, 1), l = 2w + 1. the metric codes for the vertices of the middle cycle {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w, 1), l = 1; (2l− 1, 2w − 2l + 2, 2l− 1), 2 ≤ l ≤ w; (2w + 1, 1, 2w + 1), l = w + 1; (4w − 2l + 4, 2l− 2w − 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w + 1. γ(al|r) =   (2, 2w + 1, 2), l = 1; (2l− 1, 2w − 2l + 3, 2l− 1), 2 ≤ l ≤ w; (2w + 1, 2, 2w + 1), l = w + 1; (4w − 2l + 5, 2l− 2w − 1, 4w − 2l + 6), w + 2 ≤ l ≤ 2w + 1. and γ(bl|r) =   (2, 2w, 2), l = 1; (2l, 2w − 2l + 2, 2l), 2 ≤ l ≤ w − 1; (2w, 3, 2w), l = w; (2w + 2, 2, 2w + 2), l = w + 1; (4w − 2l + 4, 2l− 2w, 4w − 2l + 5), w + 2 ≤ l ≤ 2w; (3, 2l− 2w, 4w − 2l + 5), l = 2w + 1. 202 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 at last, the metric codes for the vertices of outer cycle {sl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(sl|r) =   (2l− 1, 2w − 2l + 1, 2l), 1 ≤ l ≤ w; (2w + 1, 1, 2w + 2), l = w + 1; (4w − 2l + 3, 2l− 2w − 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w + 1. and γ(ql|r) = { (2l− 2, 2w − 2l + 2, 2l− 1), 2 ≤ l ≤ w; (4w − 2l + 4, 2l− 2w − 2, 4w − 2l + 6), w + 2 ≤ l ≤ 2w + 1. again, we find that there do not exist two vertices with the same metric codes, which suggest that β(jn) ≤ 3 i.e., the location number of the plane graph jn is less than or equal to 3. now, on assuming that β(jn) = 2, we get the same eventualities as in case(1), and similarly, the contradiction can be obtained. so, in this case, we have β(jn) = 3 as well and hence the theorem. now, in terms of independent resolving set, we have the following result: theorem 2.2. let jn be the planar graph on 7n vertices as defined above. then for every positive integer n; n ≥ 6, its independent resolving number is 3. proof. for proof, refer to theorem 2.1. in the accompanying section, we acquire the exact metric dimension of the radially symmetrical plane graph kn (see figure 3), and for each positive integer n: n ≥ 6 we prove that β(kn) = 3. 3. metric dimension of the planar graph kn the construction of the plane graph kn can be done in the following four steps: 1. construct n-copies of the complete bipartite graph (or the star) k1,5. denote the central node of each star by rl and the outer nodes of the star k1,5 by pl, ql, sl, al, and bl (1 ≤ l ≤ n). this results in a disconnected graph on 6n nodes with 5n edges (rlpl, rlql, rlsl, rlal, and rlbl for 1 ≤ l ≤ n). 2. placing new edges between these stars as qlpl+1 and blal+1 for 1 ≤ l ≤ n. this adds 2n new edges. 3. adding n new edges in each of these stars as plql for 1 ≤ l ≤ n. 4. finally, adding n new nodes {tl : 1 6 l 6 n}, and 2n new edges as sltl and tlsl+1 for 1 ≤ l ≤ n. thus, the radially symmetrical plane graph kn comprises of 7n nodes and 10n edges. it has n 7-sided cycles, n 6-sided cycles, n 3-sided cycles, and a pair of 2n-sided faces (see figure 3). for our purpose, we name the cycle generated by the set of vertices {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} in the graph, kn as the inner cycle, the cycle generated by the set of vertices {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} in the graph, kn as the middle cycle, and the cycle generated by the set of vertices {tl : 1 6 l 6 n}∪{sl : 1 6 l 6 n} in the graph, kn as the outer cycle. in the following theorem, we obtain that the minimum cardinality of resolving set for the plane graph, kn is 3 i.e., three vertices is a minimal requirement for the unique identification of all vertices in the graph kn. theorem 3.1. let kn be the planar graph on 7n vertices as defined above. then for each n ≥ 6, we have β(kn) = 3 i.e., it has metric dimension 3. 203 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 figure 3. the radially symmetrical graph kn proof. to establish this, we study the following two cases relying upon the positive integer n i.e., when n is even and when it is odd. case(1) when the integer n is even. in this case, the positive integer n can be written as n = 2w, where w ∈ n and w ≥ 3. let r = {p1,pw+1,s2}⊂ v(kn). now, to unveil that r is the resolving set for the graph kn, we consign the metric codes for each vertex of the graph kn r r regarding r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(pl|r) =   (2, 2w − 2, 2), l = 2; (2l− 2, 2w − 2l + 2, 2l− 3), 3 ≤ l ≤ w (4w − 2l + 2, 2, 2w + 1), l = w + 2; (4w − 2l + 2, 2l− 2w − 2, 4w − 2l + 6), w + 3 ≤ l ≤ 2w. and γ(ql|r) =   (1, 2w − 1, 3), l = 1; (2l− 1, 2w − 2l + 1, 2l− 2), 2 ≤ l ≤ w (4w − 2l + 1, 1, 2w), l = w + 1; (4w − 2l + 1, 2l− 2w − 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w. the metric codes for the vertices of the middle cycle {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w, 3), l = 1; (2l− 1, 2w − 2l + 2, 2l− 3), 2 ≤ l ≤ w; (2w, 1, 2w − 1), l = w + 1; (4w − 2l + 2, 2l− 2w − 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w. γ(al|r) =   (2, 2w + 1, 4), l = 1; (3, 2w − 1, 2), l = 2; (2l− 1, 2w − 2l + 3, 2l− 3), 3 ≤ l ≤ w; (2w + 1, 2, 2w − 1), l = w + 1; (4w − 2l + 3, 2l− 2w − 1, 2w + 1), l = w + 2; (4w − 2l + 3, 2l− 2w − 1, 4w − 2l + 6), w + 3 ≤ l ≤ 2w. 204 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 and γ(bl|r) =   (2, 2w, 3), l = 1; (2l, 2w − 2l + 2, 2l− 2), 2 ≤ l ≤ w − 1; (2w, 3, 2w − 2), l = w; (2w, 2, 2w), l = w + 1; (4w − 2l + 2, 2l− 2w, 4w − 2l + 5), w + 2 ≤ l ≤ 2w − 1; (3, 2l− 2w, 4w − 2l + 5), l = 2w. at last, metric codes for the vertices of outer cycle {sl : 1 6 l 6 n}∪{tl : 1 6 l 6 n} are γ(sl|r) =   (2l, 2w − 2l + 3, 2), l = 1; (2l, 2w − 2l + 3, 2l− 4), 3 ≤ l ≤ w; (2w + 1, 2, 2w − 2), l = w + 1; (4w − 2l + 3, 2l− 2w, 4w − 2l + 4), w + 2 ≤ l ≤ 2w. and γ(tl|r) =   (3, 2w, 1), l = 1; (2l + 1, 2w − 2l + 2, 2l− 3), 2 ≤ l ≤ w − 1; (2w + 1, 3, 2w − 3), l = w; (4w − 2l + 2, 3, 2w − 1), l = w + 1; (4w − 2l + 2, 2l− 2w + 1, 4w − 2l + 3), w + 2 ≤ l ≤ 2w − 1; (3, 2l− 2w + 1, 4w − 2l + 3), l = 2w. from above, we find that there do not exist two vertices with the same metric codes, which suggest that β(kn) ≤ 3 i.e., the location number of the plane graph kn is less than or equal to 3. now, so as to finish the evidence for this case, we show that β(kn) ≥ 3 by working out that there does not exist a resolving set r such that |r| = 2. on contrary, suppose that β(kn) = 2. now, from theorem 1.1, we find that the degree of basis vertices can be at most 3. but except the vertices pl, ql, sl, al, bl, and tl (1 ≤ l ≤ n), all other vertices of the radially symmetrical plane graph kn have a degree 5. then, without loss of generality, we suppose that the first resolving vertex is one of the vertices p1, q1, s1, a1, bl or t1 and other nodes lie in the inner cycle, the middle cycle, and in the outer cycle. therefore, we have the following possibilities to be discussed. resolving sets contradictions {p1,ph}, ph (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(a1|{p1,ph}) = γ(s1|{p1,ph}), a contradiction. {q1,qh}, qh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(a1|{q1,qh}) = γ(s1|{q1,qh}); and for h = w + 1, we have γ(p2|{q1,qw+1}) = γ(p1|{q1,qw+1}), a contradiction. {a1,ah}, ah (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(p2|{a1,ah}) = γ(s2|{a1,ah}), a contradiction. {b1,bh}, bh (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(p2|{b1,bh}) = γ(s2|{b1,bh}), a contradiction. {s1,sh}, sh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(r1|{s1,sh}) = γ(tn|{s1,sh}); and for h = w + 1, we have γ(t1|{s1,sw+1}) = γ(tn|{s1,sw+1}), a contradiction. {t1, th}, th (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(r1|{t1, th}) = γ(tn|{t1, th}); and for h = w + 1, we have γ(s1|{t1, tw+1}) = γ(s2|{t1, tw+1}), a contradiction. {p1,qh}, qh (1 ≤ h ≤ n) for 1 ≤ h ≤ w, we have γ(s1|{p1,qh}) = γ(a1|{p1,qh}); and for h = w + 1, we have γ(bn|{p1,qw+1}) = γ(r2|{p1,qw+1}), a contradiction. {p1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(s1|{p1,a1}) = γ(b1|{p1,a1}); when h = 2, we have γ(s2|{p1,a2}) = γ(b2|{p1,a2}); when h = 3, we have γ(s3|{p1,a3}) = γ(b3|{p1,a3}); and for 4 ≤ h ≤ w + 1, we have γ(b1|{p1,ah}) = γ(s1|{p1,ah}), a contradiction. {p1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(s1|{p1,b1}) = γ(a1|{p1,b1}); when h = 2, we have γ(sn|{p1,b2}) = γ(bn|{p1,b2}); when 3 ≤ h ≤ w, we have γ(s1|{p1,bh}) = γ(b1|{p1,bh}); and for h = w + 1, we have γ(b1|{p1,bh}) = γ(s1|{p1,bh}), a contradiction. 205 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 resolving sets contradictions {p1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(b1|{p1,s1}) = γ(a1|{p1,s1}); and for 2 ≤ h ≤ w + 1, we have γ(t1|{p1,sh}) = γ(r2|{p1,sh}), a contradiction. {p1, th}, th (1 ≤ h ≤ n) for h = 1, we have γ(b1|{p1, t1}) = γ(a1|{p1, t1}); and for 2 ≤ h ≤ w + 1, we have γ(t1|{p1, th}) = γ(r2|{p1, th}), a contradiction. {q1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(b1|{q1,a1}) = γ(s1|{q1,a1}); when h = 2, we have γ(a1|{q1,a2}) = γ(s1|{q1,a2}); when h = 3, we have γ(b3|{q1,a3}) = γ(s3|{q1,a3}); and for 4 ≤ h ≤ w + 1, we have γ(s1|{q1,ah}) = γ(b1|{q1,ah}), a contradiction. {q1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(a1|{q1,b1}) = γ(s1|{q1,b1}); when h = 2, we have γ(a2|{q1,b2}) = γ(s2|{q1,b2}); when 3 ≤ h ≤ w, we have γ(b1|{q1,bh}) = γ(s1|{q1,bh}); and for h = w + 1, we have γ(rn|{q1,bw+1}) = γ(a2|{q1,bw+1}), a contradiction. {q1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(b1|{q1,s1}) = γ(a1|{q1,s1}), when h = 2, we have γ(a2|{q1,s2}) = γ(b2|{q1,s2}); and for 3 ≤ h ≤ w + 1, we have γ(r2|{q1,sh}) = γ(q2|{q1,sh}), a contradiction. {q1, th}, th (1 ≤ h ≤ n) for h = 1, we have γ(b1|{q1, t1}) = γ(a1|{q1, t1}); when h = 2, we have γ(a2|{q1, t2}) = γ(b2|{q1, t2}); and for 3 ≤ h ≤ w + 1, we have γ(r2|{q1, th}) = γ(q2|{q1, th}), a contradiction. {a1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(p1|{a1,b1}) = γ(q1|{a1,b1}); when h = 2, we have γ(s2|{a1,b2}) = γ(q2|{a1,b2}); and for 3 ≤ h ≤ w + 1, we have γ(s1|{a1,bh}) = γ(b1|{a1,bh}), a contradiction. {a1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(p1|{a1,s1}) = γ(q1|{a1,s1}), when h = 2, we have γ(a2|{a1,s2}) = γ(p2|{a1,s2}), and for 3 ≤ h ≤ w + 1, we have γ(r2|{a1,sh}) = γ(q2|{a1,sh}), a contradiction. {a1, th}, th (1 ≤ h ≤ n) for h = 1, we have γ(p1|{a1, t1}) = γ(q1|{a1, t1}); when h = 2, we have γ(a2|{a1, t2}) = γ(p2|{a1, t2}); and for 3 ≤ h ≤ w + 1, we have γ(r2|{a1, th}) = γ(q2|{a1, th}), a contradiction. {b1,sh}, sh (1 ≤ h ≤ n) for h = 1, we have γ(p1|{b1,s1}) = γ(q1|{b1,s1}); when h = 2, we have γ(q2|{b1,s2}) = γ(p2|{b1,s2}); and for 3 ≤ h ≤ w + 1, we have γ(b2|{b1,sh}) = γ(t1|{b1,sh}), a contradiction. {b1, th}, th (1 ≤ h ≤ n) for h = 1, we have γ(p1|{b1, t1}) = γ(q1|{b1, t1}); when h = 2, we have γ(q2|{b1, t2}) = γ(p2|{b1, t2}); and for 3 ≤ h ≤ w + 1, we have γ(b2|{b1, th}) = γ(t1|{b1, th}), a contradiction. {s1, th}, th (1 ≤ h ≤ n) for 1 ≤ h ≤ w, we have γ(r1|{s1, th}) = γ(tn|{s1, th}); and for h = w + 1, we have γ(b2|{s1, tw+1}) = γ(r2|{s1, tw+1}), a contradiction. thus, from the above table, we obtain that there does not exist a resolving set consisting of two vertices for v(kn), suggesting that β(kn) = 3 in this case. case(2) when the integer n is odd. in this case, the positive integer n can be written as n = 2w + 1, where w ∈ n and w ≥ 3. let r = {p1,pw+1,s2}⊂ v(kn). now, to unveil that r is the resolving set for the graph kn, we consign the metric codes for each vertex of the graph kn r r regarding the set r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(pl|r) =   (2, 2w − 2, 2), l = 2; (2l− 2, 2w − 2l + 2, 2l− 3), 3 ≤ l ≤ w (4w − 2l + 4, 2, 2w + 1), l = w + 2; (4w − 2l + 4, 2l− 2w − 2, 4w − 2l + 8), w + 3 ≤ l ≤ 2w + 1. 206 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 and γ(ql|r) =   (1, 2w − 1, 3), l = 1; (2l− 1, 2w − 2l + 1, 2l− 2), 2 ≤ l ≤ w (4w − 2l + 3, 1, 2w), l = w + 1; (4w − 2l + 3, 2l− 2w − 1, 2w + 2), l = w + 2; (4w − 2l + 3, 2l− 2w − 1, 4w − 2l + 7), w + 3 ≤ l ≤ 2w + 1. the metric codes for the vertices of the middle cycle {rl : 1 6 l 6 n}∪{al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w, 3), l = 1; (2l− 1, 2w − 2l + 2, 2l− 3), 2 ≤ l ≤ w; (2w + 1, 1, 2w − 1), l = w + 1; (4w − 2l + 4, 3, 2w + 1), l = w + 2; (4w − 2l + 4, 2l− 2w − 1, 4w − 2l + 7), w + 3 ≤ l ≤ 2w + 1. γ(al|r) =   (2, 2w + 1, 4), l = 1; (3, 2w − 1, 2), l = 2; (2l− 1, 2w − 2l + 3, 2l− 3), 3 ≤ l ≤ w; (2w + 1, 2, 2w − 1), l = w + 1; (4w − 2l + 5, 2l− 2w − 1, 2w + 1), l = w + 2; (4w − 2l + 5, 2l− 2w − 1, 4w − 2l + 8), w + 3 ≤ l ≤ 2w + 1. and γ(bl|r) =   (2, 2w, 3), l = 1; (2l, 2w − 2l + 2, 2l− 2), 2 ≤ l ≤ w − 1; (2w, 3, 2w − 2), l = w; (2w + 2, 2, 2w), l = w + 1; (4w − 2l + 4, 2l− 2w, 2w + 2), l = w + 2; (4w − 2l + 4, 2l− 2w, 4w − 2l + 7), w + 3 ≤ l ≤ 2w; (3, 2l− 2w, 4w − 2l + 7), l = 2w + 1. at last, the metric codes for the vertices of outer cycle {sl : 1 6 l 6 n}∪{tl : 1 6 l 6 n} are γ(sl|r) =   (2l, 2w − 2l + 3, 2), l = 1; (2l, 2w − 2l + 3, 2l− 4), 3 ≤ l ≤ w; (2w + 2, 2, 2w − 2), l = w + 1; (4w − 2l + 5, 2l− 2w, 2w), l = w + 2; (4w − 2l + 5, 2l− 2w, 4w − 2l + 6), w + 3 ≤ l ≤ 2w + 1. and γ(tl|r) =   (3, 2w, 1), l = 1; (2l + 1, 2w − 2l + 2, 2l− 3), 2 ≤ l ≤ w − 1; (2w + 1, 3, 2w − 3), l = w; (4w − 2l + 4, 3, 2w − 1), l = w + 1; (4w − 2l + 4, 2l− 2w + 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w; (3, 2w + 2, 4w − 2l + 5), l = 2w + 1. 207 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 again, we find that there do not exist two vertices with the same metric codes, which suggest that β(kn) ≤ 3 i.e., the location number of the plane graph kn is less than or equal to 3. now, on assuming that β(kn) = 2, we get the same eventualities as in case(1), and similarly, the contradiction can be obtained. so, in this case, we have β(kn) = 3 as well and hence the theorem. now, in terms of independent resolving set, we have the following result: theorem 3.2. let kn be the planar graph on 7n vertices as defined above. then for every positive integer n; n ≥ 6, its independent resolving number is 3. proof. for proof, refer to theorem 3.1. in the accompanying section, we acquire the exact metric dimension of the radially symmetrical plane graph ln (see figure 4), and for each positive integer n: n ≥ 6 we prove that β(ln) = 3. 4. metric dimension of the planar graph ln the construction of the plane graph ln can be done in the following three steps: 1. construct n-copies of the complete bipartite graph (or the star) k1,5. denote the central node of each star by rl and the outer nodes of the star k1,5 by pl, ql, sl, al, and bl (1 ≤ l ≤ n). this results in a disconnected graph on 6n nodes with 5n edges (rlpl, rlql, rlsl, rlal, and rlbl for 1 ≤ l ≤ n). 2. placing new edges between these stars as qlpl+1 and blal+1 for 1 ≤ l ≤ n. this adds 2n new edges. 3. finally adding 2n new edges in each star as plql and albl for 1 ≤ l ≤ n. thus, the radially symmetrical plane graph ln comprises of 6n nodes and 9n edges. it has n 6-sided cycles, 2n 3-sided cycles, and a pair of 2n-sided faces (see figure 4). figure 4. the radially symmetrical graph ln for our purpose, we name the cycle generated by the set of vertices {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} in the graph, ln as the inner cycle, the set of vertices {rl : 1 6 l 6 n}∪{sl : 1 6 l 6 n} in the graph, ln as the set of middle vertices, and the cycle generated by the set of vertices {al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} in the graph, ln as the outer cycle. in the following theorem, we obtain that the minimum cardinality of resolving set for the plane graph, ln is 3 i.e., three vertices is a minimal requirement for the unique identification of all vertices in the graph ln. 208 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 theorem 4.1. let ln be the planar graph on 6n vertices as defined above. then for each n > 6, we have β(ln) = 3 i.e., it has metric dimension 3. proof. in order to establish this, we study the following two cases relying upon the positive integer n i.e., when n is even natural and when it is odd. case(1) when the integer n is even. in this case, the positive integer n can be written as n = 2w, where w ∈ n and w ≥ 3. let r = {p1,pw+1,s1}⊂ v(ln). now, to unveil that r is the resolving set for the graph ln, we consign the metric codes for each vertex of the graph ln r r regarding the set r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(pl|r) = { (2l− 2, 2w − 2l + 2, 2l− 1), 2 ≤ l ≤ w (4w − 2l + 2, 2l− 2w − 2, 4w − 2l + 4), w + 2 ≤ l ≤ 2w. and γ(ql|r) = { (2l− 1, 2w − 2l + 1, 2l), 1 ≤ l ≤ w (4w − 2l + 1, 2l− 2w − 1, 4w − 2l + 3), w + 1 ≤ l ≤ 2w. the metric codes for the set of middle vertices {rl : 1 6 l 6 n}∪{sl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w, 1), l = 1; (2l− 1, 2w − 2l + 2, 2l), 2 ≤ l ≤ w; (4w − 2l + 2, 2l− 2w − 1, 4w − 2l + 4), w + 1 ≤ l ≤ 2w. , and γ(sl|r) = { (2l, 2w − 2l + 3, 2l + 1), 2 ≤ l ≤ w; (4w − 2l + 3, 2l− 2w, 4w − 2l + 5), w + 1 ≤ l ≤ 2w. at last, the metric codes for the vertices of outer cycle {al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(al|r) =   (2, 2w + 1, 2), l = 1; (2l− 1, 2w − 2l + 3, 2l− 1), 2 ≤ l ≤ w; (2w + 1, 2, 2w + 1), l = w + 1; (4w − 2l + 3, 2l− 2w − 1, 4w − 2l + 4), w + 2 ≤ l ≤ 2w. and γ(bl|r) =   (2, 2w, 2), l = 1; (2l, 2w − 2l + 2, 2l), 2 ≤ l ≤ w; (4w − 2l + 2, 2l− 2w, 4w − 2l + 3), w + 1 ≤ l ≤ 2w − 1; (3, 2l− 2w, 4w − 2l + 3), l = 2w. from above, we find that there do not exist two vertices with the same metric codes, which suggest that β(ln) ≤ 3 i.e., the location number of the plane graph ln is less than or equal to 3. now, so as to finish the evidence for this case, we show that β(ln) ≥ 3 by working out that there does not exist a resolving set r such that |r| = 2. on contrary, suppose that β(ln) = 2. now, from theorem 1.1, we find that the degree of basis vertices can be at most 3. but except the vertices pl, ql, sl, al, and bl (1 ≤ l ≤ n), all other vertices of the radially symmetrical plane graph ln have a degree 5. then, without loss of generality, we suppose that the first resolving vertex is one of the vertices p1, q1, s1, a1 or bl, and 209 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 other nodes lie in the inner cycle, the set of middle nodes, and in the outer cycle. therefore, we have the following possibilities to be discussed. resolving sets contradictions {p1,ph}, ph (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(a1|{p1,ph}) = γ(s1|{p1,ph}), a contradiction. {q1,qh}, qh (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(a1|{q1,qh}) = γ(s1|{q1,qh}); and for h = w + 1, we have γ(p2|{q1,qw+1}) = γ(p1|{q1,qw+1}), a contradiction. {s1,sh}, sh (2 ≤ h ≤ n) for 2 ≤ h ≤ w + 1, we have γ(q1|{s1,sh}) = γ(b1|{s1,sh}), a contradiction. {a1,ah}, ah (2 ≤ h ≤ n) for 2 ≤ h ≤ w, we have γ(an|{a1,ah}) = γ(rn|{a1,ah}); and for h = w + 1, we have γ(b1|{a1,aw+1}) = γ(bn|{a1,aw+1}), a contradiction. {b1,bh}, bh (2 ≤ h ≤ n) for 2 ≤ h ≤ w − 1, we have γ(an|{b1,bh}) = γ(rn|{b1,bh}); when h = w, we have γ(s2|{b1,bw}) = γ(p2|{b1,bw}); and for h = w + 1, we have γ(a1|{b1,bw+1}) = γ(a2|{b1,bw+1}), a contradiction. {p1,qh}, qh (1 ≤ h ≤ n) for 1 ≤ h ≤ w, we have γ(a1|{p1,qh}) = γ(s1|{p1,qh}); and for h = w + 1, we have γ(bn|{p1,qw+1}) = γ(sn|{p1,qw+1}), a contradiction. {p1,sh}, sh (1 ≤ h ≤ n) for 1 ≤ h ≤ w − 1, we have γ(pn|{p1,sh}) = γ(rn|{p1,sh}); and for h = w,w + 1, we have γ(r2|{p1,sh}) = γ(a2|{p1,sh}), a contradiction. {p1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(pn|{p1,a1}) = γ(p2|{p1,a1}); when h = 2, we have γ(p2|{p1,a2}) = γ(a1|{p1,a2}); and for 3 ≤ h ≤ w + 1, we have γ(r1|{p1,ah}) = γ(q1|{p1,ah}), a contradiction. {p1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(rn|{p1,b1}) = γ(p2|{p1,b1}); and for 2 ≤ h ≤ w + 1, we have γ(r1|{p1,bh}) = γ(q1|{p1,bh}), a contradiction. {q1,sh}, sh (1 ≤ h ≤ n) for 1 ≤ h ≤ w − 1, we have γ(pn|{q1,sh}) = γ(rn|{q1,sh}); when h = w, we have γ(p1|{q1,sw}) = γ(r1|{q1,sw}); and for h = w + 1, we have γ(bn|{q1,sw+1}) = γ(rn|{q1,sw+1}), a contradiction. {q1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(qn|{q1,a1}) = γ(r2|{q1,a1}); when h = 2, we have γ(b1|{q1,a2}) = γ(r2|{q1,a2}); when h = 3, we have γ(b1|{q1,a3}) = γ(q2|{q1,a3}); and for 4 ≤ h ≤ w + 1, we have γ(q2|{q1,ah}) = γ(r2|{q1,ah}), a contradiction. {q1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(s1|{q1,b1}) = γ(r2|{q1,b1}); when h = 2, we have γ(b1|{q1,b2}) = γ(q2|{q1,b2}); and for 3 ≤ h ≤ w + 1, we have γ(q2|{q1,bh}) = γ(r2|{q1,bh}), a contradiction. {s1,ah}, ah (1 ≤ h ≤ n) for h = 1, we have γ(p1|{s1,a1}) = γ(q1|{s1,a1}); when h = 2, we have γ(b2|{s1,a2}) = γ(r2|{s1,a2}); and for 3 ≤ h ≤ w + 1, we have γ(q1|{s1,ah}) = γ(a1|{s1,ah}), a contradiction. {s1,bh}, bh (1 ≤ h ≤ n) for h = 1, we have γ(p1|{s1,b1}) = γ(q1|{s1,b1}); and for 2 ≤ h ≤ w + 1, we have γ(a1|{s1,bh}) = γ(q1|{s1,bh}), a contradiction. {a1,bh}, bh (1 ≤ h ≤ n) for 1 ≤ h ≤ w − 1, we have γ(rn|{a1,bh}) = γ(an|{a1,bh}); when h = w, we have γ(qn|{a1,bw}) = γ(sn|{a1,bw}); and for h = w + 1, we have γ(r2|{a1,bw+1}) = γ(qn|{a1,bw+1}), a contradiction. thus, from the above table, we obtain that there does not exist a resolving set consisting of two vertices for v(ln), suggesting that β(ln) = 3 in this case. case(2) when the integer n is odd. in this case, the positive integer n can be written as n = 2w + 1, where w ∈ n and w ≥ 3. let r = {p1,qw+1,s1}⊂ v(ln). now, to unveil that r is the resolving set for the graph ln, we consign the metric codes for each vertex of the graph ln r r regarding the set r. now, the metric codes for the vertices of inner cycle {pl : 1 6 l 6 n}∪{ql : 1 6 l 6 n} are γ(pl|r) =   (2l− 2, 2w − 2l + 3, 2l− 1), 2 ≤ l ≤ w; (2l− 2, 1, 2l− 1), l = w + 1; (4w − 2l + 4, 2l− 2w − 3, 4w − 2l + 6), w + 2 ≤ l ≤ 2w + 1. 210 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 and γ(ql|r) = { (2l− 1, 2w − 2l + 2, 2l), 1 ≤ l ≤ w; (4w − 2l + 3, 2l− 2w − 2, 4w − 2l + 5), w + 2 ≤ l ≤ 2w + 1. the metric codes for the set of middle vertices {rl : 1 6 l 6 n}∪{sl : 1 6 l 6 n} are γ(rl|r) =   (1, 2w + 1, 1), l = 1; (2l− 1, 2w − 2l + 3, 2l), 2 ≤ l ≤ w; (2w + 1, 1, 2w + 2), l = w + 1; (4w − 2l + 4, 2l− 2w − 2, 4w − 2l + 6), w + 2 ≤ l ≤ 2w + 1. and γ(sl|r) = { (2l, 2w − 2l + 4, 2l + 1), 2 ≤ l ≤ w + 1; (4w − 2l + 5, 2l− 2w − 1, 4w − 2l + 7), w + 2 ≤ l ≤ 2w + 1. at last, the metric codes for the vertices of outer cycle {al : 1 6 l 6 n}∪{bl : 1 6 l 6 n} are γ(al|r) =   (2, 2w + 2, 2), l = 1; (2l− 1, 2w − 2l + 4, 2l− 1), 2 ≤ l ≤ w + 1; (4w − 2l + 5, 3, 4w − 2l + 6), l = w + 2; (4w − 2l + 5, 2l− 2w − 2, 4w − 2l + 6), w + 3 ≤ l ≤ 2w + 1. and γ(bl|r) =   (2, 2w + 1, 2), l = 1; (2l, 2w − 2l + 3, 2l), 2 ≤ l ≤ w; (2w + 2, 2, 2w + 2), l = w + 1; (4w − 2l + 4, 2l− 2w − 1, 4w − 2l + 5), w + 2 ≤ l ≤ 2w; (3, 2l− 2w − 1, 4w − 2l + 5), l = 2w + 1. again, we find that there do not exist two vertices with the same metric codes, which suggest that β(ln) ≤ 3 i.e., the location number of the plane graph ln is less than or equal to 3. now, on assuming that β(ln) = 2, we get the same eventualities as in case(1), and similarly, the contradiction can be obtained. so, in this case, we have β(ln) = 3 as well and hence the theorem. now, in terms of independent resolving set, we have the following result: theorem 4.2. let ln be the planar graph on 7n vertices as defined above. then for every positive integer n; n ≥ 6, its independent resolving number is 3. proof. for proof, refer to theorem 4.1. 5. conclusion in this article, we determined the metric dimension of three classes of plane graphs (viz., jn, kn, and ln), which are generated by taking n-copies of the complete bipartite graph (or a star) k1,5, and are radially symmetric with the constant metric dimension. we have proved that the metric dimension of these three classes of plane graphs is finite and is independent of the number of vertices in these graphs and three vertices is a minimal requirement for the unique identification of all vertices of these 211 s. k. sharma, v. k. bhat / j. algebra comb. discrete appl. 8(3) (2021) 197–212 three classes of plane graphs. we also observed that the basis set r is independent for all of these three families of plane graphs. we now have an open problem that naturally arises from the text. open problem: characterize those classes of radially symmetrical graphs mn which are generated by taking n-copies of the complete bipartite graph (or a star) k1,5 with constant or non-constant metric dimension. acknowledgment: the authors would like to thank the referee for careful reading of the paper, remarks and suggestions to give the paper the present shape.. references [1] z. beerliova, f. eberhard, t. erlebach, a. hall, m. hoffman, m. mihalak, l. s. ram, network discovery and verification, ieee j. sel. areas commun. 24 (2006) 2168–2181. 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[13] i. tomescu, i. javaid, on the metric dimension of the jahangir graph, bull. math. soc. sci. math. roumanie 50(98) (2007) 371-376. 212 https://doi.org/10.1109/jsac.2006.884015 https://doi.org/10.1109/jsac.2006.884015 https://doi.org/10.1023/a:1025745406160 https://doi.org/10.1023/a:1025745406160 https://doi.org/10.1016/j.endm.2005.06.023 https://doi.org/10.1016/j.endm.2005.06.023 https://doi.org/10.1016/s0166-218x(00)00198-0 https://doi.org/10.1016/s0166-218x(00)00198-0 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=465948 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=2389696 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=2389696 https://doi.org/10.1016/0166-218x(95)00106-2 https://doi.org/10.1016/0166-218x(95)00106-2 https://doi.org/10.1016/0734-189x(84)90051-3 https://doi.org/10.1016/0734-189x(84)90051-3 https://doi.org/10.1287/moor.1030.0070 https://mathscinet-ams-org.ezproxy.loyno.edu/mathscinet-getitem?mr=422062 https://doi.org/10.1007/s00373-010-0988-8 https://doi.org/10.1007/s00373-010-0988-8 https://mathscinet.ams.org/mathscinet-getitem?mr=2370323 https://mathscinet.ams.org/mathscinet-getitem?mr=2370323 introduction metric dimension of the planar graph jn metric dimension of the planar graph kn metric dimension of the planar graph ln conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.17551 j. algebra comb. discrete appl. 3(3) • 127–134 received: 31 october 2015 accepted: 16 february 2016 journal of algebra combinatorics discrete structures and applications locating one pairwise interaction: three recursive constructions∗ research article charles j. colbourn, bingli fan abstract: in a complex component-based system, choices (levels) for components (factors) may interact to cause faults in the system behaviour. when faults may be caused by interactions among few factors at specific levels, covering arrays provide a combinatorial test suite for discovering the presence of faults. while well studied, covering arrays do not enable one to determine the specific levels of factors causing the faults; locating arrays ensure that the results from test suite execution suffice to determine the precise levels and factors causing faults, when the number of such causes is small. constructions for locating arrays are at present limited to heuristic computational methods and quite specific direct constructions. in this paper three recursive constructions are developed for locating arrays to locate one pairwise interaction causing a fault. 2010 msc: 05b30, 05a18, 05d99, 62k05, 68p10 keywords: locating array, covering array, detecting array 1. introduction although covering arrays have been explored as a method to reveal the presence of faults caused by interactions among components in a complex system [4, 8], they are inadequate to determine which interaction(s) account for the faulty behaviour. colbourn and mcclary [7] extend covering arrays to provide sufficient information to identify all faults when few faults each involving few factors are present. to set the stage, there are k factors f1, . . . ,fk. each factor fi has a set si = {vi1, . . . ,visi} of si possible values (levels). a test is an assignment, for each i with 1 ≤ i ≤ k, of a level from vi1, . . . ,visi to fi. a test, when executed, can pass or fail. for any subset {i1, . . . , it} ⊆ {1, . . . ,k} and levels σij ∈ sij, the set {(ij,σij ) : 1 ≤ j ≤ t} is a t-way interaction, or an interaction of strength t. thus a test on k factors ∗ the first author’s research was supported in part by the national science foundation under grant no. 1421058. the second author’s research was supported by the china scholarship council. charles j. colbourn (corresponding author); school of cidse, arizona state university, tempe az 852878809, u.s.a. (email: colbourn@asu.edu). bingli fan; department of mathematics, beijing jiaotong university, beijing, china (email: blfan@bjtu.edu.cn). 127 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 contains (covers) ( k t ) interactions of strength t. a test suite is a collection of tests; the outcomes are the corresponding set of pass/fail results. a fault is evidenced by a failure outcome for a test; however the fault is rarely due to a complete k-way interaction; rather it is the result of one or more faulty interactions of strength smaller than k covered in the test. tests are executed concurrently, so that testing is nonadaptive. we employ a matrix representation. an array a with n rows, k columns, and symbols in the ith column chosen from an alphabet si of size si is denoted as an n ×k array of type (s1, . . . ,sk). a t-way interaction in a is a choice of t columns i1, . . . , it, and the selection of a level σij ∈ sij for 1 ≤ j ≤ t, represented as t = {(ij,σij ) : 1 ≤ j ≤ t}. for such an array a = (axy) and interaction t, define ρa(t) = {r : arij = σij, 1 ≤ j ≤ t}, the set of rows of a in which the interaction is covered. for a set of interactions t , ρa(t ) = ⋃ t∈t ρa(t). let it be the set of all t-way interactions for an array of type (s1, . . . ,sk), and let it be the set of all interactions of strength at most t. consider an interaction t ∈ it of strength less than t. any interaction t ′ of strength t that contains t necessarily has ρa(t ′) ⊆ ρa(t); a subset t ′ of interactions in it is independent if there do not exist t,t ′ ∈t ′ with t ⊂ t ′. some interactions may cause faults. to formulate arrays for testing, we limit both the number of interactions causing faults and their strengths. as in [7], this leads to a variety of types of array a for testing a system with n tests and k factors having (s1, . . . ,sk) as the numbers of levels: array definition covering arrays: mca(n; t,k, (s1, . . . ,sk)) ρa(t) 6= ∅ for all t ∈it ca(n; t,k,v) ρa(t) 6= ∅ for all t ∈it and v = s1 = · · · = sk locating arrays: (d,t)-la(n; t,k, (s1, . . . ,sk)) ρa(t1) = ρa(t2) ⇔ t1 = t2 whenever t1,t2 ⊆ it, |t1| = d, and |t2| = d (d,t)-la(n; t,k, (s1, . . . ,sk)) ρa(t1) = ρa(t2) ⇔ t1 = t2 whenever t1,t2 ⊆ it, |t1| ≤ d, and |t2| ≤ d (d,t)-la(n; t,k, (s1, . . . ,sk)) ρa(t1) = ρa(t2) ⇔ t1 = t2 whenever t1,t2 ⊆ it, |t1| = d, |t2| = d, and t1 and t2 are independent (d,t)-la(n; t,k, (s1, . . . ,sk)) ρa(t1) = ρa(t2) ⇔ t1 = t2 whenever t1,t2 ⊆ it, |t1| ≤ d, |t2| ≤ d, and t1 and t2 are independent when all factors have the same number of levels v, the notation replaces v for (s1, . . . ,sk). colbourn and mcclary [7] also examine detecting arrays, which permit faster recovery than do locating arrays but in general require more tests. here we focus on locating arrays. although locating arrays have been successfully applied in applications to measurement and testing [1], few constructions are known. martínez et al. [9] develop adaptive analogues and establish feasibility conditions for a locating array to exist. in [11] and [12] the minimum number of rows in a locating array is determined when the number of factors is quite small. few direct, and no recursive, constructions are known. indeed at present unless the number of factors is small, the observation in [7] that covering arrays of higher strength provide examples of locating arrays serves as the main device for their construction. in this paper, we explore a different avenue, extending recursive constructions from covering arrays to locating arrays. we extend a covering array construction pioneered by roux [10], extended by chateauneuf and kreher [2], and further generalized in [3]. the basic strategy in each of these constructions is a “cut–and–paste” approach using covering arrays with fewer factors and the same or smaller strengths. each operates by repeating subarrays; because we want different interactions to appear in different sets of rows, such recursions for locating arrays necessitates more ingredients than for covering arrays. we focus on constructions for (1, 2)-locating arrays in which all factors have the same number v of levels. in other words, we treat the case of locating one interaction of strength at most two. 128 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 we require one further class of ingredients. let (γ,�) be a group of order k. a (k,n; λ)-difference matrix over γ is an n × kλ matrix d = (dij) with entries from γ, so that for each 1 ≤ i < j ≤ n, the multiset {di` �d−1j` : 1 ≤ ` ≤ kλ} (the difference list) contains every element of γ λ times. 2. a doubling construcion we develop a doubling construction that is reminscent of one for covering arrays of strength three in [2] generalizing that in [10]. theorem 2.1. if there exist a (1, 2)-la(n; 2,k,v) and a (1, 1)-la(m; 2,k,v) in which the set of differences modulo v between entries in two distinct columns contains all symbols, then a (1, 2)la(n + (v − 1)m; 2, 2k,v) exists. proof. let a = (aij) be a (1, 2)-la(n; 2,k,v) on symbols {0, . . .v − 1} with columns indexed by {1, . . . ,k}. let b = (bij) be a (1, 1)-la(m; 2,k,v) on symbols {0, . . .v − 1} with columns indexed by {1, . . . ,k} with the additional property that for every 1 ≤ c < c′ ≤ k and every 1 ≤ d < v, there exists a ρ for which bρc − bρc′ ≡ d (mod v). we form an (n + (v−1)m)×2k array c with columns indexed by {1, . . . ,k}×{0, 1} by juxtaposing v arrays c0, . . . ,cv−1. c0 is obtained by setting the entry in position (ρ, (c,α)) to aρc. for 1 ≤ ` < v, the entry of c` in position (ρ, (c, 0)) is bρc and the entry in position (ρ, (c, 1)) is bρc + ` mod v. let t = {((c1,α1),σ1), ((c2,α2),σ2)} and t ′ = {((c′1,α′1),σ′1), ((c′2,α′2),σ′2)} be interactions of c with ρc(t) = ρc(t ′). because a is a (1, 2)-locating array and ρa({(c1,σ1), (c2,σ2)}) = ρa({(c′1,σ′1), (c′2,σ′2)}), there are two cases to treat. c1 = c2, c′1 = c ′ 2, σ1 6= σ2, and σ′1 6= σ′2: then t = {((c1, 0),σ1), ((c1, 1),σ2)} and t ′ = {((c′1, 0),σ ′ 1), ((c ′ 1, 1),σ ′ 2)}. now ρ(t) contains at least one row index from cσ2−σ1 mod v, and ρ(t ′) contains at least one row index from cσ′2−σ′1 mod v. these agree only when σ1 = σ ′ 1, σ2 = σ ′ 2, and c1 = c ′ 1 because b is a (1, 1)-locating array. but then t = t ′. c1 = c ′ 1, c2 = c ′ 2, σ1 = σ ′ 1, and σ2 = σ ′ 2: then t = {((c1,α1),σ1), ((c2,α2),σ2)} and t ′ = {((c1,α′1),σ1), ((c2,α ′ 2),σ2)}. we treat subcases. c1 = c2: for t and t ′ to be interactions, either σ1 = σ2, or both α1 6= α2 and α′1 6= α′2. first suppose that σ1 = σ2. for 1 ≤ x < v, ρb({(c1,σ1 −xα1), (c1,σ1 −xα2)}) = ρb({(c1,σ1 −xα′1), (c1,σ1 −xα′2)}) now if α1 = α2, then ρb({(c1,σ1 − xα1)}) 6= ∅, but this is not equal to ρb({(c1,σ1 − xα′1), (c1,σ1 −xα′2)}) unless α′1 = α′2 = α1, in which case t = t ′. similarly when if α′1 = α′2, t = t ′. but when α1 6= α2 and α′1 6= α′2, again t = t ′. 129 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 so suppose that σ1 6= σ2. without loss of generality, (α1,α2) = (0, 1). for 1 ≤ ` < v, ρc`(t) = { ρb({(c1,σ1)}) if ` ≡ σ2 −σ1 (mod v) ∅ otherwise if (α′1,α ′ 2) = (0, 1), then t = t ′. so suppose that (α′1,α ′ 2) = (1, 0). then ρcσ1−σ2 (t ′) = ρb({(c1,σ2)}), and hence σ1 = σ2, which cannot be. c′1 = c ′ 2: this is symmetric to the previous case. c1 6= c2 and c′1 6= c′2: if (α1,α2) = (α′1,α′2), then t = t ′. first suppose that (α1,α2) = (0, 0) 6= (α′1,α′2). if (σ1,σ2) appears in columns (c1,c2) of b, ρc1 ({((c1, 0),σ1), ((c2, 0),σ2)}) 6= ρc1 ({((c1,α ′ 1),σ1 + α ′ 1), ((c2.α ′ 2),σ2 + α ′ 2)}), but then ρc(t) 6= ρc(t ′). if (σ1,σ2) does not appear in columns (c1,c2) of b, choose x so that (σ1 −x,σ2 −x) does appear; choose y and z so that (σ1,y) and (z,σ2) appear. then ρcx({((c1, 0),σ1), ((c2, 0),σ2)}) 6= ρcx({((c1, 1),σ1), ((c2, 1),σ2)}) ρcσ2−y ({((c1, 0),σ1), ((c2, 0),σ2)}) 6= ρcσ2−y ({((c1, 0),σ1), ((c2, 1),σ2)}) ρcσ1−z ({((c1, 0),σ1), ((c2, 0),σ2)}) 6= ρcσ1−z ({((c1, 1),σ1), ((c2, 0),σ2)}) but then ρc(t) 6= ρc(t ′). hence (α1,α2) 6= (0, 0) and, in the same way, (α′1,α′2) 6= (0, 0). next suppose that (α1,α2) = (1, 1) 6= (α′1,α′2) 6= (0, 0). choose y 6= σ2 and z 6= σ1 so that (σ1,y) and (z,σ2) appear in columns (c1,c2) of b. then ρcσ2−y ({((c1, 1),σ1), ((c2, 1),σ2)}) 6= ρcσ2−y ({((c1, 0),σ1), ((c2, 1),σ2)}) ρcσ1−z ({((c1, 1),σ1), ((c2, 1),σ2)}) 6= ρcσ1−z ({((c1, 1),σ1), ((c2, 0),σ2)}) but then ρc(t) 6= ρc(t ′). finally suppose without loss of generality that (α1,α2) = (1, 0) and (α′1,α ′ 2) = (0, 1). choose a pair (z,σ2) that appears in columns (c1,c2) of b then ρcσ1−z ({((c1, 1),σ1), ((c2, 0),σ2)}) 6= ρcσ1−z ({((c1, 0),z), ((c2, 1),σ2 + σ1 −z)}) but then ρc(t) 6= ρc(t ′). hence c is a (1, 2)-la(n + (v − 1)m; 2, 2k,v). 3. a product construction permuting symbols theorem 2.1 permutes symbols in one ingredient in order to double the number of factors. in the next construction, we also permute the symbols, but we require further ingredients in order to multiply the number of factors by v. theorem 3.1. if a (1, 2)-la(n; 2,k,v), a (1, 2)-la(r; 2,v,v), and a ca(m; 2,k,v) all exist, then a (1, 2)-la(n + m + r; 2,kv,v) exists. proof. let v = {0, . . . ,v − 1}. let a = (aij) be a (1, 2)-la(n; 2,k,v) on symbols v . b = (bij) be a ca(m; 2,k,v) on symbols v . 130 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 c = (cij) be a (1, 2)-la(r; 2,v,v) on symbols v with columns indexed by v . d be the n × kv array with columns indexed by {1, . . . ,k}× {0, . . . ,v − 1} by placing aρ,γ in entry (ρ, (γ,s)) whenever 1 ≤ ρ ≤ n, 1 ≤ γ ≤ k, and 0 ≤ s < v. e be the m ×kv array with columns indexed by {1, . . . ,k}×{0, . . . ,v − 1} by placing (bρ,γ + s) mod v in entry (ρ, (γ,s)) whenever 1 ≤ ρ ≤ m, 1 ≤ c ≤ k, and 0 ≤ s < v. f be the r×kv array with columns indexed by {1, . . . ,k}×{0, . . . ,v − 1} by placing (cρ,s + s) mod v in entry (ρ, (γ,s)) whenever 1 ≤ ρ ≤ r, 1 ≤ γ ≤ k, and 0 ≤ s < v. l be the (n + m + r) ×kv array obtained by vertically juxtaposing d, e, and f . we show that l is a (1, 2)-locating array. let t = {((c1,α1),σ1), ((c2,α2),σ2)} and t ′ = {((c′1,α′1),σ′1), ((c′2,α′2),σ′2)} be interactions of l with ρl(t) = ρl(t ′). it follows that ρa({(c1,σ1), (c2,σ2)}) = ρa({(c′1,σ′1), (c′2,σ′2)}) ρb({(c1,σ1 −α1), (c2,σ2 −α2)}) = ρb({(c′1,σ′1 −α′1), (c′2,σ′2 −α′2)}) ρc({(α1,σ1 −α1), (α2,σ2 −α2)}) = ρc({(α′1,σ′1 −α′1), (α′2,σ′2 −α′2)}), with entries of b and c computed modulo v. because a is a (1, 2)-locating array and ρa({(c1,σ1), (c2,σ2)}) = ρa({(c′1,σ′1), (c′2,σ′2)}), there are two cases to treat. c1 = c2, c′1 = c ′ 2, σ1 6= σ2, and σ′1 6= σ′2: hence ρc({(α1,σ1 −α1), (α2,σ2 −α2)}) = ρc({(α′1,σ′1 −α′1), (α′2,σ′2 −α′2)}), two subcases arise: α1 = α2, α′1 = α ′ 2, σ1 −α1 6= σ2 −α2, and σ′1 −α′1 6= σ2 −α′2: this cannot arise because then t = {((c1,α1),σ1), ((c1,α1),σ2)} is not a 2-way interaction. α1 = α ′ 1, α2 = α ′ 2, σ1 −α1 = σ′1 −α′1, and σ2 −α2 = σ′2 −α′2: then σ1 = σ′1 and σ2 = σ′2 and ρb({(c1,σ1 −α1), (c1,σ2 −α2)}) = ρb({(c′1,σ1 −α1), (c′1,σ2 −α2)}). but then c1 = c′1 and t = t ′. c1 = c ′ 1, c2 = c ′ 2, σ1 = σ ′ 1, and σ2 = σ ′ 2: hence ρb({(c1,σ1 −α1), (c2,σ2 −α2)}) = ρb({(c1,σ1 −α′1), (c2,σ2 −α′2)}), when c1 6= c2, the rows of b are partitioned into v2 nonempty sets by examining the ordered pair of symbols appearing, because b is a covering array of strength 2. therefore when c1 6= c2, α1 = α′1 and α2 = α′2, and hence t = t ′. it remains to treat the case when c1 = c2. if α1 = α2 or α′1 = α ′ 2 and both t and t ′ are interactions, we have t = t ′. so α1 6= α2, α′1 6= α′2, and ρc({(α1,σ1 −α1), (α2,σ2 −α2)}) = ρc({(α′1,σ1 −α ′ 1), (α ′ 2,σ2 −α ′ 2)}). then because c is a (1, 2)-locating array, without loss of generality α1 = α′1, α2 = α ′ 2, and hence t = t ′. consequently l is a (1, 2)-locating array. 131 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 4. a product construction permuting columns in the next construction, we combine the “cut–and–paste” approach with ideas from another main type of recursive construction, the so-called column replacement methods (see [4], for example). to do this, we permute columns in some of the ingredients, using a difference matrix to determine the column permutations. theorem 4.1. if a (1, 2)-la(n; 2,k,v) exists and k ≡ 0, 1, 3 (mod 4), a (1, 2)-la(3n; 2,k2,v) exists. proof. because k ≡ 0, 1, 3 (mod 4) and k ≥ 3, there is a (k, 3, 1)-difference matrix d = (dij) over a group γ (see, for example, [13]). suppose that γ has elements {g1, · · · ,gk} and that g1 is the group identity. assume without loss of generality that d1j = g1 and d2j = gj for 1 ≤ j ≤ k. let a = (aij) be a (1, 2)-la(n; 2,k,v) on symbols {0, . . .v − 1} with columns indexed by γ. we form three arrays c1, c2, c3 with columns indexed by γ × γ. for s ∈ {1, 2, 3}, cs has n rows, and the entry in row i and column (j,`) is ai,jd−1 s` . c is the 3n × k2 array obtained by vertical juxtaposition of c1, c2, and c3. let t = {((c1,α1),σ1), ((c2,α2),σ2)} and t ′ = {((c′1,α′1),σ′1), ((c′2,α′2),σ′2)} be interactions of c with ρc(t) = ρc(t ′), we permit ((c1,α1),σ1) = ((c2,α2),σ2) so t or t ′ may be 1-way interactions. however, we do not permit that t = ((c1,α1),σ1), ((c1,α1),σ2) but σ1 6= σ2, for then t is not an interaction at all. we must show that t = t ′. because ρc(t) = ρc(t ′) ρcs(t) = ρcs(t ′) for each 1 ≤ s ≤ 3. then for each 1 ≤ s ≤ 3, ρa({(c1d−1sα1,σ1), (c2d −1 sα2 ,σ2)}) = ρcs(t) = ρcs(t ′) = ρa({(c′1d −1 sα′1 ,σ′1), (c ′ 2d −1 sα′2 ,σ′2)}) because a is a (1, 2)-locating array, for each 1 ≤ s ≤ 3, {(c1d−1sα1,σ1), (c2d −1 sα2 ,σ2)} = {(c′1d −1 sα′1 ,σ′1), (c ′ 2d −1 sα′2 ,σ′2)} unless c1d−1sα1 = c2d −1 sα2 , c′1d −1 sα′1 = c′2d −1 sα′2 , σ1 6= σ2, and σ′1 6= σ′2. employing this equality for s = 1, without loss of generality two cases remain. c1 = c2, c′1 = c ′ 2, σ1 6= σ2, and σ′1 6= σ′2: consider the equalities when s ∈ {2, 3}. now c1d−1sα1 = c1d −1 sα2 only when α1 = α2, but then t is an interaction only when σ1 = σ2, which cannot be. similarly c′1d −1 sα′1 = c′1d −1 sα′2 only when α′1 = α ′ 2, but then t ′ is an interaction only when σ′1 = σ ′ 2, which cannot be. so because a is a locating array, {(c1d−1sα1,σ1), (c1d −1 sα2 ,σ2)} = {(c′1d −1 sα′1 ,σ′1), (c ′ 1d −1 sα′2 ,σ′2)}. without loss of generality, σ1 = σ′1 and σ2 = σ ′ 2 so c1d −1 2α1 = c′1d −1 2α′1 and c1d −1 2α2 = c′1d −1 2α′2 . then c−11 c ′ 1 = d −1 2α1 d2α′1 = d −1 2α2 d2α′2 = d −1 3α1 d3α′1 = d −1 3α2 d3α′2. then d3α′1d −1 2α′1 = d3α1d −1 2α1 and hence α1 = α′1 because d is a difference matrix. similarly d3α2d −1 2α2 = d3α′2d −1 2α′2 and hence α2 = α′2. but then t = t ′. c1 = c ′ 1, c2 = c ′ 2, σ1 = σ ′ 1, and σ2 = σ ′ 2: then for each s ∈{2, 3}, {(c1d−1sα1,σ1), (c2d −1 sα2 ,σ2)} = {(c1d−1sα′1,σ1), (c2d −1 sα′2 ,σ2)} if c1d−1sα1 = c1d −1 sα′1 , then d−1s,α1 = d −1 s,α′1 and hence α1 = α′1. but then t = t ′. 132 c. j. colbourn, b. fan / j. algebra comb. discrete appl. 3(3) (2016) 127–134 hence σ1 = σ2 and for s ∈{2, 3}, c1d−1s,α1 = c2d −1 s,α′2 and c1d −1 s,α′1 = c2d −1 s,α2 . hence c−11 c2 = d2,α′2d −1 2,α1 = d2,α2d −1 2,α′1 = d3,α′2d −1 3,α1 = d3,α2d −1 3,α′1 then d−12,α1d3,α1 = d −1 2,α′2 d3,α′2 and d −1 2,α2 d3,α2 = d −1 2,α′1 d3,α′1. because d is a difference matrix, α1 = α′2 and α2 = α ′ 1. then t = t ′. hence c is the required locating array. 5. concluding remarks theorems 2.1, 3.1, and 4.1 establish that cut–and–paste constructions provide viable methods for generating locating arrays. although the repetition inherent in methods of this type initially result in many interactions appearing in the same sets of rows, at least in the case for one interaction of strength at most two, we have shown that the symmetry from the repetition can be interrupted by adjoining further ingredients. the methods here to make locating arrays of strength two are loosely patterned on recursive constructions for covering arrays of strength three. one might hope to obtain more powerful recursive constructions by adapting the product construction for covering arrays of strength two [5], but the methods we have used do not appear to be sufficient for this. on the other hand, the theorems established here can be generalized to certain “mixed” locating arrays in which different factors have different numbers of levels. although we have not pursued it here, we also expect that the methods can generalize to the location of more than one faulty interactions at the cost of further ingredients and more cases to verify. finally further recursive constructions that exploit the methods developed for covering arrays in [4, 6] appear to be promising. acknowledgment: the authors thank violet syrotiuk for helpful discussions about this work, and thank vladimir tonchev for the opportunity to present it at the first annual kliekhandler conference. this research was completed while the second author was visiting arizona state university. he expresses his sincere thanks to the school of computing, informatics, and decision systems engineering at arizona state university for kind hospitality. references [1] a. n. aldaco, c. j. colbourn, v. r. syrotiuk, locating arrays: a new experimental design for screening complex engineered systems, sigops oper. syst. rev. 49(1) (2015) 31–40. 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[13] j. yin, cyclic difference packing and covering arrays, des. codes cryptogr. 37(2) (2005) 281–292. 134 http://dx.doi.org/10.1007/s10878-007-9082-4 http://dx.doi.org/10.1007/s10878-007-9082-4 http://dx.doi.org/10.1007/0-387-25036-0_10 http://dx.doi.org/10.1007/0-387-25036-0_10 http://dx.doi.org/10.1137/080730706 http://dx.doi.org/10.1137/080730706 http://dx.doi.org/10.1007/s11425-011-4307-5 http://dx.doi.org/10.1007/s11425-011-4307-5 http://dx.doi.org/10.1080/15598608.2012.647484 http://dx.doi.org/10.1080/15598608.2012.647484 http://dx.doi.org/10.1007/s10623-004-3991-3 introduction a doubling construcion a product construction permuting symbols a product construction permuting columns concluding remarks references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.05630 j. algebra comb. discrete appl. 4(1) • 61–74 received: 25 january 2016 accepted: 4 june 2016 journal of algebra combinatorics discrete structures and applications the nonnegative q−matrix completion problem research article bhaba kumar sarma, kalyan sinha abstract: in this paper, the nonnegative q-matrix completion problem is studied. a real n × n matrix is a q-matrix if for k ∈ {1, . . . , n}, the sum of all k × k principal minors is positive. a digraph d is said to have nonnegative q-completion if every partial nonnegative q-matrix specifying d can be completed to a nonnegative q-matrix. for nonnegative q-completion problem, necessary conditions and sufficient conditions for a digraph to have nonnegative q-completion are obtained. further, the digraphs of order at most four that have nonnegative q-completion have been studied. 2010 msc: 05c20, 05c50 keywords: digraph, partial matrix, matrix completion, nonnegative q-matrix, q-completion problem 1. introduction a partial matrix is a rectangular array of numbers in which some entries are specified while others are free to be chosen. a partial matrix m is fully specified if all entries of m are specified, i.e., if m is a matrix. let 〈n〉 = {1, . . . ,n} and m be an n × n partial matrix, i.e., one with n rows and n columns. for a subset α of 〈n〉, the principal partial submatrix m(α) is the partial matrix obtained from m by deleting all rows and columns not indexed by α. a principal minor of m is the determinant of a fully specified principal submatrix of m. a partial nonnegative (positive) matrix is a partial matrix whose specified entries are nonnegative (positive). a real n × n matrix b is a p-matrix (p0-matrix ) if every principal minor of b is positive (nonnegative). the matrix b is a q-matrix, if for each k ∈ {1, . . . ,n} the sum of all k × k principal minors of b is positive. a nonnegative q-matrix is a q-matrix whose all entries are nonnegative. the property of being p , p0 or q-matrix is invariant under permutation similarity. for a given class π of matrices (e.g., p , p0 or q-matrices) a partial π-matrix is a partial matrix for which the specified entries fulfill the requirements of a π-matrix. thus, a partial p0-matrix (partial p-matrix ) is one in which all fully specified principal minors are nonnegative (positive). similarly, a bhaba kumar sarma, kalyan sinha (corresponding author); department of mathematics, indian institute of technology guwahati, guwahat, assam 781039, india (email: bks@iitg.ernet.in, kalyansinha90@gmail.com). 61 http://dx.doi.org/10.13069/jacodesmath.05630 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 partial q-matrix is a partial matrix m in which sk(m) > 0 for every k ∈ {1,2, . . . ,n} for which all k ×k principal submatrices are fully specified. a completion of a partial matrix is a specific choice of values for the unspecified entries. a πcompletion of a partial π-matrix m is a completion of m which is a π-matrix. matrix completion problems for several classes of matrices including the classes of p and p0-matrices have been studied by a number of authors (e.g., [2, 3, 5, 7, 8, 10, 11]). in 2009, dealba et al. [4] considered the q-matrix completion problem. since the property of being a q-matrix is not inherited by principal submatrices, the authors observed that the q-matrix completion problem is substantially different from the completion problems studied earlier and attracts special attention. in their paper, it was shown that for q-matrix completion of a digraph, stratification (see section 2) of its complement is necessary and positive signing of the stratified complement of the digraph is sufficient. (here, positive signing of a digraph means a signing of each of its arcs with the property that for each even (resp. odd) cycle the product of the signs of arcs on the cycle is negative (resp. positive).) further, the authors classified all digraphs of order up to order 4 as to q-matrix completion. theorem 1.1. [4] let d be a digraph of order n that omits at least one loop. (i) if d has q-completion, then d is stratified. (ii) if n ≤ 4 and d is stratified, then d has q-completion. theorem 1.2. [4] let d 6= kn be an order n digraph that includes all loops and has q-completion. then for each k = 2,3, . . . ,n, either, (i) d has a permutation digraph of order k, or (ii) for each v ∈ v (d), d − v has a permutation digraph of order k − 1. theorem 1.3. [4] let d be a digraph such that d is stratified. if it is possible to sign the arcs of d so that the sign of every cycle is +, then d has q-completion. for an extensive survey of matrix completion problems, we refer the relevant sections in handbook of linear algebra [9] published by chapman and hall/crc press. in this paper, we make a combinatorial study of the completion problem of partial nonnegative q-matrices in which digraphs will play an important role. 2. preliminaries most of the definitions of the graph-theoretic terms used in this paper can be found in any standard reference, for example, in [1] and [6]. for our purpose, a directed graph or digraph d = (v (d),a(d)) of order n > 0 is a finite nonempty set v (d), with |v (d)| = n of objects called vertices together with a (possibly empty) set a(d) of ordered pairs of vertices (not necessarily distinct), called arcs or directed edges. sometimes, we simply write v ∈ d (resp. (u,v) ∈ d) to mean v ∈ v (d) (resp. (u,v) ∈ a(d)). if x = (u,v) is an arc in d, we say that x is incident with u and v. if x = (u,u), then x is called a loop at the vertex u. by k∗n we denote the digraph with vertex set 〈n〉 = {1, . . . ,n} and arc set 〈n〉 × 〈n〉, i.e., one with all possible arcs including loops on the vertex set 〈n〉. a digraph h is a subdigraph of the digraph d if v (h) ⊆ v (d), a(h) ⊆ a(d). if v (w) ⊆ v (d), the subdigraph induced by v (w), i.e. w , is the digraph w = (v (w),a(w)) with a(w) the set of all arcs of d between the vertices in w . the digraph w is a spanning subdigraph if v (w) = v (d). the complement of a digraph d is the digraph d, where v (d) = v (d) and (v,w) ∈ a(d) if and only if (v,w) /∈ a(d). two digraphs d1 = (v1,a1) and d2 = (v2,a2) are isomorphic, if there is a bijection ψ : v1 → v2 such that a2 = {(ψ(u),ψ(v)) : (u,v) ∈ a1}. 62 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 a (directed) u-v path p of length k ≥ 0 in d is an alternating sequence (u = v0,x1,v1, . . . , xk,vk = v) of vertices and arcs, where vi, 1 ≤ i ≤ k, are distinct vertices and xi = (vi−1,vi). then, the vertices vi and the arcs xi are said to be on p . further, if k ≥ 2 and u = v, then a u-v path is a cycle of length k. we then write ck = 〈v1,v2, . . . ,vk〉 and call ck a k-cycle in d. paths and cycles in a digraph d are considered to be subdigraphs of d in a natural way. a cycle c is even (resp. odd) if its length is even (resp. odd). a digraph d is said to be connected (resp. strongly connected) if for every pair u,v of vertices, d contains a u-v path (resp. both a u-v path and a v-u path). the maximal connected (resp. strongly connected) subdigraphs of d are called components (resp. strong components) of d. let π be a permutation of a nonempty finite set v . the digraph dπ = (v,aπ), where aπ = { ( v,π(v) ) : v ∈ v }, is called a permutation digraph. clearly, each component of a permutation digraph is a loop or a cycle. the digraph dπ is said to be positive (resp. negative) if π is an even permutation (resp. an odd permutation). it is clear that dπ is negative if and only if it has odd number of even cycles. a permutation subdigraph h (of order k) of a digraph d is a permutation digraph that is a subdigraph of d (of order k). further, h is positive (negative) if the corresponding permutation is even (odd). a digraph d is (positively) stratified if d has a (positive) permutation subdigraph of order k for every k = 2,3, . . . , |d|. by pk we denote the collection of all permutation subdigraphs of order k of k ∗ n. further, we denote by p+ k (resp. p− k ) the collection of all positive (resp. negative) permutation subdigraphs of order k of k∗n. let b = [bij] be an n × n matrix. we have detb = ∑ (sgnπ)b1π(1) · · ·bnπ(n) (1) where the sum is taken over all permutations π of 〈n〉. for a permutation digraph p of k∗n we denote the product ∏ {bij : (i,j) ∈ a(p)} by w(p,b). for k ∈ {1, . . . ,n} we denote the sum of all k × k principal minors of b by sk(b). in view of (1), we have sk(b) = ∑ p ∈p + k w(p,b) − ∑ p ∈p − k w(p,b). (2) 3. partial nonnegative q-matrices and their completions recall that a partial q-matrix m is a partial matrix such that sk(m) > 0 for every k ∈ {1, . . . ,n} for which all k ×k principal submatrices are fully specified. let m be a partial nonnegative q-matrix. if all 1 × 1 principal submatrices (i.e., all diagonal entries) in m are specified, then their sum s1(m) (the trace of m) must be positive. if all k × k principal submatrices are fully specified for some k ≥ 2, then m is fully specified and, therefore, is a nonnegative q-matrix. thus, a partial nonnegative q-matrix is characterized as follows. proposition 3.1. let m be a partial nonnegative matrix. then, m is a partial nonnegative q-matrix if and only if exactly one of the following holds: (i) at least one diagonal entry of m is not specified. (ii) all diagonal entries are specified, at least one diagonal entry is positive and m has an off-diagonal unspecified entry. (iii) all entries of m are specified and m is a q-matrix. a completion b of a partial nonnegative q-matrix m is called a nonnegative q-completion of m, if b is a nonnegative q-matrix. since any matrix which is permutation similar to a q-matrix is a q-matrix, it is evident that if a partial nonnegative q-matrix has a nonnegative q-completion, so does any partial matrix which is permutation similar to m. 63 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 any partial nonnegative matrix m with all diagonal entries unspecified has nonnegative qcompletion. a completion of m can be obtained by choosing sufficiently large values for the unspecified diagonal entries. let m = [mij] be a partial nonnegative q-matrix which contains both specified and unspecified diagonal entries. consider the principal partial submatrix m(α) of m induced by α = {i : mii is specified} ⊆ 〈n〉. in case m(α) is fully specified, m may not have a nonnegative q-completion. for example, the partial matrix m =   1 0 0 0 0 0 0 0 ∗   where ∗ denotes an unspecified entry, m(1,2) is fully specified. that m does not have a nonnegative q-completion is evident; because for any completion b of m, s3(b) = detb = 0. for the case when m(α) is not fully specified or itself a q-matrix, we have the following. theorem 3.2. let m = [mij] be a partial nonnegative q-matrix and α = {i : mii is specified}. if the principal partial submatrix m(α) of m has nonnegative q-completion, then m has nonnegative qcompletion. proof. if α = {1, . . . ,n}, then m = m(α) has a nonnegative q-completion, by the hypothesis. otherwise, without any loss of generality, we assume α = {1, . . . ,r} for some 1 ≤ r ≤ n − 1 and m = [ m11 m12 m21 m22 ] , where m11 = m(1, . . . ,r) and m22 = m(r + 1, . . . ,n). let b11 be a nonnegative q-completion of m(1, . . . ,r). then, m′ = [ b11 m12 m21 m22 ] is a partial nonnegative q-matrix, since m22 has unspecified diagonal entries. for t > 0, consider the completion b(t) = [ b11 b12 b21 b22 ] of m obtained by taking bii = t, i = r+1, . . . ,n, and bij = 0 for all other unspecified entries in m. since b11 is a nonnegative q-matrix we have si(b11) > 0 for 1 ≤ i ≤ r. now, for 1 ≤ j ≤ n, sj(b(t)) = { ( n−r j ) tj + pj(t), if j ≤ n − r, sj−n+r(b11)t n−r + pj(t), if j > n − r, where pj(t) is a polynomial in t of degree at most j − 1, if j ≤ n − r, and of degree at most n − r − 1, if j > n − r. as a consequence, for sufficiently large values of t, sj(b(t)) > 0 for 1 ≤ j ≤ n and b(t) is a nonnegative q-completion of m. the converse of theorem 3.2 is not true. the following example shows that a partial nonnegative q-matrix m may have q-completion, even when m(α) does not have. example 3.3. consider the partial matrix, m =   ∗ b12 b13 b14 b21 d2 b23 ∗ ∗ ∗ d3 b34 ∗ b42 ∗ ∗   , (3) 64 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 where ∗ denotes the unspecified entries. here, α = {2,3} and m(α) does not have a q-completion for d2 = b23 = 0 and d3 = 1. however, we show that for any choice of nonnegative values of the specified entries bij, m has nonnegative q-completions. for t > 0 consider the completion b(t) =   t b12 b13 b14 b21 d2 b23 t t t d3 b34 t b42 t t   of m. then, s1(b(t)) = 2t + d2 + d3, s2(b(t)) = t 2 + f1(t), s3(b(t)) = t 3 + f2(t), s4(b(t)) = t 4 + f3(t), where fi(t) is a polynomial in t of degree at most i, i = 1,2,3. consequently, b(t) is a nonnegative q-matrix for sufficiently large t, and therefore, m has nonnegative q-completions. 4. digraphs and nonnegative q-completions it is useful to associate a partial matrix with a digraph that describes the positions of the specified entries in the partial matrix. we say that an n×n partial matrix m specifies a digraph d = (〈n〉,a(d)), if for 1 ≤ i,j ≤ n, (i,j) ∈ a(d) if and only if the (i,j)-th entry of m is specified. for example, the partial nonnegative q-matrix m in example 3.3 specifies the digraph d1 in figure 1. b 4 b 3 b 2 b1 figure 1. the digraph d1 theorem 4.1. suppose m is a partial nonnegative q-matrix specifying the digraph d. if the partial submatrix of m induced by every strongly connected induced subdigraph of d has nonnegative q-completion, then m has nonnegative q-completion. proof. we prove the result for the case when d has two strong components h1 and h2. the general result will then follow by induction. by a relabelling of the vertices of d, if required, we have m = [ m11 m12 x m22 ] , where mii is a partial nonnegative q-matrix specifying hi, i = 1,2, and all entries in x are unspecified. by the hypothesis, mii has a nonnegative q-completion bii. consider the completion b = [ b11 b12 b21 b22 ] , 65 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 by choosing all unspecified entries in m12 and x as 0. then, for 2 ≤ k ≤ |d| we have sk(b) = sk(b11) + sk(b22) + k−1∑ r=1 sr(b11)sk−r(b22) > 0, here, we mean sk(bii) = 0 whenever k exceeds the size of bii. thus m can be completed to a nonnegative q-matrix. the proof of the following result is similar. theorem 4.2. suppose m is a partial nonnegative q-matrix specifying the digraph d. if the partial submatrix of m induced by each component of d has a nonnegative q-completion, then m has a nonnegative q-completion. that the converse of theorem 4.1 is not true can be seen from the following example. example 4.3. consider the digraph d2 in figure 2. we show that every partial nonnegative q-matrix b 1 b 2 b 3 b 4 b 5 b 6 d2 figure 2. a digraph having nonnegative q-completion specifying the digraph d2 has nonnegative q-completion. to see this consider any partial nonnegative q-matrix m = [aij] specifying d2. for t > 1 consider the completion b(t) =   t a12 0 0 a15 0 0 a22 t 0 a25 a26 0 a32 t a34 t 0 0 0 0 t a45 0 0 t 0 0 a55 a56 a61 0 0 0 0 t   of m. then, we have s1(b(t)) = 4t + a22 + a55, s2(b(t)) = 6t 2 + f1(t), s3(b(t)) = 5t 3 + f2(t), s4(b(t)) = 4t 4 + f3(t), s5(b(t)) = 3t 5 + f4(t), s6(b(t)) = t 6 + f5(t), 66 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 where fi are polynomials of degree at most i in t. it is clear that sk(b(t)) > 0, 1 ≤ k ≤ 6, for sufficiently large values of t. hence m has nonnegative q-completions. on the other hand, the vertices 1,2,5 and 6 induce a strong component h of d2. consider the partial nonnegative q-matrix m1 specifying h and with all specified entries as zero and b1 =   x11 0 0 x16 x21 0 0 0 x51 x52 0 0 0 x62 x65 x66   , any nonnegative completion of m1. then s4(b1) = −x16x21x52x65 ≤ 0, and consequently b1 is not a nonnegative q-matrix. 5. the nonnegative q-completion problem for a class π of matrices (e.g., p , p0 or q-matrices) a digraph d is said to have π-completion, if every partial π-matrix specifying d can be completed to a π-matrix. the π-matrix completion problem refers to the study of digraphs which have π-completion. we say that a digraph d has nonnegative (positive) q-completion, if every partial nonnegative (positive) q-matrix specifying d can be completed to a nonnegative (positive) q-matrix. the nonnegative (positive) q-matrix completion problem aims at studying and classifying all digraphs d which have nonnegative (positive) q-completion. example 5.1. it follows from example 4.3 that the digraph d2 in figure 2 has nonnegative q-completion. however, its strong component h induced by the vertices 1,2,5 and 6 does not have nonnegative qcompletion. in particular, this exhibits that the property of having nonnegative q-completion is not preserved to induced subdigraphs. it is clear that if a digraph d has (nonnegative, positive) q-completion, then any digraph which is isomorphic to d has (nonnegative, positive) q-completion. 5.1. sufficient conditions for nonnegative q-completion proposition 5.2. if a digraph d 6= k∗n of order n has nonnegative q-completion, then any spanning subdigraph of d has nonnegative q-completion. proof. let h be a spanning subdigraph of d. let mh be a partial nonnegative q-matrix specifying the digraph h. consider the partial matrix md obtained from mh by specifying the entries corresponding to (i,j) ∈ a(d) \ a(h) as 0. since d 6= k∗n, by proposition 3.1, md is a partial nonnegative q-matrix specifying d. let b be a nonnegative q-completion of md. clearly, b is a nonnegative q-completion of mh. for a digraph d = (v,a), a weight function on d is a real valued function φ defined on a. the triplet (v,a,φ) is then called a weighted digraph. for e ∈ a, φ(e) is called the weight of e. further, for any permutation subdigraph p of d, we denote the sum of the weights of the edges on p by φ∗(p). theorem 5.3. let d be a digraph of order n. suppose there is a weight function φ on d satisfying the following: for each k ∈ {1, . . . ,n}, there is a positive permutation subdigraph pk of order k in d such that, 67 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 b 1 b 2 b 3 b 4 d3 b1 b 2 b 3 b 4 d3 1 1 1 11 figure 3. the digraph d3 and its complement d3 (i) φ∗(pk) > φ ∗(p) for every negative permutation subdigraph p of d of size k, and (ii) φ∗(pk) > ∑ e∈s φ(e), for any subset s ⊆ a(d) with |s| ≤ k − 1. then, d has nonnegative q-completion. proof. let m = [mij] be a partial nonnegative q-matrix specifying d. for x > 1 consider the completion b(x) = [bij] of m defined by, bij = { mij, if (i,j) ∈ d xφ(e), if e = (i,j) ∈ d. for k ∈ {1, . . . ,n}, we have sk(b(x)) = ∑ p ∈p + k w(p,b(x)) − ∑ p ∈p − k w(p,b(x)). since pk ∈ p + k , w(pk,b(x)) = x φ ∗(pk) is a term with a positive coefficient in sk(b(x)). on the other hand, any p ∈ p− k is one of the following: (i) a negative permutation subdigraph of d of order k, (ii) a permutation subdigraph of k∗n with at most k − 1 arcs from d. in view of the properties of φ, we have, w(p,b(x)) = αxt, where α ≥ 0 and t < φ∗(pk). consequently, for large values of x, we have sk(b(x)) > 0 for k = 1, . . . ,n and hence d has nonnegative q-completion. example 5.4. consider the digraph d3 in figure 3. consider the weight function φ on the arcs of d3 obtained by assigning unit weights to the arcs (1,4),(4,2), (2,1) and to the loops at 3 and 4, and assigning zero weights to all other arcs. the nonzero weights have been marked in bold-faces in d3 in figure 3. we choose the following positive permutation digraphs with their respective weights in d3. p1 the loop at 3, φ ∗(p1) = 1 p2 the union of the loops at 3 and 4, φ ∗(p2) = 2 p3 the 3-cycle [1,4,2], φ ∗(p3) = 3 p4 the union of the loop at 3 and the 3-cycle [1,4,2], φ ∗(p4) = 4. further, the only even cycles in d3 are the cycles [2,3] and [1,3,4,2] of weights 0 and 2, respectively. it is clear that φ satisfies the conditions (i) and (ii) in theorem 5.3, and therefore, d3 has nonnegative q-completion. 68 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 corollary 5.5. let d be a digraph of order n such that (i) d is stratified, and (ii) d does not have an even cycle. then d has nonnegative q-completion. proof. let 2 ≤ k ≤ n. since d is stratified, d has a permutation subdigraph pk of order k. since d does not have an even cycle, d does not have any negative permutation subdigraph. thus, pk is positive. further, p2 being even, it must be composed of two loops, and therefore, d has a (positive) permutation subdigraph of order 1 as well. we define φ(e) = 1 for each e ∈ a(d). then the weight function φ in d satisfies conditions (i) and (ii) of theorem 5.3. remark 5.6. one does not expect the converse of theorem 5.3 to be true. however, the converse holds for all digraphs we have examined, including all digraphs of order at most four. that is, for each of these digraphs which have nonnegative q-completion, there is a weight function on its complement satisfying the conditions (i) and (ii) in theorem 5.3. 5.2. necessary conditions for nonnegative q-completion proposition 5.7. let d be a digraph with at least two vertices. if d has nonnegative q-completion, then d omits at least two loops. proof. suppose d omits at most one loop. let m be a partial nonnegative q-matrix specifying the digraph d with all specified entries as 0. then for any nonnegative completion b of m, s2(b) ≤ 0. the converse of the proposition 5.7 is not true. the digraph d4 in figure 4 omits 2 loops but does not have nonnegative q-completion. for example, m =   0 0 0 0 x22 x23 x31 0 x33   is a partial nonnegative q-matrix specifying d4 which has no nonnegative q-completion. in fact, a b 1 b 2 b 3 d4 figure 4. a digraph which does not have nonnegative q-completion digraph needs to satisfy stronger conditions to have nonnegative q-completion, as the following result shows. theorem 5.8. if a digraph d of order n (n ≥ 2) has nonnegative q-completion, then d is positively stratified. 69 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 proof. suppose d has nonnegative q-completion. assume d has no positive permutation digraph of order k for some k ≥ 2. if m is the partial matrix that specifies d with all specified entries zero, and b is a nonnegative completion of m, then all k × k principal minors of b are nonpositive implying that b is not a nonnegative q-matrix. corollary 5.9. let d be a digraph of order n such that |a(d)| > n(n − 1). then d does not have nonnegative q-completion. proof. if d has more than n(n − 1) arcs (including loops), then d has fewer than n2 − n(n − 1) = n arcs. thus d does not contain permutation subdigraph of order n. therefore, by theorem 5.8, d does not have nonnegative q-completion. the converse of the theorem 5.8 is not true, which can be seen from the following example. b 1 b 2 b 3 b 4 b5 d5 b 1 b 2 b 3 b 4 b 5 d5 figure 5. a digraph whose complement is positively stratified example 5.10. the complement d5 of the digraph d5 in figure 5 is positively stratified. however, we show that d5 does not have nonnegative q-completion. let m be the partial nonnegative q-matrix specifying d5 with all specified entries as zero. consider a nonnegative completion b =   0 x12 0 x14 0 0 0 0 x24 0 x31 0 d3 0 x35 0 0 x43 0 0 x51 0 0 0 d5   of m. for b to be a nonnegative q-matrix, we have s1(b) = d3 + d5 > 0, (4) s2(b) = d3d5 > 0, (5) s3(b) = x14x43x31 > 0, (6) s4(b) = d5x14x43x31 − x12x24x43x31 − x14x43x35x51 > 0, (7) s5(b) = x12x24x43x35x51 − x12x24x43x31d5 > 0. (8) clearly, the entries in b against all unspecified entries in m must be positive. now, from (7) we have d5x14x31 > x12x24x31 + x14x35x51 which yields d5x31 > x35x51. on the other hand, (8) implies that x35x51 > d5x31. hence b cannot be a nonnegative q-matrix. theorem 5.11. let d be a digraph with at least four vertices. suppose d has more than one 2-cycle and does not have a 3-cycle. if d has nonnegative q-completion, then d must omit more than three loops. 70 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 proof. for d to have nonnegative q-completion, d must omit at least two loops, by proposition 5.7. we prove that if d omits exactly three loops, then d does not have nonnegative q-completion. then, the result for the case when d omits exactly two loops will follow from proposition 5.2. suppose d omits loops at the vertices 1,2 and 3. now, we label the 2-cycles as e1, . . . ,ek (k > 1) such that the number of vertices among 1,2 and 3 that ej is incident with is in ascending order in j. let m be the partial nonnegative q-matrix specifying the digraph d with all specified entries as zero. suppose that m has a nonnegative q-completion b = [bij]. we put d1 = b11, d2 = b22, d3 = b33. for a 2-cycle e = 〈r,s〉 in d, we write w(e) = w(e,b) = brsbsr. then, by (2) we have s3(a) = d1d2d3 − (d1σ1 + d2σ2 + d3σ3), (9) where σt = ∑ { w(ei) : ei is not incident with t } , t = 1,2,3. since s3(b) > 0, (9) implies d1d2 > σ3, d2d3 > σ1, d3d1 > σ2, (10) and therefore d1d2 + d2d3 + d3d1 > σ1 + σ2 + σ3. (11) for 1 ≤ j ≤ k, let βj = ∑ { w(ei) : ei ∩ ej = ∅ and i > j } γj = ∑ { dsdt : ej is incident with none of s and t } . then, by (2) we have s4(a) = k∑ j=1 βj w(ej) − k∑ j=1 γj w(ej) = k∑ j=1 (βj − γj)w(ej). (12) however, we show that βj − γj ≤ 0 for 1 ≤ j ≤ k. then, it will follow from (12) that s4(b) ≤ 0, a contradiction to the fact that b is a q-matrix. fix j ∈ {1,2, . . . ,k}. we have γj =    d1d2 + d2d3 + d3d1, if ej is not incident with any of the vertices 1,2,3, (d1d2d3)/dt, if ej is incident with only t, t = 1,2 or 3, 0, if ej is incident with two of the vertices 1,2 and 3. if ej is not incident with any of the vertices 1,2 and 3, then from (11) we get βj ≤ k∑ i=1 w(ei) ≤ σ1 + σ2 + σ3 < d1d2 + d2d3 + d3d1 = γj. next, suppose ej is incident with exactly one vertex t in {1,2,3}. consider the case t = 1. since {ei : ei ∩ ej = ∅, i > j} ⊆ {ei : 1 is not incident with ei}, we have βj ≤ σ1. therefore, from (10) we get βj ≤ σ1 < d2d3 = γj. the cases when t = 2 and 3 are similar. finally, assume that ej is incident to two among the vertices 1,2 and 3 so that γj = 0. now, for any i > j the 2-cycle ei is incident with two vertices among 1,2 and 3. consequently, by our choice of the ordering of the 2-cycles, ei and ej have a vertex in common yielding βj = 0. therefore, our assertion that βj − γj ≤ 0 for 1 ≤ j ≤ k holds. 71 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 6. nonnegative q-completion of digraphs of small order we have examined the digraphs of order at most four to nonnegative q-completion. clearly, any digraph of order 1 (with or without a loop) has nonnegative q-completion. any digraph of order 2 without a loop has nonnegative q-completion. there are only four non-isomorphic digraphs of order 3 without loops for which the digraphs obtained by attaching a loop at any of the vertices have nonnegative q-completion. these digraphs are precisely the spanning subdigraphs of a 3-cycle. some types of digraphs of order four with respect to nonnegative q-completion are presented below. (a) let d̂6 be a digraph obtained from the digraph d6 in figure 6 by adding at most two loops at any of its vertices. then, d̂6 has nonnegative q-completion. (in fact, d̂6 satisfies the conditions of theorem 5.3.) there are 41 non-isomorphic digraphs of order 4 without loops having similar property as d6, i.e., all digraphs obtained from them by attaching at most two loops at any of the vertices have nonnegative q-completion. b1 b 2 b 3 b 4 b1 b 2 b 3 b 4 figure 6. the digraph d6 and its complement d6 (b) let d̂7 be any digraph obtained from the digraph d7 in figure 7 by attaching two loops. if d̂7 has nonnegative q-completion, then it must omit a loop at the vertex 2. in fact, in that case d̂7 satisfies the conditions of theorem 5.3. there are 66 non-isomorphic digraphs d of order 4 without loops having similar property as d7, i.e., a digraph obtained from d by attaching at most two loops at its vertices has nonnegative q-completion only when d omits a loop at a particular vertex. b1 b 2 b 3 b 4 b1 b 2 b 3 b 4 figure 7. the digraph d7 and its complement d7 (c) let d̂8 be any digraph obtained from the digraph d8 in figure 8 by adding a loop at any of the vertices. then, d̂8 does not have nonnegative q-completion (by theorem 5.11). there are 22 non-isomorphic digraphs of order 4 without loops having similar property as d8, i.e., any digraph obtained from them by attaching a loop at a vertex does not have nonnegative q-completion. 72 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 b1 b 2 b 3 b 4 b1 b 2 b 3 b 4 figure 8. the digraph d8 and its complement d8 7. comparison between q-completion, and nonnegative and positive q-completion problems in this section, we compare the nonnegative q-completion problem with the q-completion and the positive q-completion problems. proposition 7.1. if a digraph d has nonnegative q-completion, then d has positive q-completion. proof. suppose d has nonnegative q-completion. let m = [aij] be a partial positive q-matrix specifying the digraph d. then m is a partial nonnegative q-matrix specifying d. let b be a nonnegative q-completion of m. then, perturbing the zero entries in b by small positive quantities, a positive qcompletion of m can be obtained. however, the converse is not true which can be seen from the following example. example 7.2. consider the digraph d9 in figure 9. let m = [mij] be a partial positive q-matrix specifying d9. we write m =   m11 m12 x13 x21 m22 m23 m31 x32 m33   , where xij are unspecified. it is easy to see that putting sufficiently small values for xij, m can be completed to a positive q-matrix, implying that d9 has positive q-completion. however, in view of proposition 5.7, d9 does not have nonnegative q-completion, since d9 has all loops. b 1 b 2 b 3 d9 figure 9. a digraph having positive q-completion, but not nonnegative q-completion the following example shows that a digraph having q-completion may fail to have nonnegative q-completion. example 7.3. consider the digraph d10 in figure 10. let m = [mij] be a partial q-matrix specifying d10. we write, m =   x11 x12 x13 x21 m22 m23 m31 x32 m33   , 73 b. k. sarma, k. sinha / j. algebra comb. discrete appl. 4(1) (2016) 61–74 where xij are unspecified. we put x12 = −x and all other unspecified entries as x. it is easy to see that with sufficiently large values for x, m can be completed to a q-matrix, implying that d10 has qcompletion. however, in view of proposition 5.7, d10 does not have nonnegative q-completion, because it omits only one loop. b 1 b 2 b 3 d10 figure 10. a digraph having q-completion, but not nonnegative q-completion suppose d is a digraph having nonnegative q-completion. then, d is stratified and omits at least two loops. for all small digraphs (including all digraph of order 4) having these properties are seen to have q-completion. whether a stratified digraph omitting a loop necessarily have q-completion is not known (see question 2.9 in [4]). we do not know whether there is a digraph having nonnegative q completion, but not q-completion. references [1] g. chartrand, l. lesniak, graphs and digraphs, fourth edition, chapman & hall/crc, london, 2005. [2] j. y. choi, l. m. dealba, l. hogben, b. kivunge, s. nordstrom, m. shedenhelm, the nonnegative p0−matrix completion problem, electron. j. linear algebra 10 (2003) 46–59. [3] j. y. choi, l. m. dealba, l. hogben, m. s. maxwell, a. wangsness, the p0−matrix completion problem, electron. j. linear algebra 9 (2002) 1–20. [4] l. m. dealba, l. hogben, b. k. sarma, the q−matrix completion problem, electron. j. linear algebra 18 (2009) 176–191. [5] s. m. fallat, c. r. johnson, j. r. torregrosa, a. m. urbano, p−matrix completions under weak symmetry assumptions, linear algebra appl. 312(1–3) (2000) 73–91. [6] f. harary, graph theory, addison-wesley, reading, ma, 1969. [7] l. hogben, graph theoretic methods for matrix completion problems, linear algebra appl. 328(1–3) (2001) 161–202. [8] l. hogben, matrix completion problems for pairs of related classes of matrices, linear algebra appl. 373 (2003) 13–29. [9] l. hogben, a. wangsness, matrix completion problems, in handbook of linear algebra, l. hogben, editor, chapman and hall/crc press, boca raton, 2007. [10] c. r. johnson, b. k. kroschel,the combinatorially symmetric p−matrix completion problem, electron. j. linear algebra 1 (1996) 59–63. [11] c. jordon, j. r. torregrosa, a. m. urbano, completions of partial p−matrices with acyclic or non–acyclic associated graph, linear algebra appl. 368 (2003) 25–51. 74 http://dx.doi.org/10.13001/1081-3810.1095 http://dx.doi.org/10.13001/1081-3810.1068 http://dx.doi.org/10.13001/1081-3810.1303 http://dx.doi.org/10.1016/s0024-3795(00)00088-4 http://dx.doi.org/10.1016/s0024-3795(00)00299-8 http://dx.doi.org/10.1016/s0024-3795(02)00531-1 http://dx.doi.org/10.13001/1081-3810.1004 http://dx.doi.org/10.1016/s0024-3795(02)00654-7 http://dx.doi.org/10.13001/1081-3810.1095 http://dx.doi.org/10.13001/1081-3810.1068 http://dx.doi.org/10.13001/1081-3810.1303 http://dx.doi.org/10.1016/s0024-3795(00)00088-4 http://dx.doi.org/10.1016/s0024-3795(00)00299-8 http://dx.doi.org/10.13001/1081-3810.1004 http://dx.doi.org/10.1016/s0024-3795(02)00654-7 http://dx.doi.org/10.1016/s0024-3795(02)00531-1 http://dx.doi.org/10.1016/s0024-3795(02)00654-7 introduction preliminaries partial nonnegative q-matrices and their completions digraphs and nonnegative q-completions the nonnegative q-completion problem nonnegative q-completion of digraphs of small order comparison between q-completion, and nonnegative and positive q-completion problems references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1111720 j. algebra comb. discrete appl. 9(2) • 85–99 received: 20 september 2021 accepted: 16 february 2022 journal of algebra combinatorics discrete structures and applications a note on gdd(1,n,n,4;λ1,λ2) research article dinesh g. sarvate∗, dinkayehu m. woldemariam∗∗ abstract: the present note is motivated by two papers on group divisible designs (gdds) with the same block size three but different number of groups: three and four where one group is of size 1 and the others are of the same size n. here we present some interesting constructions of gdds with block size 4 and three groups: one of size 1 and other two of the same size n. we also obtain necessary conditions for the existence of such gdds and prove that they are sufficient in several cases. for example, we show that the necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. 2010 msc: 05b05, 05b30 keywords: t-designs, group divisible designs with unequal group sizes, bibds 1. introduction among all combinatorial designs, probably the most widely studied design is a balanced incomplete block design (bibd). for definitions and and background please see lindner and rodger [6]. definition 1.1. a balanced incomplete block design, bibd(v,k,λ), is an arrangement of v distinct points into b proper subsets (called blocks) of size k each, such that every point appears in exactly r blocks and every pair of distinct points occurs together in exactly λ blocks. the numbers v,b,r,k and λ are parameters of the bibd and satisfy the necessary conditions vr = bk and λ(v −1) = r(k −1) for the existence of a bibd(v,k,λ). in 1961, haim hanani [4] proved that the necessary conditions are sufficient for the existence of bibds with block size three as well as four. specifically he proved: ∗ this author thanks professor jurisich for support through departmental research funds. dinesh g. sarvate (corresponding author); college of charleston, sc 29424, usa (email: sarvated@cofc.edu). ∗∗ this author is supported by adama science and technology university research grant number astu/spr/011/19. dinkayehu m. woldemariam; adama science and technology university, adama, ethiopia (email: dinkumen@gmail.com). 85 https://orcid.org/0000-0003-3238-0389 https://orcid.org/0000-0001-7865-7430 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 theorem 1.2. a bibd(v,4,λ) exists if and only if λ ≡ 1,5 (mod 6) and v ≡ 1,4 (mod 12); λ ≡ 2,4 (mod 6) and v ≡ 1 (mod 3); λ ≡ 3 (mod 6) and v ≡ 0,1 (mod 4); λ ≡ 0 (mod 6) and v ≥ 4. group divisible designs defined below play a role in the construction of bibds as well as other designs. for example, in the construction of t-designs where instead of each pair occurring in λ blocks each t-tuple occurs in λ blocks. definition 1.3. a group divisible design, gdd(n1,n2, ...,nm,k;λ1,λ2), is a triple (x,g,b), where x is a v-set, g is a partition of x into m subsets (called groups) of size n1,n2, ...,nm respectively and b is a collection of k-subsets of x (called blocks) such that each pair of points within the same group appear together in λ1 blocks, whereas each pair of points from different groups appear together in λ2 blocks. the points in the same group are called first associate of each other and elements not in the same group are called second associates of each other. fu, rodger and sarvate [2, 3] obtained complete results on group divisible designs with m groups of size n and block size 3, namely gdd(n,n,...,n,3;λ1,λ2). in 1992, colbourn, hoffman and rees [1] proved the sufficiency of the necessary conditions for the existence of a gdd(n,n,...,n,u,3; 0,1). in 2011, pabhapote and punnim [8] studied all triples of positive integers (n1,n2,λ) for which a gdd(n1,n2,3;λ,1) exists. later, pabhapote [7] proved the existence of a gdd(n1,n2,3;λ1,λ2) for all n1 6= 2 and n2 6= 2 in which λ1 ≥ λ2. this note is specially motivated by the papers of sakda and uiyyasathian [9] and lapchinda, punnim and pabhapote [5]. in 2014, lapchinda, punnim and pabhapote [5] gave a complete solution for the existence of a group divisible design with block size 3 and 3 groups of sizes n, n and 1. in 2017, sakda and uiyyasathian, obtained complete result on group divisible designs with block size 3 and 4 groups of sizes n, n, n and 1, namely gdd(1,n,n,n,3;λ1,λ2). in this note we study the existence of a gdd(1,n,n,4;λ1,λ2) with three groups g1, g2, g3 of sizes 1, n, and n respectively. in general, when the number of groups is less than the block size the work is more involved and possibly making them harder to construct. it is well known that gdds are used as a building block for bibds, but the converse is also true, for example, an easy observation is the following result. theorem 1.4. if a bibd(n1 + n2 + ... + nm,k,λ2) and a bibd(ni,k,λ1) exist for i = 1,2, ...m, then a gdd(n1,n2, ...,nm,k;λ1 + λ2,λ2) exists. corollary 1.5. if a bibd(mn + 1,4,λ2) and a bibd(n,4,λ) exist, then a gdd(1,n,n, ...,n,4;λ1 = λ2 + λ,λ2) exists. the converse of the above corollary is not true, for example, we show in section 5 that a gdd(1,n,n,4;λ2 + λ,λ2) exist for n = 2 or n = 3 but clearly a bibd(2,4,λ) or bibd(3,4,λ) does not exist. one can find such examples for larger values of n by using the construction given in the next section. for example, the construction gives a gdd(1,7,7,4; 9,6) but a bibd(7,4,3) does not exist. another observation gives, theorem 1.6. a gdd(n1,n2, ...,nm,k;λ1,0) exists if and only if a bibd(ni,k, λ1) exist for i = 1,2, ...m. corollary 1.7. a gdd(1,n,n, ...,n,k;λ1 = λ,0) exists if and only if a bibd(n,k,λ) exists. one may notice that it is much easier to construct gdds with λ1 ≥ λ2, specially when a bibd(n,k,λ) exists. in the next section we present an important construction technique which produces gdds where λ1 is less than λ2. 86 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 2. a new construction of a gdd(1,n,n,4;λ1, λ2) a kn on gi means the vertices of the complete graph kn are labeled with the elements of gi for i = 2,3. let n be even. then the complete graph kn on g2 (respectively on g3) has a 1-factorization, say {e1,e2, ...,en−1} (respectively {f1,f2, ...,fn−1}). for x = 1,2, ...,n−1, if ex = {e1,e2, ...,en 2 } and fx = {f1,f2, ...,fn 2 }, then we can form blocks el ∪fm of size 4, for 1 ≤ l,m ≤ n2 . on the other hand when n is odd, a kn on g2 (respectively on g3) has a 2-factorization, say {e1,e2, ...,en−1 2 } (respectively {f1,f2, ...,fn−1 2 }). for x = 1,2, ..., n−1 2 , if ex = {e1,e2, ...,en} and fx = {f1,f2, ...,fn}, then we can form blocks el ∪fm of size 4, for 1 ≤ l,m ≤ n. now we define b4 = {el ∪fm : el ∈ ex and fm ∈ fx for x = 1,2, ...,n−1 and 1 ≤ l,m ≤ n2 } if n is even, and b4 = {el ∪fm : el ∈ ex and fm ∈ fx for x = 1,2, ..., n−12 and 1 ≤ l,m ≤ n } if n is odd. theorem 2.1. suppose a bibd(2n,3,λ) and a bibd(n,3,µ) exist. (a) suppose n is even and there are nonnegative integers i, j, u and v such that i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + v(n−1). (1) then there exists a gdd(1,n,n,4;iλ + jµ + unµ + vn/2, i(2n−1)λ+jµ(n−1) 2 ). (b) suppose n is odd and there are nonnegative integers i, j, u and v such that i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + 2v(n−1). (2) then there exists a gdd(1,n,n,4;iλ + jµ + unµ + vn, i(2n−1)λ+jµ(n−1) 2 ). proof. let g1 = {x}, g2 = {a1,a2, ...,an} and g3 = {b1,b2, ...,bn}. then consider the following sets: • b1 = {g1 ∪b : b is a block of bibd(2n,3,λ) on g2 ∪g3}; • b2 = {g1 ∪b : b is a block of bibd(n,3,µ) on g2 and g3}; • b3 = {{a}∪b : a ∈ gi and b is a block of bibd(n,3,µ) on gj for i,j = 2,3 and i 6= j}. in b1, every element from g2 ∪g3 comes with the point of g1 λ(2n−1) 2 times and every pair of elements from g2 ∪g3 comes λ times. in b2, every element from g2 and g3 comes with the point of g1 µ(n−1) 2 times and first associate pair from g2 and g3 comes µ times. in b3, first associate pair from g2 and g3 comes µn times and second associate pair from g2 and g3 comes µ(n−1) times. in b4, first and second associate pairs from g2 and g3 occur n and 2(n−1) times respectively if n is odd while first and second associate pairs from g2 and g3 occur n2 and n−1 times respectively if n is even. suppose we have i copies of b1, j copies of b2, u copies of b3 and v copies of b4. then the following matrix displays the replication number of each pair (a1,x), (a1,a2) and (a1,b1) in ib1, jb2, ub3 and vb4, where i, j, u and v are any nonnegative integers. for n even,   (a1,x) (a1,a2) (a1,b1) ib1 i(2n−1)λ 2 iλ iλ jb2 j(n−1)µ 2 jµ 0 ub3 0 unµ uµ(n−1) vb4 0 vn/2 v(n−1)   . 87 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 so we have a gdd(1,n,n,4;iλ + jµ + unµ + vn/2, i(2n−1)λ+jµ(n−1) 2 ) when i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + v(n−1). (3) for n odd,   (a1,x) (a1,a2) (a1,b1) ib1 i(2n−1)λ 2 iλ iλ jb2 j(n−1)µ 2 jµ 0 ub3 0 unµ uµ(n−1) vb4 0 vn 2v(n−1)   . so we have a gdd(1,n,n,4;iλ + jµ + unµ + vn, i(2n−1)λ+jµ(n−1) 2 ) when i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + 2v(n−1). (4) theorem 2.2. (a) if a bibd(2n,b,r,3,λ) exists for odd n and if r − λ = 2(n − 1)t, then a gdd(1,n,n,4;λ + nt,r) exists. (b) if a bibd(2n,b,r,3,λ) exists for even n and if r−λ = (n−1)t, then a gdd(1,n,n,4;λ+ nt 2 ,r) exists. proof. let b = {g1 ∪b : b is a block of bibd(2n,b,r,3,λ)}. then t copies of b4 along with b give the required gdd. example 2.3. as a bibd(6,40,20,3,8) exists, r−λ = 12 = 4×3. we get a gdd(1,3,3,4; 8 + 3×3 = 17,20 = (8 + 3×4)) using 3 copies of b4. essentially, b4 is a gdd on two groups of size n, where the indices depend on n odd or even. for n even, b4= gdd(n,n,4;λ1 = n2 ,λ2 = n − 1) and for n odd, b4= gdd(n,n,4;λ1 = n,λ2 = 2(n − 1)). now the replication number r for a bibd(2n,3,λ) is λ(2n−1) 2 . if we wish r − λ to be a multiple of (n−1) , say s(n−1) when n is even (respectively 2s(n−1) when n is odd), then λ = 2s(n−1) 2n−3 (respectively λ = 4s(n−1) 2n−3 ). for s = 2n−3, λ = 2(n−1) for n even (respectively λ = 4(n−1) for n odd). example 2.4. for n = 4, we have gdd(1,4,4,4; 16,21) by using blocks of a bibd(8,b,21,3,6) and s = 5 copies of b4 =gdd(4,4,4; 2,3). example 2.5. for n = 5, we have gdd(1,5,5,4; 51,72) by using blocks of a bibd(10,b,72,3,16) and s = 7 copies of b4 =gdd(5,5,4; 5,8). in general : theorem 2.6. for n even, using a bibd(2n,3,2(n−1)) and 2n−3 copies of a gdd(n,n,4; n 2 ,n−1), a gdd(1,n,n,4; 2n 2+n−4 2 ,(n−1)(2n−1)) and for n odd, using a bibd(2n,3,4(n−1)) and 2n−3 copies of a gdd(n,n,4;n,2(n−1)), a gdd(1,n,n,4; 2n2 + n−4,2(n−1)(2n−1)) exists. in the next section, we obtain some necessary conditions for the existence of a gdd(1,n,n,4;λ1,λ2). towards this aim, assuming a gdd(1,n,n,4;λ1,λ2) exists, we count the number of blocks, ri, containing a given element x of gi for i = 1,2,3, and the required number of blocks, say b, for the gdd. 88 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 3. necessary conditions suppose a gdd(1,n,n,4;λ1,λ2) exists with groups g1, g2, g3 of size 1, n, n respectively. let ri be the replication number of each element of gi for i = 1,2,3. as the size of g2 is equal to the size of g3, r2 = r3. then by counting argument, r1 = 2nλ23 and r2 = r3 = λ1(n−1)+λ2(n+1) 3 . let b be the required number of blocks for a gdd(1,n,n,4;λ1,λ2) if it exists. since 4 × b = r1 × 1 + r2 × (n + n), we have b = λ1(n 2−n)+λ2(n2+2n) 6 . as r1 and r2 must be integers, we have the following. • if n ≡ 0 (mod 3), then λ1 ≡ λ2 (mod 3). • if n ≡ 1 (mod 3), then λ2 ≡ 0 (mod 3). • if n ≡ 2 (mod 3), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3). since b must be an integer, we have the following. • if n ≡ 0,4 (mod 6), then no restriction on λ1 and λ2. • if n ≡ 1,3 (mod 6), then λ2 ≡ 0 (mod 2). • if n ≡ 2 (mod 6), then λ1 + λ2 ≡ 0 (mod 3). • if n ≡ 5 (mod 6), then 2λ1 + 5λ2 ≡ 0 (mod 6). hence some basic necessary conditions for the existence of a gdd(1,n,n,4;λ1,λ2) are • if n ≡ 0 (mod 6), then λ1 ≡ λ2 (mod 3), • if n ≡ 1 (mod 6), then λ2 ≡ 0 (mod 6), • if n ≡ 2 (mod 6), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3), • if n ≡ 3 (mod 6), then λ1 ≡ λ2 (mod 3), λ2 ≡ 0 (mod 2), • if n ≡ 4 (mod 6), then λ2 ≡ 0 (mod 3), and • if n ≡ 5 (mod 6), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 6). above necessary conditions are summarized in table 1, where “none” means the design does not exist for any n. λ1 is given in modulo 3 and λ2 is given in modulo 6. λ1\λ2 0 1 2 3 4 5 0 all n none none n even none none 1 n ≡ 1 (mod 3) n ≡ 0 (mod 6) none n ≡ 4 (mod 6) n ≡ 0 (mod 3) none 2 n ≡ 1 (mod 3) none n ≡ 0 (mod 3) n ≡ 4 (mod 6) none n ≡ 0 (mod 6) table 1. the necessary conditions for gdd(1,n,n,4;λ1,λ2) a side application of the table is that for n ≡ 3 (mod 6) instead of constructing three families: gdd(1,6m + 3,6m + 3,4; 3t,6s), gdd(1,6m + 3,6m + 3,4; 3t + 1,6s + 4) and gdd(1,6m + 3,6m + 3,4; 3t + 2,6s + 2), one needs to construct just one family gdd(1,6m + 3,6m + 3,4; 3t,6s). then the family gdd(1,6m + 3,6m + 3,4; 3t + 2,6s + 2) can be obtained by taking the blocks of a gdd(1,6m + 89 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 3,6m + 3,4; 3t,6s) and the blocks of a bibd(12m + 7,4,2). similarly, a gdd(1,6m+3,6m+3,4; 3t+1,6s+4) can be obtained by taking the blocks of a gdd(1,6m+ 3,6m + 3,4; 3(t−1),6s) and the blocks of a bibd(12m + 7,4,4) where m is any nonnegative integer. as a gdd(1,n,n,4;λ1,λ2) has 3 groups and blocks of size 4, each block contains at least one associate pair. then b ≤ [ ( n 2 ) + ( n 2 ) ]λ1 = n(n − 1)λ1. now substituting the value of b, we have the following theorem: theorem 3.1. a necessary condition for the existence of a gdd(1,n,n,4;λ1,λ2) is λ2 ≤ 5(n−1) n + 2 λ1. corollary 3.2. for the existence of a gdd(1,n,n,4,λ1,λ2), λ2 ≤ 5λ1. the blocks of a gdd(1,n,n,4;λ1,λ2), if exists, have (n2 + 2n)λ2 second associate pairs. there can be at most r1 blocks of type (1,1,2) which account for 5r1 second associate pairs, we have b − r1 ≥ (n2+2n)λ2−5r1 4 as all other blocks can have at the most 4 second associate pairs. thus, we have the following: theorem 3.3. a necessary condition for the existence of a gdd(1,n,n,4;λ1,λ2) is b ≥ (n2+2n)λ2−r1 4 . corollary 3.4. a necessary condition for the existence of a gdd(1,n,n,4;λ1,λ2) is λ2 ≤ 2(n−1) n λ1 < 2λ1. proof. substituting the values of b and r1 in b ≥ (n2+2n)λ2−r1 4 , we have λ2 ≤ 2(n−1) n λ1 < 2λ1. as a consequence of the above corollary, we have corollary 3.5. for any nonnegative integer t and any positive integers x and s, where x ≤ s following gdds do not exist. 1. a gdd(1,6t + 4,6t + 4,4; 3x + 1,6s + 3). 2. a gdd(1,3t,3t,4; 3x + 1,6s + 4). 3. a gdd(1,6t,6t,4; 3x + 2,6s + 5). 4. a gdd(1,n,n,4; 3x,6s). 4. existence of families of gdd(1,n,n,4;λ1, λ2) unless otherwise stated in this section we are assuming λ1 ≥ λ2. remark 4.1. a gdd(1,n,n,4; 0,λ2) does not exist as the number of groups is less than the block size. as a consequence of theorem 1.4 we have following theorem where unless otherwise stated λ, s and t are nonnegative integers and n > 1. theorem 4.2. if a bibd(2n+1,4,λ2) and a bibd(n,4,λ) exist, then a gdd(1,n,n,4;λ1 = λ2 +λ,λ2) exists. in particular, we have 1. a gdd(1,n,n,4; 6t,6s) exists for all n ≥ 4, where t ≥ s. 2. a gdd(1,n,n,4; 6s + 3t,6s) exists for n ≡ 0,1 (mod 4). 90 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 3. a gdd(1,n,n,4; 6t + 3s,3s) exists for n ≡ 0 (mod 2). 4. a gdd(1,n,n,4; 6t + 2s,2s) exists when n ≡ 0 (mod 3). 5. a gdd(1,n,n,4; 3t + s,s) exists for n ≡ 0 (mod 12). 6. a gdd(1,n,n,4; 3t + 2s,2s) exists for n ≡ 9 (mod 12). 7. a gdd(1,n,n,4; 6t + λ,λ) exists for n ≡ 0 (mod 6). 8. a gdd(1,n,n,4; 3t,3s) exists for n ≡ 0 (mod 4), where t ≥ s. 9. a gdd(1,n,n,4; 2t + 6s,6s) exists for n ≡ 1 (mod 6). 10. a gdd(1,n,n,4; 2t + 3s,3s) exists for n ≡ 4 (mod 6). 11. a gdd(1,n,n,4; 6s + λ,6s) exists for n ≡ 1 (mod 12). 12. a gdd(1,n,n,4; 3s + λ,3s) exists for n ≡ 4 (mod 12). case 1: λ2 ≡ 0 (mod 6) from theorem 4.2(1) we have corollary 4.3. necessary conditions are sufficient for the existence of a gdd(1, n,n,4;λ1,λ2) for λ1 ≡ 0 (mod 6) and λ2 ≡ 0 (mod 6). from theorem 4.2(2) we have corollary 4.4. necessary conditions are sufficient for the existence of a gdd(1, n,n,4;λ1,λ2) for n ≡ 5 (mod 12). in the above family λ1 −λ2 ≡ 3 (mod 6). from the necessary conditions, when λ2 ≡ 3 (mod 6) and λ1 ≡ 0 (mod 3), n has to be even. from theorem 4.2 (3), a gdd(1,n,n,4; 6t + 3,6s + 3) exists for any even n and any nonnegative integers s and t, where t ≥ s. from theorem 4.2(8), a gdd(1,n,n,4; 6t,6s+3) exists when n ≡ 0 (mod 4) for any nonnegative integers s and t, where t > s. hence we have corollary 4.5. necessary conditions are sufficient for the existence of a gdd(1, n,n,4;λ1,λ2) for n ≡ 0 (mod 4), λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3). when λ1 ≡ 2 (mod 3) and λ2 ≡ 0 (mod 6), from the necessary conditions, we have n ≡ 1 (mod 3). for n ≡ 1 (mod 3), a bibd(n,4,2) and a bibd(2n + 1,4,6) exist for n ≥ 4. from theorem 4.2(9) and (10), we have lemma 4.6. a gdd(1,n,n,4; 6t+ 2,6s) exists when n ≡ 1 (mod 3) for any nonnegative integers s and t, where t ≥ s. for n ≡ 1,4 (mod 12), a bibd(n,4,5) and a bibd(2n + 1,4,6) exist. from theorem 4.2(11) and (12), we have lemma 4.7. a gdd(1,n,n,4; 6t + 5,6s) exists when n ≡ 1,4 (mod 12) for any nonnegative integers s and t, where t ≥ s. for λ2 ≡ 0 (mod 6), and λ1 ≡ 1,2 (mod 3) from the necessary conditions, we have n ≡ 1 (mod 3). from theorem 4.2 ( 9) and (10), we have 91 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 lemma 4.8. a gdd(1,n,n,4; 6t+ 4,6s) exists when n ≡ 1 (mod 3) for any nonnegative integers s and t, where t ≥ s. from theorem 4.2 (11) and (12), we have the following: lemma 4.9. a gdd(1,n,n,4; 6t + 1,6s) exists for n ≡ 1,4 (mod 12), where s and t are nonnegative integers such that t ≥ s. for n ≡ 9 (mod 12), a bibd(n,4,3) and bibd(2n + 1,4,6) exist. so we have lemma 4.10. a gdd(1,n,n,4;λ1 = 3t,λ2 = 6s) exists when n ≡ 9 (mod 12) for any nonnegative integers s and t, where λ1 ≥ λ2. hence, we have : corollary 4.11. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,λ2) for λ1 ≥ λ2 for n ≡ 1,4 (mod 12), λ1 ≡ 1 (mod 3) and λ2 ≡ 0 (mod 6). case 2: λ2 ≡ 1 (mod 6) in this case from the necessary conditions, we have λ1 ≡ 1 (mod 3) and n ≡ 0 (mod 6). from theorem 4.2 (7): lemma 4.12. a gdd(1,n,n,4; 6t + 1,6s + 1) exists when n ≡ 0 (mod 6) for any nonnegative integers s and t, where t ≥ s. from theorem 4.2 (5): lemma 4.13. a gdd(1,n,n,4; 6t+ 4,6s+ 1) exists when n ≡ 0 (mod 12) for any nonnegative integers s and t, where t ≥ s. corollary 4.14. necessary conditions are sufficient for the existence of a gdd(1,n, n,4;λ1,λ2) for n ≡ 0 (mod 12), and λ2 ≡ 1 (mod 6). case 3: λ2 ≡ 2 (mod 6) in this case, λ1 ≡ 2 (mod 3) and n ≡ 0 (mod 3). from theorem 4.2(4) we have the following lemma. lemma 4.15. a gdd(1,n,n,4; 6t + 2,6s + 2) exists when n ≡ 0 (mod 3) for any nonnegative integers s and t, where t ≥ s. from theorem 4.2(5) and (6), we have lemma 4.16. a gdd(1,n,n,4; 6t + 5,6s + 2) exists when n ≡ 0 (mod 12) for any nonnegative integer s and t, where t ≥ s. hence we have: corollary 4.17. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,λ2) for n ≡ 0,9 (mod 12), and λ2 ≡ 2 (mod 6). case 4: λ2 ≡ 3 (mod 6) in this case from the necessary conditions, when λ1 ≡ 1,2 (mod 3), n ≡ 4 (mod 6). for n ≡ 4 (mod 6), a bibd(n,4,2), a bibd(n,4,4) and a bibd(2n+1,4,3) exist. from theorem 4.2(10), we have the following lemma: 92 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 lemma 4.18. when n ≡ 4 (mod 6), a gdd(1,n,n,4; 6t+ 1,6s+ 3) exists for any nonnegative integers s and t, where t > s and a gdd(1,n,n,4; 6t + 5,6s + 3) exists where t ≥ s. for n ≡ 4 (mod 12), a bibd(n,4,1) and a bibd(2n + 1,4,3) exist. hence we have a gdd(1,n,n, 4; 3,3), a gdd(1,n,n,4; 4,3) and a gdd(1,n,n,4; 5,3). from theorem 4.2 (12), we have corollary 4.19. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,6s+3) when n ≡ 4 (mod 12) for any nonnegative integers s and λ1, where λ1 ≥ 6s + 3. when λ1 ≡ 0 (mod 3), n ≡ 0 (mod 2), from theorem 4.2 (8), we have corollary 4.20. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,6s+3) when n ≡ 0 (mod 4) for any nonnegative integers s and λ1, where λ1 ≥ 6s + 3. case 5: λ2 ≡ 4 (mod 6) in this case from the necessary conditions, we have λ1 ≡ 1 (mod 3) and n ≡ 0 (mod 3). for n ≡ 0 (mod 12), a bibd(2n + 1,4,1), and a bibd(n,4,3) exist. hence, a gdd(1,n,n,4; 3x + 1,4) exists for x > 1. hence corollary 4.21. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1 = 3t+1,λ2 = 6s + 4) for n ≡ 0 (mod 12). similarly from theorem 4.2 (6), we have a gdd(1,n,n,4;λ1 = 3x + 4,λ2 = 4) for a nonnegative integer x, hence corollary 4.22. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1 = 3t+1,λ2 = 6s + 4) for n ≡ 9 (mod 12) where λ1 > λ2. for n ≡ 0 (mod 3), a bibd(2n + 1,4,2), a bibd(2n + 1,4,4) and a bibd(n,4,6) exist for n ≥ 4. from theorem 4.2(4), we have lemma 4.23. a gdd(1,n,n,4; 6t + 4,6s + 4) exists when n ≡ 0 (mod 3) for any nonnegative integers s and t, where t ≥ s. case 6: λ2 ≡ 5 (mod 6) here, from the necessary conditions, we have λ1 ≡ 2 (mod 3) and n ≡ 0 (mod 6). but for n ≡ 0 (mod 12), a bibd(2n + 1,4,5) and a bibd(n,4,3) exist. from theorem 4.2(5) and (6), we have lemma 4.24. a gdd(1,n,n,4; 6t+ 2,6s+ 5) exists when n ≡ 0 (mod 12) for any nonnegative integers s and t, where t > s. from theorem 4.2(7), we have lemma 4.25. a gdd(1,n,n,4; 6t + 5,6s + 5) exist when n ≡ 0 (mod 6) for any nonnegative integers s and t, where t ≥ s. hence we have corollary 4.26. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1 = 3t+2,λ2 = 6s + 5) for n ≡ 0 (mod 12) where λ1 > λ2. we have summarized main results from this section in table 2. from table 2, we have theorem 4.27. necessary conditions are sufficient for the existence of a gdd(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. 93 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 n ≡ the existence is not known for 0 (mod 12) corollary 4.3, corrolary 4.5, corollary 4.14, corollary 4.17, corollary 4.20, corollary 4.21, corollary 4.26, lemma 4.12, lemma 4.13, lemma 4.15, lemma 4.16, lemma , lemma 4.23, lemma 4.24 1 (mod 12) corollary 4.3, corollary 4.11, lemma 4.6, lemma 4.7, lemma 4.8, lemma 4.9, theorem 4.2 (11) 2 (mod 12) corollary 4.3 gdd(1,n,n,4; 3t,6s + 3) gdd(1,n,n,4; 6t + 3,6s) 3 (mod 12) corollary 4.3, lemma 4.15, lemma 4.23 gdd(1,n,n,4; 6t + 3,6s) gdd(1,n,n,4; 6t + 1,6s + 4) gdd(1,n,n,4; 6t + 5,6s + 2) 4 (mod 12) corollary 4.3, corollary 4.5, corollary 4.11, corollary 4.19, corollary 4.20, lemma 4.6, lemma 4.7, lemma 4.8, lemma 4.9, lemma 4.18 5 (mod 12) corollary 4.3, corollary 4.4 6 (mod 12) corollary 4.3, lemma 4.12, lemma 4.15, lemma 4.25 gdd(1,n,n,4; 6t + 3,6s) gdd(1,n,n,4; 3t,6s + 3) gdd(1,n,n,4; 6t + 4,6s + 1) gdd(1,n,n,4; 6t + 5,6s + 2) gdd(1,n,n,4; 6t + 1,6s + 4) gdd(1,n,n,4; 6t + 2,6s + 5) 7 (mod 12) corollary 4.3, lemma 4.6, lemma 4.8 gdd(1,n,n,4; 6t + 1,6s) gdd(1,n,n,4; 6t + 3,6s) gdd(1,n,n,4; 6t + 5,6s) 8 (mod 12) corollary 4.3, corollary 4.5, corollary 4.20, 9 (mod 12) corollary 4.3, corollary 4.17, corollary 4.22, lemma 4.10, lemma 4.15, lemma 4.23 10 (mod 12) corollary 4.3, lemma 4.6, lemma 4.8, lemma 4.18 gdd(1,n,n,4; 6t + 3,6s) gdd(1,n,n,4; 6t + 1,6s) gdd(1,n,n,4; 6t + 5,6s) gdd(1,n,n,4; 6t + 2,6s + 3) 11 (mod 12) corollary 4.3 gdd(1,n,n,4; 6t + 3,6s) table 2. for the existence of a gdd(1,n,n,4;λ1,λ2), λ1 ≥ λ2 5. specific gdds in this section, we study the existence of gdd(1,n,n,4;λ1,λ2) for specific values of the parameters. 5.1. λ1 = 1 theorem 5.1. necessary conditions are sufficient for the existence of a gdd(1,n, n,4; 1,λ2). specifically, a gdd(1,n,n,4; 1,λ2) exists when λ2 = 1 and n ≡ 0 (mod 6) and when λ2 = 0 and n ≡ 1,4 (mod 12). proof. for λ1 = 1, by corollary 3.4, λ2 < 2, hence λ2 can only be 0 or 1. a gdd(1,n,n,4; 1,1) exists for n ≡ 0 (mod 6) as a bibd(2n + 1,4,1) on g1 ∪ g2 ∪ g3 exists. a gdd(1,n,n,4; 1,0) exists for n ≡ 1,4 (mod 12) as a bibd(n,4,1) on gi for i = 2,3, where g1, g2, g3 are groups of size 1, n, n 94 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 respectively. 5.2. λ1 = 2 theorem 5.2. a gdd(1,n,n,4; 2,λ2) exists for λ2 ≤ 2, specifically when λ2 = 2 and n ≡ 0 (mod 3) and when λ2 = 0 and n ≡ 1,4 (mod 6). proof. for λ1 = 2, by corollary 3.4, λ2 < 4, hence λ2 can be 0, 1, 2 and 3. a gdd(1,n,n,4; 2,0) exists for n ≡ 1 (mod 3) as a bibd(n,4,2) on gi for i = 2,3, where g1, g2, and g3 are groups of size 1, n, and n respectively. a gdd(1,n,n,4; 2,1) does not exists for any n from the necessary conditions. a gdd(1,n,n,4; 2,2) exists for n ≡ 0 (mod 3) as a bibd(2n + 1,4,2) on g1 ∪g2 ∪g3 exists. as a gdd(1,n,n,4; 2,3) exists for n = 4 (see example 5.13) and for n = 10 (see example 6.8), we have theorem 5.3. necessary conditions are sufficient for the existence of a gdd(1,n,n,4; 2,λ2) except possibly for n ≡ 4 (mod 6), n 6= 4, 10 and λ2 = 3. 5.3. λ1 = 3 theorem 5.4. necessary conditions are sufficient for the existence of a gdd(1,n, n,4; 3,λ2). specifically, a gdd(1,n,n,4; 3,λ2) exists when λ2 = 3 and n ≡ 0 (mod 2) and when λ2 = 0 and n ≡ 0,1 (mod 4). proof. for λ1 = 3, by corollary 3.4, λ2 < 6. hence λ2 can be 0 and 3. a gdd(1,n,n,4; 3,0) exists for n ≡ 0,1 (mod 4) as a bibd(n,4,3) on gi for i = 2,3 exists. but a gdd(1,n,n,4; 3,0) does not exist for n ≡ 2,3 (mod 4) by corollary 1.7. a gdd(1,n,n,4; 3,3) exists for n ≡ 0 (mod 2) as a bibd(2n+1,4,3) on g1 ∪g2 ∪g3 exists. 5.4. n=2 when n = 2, both λ1 and λ2 are 0 modulo 3. in a gdd(1,2,2,4;λ1,λ2), there are no blocks of type (0,4) and (1,3). hence, a gdd(1,2,2,4;λ1,0) does not exist. let b1 and b2 be the number of blocks of type (1,1,2) and (2,2) for a gdd(1,2,2, 4;λ1,λ2) if it exists. then b1 + 2b2 = 2λ1 and 5b1 + 4b2 = 8λ2. hence 3b1 = 8λ2 −4λ1 as b1 = r1 = 43λ2, we have lemma 5.5. a necessary condition for the existence of a gdd(1,2,2,4;λ1,λ2) is λ1 = λ2. as a gdd(1,n,n,k;λ,λ) is a bibd(2n + 1,k,λ) and as a bibd(5,4,3) exists by theorem 1.2, we have theorem 5.6. necessary conditions are sufficient for the existence of a gdd(1,2, 2,4;λ1,λ2). 5.5. n=3 let g1 = {x}, g2 = {a,b,c} and g3 = {1,2,3}. in a gdd(1,3,3,4;λ1,λ2), there is no block of type (0,4). let b1, b2 and b3 be the number of blocks of type (1,1,2), (1,3) and (2,2) respectively. then b1 + 3b2 + 2b3 = 6λ1 and 5b1 + 3b2 + 4b3 = 15λ2. as 6λ1 = b1 + 3b2 + 2b3 ≤ 5b1 + 3b2 + 4b3 = 15λ2, we have λ1 ≤ 52λ2. also, by corollary 3.4, λ2 ≤ 4 3 λ1. hence, 95 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 lemma 5.7. a necessary condition for the existence of a gdd(1,3,3,4;λ1,λ2) is 34λ2 ≤ λ1 ≤ 5 2 λ2. in other words, 2 5 λ1 ≤ λ2 ≤ 43λ1. remark 5.8. a gdd(1,3,3,4;λ1, 0) does not exist as there are no blocks of type (0,4). case 1. λ1 < λ2. for n = 3, λ1 ≡ λ2 (mod 3). let λ2 −λ1 =3s for some positive integer s. let λ1 = 3t + i, for i = 0,1,2. then from λ2 ≤ 4λ13 , λ2 ≤ 4t+i. therefore the difference λ2−λ1 = 3s is less than or equal to t. using λ2 = 3s + λ1 and λ2 ≤ 4λ13 , the smallest λ1 will be 9s and λ2 will be 12s. hence the smallest gdd where λ2 −λ1 = 3s is a gdd(1,3,3,4; 9s,12s). for s = 1, we construct a gdd(1,3,3,4; 9,12) as follows. a relabeling construction let x = {0,1,2,3,4,5,6}, g1 = {0}, g2 = {1,2,3} and g3 = {4,5,6}. then b = {{2,4,5,6},{3,5, 6,0},{4,6,0,1},{5,0,1,2}, {6,1,2,3},{0,2,3,4}, {1,3,4,5}} is a collection of blocks of a bibd(7,4, 2) on g1 ∪g2 ∪g3 = x. we relabel elements of these blocks using different permutations (say α) on x to get following six isomorphic bibds. • α(0) = 0, α(1) = 1, α(2) = 3, α(3) = 2, α(4) = 4, α(5) = 6 and α(6) = 5. then, when we relabel the points of x using α, the blocks of b become b1 = {{3,4,6,5},{2,6,5,0},{4,5,0,1},{6, 0,1,3},{5,1,3,2},{0,3,2,4},{1,2,4,6}}. • α(0) = 0, α(1) = 2, α(2) = 1, α(3) = 3, α(4) = 6, α(5) = 5 and α(6) = 4. then the blocks of b become b2 = {{1,6,5,4},{3,5,4,0}, {6,4,0,2},{5,0,2,1},{4,2,1,3},{0,1,3,6},{2,3,6,5}}. • α(0) = 2, α(1) = 1, α(2) = 0, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 6. then the blocks of b become b3 = {{0,4,5,6},{3,5,6,2}, {4,6,2,1},{5,2,1,0},{6,1,0,3},{2,0,3,4},{1,3,4,5}}. • α(0) = 2, α(1) = 1, α(2) = 0, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 6. then the blocks of b become b4 = {{0,4,5,6},{3,5,6,2}, {4,6,2,1},{5,2,1,0},{6,1,0,3},{2,0,3,4},{1,3,4,5}}. • α(0) = 6, α(1) = 1, α(2) = 2, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 0. then the blocks of b become b5 = {{2,4,5,0},{3,5,0,6}, {4,0,6,1},{5,6,1,2},{0,1,2,3},{6,2,3,4},{1,3,4,5}}. • α(0) = 6, α(1) = 1, α(2) = 2, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 0. then the blocks of b become b6 = {{2,4,5,0},{3,5,0,6}, {4,0,6,1},{5,6,1,2},{0,1,2,3},{6,2,3,4},{1,3,4,5}}. so b∪b1 ∪ ...∪b6 gives a bibd(7,4,14) which contains 49 blocks. removing the ten blocks containing all three points 1,2,3 and all three points 4,5,6, we have a gdd(1,3,3,4; 14−5 = 9,14−2 = 12). in general, as a bibd(7,4,14t) exists for any positive integer t, we have the following result. lemma 5.9. a gdd(1,3,3,4; 14t−5s,14t−2s) for s = 0,1, · · · , t exists for any positive integer t where s = 0 gives a bibd. for example, when t = 1, we have a gdd(1,3,3,4; 14 − 5 = 9,14 − 2 = 12), and hence by using bibd(7,4,2) repeatedly, we have gdd(1,3,3,4; 9 + 2m,12 + 2m), specifically we are interested in gdd(1,3,3,4; 13,16), gdd(1,3,3,4; 15,18), gdd(1,3,3,4; 17,20), gdd(1,3,3,4; 19,22), gdd(1,3,3,4; 21,24). lemma 5.10. necessary conditions are sufficient for the existence of a gdd(1,3,3,4;λ1,λ2) for λ1 ≤ λ2. 96 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 proof. a gdd(1,3,3,4; 9,12) and a bibd(7,4,2) exist. the smallest gdd(1,3,3,4;λ1,λ2) when the difference λ2 − λ1 = 3s is gdd(1,3,3,4; 9s,12s). s copies of gdd(1,3,3,4; 9,12) and m copies of bibd(7,4,2) together give all required gdd(1,3,3,4; 9s + 2m,12s + 2m) where λ2 − λ1 = 3s. recall that for n = 3, λ2 is always even. when λ1 = λ2, necessary conditions are the same as the conditions for the existence of a bibd(7,4,λ1). case 2 λ1 > λ2. since a bibd(6,3,2) and a bibd(3,3,1) exist, we have the following from theorem 2:   (1,x) (1,2) (1,a) ib1 5i 2i 2i jb2 j j 0 ub3 0 3u 2u vb4 0 3v 4v   so when 5i + j = 2i + 2u + 4v, then we have a gdd(1,3,3,4; 2i + j + 3u + 3v,5i + j). (5) let λ1 − λ2 = 3s for some nonnegative integer s. as λ1 ≤ 5λ22 , 3s ≤ 1.5λ2. hence when the difference λ1 − λ2 = 3s, smallest value of λ2 is 2s and corresponding smallest parameter gdd will be gdd(1,3,3,4; 5s,2s). for s = 1, the required gdd is gdd(1,3,3,4; 5,2) which can be constructed from 5 by letting i = 0, j = 2, u = 1, v = 0. lemma 5.11. necessary conditions are sufficient for the existence of a gdd(1,3,3,4;λ1,λ2) for λ1 ≥ λ2. proof. a gdd(1,3,3,4; 5,2) and a bibd(7,4,2) exist. the smallest gdd(1,3,3,4;λ1,λ2) when the difference λ1−λ2 = 3s is gdd(1,3,3,4; 5s,2s). note that s copies of gdd(1,3,3, 4; 5,2) and m copies of bibd(7,4,2) together give all required gdd(1,3,3,4; 5s + 2m,2s + 2m) where the λ1 −λ2 = 3s. recall that for n = 3, λ2 is always even. when λ1 = λ2, necessary conditions are the same as the conditions for the existence of a bibd(7,4,λ1). lemma 5.10 and lemma 5.11 together complete the case for n = 3 and we have theorem 5.12. necessary conditions are sufficient for the existence of a gdd(1,3,3,4;λ1,λ2). 5.6. n = 4 example 5.13. a gdd(1,4,4,4; 2,3) with g1 = {x}, g2 = {a,b,c,d} and g3 = {1,2,3,4}. the blocks are given below in columns. x x x x x x x x a c a c a b a b a d b c b a a b b d b d c d c d 1 1 2 2 c d d c 1 1 3 3 1 1 2 2 4 4 3 3 1 2 3 4 2 2 4 4 3 3 4 4 for n = 4, from theorem 3.3, λ2 ≤ 32λ1. hence, we have the following corollary: corollary 5.14. a necessary condition for the existence of a gdd(1,4,4,4;λ1,λ2) is λ1 ≥ 23λ2. from table 1, we need to construct two families: gdd(1,4,4,4;λ1,6s) and gdd(1,4,4,4;λ1,6s+3) where s and λ1 are nonnegative integers. for the first family, by corollary 5.14, λ1 ≥ 4s. using 2s copies 97 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 of the gdd(1,4,4,4; 2,3) and λ copies of bibd(4,4,1), we have a gdd(1,4,4,4; 4s + λ,6s), for any λ. to construct a gdd(1,4,4,4;λ1,6s + 3), we observe that by corollary 5.14, λ1 ≥ 4s + 2. hence using 2s+1 copies of the gdd(1,4,4,4; 2,3) and λ copies of a bibd(4,4,1), we have gdd(1,4,4,4; 4s+ 2 + λ,6s + 3), for any nonnegative integer λ. hence we have: theorem 5.15. necessary conditions are sufficient for the existence of a gdd(1,4,4,4;λ1,λ2). 6. difference families constructions the aim of this section is to construct some examples of gdds with a difference family. in the process we make some comments to show sufficiency in certain cases. 6.1. n = 5 recall for n = 5, from the necessary conditions, λ2 ≡ 0 (mod 6). so let λ2 = 6t where t is a nonnegative integer. now from the necessary condition, we have λ1 < λ2 < 2λ1. let λ2 −λ1 =3s. then using λ2 < 2λ1, 3s < λ1. if smallest possible value of λ1 is 3s + 3, then λ2 = 6s + 3 not 0 mod 6. hence the smallest possible value of λ1 has to be 3s + 6 and λ2 = 6s + 6. for example, when s = 1 the smallest possible gdd will be gdd(1,5,5,4; 9,12). we present a difference family construction for gdd(1,5,5,4; 9,12). example 6.1. let the groups be g1 = {∞}, g2 = {1,3,5,7,9} and g3 = {0,2,4, 6,8}. difference family is {{∞,0,1,3},{∞,0,1,4},{∞,0,3,4},{∞,0,3,6},{0,5,1,2},{0,5,1,3},{0,5,1, 4},{0,5,2,3},{0,5,2,4},{0,5,3,4}}. hence, we also have gdd(1,5,5,4; 6m + 9t,6m + 12t) for nonnegative integers m and t. theorem 6.2. the necessary conditions for the existence of a gdd(1,5,5,4; 6t,6s) are sufficient for t ≥ s and the necessary conditions for the existence of a gdd(1,5,5,4; 6t + 3,6s) are sufficient for t ≥ s−1 on the other hand, let λ1 −λ2 =3s, as λ2 ≡ 0 (mod 6), a gdd(1,5,5,4; 6t,6t) and a bibd(5,4,3) exist, hence a gdd(1,5,5,4; 6t + 3s,6t) exists. theorem 6.3. the necessary conditions for the existence of a gdd(1,5,5,4;λ1,λ2) are sufficient for λ1 ≥ λ2. 6.2. n = 6 let g1 = {∞}, g2 = {1,3,5,7,9,11} and g3 = {0,2,4,6,8,10} be groups. example 6.4. the following multiset {{∞,0,2,4},{0,1,4,5},{0,2,7,8},{0,2,4,6},{0,3,6,9},{0, 3,6,9}} is a difference family for a gdd(1,6,6,4; 6,3). note that {0,3,6,9} is a short difference set and gives only three blocks. these bocks cover difference 3 and 6 pairs only once. hence, we also have a gdd(1,6,6,4;m + 6t,m + 3t) for any nonnegative integers m and t. example 6.5. the following multiset {{∞,0,1,5},{∞,0,2,3},{∞,0,3,4},{0,1,2,5},{0,1,4,5}, {0,1,7,10},{0,2,7,8},{0,3,5,10},{0,3,6,9},{0,3,6,9}} is a difference family for a gdd(1,6,6,4; 6,9). hence, we also have a gdd(1,6,6,4;m + 6t,m + 9t) for any nonnegative integers m and t. example 6.6. a gdd(1,6,6,4; 4,1) can be constructed by difference family: {{∞,0, 4,8},{0,1,4,6}, {0,2,4,6}}. 98 d. g. sarvate, d. m. woldemariam / j. algebra comb. discrete appl. 9(2) (2022) 85–99 example 6.7. the difference family for a gdd(1,6,6,4; 5,2) is {{∞,0,4,8},{∞,0,4,8},{0,1,6,7}, {0,2,4,5},{0,2,4,10},{0,3,6,9} where g1={∞}, g2={1,3,5,7,9,11} and g3={0,2,4,6,8,10}. so we have a gdd(1,6,6,4; 3t + 1,3s + 1) for t > s + 1, gdd(1,6,6,4; 3t + 2,3s + 2) for t > s + 1, gdd(1,6,6,4; 6t + 1,6s + 1), gdd(1,6,6,4; 6t + 1,6s + 1) and gdd(1,6,6,4; 6t + 3,6s + 3) for t ≥ s. 6.3. n = 10 example 6.8. g1 = {∞}, g2 = {1,3,5, ...,19} and g3 = {0,2,4, ...,18}. the difference family {{∞,0,3,6},{0,1,3,7},{0,1,8,9},{0,5,7,16},{0,5,10,15}, {0,5,10,15}} provides a gdd(1,10,10,4; 2,3). 7. summary we used interesting construction techniques to construct specific examples for gdds and obtained an important general construction for gdds with three groups of sizes 1,n,n with block size 4. we obtained necessary conditions for the existence of these gdds and proved that they are sufficient for specific values of n, specific values of λ1 and for the existence of a gdd(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. the work leads to several open problems including questions on the existence of unknown families of gdds. acknowledgment: the first author thanks professor jurisich for support through departmental research funds. the second author was supported by adama science and technology university research grant number astu/sp-r/011/19. references [1] c. j. colbourn, d. g. hoffman, r. rees, a new class of group divisible designs with block size three, j. combin. theory ser. a 59(1) (1992) 73–89. [2] h. l. fu, c. a. rodger, group divisible designs with two associate cases: n = 2 or m = 2, j. combin. theory ser. a 83(1) (1998) 94–117. [3] h. l. fu, c. a. rodger, d. g. sarvate, the existence of group divisible designs with first and second associates having block size three, ars combin. 54 (2000) 33–50. [4] h. hanani, balanced incomplete block designs and related designs, discrete math. 11 (1975) 225–369. [5] w. lapchinda, n. punnim, n. pabhapote, gdds with two associate classes with three groups of sizes 1, n and n, australas. j. combin. 58(2) (2014) 292–303. [6] c. c. lindner, c. a. rodger, design theory, 2nd edition, chapman & hall/crc, new york (2008). [7] n. pabhapote, group divisible designs with two associate classes and with two unequal groups, int. j. pure appl. math. 81(1) (2012) 191–198. [8] n. pabhapote, n. punim, group divisible designs with two associate classes and λ2 = 1, int. j. math. sci. (2011) 1–10. [9] a. sakda, c. uiyyasathian, group divisible designs gdd(n,n,n,1;λ1,λ2), australas. j. comb. 69(1) (2017) 18–28. 99 https://doi.org/10.1016/0097-3165(92)90099-g https://doi.org/10.1016/0097-3165(92)90099-g https://doi.org/10.1006/jcta.1998.2868 https://doi.org/10.1006/jcta.1998.2868 https://mathscinet.ams.org/mathscinet-getitem?mr=1742405 https://mathscinet.ams.org/mathscinet-getitem?mr=1742405 https://doi.org/10.1016/0012-365x(75)90040-0 https://mathscinet.ams.org/mathscinet-getitem?mr=3211784 https://mathscinet.ams.org/mathscinet-getitem?mr=3211784 https://doi.org/10.1201/9781315107233 https://ijpam.eu/contents/2012-81-1/16/16.pdf https://ijpam.eu/contents/2012-81-1/16/16.pdf https://www.hindawi.com/journals/ijmms/2011/148580/ https://www.hindawi.com/journals/ijmms/2011/148580/ https://mathscinet.ams.org/mathscinet-getitem?mr=3684654 https://mathscinet.ams.org/mathscinet-getitem?mr=3684654 introduction a new construction of a gdd(1,n,n,4;1, 2) necessary conditions existence of families of gdd(1,n,n,4;1, 2) specific gdds difference families constructions summary references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 1(1) • 19-27 received: 18 may 2014; accepted: 24 june 2014 doi 10.13069/jacodesmath.09554 journal of algebra combinatorics discrete structures and applications horizontal runs in domino tilings research article kamilla oliver1∗, helmut prodinger2∗∗ 1. 91502 erlangen, germany 2. department of mathematics, university of stellenbosch 7602, stellenbosch, south africa abstract: we discuss tilings of a grid (of size n×2) with dominoes of size 2×1. parameters that might be called “longest run” are investigated, in terms of generating functions and also asymptotically. extensions are also mentioned. 2010 msc: 05a15, 68r05 keywords: tiling, generating function, asymptotic, bootstrapping, mellin transform, fibonacci numbers 1. introduction donald knuth [3] included tilings of an n × 2-rectangle using 2 × 1-sized tiles (“dominoes”) as an introductory example of the use of generating functions. if tn is the number of these tilings, then tn = tn−1 + tn−2, since we have two choices to start: one vertical domino, leaving an (n − 1) × 2rectangle, or two horizontal dominos, leaving an (n − 2) × 2-rectangle. since t0 = 1 and t1 = 1, this leads to tn = fn+1 (a fibonacci number). here is one particular tiling of a 20 × 2-rectangle: denoting t the family (=set) of all tiled n× 2-rectangles, for n ≥ 0, then we can write a symbolic equation: t = + t + t ∗ e-mail: olikamilla@gmail.com, this author did most of her research while visiting the university of stellenbosch in september 2013. she thanks the department of mathematics for its hospitality. ∗∗ e-mail: hproding@sun.ac.za, this author was supported by an incentive grant of the nrf of south africa. 19 horizontal runs in domino tilings with the generating function t(z) := ∑ n≥0 tnz n, the symbolic equation translates directly into t(z) = 1 + zt(z) + z2t(z) = 1 1 −z −z2 . this equation is simple enough that, with partial fraction decomposition, one finds an explicit form tn = 1 √ 5 [( 1 + √ 5 2 )n+1 − ( 1 − √ 5 2 )n+1] . the number α := 1 + √ 5 2 is called the golden ratio; it is one of the most important constants in mathematics. in terms of asympotics it is dominating: tn ∼ 1 √ 5 ( 1 + √ 5 2 )n+1 . if one cannot resort to an explicit expression as above, one looks at the dominating singularity and writes t(z) ∼ c 1 −αz as z → 1 α . the constant c may be computed as c = lim z→ 1 α 1 −αz 1 −z −z2 = α √ 5 . in this paper, we are interested in the (consecutive) sequence of horizontal dominoes in a tiling. we are looking for the longest substructure of the type in the figure. we will indicate in section 2 how the expected value of this parameter may be computed. then, in section 3, we change our setting to tilings of n × 3 rectangles. things become more involved, but it is highly instructive to see how one has to deal with the difficulties. for more material of a similar type we refer to [8]. 2. the longest horizontal run in tilings of n×2-rectangles as the first step of our analysis, we decompose a tiled rectangle according to (maximal) runs of horizontal dominoes. we indicate this for the example from the introduction: we see here runs of 3, 1, and 2 (stacked) horizontal dominoes. so our parameter is 3 for this example. various runs of length 0 are 20 k. oliver, h. prodinger not indicated. based on this (unique!) decomposition, we introduce t <h, the family of tiled dominoes where the (maximal) run parameter is < h, we find t <h = <h( <h)∗ . for completeness, we mention that the ‘∗’ operator (kleene’s star in language theory) produces sequences. if a is a set, then a∗ = a0 ∪a1 ∪a2 ∪ ·· · , i.e., all sequences that can be formed from elements of a. this operation is also useful when one deals with generating functions. if f(z) is the generating function associated to a, so that the coefficient of zn (written as [zn]f(z)) counts the number of elements of size n in a, then 1 1−f(z) is the generating function associated to a ∗. for general background we would like to mention the book [2]. here, <h = h−1⋃ i=0 i . the lefthand side describes repetitions of with at most h− 1 elements; the righthand side splits the repetitions into the various possibilities from i = 0, 1, . . . ,h − 1 . in terms of generating functions, the last expression translates into 1 + z2 + z4 + · · · + z2(h−1) = 1 −z2h 1 −z2 . consequently, we get t<h(z) = 1 −z2h 1 −z2 1 1 − z(1−z 2h) 1−z2 = 1 −z2h 1 −z −z2 + z2h+1 . one can see from this that, as h → ∞, which means no more restrictions on the run lengths, we find again the generating function t(z) = 1 1 −z −z2 . further, t≥h(z) := t(z) −t<h(z) = 1 1 −z −z2 − 1 −z2h 1 −z −z2 + z2h+1 = 1 −z2 1 −z −z2 z2h 1 −z −z2 + z2h+1 . there is a dominant root ρh of 1 −z −z2 + z2h+1, which is close to 1/α. so we set ρh := 1/α + εh and continue 1 − 1 α −εh − 1 α2 − 2εh α + α−2h−1 ∼ 0, or εh ∼ √ 5α−2h−1. 21 horizontal runs in domino tilings we write 1 −z −z2 + z2h+1 ∼ ( 1 − z ρh ) c, and find c = lim z→ρh 1 −z −z2 + z2h+1 1 − z ρh ∼ √ 5 α . now we can read off coefficients: [zn]t≥h(z) ∼ αn+1 √ 5 − [zn] α √ 5 1 1 −z/ρh = αn+1 √ 5 − α √ 5 ρ−nh ∼ αn+1 √ 5 − α √ 5 ( 1 α + √ 5α−2h−1 )−n = αn+1 √ 5 − αn+1 √ 5 ( 1 + √ 5α−2h )−n ∼ αn+1 √ 5 − αn+1 √ 5 ( 1 − √ 5α−2h )n ∼ αn+1 √ 5 − αn+1 √ 5 e− √ 5n/α2h. normalization leads to [zn]t≥h(z) tn ∼ 1 −e− √ 5n/α2h. in order to compute the average value of our parameter, one has to sum this:∑ h≥1 ( 1 −e− √ 5n/α2h ) . it is very well understood how to get asymptotics for f(x) := ∑ h≥1 ( 1 −e−x/α 2h ) as x →∞, see [1]. one computes the mellin transform f∗(s) = ∑ h≥1 α2hs · γ(s) = − α2s 1 −α2s γ(s), valid in 〈−1, 0〉 (the fundamental strip). then one employs the inversion formula f(x) = − 1 2πi ∫ −1 2 +i∞ −1 2 −i∞ α2s 1 −α2s γ(s)x−sds, shifts the line of integration to the right and takes the residues (with a negative sign) into account. the contribution from the pole at s = 0 is 1 2 logα x + γ 2 log α − 1 2 . 22 k. oliver, h. prodinger there is also a contribution coming from the simple poles at s = kπi log α : − 1 2 log α ∑ k 6=0 γ ( kπi log α ) e−kπi·logα x. this is a periodic function of small amplitude. such functions occur frequently when one analyses run statistics, see [8]. altogether we found for the average of our parameter the asymptotic formula 1 2 logα n + 1 4 logα 5 + γ 2 log α − 1 2 −δ( √ 5n), with δ(x) := − 1 2 log α ∑ k 6=0 γ ( kπi log α ) e−kπix. let us emphasize that the paper [8] has many similar results, and concrete error terms and estimates are worked out. here, for the benefit of the reader, we suppress technical details and just stick to the main ideas. 3. tilings of n×3-rectangles let us start with an example of a 20 × 3-rectangle: it decomposes in a natural way as follows: figure 1. decomposition into blocks the individual (maximal) blocks are of 3 types, as indicated in the figure: figure 2. first type 23 horizontal runs in domino tilings figure 3. second type figure 4. third type again, we will look at figure 5. maximal sequence of horizontal dominoes present (2 layers) let the three types with a sequence of exactly i horizontal (stacked) dominoes be denoted by f i, g i, h i, then the family of dominoes with our run parameter < h can be described by t <h = (h−1⋃ i=0 f i )[(h−1⋃ i=0 g i ∪ h−1⋃ i=0 h i )(h−1⋃ i=0 f i )]∗ . in terms of generating functions (the variable z marks the length of the domino), we have the following: h−1⋃ i=0 f i −→ 1 + z2 + · · · + z2(h−1) = 1 −z2h 1 −z2 , h−1⋃ i=0 g i −→ z2 + · · · + z2h = z2(1 −z2h) 1 −z2 and h−1⋃ i=0 h i −→ z2 + · · · + z2h = z2(1 −z2h) 1 −z2 . consequently t<h(z) = 1 −z2h 1 −z2 1 1 − 2 z2(1 −z2h) 1 −z2 1 −z2h 1 −z2 = (1 −z2)(1 −z2h) 1 − 4z2 + z4 + 4z2h+2 − 2z4h+2 . 24 k. oliver, h. prodinger in the limit h →∞ (no restrictions), we find t(z) = 1 −z2 1 − 4z2 + z4 , which was already derived in [3] using a different method. these functions only depend on z2, which is clear, since a tiled n× 3-rectangle is only possible for even n. thus we set w := z2 and work with r<h(w) = (1 −w)(1 −wh) 1 − 4w + w2 + 4wh+1 − 2w2h+1 and r(w) = 1 −w 1 − 4w + w2 . there is a dominant root at w = 1/ρ with ρ = 2 + √ 3, and r(w) ∼ 3 − √ 3 6 1 1 −ρz , and so [wn]r(w) ∼ 3 − √ 3 6 ρn. there is a dominant root ωh = 1ρ + εh of 1 − 4w + w2 + 4wh+1 − 2w2h+1. one application of bootstrapping results in εh = 2 √ 3ρh+1 . from a historical point of view, it is interesting to point out that this procedure appeared for the first time in [6]. a computation that is analogous to the one in the previous section yields [wn]t≥h(z) ∼ 3 − √ 3 6 ρn − 3 − √ 3 6 ω−n ∼ 3 − √ 3 6 ρn − 3 − √ 3 6 (1 ρ + εh )−n = 3 − √ 3 6 ρn − 3 − √ 3 6 ρn ( 1 + 2 √ 3ρh )−n ∼ 3 − √ 3 6 ρn − 3 − √ 3 6 ρn ( 1 − 2 √ 3ρh )n . normalization leads to [wn]t≥h(z) [wn]t(z) ∼ 1 −e− 2n√ 3ρh . to compute the average, we need to evaluate∑ h≥1 ( 1 −e−x/ρ h ) , with x = 2n/ √ 3. the asymptotic evaluation is as before: logρ n + logρ 2 − 1 2 logρ 3 + γ log ρ − 1 2 − 1 log ρ ∑ k 6=0 γ (2kπi log ρ ) e−2πik·logρ(2n/ √ 3 ). 25 horizontal runs in domino tilings 4. the longest vertical run in tilings of n×2-rectangles for completeness, we briefly discuss how one can attack the parameter “longest vertical run.” we introduce t <h, the family of tiled dominoes where the (maximal) run parameter is < h, we find t <h = <h ( <h )∗ . in terms of generating functions, this means t<h(z) = 1 −zh 1 −z −z2 + zh+2 . we set again ρh = 1/α + εh, and find εh ∼ 1 √ 5 1 αh+2 . further, [zn]t≥h(z) tn ∼ 1 −e √ 5n/αh+1. the expected value is to be evaluated as ∑ h≥1 ( 1 −e−x/α k ) , with x = √ 5 α n. the asymptotic evaluation is logα n + 1 2 logα 5 + γ log α − 3 2 − 1 α ∑ k 6=0 γ ( 2kπi log α ) e−2πik·logα(n √ 5/α); n refers to the length of the tiled rectangle. the case of an n×3-rectangle and runs of vertical tiles can also be done, but is a bit more elaborate. it leads to a similar type of result. 5. conclusion we have demonstrated how to deal with runlength parameters: first, symbolic equations lead to explicit forms of the associated generating functions. then, one identifies the dominant singularity and finds an approximate expression for the coefficients. the average value that one wants to compute is then asymptotically described by a series. to work out asymptotics for this sum, the mellin transform [1, 2] is used. this series of operations works also in other contexts. a few references for further reading are [4, 5, 7]. references [1] p. flajolet, x. gourdon, p. dumas, mellin transforms and asymptotics: harmonic sums, theoretical computer science 144, 3-58, 1995. 26 k. oliver, h. prodinger [2] p. flajolet, r. sedgewick, analytic combinatorics, cambridge university press, cambridge, 2009. [3] r. graham, d. e. knuth, o. pathasnik, concrete mathematics, second edition, addison wesley, reading, 1994. [4] c. heuberger, h. prodinger, carry propagation in signed digit representations, european j. combin, 24(3), 293-320, 2003. [5] a. knopfmacher, h. prodinger. on carlitz compositions, european j. combin, 19(5), 579-589, 1998. [6] d. e. knuth, the average time for carry propagation, nederl. akad. wetensch. indag. math., 40(2), 238-242, 1978. [7] g. louchard, h. prodinger, asymptotics of the moments of extreme-value related distribution functions, algorithmica, 46, 431-467, 2006. [8] h. prodinger, s. wagner, bootstrapping and double-exponential limit laws, submitted, 2012. 27 introduction the longest horizontal run in tilings of n2-rectangles tilings of n3-rectangles the longest vertical run in tilings of n2-rectangles conclusion references issn 2148-838x j. algebra comb. discrete appl. 3(3) received: 11 april 2016 journal of algebra combinatorics discrete structures and applications editor’s note vladimir d. tonchev this special issue, entitled algebraic combinatorics and applications, of the journal of algebra, combinatorics, discrete structures, and applications, contains selected papers submitted by conference participants at the "algebraic combinatorics and applications: the first annual kliakhandler conference", houghton, michigan, usa, august 26 30, 2015, as well as two additional papers submitted in response to our call for papers. the conference took place on the campus of michigan technological university, and was attended by 43 researchers and graduate and postdoctoral students from usa, canada, croatia, japan, south africa, and turkey. funding for the conference was provided by a generous gift of igor kliakhandler, and a grant from the national science foundation. the conference brought together researchers and students interested in combinatorics and its applications, to learn about the latest developments, and explore different visions for future work and collaborations. over thirty talks were presented on various topics from combinatorial designs, graphs, finite geometry, and their applications to error-correcting codes, network coding, information security, quantum computing, dna codes, mobile communications, and tournament scheduling. the current special issue contains papers on covering arrays and their applications, group divisible designs, automorphism groups of combinatorial designs, covering number of permutation groups, tournaments, large sets of geometric designs, partitions, quasi-symmetric functions, resolvable steiner systems, and weak isometries of hamming spaces. department of mathematical sciences at michigan technological university, usa (email: tonchev@mtu.edu). 125 issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.59822 j. algebra comb. discrete appl. 3(2) • 97–104 received: 14 april 2015 accepted: 16 feburary 2016 journal of algebra combinatorics discrete structures and applications new results on vertex equitable labeling research article pon jeyanthi, anthony maheswari, mani vijayalakshmi abstract: the concept of vertex equitable labeling was introduced in [9]. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 1, 2, 3, · · · , q. a graph g is said to be a vertex equitable if it admits a vertex equitable labeling. in this paper, we prove that the graphs, subdivision of double triangular snake s(d(tn)), subdivision of double quadrilateral snake s(d(qn)), subdivision of double alternate triangular snake s(da(tn)), subdivision of double alternate quadrilateral snake s(da(qn)), da(qm) ⊙ nk1 and da(tm) ⊙ nk1 admit vertex equitable labeling. 2010 msc: 05c78, 05c38 keywords: vertex equitable labeling, vertex equitable graph, double triangular snake graph, double alternate triangular snake graph, double alternate quadrilateral snake graph 1. introduction all graphs considered here are simple, finite, connected and undirected. we follow the basic notation and terminology of graph theory as in [2]. a graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. there are several types of labeling and a detailed survey of graph labeling can be found in [1]. the concept of vertex equitable labeling was due to lourdusamy and seenivasan [9]. let g be a graph with p vertices and q edges and a = { 0, 1, 2, · · · , ⌈ q 2 ⌉} . a graph g is said to be vertex equitable if there exists a vertex labeling f : v (g) → a that induces an edge labeling f∗ defined by f∗(uv) = f(u)+f(v) for all edges uv such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 1, 2, 3, · · · , q, where vf (a) is the number of vertices v with f(v) = a for a ∈ a. the vertex labeling f is known as vertex equitable labeling. a graph g is said to be a vertex equitable if it admits a vertex equitable labeling. in [9] they proved that the graphs like paths, bistars b(n, n), combs, pon jeyanthi (corresponding author); research centre, department of mathematics, govindammal aditanar college for women, tiruchendur 628 215, tamilnadu, india (email: jeyajeyanthi@rediffmail.com). anthony maheswari; department of mathematics, kamaraj college of engg. and technology, virudhunagar, tamilnadu, india (email: bala_nithin@yahoo.co.in). mani vijayalakshmi; department of mathematics, department of mathematics, dr. g. u. pope college of engineering, sawyerpuram, thoothukudi district, tamil nadu, india (email: viji_mac@rediffmail.com). 97 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 cycles cn if n ≡ 0 or 3 (mod 4), k2,n, c (t) 3 for t ≥ 2, quadrilateral snakes, k2 + mk1, k1,n ∪ k1,n+k if and only if 1 ≤ k ≤ 3, ladders, arbitrary super division of any path and cycle cn with n ≡ 0 or 3 (mod 4) are vertex equitable. also they proved that the graphs k1,n if n ≥ 4, any eulerian graph with n edges where n ≡ 1 or 2 (mod 4), the wheel wn, the complete graph kn if n > 3 and triangular cactus with q ≡ 0 or 6 or 9 (mod 12) are not vertex equitable. in addition, they proved that if g is a graph with p vertices and q edges, q is even and p < ⌈ q 2 ⌉ + 2 then g is not vertex equitable. motivated by these results, we [3]-[6] proved that tp-trees, t ⊙ kn where t is a tp-trees with even number of vertices, t ◦̂pn, t ◦̂2pn, t ◦̂cn(n ≡ 0 , 3 (mod 4)), t ◦̃cn(n ≡ 0 , 3 (mod 4)), bistar b(n, n + 1), square graph of bn,nand splitting graph of bn,n, the caterpillar s(x1, x2, · · · , xn) and cn ⊙ k1, p 2n, tadpoles, cm ⊕ cn, armed crowns , [ pm; c 2 n ] , ⟨pm◦̂k1,n⟩, kc4-snakes for all k ≥ 1, generalized kcn-snakes if n ≡ 0 (mod 4), n ≥ 4 and the graphs obtained by duplicating an arbitrary vertex and an arbitrary edge of a cycle cn, total graph of pn, splitting graph of pn and fusion of two edges of a cycle cn are vertex equitable graphs. in this paper, we prove that s(d(tn)), s(d(qn)), s(da(tn)), s(da(qn)), da(qm) ⊙ nk1 and da(tm) ⊙ nk1 are vertex equitable graphs. we use the following definitions in the subsequent section. definition 1. the double triangular snake d(tn) is a graph obtained from a path pn with vertices v1, v2, · · · , vn by joining vi and vi+1 to the new vertices wi and ui for i = 1, 2, · · · , n − 1. definition 2. the double quadrilateral snake d(qn) is a graph obtained from a path pn with vertices u1, u2, · · · , un by joining ui and ui+1 to the new vertices vi, xi and wi, yi respectively and then joining vi, wi and xi, yi for i = 1, 2, · · · , n − 1. definition 3. a double alternate triangular snake da(tn) consists of two alternate triangular snakes that have a common path. that is, a double alternate triangular snake is obtained from a path u1, u2, · · · , un by joining ui and ui+1(alternatively) to the two new vertices vi and wi for i = 1, 2, · · · , n − 1. definition 4. a double alternate quadrilateral snake da(qn) consists of two alternate quadrilateral snakes that have a common path. that is, a double alternate quadrilateral snake is obtained from a path u1, u2, · · · , un by joining ui and ui+1(alternatively) to the two new vertices vi, xi and wi, yi respectively and adding the edges viwiand xiyi for i = 1, 2, · · · , n − 1. definition 5. let g be a graph. the subdivision graph s(g) is obtained from g by subdividing each edge of g with a vertex. definition 6. the corona g1 ⊙ g2 of the graphs g1 and g2 is defined as the graph obtained by taking one copy of g1 (with p vertices) and p copies of g2 and then joining the ith vertex of g1 to every vertex of the ith copy of g2. 2. main results theorem 2.1. let g1(p1, q1), g2(p2, q2), · · · , gm(pm, qm) be vertex equitable graphs with qi even (i = 1, 2, · · · , m) and ui, vi be the vertices of gi (1 ≤ i ≤ m) labeled by 0 and qi2 . then the graph g obtained by identifying v1 with u2 and v2 with u3 and v3 with u4 and so on until we identify vm−1 with um is also a vertex equitable graph. proof. first we assign the label i∑ j=1 qj 2 , 1 ≤ i ≤ m−1 to the common vertices between the two graphs gi, gi+1. then we add the number i∑ j=1 qj 2 to all the remaining vertex labels of the graph gi+1, 1 ≤ i ≤ m−1. hence the edge labels are 1, 2, · · · , q1; q1 + 1, q1 + 2, · · · , q1 + q2, q1 + q2 + 1, q1 + q2 + 2, · · · , q1 + q2 + q3; · · · ; m−1∑ j=1 qj + 1, m−1∑ j=1 qj + 2, · · · , m∑ j=1 qj. theorem 2.2. the graph s(d(tn)) is a vertex equitable graph. 98 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 b bb bb b b b b 3 2 0 5 1 5 3 4 2 figure 1. b b b b b b bb b b b b b 21 3 4 1 6 3 0 7 7 54 6 figure 2. proof. the vertex equitable labeling shown in figure 1 together with theorem 2.1 proves the result. theorem 2.3. the graph s(d(qn)) is a vertex equitable graph. proof. the vertex equitable labeling shown in figure 2 together with theorem 2.1 proves the result. theorem 2.4. the graph s(da(tn)) is a vertex equitable graph. proof. let g = s(da(tn)). let u1, u2, · · · , un be the vertices of path pn. case i. the triangle starts from u1. we construct da(tn) by joining u2i−1 and u2i to the new vertices vi, wi for 1 ≤ i ≤ ⌊ n 2 ⌋ . let v (g) = v (da(tn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪ { xi, yi, x ′ i, y ′ i|1 ≤ i ≤ ⌊ n 2 ⌋} and e(g) = e(da(tn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {xivi, x ′ iwi, viyi, wiy ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ } ∪ {u2i−1xi, u2i−1x′i, yiu2i, y ′ iu2i|1 ≤ i ≤ ⌊ n 2 ⌋ }. we consider the following two sub cases: subcase i. n is even. here |v (g)| = 5n − 1 and |e(g)| = 6n − 2. let a = {0, 1, 2, · · · , 3n − 1}. define a vertex labeling f : v (g) → a as follows: for 1 ≤ i ≤ n 2 , f(u2i−1) = 6(i − 1), f(u′2i−1) = 6i − 5, f(u2i) = f(yi) = 6i − 1, f(xi) = f(y ′ i) = 6i − 3, f(wi) = f(x ′ i) = 6i − 4, f(vi) = 6i − 2 and f(u ′ 2i) = 6i if 1 ≤ i ≤ n−2 2 . it can be verified that the induced edge labels of s(da(tn)) are 1, 2, · · · , 6n − 2 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(tn)) . subcase ii. n is odd. 99 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 b b b b b bbbb b b b b b b b b b b b b b b b b b b b b b 2 8 14 4 10 16 2 8 14 3 9 15 3 9 15 5 11 17 0 17 5 6 11 12 1 6 7 121 13 figure 3. here |v (g)| = 5n − 4 and |e(g)| = 6n − 6. let a = {0, 1, 2, · · · , 3n − 3}. define a vertex labeling f : v (g) → a as follows: we label the vertices u2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u2i, u′2i−1, u ′ 2i, vi, v ′ i, wi, w ′ i( 1 ≤ i ≤ n−1 2 ) as in sub case (i). it can be verified that the induced edge labels of s(da(tn)) are 1, 2, · · · , 6n − 6 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(tn)). case ii. the triangle starts from u2. we construct da(tn) by joining u2i and u2i+1 to the new vertices vi, wi for 1 ≤ i ≤ ⌈ n−2 2 ⌉ . let v (g) = v (da(tn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪ { xi, yi, x ′ i, y ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉} and e(g) = e(da(tn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {xivi, x ′ iwi, viyi, wiy ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ } ∪ {u2ixi, u2ix′i, u2i+1yi, u2i+1y ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ }. we consider the following two sub cases: subcase i. n is odd. here |v (g)| = 5n − 4 and |e(g)| = 6n − 6. let a = {0, 1, 2, · · · , 3n − 3}. define a vertex labeling f : v (g) → a as follows: f(u2i−1) = 6(i − 1) if 1 ≤ i ≤ ⌈ n 2 ⌉ , for 1 ≤ i ≤ ⌊ n 2 ⌋ , f(u2i) = f(u′2i−1) = 6i − 5, f(u′2i) = 6i − 4, f(wi) = f(x ′ i) = 6i − 3, f(xi) = f(y ′ i) = 6i − 2, f(yi) = 6i, f(vi) = 6i − 1. it can be verified that the induced edge labels of s(da(tn)) are 1, 2, · · · , 6n − 6 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(tn)). subcase ii. n is even. here |v (g)| = 5n − 7 and |e(g)| = 6n − 10. let a = {0, 1, 2, · · · , 3n − 5}. define a vertex labeling f : v (g) → a as follows: we label the vertices u2i−1, u′2i−1, u2i ( 1 ≤ i ≤ ⌈ n 2 ⌉) and vi, v ′ i, wi, w ′ i, xi, x ′ i, yi, y ′ i, u ′ 2i ( 1 ≤ i ≤ ⌈ n−2 2 ⌉) as in sub case (i). it can be verified that the induced edge labels of s(da(tn)) are 1, 2, · · · , 6n − 10 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(tn)). an example for the vertex equitable labeling of s(da(t6)) where the two triangles start from u1 is shown in figure 3. theorem 2.5. the graph s(da(qn)) is a vertex equitable graph. proof. let g = s(da(qn)) . let u1, u2, · · · , un be the vertices of path pn. case i. the quadrilateral starts from u1. we construct da(qn) by joining u2i−1 and u2i to the new vertices vi, wi and xi, yi respectively and then joining vi, xi and wi, yi for 1 ≤ i ≤ ⌊ n 2 ⌋ . let v (g) = v (da(qn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪{ v′i, w ′ i, x ′ i, y ′ i, zi, z ′ i|1 ≤ i ≤ ⌊ n 2 ⌋} and e(g) = e(da(qn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 100 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 1} ∪ {viv′i, vix ′ i, xizi, w ′ iwi, wiy ′ i, y ′ iyi, yiz ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ } ∪ {u2i−1v′i, u2i−1w ′ i, u2izi, u2iz ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ }. we consider the following two sub cases: subcase i. n is even. here |v (g)| = 7n − 1 and |e(g)| = 8n − 2. let a = {0, 1, 2, · · · , 4n − 1}. define a vertex labeling f : v (g) → a as follows: for 1 ≤ i ≤ n 2 , f(u2i−1) = 8(i − 1), f(u2i) = f(u′2i−1) = 8i − 1, f(xi) = f(zi) = 8i − 2, f(x′i) = 8i − 3, f(wi) = f(w ′ i) = 8i − 7, f(yi) = f(z ′ i) = 8i − 5, f(y ′ i) = 8i − 6 and f(u′2i) = 8i if 1 ≤ i ≤ n−2 2 . it can be verified that the induced edge labels of s(da(qn)) are 1, 2, · · · , 8n−2 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(qn)) . subcase ii. n is odd. here |v (g)| = 7n − 6 and |e(g)| = 8n − 8. let a = {0, 1, 2, · · · , 4n − 4}. define a vertex labeling f : v (g) → a as follows: we label the vertices u2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u2i, u′2i−1, u ′ 2i, vi, v ′ i, wi, w ′ i, xi, x′i, yi, y ′ i, zi, z ′ i ( 1 ≤ i ≤ n−1 2 ) as in sub case (i). it can be verified that the induced edge labels of s(da(qn)) are 1, 2, · · · , 8n − 8 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(qn)). case ii. the quadrilateral starts from u2. we construct da(qn) by joining u2i and u2i+1 to the new vertices vi, wi and xi, yi respectively and then joining vi, xi and wi, yi for 1 ≤ i ≤ ⌈ n−2 2 ⌉ . let v (g) = v (da(qn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪{ v′i, w ′ i, x ′ i, y ′ i, zi, z ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉} and e(g) = e(da(qn)) ∪ {uiu′i|1 ≤ i ≤ n − 1} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {viv′i, vix ′ i, xizi, w ′ iwi, wiy ′ i, xix ′ i, y ′ iyi, yiz ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ } ∪ {u2iv′i, u2iw ′ i, u2i+1zi, u2i+1z ′ i| 1 ≤ i ≤ ⌈ n−2 2 ⌉ }. we consider the following two sub cases: subcase i. n is odd. here |v (g)| = 7n − 6 and |e(g)| = 8n − 8. let a = {0, 1, 2, · · · , 4n − 4}. define a vertex labeling f : v (g) → a as follows: for 1 ≤ i ≤ ⌈ n 2 ⌉ , f(u2i−1) = 8(i − 1) , f(u′2i−1) = 8i − 7. for 1 ≤ i ≤ ⌊ n 2 ⌋ , f(u2i) = 8i − 7 , f(u′2i) = 8i, f(vi) = f(v ′ i) = 8i − 3, f(xi) = f(zi) = 8i − 1, f(x′i) = 8i − 2, f(wi) = f(w ′ i) = 8i − 6, f(yi) = f(z ′ i) = 8i − 4, f(y ′ i) = 8i − 5. it can be verified that the induced edge labels of s(da(qn)) are 1, 2, · · · , 8n − 8 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(qn)). subcase ii. n is even. let |v (g)| = 7n − 11 and |e(g)| = 8n − 14. let a = {0, 1, 2, · · · , 4n − 7}. define a vertex labeling f : v (g) → a as follows: we label the vertices u2i−1, u2i, u′2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u′2i, vi, v ′ i, wi, w ′ i, xi, x ′ i, yi, y ′ i, ( 1 ≤ i ≤ ⌈ n−2 2 ⌉) as in sub case (i). it can be verified that the induced edge labels of s(da(qn)) are 1, 2, · · · , 8n−14 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ a. hence f is a vertex equitable labeling of s(da(qn)). an example for the vertex equitable labeling of s(da(q7)) where the two quadrilaterals start from u1 is shown in figure 4. theorem 2.6. let g1(p1, q), g2(p2, q), · · · , gm(pm, q) ) be vertex equitable graphs with q odd ui, vi be vertices of gi (1 ≤ i ≤ m) labeled by 0 and ⌈ q 2 ⌉ . then the graph g obtained by joining v1 with u2 and v2 with u3 and v3 with u4 and so on until joining vm−1 with um by an edge is also a vertex equitable graph. proof. the graph g has p1 + p2 + · · · + pm vertices and mq + (m − 1) edges. let fi be the vertex equitable labeling of gi (1 ≤ i ≤ m) and let a = { 0, 1, 2, · · · , ⌈ mq+m−1 2 ⌉} . define a vertex labeling f : v (g) → a as f(x) = fi(x) + (i−1)(q+1) 2 if x ∈ gi for 1 ≤ i ≤ m . the edge labels of gi are increased by (i − 1)(q + 1) for i = 1, 2, · · · , m under the new labeling f. the bridge between the two graphs gi , gi+1 will get the label i(q + 1), 1 ≤ i ≤ m − 1. hence the edge labels of g are distinct and is {1, 2, · · · , mq + m − 1}. also |vf (a) − vf (b)| ≤ 1 for all a, b ∈ a. then the graph g is a vertex equitable graph. 101 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 b b b b b b b b b b b b b b bbb b bb b b b b b b b b bbb b bb b b b b bb b b b b b b 1 2 3 9 10 11 17 18 19 4 5 6 12 13 14 20 21 22 1 4 9 12 17 20 3 6 11 14 19 22 0 7 7 8 8 15 15 16 16 23 23 24 24 figure 4. b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b 6 7 8 11 2 3 4 5 10 9 87 13 14 151 1 2 3 4 15 14 13 11 6 12 105 0 16 figure 5. remark 2.7. [7] the graph da(qm)⊙nk1 and da(tm)⊙nk1 are vertex equitable graphs if m, n = 1, 2. theorem 2.8. the graph da(q2) ⊙ nk1 is a vertex equitable graph for n ≥ 3 proof. let g = da(q2) ⊙ nk1. let v (g) = {u1, u2, v, w, x, y} ∪ {uij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vi, wi, xi, yi|1 ≤ i ≤ n} and e(g) = {u1u2, u1v, vw, wu2, u1x, xy, yu2} ∪ {uiuij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vvi, wwi, xxi, yyi|1 ≤ i ≤ n}. here |v (g)| = 6(n+1) and |e(g)| = 6n+7. let a = { 0, 1, 2, · · · , ⌈ 6n+7 2 ⌉} . define a vertex labeling f : v (g) → a as follows: for 1 ≤ i ≤ n, f(u1i) = i, f(vi) = i+1, f(yi) = n+2+i, f(ui) = 0, f(u2) = 3n+4, f(v) = n+1, f(w) = 2(n+1), f(x) = n+2, f(y) = 2(n+2), f(u2i) = 3n+4−i if 1 ≤ i ≤ n − 1 , f(u2n) = 2n + 3, f(w1) = 1, f(wi) = 3n + 5 − i if 2 ≤ i ≤ n, f(xi) = n + i + 1 if 1 ≤ i ≤ n − 1, f(xn) = 2n + 3. it can be verified that the induced edge labels of da(q2) ⊙ nk1 are 1, 2, · · · , 6n + 7 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ a. hence f is a vertex equitable labeling of da(q2) ⊙ nk1. an example for the vertex equitable labeling of da(q2) ⊙ 4k1 is shown in figure 5. theorem 2.9. the graph da(qm) ⊙ nk1 is a vertex equitable graph for m, n ≥ 3 . proof. by theorem 2.8, da(q2) ⊙ nk1 is a vertex equitable graph. let gi = da(q2) ⊙ nk1 for 1 ≤ i ≤ m − 1. since each gi has 6n + 7 edges, by theorem 2.6, da(qm) ⊙ nk1 admits vertex equitable labeling. an example for the vertex equitable labeling of da(q6) ⊙ 4k1 is shown in figure 6. 102 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 b b b b bb b b b b b b b b b b b b bb b b bb b b b b b b b b b b b b b b b b b b b b b b b bb b b b bb b b bb b b 0 6 12 16 105 32 2822 16 21 26 7 8 9 10 2724 23 22 1 15 14 13 2120 19 18 3 4 5 2 3031 17 29 2524 23 26 7 8 11 6 1 2 4 3 31 30 27 29 15 14 11 13 17 18 20 19 figure 6. b bb b b b b b b b b b b b b b 1 2 3 8 7 6 5 6 7 2 3 4 0 9 5 4 figure 7. theorem 2.10. the graph da(t2) ⊙ nk1 is a vertex equitable graph for n ≥ 3 . proof. let g = da(t2) ⊙ nk1. let v (g) = {u1, u2, u, w} ∪ {uij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vi, wi|1 ≤ i ≤ n} and e(g) = {u1u2, u1v, vu2, u1w, wu2} ∪ {uiuij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vvi, wwi|1 ≤ i ≤ n}. here |v (g)| = 4(n + 1) and |e(g)| = 4n + 5. let a = {0, 1, 2, · · · , ⌈ 4n+5 2 ⌉ }. define a vertex labeling f : v (g) → a as follows. for 1 ≤ i ≤ n, f(u1i) = i, f(u2i) = 2n + 3 − i, f(vi) = i + 1, f(wi) = n + 1 + i, f(u1) = 0, f(u2) = 2n + 3, f(v) = n + 1, f(w) = n + 2. it can be verified that the induced edge labels of da(t2) ⊙ nk1 are 1, 2, · · · , 4n + 5 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ a. hence f is a vertex equitable labeling of da(t2) ⊙ nk1 . an example for the vertex equitable labeling of da(t2) ⊙ 3k1 is shown in figure 7. theorem 2.11. the graph da(tm) ⊙ nk1 is a vertex equitable graph for m, n ≥ 3 . proof. by theorem 2.10, da(t2) ⊙ nk1 is a vertex equitable graph. let gi = da(t2) ⊙ nk1 , 103 p. jeyanthi et al. / j. algebra comb. discrete appl. 3(2) (2016) 97–104 b b b b b b b b b b b b b b b b bbb bbb b b b b b b b b b b 5 6 7 2 3 4 161514 131211 1 2 3 6 7 8 17 16 15 10 11 12 5 4 0 9 14 13 189 figure 8. 1 ≤ i ≤ m−1. since each gi has 4n+5 edges, by theorem 2.6, da(tm)⊙nk1 admits a vertex equitable labeling. an example for the vertex equitable labeling of da(tm) ⊙ 4k1 is shown in figure 8. acknowledgment: the authors would like to thank the referees for their valuable suggestions to improve the paper. references [1] j. a. gallian, graph labeling, electron. j. combin. (2015) (dynamic survey #ds6). [2] f. harary, graph theory, addison-wesley, reading mass, 1972. [3] p. jeyanthi, a. maheswari, some results on vertex equitable labeling, open j. discrete math. 2(2) (2012) 51–57. [4] p. jeyanthi, a. maheswari, vertex equitable labeling of transformed trees, j. algorithms comput. 44(1) (2013) 9–20. [5] p. jeyanthi, a. maheswari, vertex equitable labeling of cyclic snakes and bistar graphs, j. sci. res. 6(1) (2014) 79–85. [6] p. jeyanthi, a. maheswari, m. vijayalaksmi, vertex equitable labeling of cycle and star related graphs, j. sci. res. 7(3) (2015) 33–42. [7] p. jeyanthi, a. maheswari, vertex equitable labeling of cycle and path related graphs, util. math. 98 (2015) 215–226. [8] p. jeyanthi, a. maheswari, m. vijayalakshmi, vertex equitable labeling of double alternate snake graphs, j. algorithms comput. 46 (2015) 27–34. [9] m. seenivasan, a. lourdusamy, vertex equitable labeling of graphs, j. discrete math. sci. cryptogr. 11(6) (2008) 727–735. 104 http://www.combinatorics.org/ojs/index.php/eljc/article/view/ds6/pdf http://dx.doi.org/10.4236/ojdm.2012.22009 http://dx.doi.org/10.4236/ojdm.2012.22009 http://jac.ut.ac.ir/pdf_346_b7a385d110f5ec4c2d805c82bcf3079e.html http://jac.ut.ac.ir/pdf_346_b7a385d110f5ec4c2d805c82bcf3079e.html http://dx.doi.org/10.3329/jsr.v6i1.15044 http://dx.doi.org/10.3329/jsr.v6i1.15044 http://dx.doi.org/10.3329/jsr.v7i3.22810 http://dx.doi.org/10.3329/jsr.v7i3.22810 http://jac.ut.ac.ir/pdf_357_fff6f09f5a1ff8b27a45504304da01c5.html http://jac.ut.ac.ir/pdf_357_fff6f09f5a1ff8b27a45504304da01c5.html http://www.dx.doi.org/10.1080/09720529.2008.10698401 http://www.dx.doi.org/10.1080/09720529.2008.10698401 introduction main results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.66457 j. algebra comb. discrete appl. 3(1) • 13–30 received: 17 july 2015 accepted: 24 november 2015 journal of algebra combinatorics discrete structures and applications infinitely many nonsolvable groups whose cayley graphs are hamiltonian research article dave witte morris abstract: we show there are infinitely many finite groups g, such that every connected cayley graph on g has a hamiltonian cycle, and g is not solvable. specifically, we show that if a5 is the alternating group on five letters, and p is any prime, such that p ≡ 1 (mod 30), then every connected cayley graph on the direct product a5 × zp has a hamiltonian cycle. 2010 msc: 05c25, 05c45 keywords: cayley graph, hamiltonian cycle, solvable group, alternating group 1. introduction it has been conjectured that every connected cayley graph on every finite group has a hamiltonian cycle (unless the graph has less than three vertices). in support of this conjecture, the literature provides numerous infinite families of finite groups g, for which it is known that every connected cayley graph on g has a hamiltonian cycle. (see [2] and its references for more information.) however, it seems that the union of these families contains only finitely many groups that are not solvable. this note puts an end to that unsatisfactory state of affairs: proposition 1.1. there are infinitely many finite groups g, such that every connected cayley graph on g has a hamiltonian cycle, and g is not solvable. since the alternating group a5 (of order 60) is a nonabelian simple group, and is therefore not solvable, the above is an immediate consequence of the following more specific result. proposition 1.2. if p is a prime, such that p ≡ 1 (mod 30), then every connected cayley graph on the direct product a5 ×zp has a hamiltonian cycle. dave witte morris; department of mathematics and computer science, university of lethbridge, canada (email: dave.morris@uleth.ca). 13 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 the proof is based on a case-by-case analysis of cayley graphs of the group a5. most of the hamiltonian cycles were found by computer search (using a fairly naive backtracking algorithm). remark 1.3. rather than merely groups that are not solvable, it would be much more interesting to find infinitely many finite, simple groups g, such that every connected cayley graph on g has a hamiltonian cycle. regrettably, the known methods seem to be hopelessly inadequate for this problem. 2. preliminaries definition 2.1. let s be a subset of a finite group g. the cayley graph of g with respect to the connection set s is the graph cay(g; s) whose vertices are the elements of g, and with edges g gs and g gs−1, for each g ∈ g and s ∈ s. notation 2.2. suppose s is a subset of a finite group g. for s1, . . . ,sm ∈ s ∪s−1, we use (si)mi=1 = (s1, . . . ,sm) to denote the walk in cay(g; s) that visits (in order), the vertices e, s1, s1s2, s1s2s3, . . . , s1s2 · · ·sm. we use (s1, . . . ,sm)k to denote the concatenation of k copies of the sequence (si)mi=1, and the following illustrates other notations that are often useful: (a2,b−3,si) 3 i=1 = (a,a,b −1,b−1,b−1,s1,a,a,b −1,b−1,b−1,s2,a,a,b −1,b−1,b−1,s3). notation 2.3. we use : a5 ×zp → a5 to denote the natural projection (so (x,y) = x). our argument in section 3 is based on the same outline as the proof in [3] of the following result. lemma 2.4 ([3]). every connected cayley graph on a5 has a hamiltonian cycle. the following result is the reason that the statement of proposition 1.2 assumes p ≡ 1 (mod 30). a much weaker hypothesis would suffice in all other parts of the proof. corollary 2.5. let s be a minimal generating set of a5 × zp, where p is prime, and p ≡ 1 (mod 30). if there exists a ∈ s, such that s r{a} generates a5, then cay(a5 ×zp; s) has a hamiltonian cycle. proof. since gcd(|a5|,p) = 1, the minimality of s implies that 〈s r {a}〉 = a5. (namely, since gcd(|a5|,p) = 1, we have g ∈ 〈g〉 for every g ∈ a5 × zp. therefore a5 = 〈s r{a}〉 ⊆ 〈s r {a}〉. since the minimality of s implies a /∈ 〈s r {a}〉, we conclude that 〈s r {a}〉 = a5.) from lemma 2.4, we know there is a hamiltonian cycle (si)60i=1 in cay(a5; s r {a}). since, by assumption, p − 1 is divisible by 30 = 2 · 3 · 5 (and every element of a5 has order 1, 2, 3, or 5), we know a p−1 is trivial. this means ap−1 ∈ zp (so ap−1 centralizes a5), so it is not difficult to verify that( s2i−1,a p−1,s2i,a −(p−1))30 i=1 is a hamiltonian cycle in cay(a5 ×zp; s). for completeness, we sketch the verification that the given walk is a hamiltonian cycle. since 〈s〉 = a5 × zp, and s r {a} ⊆ a5, we know that a projects nontrivially to zp. since gcd(|a5|,p) = 1, this implies zp ⊆ 〈a〉, so every element g of a5 × zp can be written (uniquely) in the form g = xar with x ∈ a5 and 0 ≤ r ≤ p− 1. since (si)60i=1 is a hamiltonian cycle in a cayley graph on a5, we have x = s1s2 · · ·sk, for some k with 0 ≤ k < 60. if k = 2i− 1 is odd, then g = xar =  i−1∏ j=1 s2j−1s2j  s2i−1ar =  i−1∏ j=1 (s2j−1a p−1s2ja −(p−1)  s2i−1ar. 14 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 (because ap−1 ∈ zp is in the center of a5 ×zp). also, if k = 2i is even, then g = xar =  i−1∏ j=1 s2j−1s2j  s2i−1s2iar =  i−1∏ j=1 s2j−1a p−1s2ja −(p−1)  s2i−1ap−1s2ia−(p−1−r). thus, we see (in either case) that g is one of the vertices on the walk (s2i−1,ap−1,s2i,a−(p−1))30i=1. this means that the walk passes through all of the vertices in cay(a5 ×zp; s). also, note that the walk has the correct length (60p) to be a hamiltonian cycle. finally, by using once again the fact that ap−1 is in the center, we see that the terminal vertex of the walk is 30∏ i=1 s2i−1a p−1s2ia −(p−1) = 30∏ i=1 s2i−1s2i = 60∏ i=1 si = e, because (si)60i=1 is a (hamiltonian) cycle. remark 2.6. for definiteness, we point out that we write our permutations on the left, so gs(i) = g ( s(i)) for g,s ∈ a5 and i ∈{1, 2, 3, 4, 5}. the remainder of this section records a few easy consequences of the following well-known, elementary observation. lemma 2.7 (“factor group lemma” [4, §2.2]). suppose • n is a cyclic, normal subgroup of g, • (si)mi=1 is a hamiltonian cycle in cay(g/n; s), and • the voltage π(si)mi=1 generates n. then (s1,s2, . . . ,sm)|n| is a hamiltonian cycle in cay(g; s). corollary 2.8 ([2, cor. 2.11]). suppose • n is a normal subgroup of g, such that |n| is prime, • the image of s in g/n is a minimal generating set of g/n, • there is a hamiltonian cycle in cay(g/n; s), and • s ≡ t (mod n) for some s,t ∈ s ∪s−1 with s 6= t. then there is a hamiltonian cycle in cay(g; s). corollary 2.9. let s be a minimal generating set of a5 × zp, such that s is a minimal generating set of a5. if every element of s has order 2, then cay(a5 ×zp; s) has a hamiltonian cycle. notation 2.10. let c = (si)mi=1 be a walk in a cayley graph cay(g; s). for s ∈ s, we use wtc (s) to denote the difference between the number of occurrences of s and the number of occurrences of s−1 in c. (this is the net weight of the generator s in c.) lemma 2.11. let s = {a1, . . . ,ak,b1, . . . ,b`} be a minimal generating set of a5 × zp, such that s is a minimal generating set of a5. assume • ` ≥ 1, and |ai| = 2 for all i, • c1, . . . ,c` are hamiltonian cycles in cay(a5; s), and 15 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 • [wtci (bj)] is the `× ` matrix whose (i,j) entry is wtci (bj). if det[wtci (bj)] 6≡ 0 (mod p), then cay(a5 ×zp; s) has a hamiltonian cycle. proof. we may assume that ai ∈ a5 for 1 ≤ i ≤ k, for otherwise corollary 2.8 applies with s = ai and t = a−1i . write bi = (bi,vi) (with vi ∈ zp) for 1 ≤ i ≤ `. since s generates a5×zp, the vector [v1, . . . ,v`] must be nonzero in (zp)`. then, since, by assumption, the matrix [wtci (bj)] is invertible over zp, this implies [wtci (bj)][v1, . . . ,v`] t 6= ~0 in (zp)`, so there is some i, such that ∑` j=1 wtci (bj) vj 6= 0 in zp. we now show that this sum is precisely the voltage πci of the walk ci, so lemma 2.7 provides the desired hamiltonian cycle in cay(a5 × zp; s). write ci = (sk)nk=1, let π = ∏n k=1 sk be the voltage of ci, and let n(s) and n′(s), respectively, be the number of occurrences of s and s−1 in ci. since ci is a (hamiltonian) cycle in cay(a5; s), we know that π is trivial, so π = ∑n k=1 s ∗ k, where s ∗ k is the projection of sk to zp. noting that (s−1)∗ = −s∗ for s ∈ s, we have π = n∑ k=1 s∗k = ∑ s∈s∪s−1 n(s)s∗ = ∑ s∈s ( n(s) −n′(s) ) s∗ = ∑ s∈s wtci (s) s ∗ = ∑̀ j=1 wtci (bj) vj. 3. proof of proposition 1.2 assumptions 3.1. let s be a minimal generating set of a5 × zp (and, in accordance with notation 2.3, let s be the image of s in a5). we may assume s is a minimal generating set of a5, for otherwise corollary 2.5 applies. we may also assume, for every element s of s with |s| = 2, that the projection of s to zp is trivial, for otherwise corollary 2.8 applies. case 1. assume s has exactly two elements. write s = {a,b}. subcase 1.1. assume |a| = 2 and |b| = 3. to simplify matters, we show that, by applying an automorphism of a5, we may assume a = (1, 2)(3, 4) and b = (2, 4, 5). first of all, we may assume a = (1, 2)(3, 4), since every element of order 2 in a5 is conjugate to this. then, in order for 〈a,b〉 to be transitive, the support of b must contain an element of each cycle of a (including the 1-cycle (5)). so we may assume b = (2, 4, 5) (after conjugating by (1, 2) and/or (3, 4), if necessary). now, we have the following hamiltonian cycle in cay(a5; s) (see note 4.1): c1 = ( (a,b 2)3, (a,b−2)3, (a,b 2,a,b−2)2 )2 . by using the fact that each left coset of 〈b〉 appears as consecutive vertices in this cycle, we will show that c2 = ( (a,b3p−1)3, (a,b−(3p−1))3, (a,b3p−1,a,b−(3p−1))2 )2 passes through all of the vertices in each left coset of 〈b〉, and is therefore a hamiltonian cycle in cay(a5× zp; s). note that b3p−1 = b2 (since |b| = 3). this implies that if we let x be the terminal vertex of the walk c2, then x is the terminal vertex of the hamiltonian cycle c1, so x is trivial. the projection of x to zp is also trivial, because wtc2 (b) = 0. therefore, the walk c2 is closed. now, since c2 has the correct length to be a hamiltonian cycle, we need only show that it passes through every element of a5 × zp. from the fact that b3p−1 = b2, we see that the vertices of c2 are precisely the same elements of a5 as the vertices of c1; that is, the walk c2 passes through every element of a5. thus, given any v ∈ a5 × zp, the walk c2 visits some vertex w with w = v; that is, v and w are in the same coset of zp. since zp ⊆〈b〉, this implies that v and w are in the same left coset of 〈b〉. also, 16 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 since there are never two consecutive appearances of a in c2, and every occurrence of b is contained in a string b3p−1, we know that c2 traverses every element of any left coset of 〈b〉 that it enters. in particular, c2 traverses every element of the left coset of w, so it passes through v. subcase 1.2. assume |a| = 2 and |b| = 5. we may assume b = (1, 2, 3, 4, 5), after conjugating by some permutation in s5. since |a| = 2, it has a fixed point, which we may assume is 5 (after conjugating by a power of b). so |a| must be either (1, 2)(3, 4), (1, 3)(2, 4), or (1, 4)(2, 3). • for a = (1, 2)(3, 4), we have the following hamiltonian cycle (see note 4.2): c = ( (a,b,a,b 4)2,a,b 2,a,b−1,a,b 4,a,b, (a,b 2)3, a,b−2,a,b 4,a,b−2, (a,b)2,a,b−1,a,b 4,a,b 2 ) . since wtc (b) = 29 6≡ 0 (mod p), theorem 2.11 applies. • for a = (1, 3)(2, 4), we have the following hamiltonian cycle (see note 4.3): c = ( a,b 4,a,b−1,a,b,a,b−1,a,b−4,a,b−2, (a,b−4,a,b 2)2, a,b−4,a,b,a,b−1,a,b−4,a,b−2,a,b−4,a,b 2 ) . since wtc (b) = −19 6≡ 0 (mod p), theorem 2.11 applies. • if a = (1, 4)(2, 3), then a normalizes b, so 〈a,b〉 6= a5, which contradicts the fact that s is a generating set. subcase 1.3. assume |a| = |b| = 3. the union of the supports of a and b must be all of {1, 2, 3, 4, 5}, since 〈a,b〉 is transitive. since each support consists of three elements, the intersection must be a single element, which we may assume is 3. then, by renumbering, we may assume the support of a is {1, 2, 3} and the support of b is {3, 4, 5}. therefore, either a or a−1 is (1, 2, 3), and either b or b−1 is (3, 4, 5). so we may assume a = (1, 2, 3) and b = (3, 4, 5). we have the following hamiltonian cycle (see note 4.4): c1 = ( b,a,b 2,a 2,b−2,a−2,b 2,a 2,b 2,a−2,b−2,a−2,b 2,a 2,b−2,a,b−2, a−2,b 2,a,b,a−1,b−2,a 2,b 2,a 2,b−2,a−2,b−1,a,b,a 2,b−1,a−2,b−1,a ) . note that wtc1 (a) = 4 and wtc1 (b) = 0. conjugation by the permutation (1, 4)(2, 5) interchanges a and b, and therefore yields a hamiltonian cycle c2 with wtc2 (a) = 0 and wtc2 (b) = 4. since det [ 4 0 0 4 ] = 16 6≡ 0 (mod p), theorem 2.11 applies. subcase 1.4. assume |a| = 3 and |b| = 5. we may assume b = (1, 2, 3, 4, 5), by replacing it with a conjugate. then the two fixed points of the 3-cycle a are either consecutive or are separated by only one element (in circular order). therefore, after conjugating by a power of b, we may assume that one of the fixed points of a is 5, and the other is either 3 or 4. hence, a is either (1, 2, 4) or (1, 2, 3) (or the inverse of one of these). • if a = (1, 2, 4), then we have the following two hamiltonian cycles (see notes 4.5 and 4.6): c1 = ( a 2,b−2,a−2,b−1,a−1,b,a 2,b−1,a−2,b,a−2,b−2,a−2,b−1,a−1,b,a 2,b−1,a 2,b, a−2,b−1,a 2,b,a 2,b−1,a−2,b,a−2,b−2,a−2,b−1,a−1,b,a 2,b−1, (a−2,b)2 ) and c2 = ( a 2,b−2,a−2,b−1,a−1,b, (a 2,b−1)2,a 2,b−2,a−2,b−1,a−1,b,a 2,b−1,a 2,b, a−2,b−1,a 2,b,a 2,b−1,a−2,b,a−2,b−2,a−2,b−1,a−1,b,a 2,b−1, (a−2,b)2 ) then [ wtc1 (a), wtc1 (b) ] = [−9,−5] and [ wtc2 (a), wtc2 (b) ] = [−1,−7]. since det [−9 −5 −1 −7 ] = 58 6≡ 0 (mod p), theorem 2.11 applies. 17 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 • if a = (1, 2, 3), then we have the following two hamiltonian cycles (see notes 4.7 and 4.8): c1 = ( a 2,b−2,a−2,b−1,a−1,b 3, (a 2,b)2,a,b−1, (a 2,b)2,a 2,b−2,a 2,b,a,b−1, a−2,b,a 2,b−1,a−2,b,a−1,b−2,a−1,b−1,a−2,b,a 2, (b−1,a−2)2,b ) and c2 = ( a 2,b−1, (a 2,b)2,a 2,b−1,a−2,b,a−2,b−1,a 2,b,a 2,b−1,a−2,b 2, (a−2,b−1)2, (a−2,b)2,a 2,b,a 2,b−1,a,b,a−2,b,a 2, (b−1,a−2)2,b ) then [ wtc1 (a), wtc1 (b) ] = [5,−1] and [ wtc2 (a), wtc2 (b) ] = [−1, 3]. since det [ 5 −1 −1 3 ] = 14 6≡ 0 (mod p), theorem 2.11 applies. subcase 1.5. assume |a| = |b| = 5. by applying an automorphism of a5, we may assume a = (1, 2, 3, 4, 5). from sylow’s theorems, we know that a5 has precisely six sylow 5-subgroups. one of them is 〈a〉, and 〈a〉 acts transitively on the other 5 by conjugation. so we may assume, after conjugating by a power of a, that 〈b〉 = 〈(1, 2, 3, 5, 4)〉. then, by replacing b with its inverse if necessary, we may assume b is either (1, 2, 3, 5, 4) or (1, 3, 4, 2, 5). • if b = (1, 2, 3, 5, 4), then we have the following hamiltonian cycle (see note 4.9): c1 = ( b,a−1,b,a,b 4,a,b 2,a−1,b 2,a−1,b−1,a−1,b,a−1,b,a,b 2,a−1,b 2,a−1,b−1, a,b−2,a−1,b−4,a−1,b−1,a,b−4,a,b−2,a−1,b−1,a,b−2,a−1,b 4,a,b−2,a ) , then [ wtc1 (a), wtc1 (b) ] = [−2, 0]. conjugating by the permutation (4, 5) interchanges a and b, and therefore yields a hamiltonian cycle c2 with [ wtc2 (a), wtc2 (b) ] = [0,−2]. since det [−2 0 0 −2 ] = 4 6≡ 0 (mod p), theorem 2.11 applies. • if b = (1, 3, 4, 2, 5), then we have the following two hamiltonian cycles (see notes 4.10 and 4.11): c1 = ( a 4,b−1,a−4,b,a 4,b−1,a 2,b,a−2,b,a−1,b,a−4,b−1, a−1,b−1,a 4,b, (a−4,b−1)2, (a 2,b)2,a 4, (b−1,a)2,b ) and c2 = ( a 4,b−1,a−4,b,a 4,b−1,a 2,b,a−1,b−1,a−4,b−1,a,b−1, a−2,b−1,a 4,b, (a−4,b−1)2, (a 2,b)2,a 4, (b−1,a)2,b ) then [ wtc1 (a), wtc1 (b) ] = [4, 0] and [ wtc2 (a), wtc2 (b) ] = [6,−4]. since det [ 4 0 6 −4 ] = −16 6≡ 0 (mod p), theorem 2.11 applies. case 2. assume s has at least three elements.. we claim that s has exactly three elements, so we may write s = {a,b,c} with |a| ≤ |b| ≤ |c|. to show this, write s = {s1, . . . ,sr}, and suppose r ≥ 4. let hi = 〈s1, . . . ,si〉 for each i, and note that, since s is a minimal generating set, we have hi−1 ( hi. since |a5| = 22 · 3 · 5 is the product of only 4 primes, we must have r = 4 and |hi : hi−1| is prime for i = 1, . . . , 4. since a4 is the only subgroup of prime index in a5 (up to conjugacy), we may assume h3 = a4. then h2 must be the sylow 2-subgroup of a4, since that is the only subgroup of prime index. so h3 = a4 is generated by s1 and s3, contradicting the minimality of s. subcase 2.1. assume |c| = 5. since a and b cannot both normalize 〈c〉 (but every proper subgroup of a5 whose order is divisible by 5 has order 5 or 10), we see that either 〈a,c〉 = a5 or 〈b,c〉 = a5, which contradicts the minimality of s. subcase 2.2. assume |a| = |b| = |c| = 3. the minimality of s implies that 〈a,b〉 6= a5, so there must be at least two elements in the intersection of the supports of a and b. the supports cannot 18 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 be equal, since a 6= b±1. therefore, the intersection of the supports consists of two points, which we may assume are 1 and 5. then we may assume a = (1, 2, 5) and b = (1, 3, 5). now, for the same reason, the support of c must contain exactly two points from the support of a and exactly two points from the support of b, and it must also contain 4 (since 4 is fixed by both a and b, but not by a5). this implies that the support of c is {1, 4, 5}, so c = (1, 4, 5)±1. therefore (perhaps after replacing c by its inverse), we have s = {(1, 2, 5), (1, 3, 5), (1, 4, 5)} = {(1,j,n) | 1 < j < n} for n = 5. so [1, app. d] provides the following hamiltonian cycle in cay(a5; s) (see note 4.12): r1 = (( (a 2,b)2,a 2,c )2 ,a,b, (b,a 2)2,c 2,a 2,c,b,a,b,c,a, (c,b 2)2, (a 2,b)2,a,c 2,a,b 2, (a,c 2)2 ) . we have [wtr1 (a), wtr1 (b), wtr1 (c)] = [29, 17, 14]. conjugating by (2, 3, 4) and (2, 3, 4) 2 cyclically permutes {a,b,c}, and therefore yields hamiltonian cycles r2 and r3, such that [wtr2 (a), wtr2 (b), wtr2 (c)] = [17, 14, 29] and [wtr3 (a), wtr3 (b), wtr3 (c)] = [14, 29, 17]. since det  29 17 1417 14 29 14 29 17   = 11, 340 = 22 · 34 · 5 · 7 6≡ 0 (mod p), theorem 2.11 applies. subcase 2.3. assume |a| = 2 and |b| = |c| = 3. we claim that we may assume s = {(12)(45), (1, 2, 3), (1, 2, 4)}. arguing as in the first paragraph of subcase 2.2 (and renumbering), we may assume b = (1, 2, 3) and c = (1, 2, 4). since 〈a,b,c〉 = a5 is transitive on {1, 2, 3, 4, 5}, we know that 5 is in the support of a. also, since 〈a,b〉 6= a5, we know that the support of b does not contain precisely one element of each of the cycles of a (and similarly for the support of c). if one of the cycles of a is disjoint from the support of b, then the cycle must be (4, 5). the support of c contains precisely one element of this cycle, and cannot be disjoint from the other cycle, so it must contain the entire cycle. the only 2-element subset of {1, 2, 3} contained in the support of c is {1, 2}, so a = (1, 2)(4, 5), as desired. we may now assume that no cycle of a is disjoint from the support of b or the support of c. (this will lead to a contradiction.) this assumption implies that the cycle (x, 5) in a must be either (1, 5) or (2, 5). we may assume it is (1, 5) (after conjugating by (1, 2) and replacing b and c by their inverses, if necessary). the other cycle either is (2, 3) (in which case, 〈a,c〉 = a5), or is either (2, 4) or (3, 4) (in which case, 〈a,b〉 = a5), which contradicts the minimality of s. this completes the proof of the claim. we have the following two hamiltonian cycles (see notes 4.13 and 4.14): c1 = ( a,c−1,a,b,a,c,a,b 2,a,b,c,b−1,a,b−2,a,c−1,a,c−1,b 2,c,b−2,a,b−2,c,a,b,c−1, b−1,a,c−1,a,b−2,a,b−1,c,a,b 2,a,b,c,b−1,a,c−1,b 2,c,a,b,c−1,a,b−1,a,b ) and c2 = ( a,c−1,a,b,a,c,a,b 2,a,b,c,b−1,a,c−1,b 2,c,a,b,c−1,b−1,a,b−2,a,c−1,b,a,b 2, a,c,a,b,c−1,b−1,a,c−1,a,c−1,b 2,c,b−2,a,b−2,c,a,b 2,a,b,c−1,a,b−1,a,b ) then [ wtc1 (b), wtc1 (c) ] = [1, 0] and [ wtc2 (b), wtc2 (c) ] = [7,−2]. since det [ 1 0 7 −2 ] = −2 6≡ 0 (mod p), theorem 2.11 applies. subcase 2.4. assume |a| = |b| = 2 and |c| = 3. we may assume c = (1, 2, 3). 19 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 • suppose a interchanges 4 and 5. this means that one of the 2-cycles in a is (4, 5). the other 2-cycle must be contained in {1, 2, 3}, so, after conjugating by a power of c, we may assume a = (1, 2)(4, 5). since 〈a,b,c〉 is transitive on {1, 2, 3, 4, 5}, the permutation b cannot have (4, 5) as one of its 2-cycles. so no 2-cycle in b is disjoint from the support of c. since 〈b,c〉 6= a5 (by the minimality of s), this implies that one of the 2-cycles must be contained in the support of c, which is {1, 2, 3}. so b fixes either 4 or 5. we may assume it is 5 that is fixed (after conjugating by (4, 5) if necessary). then b is either (1, 2)(3, 4) or (1, 3)(2, 4) or (2, 3)(1, 4). since the last two are conjugate by (1, 2) (which centralizes a and inverts b), they do not both need to be considered. – if b = (1, 2)(3, 4), we have the following hamiltonian cycle (see note 4.15): c = ( a,c−1, (a,b)2,a,c−1,a,b,c−1,b,c, (b,a)2,c, (a,b)2,a,c, (a,b)2, a,c 2,b,c−1,b, (a,b)2,c−2, ( (a,b)2,a,c−1 )2 ,c−1,b,a,c−1, (a,b)2 ) . since wtc (c) = −5 6≡ 0 (mod p), theorem 2.11 applies. – if b = (1, 3)(2, 4), we have the following hamiltonian cycle (see note 4.16): c = ( (a,b)4,c, (a,b)4,a,c−1,b,a,b,c−1, (b,a)3,c−1, b, (a,b)2,c,b,a,c−1, (b,a)4,b,c, (a,b)4,a,c−1,b ) . since wtc (c) = −2 6≡ 0 (mod p), theorem 2.11 applies. • we may now assume that neither a nor b interchanges 4 and 5. since 〈a,c〉 6= a5, and a does not interchange 4 and 5, we see that a must fix either 4 or 5. similarly for b. furthermore, a and b cannot both fix 4 (or 5), since 〈a,b,c〉 is transitive. so one of them fixes 4, and the other fixes 5. we may assume it is a that fixes 5. then we may assume a = (1, 2)(3, 4), after conjugating by a power of c. then b must be either (1, 2)(3, 5) or (1, 3)(2, 5) or (1, 5)(2, 3). however, the last two are conjugate by (1, 2) (which centralizes a and inverts c), so they do not both need to be considered. – if b = (1, 2)(3, 5), then we have the hamiltonian cycle (see note 4.17): c = ( a,c−1,a,c,b, (a,b)2,c, (a,b)2, (a,c)2,a,b,a,c−2,a,c−1,b,a, c,b,a,b,c 2, ( (a,b)2,a,c )2 , (a,c−1)2, (a,b)2,a,c−2, (a,b)2 ) . since wtc (c) = 1 6≡ 0 (mod p), theorem 2.11 applies. – if b = (1, 3)(2, 5), then we have the following hamiltonian cycle (see note 4.18): c = ( a,b,c−1, (a,b)2,a,c,b, (a,b)4,c, (a,b,a,c−1)2,b,a,b,c 2, (a,b)2,a,c, ( (b,a)2,c−1 )2 , (b,a)2,b,c−1, (b,a)2,c−1,b ) . since wtc (c) = −2 6≡ 0 (mod p), theorem 2.11 applies. subcase 2.5. assume |a| = |b| = |c| = 2. since all generators are of order 2, theorem 2.9 applies. 20 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4. details of hamiltonian cycles in a5 to aid the reader in validating the many hamiltonian cycles in a5 that appeared in the proof of theorem 1.2, this final section provides a list of the vertices in the order that they are visited by each cycle. 4.1. a hamiltonian cycle in cay(a5; a,b) with a = (1, 2)(3, 4) and b = (2, 4, 5). e a−→ (1, 2)(3, 4) b−→ (1, 2, 3, 4, 5) b−→ (1, 2, 5, 3, 4) a−→ (1, 5, 3) b−→ (1, 5, 2, 4, 3) b−→ (1, 5, 4, 2, 3) a−→ (1, 3, 2, 5, 4) b−→ (1, 3, 2) b−→ (1, 3, 2, 4, 5) a−→ (1, 4, 2, 3, 5) b −1 −→ (1, 4, 3, 5, 2) b −1 −→ (1, 4)(3, 5) a−→ (1, 2, 4, 5, 3) b −1 −→ (1, 2, 3) b−1−→ (1, 2, 5, 4, 3) a−→ (1, 5, 4) b −1 −→ (1, 5)(2, 4) b −1 −→ (1, 5, 2) a−→ (2, 5)(3, 4) b−→ (2, 3, 4) b−→ (3, 4, 5) a−→ (1, 2)(3, 5) b −1 −→ (1, 2, 3, 5, 4) b −1 −→ (1, 2, 4, 3, 5) a−→ (1, 4, 5) b−→ (1, 4)(2, 5) b−→ (1, 4, 2) a−→ (2, 4, 3) b −1 −→ (2, 5, 3) b−1−→ (2, 3)(4, 5) a−→ (1, 3, 5, 4, 2) b−→ (1, 3, 5) b−→ (1, 3, 5, 2, 4) a−→ (1, 4, 5, 2, 3) b−→ (1, 4, 2, 5, 3) b−→ (1, 4, 3) a−→ (1, 2, 4) b−→ (1, 2)(4, 5) b−→ (1, 2, 5) a−→ (1, 5)(3, 4) b −1 −→ (1, 5, 3, 4, 2) b −1 −→ (1, 5, 2, 3, 4) a−→ (1, 3)(2, 5) b −1 −→ (1, 3)(4, 5) b−1−→ (1, 3)(2, 4) a−→ (1, 4)(2, 3) b −1 −→ (1, 4, 3, 2, 5) b −1 −→ (1, 4, 5, 3, 2) a−→ (2, 4)(3, 5) b−→ (3, 5, 4) b−→ (2, 3, 5) a−→ (1, 3, 4, 5, 2) b −1 −→ (1, 3, 4) b −1 −→ (1, 3, 4, 2, 5) a−→ (1, 5)(2, 3) b−→ (1, 5, 3, 2, 4) b−→ (1, 5, 4, 3, 2) a−→ (2, 5, 4) b −1 −→ (2, 4, 5) b−1−→ e 4.2. a hamiltonian cycle in cay(a5; a,b) with a = (1, 2)(3, 4) and b = (1, 2, 3, 4, 5). e a−→ (1, 2)(3, 4) b−→ (2, 4, 5) a−→ (1, 4, 3, 5, 2) b−→ (2, 5, 4) b−→ (1, 5)(2, 3) b−→ (1, 3, 4) b−→ (1, 2, 4, 5, 3) a−→ (1, 4)(3, 5) b−→ (1, 2, 5, 4, 3) a−→ (1, 5, 4) b−→ (1, 2, 3) b−→ (1, 3, 4, 5, 2) b−→ (2, 4)(3, 5) b−→ (1, 4, 3, 2, 5) a−→ (1, 5)(2, 4) b−→ (1, 4)(2, 3) b−→ (1, 3)(4, 5) a−→ (1, 2, 3, 5, 4) b −1 −→ (1, 4, 5) a−→ (1, 2, 4, 3, 5) b−→ (1, 4)(2, 5) b−→ (1, 5, 4, 2, 3) b−→ (1, 3, 2) b−→ (3, 4, 5) a−→ (1, 2)(3, 5) b−→ (2, 5)(3, 4) a−→ (1, 5, 2) b−→ (2, 3, 4) b−→ (1, 3, 2, 4, 5) a−→ (1, 4, 2, 3, 5) b−→ (1, 3, 2, 5, 4) b−→ (1, 5, 3) a−→ (1, 2, 5, 3, 4) b−→ (1, 5, 2, 4, 3) b−→ (1, 4, 2) a−→ (2, 4, 3) b −1 −→ (1, 5, 3, 4, 2) b −1 −→ (1, 3)(2, 5) a−→ (1, 5, 2, 3, 4) b−→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) b−→ (3, 5, 4) b−→ (1, 2, 5) a−→ (1, 5)(3, 4) b−1−→ (2, 5, 3) b −1 −→ (1, 3, 5, 4, 2) a−→ (2, 3)(4, 5) b−→ (1, 3, 5) a−→ (1, 2, 3, 4, 5) b−→ (1, 3, 5, 2, 4) a−→ (1, 4, 5, 2, 3) b −1 −→ (1, 2, 4) a−→ (1, 4, 3) b−→ (1, 2)(4, 5) b−→ (2, 3, 5) b−→ (1, 3, 4, 2, 5) b−→ (1, 5, 3, 2, 4) a−→ (1, 4, 2, 5, 3) b−→ (1, 5, 4, 3, 2) b−→ e 21 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.3. a hamiltonian cycle in cay(a5; a,b) with a = (1, 3)(2, 4) and b = (1, 2, 3, 4, 5). e a−→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) b−→ (3, 5, 4) b−→ (1, 2, 5) b−→ (1, 5, 2, 3, 4) a−→ (1, 4, 3, 5, 2) b −1 −→ (1, 2, 4, 5, 3) a−→ (2, 5, 3) b−→ (1, 5)(3, 4) a−→ (1, 4, 2, 3, 5) b −1 −→ (2, 4, 5) a−→ (1, 3)(2, 5) b −1 −→ (1, 2, 3, 5, 4) b −1 −→ (1, 4, 5) b−1−→ (2, 4, 3) b −1 −→ (1, 5, 3, 4, 2) a−→ (1, 4)(3, 5) b −1 −→ (1, 3, 2, 4, 5) b −1 −→ (2, 3, 4) a−→ (1, 4, 3) b −1 −→ (1, 5, 3, 2, 4) b −1 −→ (1, 3, 4, 2, 5) b −1 −→ (2, 3, 5) b −1 −→ (1, 2)(4, 5) a−→ (1, 3, 2, 5, 4) b−→ (1, 5, 3) b−→ (1, 2)(3, 4) a−→ (1, 4)(2, 3) b −1 −→ (1, 5)(2, 4) b−1−→ (2, 5)(3, 4) b −1 −→ (1, 2)(3, 5) b −1 −→ (1, 3)(4, 5) a−→ (2, 5, 4) b−→ (1, 5)(2, 3) b−→ (1, 3, 4) a−→ (1, 4, 2) b −1 −→ (1, 5, 2, 4, 3) b −1 −→ (1, 2, 5, 3, 4) b −1 −→ (1, 3, 5) b−1−→ (2, 3)(4, 5) a−→ (1, 2, 5, 4, 3) b−→ (1, 5, 2) a−→ (1, 3, 5, 2, 4) b −1 −→ (1, 2, 3, 4, 5) a−→ (1, 4, 3, 2, 5) b −1 −→ (2, 4)(3, 5) b −1 −→ (1, 3, 4, 5, 2) b −1 −→ (1, 2, 3) b −1 −→ (1, 5, 4) a−→ (1, 3, 5, 4, 2) b −1 −→ (1, 4, 5, 2, 3) b −1 −→ (1, 2, 4) a−→ (1, 3, 2) b −1 −→ (1, 5, 4, 2, 3) b−1−→ (1, 4)(2, 5) b −1 −→ (1, 2, 4, 3, 5) b −1 −→ (3, 4, 5) a−→ (1, 4, 2, 5, 3) b−→ (1, 5, 4, 3, 2) b−→ e 4.4. a hamiltonian cycle c1 in cay(a5; a,b) with a = (1, 2, 3) and b = (3, 4, 5). e b−→ (3, 4, 5) a−→ (1, 2, 4, 5, 3) b−→ (1, 2, 4, 3, 5) b−→ (1, 2, 4) a−→ (1, 4)(2, 3) a−→ (1, 3, 4) b −1 −→ (1, 3, 5) b −1 −→ (1, 3)(4, 5) a −1 −→ (2, 3)(4, 5) a−1−→ (1, 2)(4, 5) b−→ (1, 2)(3, 5) b−→ (1, 2)(3, 4) a−→ (2, 4, 3) a−→ (1, 4, 3) b−→ (1, 4, 5) b−→ (1, 4)(3, 5) a −1 −→ (1, 5, 3, 2, 4) a −1 −→ (1, 2, 5, 3, 4) b −1 −→ (1, 2, 5) b−1−→ (1, 2, 5, 4, 3) a −1 −→ (3, 5, 4) a −1 −→ (1, 5, 4, 3, 2) b−→ (1, 5, 2) b−→ (1, 5, 3, 4, 2) a−→ (2, 4)(3, 5) a−→ (1, 4, 2, 5, 3) b −1 −→ (1, 4)(2, 5) b −1 −→ (1, 4, 3, 2, 5) a−→ (1, 5)(3, 4) b−1−→ (1, 5, 3) b −1 −→ (1, 5, 4) a −1 −→ (1, 3, 2, 5, 4) a −1 −→ (1, 2, 3, 5, 4) b−→ (1, 2, 3) b−→ (1, 2, 3, 4, 5) a−→ (1, 3, 2, 4, 5) b−→ (1, 3, 5, 2, 4) a −1 −→ (1, 5, 2, 3, 4) b −1 −→ (1, 5)(2, 3) b−1−→ (1, 5, 4, 2, 3) a−→ (1, 3, 5, 4, 2) a−→ (2, 5, 4) b−→ (2, 5, 3) b−→ (2, 5)(3, 4) a−→ (1, 5, 2, 4, 3) a−→ (1, 4, 3, 5, 2) b −1 −→ (1, 4, 5, 3, 2) b −1 −→ (1, 4, 2) a −1 −→ (1, 3)(2, 4) a−1−→ (2, 3, 4) b −1 −→ (2, 3, 5) a−→ (1, 3)(2, 5) b−→ (1, 3, 4, 2, 5) a−→ (1, 5)(2, 4) a−→ (1, 4, 2, 3, 5) b −1 −→ (1, 4, 5, 2, 3) a −1 −→ (2, 4, 5) a −1 −→ (1, 3, 4, 5, 2) b −1 −→ (1, 3, 2) a−→ e 22 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.5. a hamiltonian cycle c1 in cay(a5; a,b) with a = (1, 2, 4) and b = (1, 2, 3, 4, 5). e a−→ (1, 2, 4) a−→ (1, 4, 2) b −1 −→ (1, 5, 2, 4, 3) b −1 −→ (1, 2, 5, 3, 4) a−1−→ (3, 4, 5) a −1 −→ (1, 5, 3, 4, 2) b −1 −→ (1, 3)(2, 5) a −1 −→ (1, 4, 5, 2, 3) b−→ (1, 3, 5, 4, 2) a−→ (3, 5, 4) a−→ (1, 2, 3, 5, 4) b −1 −→ (1, 4, 5) a −1 −→ (1, 5)(2, 4) a −1 −→ (1, 2, 5) b−→ (1, 5, 2, 3, 4) a −1 −→ (2, 5)(3, 4) a −1 −→ (1, 3, 4, 5, 2) b −1 −→ (1, 2, 3) b −1 −→ (1, 5, 4) a−1−→ (2, 5, 4) a −1 −→ (1, 2)(4, 5) b −1 −→ (1, 4, 3) a −1 −→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) a−→ (2, 5, 3) a−→ (1, 5, 3, 2, 4) b −1 −→ (1, 3, 4, 2, 5) a−→ (1, 5)(3, 4) a−→ (1, 2, 3, 4, 5) b−→ (1, 3, 5, 2, 4) a −1 −→ (2, 3, 5) a −1 −→ (1, 4, 3, 5, 2) b −1 −→ (1, 2, 4, 5, 3) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 3) b−→ (1, 2)(3, 4) a−→ (2, 3, 4) a−→ (1, 3, 4) b −1 −→ (1, 5)(2, 3) a−1−→ (1, 4, 3, 2, 5) a −1 −→ (1, 3, 2, 4, 5) b−→ (1, 4)(3, 5) a −1 −→ (2, 4)(3, 5) a −1 −→ (1, 2)(3, 5) b−1−→ (1, 3)(4, 5) b −1 −→ (1, 4)(2, 3) a −1 −→ (2, 4, 3) a −1 −→ (1, 3, 2) b −1 −→ (1, 5, 4, 2, 3) a−1−→ (1, 2, 5, 4, 3) b−→ (1, 5, 2) a−→ (2, 4, 5) a−→ (1, 4)(2, 5) b −1 −→ (1, 2, 4, 3, 5) a−1−→ (1, 3, 5) a −1 −→ (1, 4, 2, 3, 5) b−→ (1, 3, 2, 5, 4) a −1 −→ (2, 3)(4, 5) a −1 −→ (1, 5, 4, 3, 2) b−→ e 4.6. a second hamiltonian cycle c2 in cay(a5; a,b) with a = (1, 2, 4) and b = (1, 2, 3, 4, 5). e a−→ (1, 2, 4) a−→ (1, 4, 2) b −1 −→ (1, 5, 2, 4, 3) b −1 −→ (1, 2, 5, 3, 4) a−1−→ (3, 4, 5) a −1 −→ (1, 5, 3, 4, 2) b −1 −→ (1, 3)(2, 5) a −1 −→ (1, 4, 5, 2, 3) b−→ (1, 3, 5, 4, 2) a−→ (3, 5, 4) a−→ (1, 2, 3, 5, 4) b −1 −→ (1, 4, 5) a−→ (1, 2, 5) a−→ (1, 5)(2, 4) b−1−→ (2, 5)(3, 4) a−→ (1, 5, 2, 3, 4) a−→ (1, 3, 4, 5, 2) b −1 −→ (1, 2, 3) b −1 −→ (1, 5, 4) a−1−→ (2, 5, 4) a −1 −→ (1, 2)(4, 5) b −1 −→ (1, 4, 3) a −1 −→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) a−→ (2, 5, 3) a−→ (1, 5, 3, 2, 4) b −1 −→ (1, 3, 4, 2, 5) a−→ (1, 5)(3, 4) a−→ (1, 2, 3, 4, 5) b−→ (1, 3, 5, 2, 4) a −1 −→ (2, 3, 5) a −1 −→ (1, 4, 3, 5, 2) b −1 −→ (1, 2, 4, 5, 3) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 3) b−→ (1, 2)(3, 4) a−→ (2, 3, 4) a−→ (1, 3, 4) b −1 −→ (1, 5)(2, 3) a−1−→ (1, 4, 3, 2, 5) a −1 −→ (1, 3, 2, 4, 5) b−→ (1, 4)(3, 5) a −1 −→ (2, 4)(3, 5) a −1 −→ (1, 2)(3, 5) b−1−→ (1, 3)(4, 5) b −1 −→ (1, 4)(2, 3) a −1 −→ (2, 4, 3) a −1 −→ (1, 3, 2) b −1 −→ (1, 5, 4, 2, 3) a−1−→ (1, 2, 5, 4, 3) b−→ (1, 5, 2) a−→ (2, 4, 5) a−→ (1, 4)(2, 5) b −1 −→ (1, 2, 4, 3, 5) a−1−→ (1, 3, 5) a −1 −→ (1, 4, 2, 3, 5) b−→ (1, 3, 2, 5, 4) a −1 −→ (2, 3)(4, 5) a −1 −→ (1, 5, 4, 3, 2) b−→ e 23 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.7. a hamiltonian cycle c1 in cay(a5; a,b) with a = (1, 2, 3) and b = (1, 2, 3, 4, 5). e a−→ (1, 2, 3) a−→ (1, 3, 2) b −1 −→ (1, 5, 4, 2, 3) b −1 −→ (1, 4)(2, 5) a−1−→ (1, 3, 5, 2, 4) a −1 −→ (1, 5, 2, 3, 4) b −1 −→ (1, 2, 5) a −1 −→ (1, 3, 5) b−→ (1, 2, 5, 3, 4) b−→ (1, 5, 2, 4, 3) b−→ (1, 4, 2) a−→ (2, 3, 4) a−→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) a−→ (3, 4, 5) a−→ (1, 2, 4, 5, 3) b−→ (1, 4, 3, 5, 2) a−→ (2, 5)(3, 4) b −1 −→ (1, 2)(3, 5) a−→ (2, 5, 3) a−→ (1, 5, 3) b−→ (1, 2)(3, 4) a−→ (2, 4, 3) a−→ (1, 4, 3) b−→ (1, 2)(4, 5) a−→ (2, 3)(4, 5) a−→ (1, 3)(4, 5) b −1 −→ (1, 4)(2, 3) b −1 −→ (1, 5)(2, 4) a−→ (1, 4, 2, 3, 5) a−→ (1, 3, 4, 2, 5) b−→ (1, 5, 3, 2, 4) a−→ (1, 4)(3, 5) b −1 −→ (1, 3, 2, 4, 5) a−1−→ (1, 2, 3, 4, 5) a −1 −→ (1, 4, 5) b−→ (1, 2, 3, 5, 4) a−→ (1, 3, 2, 5, 4) a−→ (1, 5, 4) b−1−→ (1, 4, 3, 2, 5) a −1 −→ (1, 2, 4, 3, 5) a −1 −→ (1, 5)(3, 4) b−→ (1, 2, 4) a −1 −→ (1, 3, 4) b−1−→ (1, 5)(2, 3) b −1 −→ (2, 5, 4) a −1 −→ (1, 3, 5, 4, 2) b −1 −→ (1, 4, 5, 2, 3) a −1 −→ (2, 4, 5) a−1−→ (1, 3, 4, 5, 2) b−→ (2, 4)(3, 5) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 3, 4, 2) b −1 −→ (1, 3)(2, 5) a−1−→ (2, 3, 5) a −1 −→ (1, 5, 2) b −1 −→ (1, 2, 5, 4, 3) a −1 −→ (3, 5, 4) a −1 −→ (1, 5, 4, 3, 2) b−→ e 4.8. a second hamiltonian cycle c2 in cay(a5; a,b) with a = (1, 2, 3) and b = (1, 2, 3, 4, 5). e a−→ (1, 2, 3) a−→ (1, 3, 2) b −1 −→ (1, 5, 4, 2, 3) a−→ (1, 3, 5, 4, 2) a−→ (2, 5, 4) b−→ (1, 5)(2, 3) a−→ (1, 3, 5) a−→ (1, 2, 5) b−→ (1, 5, 2, 3, 4) a−→ (1, 3, 5, 2, 4) a−→ (1, 4)(2, 5) b −1 −→ (1, 2, 4, 3, 5) a −1 −→ (1, 5)(3, 4) a −1 −→ (1, 4, 3, 2, 5) b−→ (1, 5, 4) a −1 −→ (1, 3, 2, 5, 4) a −1 −→ (1, 2, 3, 5, 4) b −1 −→ (1, 4, 5) a−→ (1, 2, 3, 4, 5) a−→ (1, 3, 2, 4, 5) b−→ (1, 4)(3, 5) a−→ (1, 2, 5, 3, 4) a−→ (1, 5, 3, 2, 4) b −1 −→ (1, 3, 4, 2, 5) a−1−→ (1, 4, 2, 3, 5) a −1 −→ (1, 5)(2, 4) b−→ (1, 4)(2, 3) b−→ (1, 3)(4, 5) a −1 −→ (2, 3)(4, 5) a−1−→ (1, 2)(4, 5) b −1 −→ (1, 4, 3) a −1 −→ (2, 4, 3) a −1 −→ (1, 2)(3, 4) b −1 −→ (1, 5, 3) a−1−→ (2, 5, 3) a −1 −→ (1, 2)(3, 5) b−→ (2, 5)(3, 4) a −1 −→ (1, 4, 3, 5, 2) a −1 −→ (1, 5, 2, 4, 3) b−→ (1, 4, 2) a−→ (2, 3, 4) a−→ (1, 3)(2, 4) b−→ (1, 4, 5, 3, 2) a−→ (3, 4, 5) a−→ (1, 2, 4, 5, 3) b −1 −→ (1, 3, 4) a−→ (1, 2, 4) b−→ (1, 4, 5, 2, 3) a −1 −→ (2, 4, 5) a−1−→ (1, 3, 4, 5, 2) b−→ (2, 4)(3, 5) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 3, 4, 2) b −1 −→ (1, 3)(2, 5) a−1−→ (2, 3, 5) a −1 −→ (1, 5, 2) b −1 −→ (1, 2, 5, 4, 3) a −1 −→ (3, 5, 4) a −1 −→ (1, 5, 4, 3, 2) b−→ e 24 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.9. a hamiltonian cycle c1 in cay(a5; a,b) with a = (1, 2, 3, 4, 5) and b = (1, 2, 3, 5, 4). e b−→ (1, 2, 3, 5, 4) a −1 −→ (1, 4, 5) b−→ (1, 2, 3) a−→ (1, 3, 4, 5, 2) b−→ (2, 4, 3) b−→ (1, 4)(3, 5) b−→ (1, 2, 5) b−→ (1, 5, 4, 2, 3) a−→ (1, 3, 2) b−→ (3, 5, 4) b−→ (1, 2, 5, 3, 4) a −1 −→ (1, 3, 5) b−→ (1, 2, 5, 4, 3) b−→ (1, 5, 3, 4, 2) a−1−→ (1, 3)(2, 5) b −1 −→ (1, 4, 2, 3, 5) a −1 −→ (2, 4, 5) b−→ (1, 4)(2, 3) a −1 −→ (1, 5)(2, 4) b−→ (1, 4, 5, 2, 3) a−→ (1, 3, 5, 4, 2) b−→ (2, 5)(3, 4) b−→ (1, 5, 3, 2, 4) a −1 −→ (1, 3, 4, 2, 5) b−→ (1, 5, 2, 4, 3) b−→ (1, 4, 5, 3, 2) a −1 −→ (1, 3)(2, 4) b −1 −→ (1, 2, 3, 4, 5) a−→ (1, 3, 5, 2, 4) b−1−→ (2, 3, 4) b −1 −→ (1, 2)(4, 5) a −1 −→ (1, 4, 3) b −1 −→ (1, 3, 2, 4, 5) b −1 −→ (1, 5, 2, 3, 4) b−1−→ (2, 5, 4) b −1 −→ (1, 2)(3, 5) a −1 −→ (1, 3)(4, 5) b −1 −→ (1, 5)(2, 3) a−→ (1, 3, 4) b−1−→ (2, 3)(4, 5) b −1 −→ (1, 5, 2) b −1 −→ (1, 4, 2, 5, 3) b −1 −→ (1, 2, 4, 3, 5) a−→ (1, 4)(2, 5) b−1−→ (2, 4)(3, 5) b −1 −→ (1, 2)(3, 4) a −1 −→ (1, 5, 3) b −1 −→ (1, 4, 3, 2, 5) a−→ (1, 5, 4) b−1−→ (2, 5, 3) b −1 −→ (1, 4, 3, 5, 2) a −1 −→ (1, 2, 4, 5, 3) b−→ (1, 4, 2) b−→ (2, 3, 5) b−→ (1, 3, 2, 5, 4) b−→ (1, 5)(3, 4) a−→ (1, 2, 4) b −1 −→ (3, 4, 5) b −1 −→ (1, 5, 4, 3, 2) a−→ e 4.10. a hamiltonian cycle c1 in cay(a5; a,b) with a = (1, 2, 3, 4, 5) and b = (1, 3, 4, 2, 5). e a−→ (1, 2, 3, 4, 5) a−→ (1, 3, 5, 2, 4) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 4, 3, 2) b−1−→ (1, 4, 2, 3, 5) a −1 −→ (2, 4, 5) a −1 −→ (1, 2)(3, 4) a −1 −→ (1, 5, 3) a −1 −→ (1, 3, 2, 5, 4) b−→ (1, 2, 4, 5, 3) a−→ (1, 4, 3, 5, 2) a−→ (2, 5, 4) a−→ (1, 5)(2, 3) a−→ (1, 3, 4) b−1−→ (1, 5, 2) a−→ (2, 3, 4) a−→ (1, 3, 2, 4, 5) b−→ (1, 2)(3, 5) a −1 −→ (1, 3)(4, 5) a−1−→ (1, 4)(2, 3) b−→ (1, 2, 5, 4, 3) a −1 −→ (1, 4)(3, 5) b−→ (1, 5, 4, 2, 3) a −1 −→ (1, 4)(2, 5) a−1−→ (1, 2, 4, 3, 5) a −1 −→ (3, 4, 5) a −1 −→ (1, 3, 2) b −1 −→ (1, 5)(2, 4) a −1 −→ (2, 5)(3, 4) b−1−→ (1, 2, 3) a−→ (1, 3, 4, 5, 2) a−→ (2, 4)(3, 5) a−→ (1, 4, 3, 2, 5) a−→ (1, 5, 4) b−→ (1, 3)(2, 4) a −1 −→ (1, 5, 2, 3, 4) a −1 −→ (1, 2, 5) a −1 −→ (3, 5, 4) a −1 −→ (1, 4, 5, 3, 2) b−1−→ (1, 3, 4, 2, 5) a −1 −→ (2, 3, 5) a −1 −→ (1, 2)(4, 5) a −1 −→ (1, 4, 3) a −1 −→ (1, 5, 3, 2, 4) b−1−→ (1, 3, 5, 4, 2) a−→ (2, 5, 3) a−→ (1, 5)(3, 4) b−→ (1, 4, 2) a−→ (2, 3)(4, 5) a−→ (1, 3, 5) b−→ (1, 5, 3, 4, 2) a−→ (2, 4, 3) a−→ (1, 4, 5) a−→ (1, 2, 3, 5, 4) a−→ (1, 3)(2, 5) b −1 −→ (1, 2, 4) a−→ (1, 4, 5, 2, 3) b −1 −→ (1, 2, 5, 3, 4) a−→ (1, 5, 2, 4, 3) b−→ e 25 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.11. a second hamiltonian cycle c2 in cay(a5; a,b) with a = (1, 2, 3, 4, 5) and b = (1, 3, 4, 2, 5). e a−→ (1, 2, 3, 4, 5) a−→ (1, 3, 5, 2, 4) a−→ (1, 4, 2, 5, 3) a−→ (1, 5, 4, 3, 2) b−1−→ (1, 4, 2, 3, 5) a −1 −→ (2, 4, 5) a −1 −→ (1, 2)(3, 4) a −1 −→ (1, 5, 3) a −1 −→ (1, 3, 2, 5, 4) b−→ (1, 2, 4, 5, 3) a−→ (1, 4, 3, 5, 2) a−→ (2, 5, 4) a−→ (1, 5)(2, 3) a−→ (1, 3, 4) b−1−→ (1, 5, 2) a−→ (2, 3, 4) a−→ (1, 3, 2, 4, 5) b−→ (1, 2)(3, 5) a −1 −→ (1, 3)(4, 5) b−1−→ (1, 4)(2, 5) a −1 −→ (1, 2, 4, 3, 5) a −1 −→ (3, 4, 5) a −1 −→ (1, 3, 2) a −1 −→ (1, 5, 4, 2, 3) b−1−→ (1, 4)(3, 5) a−→ (1, 2, 5, 4, 3) b −1 −→ (1, 4)(2, 3) a −1 −→ (1, 5)(2, 4) a −1 −→ (2, 5)(3, 4) b−1−→ (1, 2, 3) a−→ (1, 3, 4, 5, 2) a−→ (2, 4)(3, 5) a−→ (1, 4, 3, 2, 5) a−→ (1, 5, 4) b−→ (1, 3)(2, 4) a −1 −→ (1, 5, 2, 3, 4) a −1 −→ (1, 2, 5) a −1 −→ (3, 5, 4) a −1 −→ (1, 4, 5, 3, 2) b−1−→ (1, 3, 4, 2, 5) a −1 −→ (2, 3, 5) a −1 −→ (1, 2)(4, 5) a −1 −→ (1, 4, 3) a −1 −→ (1, 5, 3, 2, 4) b−1−→ (1, 3, 5, 4, 2) a−→ (2, 5, 3) a−→ (1, 5)(3, 4) b−→ (1, 4, 2) a−→ (2, 3)(4, 5) a−→ (1, 3, 5) b−→ (1, 5, 3, 4, 2) a−→ (2, 4, 3) a−→ (1, 4, 5) a−→ (1, 2, 3, 5, 4) a−→ (1, 3)(2, 5) b −1 −→ (1, 2, 4) a−→ (1, 4, 5, 2, 3) b −1 −→ (1, 2, 5, 3, 4) a−→ (1, 5, 2, 4, 3) b−→ e 4.12. a hamiltonian cycle r1 in cay(a5; a,b,c) with a = (1, 2, 5), b = (1, 3, 5), and c = (1, 4, 5). e a−→ (1, 2, 5) a−→ (1, 5, 2) b−→ (1, 3, 2) a−→ (2, 5, 3) a−→ (1, 5)(2, 3) b−→ (1, 2, 3) a−→ (1, 3)(2, 5) a−→ (1, 5, 3) c−→ (1, 4, 3) a−→ (1, 2, 5, 4, 3) a−→ (1, 5, 2, 4, 3) b−→ (2, 4, 3) a−→ (1, 4, 3, 2, 5) a−→ (1, 5, 4, 3, 2) b−→ (1, 2)(3, 4) a−→ (2, 5)(3, 4) a−→ (1, 5)(3, 4) c−→ (1, 3, 4) a−→ (1, 2, 5, 3, 4) b−→ (1, 4)(2, 5) b−→ (1, 3, 2, 5, 4) a−→ (1, 5, 3, 2, 4) a−→ (1, 4)(2, 3) b−→ (1, 2, 3, 5, 4) a−→ (1, 3, 5, 2, 4) a−→ (1, 4)(3, 5) c−→ (3, 5, 4) c−→ (1, 3, 5) a−→ (1, 2)(3, 5) a−→ (2, 3, 5) c−→ (1, 4, 2, 3, 5) b−→ (1, 5, 4, 2, 3) a−→ (1, 3)(2, 4) b−→ (2, 4)(3, 5) c−→ (1, 2, 4, 3, 5) a−→ (1, 4, 3, 5, 2) c−→ (1, 3, 5, 4, 2) b−→ (1, 5, 3, 4, 2) b−→ (1, 4, 2) c−→ (1, 2)(4, 5) b−→ (1, 3, 4, 5, 2) b−→ (1, 4, 5, 3, 2) a−→ (2, 3)(4, 5) a−→ (1, 3, 2, 4, 5) b−→ (1, 2, 4, 5, 3) a−→ (1, 4, 5, 2, 3) a−→ (1, 3)(4, 5) b−→ (3, 4, 5) a−→ (1, 2, 3, 4, 5) c−→ (1, 5, 2, 3, 4) c−→ (2, 3, 4) a−→ (1, 3, 4, 2, 5) b−→ (1, 4, 2, 5, 3) b−→ (2, 5, 4) a−→ (1, 5)(2, 4) c−→ (1, 2, 4) c−→ (2, 4, 5) a−→ (1, 4, 5) c−→ (1, 5, 4) c−→ e 26 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.13. a hamiltonian cycle c1 in cay(a5; a,b,c,c) with a = (1, 2)(4, 5), b = (1, 2, 3), and c = (1, 2, 4). e a−→ (1, 2)(4, 5) c −1 −→ (1, 5, 4) a−→ (1, 2, 5) b−→ (1, 5)(2, 3) a−→ (1, 3, 2, 5, 4) c−→ (1, 5, 4, 3, 2) a−→ (2, 5, 3) b−→ (1, 5, 3) b−→ (1, 2)(3, 5) a−→ (3, 5, 4) b−→ (1, 2, 5, 4, 3) c−→ (1, 5, 4, 2, 3) b −1 −→ (2, 5, 4) a−→ (1, 5, 2) b−1−→ (1, 3)(2, 5) b −1 −→ (2, 3, 5) a−→ (1, 3, 5, 4, 2) c −1 −→ (1, 2, 3, 5, 4) a−→ (1, 3, 5) c−1−→ (1, 4, 2, 3, 5) b−→ (1, 3, 4, 2, 5) b−→ (1, 5)(2, 4) c−→ (1, 4, 5) b −1 −→ (1, 3, 2, 4, 5) b−1−→ (1, 2, 3, 4, 5) a−→ (1, 3, 4) b −1 −→ (1, 4)(2, 3) b −1 −→ (1, 2, 4) c−→ (1, 4, 2) a−→ (2, 4, 5) b−→ (1, 4, 5, 2, 3) c −1 −→ (1, 5, 2, 4, 3) b −1 −→ (2, 5)(3, 4) a−→ (1, 5, 3, 4, 2) c−1−→ (1, 2, 5, 3, 4) a−→ (1, 5)(3, 4) b −1 −→ (1, 4, 3, 2, 5) b −1 −→ (1, 2, 4, 3, 5) a−→ (1, 4)(3, 5) b−1−→ (1, 5, 3, 2, 4) c−→ (1, 4, 5, 3, 2) a−→ (2, 4, 3) b−→ (1, 4, 3) b−→ (1, 2)(3, 4) a−→ (3, 4, 5) b−→ (1, 2, 4, 5, 3) c−→ (1, 4, 2, 5, 3) b −1 −→ (2, 4)(3, 5) a−→ (1, 4, 3, 5, 2) c−1−→ (1, 3, 5, 2, 4) b−→ (1, 4)(2, 5) b−→ (1, 5, 2, 3, 4) c−→ (1, 3, 4, 5, 2) a−→ (2, 3, 4) b−→ (1, 3)(2, 4) c −1 −→ (1, 2, 3) a−→ (1, 3)(4, 5) b −1 −→ (2, 3)(4, 5) a−→ (1, 3, 2) b−→ e 4.14. a second cycle c2 in cay(a5; a,b,c) with a = (1, 2)(4, 5), b = (1, 2, 3), and c = (1, 2, 4). e a−→ (1, 2)(4, 5) c −1 −→ (1, 5, 4) a−→ (1, 2, 5) b−→ (1, 5)(2, 3) a−→ (1, 3, 2, 5, 4) c−→ (1, 5, 4, 3, 2) a−→ (2, 5, 3) b−→ (1, 5, 3) b−→ (1, 2)(3, 5) a−→ (3, 5, 4) b−→ (1, 2, 5, 4, 3) c−→ (1, 5, 4, 2, 3) b −1 −→ (2, 5, 4) a−→ (1, 5, 2) c−1−→ (1, 4)(2, 5) b−→ (1, 5, 2, 3, 4) b−→ (1, 3, 5, 2, 4) c−→ (1, 4, 3, 5, 2) a−→ (2, 4)(3, 5) b−→ (1, 4, 2, 5, 3) c −1 −→ (1, 2, 4, 5, 3) b −1 −→ (3, 4, 5) a−→ (1, 2)(3, 4) b −1 −→ (1, 4, 3) b−1−→ (2, 4, 3) a−→ (1, 4, 5, 3, 2) c −1 −→ (1, 5, 3, 2, 4) b−→ (1, 4)(3, 5) a−→ (1, 2, 4, 3, 5) b−→ (1, 4, 3, 2, 5) b−→ (1, 5)(3, 4) a−→ (1, 2, 5, 3, 4) c−→ (1, 5, 3, 4, 2) a−→ (2, 5)(3, 4) b−→ (1, 5, 2, 4, 3) c −1 −→ (1, 3)(2, 5) b −1 −→ (2, 3, 5) a−→ (1, 3, 5, 4, 2) c −1 −→ (1, 2, 3, 5, 4) a−→ (1, 3, 5) c −1 −→ (1, 4, 2, 3, 5) b−→ (1, 3, 4, 2, 5) b−→ (1, 5)(2, 4) c−→ (1, 4, 5) b−1−→ (1, 3, 2, 4, 5) b −1 −→ (1, 2, 3, 4, 5) a−→ (1, 3, 4) b −1 −→ (1, 4)(2, 3) b −1 −→ (1, 2, 4) c−→ (1, 4, 2) a−→ (2, 4, 5) b−→ (1, 4, 5, 2, 3) b−→ (1, 3, 4, 5, 2) a−→ (2, 3, 4) b−→ (1, 3)(2, 4) c −1 −→ (1, 2, 3) a−→ (1, 3)(4, 5) b −1 −→ (2, 3)(4, 5) a−→ (1, 3, 2) b−→ e 27 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.15. a hamiltonian cycle in cay(a5; a,b,c) with a = (1, 2)(4, 5), b = (1, 2)(3, 4), and c = (1, 2, 3). e a−→ (1, 2)(4, 5) c −1 −→ (1, 3)(4, 5) a−→ (1, 2, 3) b−→ (1, 3, 4) a−→ (1, 2, 3, 4, 5) b−→ (1, 3, 5) a−→ (1, 2, 3, 5, 4) c −1 −→ (1, 5, 4) a−→ (1, 2, 5) b−→ (1, 5)(3, 4) c −1 −→ (1, 4, 3, 2, 5) b−→ (1, 5)(2, 4) c−→ (1, 4, 2, 3, 5) b−→ (1, 3, 2, 4, 5) a−→ (1, 4)(2, 3) b−→ (1, 3)(2, 4) a−→ (1, 4, 5, 2, 3) c−→ (1, 3, 4, 5, 2) a−→ (2, 3, 4) b−→ (1, 3, 2) a−→ (2, 3)(4, 5) b−→ (1, 3, 5, 4, 2) a−→ (2, 3, 5) c−→ (1, 3)(2, 5) a−→ (1, 5, 4, 2, 3) b−→ (1, 3, 2, 5, 4) a−→ (1, 5)(2, 3) b−→ (1, 3, 4, 2, 5) a−→ (1, 5, 2, 3, 4) c−→ (1, 3, 5, 2, 4) c−→ (1, 4)(2, 5) b−→ (1, 5, 2, 4, 3) c −1 −→ (2, 5)(3, 4) b−→ (1, 5, 2) a−→ (2, 5, 4) b−→ (1, 5, 4, 3, 2) a−→ (2, 5, 3) b−→ (1, 5, 3, 4, 2) c −1 −→ (1, 4, 2, 5, 3) c−1−→ (2, 4)(3, 5) a−→ (1, 4, 3, 5, 2) b−→ (2, 4, 5) a−→ (1, 4, 2) b−→ (2, 4, 3) a−→ (1, 4, 5, 3, 2) c −1 −→ (1, 2, 4, 5, 3) a−→ (1, 4, 3) b−→ (1, 2, 4) a−→ (1, 4, 5) b−→ (1, 2, 4, 3, 5) a−→ (1, 4)(3, 5) c −1 −→ (1, 5, 3, 2, 4) c −1 −→ (1, 2, 5, 3, 4) b−→ (1, 5, 3) a−→ (1, 2, 5, 4, 3) c −1 −→ (3, 5, 4) a−→ (1, 2)(3, 5) b−→ (3, 4, 5) a−→ (1, 2)(3, 4) b−→ e 4.16. a hamiltonian cycle in cay(a5; a,b,c) with a = (1, 2)(4, 5), b = (1, 3)(2, 4), and c = (1, 2, 3). e a−→ (1, 2)(4, 5) b−→ (1, 3, 2, 5, 4) a−→ (1, 5)(2, 3) b−→ (1, 2, 4, 3, 5) a−→ (1, 4)(3, 5) b−→ (1, 5, 3, 4, 2) a−→ (2, 5)(3, 4) b−→ (1, 4, 5, 2, 3) c−→ (1, 3, 4, 5, 2) a−→ (2, 3, 4) b−→ (1, 4, 3) a−→ (1, 2, 4, 5, 3) b−→ (2, 5, 3) a−→ (1, 5, 4, 3, 2) b−→ (1, 2, 3, 5, 4) a−→ (1, 3, 5) b−→ (1, 5)(2, 4) a−→ (1, 4)(2, 5) c −1 −→ (1, 3, 5, 2, 4) b−→ (1, 5, 2) a−→ (2, 5, 4) b−→ (1, 3)(4, 5) c −1 −→ (2, 3)(4, 5) b−→ (1, 2, 5, 4, 3) a−→ (1, 5, 3) b−→ (2, 4)(3, 5) a−→ (1, 4, 3, 5, 2) b−→ (1, 5, 2, 3, 4) a−→ (1, 3, 4, 2, 5) c−1−→ (1, 4, 2, 3, 5) b−→ (1, 5)(3, 4) a−→ (1, 2, 5, 3, 4) b−→ (1, 4, 5, 3, 2) a−→ (2, 4, 3) b−→ (1, 2, 3) c−→ (1, 3, 2) b−→ (1, 2, 4) a−→ (1, 4, 5) c −1 −→ (1, 3, 2, 4, 5) b−→ (1, 2, 5) a−→ (1, 5, 4) b−→ (1, 3, 5, 4, 2) a−→ (2, 3, 5) b−→ (1, 5, 2, 4, 3) a−→ (1, 4, 2, 5, 3) b−→ (3, 4, 5) a−→ (1, 2)(3, 4) b−→ (1, 4)(2, 3) c−→ (1, 3, 4) a−→ (1, 2, 3, 4, 5) b−→ (1, 4, 3, 2, 5) a−→ (1, 5, 3, 2, 4) b−→ (1, 2)(3, 5) a−→ (3, 5, 4) b−→ (1, 5, 4, 2, 3) a−→ (1, 3)(2, 5) b−→ (2, 4, 5) a−→ (1, 4, 2) c −1 −→ (1, 3)(2, 4) b−→ e 28 d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 4.17. a hamiltonian cycle in cay(a5; a,b,c) with a = (1, 2)(3, 4), b = (1, 2)(3, 5), and c = (1, 2, 3). e a−→ (1, 2)(3, 4) c −1 −→ (1, 4, 3) a−→ (1, 2, 4) c−→ (1, 4)(2, 3) b−→ (1, 3, 5, 2, 4) a−→ (1, 4, 5, 2, 3) b−→ (1, 3, 2, 4, 5) a−→ (1, 4, 2, 3, 5) b−→ (1, 3)(2, 4) c−→ (1, 4, 2) a−→ (2, 4, 3) b−→ (1, 4, 3, 5, 2) a−→ (2, 4, 5) b−→ (1, 4, 5, 3, 2) a−→ (2, 4)(3, 5) c−→ (1, 4, 2, 5, 3) a−→ (1, 5, 3, 2, 4) c−→ (1, 4)(3, 5) a−→ (1, 2, 4, 5, 3) b−→ (1, 4, 5) a−→ (1, 2, 4, 3, 5) c −1 −→ (1, 5)(3, 4) c −1 −→ (1, 4, 3, 2, 5) a−→ (1, 5)(2, 4) c−1−→ (1, 3, 4, 2, 5) b−→ (1, 5, 4, 2, 3) a−→ (1, 3, 2, 5, 4) c−→ (1, 5, 4) b−→ (1, 2, 5, 3, 4) a−→ (1, 5, 3) b−→ (1, 2, 5) c−→ (1, 5)(2, 3) c−→ (1, 3, 5) a−→ (1, 2, 3, 4, 5) b−→ (1, 3)(4, 5) a−→ (1, 2, 3, 5, 4) b−→ (1, 3, 4) a−→ (1, 2, 3) c−→ (1, 3, 2) a−→ (2, 3, 4) b−→ (1, 3, 5, 4, 2) a−→ (2, 3)(4, 5) b−→ (1, 3, 4, 5, 2) a−→ (2, 3, 5) c−→ (1, 3)(2, 5) a−→ (1, 5, 2, 3, 4) c −1 −→ (1, 4)(2, 5) a−→ (1, 5, 2, 4, 3) c −1 −→ (2, 5)(3, 4) a−→ (1, 5, 2) b−→ (2, 5, 3) a−→ (1, 5, 3, 4, 2) b−→ (2, 5, 4) a−→ (1, 5, 4, 3, 2) c−1−→ (1, 2, 5, 4, 3) c −1 −→ (3, 5, 4) a−→ (1, 2)(4, 5) b−→ (3, 4, 5) a−→ (1, 2)(3, 5) b−→ e 4.18. a hamiltonian cycle in cay(a5; a,b,c) with a = (1, 2)(3, 4), b = (1, 3)(2, 5), and c = (1, 2, 3). e a−→ (1, 2)(3, 4) b−→ (1, 4, 3, 2, 5) c −1 −→ (1, 2, 4, 3, 5) a−→ (1, 4, 5) b−→ (1, 3, 4, 5, 2) a−→ (2, 3, 5) b−→ (1, 5, 3) a−→ (1, 2, 5, 3, 4) c−→ (1, 5, 3, 2, 4) b−→ (1, 2, 3, 5, 4) a−→ (1, 3)(4, 5) b−→ (2, 4, 5) a−→ (1, 4, 3, 5, 2) b−→ (1, 5)(3, 4) a−→ (1, 2, 5) b−→ (1, 3, 2) a−→ (2, 3, 4) b−→ (1, 4, 2, 5, 3) c−→ (1, 5, 3, 4, 2) a−→ (2, 5, 3) b−→ (1, 2, 3) a−→ (1, 3, 4) c −1 −→ (1, 4)(2, 3) a−→ (1, 3)(2, 4) b−→ (2, 5, 4) a−→ (1, 5, 4, 3, 2) c −1 −→ (1, 2, 5, 4, 3) b−→ (2, 4, 3) a−→ (1, 4, 2) b−→ (1, 3, 4, 2, 5) c−→ (1, 5)(2, 4) c−→ (1, 4, 2, 3, 5) a−→ (1, 3, 2, 4, 5) b−→ (1, 2)(4, 5) a−→ (3, 5, 4) b−→ (1, 5, 2, 4, 3) a−→ (1, 4)(2, 5) c−→ (1, 5, 2, 3, 4) b−→ (1, 4)(3, 5) a−→ (1, 2, 4, 5, 3) b−→ (2, 3)(4, 5) a−→ (1, 3, 5, 4, 2) c −1 −→ (1, 5, 4, 2, 3) b−→ (2, 4)(3, 5) a−→ (1, 4, 5, 3, 2) b−→ (1, 2, 3, 4, 5) a−→ (1, 3, 5) c −1 −→ (1, 5)(2, 3) b−→ (1, 2)(3, 5) a−→ (3, 4, 5) b−→ (1, 4, 5, 2, 3) a−→ (1, 3, 5, 2, 4) b−→ (1, 5, 4) c −1 −→ (1, 3, 2, 5, 4) b−→ (1, 2, 4) a−→ (1, 4, 3) b−→ (2, 5)(3, 4) a−→ (1, 5, 2) c −1 −→ (1, 3)(2, 5) b−→ e references [1] r. gould, r. roth, cayley digraphs and (1,j,n)-sequencings of the alternating groups an, discrete math. 66(1-2) (1987) 91–102. 29 http://dx.doi.org/10.1016/0012-365x(87)90121-x http://dx.doi.org/10.1016/0012-365x(87)90121-x d. w. morris / j. algebra comb. discrete appl. 3(1) (2016) 13–30 [2] k. kutnar, d. marušič, d. w. morris, j. morris, p. šparl, hamiltonian cycles in cayley graphs whose order has few prime factors, ars math. contemp. 5(1) (2012) 27–71. [3] k. kutnar, d. marušič, d. w. morris, j. morris, p. šparl, cayley graphs on a5 are hamiltonian, unpublished appendix to [2], http://arxiv.org/src/1009.5795/anc/a5.pdf. [4] d. witte, j. a. gallian, a survey: hamiltonian cycles in cayley graphs, discrete math. 51(3) (1984) 293–304. 30 http://amc-journal.eu/index.php/amc/article/view/177/147 http://amc-journal.eu/index.php/amc/article/view/177/147 http://arxiv.org/src/1009.5795/anc/a5.pdf http://arxiv.org/src/1009.5795/anc/a5.pdf http://www.sciencedirect.com/science/article/pii/0012365x84900104 http://www.sciencedirect.com/science/article/pii/0012365x84900104 introduction preliminaries proof of proposition 1.2 details of hamiltonian cycles in a5 references issn 2148-838x j. algebra comb. discrete appl. (article-in-press) • 1–11 received: 14 february 2022 accepted: 27 january 2023 journal of algebra combinatorics discrete structures and applications on the parameters of a class of narrow sense primitive bch codes research article el mahdi mouloua, m. najmeddine abstract: the last few decades have seen an increase in the determination of the parameters of the primitive bch codes. indeed, bch codes are powerful in terms of encoding and decoding. they are applied in several fields such as: satellite communications, cryptography, compact disk drives etc, and have good structural properties. nevertheless, the dimension and the minimum distance of those codes are not known in general. in this paper, we present a class of narrow sense primitive bch codes of designed distance δ4 = (q − 1)q m−1 − 1 − q bm+3 2 c . also, we investigate their bose distance and dimension. 2010 msc: 11t71, 94b15 keywords: bch codes, cyclic codes, bose distance, cyclotomic cosets, minimum distance 1. introduction in coding theory, cyclic codes are considered as an important class of codes. they include a special subclass, discovered by bose and ray-chaudhuri in 1960 [12], and independently by hocquenghem in 1959 [2], known as bch codes. indeed, bch codes have a good error-correcting capability. in many cases, bch codes are the best linear codes, but the exact dimension and minimum weight are considered unsolved[11]. recent results on the determination of the parameters of narrow sense primitive bch codes can be found in [1, 3–8, 10, 14]. let fq be the finite field of q elements, such that q is a prime power. we recall that an [n,k,d] linear code c over fq is a k-dimensional subspace of f n q , where d is its hamming minimum distance, and n is its length. an [n,k] code c is cyclic if it is linear and if any cyclic shift of a codeword is also a e. mouloua, (corresponding author); department of mathematics, ensam–meknes, moulay ismail university, morocco (email: e.mouloua@edu.umi.ac.ma). m. najmeddine; department of mathematics, ensam–meknes, moulay ismail university, morocco (email: m.najmeddine@umi.ac.ma) 1 https://orcid.org/0000-0002-8036-9451 https://orcid.org/0000-0002-4878-1219 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 codeword, i.e., whenever ( c 0 ,c 1 , . . . ,c n−1 ) is in c then so is ( c n−1,c0, . . . ,cn−2 ) . more information about cyclic codes can be found in [16] page 121. we identify a vector ( c 0 , . . . ,c n−1 ) ∈ f n q with the polynomial c 0 + c 1 x + · · · + c n−1x n−1 in the ring rn = fq[x] < xn − 1 > . thus, we can view an [n,k] cyclic code as a principal ideal of the ring rn (see [16]). if c is not trivial there exists a unique monic polynomial g that generates the code c, g is called the (standard) generator polynomial of the code c, and g divides the polynomial xn − 1. the polynomial h defined by h(x) = xn − 1 g(x) is called the (parity) check polynomial and gives information on the dual of the cyclic code c. let δ be an integer in {0, . . . ,n− 1}, and α be a primitive n th root of unity. for any integer i such that 0 6 i 6 n− 1, let m(i)(x) denote the minimal ploynomial of αi. a cyclic code c is said to be a bch code of designed distance δ, if for some integer b > 0 its generator polynomial noted g (q,m,δ) (x) is given by : g (q,m,δ) (x) = lcm ( m(b)(x),m(b+1)(x), . . . ,m(b+δ−2)(x) ) , where lcm is the least common multiple of these minimal polynomials. therefore, g (q,m,δ) is the lowest degree monic polynomial over fq having αb,αb+1, . . . ,αb+δ−2 as zeros, and a word c is in the code if and only if c(αb) = c(αb+1) = . . . = c(αb+δ−2) = 0. in the case b = 1, we obtain the so called narrow sense bch codes. the bch codes of length n = q m −1 are called the primitive bch codes. the largest designed distance is called the bose distance and denoted by db. for more results on the bose distance of bch codes see [6]. let g̃ (q,m,δ) (x) = (x− 1)g (q,m,δ) (x). throughout this paper, we adopt the following notation : n = q m −1, δ 4 = (q−1)q m−1 −1−q bm+3 2 c , and c (q,m,δ) denotes the narrow sense primitive bch code of designed distance δ with generator polynomial g (q,m,δ) , and c̃ (q,m,δ) denotes the primitive bch code with generator polynomial g̃ (q,m,δ) (x). according to authors in [4], the code c̃ (q,m,δ) is a primitive code of designed distance δ + 1, and since dim(c̃ (q,m,δ) ) = n − deg(c̃ (q,m,δ) ), we have dim(c̃ (q,m,δ) ) = dim(c (q,m,δ) ) − 1. let ai be the number of codewords with hamming weight i, the polynomial 1 + a1z + a2z2 + · · · + anzn is called the weight enumerator of the code c (q,m,δ) . the set a1,a2, . . . ,an is called the weight distribution of the code c(q,m,δ). inspired by the results of [4], we prove that δ 4 is the fourth largest coset leader, and we study the parameters of the code c (q,m,δ 4 ) . according to [5], author determined the first largest coset leader denoted by δ 1 = (q − 1)q m−1 − 1 and examined the parameters of the code c (q,m,δ 1 ) . later, authors in [4] determined the second and the third largest coset leaders modulo n, denoted respectively δ 2 = (q − 1)q m−1 − 1 − q bm−1 2 c and δ 3 = (q − 1)q m−1 − 1 − q bm+1 2 c and studied the parameters of c (q,m,δ 2 ) and c (q,m,δ 3 ) . with their weight distributions. the remainder of this paper is organized as follows. in section 2, preliminaries and notations are introduced. section 3 is devoted to the exploration of the parameters of the codes c (q,m,δ 4 ) and c̃ (q,m,δ4) . section 4 concludes the paper. 2. notation and basic concepts in order to present the most recent results and apply them to investigate the dimension and the bose distance of our class of narrow sense primitive bch codes, we present some auxiliary results on cyclotomic cosets. more details on cyclotomic cosets and code constructions can be found in [9]. definition 2.1 ([16], page 122). the q−coset modulo n containing an element t is defined by ct = {t,tq,tq 2 , . . . , tqlt−1}, where lt is the smallest integer such that tqlt ≡ t (mod n). the smallest integer in ct is called the coset leader of ct. we denote by γ(q,m) the set of all coset leaders modulo n. 2 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 in [14], the authors give a general formula for computing the dimension of narrow sense primitive bch code. proposition 2.2 ([14], proposition 2.1). the dimension of the code c (q,m,δ) is equal to 1 + ∑ r>δ r∈γ (q,m) |cr| . cyclotomic cosets can be used to determine the bose distance of narrow sense primitive bch codes. the following proposition gives the connection between db and coset leaders. for more details about the proof we refer the readers to [13] page 3. proposition 2.3 ([13], proposition 4, page 3). the bose distance of the code c (q,m,δ) is a coset leader of a q-cyclotomic coset modulo n. furthermore if δ is a coset leader, then d b = δ. next, we present some lemmas, that we will need later. lemma 2.4 ([1], lemma 2.1). let a and b be two positive distinct integers less than or equal to n. let m−1∑ j=0 ajq j and m−1∑ j=0 bjq j be the q−adic expansions of a and b respectively. set k = min { i ∈ n : 0 ≤ i ≤ m− 1,am−1−i 6= bm−1−i } . then we have a > b if and only if am−1−k > bm−1−k. lemma 2.5. for non-negative integers a and j with 0 6 a 6 n − 1 such that a = m−1∑ i=0 aiq i and 1 6 j 6 m− 1 we have[ qja ] n = ( am−(j+1),am−(j+2),am−(j+3), . . . ,am−j+2,am−j+1,am−j ) , where [c]n = (cm−1,cm−2, . . . ,c1,c0) if c = c0 + c1q + · · · + cm−2qm−2 + cm−1qm−1 (mod n). proof. since a = a 0 + a 1 q + a 2 q 2 + · · · + a m−(j+2)q m−(j+2) + a m−(j+1)q m−(j+1) + a m−jq m−j + a m−j+1q m−j+1 + a m−j+2q m−j+2 + · · · + a m−1q m−1 , for all 1 6 j 6 m− 1, q j a = a 0 q j + a 1 q j+1 + a 2 q j+2 + · · · + a m−(j+2)q m−2 + a m−(j+1)q m−1 + a m−j + am−j+1q + am−j+2q 2 + · · · + a m−1q m−1+j (mod n). hence, [ q j a ] n = ( a m−(j+1),am−(j+2),am−(j+3), . . . ,am−j+2,am−j+1,am−j ) . lemma 2.6 ([4], lemma 5, lemma 7, lemma 12). let δ 1 = (q − 1)q m−1 − 1, δ 2 = (q − 1)q m−1 − 1 −q bm−1 2 c and δ 3 = (q − 1)q m−1 − 1 −q bm+1 2 c . then we have : 1. δ 1 is the first largest q−cyclotomic coset modulo n and |c δ 1 | = m. 2. δ 2 is the second largest q−cyclotomic coset modulo n. furthermore  ∣∣∣cδ 2 ∣∣∣ = m if m is odd, ∣∣∣cδ 2 ∣∣∣ = m 2 if m is even. 3 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 3. for m > 4, δ 3 is the third largest q−cyclotomic coset modulo n and ∣∣∣cδ 3 ∣∣∣ = m. below we give an overview about the recent results on the parameters of the codes c (q,m,δ 1 ) , c (q,m,δ 2 ) and c (q,m,δ 3 ) . theorem 2.7 ([5], theorem 13, page 5325). the two cyclic codes c (q,m,δ1) and r q (1,m)∗ are identical, and have parameters [q m −1,m+1, (q−1)qm−1−1] where r q (1,m)∗ is the first order punctured generalized reed-muller code of length n. according to authors in [16], a linear code c of length n over fq and minimum distance at least d is called optimal, if it has b q (n,d) codewords, where b q (n,d) is the largest number of codewords in the code c. there are other perspectives on optimizing a code, for more information readers can refer to [16] page 53. according to authors in [5], the code c (q,m,δ 1 ) is optimal. the main results about the parameters of the codes c (q,m,δ 2 ) , c̃ (q,m,δ 2 ) , c (q,m,δ 3 ) , and c̃ (q,m,δ 3 ) are given in [4], and according to them we have the following theorems : theorem 2.8 ([4], theorem 8, page 243). the code c̃ (q,m,δ2) has parameters [n,k̃, d̃], where n = qm − 1, δ2 = (q − 1)q m−1 − 1 −q bm−1 2 c , d̃ > δ2 + 1 and k̃ =   2m for odd m, 3m 2 for even m. in particular, afor q = 2 and m any integer, d̃ = δ2 + 1. bfor q an odd prime, d̃ = δ2 + 1. the weight distribution in the case a, b are given [4]. theorem 2.9 ([4], theorem 11, page 250). let m > 2 be an integer. the code c (q,m,δ2) has parameters [n,k,d], where n = q m − 1, δ2 = (q − 1)q m−1 − 1 −q bm−1 2 c , d > δ2 and k =   2m + 1 for odd m, 3m 2 + 1 for even m. furthermore, d = δ2 if q is prime. according to authors in [4], the codes c (q,m,δ2) and c̃ (q,m,δ2) are sometimes optimal, and sometimes have the same parameters as the best known linear codes in the tables of the best known linear codes maintained by markus grassl at http://www.codetables.de. theorem 2.10 ([4], theorem 13, page 251). let m > 4. the code c̃ (q,m,δ3) has parameters [n,k̃, d̃], where n = q m − 1, δ3 = (q − 1)q m−1 − 1 −q bm+1 2 c , d̃ > δ3 + 1 and k̃ =   3m for odd m, 5m 2 for even m. in particular, 4 http://www.codetables.de e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 awhen q = 2 for any integer m, d̃ = δ3 + 1. bif q is an odd prime, and m > 4 is even then d̃ = δ3 + 1. cif q is an odd prime, and m > 5 is odd then d̃ = δ3 + 1. the weight distributions in the cases a, b, c and d are given in [4]. theorem 2.11 ([4], theorem 15, page 254). let m > 4. the code c (q,m,δ3) has parameters [n,k,d], where n = q m − 1, δ3 = (q − 1)q m−1 − 1 −q bm+1 2 c , d > δ3 and k =   3m + 1 for odd m, 5m 2 + 1 for even m. according to authors in [4], the codes c (q,m,δ3) and c̃ (q,m,δ3) are sometimes optimal, and sometimes have the same parameters as the best known linear codes in the tables of the best linear known codes maintained by markus grassl at http://www.codetables.de. 3. parameters of the code c (q,m,δ4) in order to investigate the parameters of c (q,m,δ 4 ) , we need the following two lemmas: lemma 3.1. δ 4 is a coset leader in the q−cyclotomic coset c δ 4 and c δ 4 has cardinality m. proof. to prove the lemma, we need to distinguish two cases : ◦ case 1, m is odd. then, δ 4 = (q − 1)q m−1 − 1 −q m+3 2 = n−q m−1 −q m+3 2 = n−q m+3 2 ( 1 + q m−5 2 ) now, let us determine c δ 4 δ 4 = n−q m+3 2 (1 + q m−5 2 ) , (mod n). δ 4 q = n−qq m+3 2 (1 + q m−5 2 ) , (mod n). = n− (1 + q m+5 2 ) , (mod n). δ 4 q 2 = n−q(1 + q m+5 2 ) , (mod n). ... δ 4 q m−5 2 = n−q m−7 2 (1 + q m+5 2 ) , (mod n). δ 4 q m−3 2 = n−q m−5 2 (1 + q m+5 2 ) , (mod n). = n− (1 + q m−5 2 ) , (mod n). δ 4 q m−1 2 = n−q(1 + q m−5 2 ), (mod n). ... δ 4 qm−3 = n−q m−3 2 (1 + q m−5 2 ) , (mod n). δ 4 qm−2 = n−q m−1 2 (1 + q m−5 2 ) , (mod n). δ 4 q m−1 = n−q m+1 2 (1 + q m−5 2 ) , (mod n). 5 http://www.codetables.de e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 if i and j are two distinct integers in the set { 0, 1, . . . , m− 7 2 } , then we have n−q i (1 + q m+5 2 ) = (q − 1)q m−1 + · · · + (q − 2)q i + · · · + (q − 2)q i+ m+5 2 + · · · + (q − 1)q + (q − 1) and n−q j (1 +q m+5 2 ) = (q−1)q m−1 +· · ·+ (q−2)q j +· · ·+ (q−2)q j+ m+5 2 +· · ·+ (q−1)q + (q−1). it is clear, by lemma [2.4], we have n−q i (1 + q m+5 2 ) 6= n−q j (1 + q m+5 2 ). by the same reasoning when i and j are two distinct integers in the set {0, . . . , m+3 2 }, we have that n−q i (1 + q m−5 2 ) 6= n−q j (1 + q m−5 2 ). thus c δ 4 = { n−q i ( 1 + q m+5 2 ) : i = 0, 1, . . . , m− 7 2 }⋃{ n−q i ( 1 + q m−5 2 ) : i = 0, . . . , m + 3 2 } . it is clear that ∣∣∣cδ 4 ∣∣∣ = m. observe that δ 4 is the smallest integer in c δ 4 . indeed, in the set{ n−q i ( 1 + q m−5 2 ) : i = 0, . . . , m + 3 2 } , δ 4 is the smallest integer. let’s compare δ 4 with n−q m−7 2 (1 + q m+5 2 ). we have δ 4 − ( n−q m−7 2 (1 + q m+5 2 ) ) = q m−7 2 −q m+3 2 < 0. thus δ 4 is a coset leader modulo n. ◦ case 2, m is even. then, δ 4 = (q − 1)q m−1 − 1 −q m+2 2 = n−q m−1 −q m+2 2 = n−q m+2 2 ( 1 + q m−4 2 ) . let’s determine c δ 4 . δ 4 = n−q m+2 2 ( 1 + q m−4 2 ) , (mod n). δ 4 q = n−qq m+2 2 ( 1 + q m−4 2 ) , (mod n). = n− ( 1 + q m+4 2 ) , (mod n). δ 4 q 2 = n−q ( 1 + q m+4 2 ) , (mod n). ... δ 4 q m−8 2 = n−q m−10 2 ( 1 + q m+4 2 ) , (mod n). δ 4 q m−6 2 = n−q m−8 2 ( 1 + q m+4 2 ) , (mod n). δ 4 q m−4 2 = n−q m−6 2 ( 1 + q m+4 2 ) , (mod n). δ 4 q m−2 2 = n−q m−4 2 ( 1 + q m+4 2 ) , (mod n). = n− ( 1 + q m−4 2 ) , (mod n). δ 4 q m 2 = n−q ( 1 + q m−4 2 ) , (mod n). ... δ 4 q m−1 = n−q m 2 ( 1 + q m−4 2 ) , (mod n). 6 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 thus, c δ 4 = { n−q i (1 + q m−4 2 ) : i = 0, 1, . . . , m + 2 2 }⋃{ n−q i (1 + q m+4 2 ) : i = 0, . . . , m− 6 2 } . it is clear that ∣∣∣cδ 4 ∣∣∣ = m. observe that δ 4 is the smallest integer in the set { n−q i ( 1 + q m−4 2 ) : i = 0, . . . , m + 2 2 } . let’s compare δ 4 with n−q m−6 2 (1 + q m+4 2 ). we have δ 4 − (n−q m−6 2 (1 + q m+4 2 )) = q m−6 2 −q m+2 2 < 0. thus δ 4 is a coset leader modulo n. now we verify that δ 4 is the fourth largest coset leader modulo n. lemma 3.2. for m > 11, δ 4 is the fourth largest coset leader modulo n. proof. to prove the lemma, we need to distinguish two cases : ◦ case 1, m is odd. then, δ 4 = (q − 1)q m−1 − 1 − q m+3 2 . we verify that there is no coset leader between δ 3 and δ 4 . set s = m−1 2 . then, δ 4 = (q − 1)q m−1 − 1 −q m+3 2 = q m − 1 −q m−1 −q s+2 = (q − 1)(q m−1 + q m−2 + · · · + q + 1) −q m−1 −q s+2 = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 2)q s+2 + (q − 1)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q + (q − 1). with the same argument we have δ 3 = (q − 1)q m−1 − 1 −q m+1 2 = q m − 1 −q m−1 −q s+1 = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + (q − 2)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q + (q − 1). hence δ 3 −δ 4 = (q − 1)q s+2 + (q − 2)q s+1 − (q − 2)q s+2 − (q − 1)q s+1 = q s+2 −q s+1 = q s+1 (q − 1). since q s+1 (q−1)−1 = (q−2)q s+1 +(q−1)q s +· · ·+(q−1), any i ∈ { 1, . . . ,q s+1 (q − 1) − 1 } can be written as i = i s+1 q s+1 +i s q s +· · ·+i 1 q+i 0 , where 0 6 i s+1 6 q−2 and 0 6 i l 6 q−1; l ∈{0, . . . ,s}. let k i := δ 3 − i for all i ∈{1, . . . ,q s+1 (q − 1) − 1}. we are able to give the q−adic expansion of k i and analyze it. indeed, we have k i = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + (q − 2)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q − i s+1 q s+1 − i s q s − i s−1q s−1 − . . .− i 1 q − i 0 . 7 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 therefore, k i = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + (q − 2 − i s+1 )q s+1 + (q − 1 − i s )q s + (q − 1 − i s−1 )q s−1 + · · · + (q − 1 − i 1 )q + (q − 1 − i 0 ). now we need to verify that k i cannot be a coset leader. to this end, we consider two subcases as follows. – case 1 : q = 2. in this case, we have : k i = 2 m−2 + · · · + 2 s+2 + (1 − i s )2s + · · · + (1 − i 1 )2 + (1 − i 0 ). where i 0 ∈{0, 1}. – i 0 = 1. then k i 2 and k i are in the same q−cyclotomic coset modulo n, and since k i > k i 2 , k i cannot be a coset leader. – i 0 = 0. then, k i = 2 m−2 + 2 m−3 + · · · + 2 s+2 + (1 − i s )2 s + (1 − i s−1 )2 s−1 + · · · + (1 − i 1 ) 2 + 1. since i 6= 0, one of the i′ l s must be nonzero. let l denote the largest one such that i l = 1. thus, we have : k i = 0 × 2 m−1 + 2 m−2 + 2 m−3 + · · · + 2 s+2 + 0 × 2 s+1 + 2 s + · · · + 2 l+1 + 0 × 2 l + (1 − i l−1 )2 l−1 + · · · + (1 − i 1 )2 + 1. 2 m−1−l k i = 0 × 2 m−l−2 + 2 m−l−3 + 2 m−l−4 + · · · + 2 s+1 + · · · + 2 s−(l−1) + 2 s−(l+1) + · · · + 2 + 1 + 0 × 2 m−1 + (1 − i l−1 )2 m−2 + · · · + (1 − i 1 )2 m−l + 2 m−l−1 . = 0 × 2 m−1 + (1 − i l−1 )2 m−2 + · · · + (1 − i 1 ) 2 m−l + 2 m−l−1 + 0 × 2 m−l−2 + 2 m−l−3 + 2 m−l−4 + · · · + 2 s+2 + 2 s+1 + · · ·+ 2 s−(l−1) + 0 × 2 s−l + 2 s−(l+1) + · · · + 2 + 1. using lemma [2.4], we confirm that 2 m−1−l k i < k i . hence, k i cannot be a coset leader. in all cases there is no coset leader between δ 4 and δ 3 . – case 2 : q > 2. then, k i = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + ( q − 2 − i s+1 ) q s+1 + (q − 1 − i s ) q s + ( q − 1 − i s−1 ) q s−1 + · · · + (q − 1 − i 1 ) q + (q − 1 − i 0 ) . – if i s+1 > 1, then we verify that q m−1−(s+1) k i < k i . by lemma [2.5] we have [ q m−2−s k i ] n = ((q − 2 − i s+1 ); (q − 1 − i s ); (q − 1 − i s−1 ); . . . ; (q − 1 − i1 ); (q − 1 − i0 ); 0; (q − 2); (q − 1); . . . ; (q − 1)). since 1 6 i s+1 6 q − 2, q − 2 − i s+1 < q − 2, and by lemma [2.4], k i cannot be a coset leader. – if i l > 2 for some l with l ∈{0, . . . ,s}, let k i = (q−2)q m−1 + (q−1)q m−2 + · · ·+ (q − 1)q s+2 + (q − 2 − i s+1 )q s+1 + (q − 1 − i s )q s + (q − 1 − i s−1 )q s−1 + . . . + (q − 1 − i l )q l + . . . + (q − 1 − i 1 )q + (q − 1 − i 0 ). [q m−1−l k i ]n = ((q− 1 − il); (q− 1 − il−1 ); (q− 1 − il−2 ); . . . ; (q− 1 − i1 ); (q− 1 − i 0 ); 0; (q − 2); (q − 1); . . . ; (q − 1); (q − 2 − i s+1 ); (q − 1 − i s ); . . . ; (q − 1 − i l−1 )). since i l > 2, q−1−i l 6 q−3 < q−2, and by lemma [2.4], k i cannot be a coset leader. 8 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 – we now assume that i l ∈ {0, 1} for all 0 6 l 6 s− 1 and i s+1 = 0. since i > 1, at least one of the i′ l s must be 1. let l denote the largest one such that i l = 1. then we have : k i = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + (q − 2)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q l+1 + (q − 2)q l + (q − 1 − i l−1 )q l−1 + · · · + (q − 1 − i 1 )q + (q − 1 − i 0 ) q m−1−l k i = (q − 2)q m−2−l + (q − 1)q m−3−l + · · · + (q − 1)q s+1−l + (q − 1)q s−l + (q − 1)q s−(l+1) + · · · + (q − 1) + (q − 2)q m−1 + (q − 1 − i l−1 )q m−2 + · · · + (q − 1 − i 1 )q m−l + (q − 1 − i 0 )q m−1−l q m−1−l k i = (q − 2)q m−1 + ( q − 1 − i l−1 ) q m−2 + · · · + (q − 1 − i 1 ) q m−l + (q − 1 − i 0 ) q m−1−l + (q − 2)q m−2−l + (q − 1)q m−3−l + · · · + (q − 1)q s + · · · + (q − 1)q s−(l−1) + (q − 1)q s−l + (q − 1)q s−(l+1) + · · · + (q − 1). by lemma [2.4], k i cannot be a coset leader. – case 2, m is even. then, δ 4 = (q − 1)q m−1 − 1 −q m+2 2 . we verify that there is no coset leader between δ 3 and δ 4 . set s = m−2 2 . then, δ 4 = (q − 1)q m−1 − 1 −q s+2 = q m − 1 −q m−1 −q s+2 = (q − 1)(q m−1 + q m−2 + · · · + q + 1) −q m−1 −q s+2 = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 2)q s+2 + (q − 1)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q + (q − 1) with the same argument we have : δ 3 = (q − 1)q m−1 − 1 −q m 2 = q m − 1 −q m−1 −q s+1 = (q − 2)q m−1 + (q − 1)q m−2 + · · · + (q − 1)q s+2 + (q − 2)q s+1 + (q − 1)q s + (q − 1)q s−1 + · · · + (q − 1)q + (q − 1) it is similar to the odd case that we follow the same steps for the remainder of the proof. now we are able to investigate the parameters of the code c (q,m,δ 4 ) and the code c̃ (q,m,δ 4 ) : theorem 3.3. let m > 11. the code c̃ (q,m,δ 4 ) has parameters [n,k,d], where n = q m −1, d > δ 4 + 1 and k =   4m if m is odd, 7m 2 if m is even. proof. for the value of the dimension we apply the proposition [2.2]. finally the bch bound ensures the bound on the minimum distance. corollary 3.4. for an odd integer m, m > 11, the code c̃ (2,m,δ 4 ) has parameters [n,k,d], where n = 2 m − 1, k = 4m and d = 2 m−1 − 2 m+3 2 . 9 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 weight w number of words of weight w 0 1 2m−1 + 2 m−1 2 ( 2m−1 − 2 m−1 2 )( 151 × 22m−3 + 25 × 2m + 25 ) 2m−1 45 2m−1 − 2 m−1 2 (2m−1 + 2 m−1 2 )(151 × 22m−3 + 25 × 2m + 25)( 2 m−1 45 ) 2m−1 + 2 m+1 2 (2m−2 − 2 m−1 2 )(23 × 2m−5 + 1)(2m−1 − 1)( 2 m−1 9 ) 2m−1 − 2 m+1 2 (2m−2 + 2 m−1 2 )(23 × 2m−5 + 1)(2m−1 − 1)( 2 m−1 9 ) 2m−1 + 2 m+3 2 (2m−6 − 2 m−7 2 )(2m−3 − 1)( 2 m−1 45 ) 2m−1 − 2 m+3 2 (2 m−6 + 2 m−7 2 )(2m−3 − 1)( 2 m−1 45 ) 2 m−1 2 4m − 1 − ∑ j 6=0,2 m−1 aj table 1. weight distribution of the code c̃(2,m,δ4) proof. since m is odd, and m > 11. then the dimension of the code c̃ (2,m,δ 4 ) is 4m. according to kassami in [15] theorem 16 page 24, the code c̃ (2,m,δ 4 ) is the same as the code defined under [15] lemma 9 page 15, which has the weight distribution shown in table 1. and since there is a codeword with weight δ4. then, the minimum distance of the code c̃(2,m,δ 4 ) is exactly δ4. according to author in [15] page 24. the dual code of the code c̃ (2,m,δ 4 ) is a subcode of the dual code of the code c̃ (2,m,δ 4 ) , which has minimum distance 7. example 3.5. let (q,m) = (2, 11). then δ 4 = 895, and c̃ (q,m,δ 4 ) has parameters [2047, 44, 896], with the weight enumerator 1+3574742979584z 1056 +3805371558912z 992 +164511003360z 1088 +186445803808z 960 + 332260852z 1152 + 427192524z 896 + 9860355245375z 1024 . theorem 3.6. let m > 11. the code c (q,m,δ 4 ) has parameters [n,k,d], where n = q m − 1, d > db such that db = δ4, and k =   4m + 1 if m is odd, 7m 2 + 1 if m is even. proof. for the value of the dimension we apply the proposition [2.2], and by proposition [2.3] we find that db = δ4 since δ4 ∈ γ(q,m). finally bch bound ensures the bound on the minimum distance. conclusion and further works by this work we initiate our first reaserch in studying the parameters of narrow sense primitive bch codes. our idea in this work was inspired from the ideas proposed by authors in [4]. thus we give a similar demonstration as the one proposed in [4] to find the fourth largest coset leader modulo q m − 1, and then investigate the parameters of the code c (q,m;δ4) and the code c̃ (q,m,δ 4 ) . the investigation of the weight distribution of the code c̃ (2,m,δ 4 ) for odd m > 11 was presented by tadao kassami in [15]. in 10 e. mouloua, m. najmeddine / j. algebra comb. discrete appl. (article-in-press) (2023) 1–11 a future work we plan to study the weight distributions of the codes c (q,m;δ4) and c̃ (q,m,δ 4 ) by adopting the theory of quadratic forms over the finite field and the theory of association schemes. we also plan to attack some open problems proposed by cunsheng ding in [4] about the weight distribution of the extended codes c̃ (q,m,δ 2 ) and c̃ (q,m,δ 3 ) . acknowledgment: the authors would like to thank the reviewers for their comments and suggestions that allowed us to correct several errors and improve the readability and quality of the article. references [1] a. cherchem, a. jamous, h. lius, y. maouche, some new results on dimension and bose distance for various classes of bch codes. finite fields their appl. 65 (2020). [2] a. hocquenghem, codes correcteurs d’erreurs, chiffres (paris) 2 (1959) 147–156. [3] b. pang, s. zhu, x. kai, five families of the narrow-sense primitive bch codes over finite fields. des. codes cryptogr. 89 (2021) 2679–2696. [4] c. ding, c. fan, z. zhou, the dimension and minimum distance of two classes of primitive bch codes, finite fields appl. 340 (2017) 237–263. [5] c. ding, parameters of several classes of bch codes, ieee trans. inform. theory 61 (10) (2015) 5322–5330. [6] c. ding, x. du, z. zhou, the bose and minimum distance of a class of bch codes, ieee trans. inform. theory 61 (5) (2015) 2351–2356. [7] c. li, p. wu, and f. liu, on two classes of primitive bch codes and some related codes, ieee transactions on information theory, 65 (2019) 3830–3840. [8] e. mouloua, m. najmeddine, o. hassan, around the parameters of primitive bch codes, 2nd international conference on innovative research in applied science, engineering and technology (iraset), 2022, pp. 1–7. [9] g.g. la guardia, m.m.s. alves, on cyclotomic cosets and code constructions. linear algebra and its applications, 488 (2016) 302–319. [10] h. liu, c. ding, c. li, dimensions of three types of bch codes over gf(q), discrete math. 340 (2017) 1910–1927. [11] p. charpin, open problems on cyclic codes, in: v.s. pless, w.c. huffman (eds.), handbook of coding theory, vol. i, north-holland, 1998, pp. 963–1063 (chapter 11). [12] r. bose, d. ray-chaudhuri, on a class of error correcting binary group codes, inf. control 3 (1) (1960) 68–79. [13] s. li, c. ding, m. xiong, g. ge, narrow-sense bch codes over gf(q) with length n = qm − 1 q − 1 , ieee trans. inf. theory 63 (11) (2017) 7219–7236. [14] s. li, the minimum distance of some narrow-sense primitive bch codes, siam j. discrete math. 31 (2017) 2530–2569. [15] t. kasami, weight distributions of bose-chaudhuri-hocquenghem codes. combinatorial mathematics and its applications, 36 (1966). [16] w. c. huffman, v. pless, fundamentals of error-correcting codes. cambridge university press, 2003. 11 https://doi.org/10.1016/j.ffa.2020.101673 https://doi.org/10.1016/j.ffa.2020.101673 https://doi.org/10.1007/s10623-021-00942-z https://doi.org/10.1007/s10623-021-00942-z https://doi.org/10.1016/j.ffa.2016.12.009 https://doi.org/10.1016/j.ffa.2016.12.009 https://doi.org/10.1109/tit.2015.2470251 https://doi.org/10.1109/tit.2015.2470251 https://doi.org/10.1109/tit.2015.2409838 https://doi.org/10.1109/tit.2015.2409838 https://doi.org/10.1109/tit.2018.2883615 https://doi.org/10.1109/tit.2018.2883615 https://doi.org/10.1109/iraset52964.2022.9737805 https://doi.org/10.1109/iraset52964.2022.9737805 https://doi.org/10.1109/iraset52964.2022.9737805 https://doi.org/10.1016/j.laa.2015.09.034 https://doi.org/10.1016/j.laa.2015.09.034 https://doi.org/10.1016/j.disc.2017.04.001 https://doi.org/10.1016/j.disc.2017.04.001 https://doi.org/10.1016/s0019-9958(60)90287-4 https://doi.org/10.1016/s0019-9958(60)90287-4 https://doi.org/10.1109/tit.2017.2743687 https://doi.org/10.1109/tit.2017.2743687 https://doi.org/10.1137/16m1108431 https://doi.org/10.1137/16m1108431 http://hdl.handle.net/2142/74459 http://hdl.handle.net/2142/74459 introduction notation and basic concepts parameters of the code c(q,m,4) conclusion and further works references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.79635 j. algebra comb. discrete appl. 3(2) • 61–80 received: 5 september 2015 accepted: 19 january 2016 journal of algebra combinatorics discrete structures and applications commutative schur rings over symmetric groups ii: the case n = 6 research article amanda e. francis, stephen p. humphries abstract: we determine the commutative schur rings over s6 that contain the sum of all the transpositions in s6. there are eight such types (up to conjugacy), of which four have the set of all the transpositions as a principal set of the schur ring. 2010 msc: 20c05, 20f55 keywords: schur ring, conjugacy class, symmetric group, group algebra 1. introduction given a finite group g it is sometimes possible to characterize the schur rings over g in some way. this has been accomplished for the family of cyclic groups [3, 6–9]. in [4] we characterized some commutative schur rings over symmetric groups. in this paper we use those results to characterize certain commutative schur rings over s6. for a finite group g and x = {x1, . . . ,xk} ⊆ g, |x| = k, we let x = x1 + · · · + xk ∈ cg. we also let x−1 = {x−1 : x ∈ x}. let c2 denote the class of transpositions in the symmetric group sn. as a consequence of the main result of [4] we have: corollary 1.1. if s is a commutative schur ring over s6 containing c2, then s determines (up to conjugacy) one of the following partitions of c2: (i) c2 = c2; (ii) c2 = c1 ∪c2 ∪c3 where c1 = {(1, 2)}, c2 = {(i,j) : 3 ≤ i < j ≤ 6} and c3 = c2 \ (c1 ∪c2); amanda e. francis (corresponding author), stephen p. humphries; department of mathematics, brigham young university, provo,ut 84602, u.s.a. (email: amanda@mathematics.byu.edu, steve@mathematics.byu.edu). 61 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 (iii) c2 = c1 ∪c2 where c1 = {(1, 2), (3, 4), (5, 6)} and c2 = c2 \c1; (iv) c2 = c1 ∪c2 where c1 = {(1, 2), (1, 3), (2, 3), (4, 5), (4, 6), (5, 6)} and c2 = c2 \c1; (v) c2 = c1 ∪c2 where c1 = {(i,j) : 1 ≤ i < j ≤ 5} and c2 = c2 \c1. if h ≤ g, then the set of orbits of elements of g under the action of conjugation by elements of h gives a schur-ring that we denote s(g,h). in [4] we found all commutative schur rings over sn,n ≤ 5, that contain c2. in this paper we do the same for s6, this being our main result: theorem 1.2. the only commutative schur rings over s6 containing c2 are (up to conjugacy): (1) z(cs6); (2) s(s6,h120), where h120 = 〈(1, 4)(3, 5), (1, 4, 6, 2, 5, 3)〉∼= s5 is a 3-transitive subgroup; (3) s(s6,h), where h = a6; (4) s36 (to be constructed in §4). (5) s(s6,h), where h = s2 ×s4 ≤ s6; (6) s(s6,h), where h = s3 os2 ≤ s6; (7) s(s6,h), where h = s2 os3 ≤ s6; (8) s(s6,h), where h = s5. of these, (1)-(4) are those where c2 is a principal set of the schur ring. these schur rings have dimensions 11, 19, 12, 12, 34, 26, 34, 19, respectively we have shown in [5] that the dimension of a commutative schur-ring over g is bounded by sg :=∑r i=1 di, where the di, i ≤ r, are the irreducible character degrees of g. in [5] we showed that some (specific) groups realize this bound, while others do not. for example, we showed that s3,s4,s5 each have a commutative schur-ring of this maximal dimension. we note that ss6 = 76. then theorem 1.2 allows us to prove corollary 1.3. there is no commutative schur-ring over s6 of dimension ss6 = 76. another consequence of our main result is that there are non-schurian schur rings over s6 (see (4) above). there is of necessity a certain amount of computation involved in the proof of theorem 1.2, however we have tried to restrict such computations to: the schur-ring algorithm that is described in §2; computations of gröbner bases for certain ideals of polynomial algebras; a small number of small enumerations of possibilities. all computer computations involved in the preparation of this paper were accomplished using magma [1]. 2. schur-rings we now define the concept of a schur ring [2, 9–11]: 62 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 a schur-ring (or s-ring) over a finite group g is a subring s of cg that is constructed from a partition {γ1, γ2, . . . , γm} of the elements of g: g = γ1 ∪ γ2 ∪·· ·∪ γm, with γ1 = {id}, satisfying: (1) if 1 ≤ i ≤ m, then there is some j ≥ 1 such that γ−1i = γj; (2) if 1 ≤ i,j ≤ m, then γiγj = ∑m k=1 λijkγk, where λijk ∈ z ≥0 for all i,j,k. the γi are called the principal sets of the s-ring. an s-ring naturally gives rise to a subalgebra of cg by extending coefficients. we will usually think of s-rings as c-algebras in this way. we refer to the survey [9] for an account of recent developments and applications of the theory of schur rings. we recall the following fact (called the schur-wielandt principle, see proposition 22.3 of [11]): lemma 2.1. let s be an s-ring over a group g. if c ⊆ g satisfies c ∈ s and ∑ g∈g λgg ∈ s, then for all λ ∈ r the element ∑ g∈c δλg,λg is in s; here δ is the kronecker delta function i.e. δx,y = 0 if x 6= y and is 1 otherwise. s-ring algorithm: suppose that we have a subalgebra h of cg (|g| < ∞) and we wish to find the smallest s-ring that contains h. suppose that we start with a (ring) generating set c1, . . . ,cr for the subalgebra h ⊂ cg. for each ci partition the elements of g according to their coefficients in ci. for each such subset c of this partition add c and c−1 to your set of generators; do this for each i and consider this new set of generators, that we will denote by d1, . . . ,dt. we simplify the set d1, . . . ,dt by eliminating any c-linear dependences. now the di determine and are determined by subsets of g, so that t ≤ |g|. now consider the products didj, 1 ≤ i,j ≤ t. again we partition the terms of didj according to their coefficients, and add in c,c−1, for all sets c in the partition. simplify this generating set using any linear dependences. this describes the basic step. if, after this basic step, one has a c-basis for an s-ring, then we are done; otherwise we repeat the basic step. the process is guaranteed to terminate since |g| < ∞. it is easy to write a program to implement this algorithm (in, say, magma [1]). 3. covers of complete graphs let kn be the complete graph on n vertices. then for λ ∈ n and a graph p, a cover of λkn by p’s is a set t of subgraphs p1, . . . ,pm of kn, each of which is isomorphic to p , and such that every edge of kn occurs λ times in p1, . . . ,pm. we will need the following result in §4. lemma 3.1. any cover t of λk6 by r distinct triangles must have 5|r. further, the cases r = 5, 15 do not happen, and the case r = 10 is unique up to a permutation by an element of s6: {{1, 2, 3},{1, 2, 4},{2, 4, 5},{1, 3, 5},{1, 5, 6},{3, 4, 5},{3, 4, 6},{2, 5, 6},{2, 3, 6},{1, 4, 6}}. the situation r = 20 is where we have all the triangles in k6. proof. let t be a set of r triangles giving the cover of λk6. then counting edges we see that 3r = 15λ, so that 5|r. further, the number of triangles in k6 is 20. if r = 15, then taking the complement of t in the set of all triangles gives the case r = 5. thus we consider the cases r ∈{5, 10}, since if r = 20, then we have all the triangles. case a: r = 5. here λ = 1. then (by permuting if necessary) we can assume that {1, 2, 3},{1, 4, 5}∈ t. considering the edge {2, 4} one is forced (since λ = 1) to have {2, 4, 6} ∈ t. but now one checks that the edge {2, 5} cannot be in any triangle. thus this case does not arise. case b: r = 10. here λ = 2. we consider two cases: case b1: there is some k4 ⊂ k6, all of whose triangles are in t. here we can assume that the vertices of the k4 are 1, 2, 3, 4. then one sees that the only possibilities for extra triangles in t are 63 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 {i, 5, 6}, i = 1, 2, 3, 4 (otherwise we have edges with multiplicities greater than 2). but this makes it impossible to cover all edges and have the multiplicity of the edge {5, 6} be 2. now assume that case b1 does not happen. case b2: there is some k4 ⊂ k6, all but one of whose triangles are in t. so assume that {1, 2, 3},{1, 2, 4},{1, 3, 4} ∈ t. then, considering the edge {2, 4} we see that, up to permuting 5, 6, this forces {2, 4, 5}∈ t. considering the edge {2, 5} forces either (a) {2, 5, 6}∈ t or (b) {2, 3, 5}∈ t. assume that we have (a) {2, 5, 6}∈ t. considering the edge {1, 6} forces {2, 3, 6}∈ t. considering the edge {2, 6} forces {1, 5, 6}∈ t, and that this is the only triangle that can be in t that contains {1, 6}. thus this case cannot happen. if we have (b) {2, 3, 5}∈ t, then considering the edge {2, 6} gives a contradiction. now assume that case b1 and case b2 do not happen. then we can assume that {1, 2, 3}, {1, 2, 4} ∈ t, {1, 3, 4},{2, 3, 4} /∈ t. considering the edge {2, 4} forces {2, 4, 5} ∈ t (up to permuting 5, 6). considering the edge {1, 5} (and using the fact that case b2 does not occur) we must have {1, 3, 5},{1, 5, 6} ∈ t. considering the edge {3, 4} (and using the fact that case b2 does not occur) we must have {3, 4, 5},{3, 4, 6} ∈ t. similarly, considering the edge {5, 6} we must have {2, 5, 6} ∈ t. considering the edge {2, 6} we must have {2, 3, 6}∈ t. this leaves {1, 4, 6} as the remaining triangle. let c3 denote the class of 3-cycles in sn. in general for µ ` n we let cµ denote the class of elements of sn of cycle type µ. to each element (i,j,k) ∈ c3 there is associated the triangle {i,j,k} in k6. we let the following subset of c3 represent the set of triangles in the above lemma: c3 = {(1, 2, 3), (1, 2, 4), (2, 4, 5), (1, 3, 5), (1, 5, 6), (3, 4, 5), (3, 4, 6), (2, 5, 6), (2, 3, 6), (1, 4, 6)}. the set of triangles corresponding to the elements of c3 can be represented as the set of triangles of the hemi-icosahedron h (a polyhedral decomposition of the projective plane). see figure 1, where we have drawn the ten triangles (2-simplices) of h, and the outside edges of these 2-simplices are identified in pairs as usual. 1 2 6 5 4 3 4 5 6 figure 1. the hemi-icosahedron corresponding to c3 the automorphism group of this 2-complex is a6 ≤ s6, and it acts transitively on the ten triangles corresponding to the elements of c3. 64 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 4. the case where c2 is a principal set in this section we assume that c2 is a principal set of a commutative s-ring s. this is the most difficult case. since c2 ∈ s it follows from lemma 4.1 of [5] that z(cg) is a subring of s. it follows from this and the schur-wielandt principle that if c is a principal set of s, then c is contained in some conjugacy class, namely the class of one of its elements. the class c3. we first consider the principal sets c ⊆c3 of s, with the goal of showing c = c3. for µ ` n and α = ∑ g∈g λgg we let αµ = ∑ g∈cµ λgg. lemma 4.1. if c ⊆ c3 ⊂ sn is a principal set of s and ( c · c2 ) (2,1n−2) = λc2, then c determines a cover of λkn by triangles. moreover, we have 3 · |c| = λ · |c2|. proof. we have (i,j,k) · ((i,j) + (j,k) + (i,k)) = (i,j) + (j,k) + (i,k). thus each 3-cycle (i,j,k) ∈ c contributes (i,j) + (j,k) + (i,k) to the product c · c2. further, for each α = (i,j,k) ∈ c3, there are only three β ∈ c2 (namely β ∈ {(i,j), (j,k), (k,i)}) with αβ ∈ c2. since each (i,j) ∈ c2 occurs λ times in c · c2 we have a cover of λkn by triangles, and the first part of the result follows. by counting edges we see that 3 · |c| = λ · |c2|. if n = 6, then lemma 3.1 together with the fact that either c = c−1 or c ∩c−1 = ∅, shows that (up to conjugacy) we have one of: (i) c = c3; (ii) |c| = 20 where c = ∑ α∈c3 α + α −1; (iii) |c| = 10 where c = ∑ α∈c3 α ε(α) with ε(α) ∈{1,−1}; (iv) c = ∑ 1≤i<j<k≤6(i,j,k) ε(i,j,k), for some ε(i,j,k) ∈{1,−1}. we want to show that (i) is the only possibility. if we have (ii), then applying the s-ring algorithm to the algebra generated by z(cs6) and c gives an s-ring that is not commutative. thus (ii) does not happen. suppose that we have (iii), so that c = ∑ α∈c3 α ε(α) with ε(α) ∈{1,−1}. we will use figure 1 to determine a standard orientation (counter clockwise) for each of the triangles of c3, so that, for example the order 1, 2, 3 determines a positive orientation for the triangle {1, 2, 3} in c3. since we have (iii), every (i,j,k) ∈ c has the form αε(α), for some α ∈ c3. each such αε(α) = (i,j,k) ∈ c determines a triangle t(αε(α)) = {i,j,k} of h. in fact αε(α) = (i,j,k) ∈ c determines an orientation for t(αε(α)), this being i 7→ j 7→ k 7→ i. we denote the set of oriented triangles of h determined by c as h(c). further, each oriented triangle determines an orientation on each of its edges. an edge of h will be said to be oppositely-oriented in h(c) if that edge has different orientations in the two triangles of h(c) in which it is contained. let o(c) denote the number of oppositely-oriented edges in h(c). we note that o(c) < 15, since h has 15 edges. 65 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 lemma 4.2. (a) if (i,j,k), (r,s,t) ∈c3, then (i,j,k)(r,s,t) ∈c3 if and only if either (i) (i,j,k) = (r,s,t); or (ii) {i,j,k}∩{r,s,t} = {u,v} has size 2 and the edge u,v in (i,j,k) is oriented differently from that in (r,s,t). (b) two (oriented) triangles of h(c) that share an edge will contribute two elements of c3 to c 2 if and only if their common edge is oriented differently in the two triangles, if and only if the two triangles are oriented the same. in any other case they contribute no elements of c3. (c) ( c 2 −c−1 ) (3,13) is a sum of 2o(c) elements of c3. proof. (a) we may assume that i,j,k,r,s,t ≤ 6, and one checks a small number of cases. (b) now applying lemma 4.2 (a) to c 2 |(3,13) we see that there is the contribution that comes from∑ α∈c α 2. we note that none of these elements α2 of c3 are in c, since c∩c−1 = ∅. any other product comes from a pair α1,α2 ∈ c that satisfy lemma 4.2 (a) (ii). thus α1,α2 determine an orientation of two of the triangles of h that have opposite orientations on the common edge of these two triangles. so considering h(c) we see that two triangles of h(c) that share an edge will contribute two elements of c3 to c 2 if and only if their common edge is oriented differently in the two triangles, if and only if the two triangles are oriented the same. (c) this follows from (b) and its proof. let ω(c) be the subgraph of the 1-skeleton h(c)(1) of h(c) consisting of edges that are oppositelyoriented. let f+,f− be the number of triangles of c3 that are oriented positively or negatively (respectively) in h(c). then f+ + f− = 10 and we may assume that f+ ≥ f− (else replace c by c−1). lemma 4.3. (a) the graph ω(c) contains all the vertices of h(c)(1). (b) o(c) ∈{5, 10}. (c) f+ ≡ f− ≡ o(c) mod 2. in particular, if f− = 0, then o(c) = 12; if f− = 1, then o(c) ∈{9, 11}; if f− = 2, then o(c) ∈{6, 8, 10}; if f− = 3, then o(c) ∈{5, 7, 9}; if f− = 4, then o(c) ∈{6, 8, 10}; if f− = 5, then o(c) ∈{3, 7}; if f− = 6, then o(c) = 6. (d) if o(c) = 5, then in figure 2 we show the two possibilities for c (up to a permutation of 1, . . . , 6) where we have shaded the positively (say) oriented triangles, and where the oppositely oriented edges are drawn dashed. proof. (a) this follows, since every edge of h(c)(1) has degree 5, so that for every vertex v of h(c), at least two adjacent triangles that share the vertex v must have the same orientation in h(c). (b) we know that every principal set contained in c3 has size a multiple of 5. lemma 4.2 (c) shows that 10|o(c). but o(c) < 15 then gives the result. (c) changing the orientation of any triangle of h changes the orientation of the three edges of that triangle. it follows that f+ ≡ f− ≡ o(c) mod 2. the rest follows by looking at each particular case. (d) we use (c) to check this. 66 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 1 2 6 5 4 3 4 5 6 1 2 6 5 4 3 4 5 6 figure 2. orientations of h(c) giving o(c) = 5. thus case (iii) has been reduced to showing that the two s-rings determined by lemma 4.3 (d) do not give commutative s-rings. one checks this latter fact using the s-ring algorithm. this concludes consideration of (iii). now assume that we have (iv), so that any subset of c3 that is a principal set has size 20. thus c−1 6= c is also a principal set. thus we have c = ∑ 1≤i<j<k≤n (i,j,k)ε(i,j,k) where ε(i,j,k) ∈{1,−1}. lemma 4.4. let a,b,c ∈c3. then at most one of the eight products a±1b±1,b±1a±1 can be equal to c. proof. (a) we may assume that c = (1, 2, 3), and then one only has to check the cases where a,b ∈ c3 ∩s4. for c = (i,j,k) ∈ c we now wish to determine the coefficient of c in c 2 as a polynomial function of the ε(r,s,t). first assume that ε(i,j,k) = 1. suppose a = (r,s,t)δ1, b = (u,v,w)δ2, are distinct where δ1 = δ1,i,j,k,r,s,t,u,v,w,δ2 = δ2,i,j,k,r,s,t,u,v,w ∈ {1,−1}, r < s < t,u < v < w, and c is one of ab,ba. then by lemma 4.4 (a) the coefficient of c in (a + b)2 is 1. thus if a,b ∈ c, then they will contribute 1 to the coefficient of c in c 2 . so when a = (r,s,t)ε(r,s,t),b = (u,v,w)ε(u,v,w) ∈ c are distinct, then they will contribute 1 to the coefficient of c in c 2 if and only if ε(r,s,t) = δ1,ε(u,v,w) = δ2. thus we have proved (i) of the following result: lemma 4.5. (i) when c = (i,j,k) ∈ c, the coefficient of c = (i,j,k) in c 2 coming from the pair (r,s,t)ε(r,s,t), (u,v,w)ε(u,v,w) is (1 + δ1ε(r,s,t)) 2 · (1 + δ2ε(u,v,w)) 2 . (ii) when c = (i,j,k) /∈ c, the coefficient of c in c 2 coming from the pair (r,s,t)ε(r,s,t), (u,v,w)ε(u,v,w is (1 −δ1ε(r,s,t)) 2 · (1 − δ2ε(u,v,w)) 2 . 67 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 (iii) for any c = (i,j,k)ε(i,j,k), 1 ≤ i < j < k ≤ 6, the coefficient of c in c 2 coming from the pair a = (r,s,t)ε(r,s,t),b = (u,v,w)ε(u,v,w is (1 + δ1ε(r,s,t)ε(i,j,k)) 2 · (1 + δ2ε(u,v,w)ε(i,j,k)) 2 . proof. the proof of (ii) is similar to what we did above for (i), and (iii) is a restatement of (i) and (ii). now let p be a polynomial ring over q with generators x(i,j,k) where 1 ≤ i < j < k ≤ 6 and define, for c = (i,j,k)ε(i,j,k),a = (r,s,t)ε(r,s,t),b = (u,v,w)ε(u,v,w),a 6= b, where c is one of ab,ba. we define the polynomial s(a,b,c) = (1 + δ1x(r,s,t)x(i,j,k)) 2 · (1 + δ2x(u,v,w)x(i,j,k)) 2 . here δ1 = δ1,i,j,k,r,s,t,u,v,w,δ2 = δ2,i,j,k,r,s,t,u,v,w ∈ {1,−1}, r < s < t,u < v < w, as in the above. thus the x(i,j,k) correspond to the ε(i,j,k). we let eε : p → q be the evaluation map x(i,j,k) 7→ εi,j,k. for each c = (i,j,k)ε(i,j,k), i < j < k, we let s(c) be the sum of all s(a,b,c) where a = (r,s,t)ε(r,s,t),b = (u,v,w)ε(u,v,w) ∈ c are distinct and one of ab,ba is c. thus s(c) is a sum of nine of the s(a,b,c)s. it is clear that if c = (i,j,k)ε(i,j,k), i < j < k, and c′ = (i′,j′,k′)ε(i ′,j′,k′), i′ < j′ < k′, then eε(s(c)) = eε(s(c′)), since c is a principal set. we now define the ideal i whose generators are all s(c)−s(c′), for c,c′ as above, and all x(i,j,k)2−1 for 1 ≤ i < j < k ≤ 6. since we can always conjugate c so that ε(1, 2, 3) = 1 = ε(4, 5, 6) we also let x(1, 2, 3) − 1,x(4, 5, 6) − 1 be in i. now if there is such a subset c, then the ideal i will not be equal to p . however a gröbner basis calculation ([1]) shows that i = p , and so (iv) is not possible. thus we now have: proposition 4.6. any commutative s-ring over s6 that contains c2 as a principal set, also contains c3 as a principal set. the class c(22,12). we let c2,2 denote c(22,12). here we have: lemma 4.7. if c ⊂c2,2 satisfies c ∈ s, where s is a commutative s-ring over sn containing c2 as a principal set, then c determines a cover of λkn by pairs of disjoint edges. proof. this follows from [(i,j)(k,m) · c2](2,1n−2) = [(i,j)(k,m) · [(i,j) + (k,m) + · · · ]](2,1n−2) = (i,j) + (k,m), together with the fact that c2 is a principal set, so that [c · c2](2,1n−2) = λc2. in the above situation we have: 2|c| = λ|c2|. for each of n = 4,λ = 1; n = 5,λ = 1, and n = 5,λ = 2, one can check that there is a unique such cover. for n = 6 we have 2|c| = 15λ, so that λ must be even, and 15 divides |c|. we note that |c(22,12)| = 45, so that we can assume |c| < 45. if |c| = 30, then we replace c by c(22,12) \c, so that we can assume |c| = 15. we now show 68 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 lemma 4.8. for c ⊂c(22,12), |c| = 15, as above, the only possibilities for c are those conjugate to the following element (or its complement in c(22,12)) (1, 2)(3, 4) + (1, 2)(5, 6) + (3, 4)(5, 6) + (1, 3)(2, 5) + (1, 3)(4, 6) + (2, 5)(4, 6) + (1, 5)(2, 4) + (1, 5)(3, 6) + (2, 4)(3, 6) + (1, 6)(2, 3) + (1, 6)(4, 5) + (2, 3)(4, 5) + (1, 4)(2, 6) + (1, 4)(3, 5) + (2, 6)(3, 5). the stabilizer of this element (under conjugation by elements of s6) is h120 = 〈(1, 4)(3, 5), (1, 4, 6, 2, 5, 3)〉, so that [s6 : h120] = 6, and there are 6 conjugates of this element. further, the s-ring generated by z(cg) and the above element is s(s6,h120). proof. the last two paragraphs are simple computations once we have the first part. for the first part, let ken denote the graph with vertices the edges {i,j} of kn, and where we have an edge of ken whenever the corresponding vertices (edges of kn) are disjoint. thus each edge {i,j}−{k,m} of ken corresponds to an element (i,j)(k,m) ∈ c(22,1n−4), so that we can refer to an edge of ken by that element. now let c ∈ c(22,12) determine a cover of 2k6 by subgraphs isomorphic to k2 ∪ k2. then every vertex of ke6 is in exactly two edges. thus c determines a set of disjoint cycles γ1, . . . , γr in k e 6. each cycle γi, i ≤ r, has length at least three. we will think of the γi as subsets of c. we have: lemma 4.9. let n = 6. (i) if {i,j}−{k,m}−{r,s} represent three consecutive vertices in a cycle γu, then (up to a permutation) we have one of (a) {1, 2}−{3, 4}−{5, 6}; (b) {1, 2}−{3, 4}−{1, 5}. (ii) if {i,j}−{k,m}−{r,s}−{u,v} represent four consecutive vertices in a cycle γu, then (up to a permutation) we have one of (a) {1, 2}−{3, 4}−{1, 5}−{2, 6}; (b) {1, 2}−{3, 4}−{1, 5}−{2, 4}; (c) {1, 2}−{3, 4}−{1, 5}−{3, 4}; (d) {1, 2}−{3, 4}−{5, 6}−{3, 6}. further, in a path γu of length at least 4 with distinct non-consecutive edges {i,j}− {k,m} and {i′,j′}−{k′,m′}, the product (i,j)(k,m) · (i′,j′)(k′,m′) is not in c(22,12). (iii) if γi, γj are distinct cycles, then no term of γi · γj is in c(22,12). (iv) if γi has length greater than 4, then the number of elements of the form αβ, (α,β ∈ γi), that are in c(22,12) is exactly |γi|, and none of these elements are in c. if γi has length 4, then the number of elements of the form αβ, (α,β ∈ γi), that are in c(22,12) is 2, and none of these elements are in c. proof. (i) in such a path {i,j}−{k,m}−{r,s} we have |{r,s}∩{i,j}| = 0, 1, which give the two cases. (ii) the first part is easy to check using (i). if all the vertices of {i,j}−{k,m},{i′,j′}−{k′,m′} are distinct, then one sees that (i,j)(k,m) · (i′,j′)(k′,m′) is not in c(22,12). (iii) if γi, γj are distinct cycles with α = α1α2 ∈ γ1,β = β1β2 ∈ γj,αk,βk ∈ c2,k = 1, 2, then {α1,α2}∩{β1,β2} = ∅, since γi, γj do not share a vertex. thus no element of γi · γj is in c(22,12). 69 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 (iv) this now follows from (ii), which shows that only products of the form (i,j)(r,s) · (r,s)(u,v) will produce elements of c2,2. if we have a cycle γi of size greater than three, then lemma 4.9 (iv) shows that all cycles γj have size greater than 3. the same result shows that no cycle has length four (since then we would have a principal set in c(22,12) of size less than 15). thus (c 2 )(22,12) = 0c + d, where |d| = 15, c ∩d = ∅. we then similarly have (d 2 )(22,12) = 0d + c. thus c(22,12) \ (c ∪d) has size 15, and all of its cycles must have size 3. thus we may assume that all cycles γi of c have size 3. in this case it is easy to see that the components γi of c are as shown in figure 3 (up to conjugacy): (1,2) (5,6) (3,4) (1,3) (4,6) (2,5) (1,4) (2,6) (3,5) (1,5) (3,6) (2,4) (1,6) (4,5) (2,3) figure 3. one now finds that the s-ring determined by c and z(cs6) is s(s6,h120), and that the complement of c in c(22,12) is also a principal set (of size 30). thus we cannot have any principal sets c in s with |c| = 15 and with cycles γi of length greater than three. this completes the proof of lemma 4.8 and our discussion of the case λ = 2. for n = 6 the element c(22,12) gives a λ = 6 cover, thus if we have a cover c with λ = 4 then c(22,12) −c has λ = 2, and the s-ring generated by z(cs6) and c is the same as the s-ring generated by z(cs6) and c(22,12) −c. thus we have: proposition 4.10. any commutative s-ring s over s6 that contains c2 as a principal set, also contains c3 as a principal set, and either c2,2 is a principal set or s contains a conjugate of s(s6,h120). lastly, any principal set of s that is properly contained in c2,2 is an orbit of one of its elements under the action of a conjugate of h120. the c(23) case lemma 4.11. the only non-empty and proper subsets c of c(23) ⊂ s6 that satisfy( c · c2,2 ) (2,14) = λc2 for some λ ≥ 1 are conjugates of (i) c1 = {(1, 2)(3, 4)(5, 6), (1, 6)(2, 3)(4, 5), (1, 5)(2, 4)(3, 6), (1, 4)(2, 6)(3, 5), (1, 3)(2, 5)(4, 6)}; or (ii) c2 = c(23) \c1. further, any conjugate of (i) or (ii) generates (together with z(cs6)) an s-ring conjugate to s(s6,h120). 70 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 proof since |c(23)| = 15 we may assume that |c| ≤ 7 and that (1, 2)(3, 4)(5, 6) ∈ c. for α ∈ c(23) one sees that [α · c(22,12)](2,14) is a sum of three elements. thus 3|c| = 15λ i.e. |c| = 5λ. since |c| ≤ 7 we see that λ = 1 and |c| = 5. it is now easy to see that up to a permutation we can assume that (1, 6)(2, 3)(4, 5) ∈ c. then there are two cases: (a) (1, 4)(2, 6)(3, 5) ∈ c; and (b) (1, 4)(2, 5)(3, 6) ∈ c. either case quickly gives a unique c of size 5, these cs being conjugate. one checks that c1 and c2 are principal sets of s(s6,h120). this gives the result. this gives: proposition 4.12. let s be a commutative s-ring over s6 that contains c2 as a principal set. then either (i) c3,c(23) and c2,2 are principal sets of s, or (ii) s contains a conjugate of s(s6,h120). lastly, any principal set of s that is properly contained in c(23) is an orbit of one of its elements under the action of a conjugate of h120. the c(3,2,1) case let s be a commutative s-ring over sn,n ≥ 5, that contains c2,c3 as principal sets. for a principal set c ⊆c(3,2,1n−5) ⊂ sn we define sets di,j ⊂c3 ⊂ sn by c = ∑ 1≤i<j≤n (i,j)di,j. since [(i,j)(r,s,t) ·c3](2,1n−2) = (i,j), and c2 is a principal set of s we see that there is some λ ∈ n such that ( c · c3 ) (2,1n−2) = λc2. further, from the definition of the di,j we see that λ = |di,j| for all 1 ≤ i < j ≤ n. since [(i,j)(r,s,t) · c2](3,1n−3) = (r,s,t) and c3 is a principal set of s we also have µ ∈ n such that( c · c2 ) (3,1n−3) = µc3. but we also have ( c · c2 ) (3,1n−3) = ∑ 1≤i<j≤n di,j. thus from µc3 = ∑ 1≤i<j≤n di,j we see that for all 1 ≤ i < j ≤ n we have( n 2 ) |di,j| = ( n 2 ) λ = µ|c3| = µ n(n− 1)(n− 2) 3 . from this we obtain lemma 4.13. λ = |di,j| and 3 · |di,j| = 2µ(n− 2). since any (r,s,t) ∈ di,j must satisfy i,j /∈{r,s,t} we see that |di,j| ≤ (n− 2)(n− 3)(n− 4) 3 . 71 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 when n = 6 this gives |di,j| ≤ 8. but if n = 6, then lemma 4.13 shows that 3 divides µ so that |di,j| = 2 · µ 3 · 4 ≥ 8. it follows from these two equations that |di,j| = 8 for all 1 ≤ i < j ≤ n = 6, so that di,j is maximal, i.e. di,j = {(r,s,t)±1 : 1 ≤ r < s < t ≤ 6, i,j /∈{r,s,t}}, and c = c(3,2,1). thus we now have shown proposition 4.14. let s be a commutative s-ring over s6 that contains c2 as a principal set. then c(3,2,1) is a principal set of s, and either (i) c3,c(23),c(3,2,1) and c2,2 are principal sets of s, or (ii) s contains a conjugate of s(s6,h120). we note that in (ii) the only non-trivial conjugacy classes that do not split are c2,c3 and c(3,2,1). the c(4,12) case let c ⊆c(4,12) be a principal set. then the fact that( (i,j,k,m) · c3 ) (2,14) = (i,j) + (j,k) + (k,m) + (m,i), shows that c determines a cover of λkn by 4-cycles. thus, if [c · c3](2,14) = λc2, then 4 · |c| = 15λ. thus 15 divides |c|. now there are twelve α ∈ c(4,12) such that (1, 2, 3, 4)α is a 3-cycle. thus considering ( c · c4 ) (3,13) we see that (since |c3| = 40) there is some µ ≥ 1 such that: 12 · |c| = 40µ, (since c3 is a principal set) which shows that 10 divides |c|. thus |c| is divisible by 30. note that |c(4,12)| = 90. if |c| = 60, then we replace c by c(4,12) \c, so as to have |c| = 30. now if c−1 6= c, then c(4,12) \ (c ∪ c−1) is a principal set (since it has size 30) that is its own inverse. thus we may assume that the principal set c satisfies |c| = 30 and c = c−1. write c = α1 + α −1 1 + · · · + α15 + α −1 15 , so that (c 2 )(22,12) = 2α 2 1 + 2α 2 2 + · · · + 2α215, which shows that {α21,α22, . . . ,α215}⊂c(22,12) is a principal set of s of size 15. thus lemma 4.8 shows that such a set is conjugate to the set of 15 elements given in that lemma. call this set of 15 elements w15. we note that the map ι2 : c(4,12) →c(22,12), α 7→ α2, is a 2 to 1 surjection, where (ι2)−1((i,k)(j,m)) = {(i,j,k,m), (i,m,k,j)}, is an inverse pair. since c−1 = c we see that c is completely determined by ι2(c), which is conjugate to w15. thus c must be a conjugate of {(1, 2, 5, 4), (1, 4, 5, 2), (1, 4, 3, 6), (2, 6, 5, 4), (1, 3, 4, 5), (1, 4, 6, 5), (2, 5, 3, 4), (4.1) (2, 3, 4, 6), (3, 5, 4, 6), (1, 4, 2, 3), (1, 2, 4, 6), (1, 5, 4, 3), (3, 6, 4, 5), (2, 3, 6, 5), (1, 6, 2, 5), (2, 4, 5, 6), (1, 3, 5, 6), (2, 4, 3, 5), (1, 3, 6, 2), (1, 2, 3, 5), (1, 6, 3, 4), (2, 5, 6, 3), (1, 5, 3, 2), (1, 5, 6, 4), (1, 2, 6, 3), (1, 3, 2, 4), (2, 6, 4, 3), (1, 6, 4, 2), (1, 5, 2, 6), (1, 6, 5, 3)}. now one finds that the s-ring generated by z(cs6) and c is s(s6,h120). using the fact that any conjugate of w15 that is not equal to w15 does not commute with w15, this now easily gives: 72 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 proposition 4.15. let s be a commutative s-ring over s6 that contains c2 as a principal set. then either (i) c3,c(23),c(3,2,1),c(4,12) and c2,2 are principal sets of s, or (ii) s contains a conjugate of s(s6,h120). lastly, any principal set of s that is properly contained in c(42,12) is an orbit of one of its elements under the action of a conjugate of h120, and is either a conjugate of (4.1) or of its complement in c(4,12). the c(4,2) case there is a natural bijection c4,2 →c4, (i,j,k,m)(r,s) 7→ (i,j,k,m). so that the image of the set of elements (4.1) of c(4,12) is {(1, 2, 5, 4)(3, 6), (1, 4, 5, 2)(3, 6), (1, 4, 3, 6)(2, 5), (2, 6, 5, 4)(1, 3), (1, 3, 4, 5)(2, 6), (4.2) (1, 4, 6, 5)(2, 3), (2, 5, 3, 4)(1, 6), (2, 3, 4, 6)(1, 5), (3, 5, 4, 6)(1, 2), (1, 4, 2, 3)(5, 6), (1, 2, 4, 6)(3, 5), (1, 5, 4, 3)(2, 6), (3, 6, 4, 5)(1, 2), (2, 3, 6, 5)(1, 4), (1, 6, 2, 5)(3, 4), (2, 4, 5, 6)(1, 3), (1, 3, 5, 6)(2, 4), (2, 4, 3, 5)(1, 6), (1, 3, 6, 2)(4, 5), (1, 2, 3, 5)(4, 6), (1, 6, 3, 4)(2, 5), (2, 5, 6, 3)(1, 4), (1, 5, 3, 2)(4, 6), (1, 5, 6, 4)(2, 3), (1, 2, 6, 3)(4, 5), (1, 3, 2, 4)(5, 6), (2, 6, 4, 3)(1, 5), (1, 6, 4, 2)(3, 5), (1, 5, 2, 6)(3, 4), (1, 6, 5, 3)(2, 4)}. this case then follows by following the proof of the c4 case just completed. this then gives: proposition 4.16. let s be a commutative s-ring over s6 that contains c2 as a principal set. then either (i) c3,c(23),c(3,2,1),c(4,12),c(4,2) and c2,2 are principal sets of s, or (ii) s contains a conjugate of s(s6,h120). lastly, any principal set of s that is properly contained in c(42,2) is an orbit of one of its elements under the action of a conjugate of h120, and is either a conjugate of (4.2) or of its complement in c(4,2). the c(32) case consider a principal set c ⊆ c(32). note that if α = ab ∈ c(32) where a,b ∈ c3, and if c ∈ c3 with α · c ∈c3, then c ∈{a−1,b−1}. let c = ∑r i=1 αi,αi = aibi ∈c(32),ai,bi ∈c3. then ( c · c3 ) (3,13) = r∑ i=1 (ai + bi). since c3 is a principal set we see that r∑ i=1 (ai + bi) = λc3. since |c3| = 40 this gives 20λ = r. now r ≤ |c(32)| = 40, so that we have λ ≤ 2. if λ = 2, then r = 40 and c = c(32). so assume that λ = 1,r = |c| = 20. we may also assume (by conjugating) that (1, 2, 3)(4, 5, 6) ∈ c. we have [c · c3](3,13) = 1c3, thus for each α ∈ c3 there is a unique α∗ ∈ c3 such that αα∗ ∈ c. since (1, 2, 3)(4, 5, 6) ∈ c we cannot have (1, 3, 2)(4, 5, 6) ∈ c, and so we must have (1, 3, 2)(4, 6, 5) ∈ c. thus c = c−1. 73 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 for α = ab ∈ c(32),a,b ∈ c3, we let s(α) = {ab,a−1b,ab−1,a−1b−1}. then the s(α),α ∈ c3, partition c(32) into ten subsets s1, . . . ,s10 of size 4. then any set c ⊂ c(32) as above is a union of an inverse pair {αi,α−1i }⊂ si from each si. there are thus 2 10 possible such c’s, but only 29 if one assumes that (say) (1, 2, 3)(4, 5, 6) ∈ c of these 29 choices for c one finds that only twelve satisfy ( c 2 ) (3,3) = λc + µc(32) \c for some λ,µ ≥ 0. these twelve elements are in three conjugacy classes. of the representatives from these three conjugacy classes one checks that the subalgebra of cs6 generated by such a c and z(cs6) is only an s-ring if (up to conjugacy) we have one of the following two cases. c1 = (1, 3, 5)(2, 4, 6) + (1, 5, 3)(2, 6, 4) + (1, 6, 4)(2, 5, 3) + (1, 4, 5)(2, 3, 6) (4.3) + (1, 2, 6)(3, 5, 4) + (1, 5, 2)(3, 4, 6) + (1, 3, 4)(2, 5, 6) + (1, 6, 5)(2, 4, 3) + (1, 6, 3)(2, 4, 5) + (1, 3, 6)(2, 5, 4) + (1, 3, 2)(4, 6, 5) + (1, 5, 6)(2, 3, 4) + (1, 4, 6)(2, 3, 5) + (1, 5, 4)(2, 6, 3) + (1, 2, 5)(3, 6, 4) + (1, 4, 2)(3, 6, 5) + (1, 2, 3)(4, 5, 6) + (1, 2, 4)(3, 5, 6) + (1, 4, 3)(2, 6, 5) + (1, 6, 2)(3, 4, 5); or c2 = (1, 4, 5)(2, 3, 6) + (1, 2, 6)(3, 5, 4) + (1, 5, 2)(3, 4, 6) + (1, 6, 4)(2, 3, 5) + (1, 6, 5)(2, 3, 4) + (1, 6, 3)(2, 4, 5) + (1, 4, 3)(2, 5, 6) + (1, 3, 6)(2, 5, 4) + (1, 5, 6)(2, 4, 3) + (1, 3, 2)(4, 6, 5) + (1, 5, 4)(2, 6, 3) + (1, 5, 3)(2, 4, 6) + (1, 4, 6)(2, 5, 3) + (1, 2, 5)(3, 6, 4) + (1, 3, 4)(2, 6, 5) + (1, 4, 2)(3, 6, 5) + (1, 2, 3)(4, 5, 6) + (1, 2, 4)(3, 5, 6) + (1, 3, 5)(2, 6, 4) + (1, 6, 2)(3, 4, 5). however in each of these two cases the s-ring that one obtains is just a conjugate of s(s6,h120) again. thus we now have: proposition 4.17. let s be a commutative s-ring over s6 that contains c2 as a principal set. then either (i) c3,c(23),c(3,2,1),c(4,12),c(4,2), c2,2 and c(32) are principal sets of s, or (ii) s contains a conjugate of s(s6,h120). lastly, any principal set of s that is properly contained in c(32) is an orbit of one of its elements under the action of a conjugate of h120, and is either a conjugate of one of the elements shown in (4.3) or of its complement in c(32). the c(5,1) case consider a principal set c ⊆c5 = c(5,1). now there are five elements α ∈c2 such that (1, 2, 3, 4, 5)α ∈ c(3,2,1); since c(3,2,1) is a principal set we see that 5 · |c| = λ · |c(3,2,1)| = 120λ. this gives |c| = 24λ. also, there are five α ∈ c3 such that (1, 2, 3, 4, 5)α ∈ c(3,13). since c(3,13) is a principal set we see that 5 · |c| = µ · |c(3,13)| = 40µ. case 1: |c| = 24. here λ = 1,µ = 3, and so we have (i) [c · c2](3,2,1) = c(3,2,1) and (ii) [c · c3](3,13) = 3c(3,13). let α1, . . . ,α72 ∈c(5,1) represent the inverse pairs in c(5,1), so that c(5,1) = ∪72i=1{αi,α −1 i }. assume first that c−1 = c. then we can write c = ∑72 i=1 xi(αi + α −1 i ). here xi = 0, 1 satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 72; (b) ∑72 i=1 xi = 12; (c) (∑72 i=1 xi(αi + α −1 i ) · c2 ) (3,2,1) = c(3,2,1); (d) (∑72 i=1 xi(αi + α −1 i ) · c3 ) (3,13) = 3c3. 74 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 if we now think of the xi as indeterminates in the polynomial ring r = q[x1, . . . ,x72], then each of (a)–(d) gives relations satisfied by the xi. let i be the ideal of r generated by these relations. finding a gröbner basis for i ([1]) we see that all but five of the xi are determined (in fact by a conjugacy we can assume that one of these five is equal to zero). this leaves a small number of possibilities for c, and one checks that they are all equal to a conjugate of the h120 orbit of some element of c(5,1). further each such element, together with the class sums, generates an s-ring that is just some conjugate of s(s6,h120). this does the case |c| = 24,c = c−1. so now assume |c| = 24,c−1 6= c. then we write c = ∑72 i=1 x2i−1αi + x2iα −1 i . here xi = 0, 1 satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 144, and x2i−1x2i = 0 for 1 ≤ i ≤ 144; (b) ∑144 i=1 xi = 24; (c) (∑72 i=1(x2i−1αi + x2iα −1 i ) · c2 ) (3,2,1) = c(3,2,1); (d) (∑72 i=1(x2i−1αi + x2iα −1 i ) · c3 ) (3,13) = 3c3. finding the ideal of r = q[x1, . . . ,x144] determined by these equations gives no solutions. the above shows that if there is a c ⊂ c(5,1), |c| = 24, then the s-ring contains a conjugate of s(s6,h120), and a principal set of size 24 in c(5,1) is an orbit under the same conjugate of h120. case 2: |c| = 48. here λ = 2,µ = 6, and we aim to show that no such set exists. first note that if c 6= c−1, then d = c(5,1)\(c∪c−1) satisfies d = d−1, |d| = 48, and d is in the s-ring. if d is a union of disjoint sets of size 24, then these sets are conjugates of the sets of size 24 determined in case 1. one checks, however, that no two of these sets (there are six of them) commute, and so we see that d must also be a principal set. thus we can assume that c = c−1. then we can write c = ∑72 i=1 xi(αi + α −1 i ). here, as in case 1, the xi = 0, 1 satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 72; (b) ∑72 i=1 xi = 24; (c) (∑72 i=1 xi(αi + α −1 i ) · c2 ) (3,2,1) = 2c(3,2,1); (d) (∑72 i=1 xi(αi + α −1 i ) · c3 ) (3,13) = 6c3. let i be the ideal of r = q[x1, . . . ,x72] generated by the polynomials determined by these relations. finding a gröbner basis for i ([1]) we see that all but twelve of the xi are determined (again by a conjugacy we can assume that one of these twelve is equal to zero). choosing values in {0, 1} for these eleven variables, one checks the 211 cases, and one finds that there are only 30 (non-trivial) elements c (any other case gives i = r), and for each of these 30 cases one finds those such that ( c 2 ) (3,13) has the form mc3,m ≥ 0. there are 10 of these, and one finds that they are all conjugate. thus, if c is one of these, then we check that the s-ring generated by z(cs6) and c is commutative and has dimension 34, but does not contain c2 as a principal set. (note: each of these ten conjugate cs does give a commutative s-ring containing c2, however c2 partitions as {(1, 2), (3, 4), (5, 6)} and the complement; see corollary 1.1 (iii).) thus there are no principal sets of size 48 in c(5,1) if c2 is a principal set. case 3: |c| = 72. here λ = 3,µ = 9. assume first that c = c−1. then we can write c = ∑72 i=1 xi(αi + α −1 i ). here, as in the above, xi = 0, 1, satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 72; (b) ∑72 i=1 xi = 36; 75 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 (c) (∑72 i=1 xi(αi + α −1 i ) · c2 ) (3,2,1) = 3c(3,2,1); (d) (∑72 i=1 xi(αi + α −1 i ) · c3 ) (3,13) = 9c3; (e) (∑72 i=1 xi(αi + α −1 i ) · c4 ) (2,14) = 24c2; (f) let ( c 2 ) (3,13) = ∑ α∈c3 xαα, where xα ∈ r = q[x1, . . . ,x72]. then we have xα = xβ for all α,β ∈c3, since c3 is a principal set. here (e) is obtained in the same way as (c) and (d). also, (f) is a consequence of the fact that c3 is a principal set of s. constructing the ideal i (including all xα − xβ as in (f)) again one finds that there are 8 variables that determine the rest (7 if one uses conjugacy to set one of them to be zero). choosing values in {0, 1} for these variables gives 27 cases to consider, of which all but eleven give i = r. these eleven cs are in two conjugacy classes of sizes 1 and 10. the element c1, representing the single conjugacy class, is the orbit of some element of c(5,1) under the action of a6. let c2 be one of the ten conjugate elements. then there is a subgroup h36 of size 36 such that the orbits of h36 on c(5,1) are o1,o2,o3,o4 where |oi| = 36, i = 1, 2, 3, 4. further, these sets oi can be numbered so that c1 = o1 ∪o2, c2 = o1 ∪o3. a representative for h36 up to conjugacy is h36 = 〈(1, 6)(2, 3, 4, 5), (1, 2, 5, 4)(3, 6)〉. we note that if one uses o1 ∪o4, then one obtains a commutative s-ring of dimension 26 that contains {(1, 3), (3, 5), (1, 5), (2, 6), (4, 6), (2, 4)} and its complement in c2, as principal sets see corollary 1.1 (iii). one finds that each of the s-rings 〈z(cs6),c1〉,〈z(cs6),c2〉 has dimension 12 (recall that z(cs6) has dimension 11) where ci,c(5,1) \ ci, i = 1, 2, are principal sets of these s-rings. this concludes the case where |c| = 72,c = c−1. we will let c36 denote the element c2 in what follows, and we will denote the s-ring 〈z(cs6),c36〉 by s36. now if c 6= c−1, then we can write c = ∑72 i=1 x2i−1αi + x2iα −1 i . here xi = 0, 1 satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 144 and x2i−1x2i = 0 for 1 ≤ i ≤ 72; (b) ∑72 i=1 xi = 72; (c) (∑72 i=1(x2i−1αi + x2iα −1 i ) · c2 ) (3,2,1) = 3c(3,2,1); (d) (∑72 i=1(x2i−1αi + x2iα −1 i ) · c3 ) (3,13) = 9c3; (e) (∑72 i=1(x2i−1αi + x2iα −1 i ) · c4 ) (2,14) = 24c2; (f) let ( c 2 ) (3,13) = ∑ α∈c3 xαα, where xα ∈ r = q[x1, . . . ,x72]. then we have xα = xβ for all α,β ∈c3, since c3 is a principal set. one then finds that the ideal determined by (a)–(f) is the whole ring, so that there are no solutions in this situation. now if c ( c(5,1) is a principal set of s with |c| > 72, then there are certainly principal sets d of s with |d| < 72. however in the above considerations of cases 1,2 we have shown that the s-ring generated by d and z(cs6) does not have a principal set of size greater than 72. thus the situation |c| > 72 does not occur, and we have now proved: proposition 4.18. let s be a commutative s-ring over s6 containing c2. then any principal set c ⊂c(5,1) of s is either a conjugate of c36, or is the orbit of one of its elements under the action of a6 or of a conjugate of h120. 76 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 the c(6) case consider a principal set c ( c(6). now there are six elements α ∈c(4,12) such that (1, 2, 3, 4, 5, 6)α ∈ c(3,13); since c(3,13) is a principal set we see that 6 · |c| = λ1 · |c(3,13)| = 40λ1. this gives (i) 3|c| = 20λ1. similarly, by considering (ii) ( c · c(5,1) ) (2,14) = λ2c2, we see that 2|c| = 5λ2; (iii) ( c · c(3,2,1) ) (3,13) = λ3c3, we see that 3|c| = 10λ3; (iv) ( c · c2,2 ) (3,2,1) = λ4c(3,2,1), we see that 3|c| = 20λ4. one sees that |c| is divisible by 20. also |c(6)| = 120. let αi, i = 1, . . . , 60 be representatives for the inverse pair sets in c(6). case 1: |c| = 20. here λ1 = 3,λ2 = 8,λ3 = 6,λ4 = 3. first assume that c = c−1. then we can write c = ∑60 i=1 xi(αi + α −1 i ). here, as in the above, xi = 0, 1 satisfy (a) x2i −xi = 0 for 1 ≤ i ≤ 60; (b) ∑60 i=1 xi = 10; (c) relations for each of (i)–(iv) above; (d) let ( c 2 ) (3,13) = ∑ α∈c3 xαα, where xα ∈ r = q[x1, . . . ,x10]. then we have xα = xβ for all α,β ∈c3, since c3 is a principal set. (e) let ( c 2 · c2,2 ) (3,13) = ∑ α∈c3 xαα, where xα ∈ r = q[x1, . . . ,x10]. then we have xα = xβ for all α,β ∈c3, since c3 is a principal set. constructing the ideal i (including all xα−xβ as in (d),(e)) again one finds that there are 5 variables that determine the rest (4 if one uses conjugacy to set one of them to be zero). looking at the 24 cases one finds that there are 5 possibilities for c. these are all in a single conjugacy class, so we need only consider one of them, c say. one finds that the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). this does the case where c = c−1. now if c 6= c−1, then we can write c = ∑60 i=1 x2i−1αi + x2iα −1 i . using the same tests as in the c = c−1 case above one creates an ideal that has nine variables that determine the rest. enumerating the various possibilities for c gives ten non-conjugate elements. one checks that each of these generates cs6 as an s-ring. this shows that this case cannot occur. we further note lemma 4.19. if c1,c2 ⊆ c6 are distinct principal sets of a commutative s-ring over s6 that contains c2 as a principal set, then at most one of c1,c2 can have size 20. proof. if |c1| = 20, then the above shows that c2 is a conjugate of c1; but there are only six such conjugates and it is easy to check that no two such distinct commute. case 2: |c| = 40. here λ1 = 6,λ2 = 16,λ3 = 12,λ4 = 6. first assume that c = c−1. then we can write c = ∑60 i=1 xi(αi + α −1 i ). applying the same ideal calculation as in the |c| = 20 case one obtains an ideal having 4 variables that determine the rest. enumerating the various possibilities for c gives four elements, all of them conjugate to each other. considering one of them, c say, one finds that the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). this does the case where c = c−1. repeating the above for the situation where |c| = 40,c 6= c−1, one finds that there are no solutions. case 3: |c| = 60. here λ1 = 9,λ2 = 24,λ3 = 18,λ4 = 9. 77 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 first assume that c = c−1. then we can write c = ∑60 i=1 xi(αi + α −1 i ). applying the same ideal calculation as in the |c| = 20 case one obtains an ideal having 5 variables that determine the rest. enumerating the various possibilities for c gives six possibilities for c, and there are two conjugacy classes of such elements. if c say, represents either of these classes, one finds that the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). this does the case where c = c−1. repeating the above for the situation where |c| = 60,c 6= c−1, one finds that there are no solutions. the cases |c| > 60 are dealt with as in the situation |c| > 72 of case 2. this concludes consideration of all conjugacy classes of s6 where s has c2 as a principal set. the following is mostly a summary of what we have done: proposition 4.20. let s be a commutative s-ring over s6 containing c2 as a principal set. then (i) c(3,13) and c(3,2,1) are principal sets of s. (ii) if c ( c2,2 is a principal set of s, then |c| = 15 or |c| = 30, and c is either a conjugate of the element shown in lemma 4.8, or is c2,2 − c for such a set of size 15. there are six conjugates of each such set c. no two distinct such conjugates commute. the s-ring generated by z(cs6) and c is s(s6,h120). (iii) if c ( c(23) is a principal set of s, then |c| = 5 or |c| = 10. if |c| = 5, then c is conjugate to the element shown in lemma 4.11 (i). there are six conjugates of each such set c. no two distinct such conjugates commute. the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). (iv) if c ( c(4,12) is a principal set of s, then |c| = 30 or |c| = 60. if |c| = 30, then c is a conjugate of the element shown in (4.1), otherwise it is the complement in c(4,12) of such an element. there are six conjugates of each such set c. no two distinct such conjugates commute. the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). (v) if c ( c(4,2) is a principal set of s, then |c| = 30 or |c| = 60. if |c| = 30, then c is a conjugate of the element shown in (4.2), otherwise it is the complement in c(4,2) of such an element. there are six conjugates of each such set c. no two distinct such conjugates commute. the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). (vi) if c ( c(32) is a principal set of s, then |c| = 20 and c is one of the two elements c1, c2 shown in (4.3). now c1 and c2 commute. each of c1,c2 has six conjugates; no two distinct conjugates of each ci, i = 1, 2, commute, and each c1 g only commutes with a single conjugate of c2. for i = 1, 2, the s-ring generated by z(cs6) and ci is a conjugate of s(s6,h120). (vii) if c ( c(5,1) is a principal set of s, then either (viia) |c| = 24 and c is an orbit of an element of c(5,1) under the action of a conjugate of h120; or (viib) |c| = 120 and c is an orbit of an element of c(5,1) under the action of a conjugate of h120; or (viic) |c| = 72 and c is an a6-orbit of an element of c(5,1). (viid) |c| = 72 and c is a conjugate of c36. further, no two distinct conjugates of c, for c of type (viia) commute; no two distinct conjugates of type (viib) commute; no conjugate of type (viia) commutes with a conjugate of type (viib) unless their sum is c(5,1). any conjugate of type (viic) commutes with any conjugate of type (viia) or (viib). any element of type (viia) or (viib) generates (with z(cs6)) an s-ring which is a conjugate of s(s6,h120). any element of type (viic) generates (with z(cs6)) an s-ring which is a conjugate of s(s6,a6). (viii) if c ( c(6) is a principal set of s, then |c| ∈ {20, 40, 60}, and in each case the s-ring generated by z(cs6) and c is a conjugate of s(s6,h120). let o20,o40,o60 denote the sets of conjugates (of the sums 78 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 c) for each case, so that each oi has size six. then for i ∈ {20, 40, 60} no two distinct elements of oi commute, and for distinct i,j ∈{20, 40, 60} elements of oi,oj commute if and only if the corresponding sets are disjoint. further, the only principal elements of s(s6,h120) that c36 commutes with are c2,c3 and c(3,2,1). proof. here (i) follows from proposition 4.6 and proposition 4.14. part (ii) follows from lemma 4.8, the fact that [s6 : h120] = 6, and a calculation to show that distinct conjugates don’t commute. (iii) follows from lemma 4.11, the fact that [s6 : h120] = 6, and a calculation to show that distinct conjugates don’t commute. part (iv) follows from the proof of proposition 4.15, and a calculation to show that distinct conjugates don’t commute. part (v) follows from the discussion of the c(4,2) case, and a calculation to show that distinct conjugates don’t commute. part (vi) follows from the discussion of the c(32) class that resulted in the two cases shown in (4.3) (up to conjugacy), and a calculation to show that distinct conjugates don’t commute. parts (vii) and (viii) follow from the discussion of the c(5,1) and c(6) classes, and a calculation to show that certain conjugates of these elements don’t commute. we use this to prove: theorem 4.21. the only commutative schur rings over s6 containing c2 as a principal set are: (1) s(s6,s6); (2) s(s6,h120); (3) s(s6,a6); (4) s36. proof. the last statement of proposition 4.20 shows that such an s-ring cannot have split s6 classes that are orbits under a conjugate of h120, and also have c36 (or its complement) in it. one checks that the s-ring generated by any conjugate of s36 and s(s6,a6) is not commutative. given the fact that, according to proposition 4.20, not very many principal elements of conjugates of s(s6,h120) commute, it is now easy to prove the theorem. 5. when c2 is not a principal set we consider each case as enumerated in corollary 1.1. case (ii): c2 = c1∪c2∪c3, where c1 = {(1, 2)}; c2 = {(i,j) : 3 ≤ i < j ≤ 6}, and c2 = c2\(c1∪c2). one finds that the s-ring generated by z(cs6), c1,c2 and c3 is s(s6,s2 × s4), where s2 × s4 is the subgroup 〈(1, 2), (3, 4), (3, 4, 5, 6)〉 ≤ s6. there are 34 principal sets o1, . . . ,o34 of sizes 1, 3, 6, 8, 12, 16, 24, 48. the idea is to show that no proper, non-empty subset of each oi can be a principal set of a commutative s-ring containing s(s6,s2 ×s4). one can check this directly (using [1]) for each oi with |oi| ≤ 8. for the rest let oi = {o1, . . . ,om}. let r = q[x1, . . . ,xm] be a polynomial ring, and let e = ∑m j=1 xjoj. one considers the ideal i of r generated by all the coefficients of the elements eok − oke, these being linear polynomials in the xj. one finds that a gröbner basis for i has only elements of the form ph − puk, where the puk are free variables, and there are at most six of the puk. one chooses a subset of the puk, and puts these equal to 1, while one puts the rest equal to 0. this produces an ideal, that determines an element in cs6. one then sees whether this element (together with z(cs6)) generates a commutative s-ring. one finds that s(s6,s2 ×s4) is the only possible case. case (iii): c2 = c1 ∪c2 where c1 = {(1, 2), (3, 4), (5, 6)} and c2 = c2 \c1. 79 a. francis, s. humphries / j. algebra comb. discrete appl. 3(2) (2016) 61–80 one finds that the s-ring generated by z(cs6), c1 and c2 is s(s6,s2 o s3), where s2 o s3 is the (wreath product) subgroup 〈(1, 2), (3, 4), (5, 6), (1, 3)(2, 4), (3, 5)(4, 6)〉 ≤ s6. there are 34 principal sets o1, . . . ,o34 of sizes 1, 3, 6, 8, 12, 16, 24, 48. we note that s6 has an outer automorphism α such that α(s(s6,s2 ×s4)) = s(s6,s2 os3). thus this case follows from case (ii). case (iv): c2 = c1 ∪c2 where c1 = {(1, 2), (1, 3), (2, 3), (4, 5), (4, 6), (5, 6)} and c2 = c2 \c1. one finds that the s-ring generated by z(cs6), c1 and c2 is s(s6,s3os2), where s3os2 is the (wreath product) subgroup 〈(1, 2), (2, 3), (5, 6), (1, 4)(2, 5)(3, 6)〉 ≤ s6. there are 26 principal sets o1, . . . ,o26 of sizes 1, 4, 6, 9, 12, 18, 36, 72. the idea is to again show that no proper, non-empty subset of each oi can be a principal set of a commutative s-ring containing s(s6,s3 os2). case (v): c2 = c1 ∪c2 where c1 = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6)} and c2 = c2 \c1. one finds that the s-ring generated by z(cs6), c1 and c2 is s(s6,s5). there are 19 principal sets o1, . . . ,o19 of sizes 1, 5, 10, 15, 20, 24, 30, 40, 60, 120. we note that s6 has an outer automorphism, and that s(s6,h120) and s(s6,s5) are related by this automorphism. thus this case follows from (iv). this concludes the proof of theorem 1.2. proof of corollary 1.3 in [5] it is shown that any commutative s-ring s, of maximal dimension sg, over a group g contains z(cg). thus the principal sets of s give a partition of g that is a refinement of the partition of g by conjugacy classes. in particular, for g = s6, we must have c2 ∈ s. thus any such s-ring must be in the list given in theorem 1.2. however none of the s-rings listed in theorem 1.2 has dimension 76 = ss6. this proves corollary 1.3. references [1] w. bosma, j. cannon, magma handbook, university of sydney, 1993. [2] c. w. curtis, pioneers of representation theory: frobenius, burnside, schur, and brauer. vol. 15. american mathematical soc., 1999. [3] s. a. evdokimov, i. n. ponomarenko, on a family of schur rings over a finite cyclic group (russian), algebra i analiz. 13(3) (2001) 139–154; translation in st. petersburg math. j. 13(3) (2002) 441–451. [4] s. p. humphries, commutative schur rings over symmetric groups, j. algebraic combin. 42(4) (2015) 971–997. [5] s. p. humphries, k. w. johnson, a. misseldine, commutative schur rings of maximal dimension, comm. algebra. 43(12) (2015) 5298–5327. [6] k. h. leung, s. h. man, on schur rings over cyclic groups, israel j. math. 106(1) (1998) 251–267. [7] k. h. leung, s. h. man, on schur rings over cyclic groups, ii, j. algebra. 183(2) (1996) 273–285. [8] m. e. muzychuk, on the structure of basic sets of schur rings over cyclic groups, j. algebra. 169(2) (1994) 655–678. [9] m. muzychuk, i. ponomarenko, schur rings, european j. combin. 30(6) (2009) 1526–1539. [10] i. schur, zur theorie der einfach transitiven permutationsgruppen, 1933. [11] h. wielandt, finite permutation groups, academic press, 2014. 80 http://www.mathnet.ru/links/26300f6e5e5113706596d3dd3372a943/aa940.pdf http://www.mathnet.ru/links/26300f6e5e5113706596d3dd3372a943/aa940.pdf http://dx.doi.org/10.1007/s10801-015-0613-2 http://dx.doi.org/10.1007/s10801-015-0613-2 http://dx.doi.org/10.1080/00927872.2014.974258 http://dx.doi.org/10.1080/00927872.2014.974258 http://dx.doi.org/10.1007/bf02773471 http://dx.doi.org/10.1006/jabr.1996.0220 http://dx.doi.org/10.1006/jabr.1994.1302 http://dx.doi.org/10.1006/jabr.1994.1302 http://dx.doi.org/10.1016/j.ejc.2008.11.006 introduction schur-rings covers of complete graphs the case where c2 is a principal set when c2 is not a principal set references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.96056 j. algebra comb. discrete appl. 4(1) • 93–102 received: 28 july 2016 accepted: 25 october 2016 journal of algebra combinatorics discrete structures and applications on dna codes from a family of chain rings∗ research article elif segah oztas, bahattin yildiz, irfan siap abstract: in this work, we focus on reversible cyclic codes which correspond to reversible dna codes or reversible-complement dna codes over a family of finite chain rings, in an effort to extend what was done by yildiz and siap in [20]. the ring family that we have considered are of size 22 k , k = 1,2, · · · and we match each ring element with a dna 2k−1-mer. we use the so-called u2-adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement dna-codes. we then conclude our study with some examples. 2010 msc: 94b15, 92d10 keywords: cyclic codes, chain rings, reversible codes, dna codes 1. introduction in [3], adleman proposed a solution to an instance of the np-complete problem of the directed hamiltonian path problem, using dna molecules. this ground breaking approach of using the dna as a computational tool has led to many similar studies since its appearance. in [6] and [4], the advantages of these studies were demonstrated by a molecular program that led to breaking the data encryption system. in [11], mansuripur et al. show that dna molecules can be used as a storage medium. the dna sequences consist of four bases, namely adenine (a), thymine (t), guanine (g), cytosine (c). the governing principle in its duplication is the well known watson-crick property (wcc). according to wcc a and t bound to each other and g and c bound to each other on the opposite strands. a and g are called the complements of t and c respectively or vice versa. acting like an error-correcting code in nature, the dna, has naturally attracted the attention of coding theorists in their researches. consequently the concept of a dna-code, that is a code that has ∗ a part of this study is presented in the international conference on coding theory and cryptography (iccc2015, algeria). elif segah oztas (corresponding author), irfan siap; department of mathematics, yildiz technical university, istanbul, turkey (email: elifsegahoztas@gmail.com, irfan.siap@gmail.com). bahattin yildiz; istanbul, turkey (email: bahattinyildiz@gmail.com). 93 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 special properties (i.e. reversible and complement) akin to the dna, was introduced and has been a focal point of research in recent years. in many other studies, some constraints such as the hamming distance constraint, the reverse constraint, the reverse-complement constraint and the fixed gc-content constraint are also considered [7, 9, 10, 12]. in [1], aboluion et al. develop a new approach and extend the studies of [8] and [9]. they added the reverse-complement constraint to further prevent the unwanted hybridizations. in such works as [2, 7–9], the focus was on constructing large sets of dna codewords by using minimum hamming distance. with the four letter ambient alphabet of the dna, many of the early works on dna codes, use algebraic structures of size four to construct dna codes. to this end, we can cite [2], in which dna codes are studied over the galois field of size 4. as another possible such alphabet, siap et al. studied dna codes over the finite ring f2[u]/(u2 − 1) in [19]. in [15], dna-double pairs are used with the field f16 and optimal codes are obtained. in [17], a generalization of [15] was done to include dna 2k-bases (dna pairs) over a suitable ring. in this paper, we extend on the work done in [20], in which dna pairs were matched with the elements of the ring f2[u]/(u4 − 1) of size 16. here, we match dna 2k−1-bases (mers) {aa...aa︸ ︷︷ ︸ 2k−1 ,at...tt,...} with elements of the ring r2k = f2[u]/(u2 k − 1). the main idea of the paper is to lay out the theory behind generating reversible and reversible-complement dna codes over this ring. in doing so, we first tackle the issue of ring-reversibility versus the dna reversibility. a problem that arises in all the cases where dna k-mers (k ≥ 2) are matched with ring elements, it can be best explained in the following example: let (u1,u2,u3) ∈r4 be a codeword that corresponds to the dna string atagcc. the reverse of (u1,u2,u3) is (u3,u2,u1) and (u3,u2,u1) corresponds to ccagat. but ccagat is not the reverse of atagcc. this problem is solved by using the so-called u2-adic system. after giving the theoretical results concerning the constructions of dna codes over the ring, we give examples of dna-codes thus obtained. the rest of the work is organized as follows. in section 2, we give some brief background on the ring r2k. in section 3, we introduce the u2-adic digit system for the ring r2k and demonstrate how it can be used to solve the reversibility problem. in section 4, we give the main results about cyclic dna codes over r2k together with many examples. we finish the paper in section 5 with some concluding remarks and directions for possible future research on the related topics. 2. preliminaries we consider the finite chain ring r2k = f2[u]/(u2 k − 1) = {a0 + a1u + ... + a2k−1u2 k−1|ai ∈ f2, 1 ≤ i ≤ 2k − 1,u2 k = 1} in this work. as can easily be observed from its structure, r2k is a commutative, characteristic 2 ring of size 22 k . the chain of ideals can be listed as {0} = ((1 + u)2 k ) ⊂ ((1 + u)2 k−1) ⊂ ···⊂ ((1 + u)3) ⊂ ((1 + u)2) ⊂ (1 + u) ⊂ (1) = r2k. a linear code over r2k of length n is defined naturally as an r2k-submodule of (r2k )n. by using certain elementary row and column operations, it can be shown that any linear code c over r2k of length n is equivalent to a code whose generating matrix can be put into the following standard form:  it0 m 1 1 m 1 2 ... m 1 2k 0 (1 + u)it1 (1 + u)m 2 1 ... (1 + u)m 2 2k−1 0 0 (1 + u)2it2 ... (1 + u) 2m3 2k−2 ... ... ... ... ... ... ... ... (1 + u)2 k−1it 2k−1 (1 + u)2 k−1m2 k−1 1   , (1) 94 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 where mji ’s are matrices over r2k. as is the case with codes over finite chain rings, instead of the dimension, we can talk about the type of a code. a code with the above generator matrix is said to have type (t0, t1, t2, · · · , t2k−1) and the size of the code is given by |c| = (22 k )t0 (22 k−1)t1 (22 k−2)t2 · · ·(2)t2k−1 = 22 kt0+(2 k−1)t1+···+t2k−1. a subset c of rn 2k is called a cyclic code of length n if c is a submodule of rn 2k and if c is invariant under σ, namely σ(c) = c, where σ is the right cyclic shift on rn 2k . σ acts on rn 2k as σ(c0,c1, · · · ,cn−1) = (cn−1,c0,c1, · · · ,cn−2). (2) in studying cyclic codes, it is essential to make use of their algebraic structures by matching each codeword c = (c0,c1, · · · ,cn−1) ∈ c to a polynomial c0 +c1x+· · ·+cn−1xn−1 ∈r2k [x]. thus cyclic codes of length n in that case correspond to the ideals of the quotient ring r2k [x]/(xn − 1). as will be needed later, we include in this section, the definition of the reciprocal polynomial. for g(x) ∈ r2k [x], the reciprocal of g(x) is denoted by g(x)∗ = xdeg(g)g(1/x). a code c is said to be reversible if cr ∈ c for all c ∈ c, where cr corresponds to the reverse of c. more precisely if c = (c0,c1, · · · ,cn−1), then cr = (cn−1,cn−2, · · · ,c0). a code is said to be complement if cc ∈ c for each c ∈ c, where cc denotes a suitably defined complement of c. in general, algebraically the complement corresponds to adding a fixed constant to each of the coordinates. a code is said to be a reversible-complement code if it is both reversible and complement. dna codes are defined as codes that satisfy both the reversible and complement property. 3. u2-adic digit system for dna 2k−1-mers in [16], the u2-adic digit system was introduced for dna k-mers. we will make use of the function η from [16], which operates on dna single bases in the following form: η(a) = 0,η(t) = 1 + u,η(g) = 1, and η(c) = u. (3) the u2-adic digit system can be used for all the rings of the form r2k = f2[u]/(u2k−1), in particular for the ring r2k, that is in question in our work. the following definitions are obtained from [16] by modifying them to go along with the ring r2k: definition 3.1. every element of the ring r2k can be expressed as a linear combination of 1,u2,u4, . . . ,u2 k−2, with coefficients from {0, 1,u, 1 +u} = f2[u]/(u2−1). we can view this as a numberbase system with the digits 1s (units), u2s, u4s, u6s, u8s, ... , u2 k−2s. we call this system the u2-adic system. definition 3.2. let b1b2...b2k−1 be a dna 2 k−1-bases (2k−1-mers) where bi ∈{a,t,g,c}. this corresponds to an element in the ring r2k, where this correspondence is given by the following: ζ(b1b2...b2k−1 ) = α ∈r2k, (4) where α = η(b2k−1 )1 + η(b2k−1−1)u 2 + η(b2k−1−2)u 4 + · · · + η(b1)u(2 k−2) = 2k−1∑ t=1 η(bt)u (2k−2t). the following lemma expresses the effect of multiplying by u2: lemma 3.3. if α ∈r2k and ζ(b1b2 · · ·b2k−1 ) = α, then ζ−1(u2α) = b2 · · ·b2k−1b1. 95 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 proof. according to the structure of u2-adic digits, multiplying by u2 shifts the u2-adic digits cyclically to the left. the result then follows from the definition of ζ. for p(u) ∈ r2k, let p(u)u2 or pu2 represent the u2-adic digits of p(u) as follows: if p(u) = a01 + a1u 2 + a2u 4 + ... + a(2k−1−1)u (2k−2), then let pu2 = a(2k−1−1)...a2a1a0. denote by ε, the function ε(pu2 ) = p(u) and by [pu2 ]r, the reverse vector, namely [pu2 ]r = a0a1a2...a(2k−1−1). then by lemma 3.3 we have u2pu2 = a(2k−1−2)...a2a1a0a(2k−1−1), where ai ∈ f2[u]/(u2 − 1). example 3.4. let us consider the ring r16 = f2[u]/(u16 − 1). aaatgacc → ζ(aaatgacc) = u + u3 + u6 + u8 + u9 → 000ū10uu, where ū = 1 + u. multiplying by u2, we get u2(u + u3 + u6 + u8 + u9) = u3 + u5 + u8 + u10 + u11 → 00ū10uu0 (u2-adic digit system in r2). digits: u14 u12 u10 u8 u6 u4 u2 1 base in r2: 0 0 ū 1 0 u u 0 dna: a a t g a c c a the following theorem provides us with reverse-complement sets of dna 2k−1-mers. theorem 3.5. every ideal in r2k corresponds to a reversible-complement set of dna 2k−1-mers. proof. we know, from the ideal structure of r2k, that every ideal is principally generated by (1 + u)m for some m = 0, 1, 2, . . . 2k − 1. so let 〈(1 + u)m〉 be any ideal. we will consider two cases separately. case 1: m is even. if p = p(u) = (1 + u)m, then pu2 = 0`c1c2 · · ·c2k−1−` in the u2-adic system, where 0` corresponds to a vector of zeros of length `. we assume that c1 6= 0. now, since (1 + u)m = ((1 + u)2)m/2 = (1 + u2)m/2, and ( m/2 i ) = ( m/2 m/2−i ) , we see that (c1c2 · · ·c2k−1−`) is self-reversible, i.e., (c1c2 · · ·c2k−1−`)r = (c1c2 · · ·c2k−1−`). then, for 0 ≤ i ≤ ` we have (in u2-adic system) [ ( u2 )i ·pu2 ]r = ( u2 )`−i ·pu2 and for ` + 1 ≤ i ≤ 2k−1 − 1 we have [ ( u2 )i ·pu2 ]r = ( u2 )`+2k−1−i ·pu2. thus u2-multiples of (1 + u)m are self-reversible. and since [(u(1 + u)m)u2 ]r = u[((1 + u)m)u2 ]r, we see that u(1 + u)m is also self-reversible. but then this implies, by linearity of self-reversability, that any element in the ideal is self-reversible. case 2: m is odd. then we can write (1 + u)m = (1 + u)(1 + u)m−1 = (1 + u)m−1 + u(1 + u)m−1. we can then use case 1 for (1 + u)m−1. if we then take any polynomial q(u) = ∑2k−1−1 i=0 qiu 2i, we have ε([q(u)(1 + u)m]r) ∈ ((1 + u)m) , 96 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 since the reversibility is linear and closed under multiplying by a + ub and u2. this proves that every ideal corresponds to a reversible set of dna 2k−1-mers. according to the watson-crick complement and the correspondence we have defined the complement of an element α ∈r2k is just given by α+ 1 +u+u2 +· · ·+u2 k−1. because of the ideal structure, we know that each ideal contains the generator of the minimal ideal, which is given by (1 + u)2 k−1. so if α ∈ i for some ideal i of r2k, then α + (1 + u)2 k−1 ∈ i as well. but since (1 + u)2 k−1 = 1 + u + u2 + · · · + u2 k−1 in r2k, we see that the complement of each element in i is also in i. this proves that each ideal corresponds to a reversible-complement set of dna 2k−1-mers. let us give an illustrative example. example 3.6. consider p(u) = (1 + u)2 and j = (p(u)) in r8 = f2[u]/(u8 − 1). take q(u) = u6 + u + 1. and let us compute [(q(u)p(u))u2 ]r. first, we see that q(u)p(u) = (u6 +u+ 1)(u2 + 1) = u8 +u6 + (u+ 1)u2 +u+ 1 = u6 + (u+ 1)u2 +u → (q(u)p(u))u2 = 10ūu. notice that p(u) = u2 + 1 and so, u2 + 1 = 0 ·u6 + 0 ·u4 + 1 ·u2 + 1 → 0011. this means ` = 2. thus we have [u6p(u)u2 ] r = [(u2)3p(u)u2 ] r = (u2)`+2 k−1−3p(u) = (u2)3p(u) = u6p(u) = u6 + 1 [up(u)u2 ] r = u[(u2)0p(u)u2 ] r = u(u2)`−0p(u) = uu4p(u) = u5p(u) = u5 + u7 [p(u)u2 ] r = [(u2)0p(u)u2 ] r = (u2)`−0p(u) = (u2)2p(u) = u4p(u) = u6 + u4. hence, [(q(u)p(u))u2 ] r = [u6p(u)u2 ] r + [up(u)u2 ] r + [p(u)u2 ] r = u6 + 1 + u5 + u7 + u6 + u4 [(q(u)p(u))u2 ] r = u ·u6 + (u + 1) ·u4 + 1 = uū01. now let us verify that indeed we get the reverse dna 4-mer: [(q(u)p(u))u2 ] = u 6 + 0 ·u4 + (u + 1)u2 + u = 10ūu → ζ−1(q(u)p(u)) = gatc. [q(u)p(u)u2 ] r = u ·u6 + (u + 1) ·u4 + 0 ·u2 + 1 = uū01 → ζ−1([q(u)p(u)]r) = ctag. 4. cyclic codes for dna codes over r2k cyclic codes, have been one of the most common methods by which reversible-complement codes over different alphabets have been obtained. with the vast literature on the structural properties of cyclic codes over finite chain rings especially, this proves to be a fruitful direction to take. we will follow the footsteps of similar works to this effect. the following two theorems can easily be proven by following the similar steps as in [14] and [5]. 97 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 theorem 4.1. assume that c = (f0, (1 + u)f1, (1 + u) 2f2, ..., (1 + u) 2k−1f2k−1) generates a cyclic code of length n over r2k with f2k−1|...f3|f2|f1|f0|(xn − 1) over f2. then the dual of the code is given by c⊥ = (f̄∗2k−1, ..., (1 + u) 2k−2f̄1 ∗ , (1 + u)2 k−1f̄0 ∗ ) where f̄i = (xn − 1)/fi and if fi = 0 then fi = fi−1 is considered (f−1 = xn − 1) (0 ≤ i ≤ 2k − 1). further, if c is cyclic code of even length and f2k−1−i = f̄ ∗ i for 0 ≤ i ≤ 2 k − 1, then c is a cyclic self dual code. the following is a classical result about reversible codes: theorem 4.2. [13] the cyclic code generated by a monic polynomial g(x) is reversible if and only if g(x) is self-reciprocal where g(x)|(xn − 1). the following lemma, taken from [2] explains the basic properties of reciprocals. lemma 4.3. let f(x) and g(x) be polynomials in r2k with deg(f) ≥ deg(g). then 1. [f(x)g(x)]? = f(x)?g(x)? and 2. [f(x) + g(x)]? = f(x)? + xdeg(f)−deg(g)g(x)?. we are now ready to state the main result of this section: theorem 4.4. let c = (f0, (1 + u)f1, (1 + u)2f2, ..., (1 + u)2 k−1f2k−1) be a cyclic code over r2k with f2k−1|...f3|f2|f1|f0|(xn − 1) over f2. then the dual of the code is given by c⊥ = (f̄∗2k−1, ..., (1 + u) 2k−2f̄1 ∗ , (1 + u)2 k−1f̄0 ∗ ) with f̄i = (xn − 1)/fi. moreover, • if fi’s (f̄i’s) are self reciprocal polynomials then c (c⊥) is a reversible cyclic code and ζ−1(c) (ζ−1(c⊥)) is a reversible dna code. • if fi’s (f̄i’s) are self reciprocal polynomials and f0(f̄∗2k−1) divides (x n − 1)/(x + 1) over f2 then c (c⊥) is a reversible cyclic code with complement property and so ζ−1(c) (ζ−1(c⊥)) is a reversible complement dna code. proof. if c(x) ∈ c, then c(x) = f0a0 +(1+u)f1a1 +...+(1+u)2 k−1f2k−1a2k−1 where ai are polynomials in r2k and (deg(ai) < deg(fi−1))−deg(fi)). since f2k−1|...f3|f2|f1|f0|(xn−1), there exists non-negative integers mi = deg(f0a0) − deg(fiai), and hence by lemma 4.3, we have c(x)∗ = [f0a0 + (1 + u)f1a1 + ... + (1 + u) 2k−1f2k−1a2k−1] ∗ = (f0a0) ∗ + xm1 ((1 + u)f1a1) ∗ + ... + xm2k−1 ((1 + u)2 k−1f2k−1a2k−1) ∗ = f0(a0) ∗ + xm1 (1 + u)f1(a1) ∗ + ... + xm2k−1 (1 + u)2 k−1f2k−1(a2k−1) ∗ ⇒ c(x)∗ ∈ c. therefore, c is reversible. this means, by theorem 3.5, that ζ−1(c) is a reversible dna code. if moreover, f0 divides (xn − 1)/(x + 1), then all fi’s divide (xn − 1)/(x + 1) since f2k−1|...f3|f2|f1|f0|(xn − 1) over f2. this means (1 + u)2 k−1(xn − 1)/(x + 1) ∈ c. hence ζ−1(c) is a reversible complement code. the proofs for c⊥, being similar to the above case, are omitted here. 98 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 example 4.5. let us consider the self reciprocal polynomials f0 = x5+x4+x3+x2+x+1, f1 = x4+x2+1, f2 = x 2 +x+1, f3 = f4 = ...f7 = 0 with f2|f1|f0|(x6−1) over f2. the code c = (f0, (1+u)4f1, (1+u)6f2) is generated by the following matrix r23:  1 1 1 1 1 1 u5 + u4 + u + 1 0 u5 + u4 + u + 1 0 u5 + u4 + u + 1 0 u6 + u4 + u2 + 1 u6 + u4 + u2 + 1 u6 + u4 + u2 + 1 0 0 0 0 u6 + u4 + u2 + 1 u6 + u4 + u2 + 1 u6 + u4 + u2 + 1 0 0   . by theorem 4.4, c is a reversible cyclic code over r23 and it corresponds to a reversible-complement dna code of length 24. the dual is given by : c⊥ = (f̄∗2 , (u + 1)f̄ ∗ 2 , (u + 1) 2f̄∗1 , (u + 1) 3f̄∗1 , (u + 1) 4f̄∗0 , (u + 1) 5f̄∗0 , (u + 1) 6f̄∗0 , (u + 1) 7f̄∗0 ) which can be expressed as c⊥ = (f̄∗2 , (u + 1) 2f̄∗1 , (u + 1) 4f̄∗0 ), where f̄∗2 = 1 + x + x 3 + x4, f̄∗1 = 1 + x 2, f̄∗0 = 1 + x. then all these polynomials are self reciprocal but not all of them divide (xn − 1)/(x + 1). thus c⊥ corresponds to a reversible dna code. example 4.6. now consider self reciprocal polynomials f0 = x5 + x4 + x3 + x2 + x + 1, f1 = x + 1, f2 = f3 = f4 = ...f7 = 0 with f1|f0|(x6 − 1) over f2. the code c = (f0, (1 + u)2f1, ) is generated by the following matrix over r23:   1 1 1 1 1 1 u2 + 1 u2 + 1 0 0 0 0 0 u2 + 1 u2 + 1 0 0 0 0 0 u2 + 1 u2 + 1 0 0 0 0 0 u2 + 1 u2 + 1 0   . by the same argument as in the previous example, c is a reversible cyclic code over r23 and moreover, since the polynomials satisfy the complement properties, c corresponds to a reversible-complement dna code of length 24. the dual of c is given by c⊥ = (f̄∗1 , (u + 1) 6f̄∗0 ), where f̄ ∗ 1 = x 5 + x4 + x3 + x2 + x + 1, and f̄∗0 = 1 + x. then we again have self reciprocal polynomials since they all divide (xn −1)/(x + 1), we see that c⊥ also corresponds to a reversible-complement dna code. the next example illustrates the dna reversibility in an explicit form. example 4.7. let f0 = x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1, f1 = x6 + x3 + 1 and f1|f0|x9 −1 over f2, where we will consider the codes over r8. suppose c = ((1 + u)3f0, (1 + u)4f1), whose generator matrix can be written as follows: let ui = (1 + u)i,   v1v2 v3   =   u3 u3 u3 u3 u3 u3 u3 u3 u3u4 0 0 u4 0 0 u4 0 0 0 u4 0 0 u4 0 0 u4 0   . we can easily observe that the reverse of v2 is given by vr2 = (1 + u)v1 + v2 + v3, while v1 and v3 are self-reversible. consider the codeword c1 = (1 + u 2 + u3)v2 =(u 7 + u6 + u4 + u3 + u2 + 1, 0, 0,u7 + u6 + u4 + u3 + u2 + 1, 0, 0, u7 + u6 + u4 + u3 + u2 + 1, 0, 0) whose dna-correspondence is given by ζ−1(c1) =(t,g,t,g,a,a,a,a,a,a,a,a,t,g,t,g,a,a,a,a,a,a,a,a,t,g,t,g, a,a,a,a,a,a,a,a). 99 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 then if we take c2 = (1 + u + u2)((1 + u)v1 + v2 + v3), we see that c2 = (1 + u + u 2)vr2 =(0, 0,u 6 + u5 + u4 + u2 + u + 1, 0, 0,u6 + u5 + u4 + u2 + u + 1, 0, 0, u6 + u5 + u4 + u2 + u + 1). note that ζ−1(c2) =(a,a,a,a,a,a,a,a,g,t,g,t,a,a,a,a,a,a,a,a,g,t,g,t,a,a,a,a, a,a,a,a,g,t,g,t) and we have (ζ−1(c1))r = ζ−1(c2). next, consider e1 = uv1 + (1 + u3)v2 + (u2 + u3)v3, which is given as a vector by (u7 + u2 + u + 1,u7 + u6 + u4 + u,u4 + u3 + u2 + u,u7 + u2 + u + 1,u7 + u6 + u4 + u, u4 + u3 + u2 + u,u7 + u2 + u + 1,u7 + u6 + u4 + u,u4 + u3 + u2 + u), so that the dna correspondence is given by ζ−1(e1) =(c,a,g,t,t,g,a,c,a,g,t,c,c,a,g,t,t,g,a,c,a,g,t,c,c,a,g,t, t,g,a,c,a,g,t,c) now, if we let e2 = ((u2 + u3 + u4))vr1 + (u + u 2)vr2 + (1 + u)v r 3, which then would be written as a vector as (u7 + u5 + u4 + u2,u7 + u2 + u + 1,u7 + u6 + u4 + u,u7 + u5 + u4 + u2,u7 + u2 + u + 1, u7 + u6 + u4 + u,u7 + u5 + u4 + u2,u7 + u2 + u + 1,u7 + u6 + u4 + u), we will have ζ−1(e2) =(c,t,g,a,c,a,g,t,t,g,a,c,c,t,g,a,c,a,g,t,t,g,a,c,c,t,g,a, c,a,g,t,t,g,a,c). note that, in this case we again have (ζ−1(e1))r = ζ−1(e2). example 4.8. let us consider self reciprocal polynomials f0 = x14 +x13 +x12 +x11 +x10 +x9 +x8 +x7 + x6 + x5 + x4 + x3 + x2 + x + 1, f1 = 1 + x3 + x6 + x9 + x12, f2 = x4 + x3 + x2 + x + 1, f3 = f4 = ...f7 = 0 and f2|f1|f0|(x15 −1) over f2. the code c = ((1 + u)f0, (1 + u)3f1, (1 + u)4f2) is generated over r24 by the following matrix (ui = (1 + u)i)  u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u1 u3 0 0 u3 0 0 u3 0 0 u3 0 0 u3 0 0 0 u3 0 0 u3 0 0 u3 0 0 u3 0 0 u3 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0 0 0 0 0 0 0 0 0 u4 u4 u4 u4 u4 0 0 0   . then, c is a reversible cyclic code over r24 and it corresponds to a reversible-complement dna code of length 120 and minimum hamming distance 2. example 4.9. consider f0 = x3 + x2 + x + 1 and f1 = 1 + x2 with f1|f0|(x4 − 1) over f2. let c = ((1 + u)f0, (1 + u) 7f1) be the code over r23. then c is a cyclic code of length 4 and distance 2. ζ−1(c) is a reversible-complement dna code of length 16 and hamming distance 4. 100 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 table 1. f = 1 + x, g = 1 + x + x2. reversible cyclic codes over r23 of length 6 which correspond to reversible complement dna codes of length 24. (d(c): minimum hamming distance of the code c. d(ζ−1(c)): minimum hamming distance of the code ζ−1(c) ) the code length d(c) d(ζ−1(c)) (fg2,(1 + u)g2) 6 3 3 (fg2,(1 + u)4g2) 6 3 6 (fg2,(1 + u)2g2) 6 3 6 in table 1 we give a table of some reversible-complement codes obtained from similar constructions. we conclude this section by the following theoretical result whose proof is omitted, being similar to the above ones, with an application. theorem 4.10. let c = (f0, (1 + u)f1, (1 + u)2f2, ..., (1 + u)2 k−1f2k−1) be a cyclic code over r2k of even length. if all fi are self reciprocal polynomials and f2k−1−i = f̄ ∗ i for 0 ≤ i ≤ 2 k − 1, then c is a reversible cyclic self-dual code and ζ−1(c) is a reversible dna code. if f0|(xn − 1)/(x + 1) then ζ−1(c) is a reversible complement dna code. example 4.11. some self dual codes that correspond to reversible complement dna codes are shown in table 2: table 2. f = (x−1) and g = (x2 + x + 1) ring length generator of the code r23 16 (f 13,(u−1)3f9,(u−1)4f7,(u−1)5f3) r23 16 ((u−1) 3f9,(u−1)4f7,(u−1)5) r23 8 (f 7,(u−1)f5,(u−1)4f3,(u−1)7f) r23 6 (f 2g,(u−1)4g) 5. conclusion in this work, we solve the reversibility problem on cyclic codes with ideal structures of the ring r2k by u2-adic system. both reversible cyclic codes on the ring r2k and reversible (or reversible complement) dna codes are obtained from the same codes. the properties of dna codes with the family of cyclic dual and self dual codes have been explored. thus an extension of what was done in [20] has been established in the most general sense without puncturing. the idea that we have used can be used for different such extensions as well. however, the finite chain property of the ring, which is useful in both describing all the ideals of the ring and making it easier to study cyclic codes, has been essential in obtaining our main results. when the extension ring does not have the finite chain property, some partial results can still be obtained, however a complete characterization, similar to what we have done here, is very challenging. hence, future studies on the general case i.e reversible and complement cyclic codes over different rings such as r2k remains to be an open and interesting problem. acknowledgment: the work presented here was supported by the scientific and technological research council of turkey (tübi̇tak) (grant no: 113f071). the authors wish to express their thanks to the reviewers for their valuable remarks and comments that improved the presentation of the paper. 101 e. s. oztas et al. / j. algebra comb. discrete appl. 4(1) (2017) 93–102 references [1] n. aboluion, d. h. smith, s. perkins, linear and nonlinear constructions of dna codes with hamming distance d, constant gc–content and a reverse–complement constraint, discrete math. 312(5) (2012) 1062–1075. 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[20] b. yildiz, i. siap, cyclic codes over f2[u]/(u4 − 1) and applications to dna codes, comput. math. appl. 63(7) (2012) 1169–1176. 102 http://dx.doi.org/10.1016/j.disc.2011.11.021 http://dx.doi.org/10.1016/j.disc.2011.11.021 http://dx.doi.org/10.1016/j.disc.2011.11.021 http://dx.doi.org/10.1016/j.jfranklin.2006.02.009 http://dx.doi.org/10.1016/j.jfranklin.2006.02.009 http://dx.doi.org/10.1126/science.7973651 http://dx.doi.org/10.1126/science.7973651 https://dx.doi.org/10.1089/cmb.1999.6.53 https://dx.doi.org/10.1089/cmb.1999.6.53 http://www.ams.org/mathscinet-getitem?mr=mr2501336 http://www.ams.org/mathscinet-getitem?mr=mr2501336 https://dx.doi.org/10.1093/nar/25.23.4748 https://dx.doi.org/10.1093/nar/25.23.4748 https://dx.doi.org/10.1093/nar/25.23.4748 http://dx.doi.org/10.1016/j.tcs.2004.11.004 http://dx.doi.org/10.1016/j.tcs.2004.11.004 http://www.ams.org/mathscinet-getitem?mr=mr2014520 http://dx.doi.org/10.1021/la0112209 http://dx.doi.org/10.1021/la0112209 https://dx.doi.org/10.1089/10665270152530818 https://dx.doi.org/10.1089/10665270152530818 http://dx.doi.org/10.1016/s0019-9958(64)90438-3 https://dx.doi.org/10.1007/pl00012382 https://dx.doi.org/10.1007/pl00012382 https://dx.doi.org/10.2298/fil1303459o https://dx.doi.org/10.2298/fil1303459o https://dx.doi.org/10.1007/978-3-662-44199-2_22 https://dx.doi.org/10.1007/978-3-662-44199-2_22 http://dx.doi.org/10.1080/00207160.2014.930449 http://dx.doi.org/10.1080/00207160.2014.930449 http://dx.doi.org/10.1016/j.jfranklin.2009.07.002 http://dx.doi.org/10.1016/j.jfranklin.2009.07.002 http://dx.doi.org/10.1016/j.camwa.2011.12.029 http://dx.doi.org/10.1016/j.camwa.2011.12.029 introduction preliminaries u2-adic digit system for dna 2k-1-mers cyclic codes for dna codes over r2k conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284939 j. algebra comb. discrete appl. 4(2) • 115–122 received: 14 june 2015 accepted: 1 february 2016 journal of algebra combinatorics discrete structures and applications code–checkable group rings research article noha abdelghany, nefertiti megahed abstract: a code over a group ring is defined to be a submodule of that group ring. for a code c over a group ring rg, c is said to be checkable if there is v ∈ rg such that c = {x ∈ rg : xv = 0}. in [6], jitman et al. introduced the notion of code-checkable group ring. we say that a group ring rg is code-checkable if every ideal in rg is a checkable code. in their paper, jitman et al. gave a necessary and sufficient condition for the group ring fg, when f is a finite field and g is a finite abelian group, to be code-checkable. in this paper, we give some characterizations for code-checkable group rings for more general alphabet. for instance, a finite commutative group ring rg, with r is semisimple, is code-checkable if and only if g is π′-by-cyclic π; where π is the set of noninvertible primes in r. also, under suitable conditions, rg turns out to be code-checkable if and only if it is pseudo-morphic. 2010 msc: 16s34, 16p60 keywords: group rings, pseudo-morphic rings, a-by-b groups, checkable codes, pseudo-morphic group rings 1. introduction a code over a group ring is originally defined to be an ideal in the group algebra fg, where f is a finite field and g is a finite group. when g is cyclic, this concept characterizes the classical cyclic codes over f as, in this case, the ideals of fg ∼= f[x]/ < xn − 1 >. this concept has been first introduced by f. macwilliams [7] in 1969. in general when g is abelian, they are called abelian codes. later on 2007, hurley [4] introduced new techniques for constructing codes from encoding in group rings, for arbitrary group ring rg where r is a ring with unity and g is a finite group. codes from group-ring encoding are basically defined by considering a left r-submodule w of the group ring rg and any element u of rg, the right group-ring code c generated by u relative to w is the code defined by c = {xu : x ∈ w}. when the element u is zero-divisor (resp. unit), c is called zero-divisor (resp. unitderived) code. this method allows us to produce codes from every zero-divisor and every unit in the group ring. a zero-divisor code c is called checkable if there exists v ∈ rg such that c = {y ∈ rg : yv = 0}, that is y ∈ c if and only if yv = 0. in this case we say that v is a check element for the code c. it is noha abdelghany, nefertiti megahed (corresponding author); department of mathematics, faculty of science, cairo university, egypt (email: nmabdelghany@sci.cu.edu.eg, nefertiti@sci.cu.edu.eg). 115 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 easy to see that every cyclic code is a checkable code, but the converse is not true. this means that the concept of checkable codes is, somehow, a generalization for cyclic codes. in 2010, jitman et al [6] introduced the notion of code-checkable group rings, where a group ring is said to be code-checkable if every ideal in that group ring is a checkable code. so, the main question was about a characterization for code-checkable group rings. in the same paper [6], such a characterization for rg was given in the special case when r is a finite field and g is an abelian group. in this paper, we give a necessary and sufficient condition for a group ring rg to be code-checkable in a more general setting, when r is a finite commutative semisimple ring and g is any finite abelian group. we were also able to get rid of the condition that r has to be semisimple and give a characterization for rg to be code-checkable for any finite commutative ring r. the paper is organized as follows: in section 2, we present basic results and tools that will be used in the following sections. we start with basic structural properties for group rings specially; the relation between the group-ring elements and the matrix ring. some characterizations for principal ideal group rings are presented. we also present some notions concerning generalized morphic rings. in section 3, the notion of codes from group ring encodings, due to hurley [4], is presented. we focus on checkable codes, which are special case of codes from group ring encodings. a zero-divisor code is said to be checkable if it is a left annihilator for some element in the group ring. we also present the result due to jitman et al [6] in characterizing code-checkable group algebras. our two main results theorem 4.1 and theorem 4.5 are presented in the last section. we give necessary and sufficient conditions for group rings to be code-checkable. also, in the last section we provide example 4.6 that shows the necessity of one of the hypothesis of theorem 4.2. note: all rings are considered to be unitary. 2. preliminaries in this section we are going to describe the structure of the group rings. we also introduce some basic concepts and necessary terminologies that will be used later in this paper. 2.1. group rings and matrices in this section we describe the very important isomorphism between a group ring rg and a subring of the matrix ring mn×n(r); where n is the number of element of g. this isomorphism plays a basic role in studying the generator and the parity check matrices of certain types of codes that are going to be mentioned later. starting with a finite group g and a ring r, let {g1,g2, ...,gn} be a fixed listing of elements of the group g. then every element u in rg is written as u = ∑n i=1 ugigi, where ugi ∈ r. for u = ∑n i=1 ugigi ∈ rg define the matrix u ∈ mn×n(r) by: u =   ug−11 g1 ug−11 g2 · · · ug−11 gn ug−12 g1 ug−11 g2 · · · ug−12 gn ... ... ... ... ug−1n g1 ug−1n g2 · · · ug−1n gn   definition 2.1. define the map σ : rg → mn×n(r) by u 7→ u, for every u ∈ rg. the map σ is called left regular representation of rg. the left regular representation of rg is a monomorphism of rings. this means that restricting the codomain of σ on im(σ) will yield an isomorphism between the group ring rg and a subring of the matrix ring mn×n(r). this subring is denoted by rg(mn) and matrices in rg(mn) are called rg-matrices. 116 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 theorem 2.2. the group ring rg is isomorphic to rg(mn) as rings. when the group g is cyclic, g = {gn,g1, ...,gn−1}. any element u ∈ rg is written as u = ∑n i=1 uig i, then the associated matrix to u is of the form: u =   ugn ug1 · · · ugn−1 ugn−1 ugn · · · ugn−2 ... ... ... ... ug1 ug2 · · · ugn   because of the isomorphism σ, the elements of the group ring rg inherit many concepts and properties of matrices which turn out to be very useful in our work. definition 2.3. the transpose of an element u = ∑ g∈g ugg in rg is u t = ∑ g∈g ugg −1, or equivalently, ut = ∑ g∈g ug−1g. definition 2.4. we say that u ∈ rg is symmetric if and only if ut = u. note that this definition is consistent with the matrix definition of transpose. if we take an element u ∈ rg, then the transpose of the rg-matrix of u is again an rg-matrix and is associated to group ring element ut . that is σ(ut ) = σ(u)t = ut . 2.2. principal ideal group rings in this section we are interested in a characterization of principal ideal group rings. in the case when the ring is a finite field, the principal ideal group rings are characterized in [3]. another characterization in [2] is established in a more general setting. definition 2.5. a ring r is said to be principal ideal ring (for short: pir), if every two-sided ideal in r is a principal ideal. definition 2.6. a prime number p is said to invertible in a ring r if p.1 is an invertible element in r. otherwise p is called noninvertible. to be able to see a characterization of pir group rings, we first need the following notions about finite groups. definition 2.7. let p be a prime and g be a finite group of order n. • we say that g is a p-group if n is a power of p. • we say that g is a p′-group (here p′ does not mean another prime p′) if (n,p) = 1. the above definition can be easily generalized if we replace the prime p by a finite set of primes. that is, for a finite set of primes π and a finite group g with order n, g is said to be π-group if n is a power of primes from π while g is π′-group if n is coprime with every prime in π. for any two classes of groups; a and b, we say that a group g is a-by-b if there exists n c g such that n ∈a and g/n ∈b. so, a finite group g is called π′-by-cyclic π, if there is h c g such that h is a π′-group and g/h is cyclic and a π-group. now we are ready to introduce two results concerning principal ideal group rings. in fact, the property that the group ring rg is a pir depends on the relation between the set of noninvertible primes in r and the number of elements of g, as we shall see in theorem 2.8 and theorem 2.9. notice that, the only noninvertible prime in a finite field f is the characteristic of f. we have the following two theorems for the characterization of principal ideal group rings. 117 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 theorem 2.8. [3] let g be a finite abelian group and f a finite field of characteristic p. then fg is a pir if and only if a sylow p-subgroup of g is cyclic. theorem 2.9. [2] let r be a finite semisimple ring and g a finite group. then rg is pir if and only if g is π′-by-cyclic π, where π is the set of noninvertible primes in r. 2.3. generalized morphic rings a young topic in ring theory is being studied for the last decade, namely rings that satisfy the dual of the isomorphism theorem. that is, r ra ∼= annl(a) for every a ∈ r, in this case r is called a left morphic ring [8]. it turns out that r being morphic is equivalent to say that for all a ∈ r there exists b ∈ r such that ra = annl(b) and annl(a) = rb. later on, the notions of quasi-morphic rings and pseudo-morphic rings have been introduced by relaxing the condition on morphic rings, see definition 2.11. for more details, we refer to [8] and [1]. definition 2.10. an ideal i of a ring r is said to be (left) annihilator if i = annl(a) = {r ∈ r : ra = 0}, for some a ∈ r. definition 2.11. let r be an arbitrary ring. 1) r is called (left) morphic if for all a ∈ r there exists b ∈ r such that ra = annl(b) and annl(a) = rb. 2) r is called (left) quasi-morphic if {ra : a ∈ r} = {annl(b) : b ∈ r}. means that every (left) principal ideal is (left) annihilator ideal and vise versa. 3) r is called (left) generalized morphic if {annl(b) : b ∈ r} ⊂ {ra : a ∈ r}. means that every (left) annihilator ideal is a (left) principal ideal. 4) r is called (left) pseudo-morphic if {ra : a ∈ r} ⊂ {annl(b) : b ∈ r}. means that every (left) principal ideal is a left annihilator ideal. we get similar definitions by replacing each "left" by "right", if we drop the word "left" it means that the property is satisfied for only two-sided ideals. in our work, we found that pseudo-morphic group rings characterizes code-checkable group rings, when the group ring is finite commutative. see theorem 4.5. 3. codes from group ring encoding in this section we are going to present a construction of codes from encoding in group rings. this construction is due to hurley in [5] and it leads to two new types of codes, namely zero-divisor codes and unit-derived codes. many important codes like bch and reed-solomon turn out to be special kinds of zero-divisor codes. in the following, rg denotes a group ring, w a free left submodule of rg and u is a fixed element of rg. definition 3.1. a right group ring encoding is a map f : w → rg defined by x 7→ xu, for every x ∈ w. (the left group ring encoding maps x to ux). definition 3.2. for a right group ring encoding f, the code c derived from f is defined by c := im(f) = {xu : x ∈ w}. we say that the code c is generated by u relative to w. when u is a zero-divisor, the code c is called zero-divisor code and when u is a unit, the code c is called unit-derived code. 118 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 here w plays the role of an information set and u plays the role of the encoder. that is, u encodes the message x to the codeword xu. of course, one of the most important properties of codes is to have a way from the code c back to the information set, which is what we call a decoding algorithm. in our case, the existence of this algorithm depends on the choice of the information set w . for unit-derived codes, there is complete freedom in the choice of w , while zero-divisor codes have some restrictions placed on w . 3.1. checkable codes checkable codes are special kind of zero-divisor codes. they have one of the most important properties for a code which is to have a check matrix or a check element. a zero-divisor code c is said to be checkable if c has a single check element. that is c = {x ∈ rg : xv = 0} for some v ∈ rg. the notion of checkable codes has been discussed by hurley in [5]. where, the checkable codes have been characterized in terms of the properties of the generator element of the code. however, the name checkable codes and the notion of code-checkable group rings were first established in [6] by jitman et al. they have studied checkable codes in terms of the properties of the group ring rg. definition 3.3. let c be a zero-divisor code in the group ring rg. c is said to be a checkable code if there exists v ∈ rg such that c = {x ∈ rg : xv = 0}. in other words, c is the left annihilator, denoted by ann(v), of an element v of rg. we are interested in finding alphabets where all its two-sided ideals are checkable codes. here is the definition of such alphabets. definition 3.4. a group ring rg is said to be code-checkable if every two-sided ideal in rg is a checkable code. of course if a unit-derived code is an ideal, then it will be the whole space rg. thus, to determine whether rg is code-checkable, it suffices to consider all zero-divisor codes c where c = fgu. the following proposition and theorem which characterize when a group algebra fg is code-checkable were proved in [6], when g is a finite abelian group. proposition 3.5 may be proved along similar lines to proposition 4.1. its proof is omitted and it can be found in [[6], proposition 3.1]. proposition 3.5. let g be a finite abelian group and f be a finite field. then fg is code-checkable if and only if it is a principal ideal group ring. theorem 3.6. let g be a finite abelian group and f be a finite field of characteristic p. then the group algebra fg is code-checkable if and only if a sylow p-subgroup of g is cyclic. proof. follows immediately from theorem 2.8 and proposition 3.5. 4. code-checkable group rings the last theorem in the previous section gives a complete characterization for a group algebra to be code-checkable. our main result in this section is to give a characterization for more general alphabet group ring. if r is a semisimple commutative ring and g is a finite abelian group, we characterize when rg is a code-checkable group ring. we also give another characterization for any finite commutative group ring to be code-checkable. proposition 4.1. let r be a finite commutative ring and g a finite abelian group. then rg is codecheckable if and only if it is a pir. 119 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 proof. assume that rg is code-checkable. let i be an ideal of rg. if i is {0} or rg, then it is principal. assume that i is non-trivial. then there exists a zero-divisor v ∈ rg such that i = {x : xv = 0} = ann(v). define f : rg/ann(v) → rgv,where x + ann(v) 7→ xv, forall x ∈ rg. it can easily be seen that f is well-defined and bijection. then |rg/ann(v)| = |rgv|. now, rgv is a non-trivial ideal then, from our assumption, rgv = ann(u) for some u ∈ rg. then v = 1.v ∈ ann(u), so vu = 0. commutativity of rg implies that uv also equals zero. we claim that ann(v) = rgu. let y = xu ∈ rgu, then yv = xuv = 0. hence rgu ⊆ ann(v). we also have, |rg/ann(v)| = |rgv| and |rg/ann(u)| = |rgu|, thus |rgu| = |rg/ann(u)| = |rg|/|ann(u)| = |rg|/|rgv| = |ann(v)|. hence, i = ann(v) = rgu is a principal ideal. conversely, assume that rg is a pir. let j denote the set of all non-trivial ideals of rg. from the finiteness of rg, it follows that |j| is finite. let σ : j → j be defined by: rga 7→ ann(a). using that r is commutative, we can show that ann(rgb) = ann(b) for every b ∈ rg. if rga = rgb, then ann(a) = ann(rga) = ann(rgb) = ann(b). this implies that σ is well-defined. to show that σ is injective, assume that σ(rga) = σ(rgb), i.e. ann(a) = ann(b). since rg is pir, then ann(a) = ann(b) = rgv, for some v ∈ rg. hence, by the first part of the proof, we have rga = ann(v) = rgb. since |j| is finite and σ is injective, then σ is surjective. this implies that every non-trivial ideal of rg is a checkable code. we will use a strong result, theorem 2.9, of dorsey [[2], theorem 4.4] to complete our characterization. it gives a characterization for a group ring rg to be pir, when r is a semisimple ring and g is any finite group. using this, the complete result follows in the following theorem. theorem 4.2. let g be a finite abelian group, r a finite commutative semisimple ring and π the set of noninvertible primes in r. then the group ring rg is code-checkable if and only if g is π′-by-cyclic π. proof. follows immediately from theorem 2.9 and proposition 4.1. 4.1. the eight p-conditions in order to present our last result, we first need to recall some notions about rings. a partially ordered set p is said to satisfy the ascending chain condition (acc) if every strictly ascending sequence of elements terminates. similarly, p is said to satisfy the descending chain condition (dcc) if every strictly descending sequence of elements terminates. those two conditions are often called the finiteness conditions for the partially ordered set. a ring r is called artinian if r satisfies the dcc on the set of all ideals of r. another kind of finiteness conditions on a ring is called the eight p-conditions. here are the eight p-conditions: (i) acc on {annl(b) : b ∈ r}. (v) dcc on {annl(b) : b ∈ r}. (ii) acc on {annr(b) : b ∈ r}. (vi) dcc on {annr(b) : b ∈ r}. (iii) acc on {ra : a ∈ r}. (vii) dcc on {ra : a ∈ r}. (iv) acc on {ar : a ∈ r}. (viii) dcc on {ar : a ∈ r}. 120 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 it turns out that the eight p-conditions have some symmetry when the ring is pseudo-morphic. that is [1], if r is pseudo-morphic, then the eight p-conditions are all equivalent. since we work only on finite rings, then we don’t have to worry about any finiteness condition. in fact, any finiteness condition is satisfied for all finite rings. the following theorem by camillo and nicholson [[1], theorem 6.3.] presents the relation between r being pseudo-morphic, quasi-morphic and principal ideal ring, when r satisfies some finiteness conditions. theorem 4.3. the following conditions are equivalent for a ring r: (i) r is pseudo-morphic and satisfies any of the eight p-conditions. (ii) r is quasi-morphic and satisfies any of the eight p-conditions. (iii) r is an artinian principal ideal ring. if the ring r is finite, then it follows immediately that r is both artinian and satisfies the eight p-conditions. the following corollary follows from (i) and (iii) from the last theorem. corollary 4.4. let r be a finite ring. r is pseudo-morphic if and only if r is a pir. this allows us to present our last result for this paper. theorem 4.5. let r be a finite commutative ring and g a finite abelian group. the following are equivalent: 1) rg is code-checkable. 2) rg is a pir. 3) rg is pseudo-morphic. proof. direct consequence of proposition 4.1 and corollary 4.4 example 4.6. consider the group ring z4c2, where c2 is the cyclic group of order three. write c2 as c2 = {e,a}, then z4c2 = {0,1,2,3,a,2a,3a,1 + a,2 + a,3 + a,1 + 2a,2 + 2a,3 + 2a,1 + 3a,2 + 3a,3 + 3a}. using [[1], example 3.3], the group ring z4c2 is not a pseudo-morphic ring. therefore, by theorem 4.5, z4c2 is neither code-checkable nor pir, we show this in the following. naively, we are going to construct the multiplication table of z4c2 as in table 1. consider the ideal i = 〈2 + 2a〉 generated by 2 + 2a. by the multiplication table, i = {0,2 + 2a}. so i is a checkable code if there is a check element x ∈ z4c2 such that xy = 0 iff y ∈ i, for every y ∈ z4c2. since i has only one nonzero element, this means that x is a check element of i implies that xy = 0 only when y = 2 + 2a or y = 0. by looking at the multiplication table, there is no such an element x in z4c2. therefore, i is not checkable and hence z4c2 is not code-checkable. also, we can show that z4c2 is not a pir. consider the ideal j = 〈1 + a,1 + 3a〉 = {0,2,2a,1 + a,3 + a,2 + 2a,1 + 3a,3 + 3a}. since j does coincide with any row of our multiplication table, then j is not a principal ideal, and hence z4c2 is not a pir. finally, this example shows that the semisimplicity condition is necessary for the conclusion of theorem 4.2. in fact, the noninvertible primes of z4 is the singleton π = {2}. now, if we take the trivial subgroup h = {e} ≤ c2, then clearly h is normal in c2 with (|h|,2) = 1, so h is a π′-group. also, c2/h = c2 is cyclic and has exactly 21 elements, which means that c2/h is a cyclic π-group. therefore, c2 is π′-by-cyclic π, however z4c2 is not code-checkable. the reason that the conclusion in theorem 4.2 is not satisfied here, is because that the ring z4 is not semisimple. 121 n. abdelghany, n. megahed / j. algebra comb. discrete appl. 4(2) (2017) 115–122 table 1. multiplication table of z4c2. . 0 1 2 3 a 2a 3a 1 + a 2 + a 3 + a 1 + 2a 2 + 2a 3 + 2a 1 + 3a 2 + 3a 3 + 3a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 a 2a 3a 1 + a 2 + a 3 + a 1 + 2a 2 + 2a 3 + 2a 1 + 3a 2 + 3a 3 + 3a 2 0 2 0 2 2a 0 2a 2 + 2a 2a 2 + 2a 2 0 2 2 + 2a 2a 2 + 2a 3 0 3 2 1 3a 2a a 3 + 3a 2 + 3a 1 + 3a 3 + 2a 2 + 2a 1 + 2a 3 + a 2 + a 1 + a a 0 a 2a 3a 1 2 3 1 + a 1 + 2a 1 + 3a 2 + a 2 + 2a 2 + 3a 3 + a 3 + 2a 3 + 3a 2a 0 2a 0 2a 2 0 2 2 + 2a 2 2 + 2a 2a 0 2a 2 + 2a 2 2 + 2a 3a 0 3a 2a a 3 2 1 3 + 3a 3 + 2a 3 + a 2 + 3a 2 + 2a 2 + a 1 + 3a 1 + 2a 1 + a 1 + a 0 1 + a 2 + 2a 3 + 3a 1 + a 2 + 2a 3 + 3a 2 + 2a 3 + 3a 0 3 + 3a 0 1 + a 0 1 + a 2 + 2a 2 + a 0 2 + a 2a 2 + 3a 1 + 2a 2 3 + 2a 3 + 3a 1 3 + a a 2 + 2a 3a 1 + 3a 3 1 + a 3 + a 0 3 + a 2 + 2a 1 + 3a 1 + 3a 2 + 2a 3 + a 0 3 + a 2 + 2a 1 + 3a 0 3 + a 2 + 2a 1 + 3a 0 1 + 2a 0 1 + 2a 2 3 + 2a 2 + a 2a 2 + 3a 3 + 3a a 1 + 3a 1 2 + 2a 3 3 + a 3a 1 + a 2 + 2a 0 2 + 2a 0 2 + 2a 2 + 2a 0 2 + 2a 0 2 + 2a 0 2 + 2a 0 2 + 2a 0 2 + 2a 0 3 + 2a 0 3 + 2a 2 1 + 2a 2 + 3a 2a 2 + a 1 + a 3a 3 + a 3 2 + 2a 1 1 + 3a a 3 + 3a 1 + 3a 0 1 + 3a 2 + 2a 3 + a 3 + a 2 + 2a 1 + 3a 0 1 + 3a 2 + 2a 3 + a 0 1 + 3a 2 + 2a 3 + a 0 2 + 3a 0 2 + 3a 2a 2 + a 3 + 2a 2 1 + 2a 1 + a 3 1 + 3a 3a 2 + 2a a 3 + a 1 3 + 3a 3 + 3a 0 3 + 3a 2 + 2a 1 + a 3 + 3a 2 + 2a 1 + a 2 + 2a 1 + a 0 1 + a 0 3 + 3a 0 3 + 3a 2 + 2a 5. conclusion throughout the paper, we generalized the characterization by jitman et al [6], for a group ring rg to be code-checkable, by relaxing some of the conditions on the ring r and the group g. also, we have given example 4.6 which shows that the semisimplicity of the ring r is necessary for our characterization. in section 4, we have shown that for a group ring rg, under certain conditions, being code-checkable is equivalent to being pseudo-morphic, which is a relatively new concept for rings. acknowledgment: the authors would like to thank andré leroy for his help and comments in an earlier version. he also suggested working on pseudo-morphic rings which was the key for the last result in the paper, theorem 4.5. references [1] v. camillo, w. k. nicholson, on rings where left principal ideals are left principal annihilator, int. electron. j. algebra 17 (2015) 199–214. [2] t. j. dorsey, morphic and principal–ideal group rings, j. algebra 318(1) (2007) 393–411. [3] j. l. fisher, s. k. sehgal, principal ideal group rings, comm. algebra 4(4) (1976) 319–325. [4] p. hurley, t. hurley, module codes in group rings, proc. int. symp. information theory (isit) (2007) 1981–1985. [5] p. hurley, t. hurley, codes from zero–divisors and units in group rings, int. j. inf. coding theory (2009) 57–87. [6] s. jitman, s. ling, h. liu, x. xie, checkable codes from group rings, arxiv:1012.5498, 2010. [7] f. j. macwilliams, codes and ideals in group algebras, combinatorial mathematics and its applications (1969) 317–328. [8] w. k. nicholson, e. sánchez campos, rings with the dual of the isomorphism theorem, j. algebra 271(1) (2004) 391–406. 122 http://www.ams.org/mathscinet-getitem?mr=3310695 http://www.ams.org/mathscinet-getitem?mr=3310695 http://dx.doi.org/10.1016/j.jalgebra.2007.09.005 http://dx.doi.org/10.1080/00927877608822109 http://dx.doi.org/10.1109/isit.2007.4557511 http://dx.doi.org/10.1109/isit.2007.4557511 http://dx.doi.org/10.1504/ijicot.2009.024047 http://dx.doi.org/10.1504/ijicot.2009.024047 http://arxiv.org/abs/1012.5498 http://www.ams.org/mathscinet-getitem?mr=253797 http://www.ams.org/mathscinet-getitem?mr=253797 http://dx.doi.org/10.1016/j.jalgebra.2002.10.001 http://dx.doi.org/10.1016/j.jalgebra.2002.10.001 introduction preliminaries codes from group ring encoding code-checkable group rings conclusion references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(3) • 191-209 received: 11 may 2015; accepted: 21 august 2015 doi 10.13069/jacodesmath.36015 journal of algebra combinatorics discrete structures and applications approximate counting with m counters: a probabilistic analysis research article guy louchard1∗, helmut prodinger2∗∗ 1. université libre de bruxelles, département d’informatique, cp 212, boulevard du triomphe, b-1050 bruxelles, belgium 2. university of stellenbosch, mathematics department, 7602 stellenbosch, south africa abstract: motivated by a recent paper by cichoń and macyna [1], who introduced m counters (instead of just one) in the approximate counting scheme first analysed by flajolet [2], we analyse the moments of the sum of the m counters, using techniques that proved to be successful already in several other contexts [11]. 2010 msc: 60c05, 05a16, 68q24 keywords: approximate counting, moments, constant and fluctuating components, complex analysis, product of fourier series 1. introduction approximate counting is a technique that was first analysed by flajolet [2]; some subsequent papers [6, 12–14] added to the analysis. a counter c is kept, and each time an item arrives and needs to be counted, a random experiment is performed; if the current value of the counter is i, then with probability 2−i the counter is increased by 1, otherwise it keeps its value; at the beginning, the counter value is c = 1. after n random increments, the value of the counter is typically close to log2 n, and the cited papers contain exact and asymptotic values for average and variance. for instance, flajolet [2] gives the dominant constant part of mean and variance and the periodic part of the mean. recently, cichoń and macyna [1] used this idea as follows: instead of one counter, they keep m counters, where m ≥ 1 is an integer. for our subsequent analysis we will assume that m is fixed. when a new element arrives (and needs to be counted), it is randomly (with probability 1 m ) assigned to one ∗ e-mail: louchard@ulb.ac.be ∗∗ e-mail: hproding@sun.ac.za (corresponding author) 191 approximate counting with m counters of the m counters, and then the random experiment is performed as usual. the parameter that cichoń and macyna are interested in is the total number of changes of any counter. they also consider another strategy which is not discussed in the present paper. in other words, if we (for convenience) assume that the initial setting of a counter to the value 1 counts as a change, cichoń and macyna are interested in the sum of the values of the m counters. the paper [16] provided the first analysis of cichoń and macyna’s scheme: based on exact expressions, asymptotics for expectation and variance are derived with rice’s method. there is a price to be paid for dealing with these exact expressions, as there are computational hardships to be dealt with. let us also mention fuchs, lee and prodinger [4] who analyze this algorithm via the poisson-laplace-mellin method. in the present paper, the approach is different: going to approximations immediately, one loses exact expressions, but on the other hand the computations become much more manageable, so that one can go to higher moments, which we do here, mostly, to show the power of the method. the new interest in approximate counting that cichoń and macyna’s scheme initiated, motivated us to provide this paper: we had a long report [10] on asymptotics of the moments of extreme-value related distribution functions, but only a shortened version of it was published [11]; especially the analysis of classical approximate counting had to be left out. we present it here, together with additional material that deals with the m counters (instead of just one, as in the classical case). we also present new simplifications of products of some fourier series. let j(m,n) be the random variable (rv): “total value of the counter after n items have arrived”; we can write j(m,n) = ∑m 1 ji(n) where ji(n) is related to the ith counter. when we have only one counter, we can just write j(n). the most important motivation of the paper is to compute the asymptotic distribution and the moments of j(m,n). the asymptotic distribution is related to the extreme-value gumbel distribution function (df): exp(−exp(−x)). the moments are usually given by a constant part and a small fluctuating part. there we use laplace and mellin transforms and singularity analysis. our aim is to derive an (almost) purely mechanical computation of constant and fluctuating components, with the help of computer algebra systems (we use maple here). as an example, we provide the first four moments, (even the third moment is very rarely computed in the literature) but the treatment is completely automatic (with some human guidance of course). the fourth moment is particularly interesting: it presents a wide variety of combinatorial and mathematical constants as well as several types of fourier series (including products of them). a last but not least motivation is to simplify the analytic treatment by using only easy complex analysis: only simple poles are needed, and we do not use alternating series (so we do not need rice’s formula). a small number of analytic functions are the only tools we need. this should be compared with the complicated techniques sometimes used in previous papers. we have uniform integrability for the moments of our rv’s. to show that the limiting moments are equivalent to the moments of the limiting distributions, we need a suitable rate of convergence. this is related to a uniform integrability condition (see loève [8, section 11.4]). the total error term related to our asymptotics of moments is detailed in [11]; it is given by o(n−c), where c is some constant. another technical point of interest are the periodic oscillations that always occur in approximate counting and related questions: when one goes for higher moments, there are many extra terms coming in, making the fourier coefficients very complicated. in particular, there are high powers of some (simple) periodic functions. we present a technique to bring the fourier coefficients of these into some standard form, using residue calculus. to summarize, we had several motivations to write this paper: • show the power of the method we introduced in [11] • present, with great details, the analysis of classical approximate counting • give new simplifications of product of some fourier series 192 g. louchard, h. prodinger • analyze, with precision, approximate counting with m counters the paper is organized as follows: definitions, notations and known properties are recalled in section 2. classical approximate counter (1 counter) is analyzed in section 3. the case of m counters is considered in section 4. the asymptotic moments of j(n) are given in section 5. section 6 concludes the paper. 2. definitions, notations and known properties let us first give the notations we will need throughout the paper. they will be used in section 3, but we prefer to summarize them here instead of introducing them one by one. l := ln 2, log := log2, ε := small real > 0, α̃ := α/l, q(j) := j∏ k=1 (1 − 2−k), q(0) = 1, q := q(∞) = 0.288788095087 . . . , r(j) := (−1)j+1 2j(j+1)/2q(j) , r(0) = −1, χl := 2πil l , ρk := ∑ l 6=0 γ(χl)ψ(χl) ke−2lπi log n, k = 1, 2, 3, ρ4 := ∑ l 6=0 γ(χl)ψ(1,χl)e −2lπi log n. these functions appear in many analyses of algorithms (see, for instance, flajolet [2], flajolet and sedgewick [3], hwang et al. [5], louchard [9], louchard and prodinger [11]). the following facts will be frequently used: q(i) ≥ q, (1 −u)n = e−nu [ 1 −nu2/2 + o(nu3) ] , u ∈ ]0, 1[ . for the integer-valued rv j (from now on, we drop i and n from ji(n) in order to ease the notation; j is now related to one counter, with n items), we set p(j) := pr(j = j), p(j) := pr(j ≤ j). setting η = j − log n, we will first compute f and f such that p(j) ∼ f(η), p(j) ∼ f(η), n →∞, and, of course, f(η) = f(η) −f(η − 1). asymptotically, the distribution will be a periodic function of log n in the following sense: fix n; p(j(n) ≤ j) ∼ f(η),η = j − log n = j −blog nc−{log n}. set log n′ = log n + 1 (hence n′ = 2n, the base could of course be changed). then p(j(n′) ≤ j + 1) ∼ f(j + 1 − log n′) = f(η). the asymptotic distribution of j(n′) is the same as the one of j(n), only shifted by 1. all n with the same {log n} lead to the same points of f(η). the distribution p(j) does not converge in the weak sense, it does however converge along subsequences nm for which the fractional part of log nm is constant. this type of convergence is not uncommon in the analysis of algorithms. many examples are given in [11]. 193 approximate counting with m counters next, we must check that e ( jk ) = ∑ j jkp(j) ∼ ∑ j (η + log n)kf(η), (1) by computing a suitable rate of convergence. this is related to a uniform integrability condition (see loève [8, section 11.4].) 3. classical approximate counting (one counter). analysis of j(n) in this section, we provide the asymptotic distribution and the first four asymptotic moments of the rv j(n) based on n items. from flajolet [2, proposition 1], we have p(j) = j−1∑ k=0 2k −r(k) q(j − 1 −k) (1 − 1/2j−k)n. (2) letting η = j − log n, it is proved in [2] that f(η) = ∞∑ k=0 2k −r(k) q exp(−2−η+k). (compare also [6, 13, 14].) it has been pointed out in [9] that it has some similarities with the digital search tree distribution. actually, as noticed by s. janson (private communication), the distribution for approximate counting is the same as for unsuccessful search in digital search trees (not only asymptotically). the rate of the convergence problem is completely solved in flajolet [2]. also, we obtain by summing p(j) = − j−1∑ k=0 j∑ u=k+1 2k r(k) q(u− 1 −k) (1 − 1/2u−k)n, f(η) = − ∞∑ k=0 2k r(k) q ∞∑ i=k exp(−2−η+i). (3) we note that the algorithm can be generalized by changing the base. the analysis is quite similar, and we won’t provide details here. the first three asymptotic moments are given in our unpublished report [10]. we take the opportunity to present them here, with some complements. in particular we analyze the product of fourier series, which leads to convolutions of the coefficients. in order to show the power of the methods we use, we also give the fourth moment. using the techniques we described in our published paper [11], we proceed as follows. 3.1. some preliminary identities some preliminary identities are necessary. they appear in the asymptotic moments, but we prefer to avoid overloading sections 3.3 and 3.4. using a classical euler identity, we will derive several summation formulae. this identity is ∞∏ k=0 (1 + qkz) = ∞∑ i=0 ziqi(i−1)/2/q(i). 194 g. louchard, h. prodinger (for a simple proof, see knuth [7, ex 5.1.1–16]). it holds for general q, once the definition of q(i) it adapted by replacing 1 2 by q. set π∗ := ∞∏ k=1 (1 + qkz), π := (1 + z)π∗, σk := (k − 1)! ∞∑ i=1 qki/(1 + zqi)k. it is not hard to see that, with z = −1, q = 1/2, we get the following expansions π∗ → q, σ1 → c1, σ2 → c2, σ3 → 2!c3, σ4 → 3!c4, with the abbreviations ck := ∞∑ j=1 1 (2j − 1)k . now we want to compute sums of type uk := ∞∑ i=0 ik2ir(i). we obtain π∗ ′ = σ1π ∗, π′ = π∗ + (1 + z)σ1π ∗, and setting z = −1, q = 1/2, we derive u1 = q. similarly, set t1 := zπ′, compute t ′1, etc. with the same procedure, we obtain: u2 = −q(−1 + 2c1), u3 = q(1 − 6c1 − 3c2 + 3c21 ), u4 = −q(−1 + 14c1 + 18c2 − 18c21 + 8c3 − 12c1c2 + 4c 3 1 ), u5 = q(1 − 30c1 − 75c2 + 75c21 − 80c3 + 120c1c2 − 40c 3 1 − 30c4 + 40c1c3 + 15c22 − 30c 2 1c2 + 5c 4 1 ). (4) also u0 = 0; (5) this is equation (21) in flajolet [2]. more generally, setting z = −2k, q = 1/2 in π, we derive ∞∑ i=0 2(k+1)ir(i) = 0. (6) 3.2. slow increase property it is necessary that the functions we need decrease exponentially in the direction i∞. indeed, (see ([11] for details), some integration in the complex plane (along a rectangle) must be convergent. it will appear that all functions we use here are analytic (in some domain), depending on classical functions such as γ, ζ, ψ(k,s) (the (k + 1)-gamma function). 195 approximate counting with m counters we know that γ(s) decreases exponentially in the direction i∞: |γ(σ + it)| ∼ √ 2π|t|σ−1/2e−π|t|/2. also, we have a “slow increase property” for all functions we encounter: let s = σ + it, |ζ(s)| = o(|t|1−σ), σ < 1. we will also use the function h2(−ls), with h2(α) := ∞∑ k=0 eαk2kr(k). to analyze this function, we use the “sum splitting technique” as described in knuth [7, p. 131], and used in flajolet [2]: let σ > −1 and ρ(x) be an increasing function. for k < ρ(|s|), the contribution is bounded by ρ(|s|) sup 0≤k≤ρ(|s|) ( 1 2σk+k 2 ) = o ( 1 + ρ(|s|)2ρ(|s|) ) ; for k ≥ ρ(|s|), the contribution is bounded by ∞∑ k=ρ(|s|) 2k 2k 2 = o ( 2ρ(|s|) 22ρ(|s|) ) . choosing ρ(x) = log x insures the slow increase property. s. janson (private communication) mentioned that, with θ(z) := ∏ k≥1(1 −z/2 k), we have h2(α) = −θ(2eα) = (eα − 1)θ(eα). this easily leads to h2(α) ∼−q(α−l) + o((α−l)2), h2(α) ∼ 3q(α− 2l) + o((α− 2l)2), h2(α) ∼−21q(α− 3l) + o((α− 3l)2), (7) and similar asymptotics for α = kl,k ≥ 4. 3.3. the asymptotic moments the detailed proofs of relations (8) to (17) are given in [11]. let an (integer-valued) rv k be such that pr(k − log n ≤ η) ∼ f(η), where f(η) is the df of a continuous rv z with mean m1, second moment m2, variance σ2 and centered moments µk. assume that f(η) is either an extreme-value df or a convergent series of such and that (1) is satisfied. let ϕ(α) = e(eαz) = 1 + ∞∑ k=1 αk k! mk = e αm1λ(α), (8) say, with λ(α) = 1 + α2 2 σ2 + ∞∑ k=3 αk k! µk. (9) 196 g. louchard, h. prodinger also ϕ(α) = ∫ +∞ −∞ eαηf ′(η)dη. (10) we have here, for j, φ(α) := ∫ ∞ −∞ eαηf(η)dη = − 1 ql h2(α)γ(−α̃), <(α) < 0. but we need a larger range for α. but, by (7), we can continue φ(α) analytically for all α: all singularities of γ(−α̃) are cancelled by the successive roots of h2(α). moreover, this implies eαηf(η) → 0, η →∞, η →−∞. (11) now, we have ∫ +∞ −∞ eαη[f ′(η) −f ′(η − 1)]dη = (1 −eα)ϕ(α) = [eαη[f(η) −f(η − 1)]]∞−∞ −α ∫ +∞ −∞ eαη[f(η) −f(η − 1)]dη = [eαηf(η)] ∞ −∞ −α ∫ +∞ −∞ eαηf(η)dη = −αφ(α), by (11). this gives ϕ(α) = α eα − 1 φ(α) = − h2(α)γ(1 − α̃) q(1 −eα) . this leads to (we give only the first two expressions for mi) m1 = γ l −c1, m2 = π2 + 6γ2 6l2 − 2γc1 l −c2 + c21 −c1, µ2 = σ 2 = π2 6l2 −c1 −c2, µ3 = 2ζ(3) l3 − 2c3 − 3c2 −c1, µ4 = 3c 2 1 −c1 − 7c2 − 12c3 + 3c 2 2 − π2c1 l2 + 6c1c2 − π2c2 l2 + 3π4 20l4 − 6c4. let w, κ’s (with or without subscripts) denote periodic functions of log n, with period 1 and 0 mean (for w) or non-zero small mean (for κ) and small (of order 10−4) amplitude. actually, these functions depend on the fractional part of log n: {log n}. the moments of j(n)−log n are asymptotically given by m̃i +wi. the constant part m̃i comes from the residue of some function at 0 (see [11]) and wi comes from the residues of the same function at some complex poles. the generating function of m̃i is given by φ(α) := ∫ ∞ −∞ eαηf(η)dη = 1 + ∞∑ i=1 αi i! m̃i = ϕ(α) eα − 1 α . (12) 197 approximate counting with m counters this leads to m̃1 = 1 2 −c1 + γ l , m̃2 = π2 + 6γ2 6l2 + γ − 2γc1 l + 1 3 − 2c1 −c2 + c21. more generally, the centered moments of j(n) are asymptotically given by µi = µ̃i + κi, with the asymptotic dominant constant centered moment generating function given by θ(α) := 1 + ∞∑ k=2 αk k! µ̃k = 2 α sinh(α/2)λ(α). (13) the neglected part is of order 1/nβ with 0 < β < 1. we derive, with (4), the following result about these centered moments. theorem 3.1. the asymptotic dominant constant parts of the centered moments of j(n) are given by µ̃2 = π2 6l2 + 1 12 −c1 −c2, µ̃3 = 2ζ(3) l3 − 2c3 − 3c2 −c1, µ̃4 = 1 80 + 3c21 − 3c1 2 − 15c2 2 − 12c3 − 6c4 + 3π4 20l4 + π2 12l2 + 6c1c2 − π2c1 l2 − π2c2 l2 + 3c22. note that θ(α) = φ(α)e−αm̃1. now e(j(n) − log n)k ∼ m̃k + wk, with wk = 1 l ∑ l 6=0 υ∗k(χl)e −2lπi log n, (14) and υ∗k(s) = l φ (k)(α) ∣∣∣ α=−ls . (15) with (5), we check that we have no singularity of φ(k), k > 0, at α = 0. the fundamental strip for (15) is <(s) ∈ 〈−1, 0〉. we first obtain w1 = − 1 l ∑ l 6=0 γ(χl)e −2lπi log n; m̃1, µ̃2 and w1 are identical to the expressions given in flajolet [2]. to compute the periodic components κi (with non-zero small mean) to be added to the centered moments µ̃i, we first set m1 := m̃1 + w1. 198 g. louchard, h. prodinger now, we start from φ(α) := 1 + ∞∑ k=1 αk k! m̃k = ϕ(α) eα − 1 α . we replace m̃k by m̃k + wk, leading to φp(α) = φ(α) + ∞∑ k=1 αk k! wk. but it is easy to check that ∑ l 6=0 φ(−lχl)e −2lπi log n = 0, so we obtain φp(α) = φ(α) + ∞∑ k=0 ∑ l 6=0 φ(k)(α) ∣∣∣ α=−lχl e−2lπi log n αk k! = φ(α) + ∑ l 6=0 φ(α−lχl)e −2lπi log n. (16) finally, we compute θp(α) = φp(α)e −αm1 = 1 + ∞∑ k=2 αk k! (µ̃k + κk) = θ(α) + ∞∑ k=2 αk k! κk, (17) leading to the (exponential) generating function of κk. by expansion and taking differences, we have a result about the oscillating parts. theorem 3.2. the asymptotic oscillating parts of the centered moments of j(n) are given by κ2 = −w21 − 2γw1 l + 2 l2 ρ1, κ22 = w 4 1 + 4γ2w21 l2 + 4 l4 ρ21 + 4γw31 l − 4w21 l2 ρ1 − 8w1 l3 ρ1, κ3 = 4l2w21 + 12w1lγ + 6γ 2 −π2 2l2 w1 − 6(γ + w1l) l3 ρ1 − 3 l3 ρ2 − 3 l3 ρ4, κ4 = w1 [ −w1/2 + 12γc2 l + 12γc1 l − 3w31 + π2w1 l2 − 8ζ(3) l3 + 6c1w1 + 6w1c2 − 12γ2w1 l2 − 4γ3 l3 − 12w21γ l − γ l ] + l2 − 12c2l2 − 12c1l2 + 24γw1l + 12w21l2 + 12γ2 l4 ρ1 + 12 l4 ∑ l 6=0 e−2lπi log nψ(χl)ψ(1,χl)γ(χl) + 12(w1l + γ) l4 ρ4 + 4 l4 ∑ l 6=0 e−2lπi log nψ(2,χl)γ(χl) + 4 l4 ρ3 + 12(w1l + γ) l4 ρ2. all algebraic manipulations of this paper are mechanically performed by maple.1 1 γ′(x) = γ(x)ψ(x), γ′′(x) = γ(x)ψ(1,x) + γ(x)ψ2(x), γ′′′(x) = γ(x)ψ(2,x) + 3γ(x)ψ(1,x)ψ2(x) + γ(x)ψ3(x) etc. 199 approximate counting with m counters note that the non-zero κi mean must be added to the dominant constant parts of the centered moments. this will be considered in the next section. 3.4. the corrections products of fourier series may have a constant term, even if the factors do not. this term must be included in the dominant constant part of our moments. this is the object of the present subsection. we denote by [f]k the coefficient of e −2kπi log n in the fourier expansion of f. in [11], we have proved the following relations c1[0] := [w 2 1]0 = 1 l2 ∑ k 6=0 γ(−χk)γ(χk) = − 2 l d − 11 12 + π2 6l2 with d := ∑ l≥1 (−1)l l(2l − 1) . the coefficient c1[k] of e−2kπi log n in the fourier expansion of w21 is given by c1[k] = 1 l2 ∑ j 6=0,k γ(−χj)γ(χk + χj) = 2 l ∑ l≥1 (−1)lγ(χk + l) l!(2l − 1) + 2 l2 γ(χk) ( ψ(χk) + γ ) , c2[0] := [w 3 1]0 = 1 + 2ζ(3) l3 + 1 l d − 6 l2 d1 − 2 log 3 l − 2 l d2, with d1 = ∑ l≥1 (−1)lhl−1 l(2l − 1) , d2 = ∑ l,j≥1 (−1)l+j (l + j)(2l − 1) [ 1 2j − 1 + 1 2j+l − 1 ]( l + j j ) . this has been checked numerically and gives the tiny value −9.428177 × 10−25. the coefficient c2[k] of e−2kπi log n in the fourier expansion of w31 is given by c2[k] = −   2l3 [ 2l ∑ l≥1 (−1)l l!(2l − 1) γ(l + χk) ( ψ(l + χk) + γ ) −l2 ∑ l≥1 (−1)lγ(l + χk) l! 2l (2l − 1)2 + 1 12 ( 18ψ(1,χk) + 18 ( ψ(χk) + γ )2 − 11l2 −π2)γ(χk) ] + 2 l2 ∑ l≥1 (−1)l l!(2l − 1) [ l ∑ h≥0 (−1)h h!(2h+l − 1) γ(h + l + χk) + l ∑ h≥1 γ(l + h + χk) (−1)h h!(2h − 1) + lγ(l + χk)2 −l −lγ(l + χk) − (l− 1)!γ(χk) + γ(χk) ( ψ(χk) + γ + l 2 )] + [ π2 6l3 + 1 12l − 1 l − 2 l2 ∑ l≥1 (−1)l−1 l(2l − 1) ] γ(χk)   . 200 g. louchard, h. prodinger for a complete description of the fourier coefficients of the oscillations occurring in the third and fourth moment, we need the following expressions (note that maple splits the higher derivatives of the gamma function; if one could rework that, one could reduce the number of necessary expressions): c3[0] := [w1ρ1]0 , c4[0] := [w 4 1]0, c5[0] := [ (ρ1) 2 ] 0 , c6[0] := [ w21ρ1 ] 0 , c7[0] := [w1ρ4]0 , c8[0] := [w1ρ2]0 , c3[k] := [w1ρ1]k , c4[k] := [w 4 1]k, c5[k] := [ (ρ1) 2] k , c6[k] := [ w21ρ1 ] k , c7[k] := [w1ρ4]k , c8[k] := [w1ρ2]k . we will show now how to “compute” some fourier coefficients we need. “computing” is perhaps a very ambitious word, it might be better replaced by “rewriting”. we have w1 = − 1 l ∑ k 6=0 γ(χk)e −2πikx, and higher powers have convolutions as coefficients: w41 = 1 l4 ∑ k 6=0 ∑ k1+k2+k3+k4=k γ(χk1 )γ(χk2 )γ(χk3 )γ(χk4 )e −2πikx, and none of the k1, . . . ,k4 is allowed to be zero. the only thing that we are able to achieve is to have only one gamma-term in the (multiple) sum, where a typical term might look like∑ j1+···+jt=j c (1) j1 . . .c (t) jt γ(s)(χk + j)e −2πikx. such a representation is not a priori better than the straight-forward convolution, but we will sketch now how to achieve them. one (small) advantage is that the zeroth term can be explicitly determined, and extracted, and what is left is then oscillating around zero. as an example, we consider w := w21 − [w 2 1]0 = ∑ k 6=0 c1[k]e −2πikx, with c1[k] = 2 l ∑ l≥1 (−1)lγ(χk + l) l!(2l − 1) + 2 l2 ( γ′(χk) + γγ(χk) ) . let us discuss w2. it is clear that the convolution of w with itself contains already several terms. for instance, [w2]0 = 4 l2 ∑ j,l≥1 (−1)j+l j!(2j − 1)l!(2l − 1) ∑ k 6=0 γ(−χk + j)γ(χk + l) + 8 l3 ∑ l≥1 (−1)l l!(2l − 1) ∑ k 6=0 γ(−χk + l) ( γ′(χk) + γγ(χk) ) + 4 l4 ∑ k 6=0 ( γ′(−χk) + γγ(−χk) )( γ′(χk) + γγ(χk) ) . we would like to demonstrate how to rewrite the k-sums in these expressions. the survey paper [15] has many similar examples. the approach we found most versatile is via residue calculus. one writes a suitable function, computes all the residues in the complex plane, and the sum of them is zero. there are of course some technical subtleties, like showing that integral tends to zero for larger and larger radii, 201 approximate counting with m counters and also there are usually some series that do not converge absolutely. the suitable limit of them is the abel limit, i.e., consider a power series in x, and let x tend to a point at the boundary of convergence. here, we want to concentrate on the computational part only. the function that is suitable for the first part is l 2z − 1 γ(j −z)γ(l + z). a first contribution comes from the poles at z = χk: s1 = ∑ k∈z γ(j −χk)γ(l + χk). the next contribution stems from the poles at z = j,j + 1, . . . : s2 = (l + j − 1)!l ∑ h≥0 (−1)h−1 2j+h − 1 ( l + j + h− 1 h ) . now we look at the poles at z = −l,−l− 1, . . . : ∑ h≥0 l 2−l−h − 1 γ(j + l + h) (−1)h h! . this series does not converge absolutely. it is best to pull out the “bad” part, which leads to two contributions: s3 = (l + j − 1)!l ∑ h≥0 (−1)h−1 2l+h − 1 ( l + j + h− 1 h ) , s4 = (l + j − 1)!l ∑ h≥0 (−1)h−1 ( l + j + h− 1 h ) . as announced, we must interpret s4 as a limit: s4 = −(l + j − 1)!l lim x→1 ∑ h≥0 (−x)h ( l + j + h− 1 h ) = −(l + j − 1)!l2−l−j. altogether we found ∑ k 6=0 γ(j −χk)γ(l + χk) = −(j − 1)!(l− 1)! − (l + j − 1)!l ∑ h≥0 (−1)h−1 2j+h − 1 ( l + j + h− 1 h ) − (l + j − 1)!l ∑ h≥0 (−1)h−1 2l+h − 1 ( l + j + h− 1 h ) + (l + j − 1)!l2−l−j. now, let us look at ∑ k 6=0 γ(−χk + l) ( γ′(χk) + γγ(χk) ) . the proper function is l 2z − 1 γ(−z + l) ( γ′(z) + γγ(z) ) . 202 g. louchard, h. prodinger the poles at z = χk, k 6= 0, lead to s1 = ∑ k 6=0 γ(−χk + l) ( γ′(χk) + γγ(χk) ) . the pole at z = 0 leads to s2 = γ(l) (π2 −l2 12 − γ2 2 − lγ 2 ) − γ′(l) ( γ + l 2 ) − 1 2 γ′′(l). the poles at z = l, l + 1, . . . lead to s3 = l ∑ h≥0 1 2l+h − 1 (−1)h h! ( γ′(l + h) + γγ(l + h) ) = l ∑ h≥0 1 2l+h − 1 (−1)h(l + h− 1)! h! hl+h. the poles at z = −1,−2, . . . lead to − ∑ h≥1 l 1 − 2−h γ(h + l) (−1)hγ h! = −lγ ∑ h≥1 1 2h − 1 (−1)h(l + h− 1)! h! + lγ(1 − 2−l). therefore∑ k 6=0 γ(−χk + l) ( γ′(χk) + γγ(χk) ) = −γ(l) (π2 −l2 12 − γ2 2 − lγ 2 ) + γ′(l) ( γ + l 2 ) + 1 2 γ′′(l) −l ∑ h≥0 1 2l+h − 1 (−1)h(l + h− 1)! h! hl+h + lγ ∑ h≥1 1 2h − 1 (−1)h(l + h− 1)! h! −lγ(1 − 2−l). finally, let us consider ∑ k 6=0 ( γ′(−χk) + γγ(−χk) )( γ′(χk) + γγ(χk) ) . the function of interest is l 2z − 1 ( γ′(−z) + γγ(−z) )( γ′(z) + γγ(z) ) . the poles at z = χk, k 6= 0, lead to s1 = ∑ n 6=0 ( γ′(−χk) + γγ(−χk) )( γ′(χk) + γγ(χk) ) . the pole at z = 0 leads to s2 = − 11π4 360 − l2π2 72 − l4 720 . the poles at z = 1, 2, . . . lead to s3 = lγ ∑ h≥1 (−1)h−1hh h(2h − 1) . 203 approximate counting with m counters the poles at z = −1,−2, . . . lead to s3 = −lγ ∑ h≥1 (−1)hhh h(1 − 2−h) = lγ ∑ h≥1 (−1)h−1hh h(2h − 1) −lγ ∑ h≥1 (−1)h−1hh h . altogether ∑ k 6=0 ( γ′(−χk) + γγ(−χk) )( γ′(χk) + γγ(χk) ) = 11π4 360 + l2π2 72 + l4 720 − 2lγ ∑ h≥1 (−1)h−1hh h(2h − 1) + lγ ∑ h≥1 (−1)h−1hh h . the last alternating sum has a closed form evaluation: ∑ h≥1 (−1)hhh h = l2 2 − π2 12 . this gives us the constant term in w41: c4[0] = [w 4 1]0 = ( 2 l ∑ h≥1 (−1)h−1 h(2h − 1) − 11 12 + π2 6l2 )2 + 4 l2 ∑ j,l≥1 (−1)j+l j!(2j − 1)l!(2l − 1) × [ −(j − 1)!(l− 1)! − (l + j − 1)!l ∑ h≥0 (−1)h−1 2j+h − 1 ( l + j + h− 1 h ) − (l + j − 1)!l ∑ h≥0 (−1)h−1 2l+h − 1 ( l + j + h− 1 h ) + (l + j − 1)!l2−l−j ] + 8 l3 ∑ l≥1 (−1)l l!(2l − 1) × [ −γ(l) (π2 −l2 12 − γ2 2 − lγ 2 ) + γ′(l) ( γ + l 2 ) + 1 2 γ′′(l) −lγ(1 − 2−l) −l ∑ h≥0 1 2l+h − 1 (−1)h(l + h− 1)! h! hl+h + lγ ∑ h≥1 1 2h − 1 (−1)h(l + h− 1)! h! ] + 4 l4 [ 11π4 360 + l2π2 72 + l4 720 − 2lγ ∑ h≥1 (−1)h−1hh h(2h − 1) −lγ (l2 2 − π2 12 )] . although a few simplifications in this expression are still possible, it is clear that the complexity of the expressions does not make it attractive enough to write more similar evaluations. we can now compute the corrected values. theorem 3.3. taking the contribution of products of fourier series into account, the corrected asymp204 g. louchard, h. prodinger totic constant and oscillating parts of the centered moments of j(n) are given by µ̃2,c = µ̃2 − c1[0] = 1 −c1 −c2 + 2 d l , µ̃3,c = µ̃3 + 2c2[0] + 6 l γc1[0] − 6 l2 c3[0] = −c1 + 6ζ(3) l3 − 2c3 + 2 + 2 d l − 12 l2 d1 − 12 l2 γd − 11 2l γ + γπ2 l3 − 3c2 − 4 l d2 − 4 l log(3) − 6 l2 c3[0], µ̃4,c = µ̃4 − 1 2 c1[0] − 3c4[0] + π2 l2 c1[0] + 6c1c1[0] + 6c2c1[0] − 12 l2 γ2c1[0] − 12 l γc2[0] + 24 l3 γc3[0] + 12 l2 c6[0] + 12 l3 c7[0] + 12 l3 c8[0] κ2,c = − ∑ l 6=0 c1[l]e −2lπi log n − 2γw1 l + 2 l2 ρ1, κ3,c = 2 ∑ l 6=0 c2[l]e −2lπi log n + 6 l γ ∑ l 6=0 c1[l]e −2lπi log n + (6γ2 −π2)w1 2l2 − 6γ l3 ρ1 − 6 l2 ∑ l 6=0 c3[l]e −2lπi log n − 3 l3 ρ2 − 3 l3 ρ4, κ4,c = [ 12γc2 l + 12γc1 l − 8ζ(3) l3 − 4γ3 l3 − γ l ] w1 + l2 − 12c2l2 − 12c1l2 + 12γ2 l4 ρ1 + 12 l4 ∑ l 6=0 e−2lπi log nψ(χl)ψ(1,χl)γ(χl) + 12γ l4 ρ4 4 l4 ∑ l 6=0 e−2lπi log nψ(2,χl)γ(χl) + 4 l4 ∑ l 6=0 ρ3 + 12γ l4 ρ2 − 1 2 ∑ l 6=0 c1[l]e −2lπi log n − 3 ∑ l 6=0 c4[l]e −2lπi log n + π2 l2 ∑ l 6=0 c1[l]e −2lπi log n + 6c1 ∑ l 6=0 c1[l]e −2lπi log n + 6c2 ∑ l 6=0 c1[l]e −2lπi log n − 12 l2 γ2 ∑ l 6=0 c1[l]e −2lπi log n − 12 l γ ∑ l 6=0 c2[l]e −2lπi log n + 24 l3 γ ∑ l 6=0 c3[l]e −2lπi log n + 12 l2 ∑ l 6=0 c6[l]e −2lπi log n + 12 l3 ∑ l 6=0 c7[l]e −2lπi log n + 12 l3 ∑ l 6=0 c8[l]e −2lπi log n. note that µ̃2,c fits with the result given in [16]. 4. m counters: asymptotic independence of the m counters in this section, we analyze the asymptotic properties of the rv j(m,n). we will prove that, asymptotically, the counters are independent with n/m items each. we must analyze the random variable j(m,n) = ∑m 1 ji(n) where ji(n) has the distribution pη(j) with η is now given by νi: the number of items arriving in counter i. the quantity νi is bin[ñ, ñ(1 − 1/m))] with ñ := n/m. actually, {ν1, . . . ,νm} is given by a multinomial distribution. we know that we can construct a “box” [ñ− ñθ, ñ + ñθ]m, 1 2 < θ < 1, (18) 205 approximate counting with m counters such that, by large deviation analysis, the probability that {ν1, . . . ,νm} is outside this box is bounded by exp(−cñ2θ−1). we will analyze j(m,n) −m log(ñ) = m∑ i=1 xi, with xi := ji − log(ñ). the rate of convergence is analyzed as follows. 4.1. let us first assume that νi is exactly given by its mean ñ. as mentioned in the previous section, the rate of convergence problem is solved in flajolet [2]. 4.2. now we assume that we are inside the box (18). we drop the ˜ sign from ñ for convenience. let 0 < ε < 1. we must bound s2 := ∑ j jk ∣∣pn(j) −pn+ξnθ (j)∣∣ , with |ξ| < 1. note that n + ξnθ = n[1 + ξnθ−1]. set 1 > β > θ. • for j < β log n, we have 2−η > n1−β, |pn(j)| ≤ 1 q2 ∞∑ k=0 1 2k(k−1)/2 exp(−n1−β2k) = o(exp(−n1−β)), |pn+ξnθ (j)| ≤ 1 q2 ∞∑ k=0 1 2k(k−1)/2 exp(−n1−β(1 + ξnθ−1)2k) = o(exp(−n(1−β)(1−ε))), |pn(j) −pn+ξnθ (j)| = o(exp(−n (1−β)(1−ε))). • for β log n ≤ j < 2 log n, we have 2−η > 1 n , (1 − 1/2j)n − (1 − 1/2j)n+ξn θ = (1 − 1/2j)n(1 − (1 − 1/2j)ξn θ ) = o(nθ/2j). we use again the “sum splitting technique.” set r = √ 2 log n. 1. truncating the sum in (2) to k ≥ r leads to an error e1: e1 ≤ 1 q2 ∞∑ k=r 1 2k(k−1)/2 [ exp(− 2k n ) + exp(− 1 + ξnθ−1 n 2k) ] = o ( 1 n ) 2. the remaining sum k ≤ r leads to e2 ≤ 1 q2 r∑ k=0 1 2k(k−1)/2 [ (1 − 1/2j−k)n − (1 − 1/2j−k)n+ξn θ ] = r∑ k=0 o( nθ 2j−k ) = o( 1 n(β−θ)(1−ε) ). 206 g. louchard, h. prodinger • for j = 2 log n + x, x ≥ 0, we set r = √ 2 log n + √ 2x. so 1/2r 2/2 ≤ 2−x/n. we proceed now as in the second range 1. e1 = o ( 2−x n ) . 2. e2 ≤ 1 q2 r∑ k=0 1 2k(k−1)/2 nθ 2j−k = o ( nθ n2(1−ε)2x(1−ε) ) . now we come to s2. we get s2 = o ( (β log n)k+1 exp(−n(1−β)(1−ε)) ) + (2 log n)k+1o ( 1 n(β−θ)(1−ε) ) + o (∑ x≥0 (2 log n + x)ko (2−x n )) = o ( 1 n(β−θ)(1−ε) ) ; (not with the same ε, of course). 4.3. now we consider the case ν > n + nθ. we will show that s3 := ∑ j pν(j)j k exp(−cn2θ−1) is small. we notice that 1 < ν/n < m. but, by the rate of convergence proved in [2], ∑ j pν(j)j k is asymptotically bounded by o((log ν)k) = o((log(nm))k) and s3 is asymptotically small. 4.4. the last case to consider is 0 < ν < n−nθ. we have here 0 < ν/n < 1. the analysis proceeds like in the previous subsection. we therefore omit the details. in conclusion, as s2 and s3 are asymptotically small, we can assume that νi can be deterministically chosen as ñ for all i, and that the counters are asymptotically independent. 5. m counters: asymptotic moments if ji are iid rv, with asymptotic centered moments µk = µ̃k + κk, then j = ∑m i=1 ji has asymptotic distribution given by the convolution f(η)(m), mean m log(ñ) + mm1 and asymptotic centered moments µk(m) given by µ2(m) = mµ2, µ3(m) = mµ3, µ4(m) = m[µ4 + 3(m− 1)µ 2 2]. 207 approximate counting with m counters for µ2(m) and µ3(m), we immediately use µ2 and µ3. for µ4(m), we have µ2 = µ̃2 + κ2, so µ 2 2 = µ̃22 + κ 2 2 + 2µ̃2κ2. hence µ̃4(m) = m[µ̃4 + 3(m− 1)µ̃22], κ4(m) = m[κ4 + 3(m− 1)(κ22 + 2µ̃2κ2)]. also, we have [κ22]0 = c4[0] + 4γ2c1[0] l2 + 4c5[0] l4 + 4γc2[0] l − 4c6[0] l2 − 8c3[0] l3 , [κ22]k = c4[k] + 4γ2c1[k] l2 + 4c5[k] l4 + 4γc2[k] l − 4c6[k] l2 − 8c3[k] l3 . the corrected moments must now be computed. the interesting case is the fourth moment, since here the dependency on m is more involved: we obtain our last theorem. theorem 5.1. taking the contribution of products of fourier series into account, the asymptotic constant and oscillating parts of the corrected fourth centered moment of j are given by µ̃4,c(m) = m [ µ̃4,c + 3(m− 1)µ̃22 ] + 3m(m− 1) [ [κ22]0 − 2µ̃2c1[0] ] , κ4,c(m) = m [ κ4,c + 3(m− 1) ∑ l 6=0 [κ22]le −2lπi log n + 3(m− 1)2µ̃2κ2,c ] . 6. conclusion if we compare the approach in this paper with other ones that appeared previously, then we can notice the following. traditionally, one would stay with exact enumerations as long as possible, and only at a late stage move to asymptotics. doing this, one would, in terms of asymptotics, carry many unimportant contributions around, which makes the computations quite heavy, especially when it comes to higher moments. here, however, approximations are carried out as early as possible, and this allows for streamlined (and often automatic) computations of asymptotic distributions and higher moments. acknowledgment: we would like to thank two referees for many useful suggestions that improved the paper. references [1] j. cichoń and w. macyna, approximate counters for flash memory, 17th ieee international conference on embedded and real-time computing systems and applications (rtcsa), toyama, japan, 2011. 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[16] h. prodinger, approximate counting with m counters: a detailed analysis, theoret. comput. sci., 439, 58–68, 2012. 209 introduction definitions, notations and known properties classical approximate counting (one counter). analysis of j(n) m counters: asymptotic independence of the m counters m counters: asymptotic moments conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.27877 j. algebra comb. discrete appl. 3(3) • 195–200 received: 04 november 2015 accepted: 08 april 2016 journal of algebra combinatorics discrete structures and applications quasisymmetric functions and heisenberg doubles research article jie sun abstract: the ring of quasisymmetric functions is free over the ring of symmetric functions. this result was previously proved by m. hazewinkel combinatorially through constructing a polynomial basis for quasisymmetric functions. the recent work by a. savage and o. yacobi on representation theory provides a new proof to this result. in this paper, we proved that under certain conditions, the positive part of a heisenberg double is free over the positive part of the corresponding projective heisenberg double. examples satisfying the above conditions are discussed. 2010 msc: 16t05, 05e05 keywords: quasisymmetric function, heisenberg double, tower of algebras, hopf algebra, fock space 1. introduction symmetric functions are formal power series which are invariant under every permutation of the indeterminates ([14]). let sym denotes the ring of symmetric functions over integers. then the elementary symmetric functions form a polynomial basis of sym. the existence of comultiplication and counit gives sym a hopf algebra structure. it’s well-known (see [4]) that as hopf algebras sym is isomorphic to the grothendieck group of the abelian category c[sn]-mod, where sn is the n-th symmetric group. among different generalizations of symmetric functions, there are noncommutative symmetric functions and quasisymmetric functions. as an algebra, nsym is the free algebra z〈h1,h2, · · · 〉. as a hopf algebra, nsym is isomorphic to the grothendieck group of the abelian category hn(0)-pmod ([3], [9], [15]), where hn(0) is the 0-hecke algebra of degree n. the ring of quasisymmetric functions qsym ⊂ z[[x1,x2, · · · ]] consists of shift invariant formal power series of bounded degrees. let comp(n) denote the set of compositions of n. then the monomial quasisymmetric functions mα, where α ∈ ⋃ n∈n comp(n), form an additive basis for qsym ([14]). as hopf algebras, nsym and qsym are dual to each other under the bilinear form 〈hα,mβ〉 = δα,β. polynomial freeness of qsym was conjectured by e. ditters in 1972 in his development of the theory of formal groups ([2]). ditters conjecture was proved by m. hazewinkel ([7], [6]) combinatorially through jie sun; department of mathematical sciences, michigan technological university, houghton, mi 49931, usa (email: sjie@mtu.edu). 195 j. sun / j. algebra comb. discrete appl. 3(3) (2016) 195–200 constructing a polynomial basis for quasisymmetric functions. the explicit basis constructed by m. hazewinkel contains the elementary symmetric functions, hence qsym is free over sym. from the representation theory point of view, a. savage and o. yacobi [12] provide a new proof to the freeness of qsym over sym. for each dual pair of hopf algebras (h+,h−), one can construct the heisenberg double h = h(h+,h−) of h+. the algebra h has a natural representation on h+, called the fock space representation f. in [12] it is proved that any representation of h generated by a lowest weight vacuum vector is isomorphic to f. as an application of this stone-von neumann type theorem to h(qsym,nsym), a. savage and o. yacobi gave a different proof that qsym is free as a sym-module (proposition 9.2 in [12]). in this paper, after review some basic definitions and examples, we proved our main result theorem 3.2 which says that under certain conditions, the positive part of a heisenberg double is free over the positive part of the corresponding projective heisenberg double. examples satisfying the conditions of theorem 3.2 are discussed. 2. definitions and examples in this section, we will recall some basic definitions and examples. most of the definitions can be found in [12]. fix a commutative ring k. we use sweedler notation 4(a) = ∑ (a) a(1) ⊗ a(2) for coproducts. definition 2.1. (dual pair). we say that (h+,h−) is a dual pair of hopf algebras if h+ and h− are both graded connected hopf algebras and h± is graded dual to h∓ via a perfect hopf pairing 〈·, ·〉 : h+ ×h− → k. from a dual pair of hopf algebras, one can construct the heisenberg double. definition 2.2. (the heisenberg double, [13]). we define h(h+,h−) to be the heisenberg double of h+. as k-modules h(h+,h−) ∼= h+ ⊗ h− and we write a]x for a ⊗ x, a ∈ h+, x ∈ h−, viewed as an element of h(h+,h−). multiplication is given by (a]x)(b]y) = ∑ (x) a rx∗ (1) (b)]x(2)y =∑ (x),(b)〈x(1),b(2)〉ab(1)]x(2)y, where rx∗ (1) (b) is the left-regular action of h− on h+. for the heisenberg double h = h(h+,h−), there is a natural representation, called the fock space representation. definition 2.3. (fock space representation). the algebra h has a natural representation on h+ given by (a]x)(b) = a rx∗(b), a,b ∈ h+,x ∈ h−, which is called the lowest weight fock space representation of h and is denoted by f = f(h+,h−). it is generated by the lowest weight vacuum vector 1 ∈ h+. examples of the heisenberg double include the usual heisenberg algebra and the quasi-heisenberg algebra. when we take h+ = h− = sym, the heisenberg double h = h(sym,sym) is the usual heisenberg algebra. indeed, we can take p1,p2, · · · to be the power sums in h+ and p∗1,p∗2, · · · to be the power sums in h−, then the multiplication in h = h(sym,sym) gives the usual presentation of the heisenberg algebra: pmpn = pnpm, p ∗ mp ∗ n = p ∗ np ∗ m, p ∗ mpn = pnp ∗ m + mδm,n. when we take h + = qsym and h− = nsym, the heisenberg double h = h(qsym,nsym) is the quasi-heisenberg algebra. the natural action on qsym is the fock space representation. both the heisenberg algebra and the quasi-heisenberg algebra can be regarded as the heisenberg double associated to a tower of algebras. we now recall the definition of tower of algebras in the following. definition 2.4. let a = ⊕n∈nan be a graded algebra over a field f with multiplication ρ : a⊗a → a. then a is called a tower of algebras if the following conditions are satisfied: (ta1) each graded piece an, n ∈ n, is a finite dimensional algebra (with a different multiplication) with a unit 1n. we have a0 = f. 196 j. sun / j. algebra comb. discrete appl. 3(3) (2016) 195–200 (ta2) the external multiplication ρm,n : am ⊗an → am+n is a homomorphism of algebras for all m,n ∈ n (sending 1m ⊗1n to 1m+n). (ta3) we have that am+n is a two-sided projective am ⊗ an-module with the action defined by a · (b⊗ c) = aρm,n(b⊗ c) and (b⊗ c) ·a = ρm,n(b⊗ c)a, for all m,n ∈ n, a ∈ am+n, b ∈ am, c ∈ an. (ta4) for each n ∈ n, the pairing 〈·, ·〉 : k0(an) × g0(an) → z, given by 〈[p ], [m]〉 = dimf homan(p,m), is perfect. (note this condition is automatically satisfied if f is an algebraically closed field.) in the above definition the notation g0(an) = k0(an-mod) denotes the grothendieck group of the abelian category an-mod, and k0(an) = k0(an-pmod) denotes the grothendieck group of the abelian category an-pmod. for the rest of this section we assume that a is a tower of algebras and let g(a) = ⊕n∈ng0(an) and k(a) = ⊕n∈nk0(an). we have a perfect pairing 〈·, ·〉 : k(a) ×g(a) → z given by 〈[p ], [m]〉 = dimf homan(p,m) if p ∈ anpmod and m ∈ an-mod for some n ∈ n, and 0 otherwise. definition 2.5. (strong tower of algebras) a tower of algebras a is strong if induction is conjugate right adjoint to restriction and a mackey-like isomorphism relating induction and restriction holds. for the technical definition of the mackey-like isomorphism in definition 2.5 we refer the reader to [12] (definition 3.4 and remark 3.5). definition 2.6. (dualizing tower of algebras) a tower of algebras a is dualizing if k(a) and g(a) are dual pair hopf algebras. strong dualizing towers of algebras categorify the heisenberg double ([1]) and its fock space representation ([12] theorem 3.18). to a dualizing tower of algebras, we can define the associated heisenberg double. definition 2.7. (heisenberg double associated to a tower) suppose a is a dualizing tower of algebras. then h(a) = h(g(a),k(a)) is the associated heisenberg double and f(a) = f(g(a),k(a)) is the fock space representation of h(a). when an is the hecke algebra at a generic value of q, the associated heisenberg double is the usual heisenberg algebra. when an is the 0-hecke algebra, the associated heisenberg double is the quasi-heisenberg algebra. fix a dualizing tower of algebras a. for each n ∈ n, an-pmod is a full subcategory of an-mod. the inclusion functor induces the cartan map k(a) → g(a). let gproj(a) denote the image of the cartan map. let h− = k(a), h+ = g(a), h+proj = gproj(a), h = h(a), f = f(a). proposition 3.12 in [12] showed that h+proj is a subalgebra of h + that is invariant under the left-regular action of h−. we next recall the projective heisenberg double and its fock space in the following. definition 2.8. (the projective heisenberg double hproj) the subalgebra hproj = hproj(a) := h + proj]h − (the subalgebra of h generated by h+proj and h −) is called the projective heisenberg double associated to a. definition 2.9. (fock space fproj of hproj) the action of the algebra hproj on h+proj is called the lowest weight fock space representation of hproj and is denoted by fproj = fproj(a). it is generated by the lowest weight vacuum vector 1 ∈ h+proj. in [12] a stone-von neumann type theorem (theorem 2.11 in [12]) was proved. a consequence of this theorem tells that any representation of hproj generated by a lowest weight vacuum vector is isomorphic to fproj (see proposition 3.15 in [12]). 197 j. sun / j. algebra comb. discrete appl. 3(3) (2016) 195–200 3. main result in this section, we will first prove a lemma about the complete reducibility of an hproj-module generated by a finite set of lowest weight vacuum vectors. then we will prove our main result which says that under certain conditions, the positive part of a heisenberg double is free over the positive part of the corresponding projective heisenberg double. we will use the notations from section 2. lemma 3.1. suppose v is an hproj-module which is generated as an hproj-module by a finite set of lowest weight vacuum vectors. then v is a direct sum of copies of lowest weight fock space fproj. proof. let {vi}i∈i denote a finite generating set of v consisting of lowest weight vacuum vectors. suppose that i has minimal cardinality. suppose that zvi∩zvj 6= {0}, for some i 6= j. then nivi = njvj for some ni,nj ∈ z. let m = gcd(ni,nj). then there exists ai,aj ∈ z such that m = aini + ajnj. let w = ajvi + aivj. then w is a lowest weight vacuum vector. calculation shows that ni m w = 1 m (ajnivi + ainivj) = 1 m (ajnjvj + ainivj) = vj, nj m w = 1 m (ajnjvi + ainjvj) = 1 m (ajnjvi + ainivi) = vi. therefore, {vk}k∈i\{i,j} ∪{w} is also a generating set of v consisting of lowest weight vacuum vectors. this contradicts the minimality of the cardinality of i. thus zvi ∩zvj = {0}, for all i 6= j. by proposition 3.15(c) in [12], any representation of hproj generated by a lowest weight vacuum vector is isomorphic to fproj. thus hproj ·vi ∼= fproj as hproj-modules. by propostion 3.15(a) in [12], the only submodules of fproj are those submodules of the form nfproj for n ∈ z. therefore hproj·vi∩hproj·vj = {0} for i 6= j. the complete reducibility of v follows. theorem 3.2. let a be a dualizing tower of algebras. suppose there exists an increasing filtration {0}⊂ h+proj = (h +)(0) ⊂ (h+)(1) ⊂ (h+)(2) ⊂ ··· of hproj-submodules of h+ such that (h+)(n)/(h+)(n−1) is generated by a finite set of vacuum vectors, then h+ is free as an h+proj-module. proof. let vn = (h+)(n)/(h+)(n−1), then by lemma 3.1, vn = ⊕v∈lnh + proj · v, where ln is some collection of vacuum vectors in vn. consider the short exact sequence 0 → (h+)(n−1) → (h+)(n) → vn → 0. since vn is a free h + proj-module, the above short exact sequence split. by induction on n, we know that all (h+)(n) (n ∈ n) is free over h+proj. thus we can choose nested sets of vectors in h +: l̃0 ⊂ l̃1 ⊂ l̃2 ⊂ ··· such that for each n ∈ n, we have (h+)(n) = ⊕ ṽ∈l̃n h+proj ·ṽ. let l̃ = ⋃ n∈n l̃n, then h + = ⊕ v∈l̃h + proj ·v and h+ is free over h+proj. the main ideas used in proving lemma 3.1 and theorem 3.2 follow from lemma 9.1 and proposition 9.2 in [12]. observing that the existence of a special filtration of hproj-submodules of h+ is the key to the polynomial freeness of h+ over h+proj, we see that theorem 3.2 generalizes the result of proposition 9.2 in [12] (see the discussion of example 4.1 in section 4). 198 j. sun / j. algebra comb. discrete appl. 3(3) (2016) 195–200 4. applications in this section, we will discuss some applications of our main theorem. example 4.1. (tower of 0-hecke algebras) let a = ⊕n∈nhn(0), where hn(0) is the 0-hecke algebra of degree n. then h+ = g(a) = qsym and h− = k(a) = nsym. the associated heisenberg double is the quasi-heisenberg algebra q = h(qsym,nsym). the projective quasi-heisenberg algebra qproj is the subalgebra generated by h + proj = sym ⊂ qsym and h− = nsym. for n ∈ n, let qsym(n) := ∑ l(α)≤n qproj ·mα, where α ∈ ⋃ n∈n comp(n) and l(α) is the number of nonzero parts of α. then {0}⊂ sym = qsym (0) ⊂ qsym(1) ⊂ qsym(2) ⊂ ··· defines an increasing filtration of qproj-submodules of qsym. for α ∈⋃ n∈n comp(n) such that l(α) = n, we have rh∗m(mα) ∈ qsym (n−1) for any m > 0. so all mα with l(α) = n are lowest weight vacuum vectors in the quotient vn = qsym (n)/qsym(n−1) and generate vn. the condition of theorem 3.2 is satisfied. therefore, h+ = qsym is free over h+proj = sym. this recovers proposition 9.2 in [12]. example 4.2. (tower of 0-hecke-clifford algebras) let a = ⊕n∈nhcln(0), where hcln(0) is the 0-hecke-clifford algebra of degree n. then h+ = g(a) and h− = k(a) form a dual pair of hopf algebras. there are two main ideas used in [10] (section 3.3) to prove that (g(a),k(a)) is a dual pair of hopf algebras. first, the mackey property of 0-heckeclifford algebras guarantees that g(a) and k(a) are hopf algebras. second, it is shown that hcln(0) is a frobenius superalgebra which satisfies the conditions of proposition 6.7 in [11]. the associated heisenberg double is h(g(a),k(a)) and the corresponding projective heisenberg double is hproj. here h+ = peak∗ is the space of peak quasisymmetric functions. let θ : qsym → peak∗ be the descent-topeak map and let nα = θ(mα). then h+ is spanned by nα, where α ∈ ⋃ n∈n comp(n). for n ∈ n, let (peak∗)(n) := ∑ l(α)≤n hproj ·nα. then {0} ⊂ h+proj = (peak ∗)(0) ⊂ (peak∗)(1) ⊂ (peak∗)(2) ⊂ ··· defines an increasing filtration of hproj-submodules of peak ∗. we know that all nα with l(α) = n are lowest weight vacuum vectors in the quotient vn = (peak ∗)(n)/(peak∗)(n−1) and generate vn. the condition of theorem 3.2 is satisfied. therefore, h+ = peak∗ is free over h+proj, where h + proj is the subring of symmetric functions spanned by schur’s q-functions. this recovers proposition 4.2.2 in [10]. example 4.3. (towers of algebras related to the symmetric groups and their hecke algebras) in [8], the algebra hsn is defined to be the subalgebra of end(csn) generated by both sets of operators from c[sn] and hn(0). this algebra hsn has interesting combinatorial properties. for examples, the dimensional formula calculated in [8] finds applications in the study of symmetric functions associated with stirling permutations [5]. the author conjectures that theorem 3.2 applies to the tower of algebras ⊕n∈nhsn. the proof of this conjecture is the work in progress of the author in a forthcoming paper. acknowledgment: the author would like to thank alistair savage for explaining the paper [12]. the author would also like to thank rafael s. gonzález d’león for pointing out the algebra hsn in [8]. 199 j. sun / j. algebra comb. discrete appl. 3(3) (2016) 195–200 references [1] n. bergeron, h. li, algebraic structures on grothendieck groups of a tower of algebras, j. algebra 321(8) (2009) 2068–2084. 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accepted: 19 august 2014 doi 10.13069/jacodesmath.79879 journal of algebra combinatorics discrete structures and applications new extremal binary self-dual codes of length 68 research article abidin kaya1∗, bahattin yildiz1∗∗ 1. department of mathematics, fatih university, 34500, i̇stanbul, turkey abstract: in this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring r := f2 + uf2 + u2f2, where u3 = 1. among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. these are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. more precisely, codes with β=117, 120, 133 in w68,1 and with γ = 1, β=49, 57, 59 and codes with γ=2, β=69, 81 in w68,2 are constructed for the first time in the literature. the binary generators of these codes are available online at [7]. in addition to these, some known codes are reconstructed via this extension. the results are tabulated. 2010 msc: 94b05, 94b60, 94b65 keywords: extremal codes, codes over rings, gray maps, quadratic double-circulant codes 1. introduction the theory of self-dual codes, especially the so called extremal ones, has attracted a lot of research in the coding theory community. the connection of self-dual codes with many different fields of study such as designs, lattices and cryptography has made them of interest to many researchers. the theoretical background for extremal binary self-dual codes has been established in [1, 2, 9] and the references therein. in the aforementioned works, among other things, it was established that extremal binary self-dual codes can only have certain weight enumerators. much of the research in the study of self-dual codes has been towards finding extremal self-dual codes with new weight enumerators. starting with [3], the use of quadratic residues has been part of the armory for constructing binary self-dual codes. the method, which combines quadratic residues with the well-known methods of double circulant and bordered double circulant constructions has successfully been used to obtain many new extremal binary self-dual codes in [6] and [8]. in doing so, rings other than the binary field, which are endowed with duality and distance preserving gray maps were used. in this work we consider the construction method described above for the ring r := f2 +uf2 +u2f2, where u3 = 1. this is a non-chain extension of the binary field. we describe a lee weight and a related ∗ e-mail: akaya@fatih.edu.tr ∗∗ e-mail: byildiz@fatih.edu.tr 29 new extremal binary self-dual codes of length 68 distance-preserving gray map for the ring. unlike many of the rings studied before, the gray map is not duality-preserving. however we establish the conditions for the gray image to be self-dual. using quadratic double and bordered double circulant constructions over the ring r, we find extremal binary self-dual codes of length 66. applying extensions to these codes we find eight new binary extremal selfdual codes of length 68. more precisely, codes with β=117, 120, 133 in w68,1 and with γ = 1, β=49, 57, 59 and codes with γ=2, β=69, 81 in w68,2 are constructed for the first time in the literature. the rest of the work is organized as follows. section 2 includes a general overview on the ring r and self-dual codes. in section 3, we consider projections, lifts and duality conditions. section 4 contains the quadratic double and bordered double circulant constructions over the ring r and some extremal self-dual examples including codes of length 66. in section 5, codes of length 66 are extended, using extension theorems, to obtain new extremal binary self-dual codes of length 68. the paper ends with some concluding remarks and comments. 2. preliminaries 2.1. the structure of the ring f2 + uf2 + u2f2 with u3 = 1 the ring f2 +uf2 +u2f2 defined by the relation u3 = 1 is isomorphic to f2 [x] / 〈 x3 − 1 〉 . throughout the text the ring f2 + uf2 + u2f2 is denoted by r and it is easily observed that r ∼= f2 × f4. the ring r is not a local ring because its ideal structure is given by i0 ⊂ iu+u2,i1+u+u2 ⊂ r where; i1+u+u2 = ( 1 + u + u2 ) = { 0, 1 + u + u2 } , iu+u2 = (1 + u) = { 0,u + u2, 1 + u, 1 + u2 } . however, it is a frobenius ring as can easily be seen by the isomorphism r ∼= f2 × f4. the units in the ring r are given by { 1,u,u2 } and the non-units are{ u + u2, 1 + u, 1 + u2, 1 + u + u2 } . the ring r has two primitive idempotent elements { u + u2, 1 + u + u2 } . every element of the ring r can be written in a unique way as a + bu + cu2 = ( 1 + u + u2 ) (a + b + c) + ( u + u2 ) (a + c + (b + c) u). a linear code c of length n over the ring r is an r-submodule of rn and has a generating matrix that is permutation equivalent to g =   ik1 a b c0 (u + u2)ik2 0 (u + u2)d 0 0 ( 1 + u + u2 ) ik3 ( 1 + u + u2 ) e   . we define a gray map as follows; ϕ : rn → f3n2 a + bu + cu2 7→ ( a,b,c ) . definition 2.1. the lee weight of an element x = a + bu + cu2 ∈ r is the hamming weight of its gray image, i.e. wtl ( a + bu + cu2 ) = wth (a) + wth (b) + wth (c). an element is called even if its lee weight is even and odd otherwise. so, the elements in ideal iu+u2 are even and 1,u,u2, 1 +u+u2 are the odd elements. the lee weight of a codeword is defined to be the sum of the lee weights of its components. the minimum lee weight of a code c is denoted by wtl (c) and defined as wtl (c) = min{wtl (c) |c ∈c}. definition 2.2. a code c over r is called an even code if all the codewords have even lee weight. 30 a. kaya, b. yildiz the duality is understood in terms of the euclidean inner product; a◦ b = ∑ aibi. the dual of c is defined as c⊥ = {y ∈ rn | y ◦x = 0 for all x ∈c}. a code c is said to be self-orthogonal if c ⊆ c⊥ and self-dual if c = c⊥. a self-dual binary code is said to be type ii if all codewords have weight divisible by 4 and type i otherwise. the minimum distance d of a binary self-dual code of length n is bounded above as d ≤ 4 [n/24] + 6 if n ≡ 22 (mod 24) and d ≤ 4 [n/24] + 4, otherwise ([1, 9]). a self-dual code is called extremal if it meets the bound. 2.2. quadratic double circulant codes quadratic double circulant (qdc) codes are a generalization of quadratic residue codes and have been introduced in [3]. let p be an odd prime and qp (a,b,c) be the circulant matrix with first row r based on quadratic residues modulo p defined as r [1] = a, r [i + 1] = b if i is a quadratic residue and r [i + 1] = c if i is a quadratic non-residue modulo p. we state the special case of the main theorem from [3] where p is an odd prime; theorem 2.3. ([3]) let p be an odd prime and let qp (a,b,c) be the circulant matrix with a,b and c as the elements of the ring r. if p = 4k + 1 then qp (a,b,c) qp (a,b,c) t = qp ( a2 + 2k ( b2 + c2 ) , 2ab− b2 + k (b + c)2 , 2ac− c2 + k (b + c)2 ) if p = 4k + 3 then qp (a,b,c) qp (a,b,c) t = qp(a 2 + (2k + 1) ( b2 + c2 ) ,ab + ac + k ( b2 + c2 ) + (2k + 1) bc, ab + ac + k ( b2 + c2 ) + (2k + 1) bc). definition 2.4. ([3]) the code generated by pp (a,b,c) = ( ip qp (a,b,c) ) over the ring r is called a quadratic pure double circulant code and is denoted by pp (a,b,c). in a similar way, the code generated by bp (a,b,c | λ,β,γ) = ( ip+1 λ β ×1 γ ×1t qp (a,b,c) ) , where 1 is the all 1 vector of length p, is called a bordered quadratic double circulant code and is denoted by bp (a,b,c | λ,β,γ). 3. projections, lifts and duality conditions since the gray map introduced in section 2 does not preserve orthogonality, we start with determining the conditions when the binary image of a code over the ring r is self-orthogonal. then, a projection and related lift will be defined. the extended quadratic residue codes of parameters [24, 12, 8]2 and [48, 24, 12]2 which are unique up to equivalence are constructed as lifts of self-dual double circulant binary codes. the following example indicates that the gray image of a self-orthogonal code over the ring r is not necessarily a self-orthogonal binary code. example 3.1. let us consider the code c over the ring r of length 3 generated by( 1 + u, 1 + u2,u + u2 ) . 31 new extremal binary self-dual codes of length 68 we may easily observe that the code is self-orthogonal since (1 + u) 2 + ( 1 + u2 )2 + ( u + u2 )2 = 1 + u2 + 1 + u + u + u2 = 0. on the other hand, its binary image is the code generated by g =   1 1 0 1 0 1 0 1 10 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0   and it is not self-orthogonal since distinct rows of g are not orthogonal to each other. definition 3.2. a vector x = a + bu + cu2 over the ring r can be expressed as x = ( 1 + u + u2 )( a + b + c ) + ( u + u2 )( a + c + ( b + c ) u ) . a + b + c and a + c + ( b + c ) u are called f2 and f4-components of x, respectively. definition 3.3. a matrix over the ring r is said to be free of u if all its entries are of the form a + bu2 with a, b ∈ f2. lemma 3.4. the gray images of two vectors x = a+bu+cu2 and y = d+eu+fu2 in rn are orthogonal to each other if their f2-components are orthogonal and the product of their f4-components is free of u. proof. if the f2-components of the vectors x and y are orthogonal we have( a + b + c ) ◦ ( d + e + f ) = a◦d + a◦e + a◦f + b◦d + b◦e + b◦f + c◦d + c◦e + c◦f = 0 (1) and if the inner product of their f4-components ( a + c + ( b + c ) u ) ◦ ( d + f + ( e + f ) u ) is free of u we have a◦e + a◦f + b◦d + b◦f + c◦d + c◦e = 0, which implies a◦d + b◦e + c◦f = 0 by (1). hence ϕ (x) ◦ϕ (y) = 0. we would like to determine when the gray image of a code is self-dual. much like ring elements and vectors, a matrix g over r can also be expressed as g = g1 + ug2 + u 2g3 = ( 1 + u + u2 ) (g1 + g2 + g3) + ( u + u2 ) (g1 + g3 + (g2 + g3) u) where g1, g2 and g3 are binary matrices. g1 + g2 + g3 is called f2-component of g and denoted by gf2 and g1 + g3 + (g2 + g3) u is called f4-component of g and denoted by gf4. the following theorem characterizes codes over the ring r with self-orthogonal binary images: theorem 3.5. let c be the code generated by g over the ring r. then ϕ (c) is a self-orthogonal binary code if gf2g t f2 = 0 and gf4g t f4 and gf4 (ug) t f4 are matrices over the ring r which are free of u. proof. let g = g1 + ug2 + u2g3 then gf2g t f2 = 0 implies (g1 + g2 + g3) ( gt1 + g t 2 + g t 3 ) = 0. (2) the matrix gf4g t f4 = (g1 + g3 + (g2 + g3) u) ( gt1 + g t 3 + ( gt2 + g t 3 ) u ) is free of u implies g1g t 2 + g1g t 3 + g3g t 2 + g2g t 1 + g2g t 3 + g3g t 1 = 0 then this together with equation (2) implies g1g t 1 + g2g t 2 + g3g t 3 = 0. (3) 32 a. kaya, b. yildiz similarly, if gf4 (ug) t f4 = (g1 + g3 + (g2 + g3) u) ( gt3 + g t 2 + ( gt1 + g t 2 ) u ) is free of u then we have g1g t 1 + g1g t 2 + g3g t 1 + g2g t 3 + g2g t 2 + g3g t 3 = 0 then this together with equation (3) gives g1g t 2 + g3g t 1 + g2g t 3 = 0. (4) now consider the gray image ϕ (c) of c which is generated by g∗ =   ϕ (g)ϕ (ug) ϕ ( u2g )   =   g1 g2 g3g3 g1 g2 g2 g3 g1   . by equations (3) and (4) we have g∗ (g∗)t = 0 which implies ϕ (c) is a self-orthogonal binary code. example 3.6. the code generated by the matrix g = ( 1 0 1 + u u 0 1 u 1 + u ) is a free self-dual code over the ring r but its gray image is not self-dual. we may easily observe that gf2 = ( 1 0 0 1 0 1 1 0 ) and it generates a binary self-dual code. on the other hand, gf4 = g and (ug)f4 = ( u 0 1 1 + u 0 u 1 + u 1 ) . the matrix gf4 (ug) t f4 = ( 1 + u + u2 1 + u + u2 1 + u + u2 1 + u + u2 ) is not free of u. if c is self-orthogonal over the ring r then some of the conditions of theorem 3.5 may be relaxed. lemma 3.7. let c be a self-orthogonal code over the ring r and g be a generator matrix of c. then ϕ (c) is a binary self-orthogonal code if gf4gtf4 and gf4 (ug) t f4 are free of u. proof. let g = ( 1 + u + u2 ) gf2 + ( u + u2 ) gf4 be a generator matrix for a self-orthogonal code over r. then ggt = 0 which implies gf2g t f2 = 0. the result follows by theorem 3.5. an immediate consequence of lemma 3.7 is; lemma 3.8. let g be a generator matrix of a self-dual code c over the ring r. then ϕ (c) is a binary self-dual code if gf4g t f4 and gf4 (ug) t f4 are free of u. lemma 3.9. a self-orthogonal code c over the ring r is an even code. proof. let c be a self-orthogonal code of length n over the ring r and x be an arbitrary codeword in c. let x = a + bu + cu2 where a,b and c ∈ fn2 then x◦x = a◦a + (c◦ c) u + ( b◦ b ) u2 = 0 implies a ◦ a = 0 = b ◦ b = c ◦ c. then, wth (a), wth ( b ) and wth (c) are even since a self-orthogonal binary vector has even weight. hence, wtl (x) = wth (a) + wth ( b ) + wth (c) is even. we define a projection π : r → f2 as; π (r) = { 1 0 if r is odd if r is even. the projection π is extended componentwise and denoted by π. for a matrix g over the ring r the projection of the matrix is its f2-component; π (g) = gf2. moreover, π is a ring homomorphism. so, the following result follows; lemma 3.10. let c be a linear code over the ring r generated by matrix g then c is an even code if the rows of g are even. 33 new extremal binary self-dual codes of length 68 definition 3.11. let c be a code of length n over the ring r. the code c is said to be a lift of the binary code d if π (c) = d. the code d is called the projection of c. note that the projection of a self-dual code is a binary self-orthogonal code. on the other hand a lift of a self-dual code may not be self-dual. in the following example we construct the extended binary golay code as a lift of the [8, 4, 4]2 extended hamming code. example 3.12. take the following generator matrix of the [8, 4, 4]2 extended hamming code g =   i4 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0   . we lift g to a matrix g∗ over the ring r by keeping i4 as it is and lifting 0 to an even element and 1 to an odd element. let c∗ be the code generated by g∗ =   i4 u + u2 1 + u + u2 1 1 1 u + u2 1 + u + u2 1 1 1 u + u2 1 + u + u2 1 + u + u2 1 1 u + u2   . g∗f2 = g and g∗f4 =   i4 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1   , (ug)∗f4 =   ui4 1 + u 0 u u u 1 + u 0 u u u 1 + u 0 0 u u 1 + u   ggt = 0 and g∗f4 ( ug∗f4 )t is 4×4 circulant matrix with first row (1, 0, 1, 1) and hence is free of u. thus by theorem 3.5 ϕ (c∗) is self-dual, which turns out to be the [24, 12, 8]2 extended binary golay code. example 3.13. the binary code generated by g = (i8 | a) where a is the circulant matrix with first row ra = (0, 1, 0, 0, 0, 1, 0, 1) is a self-dual [16, 8, 4]2-code. consider a lift ra′ = ( 1 + u,u2,u + u2, 1 + u, 0, 1 + u + u2,u + u2, 1 + u + u2 ) of ra. then the code generated by g′ = (i8 | a′) where a′ is the circulant matrix with first row ra′ has a self-dual binary image that is the unique self-dual [48, 24, 12]2-code. example 3.14. the double circulant binary code generated by g = (i11 | a) where a is the 11 × 11 circulant matrix with first row ra = (1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0) is a self-dual [22, 11, 6]2-code. let ra′ = ( 1, 1, 1, 1 + u2, 1 + u2, 1 + u2, 1, 1 + u2,u + u2, 1, 1 + u2 ) be a lift of ra and c be the code over the ring r generated by g′ = (i11 | a′) where a′ is the circulant matrix with first row ra′. the binary image ϕ (c) of c which we denote by c66,0 is an extremal self-dual binary code of lengh 66. 4. quadratic double circulant codes over r in this section, we consider qdc codes over the ring r and obtain families of codes which satisfy duality conditions given in theorem 3.5. therefore we obtain self-dual binary codes as gray images of codes over the ring r. in particular, some extremal self-dual [66, 33, 12]2-codes are reconstructed. 34 a. kaya, b. yildiz theorem 4.1. let p ≡ 3 (mod 8) be an odd prime. then pp ( 0, 1 + u2, 1 + u + u2 ) and pp ( 1 + u2,u, 1 + u ) are codes over the ring r with self-dual binary images. proof. a generator matrix of pp ( 0, 1 + u2, 1 + u + u2 ) is given by g = ( ip qp ( 0, 1 + u2, 1 + u + u2 ) ) . then gf2 = ( ip qp (0, 0, 1) ) and by theorem 2.3 we have qp (0, 0, 1) qp (0, 0, 1) t = qp (1, 0, 0) = ip. so gf2 (gf2 ) t = 0. analogously, gf4 = ( ip qp (0,u, 0) ) by theorem 2.3 we have qp (0,u, 0) qp (0,u, 0) t = qp ( u2, 0, 0 ) = u2ip. therefore, gf4 (gf4 ) t = ( 1 + u2 ) ip which is free of u. now, we need to show that gf4 (ug) t f4 is also free of u where (ug)f4 = ( uip qp (0, 1 + u, 0) ) . we have qp (0,u, 0) qp (0, 1 + u, 0) t = uqp (0, 1, 0) (1 + u) qp (0, 1, 0) = ( u + u2 ) ip by theorem 2.3, which implies gf4 (ug) t f4 = uip + ( u + u2 ) ip = u 2ip which is free of u. hence, by theorem 3.5 the binary image of pp ( 0, 1 + u2, 1 + u + u2 ) is self-orthogonal. since ∣∣pp (0, 1 + u2, 1 + u + u2)∣∣ = 8p it has a self-dual binary image. the same can be done for pp ( 1 + u2,u, 1 + u ) . theorem 4.2. let p be a prime with p ≡ 3 (mod 8). then the code bp ( 1,u + u2, 1 + u + u2 | 0, 1, 1 ) is a self-dual code over the ring r and its gray image is a binary self-dual type ii code. proof. let bi denote the i-th row of the matrix g = bp ( 1,u + u2, 1 + u + u2 | 0, 1, 1 ) = ( ip+1 0 1 1t qp ( 1,u + u2, 1 + u + u2 ) ) . then b1 ◦ b1 = 0 since p is odd. if p = 8k + 3 then for 2 ≤ i ≤ p + 1 we have b1 ◦ bi = 1 + p− 1 2 ( u + u2 ) + p− 1 2 ( 1 + u + u2 ) = 1 + (4k + 1) = 0. so by theorem 2.3 we have qp ( 1,u + u2, 1 + u + u2 ) qp ( 1,u + u2, 1 + u + u2 )t = qp ( 1 + u + u2 + 1 + u + u2,u + u2 + 1 + u + u2 + 0,u + u2 + 1 + u + u2 + 0 ) = qp (0, 1, 1) , which implies bi ◦ bj = 0 for 2 ≤ i ≤ j ≤ q + 1. hence the code is self-dual. on the other hand, the binary code generated by gf2 = π (g) = bp (1, 0, 1 | 0, 1, 1) generates a self-dual binary code since qp (1, 0, 1) qp (1, 0, 1) t = qp (0, 1, 1). so, gf2 (gf2 ) t = 0. gf4 = bp (1, 1, 0 | 0, 1, 1) is a binary matrix and moreover gf4 (gf4 ) t = 0 since qp (1, 1, 0) qp (1, 1, 0) t = qp (0, 1, 1). (ug)f4 = ( uip+1 0 u1 u1t qp (u,u, 0) ) = u (gf4 ) and gf4 (ug) t f4 = ugf4 (gf4 ) t = u0 = 0. hence by theorem 3.5 the binary image of the code is selfdual. in addition, wt (b1) = 8k + 4 and wt (bi) = 3 + (4k + 1) 2 + (4k + 1) 3 = 20k + 8 for 2 ≤ i ≤ q + 1 which implies that the gray image is type ii. 35 new extremal binary self-dual codes of length 68 an extremal self-dual binary code of length 66 has a weight enumerator in one of the following forms ([2]): w66,1 = 1 + (858 + 8β) y 12 + (18678 − 24β) y14 + · · · , 0 ≤ β ≤ 778, w66,2 = 1 + 1690y 12 + 7990y14 + · · · and w66,3 = 1 + (858 + 8β) y12 + (18166 − 24β) y14 + · · · , 14 ≤ β ≤ 756. recently, 24 new codes in w66,3 are constructed in [6]. for a list of known codes we refer to [6] and references therein. the code c66,0 in example 3.14 has weight enumerator β = 0 in w66,1. we complete this section by giving some examples of qdc codes over the ring r in table 1. the binary images of two of the codes are extremal self-dual binary codes with weight enumerators β = 0 and 66 in w66,1. table 1. examples of quadratic double circulant codes over the ring r p code over r gray image remark 3 p3 ( 1 + u2,u,1 + u ) [18,9,4] 2 3 p3 ( 0,1 + u2,1 + u + u2 ) [18,9,4] 2 3 b3 ( 0,u,1 + u2 | 1 + u + u2,1 + u,1 + u ) [24,12,6] 2 11 p11 ( 1 + u2,u,1 + u ) [66,33,12] 2 β = 0 in w66,1 11 p11 ( 0,1 + u2,1 + u + u2 ) [66,33,12] 2 β = 66 in w66,1 11 b11 ( 1,u + u2,1 + u + u2 | 0,1,1 ) [72,36,12] 2 type ii 19 p19 ( 1 + u2,u,1 + u ) [114,57,16]2 19 p19 ( 0,1 + u2,1 + u + u2 ) [114,57,16]2 19 b19 ( 1,u + u2,1 + u + u2 | 0,1,1 ) [120,60,16]2 type ii 5. new extremal binary self-dual codes of length 68 an efficient method to construct self-dual binary codes of length n+ 2 is to extend a self-dual binary code of length n. such an extension method for an arbitrary generator matrix of the code is used in [5]. such extension methods for binary rings are introduced in [8] and a substantial number of new extremal binary self-dual codes of length 68 are obtained. in this section, extension is used for extremal self-dual binary codes of length 66 which were constructed in sections 3 and 4. as a result, we were able to obtain eight extremal self-dual binary codes of length 68 with new weight enumerators. the weight enumerator of an extremal binary self-dual code of length 68 is characterized in [2] as follows: w68,1 = 1 + (442 + 4β) y 12 + (10864 − 8β) y14 + · · · , 104 ≤ β ≤ 1358, w68,2 = 1 + (442 + 4β) y 12 + (14960 − 8β − 256γ) y14 + · · · where 0 ≤ γ ≤ 11 and 14γ ≤ β ≤ 1870−32γ. tsai et al. constructed new extremal self-dual binary codes of lengths 66 and 68 in [10]. together with the codes obtained in [10] the existence of codes in w68,1 are known for β =104, 122, 125,. . . ,132, 134,. . . ,168, 170,. . . ,232, 234, 235, 236, 241, 255, 257,. . . ,269, 302, 328,. . . , 336, 338, 339, 345, 347, 355, 401. we construct codes with weight enumerators β =117, 120 and 133 in w68,1 which are listed in table 2. recently, new codes in w68,2 are obtained in [6, 8] together 36 a. kaya, b. yildiz with these, codes exists for w68,2 when γ = 0, β = 44, . . . , 154 or β ∈{2m|m = 17, 20, 88, 102, 119, 136 or 78 ≤ m ≤ 86} ; γ = 1, β = 60, . . . , 160 or β ∈{2m|m = 27, 28, 29, 95, 96 or 81 ≤ m ≤ 89} ; γ = 2, β = 65, 68, 71, 77, 159 or β ∈{2m|37 ≤ m ≤ 68, 70 ≤ m ≤ 81} or β ∈ {2m + 1|42 ≤ m ≤ 69, 71 ≤ m ≤ 77} ; γ = 3, β = 101, 117, 123, 127, 133, 137, 141, 145, 147, 149, 153, 159, 193 or β ∈ {2m|m = 44,45,48,50,51,52,54, . . . ,58,61,63, . . . ,66,68, . . . ,72,74,77, . . . ,81,88,94,98} ; γ = 4, β ∈{2m|m = 51, 55, 58, 60, 61, 62, 64, 65, 67, . . . ,71, 75, . . . , 78, 80} and γ = 6 with β ∈{2m|m = 69, 77, 78, 79, 81, 88} . we construct codes with weight enumerators γ = 1 and β =49, 57, 59, 67, 69, 71 and codes with weight enumerators γ = 2 and β =69, 81 in w68,2 that are given in table 3 and example 5.2. the following is an extension theorem that is true for all commutative rings a of characteristic 2. theorem 5.1. ([8]) let c be a self-dual code over a of length n and g = (ri) be a k × n generator matrix for c, where ri is the i-th row of g, 1 ≤ i ≤ k. let c be a unit in a such that c2 = 1 and x be a vector in an with x ◦x = 1. let yi = ri ◦x for 1 ≤ i ≤ k. then the following matrix g∗ =   1 0 x y1 cy1 r1 ... ... ... yk cyk rk   , generates a self-dual code c∗ over a of length n + 2. example 5.2. when we apply the extension in theorem 5.1 to c66,0 in example 3.14 with x = (000101000011111010001111011100110101001110001101101011001111111111) in other words, when we consider the code generated by  1 0 x y1 y1 ... ... ϕ (g′) y33 y33   we obtain an extremal binary self-dual code of length 68 with an automorphism group of order 10 and weight enumerator γ = 1, β = 49 in w68,2. note that this is the first extremal binary self-dual code with this weight enumerator. in a similar way, the extension is applied to the gray image of p11 ( 1 + u2,u, 1 + u ) in table 1 and seven new extremal self-dual binary codes of length 68 are obtained. these are listed in tables 2 and 3. remark 5.3. recently, in [6] by using a different method codes with weight enumerators γ = 1 and β = 67, 69, 71 in w68,2 are constructed for the first time in literature. since the method used here is different we list them in table 3. 6. conclusion in this work, we applied the quadratic double and bordered double circulant constructions over the ring r := f2[u]/ ( u3 − 1 ) to obtain extremal binary self-dual codes. applying extension theorems to the 37 new extremal binary self-dual codes of length 68 table 2. extremal binary self-dual codes in w68,1 as extensions of ϕ ( p11 ( 1 + u2,u,1 + u )) x β 011011011011011001001110100111000100000110010000010000000011100111 117 011011001001011000010101111011101111010101011011010110100011111011 120 011100000001010000000001111011101111001011111010100001100101110100 133 table 3. extremal binary self-dual codes in w68,2 as extensions of ϕ ( p11 ( 1 + u2,u,1 + u )) x γ β 100010010101000101110011011111100101100000000001011010000100111111 1 57 000010001001100101011111101000110001100010010111111111011011110001 1 59 001010111001001111111010010100100010101000001010110101010101001010 1 67 001100101010011110110111001000000111101110100111100101110101001111 1 69 111000000010100001010000101001101110101011100110110011101110001100 1 71 100010101100011000011001101100100101110110101111001011111001101001 2 69 000111000010101101101111110010001000100011010010110101101100110101 2 81 extremal self-dual binary codes of length 66 obtained from the ring r we were able to find eight new extremal self-dual binary codes of length 68 updating the list of all known such codes. the methods we have used have proven to be useful in many works in the literature of self-dual codes. we believe they could be applied to other rings and structures such as z4. acknowledgements the authors would like to thank prof. i̇rfan şiap for his suggestions and the anonymous referees for their remarks which improved the manuscript considerably. references [1] j. h. conway, n. j. a. sloane, a new upper bound on the minimal distance of self-dual codes, ieee trans. inform. theory, 36(6), 1319-1333, 1990. [2] s. t. dougherty, t. a. gulliver, m., harada, extremal binary self dual codes, ieee trans. inform. theory, 43(6), 2036-2047, 1997. [3] p. gaborit, quadratic double circulant codes over fields, journal of combinatorial theory series a, 97(1), 85-107, 2002. [4] w.c. huffman, v. pless, fundamentals of error correcting codes, cambridge university press, 2003. [5] j.-l. kim, new extremal self-dual codes of lengths 36, 38 and 58, ieee trans. inf. theory, 47(1), 386-393, 2001. [6] a. kaya, b. yildiz, i̇. şiap, new extremal binary self-dual codes from f4 + uf4-lifts of quadratic 38 a. kaya, b. yildiz double circulant codes over f4, available online at http://arxiv.org/abs/1405.7147 [7] a. kaya, b. yildiz, binary generator matrices of new extremal self-dual binary codes of length 68, available online at http://www.fatih.edu.tr/~akaya/binary/68u31.txt [8] a. kaya, b. yildiz, extension theorems for self-dual codes over rings and new binary self-dual codes, available online at http://arxiv.org/abs/1404.0195. [9] e. m. rains, shadow bounds for self dual codes, ieee trans. inf. theory, 44(1), 134-139, 1998. [10] h.-p. tsai, p.-y. shih, r.-y. wuh, w.-k. su, c.-h. chen, construction of self-dual codes, ieee trans. inform. theory, 54(8), 3826-3831, 2008. 39 introduction preliminaries projections, lifts and duality conditions quadratic double circulant codes over r new extremal binary self-dual codes of length 68 conclusion acknowledgements references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.24447 j. algebra comb. discrete appl. 4(1) • 1–11 received: 5 april 2015 accepted: 10 april 2016 journal of algebra combinatorics discrete structures and applications on the matching polynomial of hypergraphs∗ research article zhiwei guo, haixing zhao, yaping mao abstract: the concept of the matching polynomial of a graph, introduced by farrell in 1979, has received considerable attention and research. in this paper, we generalize this concept and introduce the matching polynomial of hypergraphs. a recurrence relation of the matching polynomial of a hypergraph is obtained. the exact matching polynomials of some special hypergraphs are given. further, we discuss the zeros of matching polynomials of hypergraphs. 2010 msc: 05c31, 05c65, 05c70 keywords: hypergraph, matching polynomial, line-graph, independence polynomial 1. introduction hypergraphs are systems of finite sets. with the development of computer science, this branch of mathematics has developed rapidly during the late twentieth century. hypergraphs model more general types of relations than graphs. for more results on hypergraph, we refer to [1, 2, 5, 11–13]. the matching polynomial of a graph was introduced by farrell in 1979. for more details on the matching polynomial of a graph, we refer to [3, 4, 6, 7, 9, 15, 21]. an h-uniform hypergraph is a pair h = (v ; e), where v = v (h) is a finite set of vertices, and e = e(h) ⊆ ( v h ) is a family of h-element subsets of v called hyperedges. all hypergraphs considered in this paper are h-uniform hypergraphs. a simple hypergraph is a hypergraph h = (v ; e) such that ei ⊆ ej if and only if i = j. a hypergraph is linear if it is simple and |ei ∩ej| ≤ 1, for all ei, ej ∈ e where ∗ supported by the national science foundation of china (nos. 61164005, 11161037, 11101232, 61440005 and 11461054) and the program for changjiang scholars and innovative research team in universities (no. irt1068), the research fund for the chunhui program of ministry of education of china (no. z2014022) and the nature science foundation from qinghai province (nos. 2013-z-y17, 2014-zj-721, 2015-zj-905 and 2014-zj-907). zhiwei guo, yaping mao (corresponding author); department of mathematics, qinghai normal university, xining, qinghai 810008, china (email: guozhiweic@yahoo.com, maoyaping@ymail.com). haixing zhao; school of computer, qinghai normal university, xining, qinghai 810008, china (email: h.x.zhao@163.com). 1 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 i 6= j. given v ′ ⊆ v , the subhypergraph h′ is the hypergraph h′ = (v ′, e′ = (ej)j∈j) such that for all ej ∈ e′, ej ⊆ v ′. a matching of h consists of isolated vertices and hyperedges only. a k-matching of h contains k disjoint hyperedges in e(h). let m be a k-matching in h and let us assign “weights” w1 and w2 to each vertex and edge, respectively in m. let us associate the weight wn−kh1 w k 2 with m. then if ak is the number of kmatchings in h, the total weight of the k-matchings in h will be a akw n−kh 1 w k 2. by summing the weights of all the k-matchings in h, for all possible values of k, we will obtain a polynomial in w1 and w2. this polynomial is called the matching polynomial of h. denote by it m(h; w), i.e. m(h; w) = bn/hc∑ k=0 akw n−kh 1 w k 2, where w = (w1, w2) is called the weight vector associated with the matching polynomial. if we put w1 = w2 = w, then the resulting polynomial in w will be called the simple matching polynomial of h, m(h; w) = bn/hc∑ k=0 akw n−k(h−1). the matching polynomials of hypergraphs may also be regarded as a special type of multivariate hyperedge elimination chromatic polynomials first investigated by white [20]. this paper is organized as follows. in section 2, we present a well-known fundamental theorem for the matching polynomials of hypergraphs which we use in section 3, to obtain recursively the formulae for the matching polynomials of some classes of hypergraphs. in general, these formulae are intractable, giving our choice of these particular hypergraphs which are much easier to manipulate. we hope these results can inspire further computations of these polynomials. in the last section, we discuss the zeros of these polynomials when compared to their graphs counterparts. 2. the fundamental theorem for matching polynomials firstly, we define two graphs operations which will be used later. deletion of hyperedge e, i.e. e is removed and the vertices of e are still retained. contraction of hyperedge e, i.e. e is removed and all vertices of e and hyperedges adjacent to e are removed. since matchings consist of isolated vertices and hyperedges, the inclusion of an hyperedge in a matching implies the exclusion of all hyperedges adjacent to it. by partitioning the set of matchings in the hypergraph into two classes: (i) those containing a given hyperedge e and (ii) those not containing e, the following result is immediate. theorem 2.1. let h be a hypergraph containing a hyperedge e. let h′ be the hypergraph obtained from h by deleting e, and h′′ be the hypergraph obtained from h by contracting e. then m(h; w) = m(h′; w) + w2m(h ′′; w). the fundamental recurrence relation of theorem 2.1 firstly appeared in [7], for graphs and it is widely used. other versions of this recursion also appeared in [18]. proposition 2.2. let h be a hypergraph, and let h1 and h2 be two subhypergraphs of h such that v (h1)∩v (h2) = ∅ and h1 ∪h2 = h. then m(h; w) = m(h1; w) ·m(h2; w). the line-graph of h is the graph l(h) = (v ′; e′) such that: (i) v ′ = e when h is without repeated hyperedge; 2 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 (ii) {ei, ej}∈ e′(i 6= j) if and only if ei ∩ej 6= ∅. the first formal definition of the independence polynomial appears to be due to gutman and harary [10]. the independence polynomial of a graph g will be denoted by i(g, x), i.e. i(g; x) = α(g)∑ k=0 akx k, where ak is the number of independence sets of g with exactly k vertices. for more details on the independence polynomial, we refer to [14, 16, 17, 19]. by the above definition, one can see that the coefficients of the matching polynomial of a simple hypergraph and the coefficients of the independence polynomial of its line-graph are identical. since the line-graph of a hypergraph h is a simple graph, it follows that the matching polynomial of a hypergraph can be obtained by the known independence polynomial of its line-graph. the following proposition gives us a method to calculate the matching polynomial of a given hypergraph. note that i(l(h); x) is the independence polynomial of l(h). then the following proposition is immediate. proposition 2.3. let i(l(h); x) = α(l(h))∑ k=0 akx k be the independence polynomial of l(h). then the matching polynomial of h is m(h; w) = bn/hc∑ k=0 akw n−kh 1 w k 2. 3. matching polynomials of some special hypergraphs in this section, we discuss the matching polynomials of hyperpaths, hyperstars, hypercycles, complete h-uniform hypergraphs and two new hypergraphs hw` and hs`. 3.1. matching polynomials of hyperpaths and hyperstars a hyperpath in h from x to y, is a vertex-hyperedge alternating sequence: x = x1, e1, x2, e2, . . . , xs, es, xs+1 = y such that • x1, x2, . . . , xs, xs+1 are distinct vertices with the possibility that x1 = xs+1; • e1, e2, . . . , es are distinct hyperedges; • xi, xi+1 ∈ ei, for all i ∈{1, 2, . . . , s}. if x = x1 = xs+1 = y, then the hyperpath is called a hypercycle. here and throughout, the length is equal to the number of hyperedges. let p` be a linear hyperpath with length of `. we can apply theorem 2.1 to p` by operating for a terminal hyperedge. this immediately yields the recurrence relation. proposition 3.1. let p` be a linear hyperpath with length of ` (` > 1). then m(p`; w) = w (h−1) 1 m(p`−1; w) + w (h−2) 1 w2m(p`−2; w). (1) 3 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 furthermore, the following expression of m(p`; w) can be obtained. theorem 3.2. let p` be a linear hyperpath with length of ` (` > 1). then m(p`; w) = bn/hc∑ k=0 ( ` + 1−k k ) w (n−kh) 1 w k 2, (2) where n denotes the order of p`. proof. by proposition 3.1, we have m(p`; w) = w (h−1) 1 m(p`−1; w) + w (h−2) 1 w2m(p`−2; w). for convenience, we use p(i) to denote m(pi; w). then p(0) = w1, p(1) = w2 + w h 1 . we can obtain the generating function for m(p`; w) from equation (1). let g(x) be the generating function of m(p`; w). g(x) = p(0) + p(1)x + p(2)x 2 + p(3)x 3 + . . . −w(h−1)1 xg(x) = −w (h−1) 1 p(0)x−w (h−1) 1 p(1)x 2 − w(h−1)1 p(2)x3 + . . . −w(h−2)1 w2x2g(x) = −w (h−2) 1 w2p(0)x 2 −w(h−2)1 w2p(1)x3 + . . . by equation (1), we can obtain the generating function, i.e. g(x) = w1 + w2x 1−w(h−1)1 x−w (h−2) 1 w2x 2 . (3) by expanding equation (3) as the sum of an infinite geometrical progression and extracting the coefficient of x`, we can obtain the matching polynomials of hyperpath p`, i.e. m(p`; w) = bn/hc∑ k=0 ( ` + 1−k k ) w (n−kh) 1 w k 2, where n denotes the order of p`. remark 3.3. in fact, we can also prove theorem 3.2 by the idea from proposition 2.3. note that the matching polynomial of a hypergraph and the independence polynomial of its line-graph are identical. observe that the line-graph of p` is the path p with ` vertices. song et al. [17] obtained the independence polynomial of the path p with n vertices, i.e. i(pn; x) = b(n+1)/2c∑ s=0 ( n + 1−s s ) xs. a hyperstar s is a family of hyperedges containing a given vertex. let s` be a linear hyperstar with ` edges. since any two edges of s are intersecting, a matching of a hyperstar has at most one edge. so, we can obtain the matching polynomials of hyperstars. proposition 3.4. the matching polynomial of s` is m(s`; w) = 1 + `w (n−h) 1 w2, where n denotes the order of s`. 4 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 3.2. matching polynomials of hypercycles let c` be a linear hypercycle with length of `. we apply theorem 2.1 to c` by operating for an arbitrarily hyperedge. this immediately yields the recurrence relation. theorem 3.5. let c` be a linear hypercycle with length of ` (` > 2). then m(c`; w) = w (h−2) 1 m(p`−1; w) + w 2(h−2) 1 w2m(p`−3; w). (4) theorem 3.6. let c` be a linear hypercycle with length of ` (` > 2). then m(c`; w) = w n 1 + bn/hc∑ k=1 ` k ( `−1−k k −1 ) w (n−kh) 1 w k 2, where n denotes the order of c`. proof. from theorem 3.5, we have m(c`; w) = w (h−2) 1 m(p`−1; w) + w 2(h−2) 1 w2m(p`−3; w). by equation (2), we can obtain the expression of m(p`−1; w) and m(p`−3; w) as follows. m(p`−1; w) = b[(`−1)·(h−1)+1]/hc∑ k=0 ( `−k k ) w ([(`−1)·(h−1)+1]−kh) 1 w k 2, m(p`−3; w) = b[(`−3)·(h−1)+1]/hc∑ k=0 ( `−2−k k ) w ([(`−3)·(h−1)+1]−kh) 1 w k 2. therefore, we have m(c`; w) = w (h−2) 1 b[(`−1)·(h−1)+1]/hc∑ k=0 ( `−k k ) w ([(`−1)·(h−1)+1]−kh) 1 w k 2 +w 2(h−2) 1 w2 b[(`−3)·(h−1)+1]/hc∑ k=0 ( `−2−k k ) w ([(`−3)·(h−1)+1]−kh) 1 w k 2 = w [(`−1)·(h−1)+1]+h−2 1 + b[(`−1)·(h−1)+1]/hc∑ k=1 ( `−k k ) w ([(`−1)·(h−1)+1]−kh+h−2) 1 w k 2 + b[(`−3)·(h−1)+1]/hc∑ k′=1 ( `−1−k′ k′ −1 ) w ([(`−3)·(h−1)+1]−k′h+3h−4) 1 w k′ 2 = w `(h−1) 1 + b`(h−1)/hc∑ k=1 ( ( `−k k ) + ( `−1−k k −1 ) )w `(h−1)−kh 1 w k 2 = wn1 + bn/hc∑ k=1 ` k ( `−k −1 k −1 ) wn−kh1 w k 2 furthermore, we obtain the recurrence relation involving the matching polynomials of hypercycles only. 5 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 theorem 3.7. let c` be a linear hypercycle with length of ` (` ≥ 5). then m(c`; w) = w (h−1) 1 m(c`−1; w) + w (h−2) 1 w2m(c`−2; w). proof. from theorem 3.5, we have m(c`; w) = w (h−2) 1 m(p`−1; w) + w 2(h−2) 1 w2m(p`−3; w) (` ≥ 3). (5) by proposition 3.1, we can obtain the following equalities: m(p`−1; w) = w (h−1) 1 m(p`−2; w) + w (h−2) 1 w2m(p`−3; w) (6) and m(p`−3; w) = w (h−1) 1 m(p`−4; w) + w (h−2) 1 w2m(p`−5; w). (7) by equations (5), (6) and (7), we have m(c`; w) = w (h−1) 1 [w (h−2) 1 m(p`−2; w) + w 2(h−2) 1 w2m(p`−4; w)] +w (h−2) 1 w2[w (h−2) 1 m(p`−3; w) + w 2(h−2) 1 w2m(p`−5; w)]. from theorem 3.5, we have m(c`; w) = w (h−1) 1 m(c`−1; w) + w (h−2) 1 w2m(c`−2; w). remark 3.8. in fact, we can also prove theorem 3.6 by the idea from proposition 2.3. note that the coefficients of the matching polynomial of a simple hypercycle and the coefficients of the independence polynomial of its line-graph are identical. observe that the line-graph of a cycle c` is also a cycle with ` vertices. song et al. [17] obtained the independence polynomials of the cycle c with n vertices, i.e. i(cn; x) = 1 + bn/2c∑ s=1 n s ( n−1−s s−1 ) xs. 3.3. matching polynomial of a complete h-uniform hypergraphs a complete h-uniform hypergraph is a hypergraph which has all h-subsets of v as hyperedges. let khn be a complete h-uniform hypergraph. we can obtain the matching polynomial of a complete h-uniform hypergraph by applying the principle of combinatorial enumeration. proposition 3.9. let khn be a complete h-uniform hypergraph with n vertices. then m(khn; w) = bn/hc∑ k=0 akw (n−kh) 1 w k 2, where a0 = 1, ak = ∏k−1 i=0 ( n−ih h ) k! (k ≥ 1). 6 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 proof. we choose h vertices from the vertex set of khn to form the first hyperedge e1 of the k-matching. note that we have ( n h ) ways to do it. there are n − h remaining vertices in khn −{e1}. we continue to choose h vertices from the vertex set of khn −{e1} to form a new hyperedge of the k-matching, say e2. note that we have ( n−h h ) ways to do it. there are n−2h remaining vertices in khn −{e1, e2}. continue the process by the same method. then we can find e1, e2, . . . , ek such that ei ∩ ej = ∅ (1 ≤ i 6= j ≤ k). since the hyperedges of a k-matching are unordered, the coefficient ak of m(khn; w) is∏k−1 i=0 ( n−ih h ) k! for the special case, a0 = 1. 3.4. matching polynomials of hw` and its dual a centipede is a tree denoted by wn = (a ∪ b, e) (n ≥ 1), where a ∪ b is its vertex set, a = {a1, . . . , an}, b = {b1, . . . , bn}, and the edge set e = {aibi : 1 ≤ i ≤ n}∪{bibi+1 : 1 ≤ i ≤ n−1} (see figure 1(a)). b1 b2 b3 bn−1 bn a1 a2 a3 an−1 an (a) e1 e2 eℓ−1 eℓ e′1 e ′ 2 e′ℓ−1 e ′ ℓ (b) figure 1. (a) centipede wn, (b) hypergraph hw`. we define a new hypergraph hw` = (v ; e1 ∪ e2) such that ei ∈ e1, ei ′ ∈ e2 and ei ∩ei ′ 6= ei ∩ei+1 6= ∅, where 1 ≤ i ≤ `−1 (see figure 1(b)). lemma 3.10. [14] let i(w`; x) be the independence polynomial of w`. then i(w`; x) = ∑̀ s=0 tsx s, where ts = s∑ j=0 ( `− j `−s )( ` + 1− j j ) , s ∈{0, 1, . . . , `}. proposition 3.11. let m(hw`; w) be the matching polynomial of hw` with n vertices. then m(hw`; w) = bn/hc∑ k=0 akw (n−kh) 1 w k 2, 7 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 where ak = k∑ j=0 ( `− j `−k )( ` + 1− j j ) . proof. observe that the line-graph of hw` is a centipede w`. from lemma 3.10, we have i(w`; x) = ∑̀ s=0 tsx s, where ts = s∑ j=0 ( `− j `−s )( ` + 1− j j ) , s ∈{0, 1, . . . , `}. by proposition 2.3 and the above formula, we can obtain the matching polynomial of hw`, i.e. m(hw`; w) = bn/hc∑ k=0 akw (n−kh) 1 w k 2, where ak = k∑ j=0 ( `− j `−k )( ` + 1− j j ) . remark 3.12. according to our discussion, we observe that the coefficients of m(p`; w) are equivalent to those of a simple path of length `. likewise, the coefficients of m(c`; w) are the same as those of a simple cycle of length `. however, we are unable to find an equivalent matching polynomial for hypergraph hw`, giving the importance of this results and the next one. let k` = (v, e) denote a simple complete graph with ` vertices. set v (k`) = {vi : 1 ≤ i ≤ `}. let k∗` be a simple graph obtained from k` by adding ` new vertices {u1, u2, · · · , u` and the edges {uivi : 1 ≤ i ≤ `}. we now define a new hypergraph hs` = (v ; e1 ∪e2) satisfying the following conditions (see figure 2). (1) e1 = {e1, e2, · · · , e`} and e2 = {e′1, e′2, · · · , e′`}; (2) the hypergraph induced by the edges in e1 is a linear hyperstar; (3) for ei ∈ e1 and e′j ∈ e2, ei ∩ej ′ 6= ∅ for i = j, ei ∩ej ′ = ∅, for i 6= j. lemma 3.13. [14] let i(k∗` ; x) be the independence polynomial of k ∗ ` , then i(k∗` ; x) = (1 + x) `−1 · [1 + (` + 1)x]. proposition 3.14. let m(hs`; w) be the matching polynomial of hs`, then m(hs`; w) = (`−1)wn1 + bn/hc∑ k=1 [ ( `−1 s ) + ( `−1 s−1 ) ]w (n−kh) 1 w k 2. 8 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 e1 e2 eℓ−1 eℓ e ′ 2 e ′ 1 e ′ ℓ−1 e ′ ℓ figure 2. hypergraph hs`. proof. we know that the line-graph of hs` is a k∗` . from lemma 3.13, we have i(k ∗ ` ; x) = (1 + x)`−1 · [1 + (` + 1)x]. by simplifying the above equation, we obtain the following equation. i(k∗` ; x) = (`−1) + ∑̀ s=1 [( `−1 s ) + ( `−1 s−1 )] xs. by the propostion 2.3 and the above equation, we can obtain the matching polynomial of hs`, i.e. m(hs`; w) = (`−1)wn1 + bn/hc∑ k=1 [( `−1 s ) + ( `−1 s−1 )] w (n−kh) 1 w k 2. 4. zeros of matching polynomials of a hypergraphs it is well known that the zeros of matching polynomial of any connected graph are all real numbers [8]. however, we observe from an example that this assertion does not hold true for hypergraphs, in general. to the best of our knowledge, this case is unknown. consider the hypergraph h as shown in figure 3(a). observe that the line-graph of h is the next simple graph l(h) (see figure 3(b)). v1 v2 v3 v4 v5v6 v7 e1 e2 e3 e4 e5e6 e7 (a) (b) figure 3. (a) hypergraph h, (b) the line-graph of h. proposition 4.1. [14] let g = (v, e) be a graph, u ⊂ v be such that g[u] is a complete subgraph of g and i(g, x) be the independence polynomial of g. then i(g, x) = i(g−u, x) + x ∑ v∈u i(g−n[v], x), where n(v) = {w : vw ∈ e(g)}, n[v] = n(v) ⋃ {v}. 9 z. guo et al. / j. algebra comb. discrete appl. 4(1) (2017) 1–11 proposition 4.2. [19] let en be an edgeless graph on n vertices, i(en, x) be the independence polynomial of en. then i(en, x) = (1 + x) n. from proposition 4.1 and proposition 4.2, we obtain the independence polynomial of l(h), i.e. i(l(h), x) = x5 + 5x4 + 12x3 + 14x2 + 7x + 1. by proposition 2.3, we obtain the matching polynomial of h, i.e. m(h; w) = w15 + 7w13 + 14w11 + 12w9 + 5w7 + w5. using the matlab software, we obtain from m(h; w) the following zeros: 0, 0, 0, 0, 0, -0.0000 + 2.0893i, -0.0000 2.0893i, -0.2839 + 0.6309i, -0.2839 0.6309i, 0.2839 + 0.6309i, 0.2839 0.6309i, 0.0000 + 1.0000i, 0.0000 1.0000i, -0.0000 + 1.0000i, -0.0000 1.0000i. thus, we can obtain the fact that the zeros of a matching polynomial of a hypergraph are not necessarily all real numbers. acknowledgment: the authors are very grateful to the editor and the referees’ valuable comments and suggestions, which helped to improve the presentation of this paper. references [1] n. alon, j. kim, j. spencer, nearly perfect matchings in regular simple hypergraphs, j. isr. j. math. 100(1) (1997) 171–187. 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bouchat, tricia muldoon brown abstract: for a rooted tree γ, we consider path ideals of γ, which are ideals that are generated by all directed paths of a fixed length in γ. in this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. in particular, we provide a class of subforests of γ that are in one-to-one correspondence with the multi-graded betti numbers of the path ideal as well as providing a method for determining the projective dimension and the castelnuovo-mumford regularity of a given path ideal. 2010 msc: 05e40 keywords: betti numbers, path ideals, rooted trees, monomial ideals 1. introduction monomial ideals have been studied extensively in the literature and many applications of monomial ideals have been explored. in this paper, we consider a specific class of monomial ideals, known as path ideals. path ideals are a type of squarefree monomial ideals that are generated in a fixed degree. edge ideals, a specific class of path ideals, were first studied by conca and de negri in [4]. given a graph γ = (v,e) having vertex set v and edge set e, we can form the path ideal of length (t− 1) associated to γ by considering the ideal generated by the monomials corresponding to all (t− 1) length paths in γ. if v = {x1, . . . ,xn}, this ideal is considered in the polynomial ring r := k[x1, . . . ,xn], where k is a field. in [10], nagel and reiner showed in proposition 6.1 that the betti numbers of edge ideals of arbitrary graphs can be as complicated as desired. in particular, this proposition implies that the betti numbers of the edge ideal of an arbitrary graph can depend on the choice of the field, k. for the case of path ideals of directed, rooted trees bouchat, há, and o’keefe showed in theorem 2.7 of [3] that the betti rachelle r. bouchat; indiana university of pennsylvania (email: rbouchat@iup.edu). tricia muldoon brown (corresponding author); armstrong state university (email: patricia.brown@armstrong.edu). 23 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 x8 x4 x5 x6 x7 x2 x3 x1 i2(γ) = (x1x2, x2x4, x4x8, x2x5, x1x3, x3x6, x3x7) i3(γ) = (x1x2x4, x2x4x8, x1x2x5, x1x3x6, x1x3x7) i4(γ) = (x1x2x4x8) figure 1. an example of a graph, γ, and its associated path ideals numbers of a path ideal associated to a rooted tree are independent of the choice of the field, k. in this paper, we will restrict our focus to the study of path ideals of rooted trees. recall that a tree is a simple, connected graph containing no loops or multi-edges. then a rooted tree is a tree together with a fixed vertex, called the root. it is natural to consider a rooted tree as a directed graph in which all edges are assigned the direction going away from the root. it should be noted that xi will denote both the vertex in the graph γ as well as the monomial in the polynomial ring r. definition 1.1. given a directed, rooted tree γ and t ≥ 2, the path ideal of length (t− 1) of γ is: it(γ) := (xi1xi2 · · ·xit | {xi1,xi2, . . . ,xit} forms a directed path in γ) . thus, for a given directed, rooted tree γ there can be more than one path ideal associated to γ as illustrated in figure 1. for a given tree γ, we can successively remove any leaves that occur at level less than (t− 1) when considering the path ideal it(γ), as these vertices cannot contribute to a minimal generator in it(γ). a tree that has had successive removal of all leaves ocurring at level less than (t− 1) will be said to be in clean form. the minimal free resolutions of path ideals of trees have been studied by many authors, including há and van tuyl in [7], katzman in [8], and kummini in [9]. the basic tools and decompositions that will be used in this paper were introduced by faridi and alilooee in [1]. the aim of this paper is to give a constructive description of the multi-graded betti numbers for path ideals of rooted trees. this constructive description of the betti numbers corresponding to a path ideal of a rooted tree will also provide a method to compute the projective dimension as well as the castelnuovo-mumford regularity. 2. basic definitions we begin with the background concepts from commutative algebra. let m be a finitely generated graded s-module. associated to m is a minimal free resolution, which is of the form 0 → ⊕ a s(−a)βp,a(m) δp−→ ⊕ a s(−a)βp−1,a(m) δp−1−→ ··· δ1−→ ⊕ a s(−a)β0,a(m) → m → 0 where the maps δi are exact and where s(−a) denotes the translation of s obtained by shifting the degree of elements of s by a ∈ (n∪{0})n. the numbers βi,a(m) are called the multi-graded betti numbers (or nn-graded betti numbers) of m, and they correspond to the number of minimal generators of degree a occurring in the ith-syzygy module of m. it should be noted that the graded betti numbers (or n-graded betti numbers) of m can be defined as βi,j(m) := ⊕ a1+···+an=j βi,a(m) where a = (a1, . . . ,an). 24 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 there are two invariants corresponding to the minimal free resolution of m which measure the “size” of the resolution. definition 2.1. let m be a finitely generated graded s-module. 1. the projective dimension of m, denoted pd(m), is the length of the minimal free resolution associated to m. 2. the castelnuovo-mumford regularity (or regularity), denoted reg(m), is reg(m) := max{j − i | βi,j(m) 6= 0}. we also need the following definitions from simplicial topology. definition 2.2. 1. an abstract simplicial complex, ∆, on a vertex set x = {x1, . . . ,xn} is a collection of subsets of x satisfying: (a) {xi}∈ ∆ for all i, and (b) f ∈ ∆, g ⊂ f =⇒ g ∈ ∆. the elements of ∆ are called faces of ∆, and the maximal faces (under inclusion) are called facets of ∆. the simplicial complex ∆ with faces f1, . . . ,fs will be denoted by 〈f1, . . . ,fs〉. 2. if ∆ is an abstract simplicial complex, then a face τ ∈ ∆ is called a free face if it is contained in a unique facet of ∆. 3. for any y ⊆ x, an induced subcollection of ∆ on y, denoted by ∆y, is the simplicial complex whose vertex set is a subset of y and whose facet set is given by {f | f ⊆y and f is a facet of ∆}. 4. if f is a face of ∆ = 〈f1, . . . ,fs〉, the complement of f in ∆ is given by fcx = x \f and ∆cx = 〈(f1) c x , . . . , (fs) c x〉. it should be noted that if f1, . . . ,fs are facets in definition 2.2(1), then 〈f1, . . .fs〉 is a minimal representation of ∆. in particular, the complementary complex ∆cx , described in the fourth part of definition 2.2, is heavily utilized within this paper. definition 2.3. 1. let ∆ be a simplicial complex with vertex set x1, . . . ,xn. then the facet ideal of ∆ is defined as i(∆) := (∏ x∈f x | f is a facet of ∆ ) . moreover, i is an ideal in the polynomial ring r := k[x1, . . . ,xn]. 2. let i be an ideal in r := k[x1, . . . ,xn] minimally generated by square-free monomials m1, . . . ,ms. the facet complex ∆(i) associated to i has vertex set {x1, . . . ,xn} and is defined by ∆(i) := 〈f1, . . . ,fs〉 where fi = {xj | xj|mi , 1 ≤ j ≤ n} for 1 ≤ i ≤ s. 25 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 the above constructions provide a one-to-one correspondence between simplicial complexes on the vertex set {x1, . . . ,xn} and the square-free monomial ideals in r := k[x1, . . . ,xn]. we illustrate this construction with the following example. example 2.4. consider the path ideal i3(γ) = (x1x2x3,x2x3x4,x2x3x5) associated to the tree, γ, pictured below. then the facet complex associated to γ with respect to t = 3 is γcvert(γ) = 〈(x1x2x3) c vert(γ), (x2x3x4) c vert(γ), (x2x3x5) c vert(γ)〉 = 〈x4x5,x1x5,x1x4〉 x4 x3 x2 x1 x5 x5 x1 x4 γ γc vert(γ) the dimension of the reduced homology group of the complementary complex of a path ideal is instrumental in determining the betti numbers of the corresponding minimal free resolution. we will use a corollary of theorem 2.8 of alilooee and faridi in [1] to help determine the multi-graded betti numbers for path ideals of rooted trees. this corollary, found below, appears in [2] as corollary 2.11. corollary 2.5. let s = k[x1, . . . ,xn] be a polynomial ring over a field k, and let i be a pure, squarefree monomial ideal in s. then the multi-graded betti numbers of i are given by βi,a(i) = dimk h̃i−1 ( γcvert(γ) ) for i ≥ 1 where γ is an induced subcollection of ∆(i) with vert(γ) = {xi | ai = 1} where a = (a1, . . . ,an). it should be noted that the betti numbers β0,a(i) correspond to the minimal generating set of i. finally, we will rely upon lemma 4.1 in [1] of alilooee and faridi, which is a tool that allows us to determine the ith reduced homology group of the complementary complex using the (i − 1)st reduced homology group of a smaller complex, which we call the deleted complementary complex, that is, lemma 2.6 ([lemma 4.1, alilooe and faridi]). suppose γ is a tree generated by the paths p1,p2, . . . ,pk and suppose p1 ∩ (p2 ∪p3 ∪·· ·∪pk) 6= ∅. then dimk h̃i (〈pc1 ,p c 2 , . . . ,p c k〉) = dimk h̃i−1 ( 〈(p2)cvert(γ)\p1, (p3) c vert(γ)\p1, . . . , (pk) c vert(γ)\p1〉 ) . it should further be noted that in corollary 6.1 of [5], it is shown that the independence complex of graph forests are simple-homotopy equivalent to a vertex or to a sphere. thus by corollary 2.5, the multi-graded betti numbers of path ideals of directed, rooted trees correspond via simple-homotopy equivalence to a vertex or to a sphere. we conclude this section with some graph theoretic definitions before stating our three line graph constructions on directed, rooted trees. definition 2.7. let γ be a rooted tree with root x and vertex set vert(γ), and let y ∈ vert(γ). 1. the level of y, denoted level(y), is the length of the unique path in γ from x to y. the height of γ is the maximal level over all vertices in γ. 2. the parent vertex of the non-root vertex y is the unique vertex z such that yz forms an edge in γ and level(z) = level(y) − 1. 26 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 3. a subgraph h of γ is called an induced subgraph if for every pair of vertices x,y ∈ h the following condition holds: if {x,y} is an edge of γ, then it is also an edge of h. now we can define the essential constructions for this paper. starting with a tree γ which is composed of generating paths of length (t− 1) in the set {p1,p2, . . . ,pk}, we assume γcvert(γ) is not contractible, that is, there exists an i ≥ 0 for which dimk h̃i ( γc vert(γ) ) = 1. we create a new tree γ′ such that all edges and vertices of γ are also contained in γ′ as follows. definition 2.8. let γ be a tree with vertex set vert(γ)and edge set e(γ) and assume the vertex set {y1,y2, . . . ,yt+1}∩ vert(γ) = ∅. 1. let γ′ be the tree whose vertex set is vert(γ′) = vert(γ) ∪{y1,y2, . . . ,yt+1} and whose edge set is e(γ′) = e(γ)∪{(x0,yt+1), (yt+1,yt), (yt,yt−1), . . . , (y2,y1)} for some vertex x0 ∈ vert(γ). we say γ′ is a glue of γ. 2. let γ′ be the tree whose vertex set is vert(γ′) = vert(γ) ∪{y1,y2, . . . ,yn} for some 1 ≤ n ≤ t and whose edge set is e(γ′) = e(γ) ∪{(x0,yn), (yn,yn−1), (yn−1,yn−2), . . . , (y2,y1)} for some vertex x0 ∈ vert(γ). if level(y1) = level(x`) for at least one leaf vertex x` ∈ γ which lies on a directed path from x0, we say γ′ is a split of γ. 3. let γ′ be the tree whose vertex set is vert(γ′) = vert(γ) ∪{y1,y2, . . . ,yn} for some 1 ≤ n ≤ t and whose edge set is e(γ′) = e(γ) ∪{(x0,yn), (yn,yn−1), (yn−1,yn−2), . . . , (y2,y1)} for some vertex x0 ∈ vert(γ). if level(y1) 6= level(x`) for at least one leaf vertex x` ∈ γ which lies on a directed path from x0, we say γ′ is a contraction of γ. remark 2.9. we will also use the terms glue and split as constructions on the empty graph. consider t ≥ 2. starting with the empty graph, a split will result in the line graph having exactly t vertices, and a glue from the empty graph will result in the line graph having exactly t+ 1 vertices. it should be noted that adding less than t new vertices to the empty graph will result in a graph having no minimal generators for the ideal it(γ). definition 2.8 along with lemma 2.6 will allow us to begin with a tree whose complementary complex is homotopic to a sphere in some dimension, adjoin a line graph, and consequently determine the homotopy type of the resulting complementary simplicial complex of the modified tree. in particular, we will show that these three cases, glue, split, and contraction, are the only needed constructions and consequently the new simplicial complex will be homotopic to a sphere in either one or two dimensions higher, or it will be contractible. 3. constructions and betti numbers in this section, we will explore how a glue, split, and contraction of a directed graph γ will determine the non-zero betti number of top dimension in the corresponding minimal free resolution of the path ideal of the newly constructed graph, γ′ and consequently will determine the projective dimension. when we create a new tree, γ′, from γ by adding n new vertices {y1,y2, . . . ,yn} through a glue, split, or contraction, at most n new generating paths of length (t − 1) will be added to the path ideal of it(γ); depending on the level of the vertex y1. let ei be the unique directed path of length (t − 1) which terminates in vertex yi for 1 ≤ i ≤ j where j = min{level(y1) − t + 2,n}. the number j of new generating paths depends on the level of x0 in the tree γ and the number of vertices added, as there may not exist a path of length (t− 1) which terminates in vertex yi. thus γ′ is generated by the paths {p1, . . . ,pk,e1, . . . ,ej}. in each of the following cases we consider the deleted complementary complex that was described in section 2. since e1 always contains a leaf of γ′, the conditions of lemma 2.6 are satisfied and γ′ c vert(γ′)\e1 = 〈(p1) c vert(γ′)\e1, . . . , (pk) c vert(γ′)\e1, (e2) c vert(γ′)\e1, . . . , (ej) c vert(γ′)\e1〉. 27 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 we begin with the case of a glue. proposition 3.1. let γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which βi,a(it(γ)) = 1, where a = (a1, . . . ,am) and aj = 1 if and only if xj ∈ γ. let γ′ be a glue of γ, then βi+2,a′(it(γ ′)) = 1 where a′ = (a1, . . . ,am+t+1) and aj = 1 if and only if xj ∈ γ′. before the proof, we illustrate the deletion methods of lemma 2.6 in the case of a glue with the following example. example 3.2. consider the tree γ1, occurring as a glue of γ: x4 x3 x1 x5 y4 y3 y2 y1 x2 then using the deletion method of lemma 2.6, we set e1 = {y1,y2,y3}, e2 = {y2,y3,y4}, e3 = {x2,y3,y4}, e4 = {x1,x2,y4}, e5 = {x1,x2,x3}, e6 = {x2,x3,x4}, and e7 = {x2,x3,x5}. it follows that: dimk h̃i ( (γ1) c vert(γ1) ) = dimk h̃i−1〈(e2)cvert(γ1)\e1, . . . , (e7) c vert(γ1)\e1〉 = dimk h̃i−1〈x1x2x3x4x5,x1x3x4x5,x3x4x5,x4x5y4,x1x5y4,x1x4y4〉 = dimk h̃i−1〈x1x2x3x4x5,x4x5y4,x1x5y4,x1x4y4〉 after identifying the vertices x2 and x3, the simplicial complex can be visualized as, x1 x4 x2x3 x5 y4 where the tetrahedron labeled by x1x2x3x4x5 is solid, but the tetrahedron labeled by x1x4x5y4 is hollow. thus dimk h̃2(〈x1x2x3x4x5,x4x5y4,x1x5y4,x1x4y4〉) = 1, and thus, by lemma 2.6, we have dimk h̃3 ( (γ1) c vert(γ1) ) = 1. 28 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 proof of proposition 3.1. let γ′ be a glue of the tree γ in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which βi,a(it(γ)) = 1, where a = (a1, . . . ,am) and ai = 1 if and only if xi ∈ γ. further, let e1,e2, . . . ,ej be the new generating paths of length t−1 that are in γ′ but not in γ, where e1 is the directed path e1 = yt . . .y2y1. to proceed, we will delete the path e1 and apply lemma 2.6. first, the sole vertex in the set vert(e2) \ vert(e1) is yn, so 〈(e2)cvert(γ′)\e1〉 = 〈(yn) c vert(γ′\e1)〉. further yn is a vertex contained in every generating path e3,e4, . . .ej, thus 〈(yn)cvert(γ′\e1)〉 contains 〈(ei)cvert(γ′)\e1〉 for all 3 ≤ i ≤ j. therefore, γ′ c vert(γ′)\e1 = 〈(p1) c vert(γ′)\e1, . . . , (pk) c vert(γ′)\e1, (yn) c vert(γ′\e1)〉. we observe, for all 1 ≤ i ≤ k, the complex 〈(pi)cvert(γ′\e1)〉 is equal to the simplicial join 〈(pi)cvert(γ) ∗yn〉, and thus yn must be a vertex of the simplex 〈(pi) c vert(γ′)\e1〉. therefore the complex 〈(p1)cvert(γ′)\e1, . . . , (pk) c vert(γ′)\e1〉 is exactly the cone of yn over γ c vert(γ) . in particular, because of our assumption of a non-zero betti number in dimension i ≥ 0, there exists a union of faces of γc vert(γ) that is homotopic to an i-dimensional sphere, and thus the simplicial complex 〈(p1)cvert(γ′)\e1, . . . , (pk) c vert(γ′)\e1〉 is homotopic to a cone over an i-dimensional sphere. the complex 〈(yn)cvert(γ′\e1)〉 is the simplex whose vertices are all the vertices in vert(γ). any vertex in vert(γ) that in not contained in the resulting i-dimensional sphere is a free face of the complex γc vert(γ) as well as of the complex γ′cvert(γ′)\e1, and hence can be collapsed into the face not containing the vertex yn. in particular it can be retracted to the face whose boundary is contained in the union of faces of γc vert(γ) that is homotopic to an i-dimensional sphere. therefore γ′cvert(γ′)\e1 is homotopic to the cone over an open i-dimensional sphere which has been filled with the simplex 〈(yn)cvert(γ′\e1)〉, thus creating a sphere of dimension i. we now have dimk h̃i+1((γ ′)cvert(γ′)) = dimk h̃i((γ ′)cvert(γ′)\e1 ) = 1. then the complex (γ′)c vert(γ′) is homotopic to a sphere of dimension i + 1; that is, of dimension two more than the sphere describing the complementary complex of γ and further βi+2,a′(it(γ ′)) = 1 where a′ = (a1, . . . ,am′) and ai = 1 if and only if xi ∈ γ′. next, we consider the split construction from a given tree. proposition 3.3. let γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which βi,a(it(γ)) = 1, where a = (a1, . . . ,am) and aj = 1 if and only if xj ∈ γ. let γ′ be a split of γ, then βi+1,a′(it(γ ′)) = 1 where a′ = (a1, . . . ,am′) and aj = 1 if and only if xj ∈ γ′. as before, we illustrate the deletion methods of lemma 2.6 in the case of a split with the following example before providing the proof. example 3.4. consider the tree γ2, occuring as a split of γ: x4 x3 x1 x5 y2 y1 x2 29 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 using the deletion method, we set e1 = {x2,y1,y2}, e2 = {x1,x2,y2}, e3 = {x1,x2,x3}, e4 = {x2,x3,x4}, and e5 = {x2,x3,x5}. it follows that dimk h̃i ( (γ2) c vert(γ2) ) = dimk h̃i−1〈(e2)cvert(γ2)\e1, . . . , (e5) c vert(γ2)\e1〉 = dimk h̃i−1〈x3x4x5,x4x5,x1x5,x1x4〉 = dimk h̃i−1〈x3x4x5,x1x5,x1x4〉. the complex can be visualized as: x5 x1 x4x3 hence dimk h̃1(〈x3x4x5,x1x5,x1x4〉) = 1, and by lemma 2.6 dimk h̃2 ( (γ2) c vert(γ2) ) = 1. proof of proposition 3.3. let γ be a tree in clean form such that βi,a(it(γ)) = 1 for some i ≥ 0 where a = (a1, . . . ,am) and ai = 1 if and only if x ∈ vert(γ), and let γ′ be a split of γ. then γ′ is formed from γ by attaching a line graph with vertices yn, . . . ,y1 where n ≤ t via the edge {x0,yn} for some x0 ∈ vert(γ). further there exists a leaf vertex x` of γ that is on a directed path from x0 such that level(x`) = level(y1). let ei be the length t− 1 path in γ′ terminating at vertex yi while p1, . . . ,pk will denote the paths of length t− 1 in γ. if e1 ∩ vert γ = ∅, the result follows directly from lemma 2.6. suppose to the contrary, and let x be a vertex in the intersection of e1 and vert(γ). we note for any 1 ≤ i ≤ k, if x ∈ (pi)cvert(γ), then x` ∈ (pi)cvert(γ). this is because level(x`) − level(x) ≤ t, so all paths containing x` must also contain x. thus every face in γc vert(γ) containing x must also contain x`. hence, vertex x is a free vertex in the complex γc vert(γ) which may be retracted onto x`, and the complex γcvert(γ) is collapsed onto the complex 〈(p1)cvert γ′\e1, . . . , (pk) c vert γ′\e1〉⊆ (γ ′)c vert(γ′\e1). as retractions preserve homotopy, we have γcvert(γ) ∼〈(p1) c vert γ′\e1, . . . , (pk) c vert γ′\e1〉. if γ′ only contains one additional path when compared to γ, then the paths making up γ′ are e1,p1, . . . ,pk. it follows that βi+1,a′ (it(γ ′)) = dimk h̃i ( (γ′)cvert(γ′) ) = dimk h̃i−1 ( 〈(p1)cvert γ′\e1, . . . , (pk) c vert γ′\e1〉 ) where a′ = (a1, . . . ,am′) such that ai = 1 if and only if xi ∈ γ′. it follows that βi,a (it(γ)) = dimk h̃i−1 ( (γ)cvert(γ) ) = 1 and hence the claim is proven. now assume that γ′ contains two or more new paths when compared to γ. it follows that βi+1,a′ (it(γ ′)) = dimk h̃i ( (γ′)c vert(γ′) ) = dimk h̃i−1 ( 〈(e2)cvert γ′\e1, . . . , (ej) c vert γ′\e1, (p1) c vert γ′\e1, . . . , (pk) c vert γ′\e1〉 ) . let x′ denote the sole vertex in the set vert(e2) − vert(e1). then (e2)cvert(γ′)\e1 = (x ′)c vert(γ′)\e1. furthermore, notice that (ei)cvert(γ′)\e1 ⊆ (e2) c vert(γ′)\e1 for all 3 ≤ i ≤ j. hence, βi+1,a′ (it(γ ′)) = dimk h̃i−1 ( 〈(x′)cvert γ′\e1, (p1) c vert γ′\e1, . . . , (pk) c vert γ′\e1〉 ) . 30 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 furthermore, because the complex 〈(p1)cvert γ′\e1, . . . , (pk) c vert γ′\e1〉 is just the complex formed by retracting the vertices in vert(e1) ∩ vert(γ) onto the leaf vertex x` in the complex γcvert(γ), it is left to check that the face (x′)c vert(γ′)\e1 of (γ ′)c vert(γ′\e1) does not close the i-dimensional hole that we know by assumption is in γc vert(γ) . suppose to the contrary, when the simplex (x′)c vert(γ′)\e1 is inserted in the complex 〈(p1)cvert γ′\e1, . . . , (pk) c vert γ′\e1〉, the newly formed complex (γ ′)c vert(γ′\e1) is contractible. first observe that the height of the subtree rooted at x′ is t. we first assume that γ′ has exactly two leaves, x` and y1 on directed paths from x′. therefore, without loss of generality, we assume the directed path from x′ to x` was glued onto some subtree of γ whose leaves are all at a level less than level(x`). we know from the proof of proposition 3.1 that in this case, the vertex x′ is on the boundary of the complex homotopic to the i-dimensional sphere formed by the glue. hence, x′ is not a free vertex and cannot be retracted onto another vertex in the complex. therefore, the simplex (x′)c vert(γ′)\e1 cannot fill this hole and the thus the complex (γ′)c vert(γ′\e1) is not contractible. by extension, if γ ′ has more than two leaves on directed paths from x′, each additional split still has a face determined by the complement of x′, and hence the complex cannot be contractible. in the final proposition of this section, we consider a contraction of a given tree. proposition 3.5. let γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which βi,a(it(γ)) = 1, where a = (a1, . . . ,am) and aj = 1 if and only if xj ∈ γ. let γ′ be a contraction of γ, then βi,a′(it(γ ′)) = 0 for all i where a′ = (a1, . . . ,am′) and aj = 1 if and only if xj ∈ γ′. we precede the proof with an example illustrating the deletion methods of lemma 2.6 in the case of a contraction. example 3.6. consider the tree γ3, occurring as a contraction of γ: x4 x2 x1 x5 y2 y1 x3 then using the deletion method, we set e1 = {x3,y1,y2}, e2 = {x2,x3,y2}, e3 = {x1,x2,x3}, e4 = {x2,x3,x4}, and e5 = {x2,x3,x5}. it follows that: dimk h̃i ( (γ3) c vert(γ3) ) = dimk h̃i−1〈(e2)cvert(γ3)\e1, . . . , (e5) c vert(γ3)\e1〉 = dimk h̃i−1〈x1x4x5,x4x5,x1x5,x1x4〉 = dimk h̃i−1〈x1x4x5〉 the complex can be visualized as: x5 x1 x4 31 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 notice that dimk h̃i(〈x1x4x5〉) = 0 for all i. hence, by lemma 2.6 βi,a (it(γ)) = 0 for all i. proof of proposition 3.5. let γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which βi,a(it(γ)) = 1, where a = (a1, . . . ,am) and ai = 1 if and only if xi ∈ γ and let γ′ be a contraction of γ, also in clean form. (if γ′ is not in clean form, the vertices lying on the paths of length less than t− 1 do not contribute to the minimal free resolution and hence trivially βi,a’(it(γ′)) = 0 for all i where a′ = (a1, . . . ,am′) and ai = 1 if and only if xi ∈ γ′.) assume e1,e2, . . . ,ej are the new paths of length (t−1) found in γ′ but not in γ. in this final case, we still assume 1 ≤ n ≤ t, but now let level(y1) 6= level(x) for all leaf vertices x of γ which lie on a directed path from x0. first we look at the situation where level(y1) < level(x`) for some leaf vertex x`. as before we want to apply lemma 2.6, but this time we delete the path p = xtxt−1 · · ·x1 which terminates at the leaf x` ∈ γ which is on a directed path from x0. let xt+1 be the unique parent vertex of the initial vertex xt of the path p , which must exist because γ′ is in clean form. then xt+1xt · · ·x2 = pm was a generating path of γ for some m, and so 〈(pm)cvert(γ′)\p〉 = 〈(xt+1) c vert(γ′)\p〉 is a part of the deleted complementary complex γ′ c vert(γ′)\p = 〈(p1) c vert(γ′)\p , . . . , (pk) c vert(γ′)\p , (e1) c vert(γ′)\p , . . . , (ej) c vert(γ′)\p〉. any other generating paths that contain xt+1, will have complementary complexes contained in the simplex 〈(xt+1)cvert(γ′)\p〉, and because level(y1) < level(x1), this includes all of the new generators e1,e2, . . . ,ej, so γ′ c vert(γ′)\p = 〈(p1) c vert(γ′)\p , . . . , (pk) c vert(γ′)\p〉. now, the vertex y1 is a vertex of the simplex 〈xct+1〉, but y1 is also a vertex of every simplex generated from the original paths pc1 ,p c 2 , . . . ,p c k in γ. thus (γ ′)c vert(γ′)\p is homotopic to a cone of the vertex y1 over some simplicial complex and hence is contractible. on the other hand, if the level of y1 is greater than the level of some leaf vertex x`, apply lemma 2.6 and delete the unique path e1 of length t−1 terminating at y1. in this final situation, we must be careful of the case where x0 is a leaf. if x0 is not a leaf, as above, let xt+1 be the parent of the initial vertex of the path e1. proceeding as before we see γ′ c vert(γ′)\e1 = 〈(p1) c vert(γ′)\e1, . . . , (pk) c vert(γ′)\e1〉 and that 〈(pm)cvert(γ′)\e1〉 = 〈(xt+1) c vert(γ′)\p〉 for some m. this time x` is a vertex of the simplex 〈(xt+1)cvert(γ′)\p〉 as well as of every other simplex 〈(pj) c vert(γ′)\e1〉 not contained in 〈(xt+1) c vert(γ′)\p〉. therefore (γ′)c vert(γ′)\e1 is a cone over the vertex x`, and thus is contractible. if x0 is a leaf, then y1y2 · · ·yn is a part of a longer line graph containing x0. in this case the attachment of this longer line graph should be considered as whole: as either a glue or a contraction with a different attaching vertex x0 which is not a leaf. in the final section, we state our main result and apply the constructions above to an example, completely determining the betti numbers of the minimal free resolution of its path ideal. 4. minimal free resolutions we observe, any tree γ′ can be constructed from one of its subtrees, γ, through a sequence of line graph attachments. therefore, we only need to consider the situation where γ′ differs from γ by exactly one line graph. if the line graph consists of an attachment of more than t + 1 vertices, we further decompose the attachment into a sequence of some number of glues as well as possibly either a split or a contraction. we will also impose order on the creation of γ′ such that when a new line graph is adjoined, the level of x0 (i.e. the level of the glue, split, or contraction) is greater than ht(γ) − t. thus, under this 32 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 model, any directed graph γ′ containing the directed graph γ as a subgraph, could have been created from γ through a sequence of glues, splits, and contractions. by extension and given the conventions of a glue and split to an empty graph, any clean graph can be created from a sequence of glues, splits, and contractions starting with an empty graph for some t ≥ 2. as the empty graph trivially satisfies the conditions on propositions 3.1, 3.3, and 3.5, we have that all clean, directed graphs can be constructed from a sequence of glues, splits, and contractions. given a subforest λ of γ, let g(λ) and s(λ) denote the number of glues and splits, respectively, required to construct λ from an empty graph. then we have the following theorem. theorem 4.1. let γ be a directed, rooted tree, and let t ≥ 2. then βi,a (it(γ)) = 1 precisely when {xj | aj 6= 0} corresponds to the vertex set of an induced subforest, λ, of γ constructed from the empty graph by a sequence of only glues and splits. moreover, in this case i = 2g(λ) + s(λ) − 1. proof. since multigraded betti numbers correspond to induced subforests of γ, the result follows immediately from propositions 3.1, 3.3, and 3.5. we illustrate theorem 4.1 with an example. example 4.2. consider the tree γ depicted below. the graded minimal free resolution of i3(γ) is also provided below and was obtained using macaulay 2 (see [6]). x1 x2 x3 x4 x5 x6 x8x7 x9 x10 i3(γ) = (x1x2x3,x2x3x5,x3x5x7,x3x5x8,x5x7x10,x1x2x4,x2x4x6,x4x6x9) 0 → r3(−8) → r(−5) ⊕ r(−6) ⊕ r8(−7) → r8(−4) ⊕ r(−5) ⊕ r5(−6) → r8(−3) → i3(γ) → 0 the betti numbers, β0,3(i3(γ)) correspond to all paths in γ of length 2, which are all induced subforests of γ formed as a split from the empty graph. the betti numbers, β1,j(i3(γ)), correspond to all induced subtrees of γ formed either by two splits from the empty graph or by one glue from the empty graph. hence, these betti numbers correspond to induced subforests of γ of the following form: two splits two splits two splits one glue (2 of this type) (1 of this type) (5 of this type) (6 of this type) the betti numbers, β2,j(i3(γ)), correspond to all induced subforests of γ formed either by three splits from the empty graph or by one glue and one split from the empty graph. 33 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 one glue & one split one glue & one split one glue & one split one glue & one split (1 of this type) (1 of this type) (2 of this type) (1 of this type) one glue & one split one glue & one split three splits (1 of this type) (3 of this type) (1 of this type) lastly, the betti numbers, β3,j(i3(γ)), correspond to all induced subforests of γ formed from either four splits, from one glue and two splits, or from two glues. the only possible induced subforests of these types in γ are depicted below: one glue & two splits two glues two glues (1 of this type) (1 of this type) (1 of this type) it should be noted that the vertex sets of the induced subforests correspond to the multi-graded betti numbers in the minimal free resolution of i3(γ). theorem 4.1 also implicitly describes the projective dimension and regularity of the it(γ). we have the following corollaries. corollary 4.3. let γ be a directed, rooted tree and let t ≥ 2, and let c := {λ | λ is an induced subforest of γ constructible from the empty graph using only splits and glues}. then the projective dimension of it(γ) is given by pd(it(γ))) = max{2g(λ) + s(λ) | λ ∈c}− 1. and the castelnuovo-mumford regularity of it(γ) is given by reg(it(γ)) = max{|vert(λ)|− (2g(λ) + s(λ)) | λ ∈c} + 1. 34 r. bouchat, t. m. brown / j. algebra comb. discrete appl. 4(1) (2017) 23–35 references [1] a. alilooee, s. faridi, on the resolution of path ideals of cycles, commun. algebra 43(12) (2015) 5413–5433. [2] r. r. bouchat, t. m. brown, multi–graded betti numbers of path ideals of trees, to appear in j. algebra appl. [3] r. bouchat, a. o’keefe, h. tài hà, path ideals of rooted trees and their graded betti numbers, j. combin. theory ser. a 118(8) (2011) 2411–2425. [4] a. conca, e. de negri, m–sequences, graph ideals, and ladder ideals of linear type, j. algebra 211(2) (1999) 599–624. [5] r. ehrenborg, g. hetyei, the topology of the independence complex, european j. combin. 27(6) (2006) 906–923. [6] d. grayson, m. e. stillman, macaulay 2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/macaulay2/. [7] h. tài hà, a. van tuyl, monomial ideals, edge ideals of hyper graphs, and their graded betti numbers, j. algebraic combin. 27(2) (2008) 215–245. [8] m. katzman, characteristic–independence of betti numbers of graph ideals, j. combin. theory ser. a 113(3) (2006) 435–454. [9] m. kummini, regularity, depth and arithmetic rank of bipartite edge ideals, j. algebraic combin. 30(4) (2009) 429–445. [10] u. nagel, v. reiner, betti numbers of monomial ideals and shifted skew shapes, electron. j. combin. 16(2) (2009) 1–59. 35 http://dx.doi.org/10.1080/00927872.2013.837170 http://dx.doi.org/10.1080/00927872.2013.837170 http://dx.doi.org/10.1142/s0219498817500189 http://dx.doi.org/10.1142/s0219498817500189 http://dx.doi.org/10.1016/j.jcta.2011.06.007 http://dx.doi.org/10.1016/j.jcta.2011.06.007 http://dx.doi.org/10.1006/jabr.1998.7740 http://dx.doi.org/10.1006/jabr.1998.7740 http://dx.doi.org/10.1016/j.ejc.2005.04.010 http://dx.doi.org/10.1016/j.ejc.2005.04.010 http://www.math.uiuc.edu/macaulay2/ http://www.math.uiuc.edu/macaulay2/ http://dx.doi.org/10.1007/s10801-007-0079-y http://dx.doi.org/10.1007/s10801-007-0079-y http://dx.doi.org/10.1016/j.jcta.2005.04.005 http://dx.doi.org/10.1016/j.jcta.2005.04.005 http://dx.doi.org/10.1007/s10801-009-0171-6 http://dx.doi.org/10.1007/s10801-009-0171-6 http://www.ams.org/mathscinet-getitem?mr=2515766 http://www.ams.org/mathscinet-getitem?mr=2515766 introduction basic definitions constructions and betti numbers minimal free resolutions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284931 j. algebra comb. discrete appl. 4(2) • 181–188 received: 15 june 2015 accepted: 22 february 2016 journal of algebra combinatorics discrete structures and applications essential idempotents and simplex codes∗ research article gladys chalom, raul a. ferraz, c. polcino milies abstract: we define essential idempotents in group algebras and use them to prove that every mininmal abelian non-cyclic code is a repetition code. also we use them to prove that every minimal abelian code is equivalent to a minimal cyclic code of the same length. finally, we show that a binary cyclic code is simplex if and only if is of length of the form n = 2k − 1 and is generated by an essential idempotent. 2010 msc: 16s34, 20c05, 94b15 keywords: group code, essential idempotent, simplex code 1. introduction let fq be a finite field with q elements and m a positive integer. we recall that a linear code of length n over fq is any proper subspace c of fnq . given a vector v = (a0,a1, . . . ,an−2,an−1) ∈c, its shift is the vector v1 = (an−1,a0,a1, . . . ,an−2). a linear code c is cyclic if, for every vector v ∈ c, its shift also belongs to c. notice that the definition implies that if a vector v1 = (a0,a1, . . . ,an−2,an−1) is in c, then every vector obtained as a circular permutation of v is also in c. the map ψ : fnq → fq[x]/〈xn−1〉 given by ψ(a0,a1, . . . ,am−2,am−1) = a0+a1x+· · · ,am−2xm−2+ am−1x m−1 is an isomorphism of linear spaces and it is easy to see that a code c of length n over fq is cyclic if and only its image ψ(c) is an ideal of the ring fq[x]/〈xn − 1〉. since this ring is isomorphic to the group algebra of a cyclic group c, of order n, over fq, we can think of cyclic codes as ideals in the group algebra fqc. more generally, an abelian code over fq is any ideal in the group algebra fqa of a finite abelian group a. these codes were introduced independently by s.d. berman [1] and macwilliams [8]. ∗ this work was supported by fapesp proc. 2009/52665-0 and cnpq 300243/79-0. gladys chalom, raul a. ferraz; universidade de sao paulo, brazil (email: agchalom@ime.usp.br, raul@ime.usp.br). c. polcino milies (corresponding author); universidade de sao paulo and univeersidade federal do abc, brazil (email: polcino@ime.usp.br). 181 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 since in the case when char(f) |6 |a| the group algebra fa is semisimple and all ideals are direct sums of the minimal ones, it is only natural to study minimal abelian code or equivalently, primitive idempotents and these has been done by several authors (see, for example [6], [5], [4] [9] [12] [14]). also, sabin and lomonaco [15] have shown that central codes in metacyclic group algebras are equivalent to abelian codes. in what follows, we shall show that these are not better than minimal cyclic codes. first, we prove in corollary 2.5 that every minimal abelian non-cyclic code is a repetition code. then, we show, in theorem 3.3, that every minimal abelian code is equivalent to a minimal cyclic code of the same length. finally, in section § 4, we show that essential idempotents can be used to characterize simplex codes. 2. basic facts throughout this paper all groups will be finite, and we shall always assume that the fields f considered are such that char(f) |6 |g|. for an element α in the group algebra fg, the hamming weight of α is the number of elements in its support; i.e., if α = ∑ g∈g αgg, then ω(α) = |{g ∈ g | αg 6= 0}|. given an ideal i ⊂ fg the weight distribution of i is the map which assigns, to each possible weight t, the number of elements of i having weight t. let h 6= {1} be a normal subgroup of a group g. then ĥ = 1 |h| ∑ h∈h h. is a central idempotent of fg and fg = fg · ĥ ⊕fg · (1 − ĥ). remark 2.1. as shown in [11, proposition 3.6.7] we have that fg · (1 − ĥ) = ∆(g,h), the kernel of the natural projection π : fg → f[g/h] and it is easy to see that fg · ĥ ∼= f[g/h] via the map ψ : fg · ĥ → f[g/h] defined by g · ĥ 7→ gh ∈ g/h. also, if α ∈ fa · ĥ, taking a transversal t of h in a we can rewrite α as α = ∑ t∈t ∑ h∈h αthth. as α ∈ fa · ĥ, we have that α = αĥ = ∑ t∈t ∑ h∈h αththĥ = ∑ t∈t ∑ h∈h αthtĥ. so αth = αth′ for every h,h′ ∈ h and, setting αt = αth,∀h ∈ h we can write α = |h| ∑ t∈t αttĥ. (1) since ĥ is central, it is a sum of primitive central idempotents called its constituents. given a primitive idempotent e we have that either eĥ = e or eĥ = 0, depending on wheather e is, or is not, a constituent of h. 182 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 in the first case, e is an element in the ideal fgĥ, and thus an element α in fge is also in fgĥ. so, it can be written as in eqution 1. writing t = {t1, t2, . . . , td} and h = {h1,h2, . . . ,hm}, the explicit expression of α is α = α1t1h1 + α1t1h2 + · · · + α1t1hm + · · · + αdtdh1 + αdtdh2 + · · · + αdtdhm. in terms of coding theory, this means that the code given by the minimal ideal fge is a repetition code. we shall be interested in idempotents that are not of this type. definition 2.2. a primitive idempotent e in the group algebra fg, is an essential idempotent if e · ĥ = 0, for every subgroup h 6= (1) in g. a minimal ideal of fg is called an essential ideal if it is generated by an essential idempotent and non essential otherwise. notice that, if e is a central idempotent, then the map π : g → ge, given by π(g) = g ·e is a group epimorphism. we can use this map to characterize essential idempotents. proposition 2.3. let e ∈ fg be a primitive central idempotent. then e is essential if and only if the map π : g → ge, is a group isomorphism. proof. let e be essential and assume, by way of contradiction, that π is not a monomorphism. then, there exists 1 6= g ∈ g such that π(g) = ge = e, hence also gie = e for every positive integer i. thus 〈̂g〉 ·e = e, a contradiction. conversely, assume that e is not essential. then, there exists h 6= (1) such that eĥ = e. for every h ∈ h, we have that h · e = h · eĥ = ĥ · e = e. hence h ⊂ ker(π) and thus π is not injective. consequently, if π is an isomorphism, e is essential. corollary 2.4. if g is abelian and fg contains an essential idempotent, then g is cyclic. proof. if e ∈ fg is essential, then g ∼= ge ⊂ fge. as g is abelian, a simple component fge of fg is a field and ge, being a finite subgroup contained in it, is cyclic. in terms of coding theory, the result above gives the following. corollary 2.5. let a be an abelian non-cyclic group. then, for any finite field fq, all the minimal codes of fqa are repetition codes. we wish to show, on the other hand, that if g is a cyclic group, then fg always contains an essential idempotent. to do so, assume that g is cyclic of order n = pn11 · · ·p nt t . then, g can be written as a direct product g = c1 ×···×ct, where ci is cyclic, of order pnii , 1 ≤ i ≤ t. let ki be the minimal subgroup of ci; i.e. the unique subgroup of order pi in ci and denote by ai a generator of this subgroup, 1 ≤ i ≤ t. set e0 = (1 − k̂1) · · ·(1 − k̂t) =  1 − (1 + a1 + · · · + apn1−111 ) p1   · · ·  1 − (1 + at + · · · + apnt−1tt ) pt   . then e0 is a central idempotent and we claim that it is non zero. in fact, it is easy to see that the coefficient of a1 · · ·at in this expression is (−1)t(1/p1) · · ·(1/pt) so, e0 6= 0. theorem 2.6. let g be a cyclic group. then, a primitive idempotent e ∈ fg is essential if and only if e ·e0 = e. 183 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 proof. let e ∈ fg be essential. then, in particular, ek̂i = 0, so e(1 − k̂i) = e, 1 ≤ i ≤ t. hence e ·e0 = e(1 − k̂1)(1 − k̂2) · · ·(1 − k̂t) = e · (1 − k̂2) · · ·(1 − k̂t) = · · · = e. conversely, if e is not essential then there exists a subgroup h of g such that e · ĥ = e. there exists a minimal subgroup ki ⊂ h, and for this subgroup we have that ĥ(1 − k̂i) = ĥ − ĥk̂i = 0. consequently ĥ ·e0 = 0. hence: e ·e0 = (e · ĥ) ·e0 = e · (ĥ ·e0) = 0. remark 2.7. notice that the previous theorem actually shows that e0 is the sum of all the essential idempotents of fg and, consequently, the simple components of the ideal fge0 are precisely the essential ideals. since e0 is non zero, we have the following. corollary 2.8. let g be a cyclic group and f a field such that char(f) |6 |g|. then, fg always contains an essencial idempotent. 3. the equivalence let f be a field, a be a finite abelian group such that char(f) |6 |a| and e 6= â an idempotent in fa. set he = {g ∈ g | ge = e}. (2) clearly, he is the unique maximal subgroup of g is such that hee = e and it can be shown easily that he = g if and only if e = ĝ, the principal idempotent of fg. actually, he is the kernel of the irreducible representation associated to the simple component fg ·e. theorem 3.1. let e 6= â be a primitive idempotent of fa and ψ the natural projection defined in remark 2.1. then, the element ψ(e) is an essential idempotent of f[a/he]. proof. let k 6= 1 be a non-trivial subgroup of a/he. then, it is of the form k = k/he where k 6= he is a subgroup of a containing he. let t be a transversal of he in k. then k̂ = 1 |k| ∑ k∈k k = |he| |k| ∑ t∈t tĥe = 1 |k| ∑ t∈t tĥe. as ψ(ĥe)=1, we have ψ(k̂) = 1 |k| ∑ t∈t ψ(t) = 1 |k| ∑ t∈t the = k̂. then ψ(e) ·k = ψ(e) ·ψ(k̂) = ψ(e · k̂) = 0, as k 6⊂ he. corollary 3.2. let e 6= â be a primitive idempotent of fa. then, the factor group a/he is cyclic. 184 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 proof. this fact follows immediately from proposition 3.1 above and corollary 2.4. let g1 and g2 denote two finite groups of the same order, f a field, and let γ : g1 → g2 be a biijection. denote by γ : fg1 → fg2 its linear extension to the corresponding group algebras. clearly, γ is a hamming isometry; i.e., elements corresponding under this map have the same hamming weight. ideals i1 ⊂ fg1 and i2 ⊂ fg2 such that γ(i1) = i2 are thus equivalent, in the sense that they have the same dimension and the same weight distribution. in this case, the codes i1 and i2 are said to be permutation equivalent and were called combinatorially equivalent in [15]. in what follows, we will show that every minimal ideal of an abelian group algebra fa is permutation equivalent to a minimal ideal of the group algebra of a cyclic group of the same length. let a be an abelian group of order n, f a field, i a minimal ideal of fa and e be the primitive idempotent generating i. let he be the subgroup defined in (2) and let c be a cyclic group of the same order n. if he = a then e = â and any bijection σ : a → c is such that i = fae = fe is mapped to fcĉ, so i is equivalent to fcĉ. so, assume that he 6= a. since the order d of the factor group a/he is a divisor of n, there exists a unique subgroup k of c such that |a/he| = |c/k| = d and, as they are both cyclic groups, we have that a/he ∼= c/k. (3) so, also f[a/he] ∼= f[c/k] and fa · ĥe ∼= f[a/he] ∼= f[c/k] ∼= fc · k̂. denote by µ : f[a/he] → f[c/k] and θ : fa · ĥe → fc · k̂ realizing these isomorphisms. let a ∈ a be an element such that a = ahe is a generator of a/he. then {1,a,a2, . . . ,ad−1} is a transversal of he in a and we can write a = {aih | 0 ≤ i ≤ d− 1, h ∈ he}. similarly, if t ∈ c is such that t = tk = µ(ahe), it is a generator of c/k and we can write c = {tik | 0 ≤ i ≤ d− 1, k ∈ k}. as he and k have the same order, we can choose a bijection f : he → k and define a map η : a → c by η(aih) = tif(h), for all aih ∈ a. given an element α ∈ fa · ĥe, using formula (1) we can write it in the form α = |he| d−1∑ i=0 αia i · ĥe, and, taking into account that |he| = |k|, we compute θ(α) = µ ( |he| d−1∑ i=0 αiā i ) = |k| d−1∑ i=0 αit ik̂ = d−1∑ i=0 αit i ( ∑ h∈he f(h) ) . comparing the expressions of α and θ(α) it is clear that the linear extension of η : a → c to fa coincides with θ on fa · ĥe, and it is such that θ(fa · ĥe) = fc · k̂. notice that fa · e ⊂ fa · ĥe and thus e is primitive also in fa · ĥe, so e′ = θ(e) is primitive in fc · k̂. 185 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 we claim that e′ is also primitive in fc. in fact, assume that e′ = f1 + f2, with f1,f2 orthogonal idempotents in fc. then e′ ·k̂ = f1k̂ + f2k̂ is a decomposition of e′ in fc ·k̂. hence, either f1k̂ = 0 and f2k̂ = e′ or vice-versa. so either f1 = f1e′ = f1e′k̂ = 0 or, in a similar way, f2 = 0. hence, we have shown the following. theorem 3.3. every minimal ideal in the semisimple group algebra fa of a finite abelian group a is permutation equivalent to a minimal ideal in the group algebra fc of a cyclic group c of the same order. it should be noted that non minimal abelian codes can actually be more convenient than the cyclic ones (see, for example, [7], [10] and [13]). 4. binary simplex codes set r = dimfqc and let b = {v1,v2, . . . ,vr} be a basis of c over fq. we shall denote by gb the generating matrix of c; i.e. the matrix whose rows are the componentes of the vectors of b, when written in the basis b of fqc. we start with a very simple remark. lemma 4.1. the matrix gb contains no zero columns. proof. in fact, assume that the jth column of the given matrix is zero. then, the jth component of every vector in c is equal to 0. take any non-zero vector v ∈c. so it has at least one component which is equal to 1. since every shift of v is in c, there exists a vector in c whose jth component is equal to 1, a contradiction. notice also that, if a matrix gb has two columns, in positions i and j, say, that are equal to one another, then the ith component of every vector in c is equal to its jth component. hence, we have shown the following. lemma 4.2. the columns of a matrix gb are pairwise different for a given basis b of c, if and only if the columns of generating matrices are pairwise different, for every basis of c. set f = fq · e. then f is an isomorphic copy of fq contained in fqc · e. notice that, since g is a generator of c, we have that f[ge] = fqc ·e. then, the evaluation mapping ϕ : f[x] → fqc ·e given by ϕ(f) = f(ge), for all f ∈ f[x], is an epimorphism and we have that fqc ·e ∼= f[x] ker(ϕ) . notice that ker(ϕ) = 〈h〉, for some h ∈ f[x], which is a polinomial of minimal degree having ge as a root. since dimffqc ·e = dimfq fqc ·e = r we have that the degree of h is precisely r and there exist coeficients b0, . . . ,br−1 in f such that gre = br−1g r−1e + · · · + b0. this clearly implies that b0 = {e,ge, . . . ,gr−1e} is a basis of fqc ·e over f and also over fq. moreover, we have the following. theorem 4.3. for every basis b of c the generating matrix gb has pairwise different columns if and only if the mapping ψ : c → c ·e is an isomorphism. 186 g. chalom et al. / j. algebra comb. discrete appl. 4(2) (2017) 181–188 proof. notice that fqc ·e is of dimension r over fq, it contains qr elements. suppose, by way of contradiction, the ψ is not injective. then, there exists an index j, 0 < j < n, such that e = gje. write e = a0 + a1g + · · · + an−1gn−1. then gje = a0gj + a1gj+1 + · · · + an−1gj+n−1. since e = gje, we have that ai = ai+j, 0 ≤ i ≤ n − 1, where the subindexes are taken modulo n. this shows that if g denotes the generating matrix with respect to the basis b then, the first column of g is equal to its jth column. conversely, assume that ψ is an isomorphism and, by way of contradiction, that there exists a basis whose corresponding generating matrix has two equal columns, in positions i and j, say. in view of lemma 4.2 we can assume, without loss of generality, that this basis is precisely the basis b0. this means that ai+t = aj+t or, equivalently, that at = at+j−i, for all t, where indexes are taken, again, modulo n. this readily implies that e = gj−ie. recall that a binary linear code of dimension k and length n is called simplex if a generating matrix for the code contains all possible non zero columns of length k. since these are 2k − 1 in number, this matrix must be of size k × (2k − 1) so, we must have n = 2k − 1. theorem 4.4. let c be a binary linear code of dimension k and length n = 2k −1. then c is a simplex code if and only if it is essential. proof. assume that c is simplex. since all its columns are different, it follows from theorem 4.3 that the mapping ψ : c → c · e is an isomorphism. to prove thar e is an essential idempotent we are only left to prove that it is primitive. notice that c ·e ⊂ f2c ·e\{0}. since f2c ·e = c is of dimension k over f2, it contains 2k elements. hence |f2c ·e\{0}| = n = |c|, showing that actually c ·e = f2c ·e\{0}. this means that every non zero element in fc ·e is invertible and thus, it is a field. consequently, e is primitive. as a consequence, and taking the results in [3] into account, we can state the following. corollary 4.5. every binary linear code of constant weight is a repetition of a code generated by an essential idempotent. references [1] s. d. berman, semisimple cyclic and abelian codes. ii, kibernetika 3(3) (1967) 21–30. 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[15] r. e. sabin, s. j. lomonaco, metacyclic error–correcting codes, appl. algebra engrg. comm. comput. 6(3) (1995) 191–210. 188 http://dx.doi.org/10.1016/0097-3165(79)90066-9 http://dx.doi.org/10.1016/0097-3165(79)90066-9 http://dx.doi.org/10.1109/tit.2013.2275111 http://dx.doi.org/10.1109/tit.2013.2275111 http://dx.doi.org/10.1007/3-540-16767-6_48 http://dx.doi.org/10.1007/3-540-16767-6_48 http://dx.doi.org/10.1007/3-540-16767-6_48 http://dx.doi.org/10.1007/bf01268659 http://dx.doi.org/10.1007/bf01268659 http://dx.doi.org/10.1007/3-540-56686-4_50 http://dx.doi.org/10.1007/3-540-56686-4_50 http://dx.doi.org/10.1007/3-540-56686-4_50 http://dx.doi.org/10.1007/bf01195337 http://dx.doi.org/10.1007/bf01195337 introduction basic facts the equivalence binary simplex codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284947 j. algebra comb. discrete appl. 4(2) • 131–140 received: 12 june 2015 accepted: 17 february 2016 journal of algebra combinatorics discrete structures and applications codes over an infinite family of algebras research article irwansyah∗, intan muchtadi-alamsyah∗∗, ahmad muchlis, aleams barra∗∗, djoko suprijanto abstract: in this paper, we will show some properties of codes over the ring bk = fp[v1, . . . , vk]/(v2i = vi, ∀i = 1, . . . , k). these rings, form a family of commutative algebras over finite field fp. we first discuss about the form of maximal ideals and characterization of automorphisms for the ring bk. then, we define certain gray map which can be used to give a connection between codes over bk and codes over fp. using the previous connection, we give a characterization for equivalence of codes over bk and euclidean self-dual codes. furthermore, we give generators for invariant ring of euclidean self-dual codes over bk through macwilliams relation of hamming weight enumerator for such codes. 2010 msc: 11t71 keywords: gray map, equivalence of codes, euclidean self-dual, hamming weight enumerator, macwilliams relation, invariant ring 1. introduction codes over finite rings has been an interesting topic in algebraic coding theory since the discovery of codes over z4, see [4]. an example of finite rings which has interesting properties is the ring ak = f2[v1, . . . ,vk], where v2i = vi, for 1 ≤ i ≤ k, because it has two gray maps which relate codes over such ring and binary codes, see [2]. this ring also has non-trivial automorphisms which can be used to define skew-cyclic codes, for example in [1], skew-cyclic codes over the ring a1 = f2 + vf2, where v2 = v, which give some optimal euclidean and hermitian self-dual codes. furthermore, abualrub et al. show that skew-cyclic codes over a1 have a connection to left submodules over a skew-polynomial ring and ∗ the author is supported by beasiswa unggulan bpkln direktorat jenderal pendidikan tinggi. ∗∗ the authors are supported by hibah desentralisasi dikti 2016. irwansyah (corresponding author); algebra research group, institut teknologi bandung, bandung, indonesia, and department of mathematics, universitas mataram, mataram, indonesia (email: irw@unram.ac.id). intan muchtadi-alamsyah, ahmad muchlis, aleams barra; algebra research group, institut teknologi bandung, bandung, indonesia (email: ntan@math.itb.ac.id, muchlis@math.itb.ac.id, barra@math.itb.ac.id). djoko suprijanto; combinatorial research group, institut teknologi bandung, bandung, indonesia (email: djoko@math.itb.ac.id). 131 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 give skew-polynomial generators for these codes. in [6], skew-cyclic codes over the ring a1 have been characterized using a gray map. this characterization gives a way to construct skew-cyclic codes over the ring a1 from binary cyclic or quasi-cyclic codes, and also gives decoding algorithm for some codes over such ring. meanwhile, gao [3] consider skew-cyclic codes over the ring b1 = fp + vfp, where v2 = v, and found that these codes are equivalent to either cyclic codes or quasi-cyclic codes. using this connection, gao is able to give an enumeration for skew-cyclic codes which are constructed using an automorphism with order relatively prime to the length of the codes. in this paper, we consider codes over the ring bk = fp[v1, . . . ,vk], where v2i = vi for 1 ≤ i ≤ k, which is a generalization of the ring ak in [2] and b1 in [3]. we study its maximal ideals, automorphisms, equivalence codes, and euclidean self-dual codes over these rings, including the generators for its invariant ring. this paper is organized as follows: section 2 describes some properties of the ring bk such as maximal ideals and automorphisms. meanwhile, in section 3, we describe a gray map for the ring bk, and we characterize linear codes and equivalent codes over the ring bk. finally, in section 4, we characterize euclidean self-dual codes, give the shape of macwilliams relation and generators of invariant rings for euclidean self-dual codes. 2. the ring bk as we readily see, the ring bk forms a commutative algebra over prime field fp. let ω = {1, 2, . . . ,k} and 2ω is the collection of all subsets of ω. also, let wi be an element in the set {vi, 1−vi}, for 1 ≤ i ≤ k. then, we will prove the following observation. lemma 2.1. ω ∈ bk is a zero divisor if and only if ω ∈ 〈w1,w2, . . . ,wk〉. proof. (⇐=) it is clear that, vi(1−vi) = 0, for all i = 1, . . . ,k. therefore, if ω ∈ 〈w1,w2, . . . ,wk〉, then it is a zero divisor in bk. (=⇒) consider the equation, (α + βvk)(γ + �vk) = a + bvk given α + βvk,a + bvk ∈ bk, for some α,β,a,b ∈ bk−1. we have γ = aα−1 and � = (b−βa)(α(β + α))−1. therefore, if a + bvk = 1, then γ = 1 and � = −β(α(β + α))−1. which implies, α + βvk is a unit if and only if α and α + β are also units. considering this observation for elements in bk−1,bk−2, . . . ,b1, we have α + βv ∈ b1 is a unit if and only if α,α + β ∈ fp are non zero elements. since, every element in finite commutative ring is either a unit or a zero divisor, we can see that the only zero divisors in b1 are the elements in the ideals generated by βv or α(1 − v). by generalizing this result recursively, we have the intended conclusion. also, we can easily show that i = 〈w1,w2, . . . ,wk〉 is a maximal ideal in bk. lemma 2.2. let i = 〈w1,w2, . . . ,wk〉. then i is a maximal ideal in bk. proof. consider quotient ring bk/i. if vi ∈ i, then 1 − vi ≡ 1 mod i, and if 1 − vi ∈ i, then vi = 1 − (1 − vi) ≡ 1 mod i. consequently, bk/i is a field. so, i is a maximal ideal. moreover, bk/i ∼= fp. the following lemma is needed to prove proposition 2.4. lemma 2.3. αp = α, for all α ∈ bk. 132 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 proof. let α = ∑ a⊆{1,...,k}αava, for some αa ∈ fp, where va = ∏ j∈a vj. then, consider αp = p∑ i=0 ( p i ) αia1va1   ∑ a6=a1 αava  p−i = αa1va1 +   ∑ a 6=a1 αava  p since fp has characteristic p and βp−1 = 1 for all β ∈ fp. if we continue this procedure, then we have αp = α. the following result shows that the ring bk is a principal ideal ring. proposition 2.4. let i = 〈α1, . . . ,αm〉 be an ideal in bk, for some α1, . . . ,αm ∈ bk. then, i = 〈 ∑ a⊆{1,...,m},a 6=∅ (−1)|a|+1( ∏ j∈a αj) p−1〉. proof. consider αi ∑ a⊆{1,...,m},a 6=∅(−1) |a|+1( ∏ j∈a αj) p−1. for any a ⊆{1, . . . ,m}, if i ∈ a, then αi(−1)|a|+1( ∏ j∈a αj) p−1 = (−1)|a|+1αi( ∏ j∈a−{i} αj) p−1 since αpi = αi by lemma 2.3. consequently, there is a unique a ′ = a−{i}⊆{1, . . . ,m}, such that αi  (−1)|a|+1(∏ j∈a αj) p−1 + (−1)|a ′|+1( ∏ j∈a αj) p−1   = 0. otherwise, if i 6∈ a, then there is a unique a′′ = a∪{i}⊆{1, . . . ,m} such that αi  (−1)|a|+1(∏ j∈a αj) p−1 + (−1)|a ′′|+1( ∏ j∈a αj) p−1   = 0. so, every term will be vanish except αiα p−1 i = αi. therefore, i ⊆〈 ∑ a⊆{1,...,m},a6=∅ (−1)|a|+1( ∏ j∈a αj) p−1〉. it is clear that 〈 ∑ a⊆{1,...,m},a 6=∅ (−1)|a|+1( ∏ j∈a αj) p−1〉⊆ i. thus, i = 〈 ∑ a⊆{1,...,m},a6=∅(−1) |a|+1( ∏ j∈a αj) p−1〉. the following proposition shows that the ideal in lemma 2.2 is the only maximal ideal in bk. proposition 2.5. an ideal i in bk is maximal if and only if i = 〈w1,w2, . . . ,wk〉. proof. (⇐=) it is clear by lemma 2.2. (=⇒) let j be a maximal ideal in bk. by proposition 2.4, bk is a principal ideal ring. then, let j = 〈ω〉, for some ω ∈ bk. note that, ω is not a unit in bk, so it is a zero divisor. by lemma 2.1, ω is an element of some mi = 〈w1,w2, . . . ,wk〉, which means j ⊆ mi. consequently, j = mi, because j is a maximal ideal. 133 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 using the above result, we have the following lemmas. lemma 2.6. the ring bk can be viewed as an fp-vector space with dimension 2k whose basis consists of elements of the form ws = ∏ i∈s wi, where s ∈ 2 ω. proof. as we can see, every element a ∈ bk can be written as a = ∑ s∈2ω αsvs, for some αs ∈ fp, where vs = ∏ i∈s vi and v∅ = 1. so, bk is a vector space over fp whose basis consists of elements of the form vs = ∏ i∈s vi, where v∅ = 1 and there are ∑k j=0 ( k j ) = 2k elements of basis. now, we will show that the set {1,ws2, . . . ,ws2k} is also a basis. consider, α1 + α2ws2 + · · · + α2kws2k = 0 for some αi ∈ fp, for all i = 1, . . . , 2k, which gives, −α1 = α2ws2 + · · · + α2kws2k . if α1 6= 0, then ξ1 = ( α2ws2 + · · · + α2kws2k ) is a unit, a contradiction to the fact that ξ1 ∈ 〈w1, . . . ,wk〉. so, α1 = 0, which means, − ( α2ws2 + · · · + αk+1wsk+1 ) = αk+2wsk+2 + · · · + α2kws2k . if ( α2ws2 + · · · + αk+1wsk+1 ) 6= 0, then it is a contradiction to the fact that |sj| ≥ 2, for all j = k + 2, . . . , 2k. consequently, ( α2ws2 + · · · + αk+1wsk+1 ) = 0. we have to note that, the set with elements of the ws, where s ∈ 2ω, is also linearly independent over fp, because sk is a vector space over fp with element of basis are of the form vs, where s ⊆ ω. therefore, ( α2ws2 + · · · + αk+1wsk+1 ) = 0 gives α2 = · · · = αk+1 = 0. by continuing this process, we have α1 = · · · = α2k = 0, which means they are linearly independent over fp. lemma 2.7. the ring bk has characteristic p and cardinality p2 k . proof. it is immediate since characteristic of fp is p, and bk can be viewed as a fp-vector space with dimension ∑k i=0 ( k i ) = 2k. so, |bk| = p2 k . the following theorem characterizes the shape of automorphisms in the ring bk. theorem 2.8. let θ be an endomorphism in bk. then, θ is an automorphism if and only if θ(vi) = wj, for every i ∈ ω, and θ, when restricted to fp, is an identity map. proof. (=⇒) let j = 〈v1, . . . ,vk〉 and jθ = 〈θ(v1), . . . ,θ(vk)〉. consider the map λ : bk j → bk jθ a + j 7→ θ(a) + jθ we can see that the map λ is a ring homomorphism. for any a,b ∈ bk/j where λ(a) = λ(b), let a = a1 + j and b = b1 + j for some a1,b1 ∈ bk. as we can see, θ(a1 − b1) ∈ jθ, so a1 − b1 ∈ j. consequently, a − b = 0 + j, which means a = b, in other words, λ is a monomorphism. moreover, for any a′ ∈ bk/jθ, let a′ = a2 + jθ for some a2 ∈ bk, then there exists a = θ−1(a2) + j such that λ(a) = a′. therefore, fp ' bk/j ' bk/jθ, which implies jθ is also a maximal ideal. by proposition 2.5, jθ = 〈w1, . . . ,wk〉, where wi ∈{vi, 1 −vi} for 1 ≤ i ≤ k. by proposition 2.4, jθ = 〈 ∑ a⊆ω,a 6=∅ (−1)|a|+1( ∏ j∈a wj) p−1〉 = 〈 ∑ a⊆ω,a 6=∅ (−1)|a|+1( ∏ j∈a θ(vj)) p−1〉 which means, ∑ a⊆ω,a 6=∅(−1) |a|+1( ∏ j∈a wj) p−1 and ∑ a⊆ω,a6=∅(−1) |a|+1( ∏ j∈a θ(vj)) p−1 are associate. therefore, θ(vi) = βwj for some unit β which satisfies ( β|a| )p−1 = β, for all a 6= ∅. consequently, we 134 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 have βp−1 = β, but by lemma 2.3, βp = β. since β is a unit, we have that βp−1 = 1. therefore, β must be equal to 1. moreover, since θ is an automorphism, θ(vi) 6= θ(vj) whenever i 6= j. also, since the only automorphism in fp is identity map, we have the conclusion. (⇐=) suppose that θ(vi) = wj, and θ(vi) 6= θ(vj) whenever i 6= j. by lemma 2.6, we can see that θ is also an automorphism. now, we have to note that every element a in bk can be written as a = ∑ s∈2ω αsws for some αs ∈ fp, where ws = ∏ i∈s wi. define a map ϕ as follows. ϕ : bk → f2 k p a = ∑2k i=1 αsiwsi 7→ ( ∑ s⊆s1 αs, ∑ s⊆s2 αs, . . . , ∑ s⊆s 2k αs) we can show that this map ϕ is a bijection map. furthermore, this map can be extended n tuples of bk as follows. ϕ : bnk → f n2k p (a1, . . . ,an) 7→ (ϕ(a1), . . . ,ϕ(an)). since ϕ is a bijection map, we also have ϕ is a bijection map. we have to note that, the map ϕ is a permutation, based on the choice of subsets si ∈ 2ω, of gray maps in [2]. 3. codes over the ring bk a subset c ⊆ bnk is called code over bk of length n. if c is a bk-submodule of b n k , then c called linear code. the following proposition gives a characterization of bk-linear codes using the map ϕ. proposition 3.1. c is a linear code over bk if and only if there exist linear codes c1, . . . ,c2k over fp such that c = ϕ−1(c1, . . . ,c2k ). proof. (=⇒) since ϕ is a bijection, there exist c1, . . . ,c2k such that c = ϕ−1(c1, . . . ,c2k ). now, we only need to show that ci is a linear code over fp for all i = 1, . . . , 2k. for any ci, let c1 and c2 be two codewords in ci. for l = 1, 2, let cl = (α (l) 1 , . . . ,α (l) n ), for some α (l) j in fp. consider c′l = ϕ −1(0, . . . ,0,λlcl,0,0) = ( ϕ−1(0, . . . , 0,λlα (l) 1 , 0, . . . , 0), . . . ,ϕ −1(0, . . . , 0,λlα (l) n , 0, . . . , 0) ) = ( λlα (l) 1 ( wsl − ∑ j∈{1,...,k}−sl wsl∪{j} ) , . . . ,λlα (l) n ( wsl − ∑ j∈{1,...,k}−sl wsl∪{j} )) , for any λl in f×p for all l = 1, 2. the last equality holds since ϕ  α(l)t  wsl − ∑ j∈{1,...,k}−sl wsl∪{j}     = (0, . . . , 0,α(l)t , 0, . . . , 0) for all 1 ≤ t ≤ n. since c = ϕ−1(c1, . . . ,c2k ), we have c′l is in c for all l = 1, 2, and c ′ 1 + c ′ 2 is also in c. then, consider 135 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 ϕ(c′1 + c ′ 2) =   0 · · · 0 · · · 0 ... · · · ... · · · ... 0 · · · 0 · · · 0 λ1α (1) 1 + λ2α (2) 1 · · · λ1α (1) l + λ2α (2) l · · · λ1α (1) n + λ2α (2) n 0 · · · 0 · · · 0 ... · · · ... · · · ... 0 · · · 0 · · · 0   . hence, λ1c1 + λ2c2 is also in ci. (⇐=) take any two codewords c3 and c4 in c. let c3 = ( ∑ s∈2ω α (1) s ws, . . . , ∑ s∈2ω α (n) s ws ) and c4 = ( ∑ s∈2ω β (1) s ws, . . . , ∑ s∈2ω β (n) s ws ) , for some αi,βi in fp, where i = 1, . . . , 2k. for any λ3 and λ4 in f×p we have ϕ(λ3c3 + λ4c4) =   λ3α (1) s1 + λ4β (1) s1 · · · λ3α (n) s1 + λ4β (n) s1 λ3 ∑ s⊆s2 α (1) s + λ4 ∑ s⊆s2 β (1) s · · · λ3 ∑ s⊆s2 α (n) s + λ4 ∑ s⊆s2 β (n) s ... ... ... λ3 ∑ s⊆s 2k α (1) s + λ4 ∑ s⊆s 2k β (1) s · · · λ3 ∑ s⊆s 2k α (n) s + λ4 ∑ s⊆s 2k β (n) s   is also in (c1, . . . ,c2k ) , since ci is a linear code for every i = 1, . . . , 2k. therefore, λ3c3 + λ4c4 is also in c. now, following [5], we define permutation equivalence of codes as follows. definition 3.2. two codes are permutation equivalent if one can be obtained from the other by permuting the coordinates. using definition 3.2, we can define the following notion of equivalence between two codes. definition 3.3. two codes c and c′ over bk are equivalent if either they are permutation-equivalent or c is permutation equivalent to the code θ(c′) for some automorphism θ in bk, i.e. the code θ(c′) obtained from c′ by changing α with θ(α) in all coordinates. note that, the above definition is similar to the one in [5]. now, let πθ be a permutation on 2k tuples of fp induced by automorphism θ. then we have (πθ ◦ϕ) (c) = ϕ(θ(c)) (1) for any c ∈ bnk . then, we have the following characterization. theorem 3.4. let c and c′ be two codes over bk. then, c and c′ are equivalent if and only if there exists a permutation which sends (c1, . . . ,c2k ) to (c ′ 1, . . . ,c ′ 2k ) or to (πθ(c′1), . . . , πθ(c ′ 2k )). 136 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 proof. (=⇒) let c = ϕ−1(c1, . . . ,c2k ) and c′ = ϕ−1(c′1, . . . ,c′2k ), where ci and c ′ i are codes over fp, for all 1 ≤ i ≤ 2k. if there exists an automorphism θ such that c is permutation equivalent to θ(c′), then by equation 1, we have c = ϕ−1(c1, . . . ,c2k ) is permutation equivalent to ( πθ(c ′ 1), . . . , πθ(c ′ 2k ) ) . (⇐=) if there exists a permutation which sends (c1, . . . ,c2k ) to (πθ(c ′ 1), . . . , πθ(c ′ 2k )) , for some bijective map πθ, then we can have the automorphism θ using the equation 1. 4. invariant ring in this section, we describe some aspect of euclidean self-dual codes as well as macwiliams identity and invariant ring. related to euclidean self-dual codes over the ring bk, we have the following result. proposition 4.1. let c = ϕ−1(c1,c2, . . . ,c2k ), for some p-ary codes c1, . . . ,c2k. then, c is eulidean self-dual codes over bk if and only if ci is also euclidean self-dual codes, for 1 ≤ i ≤ 2k. proof. (=⇒) for any ci ∈ ci, let ci = (α (0) si , . . . ,α (n−1) si ), for some α(j)si ∈ fp, where 0 ≤ j ≤ n− 1. let c = ϕ−1(0, . . . , 0,ci, 0, . . . , 0) ∈ c, then we have 〈c,c′〉 = 0 for every c′ ∈ c. to make the representation for any element in the ring bk easier, we will use the basis whose elements are of the form vs, for all s ⊆{1, 2, . . . ,k}. now, let c′ =  β(0)si vsi + ∑ s∈2ω,s 6=si β (0) s vs, . . . ,β (n−1) si vsi + ∑ s∈2ω,s 6=si β (n−1) s vs   . consider, c = ϕ−1(0, . . . , 0,ci, 0, . . . , 0) = ( α (0) si (vsi − ∑ j∈{1,...,k}−si vsi∪{j}), . . . ,α (n−1) si (vsi − ∑ j∈{1,...,k}−si vsi∪{j}) ) . since 〈c,c′〉 = 0 for every c′ ∈ c and v2s = vs for every s ∈ 2 ω, we have n−1∑ j=0  α(j)si β(j)si vsi − ∑ j∈{1,...,k}−si α (j) si β (j) si∪{j} vsi∪{j} = 0   . consequently, ∑n−1 j=0 α (j) si β (j) si = 0. take any c′i ∈ ci. let c ′ i = (γ (0) si , . . . ,γ (n−1) si ), for some γ(j)si ∈ fp, where 0 ≤ j ≤ n − 1. since c′ = ϕ−1(0, . . . , 0,ci, 0, . . . , 0) ∈ c, we have 〈c,c′〉 = 0. so 〈ci,c′i〉 = n−1∑ j=0 α (j) si γ (j) si = 0. therefore ci ⊆ c⊥i . for any c1 ∈ c⊥i , let c1 = (ζ0, . . . ,ζn−1) for some ζj ∈ fp, where 0 ≤ j ≤ n− 1. since 〈c1,ci〉 = 0, we have ∑n−1 j=0 ζjα (j) si = 0. we can see that c′1 = ϕ −1(0, . . . , 0,c1, 0, . . . , 0) = ( ζ0(vsi − ∑ j∈{1,...,k}−si vsi∪{j}), . . . ,ζn−1(vsi − ∑ j∈{1,...,k}−si vsi∪{j}) ) . 137 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 now, since ∑n−1 j=0 ζjα (j) si = 0, we also have 〈c′1,c2〉 = 0 for every c2 ∈ c. remember that c = c⊥, which gives c′1 ∈ c. so, c1 ∈ ci, or in other words c⊥i ⊆ ci. thus, ci is a euclidean self-dual code, for all i = 1, . . . , 2k. (⇐=) take any c1,c2 ∈ c. for every i = 1, 2, let ci =   ∑ s⊆{1,...,k} c (i,0) s , . . . , ∑ s⊆{1,...,k} c (i,n−1) s   , for some c(i,j)s ∈ fp, where i = 1, 2, and j = 0, . . . ,n− 1. consider, ϕ(ci) = (∑ s⊆s1 c (i,0) s , . . . , ∑ s⊆s1 c (i,n−1) s , . . . . . . , ∑ s⊆sl c (i,0) s , . . . , ∑ s⊆sl c (i,n−1) s , . . . . . . , ∑ s⊆s 2k c (i,0) s , . . . , ∑ s⊆s 2k c (i,n−1) s ) , where i = 1, 2. since cl is a euclidean self-dual code, for all l = 1, . . . , 2k, we have 〈c1,c2〉 = ∑n−1 j=0 ∑ sl∈2ω ∑ s⊆sl c (1,j) s c (2,j) s vs = 0. so, c ⊆ c⊥. now, take any c3 ∈ c⊥. since 〈c3,c〉 = 0 for all c ∈ c, we have n−1∑ j=0 ∑ s⊆sl c (1,j) s c (2,j) s vs = 0, for all s ∈ 2ω. remember that cl is a euclidean self-dual code, for all l = 1, 2, . . . , 2k, which give n−1∑ j=0 ∑ s⊆sl c (1,j) s c (2,j) s vs = 0, for all s ∈ 2ω, and moreover c3 ∈ c. so, c⊥ ⊆ c. therefore, c is a euclidean self-dual code. the following lemma gives macwilliams identity for codes over the ring bk. lemma 4.2. the macwilliams identity for hamming weight enumerators for codes over bk is : wc⊥(x,y ) = 1 |c| wc(x + (p 2k − 1)y,x −y ) (2) proof. the identity follows from [7, theorem 8.3] and proposition 4.1. as we can see from lemma 4.2, macwilliams identity gives a transformation between polynomial representing a code and polynomial representing its corresponding dual code. we have to note that if c is an euclidean self-dual code, then the weight enumerator of c is invariant under this transformation. the above transformation can be formulated as an action ’◦’ by a matrix group g generated by matrices t =   1p2k−1 p2k−1p2k−1 1 p2 k−1 −1 p2 k−1   and d = ( −1 0 0 −1 ) . the action of any g = ( a1 a2 a3 a4 ) ∈ g to a polynomial f(x,y ) is written as g ◦f(x,y ) = f(a1x + a2y,a3x + a4y ). 138 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 note that the matrix t is derived from the identity in lemma 4.2 and the matrix d is derived from the condition that n is always even. also, it is easy to see that g = {i,d,t,−t}. formally, we have the following result. lemma 4.3. if wc(x,y ) is a hamming weight enumerator for an euclidean self-dual code c over bk, then wc(x,y ) is invariant under the action of g. let rg be a set of all polynomials in two variables which are invariant under the action ◦ of g. we can easily prove that rg is a ring, and by the above lemma we can see that every hamming weight enumerator of euclidean self-dual codes must be inside rg. this ring rg called invariant ring for euclidean self-dual codes over bk. the following theorem gives generators for rg. theorem 4.4. invariant ring of g is generated by wc0 (x,y) = x 2 + (p2 k − 1)y2 and f̃(x,y) = 1 4  2p2k−1 + 2 p2 k x2 + 4 ( p2 k − 1 ) p2 k−1 xy + 2(p2 k − 1)2 p2 k−1 y 2   . proof. consider the molien series, φ(λ) = 1|g| ∑ a∈g 1 det(i−λa) = 1 4 ( 1 (1+λ)2 + 1 (1−λ)2 + 2 (1−λ2) ) = 1 (1−λ2)2 = 1 + 2λ2 + 3λ4 + 4λ6 + 5λ8 + · · · + nλ2(n−1) + . . . we can see that, the invariant ring generated by two invariants of degree 2. consider the weight enumerator for self-dual code c0 = {cc|∀c ∈ ak} i.e. wc0 (x,y) = x 2 + (p2 k − 1)y2. this weight enumerator is of degree 2 and invariant under the action of g. so, this weight enumerator is one of the generator. we use averaging method to find the other one. let f(x) = x2, then by averaging method, we have f̃(x,y) = 1 4  2p2k−1 + 2 p2 k x2 + 4 ( p2 k − 1 ) p2 k−1 xy + 2(p2 k − 1)2 p2 k−1 y 2   f̃(x,y) are algebraically independent. references [1] t. abualrub, n. aydin, p. seneviratne, on θ−cyclic codes over f2 + vf2, australas. j. combin. 54 (2012) 115–126. [2] y. cengellenmis, a. dertli, s. t. dougherty, codes over an infinite family of rings with a gray map, des. codes cryptogr. 72(3) (2014) 559–580. [3] j. gao, skew cyclic codes over fp + vfp, j. appl. math. inform. 31(3–4) (2013) 337–342. 139 http://www.ams.org/mathscinet-getitem?mr=3013244 http://www.ams.org/mathscinet-getitem?mr=3013244 http://dx.doi.org/10.1007/s10623-012-9787-y http://dx.doi.org/10.1007/s10623-012-9787-y http://www.ams.org/mathscinet-getitem?mr=3098892 irwansyah et al. / j. algebra comb. discrete appl. 4(2) (2017) 131–140 [4] a.r. hammons, p. v. kumar, a. r. calderbank, n. j. a. sloane, p. sole, the z4–linearity of kerdock, preparata, goethals and related codes, ieee trans. inform. theory 40(2) (1994) 301–319. [5] w. huffman, v. pless, fundamentals of error correcting codes, cambridge university press, 2003. [6] irwansyah, i. muchtadi–alamsyah, a. muchlis, a. barra, d. suprijanto, construction of θ–cyclic codes over an algebra of order 4, proceeding of the third international conference on computation for science and technology (iccst–3), atlantis press, 2015. [7] j. wood, duality for modules over finite rings and applications to coding theory, amer. j. math. 121(3) (1999) 555–575. 140 http://dx.doi.org/10.1109/18.312154 http://dx.doi.org/10.1109/18.312154 http://dx.doi.org/10.2991/iccst-15.2015.27 http://dx.doi.org/10.2991/iccst-15.2015.27 http://dx.doi.org/10.2991/iccst-15.2015.27 http://dx.doi.org/10.1353/ajm.1999.0024 http://dx.doi.org/10.1353/ajm.1999.0024 introduction the ring bk codes over the ring bk invariant ring references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327391 j. algebra comb. discrete appl. 5(1) • 1–4 received: 27 december 2016 accepted: 6 april 2017 journal of algebra combinatorics discrete structures and applications ternary maximal self-orthogonal codes of lengths 21, 22 and 23 research article makoto araya, masaaki harada, yuichi suzuki * abstract: we give a classification of ternary maximal self-orthogonal codes of lengths 21, 22 and 23. this completes a classification of ternary maximal self-orthogonal codes of lengths up to 24. 2010 msc: 94b05 keywords: ternary code, self-dual code, self-orthogonal code 1. introduction a ternary [n, k] code c is a k-dimensional vector subspace of fn3 , where f3 denotes the finite field of order 3. all codes in this note are ternary. the parameters n and k are called the length and the dimension of c, respectively. the weight of a vector x ∈ fn3 is the number of non-zero components of x. a vector of c is a codeword of c. the minimum non-zero weight of all codewords in c is called the minimum weight of c. two codes c and c′ are equivalent if there is a (0, 1,−1)-monomial matrix p with c′ = c · p = {xp | x ∈ c}, and inequivalent otherwise. the automorphism group aut(c) of c is the group of all (0, 1,−1)-monomial matrices p with c = c ·p . the dual code c⊥ of a code c of length n is defined as c⊥ = {x ∈ fn3 | x · y = 0 for all y ∈ c}, where x · y is the standard inner product. a code c is self-dual if c = c⊥, and c is self-orthogonal if c ⊂ c⊥. a self-dual code of length n exists if and only if n ≡ 0 (mod 4). a self-orthogonal code c is maximal if c is the only self-orthogonal code containing c. a self-dual code is automatically maximal. the dimension of a maximal self-orthogonal code of length n is a constant depending only on n. more precisely, a maximal self-orthogonal code of length n has dimension (n−1)/2 if n is odd, n/2−1 if n ≡ 2 (mod 4) (see [8]). makoto araya (corresponding author); department of computer science, shizuoka university, hamamatsu 432–8011, japan (email: araya@inf.shizuoka.ac.jp). masaaki harada; research center for pure and applied mathematics, graduate school of information sciences, tohoku university, sendai 980–8579, japan (email: mharada@m.tohoku.ac.jp). yuichi suzuki; hitachi systems, ltd., 1–2–1, osaki, shinagawa-ku, tokyo, 141–0032, japan. * this author carried out his work at yamagata university. 1 http://orcid.org/0000-0002-9935-038x http://orcid.org/0000-0002-2748-6456 m. araya et al. / j. algebra comb. discrete appl. 5(1) (2018) 1–4 a classification of maximal self-orthogonal codes of lengths up to 12, lengths 13, 14, 15, 16 and lengths 17, 18, 19, 20 was done in [8], [2] and [9], respectively (see [4] for lengths 18 and 19). in this note, we give a classification of maximal self-orthogonal codes of lengths 21, 22 and 23. the mass formula is used to verify that our classification is complete. since a classification of self-dual codes of length 24 was done in [3], our result completes a classification of maximal self-orthogonal codes of lengths up to 24. 2. classification results let c be a code of length n and let s = {i1, i2, . . . , ij} be a subset of {1, 2, . . . , n}. a shortened code of c is the set by selecting only the codewords of c having zeros in each of the coordinate positions i1, i2, . . . , ij and deleting these components. throughout this note, we denote the code by c(s). all maximal self-orthogonal codes of lengths 4m + 1, 4m + 2, 4m + 3 can be obtained from self-dual codes of length 4m + 4 as shortened codes (see [2]). for length 24, there are 338 inequivalent self-dual codes, two of which have minimum weight 9, 166 of which have minimum weight 6 and 170 of which have minimum weight 3 [3] and [7]. from the 338 self-dual codes c of length 24, we found maximal self-orthogonal codes of lengths 23 and 22, which must be checked further for equivalences, as shortened codes c(s) by considering all sets s with |s| = 1 and 2, respectively. this computer calculation was done by using the magma [1] function shortencode. then we determined the equivalence or inequivalence of two codes among the maximal self-orthogonal codes. this calculation was done by the magma function isisomorphic. then we have 13625 and 2005 inequivalent maximal self-orthogonal codes of lengths 22 and 23, respectively. note that the dimensions of maximal self-orthogonal codes of lengths 21 and 22 are 10. the 126 codes among the 13625 maximal selforthogonal codes of length 22 have a zero coordinate. hence, 216 inequivalent maximal self-orthogonal codes of length 21 are obtained, as shortened codes. we denote by c(n, d) the set of the inequivalent maximal self-orthogonal codes of length n and minimum weight d for (n, d) = (21, 3), (21, 6), (22, 3), (22, 6), (22, 9), (23, 3), (23, 6) and (23, 9). in addition, we define subsets of c(n, d): c(n, d, d′) = {c ∈c(n, d) | d(c⊥) = d′}, where d(c) denotes the minimum weight of c. the numbers |c(n, d, d′)| are listed in table 1. as a check, in order to verify that c(n, d) contains no pair of equivalent codes for the above (n, d), we employed the following method obtained by applying the method given in [6, section 2]. let c be a code of length n. suppose that t is a positive integer such that the codewords of weight t generate c. let at denote the number of codewords of weight t in c. we expand each codeword of c into a binary vector of length 2n by mapping the elements 0, 1 and 2 of f3 to the binary vectors (0, 0), (0, 1) and (1, 0), respectively. in this way, we have an at × 2n binary matrix m(c, t) composed of the binary vectors obtained from the at codewords of weight t in c. then, from m(c, t), we have an incidence structure d(c, t) having 2n points. this calculation was done by using the magma function incidencestructure. if c and c′ are equivalent, then d(c, t) and d(c′, t) are isomorphic. by the magma function isisomorphic, we verified that the incidence structures d(c, t) are non-isomorphic for the above (n, d). this shows that c(n, d) contains no pair of equivalent codes for the above (n, d). the number of distinct maximal self-orthogonal codes of length n is known [8] as: n(n) = {∏(n−1)/2 i=1 (3 i + 1) if n is odd,∏n/2 i=2(3 i + 1) if n ≡ 2 (mod 4). we calculated the following values: t(n, d, d′) = ∑ c∈c(n,d,d′) 2n ·n! |aut(c)| . the results are listed in table 1. the automorphism groups of the codes were calculated by the magma function automorphismgroup. we remark that the automorphism group of a code c is isomorphic to the 2 m. araya et al. / j. algebra comb. discrete appl. 5(1) (2018) 1–4 table 1. |c(n, d, d′)| and t(n, d, d′). (n, d, d′) |c(n, d, d′)| t(n, d, d′) (21, 3, 1) 18 37261233666612695040000 (21, 3, 3) 129 22666803510606607679488000 (21, 6, 1) 6 156912620925725599334400 (21, 6, 4) 59 221566090068991210527129600 (21, 6, 6) 4 28572125748609278803968000 (22, 3, 1) 147 499079550803678108594176000 (22, 3, 2) 671 8999173098190687835078656000 (22, 3, 3) 3606 397450658156464202444177408000 (22, 6, 1) 69 5504766786817393746876825600 (22, 6, 2) 458 116255553756749319466332979200 (22, 6, 4) 6528 8198363298466655101459523174400 (22, 6, 5) 2142 3362889158614819168464981196800 (22, 9, 7) 4 353580056139039825199104000 (23, 3, 2) 153 23004306466349702422944153600 (23, 3, 3) 728 1838692744522339728778225254400 (23, 6, 2) 63 253139874407411695203070771200 (23, 6, 5) 1059 46245009828325897079698017484800 (23, 9, 8) 2 1414320224556159300796416000 stabilizer of {{1, 2},{3, 4}, . . . ,{2n−1, 2n}} inside of the automorphism group of the incidence structure d(c, t). in order to verify the correctness of the above calculations of the automorphism groups, we also calculated the stabilizers for d(c, t). this was done by the magma function stabilizer. finally, as a check, we verified the mass formula: n(21) =t(21, 3, 1) + t(21, 3, 3) + t(21, 6, 1) + t(21, 6, 4) + t(21, 6, 6), n(22) =t(22, 3, 1) + t(22, 3, 2) + t(22, 3, 3) + t(22, 6, 1) + t(22, 6, 2) + t(22, 6, 4) + t(22, 6, 5) + t(22, 9, 7), n(23) =t(23, 3, 2) + t(23, 3, 3) + t(23, 6, 2) + t(23, 6, 5) + t(23, 9, 8). the mass formula shows that there is no other maximal self-orthogonal code of lengths 21, 22 and 23. we summarize a classification of maximal self-orthogonal codes of lengths 21, 22 and 23. proposition 2.1. (1) up to equivalence, there are 216 maximal self-orthogonal codes of length 21, 147 of which have minimum weight 3 and 69 of which have minimum weight 6. (2) up to equivalence, there are 13625 maximal self-orthogonal codes of length 22, 4424 of which have minimum weight 3, 9197 of which have minimum weight 6 and 4 of which have minimum weight 9. (3) up to equivalence, there are 2005 maximal self-orthogonal codes of length 23, 881 of which have minimum weight 3, 1122 of which have minimum weight 6 and 2 of which have minimum weight 9. remark 2.2. generator matrices of all the maximal self-orthogonal codes of lengths 21, 22 and 23 can be obtained electronically from [5]. acknowledgment: the first and second authors would like to thank ken saito for helpful discussions on the classification method given in [6]. the first author would also like to thank the graduate school of information sciences, tohoku university for the hospitality during his visit in august and december 2016. this work is supported by jsps kakenhi grant numbers 15h03633 and 15k04976. 3 m. araya et al. / j. algebra comb. discrete appl. 5(1) (2018) 1–4 references [1] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symb. comput. 24(3–4) (1997) 235–265. 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[9] v. pless, n. j. a. sloane, h. n. ward, ternary codes of minimum weight 6 and the classification of the self–dual codes of length 20, ieee trans. inform. theory 26(3) (1980) 305–316. 4 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1109/tit.1979.1056047 https://doi.org/10.1109/tit.1979.1056047 https://doi.org/10.1016/j.jcta.2008.11.011 https://doi.org/10.1016/j.jcta.2008.11.011 https://doi.org/10.1002/jcd.20295 https://doi.org/10.1002/jcd.20295 http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/mso3.htm http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/mso3.htm http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/mso3.htm http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/mso3.htm https://doi.org/10.1109/tit.1981.1056328 https://doi.org/10.1109/tit.1981.1056328 http://dx.doi.org/10.1137/0131058 http://dx.doi.org/10.1137/0131058 https://doi.org/10.1109/tit.1980.1056195 https://doi.org/10.1109/tit.1980.1056195 introduction classification results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284966 j. algebra comb. discrete appl. 4(2) • 197–206 received: 12 june 2015 accepted: 29 february 2016 journal of algebra combinatorics discrete structures and applications correspondence between steganographic protocols and error correcting codes research article m’hammed boulagouaz, mohamed bouye abstract: in this work we present a correspondence between the steganographic systems and error correcting codes. we propose a new steganographic protocol based on 3-error-correcting primitive bch codes. we show that this new protocol has much better parameters than protocols which we get from hamming codes or from the 2-error-correcting primitive bch codes, for high levels of incorporation. 2010 msc: 94a60, 94b29, 94b40 keywords: steganographic protocol, hamming distance, linear correcting code, covering radius, parity matrix, bch code 1. introduction the etymology of steganography is composed of two greek words: "stego" which means secret and "graphia" writing. steganography is then the art of secret writing. more generally, we call steganography, the art of hide information in a carrier, so that is hidden information from to furtive eyes of anyone other than the sender and the recipient. the field of information security grows and grows increasingly in recent spent decades. the development of systems for the protection of information has taken great interest as a subject of scientific research. cryptographic protocols is the best acknowledged of such systems. however, new ideas emerged with force in recent years. steganography is an old area that thanks to recent progress begins to play an important role as an alternative and more generally as a complement of cryptography. the advantage of steganography over cryptography is that steganography protects information like cryptography and it protects also communicating sources. error-correcting codes are important tools for the design of algorithms for steganography. they are used to hide information in an image and also to extract hidden information from the modified image. m’hammed boulagouaz (corresponding author), mohamed bouye; king khalid university, faculty of sciences, b.p. 9004, abha, saudi arabia (email: boulag@rocketmail.com, medeni.doc@gmail.com). 197 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 this technique is a method of decoding by syndrome, using the theory of error correcting codes, especially linear codes (see [2, 3]). steganography has taken the concepts associated to error correcting codes and to decoding by syndrome but the use has been reversed. this work is organized as follows. sections 2 and 3 present the correspondence between the errorcorrecting codes and steganographic protocols. in section 4, we recall some properties of linear steganographic protocols. we build our approach in section 5. 2. steganographic protocol definition 2.1. let n,k and ρ are three positive integers such that k ≤ n. a steganographic protocol s over an alphabet a to hide messages of length k (secret words) in words of length n (cover words) by modifying at most ρ coordinates(covering radius) is a pair of maps s = (e,r) satisfying: • e : an ×ak → an, • r : an → ak, • ∀(x,s) ∈ an ×ak : r(e(x,s)) = s, • ∀(x,s) ∈ an ×ak : dh(x,e(x,s)) ≤ ρ, n,k and ρ are the parameters of the steganographic protocol s. maps e and r are respectively called embedding and extraction maps of the steganographic protocol s. such a protocol is called a (n,k,ρ)-steganographic protocol s. example 2.2. let s and x be the secret word to hide and the cover word respectively. we may suppose that those two words are a sequences of symbols of a finite alphabet a. let be s = (s1,s2, · · · ,sk) and x = (x1,x2, · · · ,xn), so s ∈ ak and x ∈ an. let consider the two following maps : e : an ×ak → an and r : an → ak defined by e((x1,x2, · · · ,xn), (s1,s2, · · · ,sk)) = (x1,x2,x3, · · · ,xn−k,s1,s2, · · · ,sk) and r(x1,x2,x3, · · · ,xn) = (xn−k,x(n−k)+1, · · · ,xn). then (e,r) is a (n,k,k)-steganographic protocol over a. definition 2.3. we call radius of a protocol (e,r) the number ρ := max{d(x,e(x,s)),s ∈ ak,x ∈ an}. remark 2.4. a good protocol (n,k,ρ)-steganographic (e,r) must satisfies two main requirements: 1. the two maps e and r are effective, 2. (n,k,ρ) are good parameters such that: 198 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 • k n is the greatest possible, • ρ n is the smallest possible. definition 2.5. a good protocol steganographic (e,r) of length n is said to be proper if : the embedding map e is such that: e(x,s) is the nearest element to x belonging to r−1(s) = {y ∈ an/r(y) = s} (with respect to the hamming distance over an, where a denote the alphabet of the protocol). proposition 2.6. if a steganographic protocol (e,r) is proper then the covering radius ρ is given by: ρ = max{d(x,r−1(s))/s ∈ ak and x ∈ an}. proof. since the protocol is proper then : ∀x ∈ an and ∀s ∈ ak if e(x,s) = v then dh(x,v) = min{d(x,y)/y ∈ r−1(s)} = d(x,r−1(s)). since that ρ := max{d(x,e(x,s))/s ∈ ak,x ∈ an} we have ρ = max{d(x,r−1(s)),s ∈ ak and x ∈ an}. 2.1. the steganographic protocol f5 the protocol f5 over the galois field f2 permits to hide messages of length k (secret words) in words (cover words) of length n = 2k − 1 by changing more than one of them (i.e. protocol of type (2k − 1,k, 1) ). let < m >2 be the binary expression of m with k bits ( so we can consider that < m >2 is in fk2 ). conversely, for z ∈ fk2 let < z >10 be the integer which has z as binary expression, then 1 ≤ < z >10 ≤ 2k − 1. finally, let ei be the ith vector of the canonical basis of f2 k−1 2 ; e0 = 0. let consider i) the map γ : f2 k−1 2 ×f k 2 → n (x,s) → γ(x,s) =< s + ∑2k−1 i=1 xi < i >2>10. ii) maps e and r defined by : e : f2 k−1 2 ×f k 2 → f 2k−1 2 , (x,s) → e(x,s) = x + eγ(x,s). r : f2 k−1 2 → f k 2, x → r(x) = ∑2k−1 i=1 xi < i >2. let show that (e,r) is a steganographic protocol, or let show that r(e(x,s)) = s, for any s ∈ fk2 and for any x ∈ f2 k−1 2 . indeed, 1. r(e(x,s)) = r(x + eγ(x,s)), we put j = γ(x,s) =< s + ∑2k−1 i=1 xi < i >2>10 then < j >2= s + 2k−1∑ i=1 xi < i >2: (∗) 2. r(x+ej) = r(x1,x2, · · · ,xj+1, · · · ,xn) = ∑2k−1 i=1,i6=j xi < i >2 +xj+1 < j >2, changing < j >2 by its expression given in (∗) we obtain : r(x+ej) = s so r(e(x,s)) = s. therefore f5 is a steganographic protocol. remark 2.7. • insert a message s by f5 in a covering x consists to change the coordinate number γ(x,s). • extraction consists to add all products of each component to the value of the binary expression of the index. i.e : r(x) = ∑2k−1 i=1 xi < i >2 199 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 2.2. example of application for f5 for n = 7, k = 3, how to insert s = 011 in x = 1100101. i.e. how to calculate e(011, 1100101). γ(1100101, 011) =< 011 + ∑7 i=1 xi < i >2>10 =< 011 + (1.(001) + 1.(010) + 1.(101) + 1.(111)) >10=< 010 >10= 2. since s = 011 and x = 1100101 then to insert s in x consists in changing the position number 2 of x from 1 to 0, i.e. e(x,s) = e(1100101, 011) = 1000101 = v. how to extract the message hidden s in the message v = 1000101? i.e. how to calculate r(1000101). by applying the second point of the previous remark we get that r(v) = r(1000101) = (1.(001) + 1.(101) + 111) = 011 = s. 3. codes defined by a steganographic protocol recall that a correcting code of length n on alphabet a is a subset of an, and the covering radius ρ of a correcting code satisfies ∀x = (x1, ...,xn) ∈ an : dh(x,c) = minc∈cdh(x,c) = minc=(c1,...,cn)∈c|{i : xi 6= ci}|≤ ρ. we denote (n, | c |)ρ the parameters of a such code. when c is linear over fq of cardinality | c |= qk with k is the dimension of the code, we use the notation [n,k]ρ to say that the parameters of the linear code c are (n,qk)ρ. let υ = (e,r) be a steganographic protocol. the protocol υ define a collection fυ of correcting codes defined by: fυ = {cs := r−1(s),s ∈ ak} to decode a word x ∈ an according to the code cs = r−1(s) of the collection fυ, we proceed in this way: if ρ is the radius of υ then there exists a word x′ satisfying: d(x,x′) ≤ ρ and r(x′) = s. then r(e(x,s)) = s which means that e(x,s) is a word decoding x relative to the code cs. 3.1. construction of steganographic protocols to build a steganographic protocol of parameters (n,k,ρ) on an alphabet a, one way is to start by building a surjective map r : an → ak, which map r define a family fr := {cs = r−1(s) | s ∈ ak} of codes on a of length n. then the embedding map e such that (e,r) is a steganographic protocol of parameters (n,k,ρ) is defined by: for s ∈ ak denote e(s,−) the decoding map, defined by the nearest word, associated to the code cs = r −1(s). then e(x,s) = x′ ∈ cs, with dh(x,x′) = dh(x,cs) := min{dh(x,y),y ∈ cs}. example 3.1. to build a steganographic protocol of parameters (3, 2, 2) on f2, start with given a surjective function r : f32 → f22. if r : f32 → f22 is such that: r−1(00) = {000, 100} r−1(01) = {010, 001} r−1(10) = {110, 100} r−1(11) = {101, 111}, 200 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 then c00 = {000, 100}; c01 = {010, 001} ; c10 = {110, 100} ; c11 = {101, 111}. so, c00 = r−1(00) ; c01 = r−1(01) ; c10 = r−1(10) ; c11 = r−1(11). therefore e(000,−) : f32 → f32 satisfies e(000, 00) = 000, since d(000,x′) = d(000,c00) = 0. more generally: e : f32 ×f22 → f32 (x,s) → e(x,s) = d(x,cs), if s = 10 and x = 101 then we have d(101,c10) = d(101,{110, 100}) = 2 and x′ can be 110 or 011, since d(110, 101) = 2 and (011, 101) = 2. proposition 3.2. the best steganographic protocol of parameters (n,k,ρ) and of extraction map r on an alphabet a, is one that has as embedding map the map e defined by: (∀x ∈ an)(∀s ∈ ak)[e(x,s) = x′ with dh(x,x′) = dh(x,r−1(s))]. lemma 3.3. a map r : an → ak is an extraction map of a [n,k]-steganographic protocol if and only if r is surjective. proof. indeed if r : an → ak is an extraction function of a [n,k]-steganographic protocol then for all x ∈ an we have r ◦ e(x,.) = iak so r is surjective. if r : an → ak is surjective then from the subsection, construction of steganographic protocols, there exists an embedding map e such that (e,r) is a steganographic protocol of parameters (n,k,ρ). for all t ∈ ir and for all x0 ∈ an put b(x0, t) = {y ∈ /dh(x0,y) ≤ t}. lemma 3.4. for all (n,k,ρ)-steganographic protocol (e,r) over an alphabet a and for all x0 ∈ an the map r/b(x0,ρ), the restriction of r to the ball b(x0,ρ), is surjective. or, r/b(x0,ρ) : b(x0,ρ) → ak y → r(y), is a surjective map. proof. the map r/b(x0,ρ) is well defined. let consider s ∈ ak, since e is the embedding map of the (n,k,ρ)-steganographic protocol (e,r), then d(x0,e(s,x0)) ≤ ρ. let y = e(x0,s) then d(x0,y) ≤ ρ and r(y) = r(e(x0,s)) = s. that proves the existence of y ∈ b(x0,ρ) such that r/b(x0,ρ)(y) = s. it is know that the cardinal of a ball b(x0,ρ) of an is independent from it’s center x0 and it is often denoted by vq(n,ρ) if q is the cardinal of the alphabet a. corollary 3.5. for all (n,k,ρ)-steganographic protocol over an alphabet a with q elements we have : qk ≤ vq(n,ρ) proof. the map r/b(x0,ρ) is surjective and then we have card(fkq ) ≤ bn(n,ρ) = vq(n,ρ). therefore qk ≤ vq(n,ρ). 201 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 4. linear steganographic protocol definition 4.1. a steganographic protocol υ = (e,r) is called linear if the extraction map r is linear. consequence: c0 := ker(r) = r −1(0fkq ) is a subspace vector of the fq-space vector f n q . so it is a linear code of length n and its dimension is n−k. c0 is called the principal code associated to the steganographic protocol υ. so fυ = {cs | s ∈ fkq} = f n q /c0 := {x + c0 | x ∈ f n q}. definition 4.2. we call an extraction matrix of a steganographic protocol a control matrix of its principal code. or, an extraction matrix of a steganographic protocol is a matrix associated to the extraction linear map e of this protocol. proposition 4.3. for each linear steganographic protocol υ of parameters (n,k,ρ) correspond a linear [n,n−k]-code c(= c0) which has as a control matrix the control matrix of the extraction map of υ. proposition 4.4. for each linear code c of, length n, dimension n−k and a control matrix h, correspond an appropriate linear steganographic protocol which has, the mapping r associated to h as an extraction map and (n,k,ρ) as parameters, where ρ denote the covering radius of this code c. proof. for each a ∈ fnq the map (translation): τa : fnq → fnq x → τa(x) = x + a is a bijection which conserve the hamming distance (isometry). indeed it is clear that τa is bijective and that d(x,y) = d(τa(x),τa(y)) = d(x + a,y + a) since the code is linear d(x,y) = wt(x−y). from where d(τa(x),τa(y)) = wt(τa(x)−τa(y)) = wt(x−y) = d(x,y). more r is onto (because its associated matrix is a control matrix of a linear code) therefore lemma 3.3 implies that there is a [n,k]-linear steganographic protocol υ which has an extraction map r such that cυ : c0 = c. it is also true that for all s ∈ fkq there is ys ∈ fnq such that cs = ys + c therefore r(ys) = s. ρ := max{d(x,c) | x ∈ fnq} = max{d(x,c0) | x ∈ fnq} = max{d(x + ys,c0 + ys) | x ∈ fnq} = max{d(τys (x),cs) | x ∈ fnq} = max{d(x′,cs) | x′ ∈ fnq} = ρs . so ρ(υ) = maxs∈fkq ρs = maxρ = ρ. 5. construction of linear protocols to construct a (n,k) linear steganographic protocol υ, just consider a matrix h of type (k × n) and rank k. this matrix h is the extraction matrix of a protocol υ and at the same time the extraction matrix of the map of this protocol υ. it is also true that h is the control matrix of the code cυ associated to the protocol υ. 5.1. decoding a linear code by syndrome let c be a [n,n − k]linear code of control matrix h ∈ mk×n(fq) for each x ∈ fnq , we define the syndrome of x by the quantity s(x) = x×ht. the decoding method by syndrome associated to the code c consists to : if y is a received word and if iy is the word of smallest weight in y + c, then y is decoded by x = y − iy. 202 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 5.2. embedding algorithm associated to a linear steganographic protocol let r be the extraction map of a linear steganographic protocol υ and let cυ = r−1(0), be "the principal linear code associated to υ". to calculate e(x,s),(if x is a covering word and s is a message to hide in x) if e denote the embedding map associated to the protocol υ, one way is : needed : a coset decoding algorithm: input a syndrome u, output: a coset leader lu. input : a cover x of size n and a message s of size k. output : x′ = e(x,s), a steganographic cover of s with distortion d(x,x′) as small as possible. • compute u := r(x) −s, • set c := x− lu, • e(x,s) := c. then e is the embedding map of the system υ. indeed : r(e(x,s)) = r(c) = r(x− lu) = r(x) −r(lu) = u + s−u = s remark 5.1. the linear codes which has a fast and effective decoding algorithm, can be used to construct perform embedding algorithm. so these correcting codes are good tools to construct steganographic protocol. the terminals of the parameters of these codes can be translated into terminal parameters of their protocols steganographic associates. all control matrices associated with a linear code give steganographic protocols of same parameters (n,k,ρ). 5.3. the average symbols modified by a protocol upon embedding a message by steganographic protocol υ, the number of bits modified in a cover word is bounded but not determined, by the covering radius ρ. definition 5.2. let [υ = (e,r)] be a (n,k)-protocol on an alphabet a of cardinal q with all secret words s and cover words x have the same probability. we call average symbols modified by υ the umber α(υ) given by the formula : α(υ) = 1 qk+n ∑ s∈ak ∑ x∈an d(x,e(x,s)) lemma 5.3. let υ = (e,r) be an appropriate (n,k)-steganographic protocol on fq associate to the linear code c then : α(υ) = 1 qn ∑ x∈fnq d(x,c) proof. since e(x,s) = y then d(x,y) = inf{d(x,c) | c ∈ cs}. let ls be a word of smallest weight in cs then y = x + ls et e(x,s) = x + ls, so d(x,e(x,s)) = d(x,x + ls), and since d(x,x + ls) = d(x,ls + c) then d(x,e(x,s)) = d(x− ls,c), from where ∑ x∈fnq d(x,ls + c) = ∑ x∈fnq d(x− ls,c) = ∑ z∈fnq d(z,c). therefore α(υ) = 1 qn+k ∑ s∈fkq ∑ z∈fnq d(z,c) = qk qn+k ∑ z∈fnq d(z,c) = 1 qn ∑ z∈fnq d(z,c) 203 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 lemma 5.4. let υ = (e,r) be a (n,k)-steganographic protocol on fq associated to a linear code c. for i = 0, 1, · · · ,n denote αi the number of representatives of the i weight classes compared to the code c then α(υ) = 1 qk ∑n i=1 i.αi proof. we have d(x,e(x,s)) = d(x,x + ls) = w(ls) so,∑ s∈fkq ∑ x∈fnq d(x,e(x,s)) = ∑ s∈fkq ∑ x∈fnq w(ls) = q n ∑ s∈fkq w(ls), or, 1 qn+k ∑ s∈fkq ∑ z∈fnq d(x,e(x,s)) = 1 qk ∑ s∈fkq w(ls) = 1 qk (w(ls1 ),w(ls2 ), · · · ,w(lsqk )) 6. comparative tables in this section we present a new family of linear steganographic systems from the family of binary codes, 3-error-correcting primitive bch codes and we give a comparative study in tables, between the new family, the family of systems presented by c. manuera in [7] and one studied by a. westfeld in [8]. a. westfeldt in [8], studied the family of linear steganographic systems f5, associated to the family of hamming binary linear error correcting codes. to a code c of this family of codes, the dimension is 2m − m − 1 and a such code is of parameters [2m − 1, 2m − m − 1, 1]. the parameters of a linear steganographic system of this family are of the form (2m − 1, 2m, 3). from where the following table: table 1. the parameters of f5 [8], ρ = 1, t = 1. code m n k ′ = n − k ρ k ′ n ρ n (n,k ′ ,ρ) hamming 4 15 4 1 0.266 0.066 (15,4,1) hamming 5 31 5 1 0.161 0.032 (31,5,1) hamming 6 63 6 1 0.095 0.015 (63,6,1) hamming 7 127 7 1 0.055 0.007 (127,7,1) hamming 8 255 8 1 0.031 0.003 (255,8,1) c. munuera in [7], presents the family of binary linear error correcting primitive bch codes, consisting of the quasi perfect codes which are 2-correcting and of length n = 2m − 1. because these codes are quasi-perfect then their radius of coverage is 3. for a code c of this family of codes, its dimension is 2m − 2m− 1 and a such code is of parameters [2m − 1, 2m − 2m− 1, 5]. thus each code c of this family of codes, is associated with a linear steganographic protocol of parameters (2m − 1, 2m, 3). from where the following table: table 2. bch codes [7], ρ = 3, t = 2. code m n k ′ = n − k ρ k ′ n ρ n (n,kk ′ ,ρ) bch 4 15 8 3 0.533 0.2 (15,8,3) bch 5 31 10 3 0.322 0.096 (31,10,3) bch 6 63 12 3 0.190 0.047 (63,12,3) bch 7 127 14 3 0.11 0.023 (127,14,3) bch 8 255 16 3 0.062 0.011 (255,16,3) the family of binary linear error correcting codes we offer to study is that of the 3error-correcting primitive bch codes. each code of this family is of length n = 2m − 1 for a non-zero integer m, the covering radius of a such code is 5 ( [1], [4] and [5]). also for a such code c the number of the elements of fn2 /c is equal to: 204 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 23m, if m ≥ 5 and then c has [2m − 1, 2m − 1 − 3m, 3] as parameters. 2 5m 2 if m = 4 and then has [15, 5, 3] as parameters, ( mac williams and sloane [[6], p.262]. thus for each code c of this family is associated to a linear steganographic protocol of parameters: (2m − 1, 3m, 5), if m ≥ 5. (15, 10, 3), if m = 4. from where the following table: table 3. bch primitive codes, ρ = 5, t = 3. code m n k ′ = n − k ρ k ′ n ρ n (n,k ′ ,ρ) bch 4 15 10 5 0.6666 0.333 (15,10,5) bch 5 31 16 5 0.516 0.161 (31,16,5) bch 6 63 45 5 0.714 0.079 (63,45,5) bch 7 127 106 5 0.834 0.039 (127,106,5) bch 8 255 231 5 0.905 0.119 (255,231,5) 7. conclusion in this work, we presented a correspondence between the steganographic systems and error-correcting codes, and we have explained that this correspondence is one to one between linear steganographic systems and linear error correcting codes. we gave some relationships between the parameters and concepts associated to linear steganographic systems, and those of the linear error correcting codes to which they correspond (control matrix covering radius, ...). as the error correcting codes are quite well known, we showed that this correspondence can be used to build good steganographic protocols and to study their properties. we also presented a new steganography based on primitive 3-error correcting bch codes. our new protocol, shows that we are able to improve some previous results and suggest new sets of parameters for binary linear steganographic systems. in particulary for steganographic systems of high incorporation rate. references [1] e. assmus, h. mattson, some 3–error–correcting bch codes have covering radius 5, ieee trans. inform. theory 22(3) (1976) 348–349. [2] r. crandall, some notes on steganography, available at http://dde.binghamton.edu/download/ crandall_matrix.pdf, 1998. [3] j. fridrich, d. soukal, matrix embedding for large payloads, ieee trans. inf. forensics security 1(3) (2006) 390–395. [4] t. helleseth, all binary 3–error–correcting bch codes of length 2m−1 have covering radius 5, ieee trans. inform. theory 24(2) (1978) 257–258. [5] j. van der horst, t. berger, complete decoding of triple–error–correcting binary bch codes, ieee trans. inform. theory 22(2) (1976) 138–147. [6] f. j. mac williams, n. sloane, the theory of error correcting codes, amsterdam, netherlands, north–holland, 1966. 205 http://dx.doi.org/10.1109/tit.1976.1055556 http://dx.doi.org/10.1109/tit.1976.1055556 http://dde.binghamton.edu/download/crandall_matrix.pdf http://dde.binghamton.edu/download/crandall_matrix.pdf http://dx.doi.org/10.1109/tifs.2006.879281 http://dx.doi.org/10.1109/tifs.2006.879281 http://dx.doi.org/10.1109/tit.1978.1055847 http://dx.doi.org/10.1109/tit.1978.1055847 http://dx.doi.org/10.1109/tit.1976.1055530 http://dx.doi.org/10.1109/tit.1976.1055530 m. boulagouaz, m. bouye / j. algebra comb. discrete appl. 4(2) (2017) 197–206 [7] c. munuera, steganography and error–correcting codes, signal process. 87(6) (2007) 1528–1533. [8] a. westfeld, f5—a steganographic algorithm, lecture notes in comput. sci. 2137 (2001) 289–302. 206 http://dx.doi.org/10.1016/j.sigpro.2006.12.008 http://dx.doi.org/10.1007/3-540-45496-9_21 introduction steganographic protocol codes defined by a steganographic protocol linear steganographic protocol construction of linear protocols comparative tables conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284954 j. algebra comb. discrete appl. 4(2) • 155–164 received: 12 june 2015 accepted: 20 february 2016 journal of algebra combinatorics discrete structures and applications strongly nil ∗-clean rings research article abdullah harmancı, huanyin chen, a. çiğdem özcan abstract: a ∗-ring r is called strongly nil ∗-clean if every element of r is the sum of a projection and a nilpotent element that commute with each other. in this paper we investigate some properties of strongly nil ∗rings and prove that r is a strongly nil ∗-clean ring if and only if every idempotent in r is a projection, r is periodic, and r/j(r) is boolean. we also prove that a ∗-ring r is commutative, strongly nil ∗-clean and every primary ideal is maximal if and only if every element of r is a projection. 2010 msc: 16w10, 16u99 keywords: rings with involution, strongly nil ∗-clean ring, ∗-boolean ring, boolean ring 1. introduction let r be an associative ring with unity. a ring r is called strongly nil clean if every element of r is the sum of an idempotent and a nilpotent that commute. these rings were first considered by hirano-tominaga-yakub [9] and refered to as [e-n]-representable rings. in [7], diesl introduces this class and studies their properties. the class of strongly nil clean rings lies between the class of boolean rings and strongly π-regular rings (i.e. for every a ∈ r, an ∈ ran+1 ∩an+1r for some positive integer n) [7, corollary 3.7]. an involution of a ring r is an operation ∗ : r → r such that (x + y)∗ = x∗ + y∗, (xy)∗ = y∗x∗ and (x∗)∗ = x for all x,y ∈ r. a ring r with an involution ∗ is called a ∗-ring. an element p in a ∗-ring r is called a projection if p2 = p = p∗ (see [2]). recently the concept of strongly clean rings were considered for any ∗-ring. vaš [12] calls a ∗-ring r strongly ∗-clean if each of its elements is the sum of a projection and a unit that commute with each other (see also [10]). in this paper, we adapt strongly nil cleanness to ∗-rings. we call a ∗-ring r strongly nil ∗-clean if every element of r is the sum of a projection and a nilpotent element that commute. the paper consists abdullah harmancı (corresponding author), a. çiğdem özcan; hacettepe university, department of mathematics, 06800 beytepe, ankara, turkey (email: harmanci@hacettepe.edu.tr, ozcan@hacettepe.edu.tr). huanyin chen; hangzhou normal university, department of mathematics, 310036, hangzhou, china (email: huanyinchen@aliyun.com). 155 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 of three parts. in section 2, we characterize the class of strongly nil ∗-clean rings in several different ways. for example, we show that a ring r is a strongly nil ∗-clean ring if and only if every idempotent in r is a projection, r is periodic, and r/j(r) is boolean. also, if r is a commutative ∗-ring and r[i] = {a + bi | a,b ∈ r, i2 = −1 }, then with the involution ∗ defined by (a + bi)∗ = a∗ + b∗i, the ring r[i] is strongly nil ∗-clean if and only if r is strongly nil ∗-clean. foster [8] introduced the concept of boolean-like rings as a generalization of boolean rings. in section 3, we adapt the concept of boolean-like rings to rings with involution and prove that a ∗-ring r is ∗-boolean-like if and only if r is strongly nil ∗-clean and αβ = 0 for all nilpotent elements α, β in r. in the last section, we investigate submaximal ideals (see [11]) of strongly nil ∗-clean rings. we also define ∗-boolean rings as ∗-rings over which every element is a projection and characterize them in terms of strongly nil ∗-cleanness. as a corollary, we get that r is a boolean ring if and only if r is commutative, strongly nil clean and every primary ideal of r is maximal. other characterizations of boolean rings by means of (strongly) nil clean rings can be found in [7]. throughout this paper all rings are associative with unity (unless otherwise noted). we write j(r), n(r) and u(r) for the jacobson radical of a ring r, the set of all nilpotent elements in r and the set of all units in r, respectively. the ring of all polynomials in one variable over r is denoted by r[x]. 2. characterization theorems the main purpose of this section is to provide several characterizations of strongly nil ∗-clean rings. first recall some definitions. a ring r is called uniquely nil clean if, for any x ∈ r, there exists a unique idempotent e ∈ r such that x − e ∈ n(r) [7]. if, in addition, x and e commute, r is called uniquely strongly nil clean [9]. strongly nil cleanness and uniquely strongly nil cleanness are equivalent by [9, theorem 3]. analogously, for a ∗-ring, we define uniquely strongly nil ∗-clean rings by replacing “idempotent" with “projection" in the definition of uniquely strongly nil clean rings. we will use the following lemma frequently. lemma 2.1. [10, lemma 2.1] let r be a ∗-ring. if every idempotent in r is a projection, then r is abelian, i.e. every idempotent in r is central. proposition 2.2. let r be a ∗-ring. then the following are equivalent. (i) r is strongly nil ∗-clean; (ii) r is strongly nil clean and every idempotent in r is a projection; (iii) r is uniquely strongly nil ∗-clean. proof. (i) ⇒ (ii) assume that r is strongly nil ∗-clean. then r is strongly ∗-clean as can be seen in the proof of [7, proposition 3.4], i.e. if x ∈ r, there exist a projection e and a nilpotent w in r such that x− 1 = e + w and ew = we. this gives that x = e + (1 + w) where e is a projection, 1 + w is invertible and e(1 + w) = (1 + w)e. now, by [10, theorem 2.2], every idempotent in r is a projection and central. hence r is uniquely nil clean by [9, theorem 3]. (ii) ⇒ (iii) if r is uniquely nil clean, then r is uniquely strongly nil clean by lemma 2.1. hence r is uniquely strongly nil ∗-clean. (iii) ⇒ (i) clear. we note that the condition “every idempotent in r is a projection" in proposition 2.2 is necessary as the following example shows. 156 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 example 2.3. let r = {( 0 0 0 0 ) , ( 1 0 0 1 ) , ( 1 1 0 0 ) , ( 0 1 0 1 )} where 0, 1 ∈ z2. define ∗ : r → r,( a b c d ) 7→ ( a + b b a + b + c + d b + d ) . then r is a commutative ∗-ring with the usual matrix addition and multiplication. in fact, r is boolean. thus, for any x ∈ r, there exists a unique idempotent e ∈ r such that x− e ∈ r is nilpotent. but it is not strongly nil ∗-clean because the only projections are the trivial projections and there does not exist a projection e in r such that ( 1 1 0 0 ) −e is nilpotent. in [9, theorem 3], it is proved that r is strongly nil clean if and only if n(r) is an ideal and r/n(r) is boolean. also, r is uniquely nil clean if and only if r is abelian, n(r) is an ideal and r/n(r) is boolean [9, theorem 4]. so if we adapt these results to rings with involution, immediately we have the following proposition by using proposition 2.2. proposition 2.4. let r be a ∗-ring. then r is strongly nil ∗-clean if and only if (1) every idempotent in r is a projection; (2) n(r) forms an ideal; (3) r/n(r) is boolean. a ring r is called strongly j-∗-clean if for any x ∈ r there exists a projection e ∈ r such that x − e ∈ j(r) and ex = xe [6], equivalently, for any x ∈ r there exists a unique projection e ∈ r such that x− e ∈ j(r) [6, theorem 3.2]. we call r uniquely nil ∗-clean ring if for any a ∈ r, there exists a unique projection e ∈ r such that a−e ∈ n(r). proposition 2.5. let r be a ∗-ring. then the following are equivalent. (i) r is strongly nil ∗-clean; (ii) r is strongly j-∗-clean and j(r) is nil; (iii) r is uniquely nil ∗-clean and j(r) is nil. proof. (i) ⇒ (ii) suppose that r is strongly nil ∗-clean. in view of proposition 2.4, n(r) forms an ideal of r, and this gives that n(r) ⊆ j(r) (see also [7, proposition 3.18]). by [7, proposition 3.16], j(r) is nil, and so n(r) = j(r). hence r is strongly j-∗-clean. (ii) ⇒ (i) is obvious. (i) and (ii) ⇒ (iii) since r is strongly j-∗-clean, there exists a unique projection e ∈ r such that x−e ∈ j(r) by [6, theorem 3.2]. since j(r) = n(r), r is uniquely nil ∗-clean. (iii) ⇒ (ii) since j(r) ⊆ n(r), r is strongly j-∗-clean rings by [6, theorem 3.2]. from proposition 2.5 and [6, proposition 2.1], it follows that {strongly nil ∗-clean}⊂{strongly j-∗-clean}⊂{strongly ∗-clean}. the first inclusion is strict because, for example, the power series ring z2[[x]] with the identity involution is strongly j-∗-clean but not strongly nil ∗-clean by [4, example 2.5(5)]. the second inclusion is also strict by [6, example 2.2(2)]. we should note that a strongly nil clean ring may not be strongly j-clean (see [4, example on p. 3799]). hence strongly nil clean and strongly nil ∗-clean classes have different behavior when compared to classes of strongly j-clean and strongly j-∗-clean classes respectively. 157 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 lemma 2.6. let r be a ∗-ring. then r is strongly nil ∗-clean if and only if (1) every idempotent in r is a projection; (2) j(r) is nil; (3) r/j(r) is boolean. proof. assume that (1), (2) and (3) hold. for any x ∈ r, x + j(r) = x2 + j(r). as j(r) is nil, every idempotent in r lifts modulo j(r). thus, we can find an idempotent e ∈ r such that x − e ∈ j(r) ⊆ n(r). by lemma 2.1, xe = ex, and so the result follows. the converse is by propositions 2.4 and 2.5. recall that a ring r is periodic if for any x ∈ r, there exist distinct m,n ∈ n such that xm = xn. with this information we can now prove the following. theorem 2.7. let r be a ∗-ring. then r is strongly nil ∗-clean if and only if (1) every idempotent in r is a projection; (2) r is periodic; (3) r/j(r) is boolean. proof. suppose that r is strongly nil ∗-clean. by virtue of lemma 2.6, every idempotent in r is a projection and r/j(r) is boolean. for any x ∈ r, x − x2 ∈ n(r). write (x − x2)m = 0, and so xm = xm+1f(x), where f(x) ∈ z[x]. according to herstein’s theorem (cf. [3, proposition 2]), r is periodic. conversely, j(r) is nil as r is periodic. therefore the proof is completed by lemma 2.6. proposition 2.8. a ∗-ring r is strongly nil ∗-clean if and only if (1) r is strongly ∗-clean; (2) n(r) = {x ∈ r | 1 −x ∈ u(r)}. proof. suppose that r is strongly nil ∗-clean. by the proof of proposition 2.5, n(r) = j(r). since r is strongly j-∗-clean, n(r) = {x ∈ r | 1 −x ∈ u(r)} by [6, theorem 3.4]. conversely, assume that (1) and (2) hold. let a ∈ r. then we can find a projection e ∈ r such that (a − 1) − e ∈ u(r) and e(a − 1) = (a − 1)e. that is, (1 − a) + e ∈ u(r). as 1 − (a − e) ∈ u(r), by hypothesis, a−e ∈ n(r). in addition, ea = ae. accordingly, r is strongly nil ∗-clean. let r be a ∗-ring. define ∗ : r[x]/(xn) → r[x]/(xn) by a0 + a1x + · · · + an−1xn−1 + (xn) 7→ a∗0 + a ∗ 1x + · · · + a∗n−1xn−1 + (xn). then r[x]/(xn) is a ∗-ring (cf. [10]). corollary 2.9. let r be a ∗-ring. then r is strongly nil ∗-clean if and only if so is r[x]/(xn) for every n ≥ 1. proof. one direction is obvious. conversely, assume that r is strongly nil ∗-clean. clearly, n ( r[x]/(xn) ) = {a0 + a1x + · · · + an−1xn−1 + (xn) | a0 ∈ n(r),a1, · · · ,an−1 ∈ r}. in view of proposition 2.8, n ( r[x]/(xn) ) = {a0 + a1x + · · · + an−1xn−1 + (xn) | 1 −a0 ∈ u(r),a1, · · · ,an−1 ∈ r}. also note that r is abelian. thus, it can be easily seen that every element in r[x]/(xn) can be written as the sum of a projection and a nilpotent element that commute. 158 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 let r be a commutative ∗-ring and consider the ring r[i] = {a + bi | a,b ∈ r, i2 = −1 } and i commutes with elements of r. then r[i] is a ∗-ring, where the involution is ∗ : r[i] → r[i], a + bi 7→ a∗ + b∗i. note that if x and y are idempotent elements that commute, then (x−y)3 = x−3xy+3xy−y = x−y. this argument will also be used in lemma 4.6. proposition 2.10. let r be a commutative ∗-ring. then with the involution (a + bi)∗ = a∗ + b∗i, r[i] is strongly nil ∗-clean if and only if r is strongly nil ∗-clean. proof. suppose that r[i] is strongly nil ∗-clean. then every idempotent in r is a projection. since r is commutative, n(r) forms an ideal. for any a ∈ r, we see that a−a2 ∈ n ( r[i] ) , and so a−a2 ∈ n(r). thus, r/n(r) is boolean. therefore r is strongly nil ∗-clean by proposition 2.4. conversely, assume that r is strongly nil ∗-clean. as r is commutative, n ( r[i] ) forms an ideal of r[i]. let a + bi ∈ r[i] be an idempotent. then we can find projections e,f ∈ r and nilpotent elements u,v ∈ r such that a = e + u, b = f + w. then a − a∗,b − b∗ ∈ n(r). this shows that (a + bi) − (a + bi)∗ = (a−a∗) + (b− b∗)i ∈ n ( r[i] ) . as a + bi, (a + bi)∗ ∈ r[i] are idempotents, we see that ( (a + bi)−(a + bi)∗ )3 = (a + bi)−(a + bi)∗ by the above argument. hence, ( (a + bi)−(a + bi)∗ )( 1− ((a + bi) − (a + bi)∗)2 ) = 0, therefore (a + bi) − (a + bi)∗ = 0. that is, a + bi ∈ r[i] is a projection. since r is strongly nil ∗-clean, it follows from proposition 2.4 that 2−22 ∈ n(r), and so 2 ∈ n(r). for any a + bi ∈ r[i], it is easy to verify that (a + bi) − (a + bi)2 = (a−a2) − 2abi + bi− b2i2 ≡ b2 + bi ≡ b + bi ( mod n ( r[i] )) . this shows that ( (a+bi)−(a+bi)2 )2 ≡ 2b2i ≡ 2b ≡ 0 (mod n(r[i])). hence, (a+bi)−(a+bi)2 ∈ n(r[i]). that is, r[i]/n ( r[i] ) is boolean. according to proposition 2.4, we complete the proof. 3. ∗-boolean like rings in this section, we consider a subclass of strongly nil ∗-clean rings consisting of rings which we call ∗-boolean-like. first recall that a ring r is called boolean-like if it is commutative with unit and is of characteristic 2 with ab(1 +a)(1 +b) = 0 for every a,b ∈ r [8]. any boolean ring is clearly a boolean-like ring but not conversely (see [8]). any boolean-like ring is uniquely nil clean by [8, theorem 17]. also, r is boolean-like if and only if (1) r is a commutative ring with unit; (2) it is of characteristic 2; (3) it is nil clean; (4) ab = 0 for every nilpotent element a,b in r [8, theorem 19]. definition 3.1. a ∗-ring r is said to be ∗-boolean-like provided that every idempotent in r is a projection and (a−a2)(b− b2) = 0 for all a,b ∈ r. the following is an example of a ∗-boolean-like ring. example 3.2. let r = { ( a b c a ) | a,b,c ∈ z2}. define ( a b c a ) + ( a′ b′ c′ a′ ) = ( a + a′ b + b′ c + c′ a + a′ ) ,( a b c a )( a′ b′ c′ a′ ) = ( aa′ ab′ + ba′ ca′ + ac′ aa′ ) and ∗ : r → r, ( a b c a ) 7→ ( a c b a ) . then r is a ∗-ring. let ( a b c a ) ∈ r be an idempotent. then a = a2 and (2a−1)b = (2a−1)c = 0. as (2a−1)2 = 1, we see that b = c = 0, and so the set of all idempotents in r is { ( 0 0 0 0 ) , ( 1 0 0 1 ) }. thus, every idempotent in r is a projection. for any a,b ∈ r, we see that (a−a2)(b −b2) = ( 0 ∗ ∗ 0 )( 0 ∗ ∗ 0 ) = 0. therefore r is ∗-boolean-like. 159 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 theorem 3.3. let r be a ∗-ring. then r is ∗-boolean-like if and only if (1) r is strongly nil ∗-clean; (2) αβ = 0 for all nilpotent elements α,β ∈ r. proof. suppose that r is ∗-boolean-like. then every idempotent in r is a projection; hence, r is abelian. for any a ∈ r, (a − a2)2 = 0, and so a2 = a3f(a) for some f(t) ∈ z[t]. this implies that r is strongly π-regular, and so it is π-regular. it follows from [1, theorem 3] that n(r) forms an ideal. further, a−a2 ∈ n(r). therefore r/n(r) is boolean. according to proposition 2.4, r is strongly nil ∗-clean. for any nilpotent elements α,β ∈ r, we can find some m,n ∈ n such that αm = βn = 0. since α2 = α3g(α) for some g(t) ∈ z[t], α2 = 0. likewise, β2 = 0. this shows that αβ = (α−α2)(β−β2) = 0. conversely, assume that (1) and (2) hold. by proposition 2.4, every idempotent is a projection, and for any a ∈ r, a − a2 is nilpotent. hence for any a,b ∈ r, (a − a2)(b − b2) = 0. therefore r is ∗-boolean-like. corollary 3.4. ∗-boolean-like rings are commutative rings. proof. let x,y ∈ r. in view of theorem 3.3, x − e and y − f are nilpotent for some projections e,f ∈ r. again by theorem 3.3, (x−e)(y −f) = 0 = (y −f)(x−e). since r is abelian, it follows that xy = yx. hence r is commutative. example 3.5. let r be the ring { ( 0 0 0 0 ) , ( 1 0 0 1 ) , ( 0 1 1 0 ) , ( 1 1 0 0 ) , ( 0 0 1 1 ) , ( 1 0 1 0 ) , ( 0 1 0 1 ) , ( 1 1 1 1 ) }, where 0, 1 ∈ z2. define ∗ : r → r,a 7→ at , the transpose of a. then r is a ∗-ring in which (a−a2)(b− b2) = 0 for all a,b ∈ r. further, αβ = 0 for all nilpotent elements α,β ∈ r. but r is not ∗-boolean-like. we end this section with an example showing that strongly nil clean rings need not be strongly nil ∗-clean. example 3.6. consider the ring r = {( a 2b 0 c ) | a,b,c ∈ z4 } . then for any x,y ∈ r, (x − x2)(y − y2) = 0. obviously, r is not commutative. this implies that r is not a ∗-boolean-like ring for any involution ∗. accordingly, r is not strongly nil ∗-clean for any involution ∗; otherwise, every idempotent in r is a projection, a contradiction (see lemma 2.1). we can also consider the involution ∗ : r → r, ( a 2b 0 c ) 7→ ( c −2b 0 a ) and the idempotent ( 0 0 0 1 ) which is not a projection. on the other hand, since (x−x2)2 = 0 and so x−x2 ∈ n(r) for all x ∈ r, we get that r is strongly nil clean by [9, theorem 3]. 4. submaximal ideals and ∗-boolean rings an ideal i of a ring r is called a submaximal ideal if i is covered by a maximal ideal of r. that is, there exists a maximal ideal i1 of r such that i $ i1 $ r and for any ideal k of r such that i ⊆ k ⊆ i1, we have i = k or k = i1. this concept was initially introduced to study boolean-like rings (cf. [11]). a ∗-ring r is called a ∗-boolean ring if every element of r is a projection. the purpose of this section is to characterize submaximal ideals of strongly nil ∗-clean rings, and ∗-boolean rings by means of strongly nil ∗-cleanness. we begin with the following lemma. 160 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 lemma 4.1. let r be strongly nil ∗-clean. then an ideal m of r is maximal if and only if (1) m is prime; (2) for any a ∈ r,n ≥ 1, an ∈ m implies that a ∈ m. proof. suppose that m is maximal. obviously, m is prime. let a ∈ r and an ∈ m. if a 6∈ m, rar + m = r. thus, rar = r where r = r/m and a = a + m. clearly, r is an abelian clean ring, and so it is an exchange ring by [5, theorem 17.2.2]. this implies that r/m is an abelian exchange ring. as in the proof of [5, proposition 17.1.9], there exists a nonzero idempotent e ∈ r such that e ∈ ar and 1 − e ∈ (1 − a)r. since rer is a nonzero ideal of simple ring r, rer = r. thus 1 − e ∈ m. hence, 1 −ar ∈ m for some r ∈ r. this implies that an−1 −anr ∈ m, and so an−1 ∈ m. by iteration of this process, we see that a ∈ m, as required. conversely, assume that (1) and (2) hold. assume that m is not maximal. then we can find a maximal ideal i of r such that m $ i $ r. choose a ∈ i while a 6∈ m. by hypothesis, there exists a projection e ∈ r and a nilpotent u ∈ r such that a = e + u. write um = 0. then um ∈ m. by hypothesis, u ∈ m. this shows that e 6∈ m. clearly, r is abelian. thus er(1−e) ⊆ m. as m is prime, we deduce that 1 −e ∈ m. as a result, 1 −a = (1 −e) −u ∈ m, and so 1 = (1 −a) + a ∈ i. this gives a contradiction. therefore m is maximal. let r be a strongly nil ∗-clean ring, and let x ∈ r. then there exists a unique projection e ∈ r such that x−e ∈ n(r). we denote e by xp and x−e by xn. lemma 4.2. let i be an ideal of a strongly nil ∗-clean ring r, and let x ∈ r be such that x 6∈ i. if xp 6∈ i, then there exists a maximal ideal j of r such that i ⊆ j and x 6∈ j. proof. let ω = {k | k is an ideal in r, i ⊆ k,xp 6∈ k}. then ω 6= ∅. given k1 ⊆ k2 ⊆ ··· in ω, we set q = ∞⋃ i=1 ki. then q is an ideal of r. if q 6∈ ω, then xp ∈ q, and so xp ∈ ki for some i. this gives a contradiction. thus, ω is inductive. by using zorn’s lemma, there exists an ideal l of r which is maximal in ω. let a,b ∈ r such that a,b 6∈ l. by the maximality of l, we see that rar+l,rbr+l 6∈ ω. this shows that xp ∈ (rar + l)∩(rbr + l). hence, xp = x2p ∈ rarbr + l. this yields that arb * l; otherwise, xp ∈ l, a contradiction. hence, l is prime. assume that l is not maximal. then we can find a maximal ideal m of r such that l $ m $ r. clearly, r is abelian. by the maximality, we see that xp ∈ m, and so 1 − xp 6∈ m. this implies that 1 − xp 6∈ l. as xp r(1 − xp ) = 0 ⊆ l, we have that xp ∈ l, a contradiction. therefore l is a maximal ideal, as asserted. proposition 4.3. let r be strongly nil ∗-clean. then the intersection of two maximal ideals is submaximal and it is covered by each of these two maximal ideals. further, there is no other maximal ideals containing it. proof. let i1 and i2 be two distinct maximal ideals of r. then i1∩i2 $ i1. suppose i1∩i2 ⊆ l $ i1. then we can find some x ∈ i1 while x 6∈ l. write xnn = 0. then x n n ∈ i1. in light of lemma 4.1, xn ∈ i1. likewise, xn ∈ i2. thus, xn ∈ i1 ∩ i2 ⊆ l. this shows that xp 6∈ l. by virtue of lemma 4.2, there exists a maximal ideal m of r such that l ⊆ m and x 6∈ m. hence, i1∩i2 ⊆ m and i1 6= m. if i2 6= m, then i2 +m = r. write t+y = 1 with t ∈ i2,y ∈ m. then for any z ∈ i1, z = zt+zy ∈ i1∩i2 +m = m, and so i1 = m. this gives a contradiction. thus i2 = m, and then l ⊆ m ⊆ i2. as a result, l ⊆ i1∩i2, and so i1 ∩ i2 = l. therefore i1 ∩ i2 is a submaximal ideal of r. we claim that i1 ∩ i2 is semiprime. if k2 ⊆ i1 ∩ i2, then for any a ∈ k, we see that a2 ∈ i1 ∩ i2. in view of lemma 4.1, a ∈ i1 ∩ i2. this implies that k ⊆ i1 ∩ i2. hence, i1 ∩ i2 is semiprime. therefore i1 ∩ i2 is the intersection of maximal ideals containing i1 ∩ i2. assume that k is a maximal ideal of r such that i1∩i2 ⊆ k. if k 6= i1,i2, then i1+k = i2+k = r. this implies that i1 ∩ i2 + k = r, and so k = r, a contradiction. thus, k = i1 or k = i2, and so the proof is completed. 161 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 we call a local ring r absolutely local provided that for any 0 6= x ∈ j(r), j(r) = rxr. corollary 4.4. let r be strongly nil ∗-clean, and let i be an ideal of r. then i is a submaximal ideal if and only if r/i is boolean with four elements or r/i is absolutely local. proof. let i be a submaximal ideal of r. case i. i is contained in more than one maximal ideal. then i is contained in two distinct maximal ideals of r. since i is submaximal, there exists a maximal ideal l of r such that i is covered by l. thus, we have a maximal ideal l′ such that l′ 6= l and i $ l′. hence, i ⊆ l∩l′ ⊆ l. clearly, l∩l′ 6= l as l + l′ = r, and so i = l ∩ l′. in view of proposition 4.3, there is no maximal ideal containing i except for l and l′. this shows that r/i has only two maximal ideals covering {0 + i}. for any a ∈ r, it follows from proposition 2.4 that a−a2 ∈ r is nilpotent. write (a−a2)n = 0. then (a−a2)n ∈ l. according to lemma 4.1, a−a2 ∈ l. likewise, a−a2 ∈ l′. thus, a−a2 ∈ l∩l′, and so a−a2 ∈ i. this shows that r/i is boolean. therefore r/i is boolean with four elements. case ii. suppose that i is contained in only one maximal ideal l of r. then r/i has only one maximal ideal l/i. clearly, r is an abelian exchange ring, and then so is r/i. let e ∈ r/i be a nontrivial idempotent. then i ⊆ i + rer ⊆ l or i + rer = r. likewise, i ⊆ i + r(1 − e)r ⊆ l or i + r(1−e)r = r. this shows that i + rer = r or i + r(1−e)r = r thus, (r/i)(e + i)(r/i) = r/i or (r/i)(1−e + i)(r/i) = r/i, a contradiction. therefore all idempotents in r/i are trivial. it follows from [5, lemma 17.2.1] that r/i is local. for any 0 6= x ∈ l/i, we see that 0 6= i ⊆ rxr ⊆ l. as i is submaximal, we deduce that l = rxr. therefore r is absolutely local. conversely, assume that r/i is boolean with four elements. then r/i has precisely two maximal ideals covering {0 + i}, and so r has precisely two maximal ideals covering i. thus, we have a maximal ideal l such that i $ l. if i ⊆ k ⊆ l. then k = i or k is maximal, and so k = l. consequently, i is submaximal. assume that r/i is absolutely local. then r/i has a unique maximal ideal l/i. hence, l is a maximal ideal of r such that i $ l. assume that i $ k ⊆ l. choose a ∈ k while a 6∈ i. then l = rar ⊆ k, and so k = l. therefore i is submaximal, as required. corollary 4.5. let r be strongly nil ∗-clean. if i1 and i2 are distinct maximal ideals of r, then r/(i1 ∩ i2) is boolean. proof. since i1/(i1 ∩i2) and i2/(i1 ∩i2) are distinct maximal ideals, r/(i1 ∩i2) is not local. in view of proposition 4.3, i1 ∩ i2 is a submaximal ideal of r. therefore corollary 4.4 yields the proof. recall that an ideal i of a commutative ring r is primary provided that for any x,y ∈ r, xy ∈ i implies that x ∈ i or yn ∈ i for some n ∈ n. clearly, every maximal ideal of a commutative ring is primary. we end this article by giving the relation between strongly nil ∗-clean rings and ∗-boolean rings. lemma 4.6. let r be a commutative strongly nil ∗-clean ring. then the intersection of all primary ideals of r is zero. proof. let a be in the intersection of all primary ideal of r. assume that a 6= 0. let ω = {i | i is an ideal of r such that a 6∈ i}. then ω 6= ∅ as 0 ∈ ω. given any ideals i1 ⊆ i2 ⊆ ··· in ω, we set m = ∞⋃ i=1 ii. then m ∈ ω. thus, ω is inductive. by using zorn’s lemma, we can find an ideal q which is maximal in ω. it will suffice to show that q is primary. if not, we can find some x,y ∈ r such that xy ∈ q, but x 6∈ q and yn 6∈ q for any n ∈ n. this shows that a ∈ q+(x), and so a = b+cx for some b ∈ q,c ∈ r. since r is strongly nil ∗-clean, it follows from theorem 2.7 that there are some distinct k,l ∈ n such that yk = yl. say k > l. then yl = yk = yl+1yk−l−1 = ylyyk−l−1 = yl+2y2(k−l−1) = · · · = y2lyl(k−l−1). hence, yl(k−l) = yl(yl(k−l−1)) = y2ly2l(k−l−1) = ( yl(k−l) )2 . choose s = l(k − l). then ys is an idempotent. write y = yp + yn. then ys − yp = (yp + yn )s − yp = yn ( syp + · · · + ys−1n ) ∈ n(r). as r is a commutative ring, we see that (ys − yp )3 = ys − yp . this implies that ys = yp . since xy ∈ q, we have that xys ∈ q, and so xyp ∈ q. it follows from a = b + cx that ayp = byp + cxyp ∈ q. clearly, ys 6∈ q, and so a ∈ q + (yp ). write a = d + ryp for some d ∈ q,r ∈ r. we see that ayp = dyp + ryp , 162 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 and so ryp ∈ q. this implies that a ∈ q, a contradiction. therefore q is primary, a contradiction. consequently, the intersection of all primary ideals of r is zero. theorem 4.7. let r be a ∗-ring. then r is a ∗-boolean ring if and only if (1) r is commutative; (2) every primary ideal of r is maximal; (3) r is strongly nil ∗-clean. proof. suppose that r is a ∗-boolean ring. clearly, r is a commutative strongly nil ∗-clean ring. let i be a primary ideal of r. if i is not maximal, then there exists a maximal ideal m such that i $ m $ r. choose x ∈ m while x 6∈ i. as x is an idempotent, we see that xr(1−x) ⊆ i, and so (1−x)m ∈ i ⊂ m for some m ∈ n. thus, 1 −x ∈ m. this implies that 1 = x + (1 −x) ∈ m, a contradiction. therefore i is maximal, as required. conversely, assume that (1), (2) and (3) hold. clearly, every maximal ideal of r is primary, and so j(r) = ⋂ {p | p is primary}. in view of lemma 4.6, j(r) = 0. hence every element is a projection i.e. r is ∗-boolean. corollary 4.8. a ring r is a boolean ring if and only if (1) r is commutative; (2) every primary ideal of r is maximal; (3) r is strongly nil clean. proof. choose the involution as the identity. then the result follows from theorem 4.7. acknowledgment: the authors are grateful to the referee for his/her careful reading and valuable comments on this manuscript. this research was supported by the scientific and technological research council of turkey (tubitak-2221 visiting scientists fellowship programme). huanyin chen is thankful for the support by the natural science foundation of zhejiang province (no. ly17a010018). references [1] a. badawi, on abelian π–regular rings, comm. algebra 25(4) (1997) 1009–1021. [2] s. k. berberian, baer ∗–rings, springer-verlag, heidelberg, london, new york, 2011. [3] m. chacron, on a theorem of herstein, canad. j. math. 21 (1969) 1348–1353. [4] h. chen, on strongly j–clean rings, comm. algebra 38(10) (2010) 3790–3804. [5] h. chen, rings related stable range conditions, series in algebra 11, world scientific, hackensack, nj, 2011. [6] h. chen, a. harmancı a. ç. özcan, strongly j–clean rings with involutions, ring theory and its applications, contemp. math. 609 (2014) 33–44. [7] a. j. diesl, nil clean rings, j. algebra 383 (2013) 197–211. [8] a. l. foster, the theory of boolean–like rings, trans. amer. math. soc. 59 (1946) 166–187. [9] y. hirano, h. tominaga, a. yaqub, on rings in which every element is uniquely expressable as a sum of a nilpotent element and a certain potent element, math. j. okayama univ. 30 (1988) 33–40. [10] c. li, y. zhou, on strongly ∗–clean rings, j. algebra appl. 10(6) (2011) 1363–1370. 163 http://dx.doi.org/10.1080/00927879708825906 http://www.ams.org/mathscinet-getitem?mr=262295 http://dx.doi.org/10.1080/00927870903286835 http://www.ams.org/mathscinet-getitem?mr=3204350 http://www.ams.org/mathscinet-getitem?mr=3204350 http://dx.doi.org/10.1016/j.jalgebra.2013.02.020 http://dx.doi.org/10.2307/1990316 http://www.ams.org/mathscinet-getitem?mr=976729 http://www.ams.org/mathscinet-getitem?mr=976729 http://dx.doi.org/10.1142/s0219498811005221 a. harmancı et al. / j. algebra comb. discrete appl. 4(2) (2017) 155–164 [11] v. swaminathan, submaximal ideals in a boolean–like rings, math. sem. notes kobe univ. 10(2) (1982) 529–542. [12] l. vaš, ∗–clean rings; some clean and almost clean baer ∗–rings and von neumann algebras, j. algebra 324(12) (2010) 3388–3400. 164 http://www.ams.org/mathscinet-getitem?mr=704940 http://www.ams.org/mathscinet-getitem?mr=704940 http://dx.doi.org/10.1016/j.jalgebra.2010.10.011 http://dx.doi.org/10.1016/j.jalgebra.2010.10.011 introduction characterization theorems *-boolean like rings submaximal ideals and *-boolean rings references issn 2148-838x j. algebra comb. discrete appl. 4(2) received : 29 december 2016 journal of algebra combinatorics discrete structures and applications editor’s note andré leroy this special issue contains papers written by participants to the conference "noncommutative rings and their applications" that was held in lens (france) at the science faculty of the université d’artois. the meeting gave the experts from different domains the opportunity to exchange their views, share their research, and learn from one another new results and problems in a friendly atmosphere. they were 55 researchers, graduate and postdoctoral students from usa, canada, poland, czech republic, italy, spain, germany, portugal, egypt, senegal, new zeland, south africa, algeria, turkey, mexico, brazil, uruguay, indonesia...and france ! the interplay between ring and coding theory was emphasized by the very nice and interesting course "linear codes from the axiomatic point of view" given by jay wood. the four invited speakers : frédérique oggier, christophe reutenauer, angel del rio and irfan siap contributed greatly to the success of this conference. the topics of the jay wood’s course and the 39 talks presented at the conference are well represented by this special issue of jacodesmath. they cover pure ring theory such as nil *-clean rings, radical classes or algebras such as grassman algebras and steenrod algebras and many papers relating these subjects with coding theory such as macwilliams extension theorem, dual codes, lee and hamming weight. so this volume will be particularly useful for mathematicians at the confluent of these two branches. this meeting was supported by the laboratoire de mathématiques de lens (lml), by different bodies from the université d’artois (ri, bqr), as well as by a regional organization (the fédération des laboratoires de mathematiques du nord pas de calais). we would like to thank all the participants for their efforts and enthusiasm. a very warm thanks to the colleagues who kindly agreed to referee the papers. their expertise, promptitude, and professionalism improved the quality of the articles in this volume. many thanks are due to the editorial staff of the jacodesmath journal for the very efficient way they managed the process of preparing and publishing these proceedings. department of mathematics, university of artois, france (email: andre.leroy@univ-artois.fr). 103 issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327360 j. algebra comb. discrete appl. 4(3) • 227–234 received: 14 may 2016 accepted: 27 september 2016 journal of algebra combinatorics discrete structures and applications some new ternary linear codes∗ research article rumen daskalov, plamen hristov abstract: let an [n, k, d]q code be a linear code of length n, dimension k and minimum hamming distance d over gf(q). one of the most important problems in coding theory is to construct codes with optimal minimum distances. in this paper 22 new ternary linear codes are presented. two of them are optimal. all new codes improve the respective lower bounds in [11]. 2010 msc: 94b05, 94b65 keywords: ternary linear codes, construction x 1. introduction let an [n,k,d]q code be a linear code of length n, dimension k and minimum hamming distance d over a finite field gf(q). one of the most important and fundamental problems in coding theory is to find the optimal values of the parameters of a linear code. this optimization problem can be formulated in a couple of ways. for example, for fixed q,n and k we may wish to maximize the minimum distance d; or for given q,k and d to minimize the block length n. let dq(n,k) denote the largest value of d for which there exists an [n,k,d]q code, and nq(k,d) be the smallest value of n for which there exists an [n,k,d]q code. then an [nq(k,d),k,d]q code is called lengthoptimal and an [n,k,dq(n,k)]q code is called distance-optimal. both length-optimal and distance-optimal codes are called optimal codes. the problem of finding the parameters of optimal codes is a very difficult one and has two aspects one involves the construction of new codes with better minimum distances and the other is proving the nonexistence of codes with given parameters. it has been solved only over small finite fields for small dimensions and co-dimensions. computer search is often used in looking for codes with better minimum distances, but it is a well known fact that computing the minimum distance of a linear code is an np-hard problem [15]. since it is ∗ this work was partially supported by the bulgarian ministry of education and science under contract in tu– gabrovo. rumen daskalov (corresponding author), plamen hristov; department of mathematics, technical university of gabrovo, bulgaria (email: daskalov@tugab.bg, plhristov9@gmail.com). 227 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 not possible to carry out exhaustive searches for linear codes with large dimension, it is natural to focus one’s effort on subclasses of linear codes, having rich mathematical structures. quasi-cyclic (qc) codes are known to have such a structure and it has been shown in recent years that this subclass contains many new good linear codes ([1, 4–10, 12–14] and [e. metodieva, n. daskalova, generating generalized necklaces and new quasi-cyclic codes, in preparation, 2017]). grassl [11] maintains a table with lower and upper bounds on minimum distances of linear codes over small finite fields gf(q) (q ≤ 9). when the constructed code has a minimum distance equal to the upper bound, it is optimal and there is no place for improvement in the table. when there is a gap between the minimum distance of the best-known code and the upper bound on the minimum distance, this is indicated in the table by listing both values dl and du. many of the best-known codes in these tables are qc codes. a code that attains a lower bound in the table is called a good code. a code that improves a lower bound in the table will be called a new code. another online table of linear codes is also maintained by chen. chen’s table [3] contains only good and best-known quasi-cyclic and quasi-twisted codes (q ≤ 13). these two databases are updated when new codes are discovered. the remainder of the paper is organized as follows. in section 2, some basic definitions and facts on qc codes are presented. in section 3, sixteen good one-generator qc codes (p ≥ 2) are constructed using an algebraic-combinatorial computer search. in section 4 (theorem 4.1), we use the codes presented in section 3, along with construction x, to obtain seventeen new ternary linear codes. in theorem 4.2 five new codes are also presented. 2. quasi-cyclic codes a code c is said to be quasi-cyclic (qc or p-qc) if a cyclic shift of a codeword by p positions results in another codeword. a cyclic shift of an m-tuple (x0,x1, . . . ,xm−1) is the m-tuple (xm−1,x0, . . . ,xm−2). the blocklength n of a p-qc code is a multiple of p, so that n = pm. a matrix b of the form b =   b0 b1 b2 · · · bm−2 bm−1 bm−1 b0 b1 · · · bm−3 bm−2 bm−2 bm−1 b0 · · · bm−4 bm−3 ... ... ... ... ... b1 b2 b3 · · · bm−1 b0   , (1) is called a circulant matrix. a class of qc codes can be constructed from m×m circulant matrices. in this case, the generator matrix g can be represented as g = [b1, b2, ... , bp] , (2) where bi is a circulant matrix. the algebra of m×m circulant matrices over gf(q) is isomorphic to the algebra of polynomials in the ring gf(q)[x]/(xm−1), with b being mapped to the polynomial, b(x) = b0+b1x+b2x2+· · ·+bm−1xm−1, formed from the entries in the first row of b. the bi(x)’s associated with a qc code are called the defining polynomials. if the defining polynomials bi(x) contain a common factor which is also a factor of xm −1, then the qc code is called degenerate. the dimension k of the qc code is equal to the degree of h(x), where h(x) = xm −1 gcd{xm −1,b0(x),b1(x), · · · ,bp−1(x)} . (3) 228 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 if the polynomial h(x) has degree m, the dimension of the code is m, and (2) is a generator matrix. if deg(h(x)) = k < m, a generator matrix for the code can be constructed by deleting m − k rows of (2). let the defining polynomials of the code c have the following form d1(x) = g(x), d2(x) = g(x)f2(x), · · · , dp(x) = g(x)fp(x), (4) where g(x)|(xm − 1),g(x),fi(x) ∈ gf(q)[x]/(xm − 1), (fi(x),(xm − 1)/g(x)) = 1 and deg fi(x) < m−deg g(x) for all 1 ≤ i ≤ p. then c is a degenerate, one-generator qc code having n = mp, and k = m−deg g(x) (see [14]). in our constructions we will use the following well-known theorems. theorem 2.1 (construction x). [2] given an [n,k,d]q code c1, and an [n,k− l,d + s]q subcode c2, we can construct an [n+a,k,d+s]q code c when we have an [a,l,s]q code c3 (by appending codewords from the latter code to cosets of the second code in the first code). theorem 2.2 (construction xx). [2] let an [n,k,d]q code c have two subcodes c1 and c2 of dimensions k−k1 and k−k2 and append tails from a [ai,ki,δi]q code to the codewords of c , where the two tails of codewords correspond to the coset of ci(i = 1,2) it is in. if c1, c2 and c1 ⋂ c2 have minimum distance d1, d2 and d0, respectively, then there exists an [n+a1 +a2,k,min(d0,d1 +δ2,d2 +δ1,d+δ1 +δ2)]q code. 3. good qc codes in this section sixteen good one-generator qc codes (p ≤ 4) are constructed using a non-exhaustive algebraic-combinatorial computer search, similar to that in [1, 4–6, 8–10, 14]. an important feature of these codes is that they have good subcodes and can be used for construction x. we have restricted our search to one-generator qc codes with defining polynomials of the form (4). example 3.1. : let m = 35. the factorization of the polynomial x35 −1 over gf(3) is x35 −1 = 5∏ i=1 pi(x), where p1(x) = x 12 + x10 + 2x8 + x7 + x5 + 2x4 + x3 + 2x2 + 2x + 1 p2(x) = x 12 + 2x11 + 2x10 + x9 + 2x8 + x7 + x5 + 2x4 + x2 + 1 p3(x) = x 6 + x5 + x4 + x3 + x2 + x + 1 p4(x) = x 4 + x3 + x2 + x + 1 p5(x) = x + 2. let the dimension k = 17 . then the degree of the polynomials g(x) has to be 18. taking the product of two of the polynomials above, one of degree 12 and one of degree 6, we obtain two polynomials g(x) of degree 18. we choose g(x) = x18 + x17 + 2x16 + 2x15 + x14 + 2x13 + 2x12 + 2x11 + x10 + x9 + 2x4 + 2x2 + 1, and then we search for f2(x). the polynomial f2(x) = x9 + 2x8 + 2x7 + x6 + x4 + x3 + 1 yields a [70,17,29]3 quasi-cyclic code. afterwards, we search for f3(x) and f4(x) in succession. the polynomial f3(x) = x 10 +x8 +2x6 +x5 +x3 +2x+2 leads to a [105,17,48]3 code and f4(x) = x8 +2x4 +2x3 +x2 +2 gives a [140,17,69]3 code. in the following theorems the defining polynomials are listed with the lowest degree coefficient on the left. 229 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 theorem 3.2. there exist one-generator quasi-cyclic codes with parameters: [160,12,90]3, [104,13,54]3, [120,13,62]3, [156,13,86]3, [182,13,102]3, [224,13,127]3, [160,14,87]3, [48,15,18]3, [160,16,84]3, [52,17,19]3, [105,17,48]3, [123,17,59]3, [140,17,69]3, [111,19,49]3, [104,20,45]3 [170,20,82]3. proof. the coefficients of the defining polynomials of the codes are as follows: a [160,12,90]3 code (m = 80,p = 2): 21021012201011010101000101121011012110020122220021112210200212100111100000000000, 11202022111110110221012101101122200120220001001120211021000110020101102012110000; a [104,13,54]3 code (m = 26,p = 4): 22000102100211000000000000, 22212202222121122101000000, 10221201011011101102210000, 21122112010222002201000000; a [120,13,62]3 code (m = 40,p = 3): 1102201002011122212010121021000000000000, 2001110011212210020202220120121210000000, 2211012011110100012021202001210120010000; a [156,13,86]3 code (m = 52,p = 3): 2012002102201121122202110200200022200021000000000000, 1111211022101012120102220001022012010102022021000000, 1021020221112201002220221102211002022222012200010000; a [182,13,102]3 code (m = 91,p = 2): 12121202201122111220201000101011000200222002000110101000102022111221102202121210 00000000000, 12011200200212120120011020020222110202200002101112002122011221020211201021222100 20112100000; a [224,13,127]3 code (m = 56,p = 4): 22120222120002200111020212101101012000020221000000000000, 21100120002100102020210110221201212000102201100210100000, 22102211102110121220010210100222200112210222112012010000, 12120121221021102201021101002021211201202002222112101000; a [160,14,87]3 code (m = 80,p = 2): 11010200020021110210121201112212021120120202011111222020121210112010000000000000, 21211111201022111210011200101201021111222022210201212111211121022102200100100000; a [48,15,18]3 code (m = 16,p = 3): 2100000000000000, 2022201212010000, 2011012221000000; a [160,16,84]3 code (m = 80,p = 2): 21122202101002000221000021202201102222121011212120112222222221111000000000000000, 11212010202020100022102121021011100212100111210201122001110012101112012110000000; a [52,17,19]3 code (m = 26,p = 2): 21101001110000000000000000, 22120102122100222001210000; a [105,17,48]3 code (m = 35,p = 3): 10202000011222122110000000000000000, 10210201001210110110202100010000000, 22121121212010222210222120011000000; a [123,17,59]3 code (m = 41,p = 3): 10111221201020102122111010000000000000000, 21012100122202021022002102122101000000000, 10001020002202222112112100212221100100000; a [140,17,69]3 code (m = 35,p = 4): 10202000011222122110000000000000000, 10210201001210110110202100010000000, 22112002222210021120212112100000000, 20021202112020221122002110000100000; a [111,19,49]3 code (m = 37,p = 3): 1020220100010220201000000000000000000, 2110000221222002121020122210000000000, 2122202011110212101012201121000000000; a [104,20,45]3 code (m = 52,p = 2): 2001111020002211212012002112211010000000000000000000, 2100222212021021100112122012201100020010110000000000; a [170,20,82]3 code (m = 85,p = 2): 230 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 2202110000022200100210120122220000112001100102212020020200020100010000000000000000000, 1111002221101112100111212110102011001121001001222102000022221210102102000110000000000. 4. the new codes let us look at an example, related to the next theorem 4.1. we will show how the [151,17,75]3 code c is constructed. a generator matrix of a code c has the form g =   g2 0 ∗ g3   , where g2 and g3 are generator matrices of the codes c2 and c3 respectively, and (∗) denotes l linearly independent codewords of a code c1. the generator matrix g2 of the [140,12,75]3 subcode c2 has the same first row as the row given in theorem 4.1. the generator matrix of the [11,5,6]3 code is g3 =   10122002010 01012200201 10101220020 01010122002 20101012200   and the five independent codewords from the [140,17,69]3 code c1 are: 1020200001122212211000000000000000010210201001210110110202100010000000221120022222100211202 1211210000000020021202112020221122002110000100000, 0102020000112221221100000000000000001021020100121011011020210001000000022112002222210021120 2121121000000002002120211202022112200211000010000, 0010202000011222122110000000000000000102102010012101101102021000100000002211200222221002112 0212112100000000200212021120202211220021100001000, 0001020200001122212211000000000000000010210201001210110110202100010000000221120022222100211 2021211210000000020021202112020221122002110000100, 0000102020000112221221100000000000000001021020100121011011020210001000000022112002222210021 1202121121000000002002120211202022112200211000010; the weight enumerator of the [151,17,75]3 code is 01 75784 762664 787728 7920870 8140824 82111524 84175574 85491414 87610190 881651468 901659156 914416804 933644426 949032618 966278202 9714388718 998407602 10017626234 1028667680 10316568328 1056850326 10611875360 1084084836 1096379366 1111839922 1122579332 114607642 115760172 117143598 118162324 12024976 12123782 1233150 1242198 12684 127266 13212 1358. theorem 4.1. there exist new ternary linear codes, having parameters: [168,12,96]3, [110,13,57]3, [114,13,60]3, [122,13,64]3, [157,13,87]3, [184,13,104]3, [225,13,128]3, [170,14,93]3, [49,15,19]3, [165,16,86]3, [53,17,20]3, [106,17,49]3, [124,17,60]3, [151,17,75]3, [112,19,50]3, [110,20,48]3, [171,20,83]3. proof. all of the codes are obtained by construction x. the parameters of the codes, related to theorem 2.1, are given in table 1. the defining polynomials of the subcodes c2 are given for clearness. the coefficients of these polynomials are as follows: a [160,10,96]3 code: 21201222020021120202010102102221122201120020112221020002002211221112211000000000, 11011012002221211020222222112100122121122201011101110101210111120002110002201100; a [104,10,57]3 code: 21002111121000201000000000, 21200212021001111212221000, 11121000000010111002211110, 20011221200122201101021000; a [104,9,60]3 code: 231 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 table 1. the new codes. code c1 subcode c2 code c3 new code c [160,12,90]3 [160,10,96]3 [8,2,6]3 [168,12,96]3 [104,13,54]3 [104,10,57]3 [6,3,3]3 [110,13,57]3 [104,13,54]3 [104,9,60]3 [10,4,6]3 [114,13,60]3 [120,13,62]3 [120,12,64]3 [2,1,2]3 [122,13,64]3 [156,13,86]3 [156,12,87]3 [1,1,1]3 [157,13,87]3 [182,13,102]3 [182,12,104]3 [2,1,2]3 [184,13,104]3 [224,13,127]3 [224,12,128]3 [1,1,1]3 [225,13,128]3 [160,14,87]3 [160,10,93]3 [10,4,6]3 [170,14,93]3 [48,15,18]3 [48,14,19]3 [1,1,1]3 [49,15,19]3 [160,16,84]3 [160,12,86]3 [5,4,2]3 [165,16,86]3 [52,17,19]3 [52,16,20]3 [1,1,1]3 [53,17,20]3 [105,17,48] 3 [105,16,49]3 [1,1,1]3 [106,17,49]3 [123,17,59]3 [123,16,60]3 [1,1,1]3 [124,17,60]3 [140,17,69]3 [140,12,75]3 [11,5,6]3 [151,17,75] 3 [111,19,49]3 [111,18,50]3 [1,1,1]3 [112,19,50]3 [104,20,45]3 [104,17,48]3 [6,3,3]3 [110,20,48]3 [170,20,82]3 [170,19,83]3 [1,1,1]3 [171,20,83]3 21012011002110011100000000, 21210221012001202202000100, 20102202120100012001122212, 21211221200221021211002002; a [120,12,64]3 code: 1022011202110010021112112222200000000000, 2101002010121022021212001111112122000000, 2020211110002120011122112101122111012000; a [156,12,87]3 code: 2012002102201121122202110200200022200021000000000000, 1111211022101012120102220001022012010102022021000000, 1021020221112201002220221102211002022222012200010000; a [182,12,104]3 code: 11212112011010200101211200121210200210200102100102121200122120200102022012212 12200000000000, 11110110210221211111010221021200202212010002221001102210110102221220111222100 22021101220000; a [224,12,128]3 code: 20211200211002010100221221221021211100021202200000000000, 22020111002220122121222102202111121100122011020222120000, 20222020022202112101012222120200010101022200201111112000, 11211112102222022011222021202122120111112102000201221200; a [160,10,93]3 code: 11122201000222022201201101000220220102212122011021220121120110122122202000000000, 21121101202212200111020011012200211220220120200112201111112111120221012211121020; a [48,14,19]3 code: 2010000000000000,2221121000211000,2212110110100000; a [160,12,86]3 code: 22202221200021221101211122002221022000100211020211121021222221212010100000000000, 10220111202202200200120011220201221200120211101221110011212112001121200212201000; a [52,16,20]3 code: 22021201002000000000000000, 20211122210220200101122000; a [105,16,49]3 code: 232 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 12212100010100210202000000000000000, 12222211201122102102212220012000000, 20212012121112200022200211010200000; a [123,16,60]3 code: 12100102111221122210200212000000000000000, 22211220110012122220102222210221200000000, 12001221002012000201201220221002020120000; a [140,12,75]3 code: 10202201021222010222112200000000000, 10210102202200101012101001022000200, 22112111102221102020001210012212000, 20021002200002101220121220012000002; a [111,18,50]3 code: 1221201120012201211200000000000000000, 2202000202100102212221110022000000000, 2210012110002221221211011012200000000; a [104,17,48]3 code 2202221011012100122001220021020200020000000000000000, 2011111211011102222020222121112012222021111220000000; a [170,19,83]3 code: 2110101121021000201010101101102012121111122020201112111111212000121111000000000000000, 1201101201202112021100201112201011120012211212121211202221020020211112202022221000000. theorem 4.2. there exist new ternary linear codes with parameters: [39,12,18]3, [48,13,21]3, [66,18,27]3 and [106,14,54]3. proof. 1. there exist a [36,12,15]3 qc code c, having the following defining polynomials: 112122201101, 222010220221, 222001010001. this code is triple extendible. by adding the next three columns (110110110110)>, (101101101101)>, (011011011011)> to the generator matrix of c, we get a new self-orthogonal optimal [39,12,18]3 code. the weight distribution of the new code is 01 188034 2148204 24169415 27204464 3090090 3310998 36234. the shortened [38,11,18]3 code is also optimal. 2. there exist a [48,12,21]3 qc code c, having the following defining polynomials: 102100112112, 102121001210, 111002112002, 111210200210. by adding the next row 000000000000000000000000111111111111222222222222 to the generator matrix of code c, we get a new [48,13,21]3 code. 3. first a new [64,18,25]3 code d of type (4) has been constructed. it has defining polynomials d1(x) = g(x) = (x8 + 2x4 + 2)(x4 + 2x2 + 2)(x2 + 2x + 2) and d2(x) = g(x).f2(x), where f2(x) = x 13 + x10 + 2x8 + x7 + 2x6 + x5 + x4 + 2x3 + 2x2 + 2x + 1, i.e. the defining polynomials are 21011021012211100000000000000000, 22011120101001120212221001110000. this code is double extendible. by adding the next columns (101010101010101010)>, (01010101010101010)> to the generator matrix of d we get a new [66,18,27]3 code. 4. a new [106,14,54]3 code has been constructed by construction xx (theorem 2.2), where c is a [104,14,52]3 code, c1 is a [104,13,53]3 code, c2 is a [104,13,53]3 code, c1 ⋂ c2 is a [104,12,54]3 code and a1 = a2 = 1, k1 = k2 = 1, δ1 = δ2 = 1. the defining polynomials of c, c1, c2 and c1 ⋂ c2 are as follows: 12000020022110000000000000, 22212202200212210221000000, 20122211021022101000000000, 20101021210012001222101100; 11100021020202000000000000, 20021012010221022202200000, 21110020222220221200000000, 21121222122011101100221020; 10200022021021000000000000, 21100122120200101210100000, 22101102120121011100000000, 22111120001010201011011210; 12210020122222200000000000, 22020110211210121122120000, 20221022211112210020000000, 20200011001212211210210122; a generator matrix of the new code has the next form g =   g ? 0 ∗ g3   , where g? is a generator 233 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 r. daskalov, p. hristov / j. algebra comb. discrete appl. 4(3) (2017) 227–234 matrix of c1 ⋂ c2, g3 = ( 1 1 1 2 ) and ∗ denotes the next two linearly independent codewords of a code c : 1200002002211000000000000022212202200212210221000000201222110210221010000000002010102121001 2001222101100, 0120000200221100000000000002221220220021221022100000020122211021022101000000000201010212100 1200122210110 . references [1] n. aydin, i. siap, d. ray-chaudhuri, the structure of 1–generator quasi–twisted codes and new linear codes, des. codes cryptogr. 24(3) (2001) 313–326. [2] a. e. brouwer, bounds on the size of linear codes, in handbook of coding theory, v.s. pless, w.c. huffman, r.a. brualdi(eds), elsevier amsterdam, 1998. [3] e. z. chen, database of quasi–twisted codes, available at http://www.tec.hkr.se/∼chen/ research/codes/searchqc2.htm [4] e. z. chen, a new iterative computer search algorithm for good quasi–twisted codes, des. codes cryptogr. 76(2) (2015) 307–323. [5] e. chen, n. aydin, a database of linear codes over f13 with minimum distance bounds and new quasi–twisted codes from a heuristic search algorithm, j. algebra comb. discrete appl. 2(1) (2015) 1–16. [6] e. chen, n. aydin, new quasi–twisted codes over f11−minimum distance bounds and a new database, j. inf. optim. sci. 36(1–2) (2015) 129–157. [7] r. n. daskalov, t. a. gulliver, new good quasi–cyclic ternary and quaternary linear codes, ieee trans. inform. theory 43(5) (1997) 1647–1650. [8] r. daskalov, p. hristov, new one–generator quasi–cyclic codes over gf(7), problemi peredachi informatsii 38(1) (2002) 59–63. english translation: probl. inf. transm. 38(1) (2002) 50–54. [9] r. daskalov, p. hristov, new quasi–twisted degenerate ternary linear codes, ieee trans. inform. theory 49(9) (2003) 2259–2263. [10] r. daskalov, p. hristov, e. metodieva, new minimum distance bounds for linear codes over gf(5), discrete math. 275(1–3) (2004) 97–110. [11] m. grassl, linear code bound [electronic table; online], available at http://www.codetables.de. [12] p. p. greenough, r. hill, optimal ternary quasi–cyclic codes, des. codes cryptogr. 2(1) (1992) 81–91. [13] t. a. gulliver, p. r. j. ostergard, improved bounds for ternary linear codes of dimension 7, ieee trans. inform. theory 43(4) (1997) 1377–1381. [14] i. siap, n. aydin, d. ray-chaudhury, new ternary quasi–cyclic codes with better minimum distances, ieee trans. inform. theory 46(4) (2000) 1554–1558. [15] a. vardy, the intractability of computing the minimum distance of a code, ieee trans. inform. theory 43(6) (1997) 1757–1766. 234 http://orcid.org/0000-0001-7441-4757 http://orcid.org/0000-0002-7350-4061 http://dx.doi.org/10.1023/a:1011283523000 http://dx.doi.org/10.1023/a:1011283523000 http://www.tec.hkr.se/~chen/research/codes/searchqc2.htm http://www.tec.hkr.se/~chen/research/codes/searchqc2.htm http://dx.doi.org/10.1007/s10623-014-9950-8 http://dx.doi.org/10.1007/s10623-014-9950-8 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.13069/jacodesmath.36947 http://dx.doi.org/10.1080/02522667.2014.961788 http://dx.doi.org/10.1080/02522667.2014.961788 http://dx.doi.org/10.1109/18.623167 http://dx.doi.org/10.1109/18.623167 http://dx.doi.org/10.1023/a:1020094206873 http://dx.doi.org/10.1023/a:1020094206873 https://doi.org/10.1109/tit.2003.815798 https://doi.org/10.1109/tit.2003.815798 http://dx.doi.org/10.1016/s0012-365x(03)00126-2 http://dx.doi.org/10.1016/s0012-365x(03)00126-2 http://www.codetables.de http://dx.doi.org/10.1007/bf00124211 http://dx.doi.org/10.1007/bf00124211 https://doi.org/10.1109/18.605613 https://doi.org/10.1109/18.605613 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.641542 https://doi.org/10.1109/18.641542 introduction quasi-cyclic codes good qc codes the new codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.369864 j. algebra comb. discrete appl. 5(1) • 45–49 received: 11 february 2017 accepted: 1 september 2017 journal of algebra combinatorics discrete structures and applications no macwilliams duality for codes over nonabelian groups research article m. ryan julian jr. abstract: dougherty, kim, and solé [3] have asked whether there is a duality theory and a macwilliams formula for codes over nonabelian groups, or more generally, whether there is any subclass of nonabelian groups which have such a duality theory. we answer this in the negative by showing that there does not exist a nonabelian group g with a duality theory on the subgroups of gn for all n. 2010 msc: 94b60, 20e15 keywords: dual code, subgroup lattice, macwilliams identity, iwasawa group 1. introduction for codes over finite fields, c ⊂ fnq , the usual inner product for vectors over fnq produces a well established duality theory between c and c⊥ = {x ∈ fnq |〈x,c〉 = 0 ∀c ∈ c}. furthermore, the weight enumerator polynomials for a pair of dual codes are related by the famous macwilliams identity, which for codes over f2 takes the form w(c⊥; x,y) = 1|c|w(c; y−x,y +x). there are a number of generalizations of this result that cover different types of weight enumerators as well as codes over different algebraic objects, including abelian groups. dougherty, kim, and solé [3] asked whether it is possible to extend these results to nonabelian groups. in particular, they asked “is there a subclass of nonabelian groups for which a duality and a macwilliams formula exist?” we will answer this question under the assumption that a code over a group g is defined to be a subgroup of gn, in analogy to the usual definition of codes over abelian groups. our approach will refrain from choosing a particular definition of a dual code, and instead we will determine whether any suitable choice exists that can provide a duality satisfying our definition 2.1. it remains open whether there is some other definition of a code over a group g that might better generalize the existing theory of codes over abelian groups, allowing duality to rely on something other than just the subgroup structure. when they posed the question, dougherty, kim, and solé [3] had already identified one particular difficulty in finding a duality theory for nonabelian groups. they observed that the subgroups of the m. ryan julian jr.; university of wisconsin madison, united states (email: mrjulian@math.wisc.edu). 45 http://orcid.org/0000-0002-6117-1415 m. r. julian / j. algebra comb. discrete appl. 5(1) (2018) 45–49 quaternion group {±1,±i,±j,±k} do not form a self-dual subgroup lattice. this group has three subgroups of order 4, namely {±1,±i}, {±1,±j}, and {±1,±k}, but only one subgroup of order 2, {±1}. so if we expect codes over g and their duals to be subgroups of gn with the property that |c||c⊥| = |g|n, then we would need to restrict our attention to some subclass of nonabelian groups that does not include the quaternions. pursuing this line of thought, instead of looking for an inner product that might produce a duality theory, as is done for finite fields and abelian groups, we focus on the structures of subgroup lattices. if a nonabelian group g is to have a duality theory for the subgroups of gn, it would be necessary for the subgroup lattice of gn to be self-dual. fortunately, subgroup lattices with duals have already been classified. in the 1950’s, suzuki determined which finite solvable groups have duals [6], and zacher was able to prove ten years later that all finite groups with duals are solvable [7]. we will apply this classification to show that if g is a finite nonabelian group with a self-dual subgroup lattice, then g×g will not have a self-dual subgroup lattice. as a corollary, there cannot exist any nonabelian finite group g with the property that gn has a self-dual subgroup lattice for all n. while we can define a code over a nonabelian group g to be a subgroup of gn, there is no subclass of nonabelian groups that will support a duality theory with this definition. 2. self-dual subgroup lattices since our approach to this problem takes us into the theory of subgroup lattices, we will need to begin with a quick tour through the relevant terminology. definition 2.1. let l(g) denote the subgroup lattice of a group g. we say g has a dual group ḡ if there exists a bijective map δ : l(g) → l(ḡ) such that for all h,k ∈ l(g), h ≤ k if and only if δ(k) ≤ δ(h). we are interested in groups that are self-dual, i.e. groups g with a dual ḡ ∼= g. observe that if gn was self-dual, then by defining a code over g to be a subgroup of gn, the duality map for gn would take a code c to a dual code c⊥. this is the same arrangement that works to produce dual codes over finite fields and abelian groups, but in those cases the duality map can be produced by appealing to an inner product. without an inner product structure to fall back on for nonabelian groups, we instead depend on studying the subgroup lattices. we will make use of the classification of finite groups with duals given in chapter 8 of schmidt’s book on subgroup lattices [5], but the classification requires a bit of specialized terminology. definition 2.2. a finite group g is called a p-group if it is: 1. an elementary abelian group of prime power order, or 2. a group which can be decomposed as g = spsq, where sp is a p-sylow subgroup which is an abelian p-group, sq is a cyclic q-sylow subgroup of order q, sq =< b >, and for any element a of sp we have bab−1 = ar, r 6≡ 1, rq ≡ 1 (mod p). definition 2.3. a modular lattice is a lattice that satisfies the condition x ≤ b implies x ∨ (a ∧ b) = (x ∨ a) ∧ b for any lattice element a, where ∨ and ∧ are the join and meet operations on the lattice. groups with modular subgroup lattices are called iwasawa groups. since p-groups with modular subgroup lattices have already been classified by iwasawa [4], we will not actually need to work much with this definition, but we include it for completeness. definition 2.4. a hamiltonian group is a nonabelian group g such that every subgroup of g is normal. 46 http://orcid.org/0000-0002-6117-1415 m. r. julian / j. algebra comb. discrete appl. 5(1) (2018) 45–49 the smallest example of a hamiltonian group is the quaternion group of order 8 that we met earlier, and, in fact, dedekind showed that every hamiltonian group is the direct product of the quaternion group with some other abelian group [2]. we have already seen that the subgroup lattice of the quaternions is not self-dual. so it is perhaps unsurprising that in our attempts to understand self-dual subgroup lattices we will be applying a theorem [5] that instructs us to focus our attention on non-hamiltonian groups. theorem 2.5. a finite group g has a dual if and only if g is a direct product of finite coprime groups gλ such that each gλ is a p-group or a non-hamiltonian p-group with a modular subgroup lattice. in particular, if g is self-dual, it must satisfy this condition. to use this classification to understand codes over g (e.g. subgroups of gn), we must understand the direct products of p-groups and of nonhamiltonian p-groups with modular subgroup lattices with themselves. we will examine these first before tackling the main theorem. 3. direct products lemma 3.1. if g is a nonabelian p-group, then g×g is not a p-group. proof. from the definition above, a nonabelian p-group must be the semidirect product of an elementary abelian group of order pk and a cyclic group of prime order q (along with some additional structure). in particular, |g| = pkq, so |g × g| = p2kq2. then g × g cannot be a p-group, since the order of nonabelian p-groups must be of the form p`q for primes p and q. to handle the case where g is a non-hamiltonian p-group with a modular subgroup lattice, we will apply a theorem of iwasawa [4]. theorem 3.2. if g is a non-hamiltonian nonabelian p-group with a modular subgroup lattice, then g contains an abelian normal subgroup n such that g/n =< q > is cyclic and for all n ∈ n, q−1nq = n1+p s , where s ≥ 1 (s ≥ 2 if p = 2). in fact, to show that this property cannot hold for both g and g × g, we actually only need an abelian normal subgroup with a cyclic quotient. the further structure described in iwasawa’s theorem is unnecessary for the following lemma. lemma 3.3. if g is a nonabelian group, then g and g×g cannot both be non-hamiltonian p-groups with modular subgroup lattices. proof. suppose that both g and g×g are non-hamiltonian p-groups with modular subgroup lattices. let n be an abelian normal subgroup of g×g such that (g×g)/n is cyclic, and let p1 and p2 be the standard projections g×g → g. observe that if p1(n) 6= g and p2(n) 6= g, then (g×g)/n cannot be cyclic, since (g × g)/n would have a non-cyclic quotient. in particular, since n ⊆ p1(n) × p2(n) and p1(n) and p2(n) are normal in g, we have ((g×g)/n) / ((p1(n) ×p2(n))/n) ∼= (g×g)/(p1(n) ×p2(n)) ∼= (g/p1(n)) × (g/p2(n)), and since p1(n) 6= g and p2(n) 6= g, we claim that (g/p1(n)) × (g/p2(n)) is not cyclic. to confirm this claim, we first observe that g/p1(n) and g/p2(n) are p-groups of orders pa and pb, respectively for some integers a and b. since p1(n) 6= g and p2(n) 6= g, we know that a,b > 0. then (g/p1(n))×(g/p2(n)) is a group of order papb = pa+b. suppose that g1 ∈ g/p1(n) and g2 ∈ g/p2(n) with |g1| = pc and |g2| = pd. then (g1,g2) ∈ (g/p1(n)) × (g/p2(n)) and |(g1,g2)| = lcm(|g1|, |g2|) = lcm(pc,pd) = pmax(c,d). since c ≤ a, d ≤ b, and a,b > 0, we have that pmax(c,d) < pa+b, so (g1,g2) cannot 47 http://orcid.org/0000-0002-6117-1415 m. r. julian / j. algebra comb. discrete appl. 5(1) (2018) 45–49 generate (g/p1(n)) × (g/p2(n)). since no element of (g/p1(n)) × (g/p2(n)) can generate the entire group, (g/p1(n)) × (g/p2(n)) cannot be cyclic. so, for some i ∈ {1, 2}, pi(n) = g. but this contradicts the existence of such an abelian normal subgroup n, since n was chosen to be abelian while g is nonabelian and abelian groups cannot project onto nonabelian groups. there is already a result in the literature [1] that states the following: a nonabelian non-hamiltonian p-group p = p1 ×p2 is an iwasawa group if and only if p1 and p2 are iwasawa and, for i = 1 or i = 2, pi is abelian such that exp(pi) ≤ ps and s is the integer that comes from the nonabelian factor pj for j 6= i. our second lemma would be a direct corollary of this result. unfortunately, the published proof is incorrect, since after establishing elements of p such that xk1a ` 1a ` 2 = x1a 1+ps1 1 a2, the author claims without any further justification that this implies that either ` = 1 or ` = 1 + ps1. without this step, there remains a gap in the published proof, so our proof above can be considered a partial correction of that result. 4. the main theorem with these lemmas in hand, we are now ready to prove the main theorem. theorem 4.1. if g is a finite nonabelian group with a self-dual subgroup lattice, then g×g does not have a self-dual subgroup lattice. proof. first, if g is a finite nonabelian group with a self-dual subgroup lattice, then by theorem (2.5) g can be expressed as g = ⊕ gλ, where the gλ are coprime and each gλ is either a p-group or a non-hamiltonian p-group with a modular subgroup lattice. then g×g = ⊕ (gλ ×gλ), and since the gλ×gλ are still coprime, it suffices to check whether it is possible for each gλ×gλ to still be a p-group or a non-hamiltonian p-group with a modular subgroup lattice. by theorem (3.1), we know that this will not work if gλ is a p-group, and by theorem (3.3), we know that this will also not work if gλ is a non-hamiltonian p-group with a modular subgroup lattice. thus we can conclude that if g is a finite nonabelian group with a self-dual subgroup lattice, then g × g cannot also have a self-dual subgroup lattice. 5. conclusion this result shows that there does not exist any finite nonabelian group g so that gn has a duality theory for all n. so if we define a code over a nonabelian group g to be a subgroup of gn, then our coding theory over nonabelian groups cannot have a macwilliams-type duality theory, and there is no subclass of nonabelian groups where a duality theory could be recovered. some variations of dougherty, kim, and solé’s question remain open. for example, one could change our definition so that codes over g are not restricted to be subgroups of gn, and ask whether some other collection of codes could produce meaningful self-dual lattices. another possibility would be to relax our demands for the duality map. although our work goes a long way towards describing the boundary between classes of codes with and without duality, the most general form of dougherty, kim, and solé’s question, “find the largest class of algebraic structures a for which a duality and macwilliams relations hold” remains a target for future research. 48 http://orcid.org/0000-0002-6117-1415 m. r. julian / j. algebra comb. discrete appl. 5(1) (2018) 45–49 references [1] j. chifman, note on direct products of certain classes of finite groups, commun. algebra 37(5) (2009) 1831–1842. [2] r. dedekind, ueber gruppen, deren sämmtliche theiler normaltheiler sind, math. ann. 48(4) (1897) 548–561. [3] s. dougherty, j.-l. kim, p. solé, open problems in coding theory, contemp. math. 634 (2015) 79–99. [4] k. iwasawa, über die endlichen gruppen und die verbände ihrer untergruppen, j. fac. sci. imp. univ. tokyo. sect. i. 4 (1941) 171–199. [5] r. schmidt, subgroup lattices of groups, walter de gruyter, berlin, 1994. [6] m. suzuki, on the lattice of subgroups of finite groups, trans. amer. math. soc. 70(2) (1951) 345–371. [7] g. zacher, caratterizzazione dei gruppi immagini omomorfe duali di un gruppo finito, rend. sem. mat. univ. padova 31 (1961) 412–422. 49 http://orcid.org/0000-0002-6117-1415 http://dx.doi.org/10.1080/00927870802070322 http://dx.doi.org/10.1080/00927870802070322 http://doi.org/10.1007/bf01447922 http://doi.org/10.1007/bf01447922 http://www.ams.org/mathscinet-getitem?mr=3307391 http://www.ams.org/mathscinet-getitem?mr=5721 http://www.ams.org/mathscinet-getitem?mr=5721 http://www.ams.org/mathscinet-getitem?mr=1292462 http://dx.doi.org/10.2307/1990375 http://www.ams.org/mathscinet-getitem?mr=140566 http://www.ams.org/mathscinet-getitem?mr=140566 introduction self-dual subgroup lattices direct products the main theorem conclusion references issn 2148-838x j. algebra comb. discrete appl. 10(1) • 61–71 received: 9 april 2022 accepted: 9 july 2022 journal of algebra combinatorics discrete structures and applications hamiltonicity after reversing the directed edges at a vertex of a cartesian product research article dave witte morris abstract: let ~cm and ~cn be directed cycles of length m and n, with m, n ≥ 3, and let p( ~cm � ~cn) be the digraph that is obtained from the cartesian product ~cm� ~cn by choosing a vertex v, and reversing the orientation of all four directed edges that are incident with v. (this operation is called “pushing” at the vertex v.) by applying a special case of unpublished work of s. x. wu, we find elementary numbertheoretic necessary and sufficient conditions for the existence of a hamiltonian cycle in p( ~cm � ~cn). a consequence is that if p( ~cm � ~cn) is hamiltonian, then gcd(m, n) = 1, which implies that ~cm � ~cn is not hamiltonian. this final conclusion verifies a conjecture of j. b. klerlein and e. c. carr. 2010 msc: 05c45, 05c20, 05c76 keywords: hamiltonian cycle, cartesian product, directed cycle, pushing at a vertex, reverse edges 1. preliminaries notation 1.1. for m, n, i, j ∈ z (with m 6= 0 and n 6= 0), we define the integer am,n(i, j) by the following conditions: am,n(i, j) ≡ i (mod m), am,n(i, j) ≡ j (mod n), and 1 ≤ am,n(i, j) ≤ lcm(m, n). the integer is unique, if it exists. by the chinese remainder theorem, am,n(i, j) does exist whenever gcd(m, n) = 1 (or, more generally, whenever i ≡ j (mod gcd(m, n))). notation 1.2. we use ~cm to denote a directed cycle of length m. definition 1.3 ([6, p. 88]). if x is a digraph that is vertex-transitive, then the digraph p(x) is constructed from x by choosing a vertex v, and reversing the orientation of each directed edge that is incident with v. (this operation is called “pushing” at the vertex v [6, 7].) since x is vertex-transitive, the isomorphism class of the resulting digraph is independent of the choice of v. dave witte morris; department of mathematics and computer science, university of lethbridge, lethbridge, alberta, t1k 6r4, canada (dmorris@deductivepress.ca). 61 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 definition 1.4 ([4, pp. 35 and 421]). recall that the cartesian product x �y of two digraphs x and y is the digraph whose vertex set is v (x)×v (y ), with a directed edge from (x1, y1) to (x2, y2) if and only if either • x1 = x2, and there is a directed edge from y1 to y2 in y , or • y1 = y2, and there is a directed edge from x1 to x2 in x. 2. statement of the main result this note explains that a special case of unpublished work of s. x. wu [11] (or slightly later published work of s. j. curran et al. [1]) provides the following elementary number-theoretic necessary and sufficient conditions for p(~cm � ~cn) to be hamiltonian. proposition 2.1. let ~cm and ~cn be directed cycles of length ≥ 3. the digraph p(~cm � ~cn) has a hamiltonian cycle if and only if (1) gcd(m, n) = 1, (2) min{am,n(0,−2), am,n(−2, 0)} < min { am,n(0,−1), am,n(−1, 0) } , and (3) gcd ( am,n(0,−4) m , am,n(−4, 0) n ) = 1. if gcd(m, n) = 1, then it is well known (and easy to see) that ~cm � ~cn is not hamiltonian [3, thm. 28.1, p. 510]. therefore, the proposition has the following consequence, which was conjectured by j. b. klerlein and e. c. carr [6, p. 94]: corollary 2.2. if ~cm � ~cn is hamiltonian (and m, n ≥ 3), then p(~cm � ~cn) is not hamiltonian. remarks 2.3. (1) proposition 2.1 requires m and n to be at least 3. the remaining case was settled by j. b. klerlein and e. c. carr [6, thm. 6]: p(~c2 � ~cn) is hamiltonian if and only if n ∈ {2, 3}. (since ~c2 � ~c2 and p(~c2 � ~c2) are hamiltonian, it is clear that corollary 2.2 would be false if it allowed the case where m = n = 2.) (2) j. b. klerlein and e. c. carr also determined whether p(cm � cn) is hamiltonian in certain other special cases. in particular, they [6, thm. 7] proved a much more concrete form of the case m = 3 of proposition 2.1: p(c3 � cn) is hamiltonian if and only if n ≡ 2 (mod 3). (3) the conditions in proposition 2.1 are so efficient that they can be checked by a computer in seconds, even if m and n have 100,000 digits. this can be verified by using the sample code in fig. 1. 3. proof of the main result we assume that the vertices of ~cm � ~cn are identified in the natural way with the elements of the abelian group zm ×zn. notation 3.1 (cf. [11, p. 2]). (1) for a, b ∈ z+, the rectangle ra,b is the subset {0, 1, . . . , a−1}×{0, 1, . . . , b−1} of v (~cm � ~cn). 62 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 def is_pcmxcn_hamiltonian(m, n): r"""return ‘true‘ if ‘p(c_m x c_n)‘ has a hamiltonian cycle (otherwise return ‘false‘).""" m = integer(m) n = integer(n) if min(m, n) < 3: raise notimplementederror("m and n must be at least 3") if gcd(m, n) != 1: return false def a(i, j): return crt(i, j, m, n) if min( a(0, -2), a(-2, 0) ) > min( a(0, -1), a(-1, 0) ): return false return gcd( a(0, -4) // m, a(-4, 0) // n ) == 1 figure 1. a sagemath program that implements proposition 2.1. (this program can be run online at https://cocalc.com.) for example, is_pcmxcn_hamiltonian(3,5) returns true because the digraph p( ~c3 � ~c5) is hamiltonian (see remarks 2.3(2)). (2) we use (~cm � ~cn) r ra,b to denote the digraph that is obtained from ~cm � ~cn by deleting all of the vertices in ra,b (and also deleting all of the directed edges that are incident with this set). the following simple observation is crucial: lemma 3.2. for m, n ≥ 3, the digraph p(~cm � ~cn) is hamiltonian if and only if (~cm � ~cn) r r2,2 is hamiltonian. proof. (⇒) figure 2(a) shows a part of p(~cm � ~cn) with the pushed vertex v at its centre. (all nine vertices in the figure are distinct, because m, n ≥ 3.) note that the vertices v + (1, 0) and v + (0, 1) have only one in-edge, and the vertices v − (1, 0) and v − (0, 1) have only one out-edge. this implies that the hamiltonian cycle must traverse these four directed edges (which are dark in the figure). the out-edges of v go to v − (1, 0) and v − (0, 1). by symmetry (i.e., by interchanging m and n if necessary), we may assume without loss of generality that the hamiltonian cycle uses the (white) directed edge from v to v − (1, 0). then the hamiltonian cycle cannot use the edge from v + (0, 1) to v (because that would create a 4-cycle), so it must use the other in-edge of v, which is the (grey) directed edge from v + (1, 0) to v. also, the hamiltonian cycle cannot use the (striped) directed edge from v−(1, 1) to v − (1, 0) (because it already uses a different in-edge of v − (1, 0)), so it must use the other out-edge of v − (1, 1), which goes to v − (0, 1) (and is grey in the figure). now, assuming without loss of generality that r2,2 consists of the four white vertices in the bottom right of the picture, we can construct a hamiltonian cycle in (~cm � ~cn) r r2,2 by deleting the edges in the walk v − (1, 1), v − (0, 1), v + (1,−1), v + (1, 0), v, v − (1, 0), and inserting the (striped) directed edge from v − (1, 1) to v − (1, 0). (⇐) figure 2(b) shows the same portion of p(~cm � ~cn), centred at the pushed vertex v, with the (white) vertices of the rectangle r2,2 in the bottom right corner again. note that p(~cm � ~cn) r r2,2 = (~cm � ~cn) r r2,2, so, by assumption, there is a hamiltonian cycle in p(~cm � ~cn) r r2,2. it must use all of the directed edges that are dark or striped in this picture, because v − (1, 1) and v − (1, 0) have only one out-edge 63 https://cocalc.com dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 v (a) v (b) figure 2. two drawings centred at the pushed vertex v. that has not been deleted, and the vertices v + (0, 1) and v + (1, 1) have only one in-edge that has not been deleted. then we can construct a hamiltonian cycle in p(~cm � ~cn) by reversing the process in the previous part of the proof: delete the (striped) edge from v − (1, 1) to v − (1, 0), and replace it with the walk v − (1, 1), v − (0, 1), v + (1,−1), v + (1, 0), v, v − (1, 0), whose edges are grey in the picture. hence, the existence of a hamiltonian cycle in p(~cm � ~cn) is characterized by the case a = b = 2 of the following result: theorem 3.3 (s. x. wu [11, cor. 11]). the digraph (~cm � ~cn) r ra,b is hamiltonian if and only if either the following conditions are satisfied, or they are satisfied after interchanging m and n, and also interchanging a and b: am,n(−a, 0) exists, am,n(−a, 0) = min { am,n(−a,−b), am,n(−a,−b + 1), am,n(−a,−b + 2), . . . , am,n(−a, 0), am,n(−a,−b), am,n(−a + 1,−b), am,n(−a + 2,−b), . . . , am,n(0,−b) } (where any terms in the minimum that do not exist are simply ignored), and gcd ( n− b− b ⌊ am,n(−a, 0) m ⌋ , b am,n(−a, 0) n ) = 1. remarks 3.4. (1) s. x. wu showed that if (~cm � ~cn) r ra,b has a hamiltonian cycle, then it is unique (see lemma 5.4 below). it follows that if p(~cm � ~cn) is hamiltonian (and m, n ≥ 3), then p(~cm � ~cn) has exactly two hamiltonian cycles. one hamiltonian cycle will be constructed in the proof of lemma 3.2, and the other is constructed by interchanging ~cm and ~cn in this proof (or, in other words, by reflecting fig. 2(b) across the line y = x). (2) a very different formulation of the conditions in the statement of theorem 3.3 was proved by s. j. curran et al. [1, thm. 4.3], as a special case of a more general version [1, thm. 4.2] that applies to all 2-generated cayley digraphs on finite abelian groups, not only cartesian products of directed cycles. we need only the special case where a = b = 2, which can be restated as follows (see section 4 or section 5): 64 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 corollary 3.5. for m, n ≥ 3, the digraph (~cm � ~cn) r r2,2 is hamiltonian if and only if: (1) gcd(m, n) = 1, (2) min{am,n(0,−2), am,n(−2, 0)} < min { am,n(0,−1), am,n(−1, 0) } , and (3) gcd ( am,n(0,−4) m , am,n(−4, 0) n ) = 1. proof of proposition 2.1. combine lemma 3.2 and corollary 3.5. 4. proof of the corollary 3.5 from the theorem 3.3 we now explain how to derive corollary 3.5 from theorem 3.3. (alternatively, the corollary could also be derived from the work of s. j. curran et al. [1, thm. 4.3], or see section 5 for a direct proof that does not assume familiarity with [1] or [11].) actually, we prove only (⇒) in this section, but the argument is reversible. the conclusions of the corollary are symmetric under interchanging m and n, so we may assume that the conditions in the statement of theorem 3.3 hold. for a = b = 2, this means: am,n(−2, 0) = min { am,n(−2,−2), am,n(−2,−1), am,n(−2, 0), am,n(−2,−2), am,n(−1,−2), am,n(0,−2) } (1) and gcd ( n−2−2 ⌊ am,n(−2, 0) m ⌋ , 2 am,n(−2, 0) n ) = 1. (2) (1) note that n must be odd. (otherwise, both terms in the gcd of (2) are even, which contradicts the fact that the gcd is 1.) also, since am,n(−2, 0) exists, we know that gcd(m, n) ∈ {1, 2}. from the fact that n is odd, we conclude that gcd(m, n) = 1. (2) since am,n(i−1, j −1) = am,n(i, j)−1 (unless i ≡ j ≡ 0 (mod lcm(m, n))), (3) we have am,n(−2,−1) = am,n(−1, 0)−1 and am,n(−1,−2) = am,n(0,−1)−1. (4) therefore, we see from (1) that (2) holds. (for reversing the argument, note that am,n(−2,−2) = mn−2, so it is always true that am,n(−2, 0) < am,n(−2,−2), and also note that the inequality am,n(−2, 0) < am,n(0,−2) can be achieved by interchanging m and n if it does not already hold.) (3) note that am,n(−2, 0) + am,n(0,−2) = mn−2, because the left-hand side is congruent to −2 modulo both m and n (and we know from (1) that m and n are relatively prime). since (by (1)) we have am,n(−2, 0) < am,n(0,−2), this implies that am,n(−2, 0) < mn 2 −1, 65 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 so am,n(−4, 0) = 2 am,n(−2, 0). therefore, we have 2 am,n(−2, 0) n = am,n(−4, 0) n . hence, in order to establish that (2) is the same as conclusion (3) of the corollary, all that remains is to show n−2−2 ⌊ am,n(−2, 0) m ⌋ = am,n(0,−4) m . since am,n(−2, 0) + 2 is a multiple of m, we see that the left-hand side is n−2−2 ( am,n(−2, 0) + 2 m −1 ) = n−2 am,n(−2, 0) + 2 m = mn−2 am,n(−2, 0)−4 m . also note that mn−2 am,n(−2, 0)−4 > mn−2 (mn 2 −1 ) −4 = −2. it is therefore easy to see that mn−2 am,n(−2, 0)−4 = am,n(0,−4) (because the two sides are congruent modulo both m and n), which completes the proof. 5. direct proof of the corollary 3.5 for completeness (since [11] was never published), and because some readers may find it instructive, we sketch a direct proof of corollary 3.5 that is based on s. x. wu’s proof [11, §4] of theorem 3.3. (the same ideas apply to the general case of theorem 3.3, but the details are more complicated.) we begin with two definitions and some lemmas. definition 5.1 ([9]). a spanning subdigraph h of a digraph x is a vertex-disjoint cycle cover if h is a vertex-disjoint union of directed cycles. (equivalently, the invalence and outvalence of every vertex of h is 1.) definition 5.2 (cf. [5, p. 82]). assume h is a vertex-disjoint cycle cover of (~cm � ~cn) r r2,2. let v be a vertex of h, and let s ∈ {(1, 0), (0, 1)}. we say that v travels by s if the out-edge of v is the directed edge from v to v + s. the arguments in this section utilize basic properties of the “arc-forcing subgroup” 〈(1,−1)〉 [10, §2.3] that were discovered by r. a. rankin [8, lem. 1] and d. housman [5, pp. 82–83]. the specific facts that we need are recorded in the following lemma. lemma 5.3 (cf. [11, p. 2] or [1, rem. 2.2]). let h be a vertex-disjoint cycle cover of (~cm � ~cn) r r2,2. for every vertex v of h: (1) if v travels by (1, 0), and v + (1,−1) /∈ r2,2, then v + (1,−1) also travels by (1, 0). (2) if v travels by (1, 0), and neither v − (1,−1) nor v + (0, 1) is in r2,2, then v − (1,−1) also travels by (1, 0). 66 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 (3) if v travels by (0, 1), and v − (1,−1) /∈ r2,2, then v − (1,−1) also travels by (0, 1). (4) if v travels by (0, 1), and neither v + (1,−1) nor v + (1, 0) is in r2,2, then v + (1,−1) also travels by (0, 1). proof. (1, 3) by symmetry, it suffices to prove (1). for convenience, let w = v + (1, 0). since h is a vertex-disjoint cycle cover, we know that h cannot have both a directed edge from v to w and a directed edge from v + (1,−1) to w (because the invalence of w cannot be greater than 1). hence, v + (1,−1) cannot travel by (0, 1). however, v+(1,−1) is a vertex of h (because, by assumption, it is not in r2,2), so it must have some out-edge. we conclude that it travels by (1, 0), since that is the only other possibility. (2, 4) by symmetry, it suffices to prove (2). for convenience, let w = v + (0, 1). since v travels by (1, 0), it does not travel by (0, 1), so h does not contain the directed edge from v to w. since w must have an in-edge, this implies that h has the directed edge from v − (1,−1) to w (since that is the only other possibility). this means that v − (1,−1) travels by (1, 0). lemma 5.4 (wu [11, lem. 1] or [1, lem. 2.3]). for m, n ≥ 2, the digraph (~cm � ~cn)rr2,2 has no more than one vertex-disjoint cycle cover. proof. let h be a vertex-disjoint cycle cover. we claim that every coset of the subgroup 〈(1,−1)〉 contains at least one element of r2,2. suppose not, so we may let v + 〈(1,−1)〉 be a coset that does not intersect the set r2,2. by symmetry, we may assume, without loss of generality, that v travels by (1, 0). then, by repeated application of lemma 5.3(1), we conclude that every element of this coset travels by (1, 0). since the terminal endpoint of every directed edge of h must be a vertex of h, this implies that every element of the coset v + (1, 0) + 〈(1,−1)〉 is a vertex of h. in other words, this coset does not intersect the set r2,2. by repeating this argument, we conclude, for every k ∈ z+, that the coset v + (k, 0) + 〈(1,−1)〉 does not intersect r2,2. however, the union of these cosets is all of ~cm � ~cn. we conclude that r2,2 has no elements, which is a contradiction. the claim implies that every vertex of h is contained in set of the form iv,k = {v, v + (1,−1), v + 2(1,−1), . . . , v + k(1,−1)}, such that (a) either v − (1,−1) ∈ r2,2 or v + (0, 1) ∈ r2,2, (b) either v + (k + 1) (1,−1) ∈ r2,2 or v + (k + 1) + (1, 0) ∈ r2,2, but (c) no element of iv,k is in r2,2, and (d) for 1 ≤ j < k, neither v + j(1,−1) + (1, 0) nor v + j(1,−1) + (1, 0) is in r2,2. from (c) and (d) (combined with lemma 5.3) and induction, we see that either every element of iv,k travels by (1, 0), or every element of iv,k travels by (0, 1). to complete the proof, we will show that there is no choice about whether these vertices travel by (1, 0) or by (1, 0): it is uniquely determined for each v. first of all, if v + (0, 1) ∈ r2,2, then v cannot travel by (0, 1), so it must travel by (1, 0); hence, every vertex in iv,k must travel by (0, 1). on the other hand, if v + (0, 1) /∈ r2,2, then (by (a)) we must have v − (1,−1) ∈ r2,2, so the vertex v − (1,−1) is not in h, and therefore cannot travel by (1, 0). since v + (0, 1) must have an in-edge, we conclude that v travels by (0, 1); hence, every vertex in iv,k must travel by (0, 1). 67 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 lemma 5.5 (cf. [11, thm. 10]). for m, n ≥ 3, the digraph (~cm � ~cn) r r2,2 has a vertex-disjoint cycle cover if and only if am,n(−2, 0) and am,n(0,−2) exist, and min { am,n(−2, 0), am,n(0,−2) } < min { am,n(0,−1), am,n(−1, 0) } . furthermore, if the digraph does have a vertex-disjoint cycle cover, then the number of vertices that travel by (1, 0) in this subdigraph is exactly twice the left-hand side of the above inequality. proof. (⇒) if a vertex u travels by (1, 0), and we let r(u) = min{k ∈ z+ | u + k(1,−1) ∈ r2,2 }, (5) then it follows from lemma 5.3(1) (and induction on k) that u+k(1,−1) travels by (1, 0) for 0 ≤ k < r(u). this implies u + k(1,−1) + (1, 0) /∈ r2,2 for 0 ≤ k < r(u). in particular, we can apply this with u = (1,−1), since this vertex travels by (1, 0) because u+(0, 1) = (1, 0) ∈ r2,2. we have r2,2 = { (0, 1), (1, 1), (0, 0), (1, 0) } = { (1,−1) + (−1, 2), (1,−1) + (0, 2), (1,−1) + (−1, 1), (1,−1) + (0, 1) } (6) = { u + am,n(−1,−2) · (1,−1), u + am,n(0,−2) · (1,−1), u + am,n(−1,−1) · (1,−1), u + am,n(0,−1) · (1,−1) } , so r(u) = min{am,n(−1,−2), am,n(0,−2), am,n(−1,−1), am,n(0,−1)}. (7) also, since u + am,n(−2,−1) · (1,−1) + (1, 0) = (1,−1) + (−2, 1) + (1, 0) = (0, 0) ∈ r2,2 and u + am,n(−2,−2) · (1,−1) + (1, 0) = (1,−1) + (−2, 2) + (1, 0) = (0, 1) ∈ r2,2, we know that u + am,n(−2,−1) · (1,−1) and u + am,n(−2,−2) · (1,−1) do not travel by (1, 0), so r(u) ≤ min { am,n(−2,−1), am,n(−2,−2)}. (8) since am,n(−2,−2) = lcm(m, n)−2 is very large, it is almost entirely irrelevant in (8), but it does imply that am,n(−1,−1) = lcm(m, n) − 1 is not the only term that exists in the right-hand side of (7). this implies that gcd(m, n) ∈{1, 2}, so am,n(−2, 0) and am,n(0,−2) exist. we may now assume that am,n(0,−1) and (equivalently) am,n(−1, 0) exist, for otherwise the inequality in the statement of the lemma is vacuously true. we may also assume (by interchanging m and n if necessary) that min { am,n(0,−1), am,n(−1, 0) } = am,n(−1, 0). thus, we see from (4) that (8) is equivalent to the condition that am,n(0,−2) < am,n(−1, 0). this establishes the inequality in the statement of the lemma. (⇐) let u = (1,−1) and assume, without loss of generality, that am,n(−1, 0) < am,n(0,−1). 68 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 since am,n(−1, 0) + am,n(0,−1) = am,n(−1,−1) = lcm(m, n)−1, this implies that am,n(−1, 0) < lcm(m, n)/2, so am,n(−2, 0) = 2 am,n(−1, 0) > am,n(−1, 0). so we see from the assumption of this direction of the proof that am,n(0,−2) < am,n(−1, 0) = min { am,n(−1, 0), am,n(0,−1), am,n(−2, 0) } . (9) we then conclude from (6) (and the definition of r(u) in (5)) that r(u) = am,n(0,−2) (10) and (using (3)) that u + k(1,−1) + (1, 0) /∈ r2,2 for 0 ≤ k < r(u). let u′ = u− (1, 0) = (0,−1). we claim that r(u′) = r(u). (11) to see this, first note that u′ + r(u) · (1,−1) = (0,−1) + am,n(0,−2) · (1,−1) = (0,−1) + (0, 2) = (0, 1) ∈ r2,2, so r(u′) ≤ r(u). on the other hand, we have r2,2 = { (0, 1), (1, 1), (0, 0), (1, 0) } = { (0,−1) + (0, 2), (0,−1) + (1, 2), (0,−1) + (0, 1), (0,−1) + (1, 1) } = { u′ + am,n(0,−2) · (1,−1), u′ + am,n(1,−2) · (1,−1), u′ + am,n(0,−1) · (1,−1), u′ + am,n(1,−1) · (1,−1) } , so r(u′) = min { am,n(0,−2), am,n(1,−2), am,n(0,−1), am,n(1,−1) } = min { am,n(0,−2), am,n(0,−3) + 1, am,n(0,−1), am,n(0,−2) + 1 } from (9), we know that the only value in this minimum that could possibly be smaller than r(u) = am,n(0,−2) is am,n(0,−3) + 1. however, we have am,n(0,−2) + am,n(0,−1) < am,n(−1, 0) + am,n(0,−1) = lcm(m, n)−1 < lcm(m, n), so am,n(0,−3) = am,n(0,−2) + am,n(0,−1) > am,n(0,−2) = r(u). this completes the proof of the claim. note that, for 0 ≤ k < r(u), we have u′ + k(1,−1) + (1, 0) = u + k(1,−1) /∈ r2,2. also note that u and u′ are the only vertices in (~cm � ~cn) r r2,2 that cannot travel by (0, 1). therefore, we can construct a spanning subdigraph h in which a vertex travels by (1, 0) if it is in the set{ v + k(1,−1) ∣∣∣∣ v ∈{u, u′},0 ≤ k < r(u) } 69 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 and travels by (0, 1) otherwise. by construction (and (11)), if a vertex v travels by (1, 0), and v+(1,−1) /∈ r2,2, then v + (1,−1) also travels by (1, 0). hence, no vertex has invalence 2, so the in-degree (and outdegree) of every vertex of h is 1, which means that h is the desired vertex-disjoint cycle cover. furthermore, we know from the construction of h (together with (9) and (10)) that the number of vertices that travel by (1, 0) is as specified in the final sentence of the statement of the lemma. since h is the only vertex-disjoint cycle cover (see lemma 5.4), this completes the proof. direct proof of corollary 3.5. lemma 5.5 provides necessary conditions for the existence of a hamiltonian cycle (in particular, 2.1(2) must hold), but they is not sufficient, because we need an additional condition that determines whether the cycle cover is a single cycle, rather than a union of several cycles. this condition is provided by the “knot class,” which is a topological concept that was introduced into the study of cartesian products of directed cycles by s. j. curran [2, §4]. namely, let h be the vertex-disjoint cycle cover, and suppose the number of vertices of h that travel by (1, 0) is x, and the number that travel by (0, 1) is y. then x/m and y/n are integers, and the knot class of h is defined to be the ordered pair (x/m, y/n) [2, rem. 4.5]. the theory [2, prop. 4.12(a)] tells us that h consists of a single cycle if and only if gcd(x/m, y/n) = 1. (12) we now use (12) to show that 2.1(1) is a necessary condition for h to be a hamiltonian cycle. the key is to notice that if gcd(m, n) 6= 1, then since lemma 5.5 tells us that am,n(−2, 0) exists, we must have gcd(m, n) = 2, so m and n are even. however, if we assume, without loss of generality, that am,n(0,−2) < am,n(−2, 0), then the last sentence of lemma 5.5 tells us that x = 2 am,n(0,−2). since the number of vertices of h is mn−4, this implies y = mn−4−x = mn−4−2 am,n(0,−2) = mn−2 am,n(2, 0). since m is even, it is now obvious that x m = 2 am,n(0,−2) m and y n = m−2 am,n(2, 0) n are even. hence, gcd(x/m, y/n) 6= 1, so we see from (12) h is not a hamiltonian cycle. to complete the proof, we now consider the situation where 2.1(1) holds, which means that gcd(m, n) = 1. since x/m and y/n are integers, we know that x ≡ 0 (mod m) and y ≡ 0 (mod n). also, since the number of vertices of h is mn−4, we know that x + y = mn−4 ≡−4 (mod mn). combining these congruences tells us that x ≡−4 (mod n) and y ≡−4 (mod m). so x = am,n(0,−4) and y = am,n(−4, 0). we conclude from (12) that h consists of a single cycle (and is therefore a hamiltonian cycle) if and only if the condition in 2.1(3) holds. 70 dave witte morris / j. algebra comb. discrete appl. 10(1) (2023) 61–71 references [1] s. j. curran, m. n. ferencak, c. j. morgan, j. w. thompson, the hamiltonicity of a cayley digraph of a modified abelian group on two generators, congr. numerantium 188 (2007) 75–95. [2] s. j. curran, d. witte, hamilton paths in cartesian products of directed cycles, cycles in graphs, edited by b.r. alspach and c.d. godsil, north-holland publishing co. 115 (1985) 35–74. [3] j. gallian, contemporary abstract algebra, 10th ed. crc press, boca raton, fl (2021). [4] r. hammack, w. imrich, s. klavžar, handbook of product graphs, 2nd ed. crc press, boca raton, fl (2011). [5] s.housman, enumeration of hamiltonian paths in cayley diagrams, aequationes math. 23 (1981) 80–97. [6] j.b. klerlein, e. c. carr, hamiltonicity in cn × cm after a single push, congr. numerantium 190 (2008) 87–96. [7] w. f. klostermeyer, pushing vertices and orienting edges, ars combin. 51 (1999) 65–75. [8] r .rankin, a campanological problem in group theory, proc. cambridge philos. soc. 44 (1948) 17–25. [9] wikipedia, vertex cycle cover, (2022, november 19). [10] d. witte, j. a. gallian, a survey: hamiltonian cycles in cayley graphs, discrete math. 51 (1984) 293–304. [11] s. x. wu, cycles in the cartesian product of two directed cycles, 2007 jmm2007. 71 https://mathscinet.ams.org/mathscinet-getitem?mr=2408756 https://mathscinet.ams.org/mathscinet-getitem?mr=2408756 https://doi.org/10.1016/s0304-0208(08)72996-7 https://doi.org/10.1016/s0304-0208(08)72996-7 https://doi.org/10.1201/9781003142331 https://doi.org/10.1201/b10959 https://doi.org/10.1201/b10959 https://doi.org/10.1007/bf02188014 https://doi.org/10.1007/bf02188014 https://mathscinet.ams.org/mathscinet-getitem?mr=2489794 https://mathscinet.ams.org/mathscinet-getitem?mr=2489794 https://zbmath.org/?q=an:0977.05057 https://doi.org/10.1017/s030500410002394x https://doi.org/10.1017/s030500410002394x https://en.wikipedia.org/wiki/vertex_cycle_cover https://doi.org/10.1016/0012-365x(84)90010-4 https://doi.org/10.1016/0012-365x(84)90010-4 https://www.jointmathematicsmeetings.org/meetings/national/jmm/1023-05-790.pdf preliminaries statement of the main result proof of the main result proof of the corollary 3.5 from the theorem 3.3 direct proof of the corollary 3.5 references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.41075 j. algebra comb. discrete appl. 3(3) • 177–186 received: 03 november 2015 accepted: 04 january 2016 journal of algebra combinatorics discrete structures and applications the part-frequency matrices of a partition research article william j. keith abstract: a new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by glaisher’s bijection. we prove results that suggest surprising usefulness for such a simple tool, including the existence of a related statistic that realizes every possible ramanujan-type congruence for the partition function. to further exhibit its research utility, we give an easy generalization of a theorem of andrews, dixit and yee [1] on the mock theta functions. throughout, we state a number of observations and questions that can motivate an array of investigations. 2010 msc: 05a17, 11p83 keywords: partitions, partition rank, glaisher’s bijection 1. introduction a nonincreasing sequence of positive integers (λ = (λ1,λ2, . . . ,λi) such that ∑ λj = n is a partition of n, denoted λ ` n. we say that n is the weight of the partition, or write |λ| = n. denote the number of partitions of n by p(n). their generating function is p(n) := ∞∑ n=0 p(n)qn = ∞∏ i=1 1 1−qi . one of the first and most useful theorems learned by a student of partition theory is euler’s result that the number of partitions of n into odd parts (e.g., 3 + 1 or 1 + 1 + 1 + 1 for n = 4) is equal to the number of partitions of n into distinct parts (4 or 3 + 1). this enumeration theorem, interpreted in generating functions, gives the q-series identity ∞∏ k=1 1 1−q2k−1 = ∞∏ k=1 (1 + qk). william j. keith; department of mathematics, michigan tech university, usa (email: wjkeith@mtu.edu). 177 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 glaisher’s bijection gives a direct mapping between the two sets: if in a partition into odd parts the part i appears ai,020 + ai,121 + . . . times, ai,` ∈ {0,1}, then construct the distinct parts ai,` i · 2`. the reverse map collects parts with common odd factor i. the identity and glaisher’s map both generalize in the obvious way to partitions into parts not divisible by m and partitions in which parts appear fewer than m times. observe that for each m, glaisher’s map can be extended to an involution on all partitions: construct a list of matrices mj indexed by the numbers j not divisible by m. enumerate columns and rows starting with 0. if the part n = jmk appears an,0m0 +an,1m1 +an,2m2 + . . . times, then assign row k of matrix mj to be ak,1 ak,2 ak,3 . . . := an,0 an,1 an,2 . . .. for example, if m = 2, the partition (20,5,5,4,2,2,1,1,1,1,1) of 43 would be depicted (with all other entries 0): 1 1 0 1 0 1 0 1 0 0 3 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 1 0 0 now it is easy to see that glaisher’s map is simply transposition of this sequence of matrices, restricted to partitions into odd or distinct parts, where either the first row or the first column are the only nonzero entries. once these matrices are built, a wide array of transformations suggest themselves. we will consider several in this paper. the crucial observation is that any bijection which preserves the sums in the sw-ne antidiagonals of the matrices will preserve the weights of the partitions under consideration. for an object so easily related to such a classical theorem in partitions, this simple idea seems to have a surprising range of possible applications. in section 2 we consider the action of rotating antidiagonals through their length, and in doing so find several theorems including new statistics that realize any of the known congruences for the partition function, including ramanujan’s congruences mod 5, 7, and 11. in section 3 we generalize a theorem from andrews’, dixit’s and yee’s paper [1] related to q-series identities concerning partitions into odd and even parts which arises from third order mock theta functions. throughout, we mention questions of possible research interest which arise in connection with these ideas. 2. a congruence statistic: the orbit size a second famous result in partition theory is the set of congruences for the partition function observed by ramanujan, the first examples of which are p(5n + 4) ≡ 0 (mod 5) p(7n + 5) ≡ 0 (mod 7) p(11n + 6) ≡ 0 (mod 11). in fact for any α,β,γ ≥ 0, there is a δ such that p(5α7b β+1 2 c11γn + δ) ≡ 0 (mod 5α7β11γ). the first two congruences can be combinatorially realized by dyson’s rank of a partition, which is simply the largest part minus the number of parts. the number of partitions λ of 5n + 4 is divided into 178 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 five equal classes of size p(5n+4) 5 by the residue mod 5 of the rank of λ, and likewise seven classes by the residue of 7n + 6. a more important statistic, the crank, realizes all the congruences listed, and all of the other known congruences for the partition function, although the classes so constructed divide p(an+b) into classes of possibly unequal sizes divisible by c for the various progressions p(an+b) ≡ 0 (mod c). the crank has proven to be of great use in partition theory (see for instance [7], [6], [2] for just a few papers employing and illuminating this statistic). although combinatorially defined, proofs for properties of the crank typically require and illuminate deep and beautiful properties of modular forms. we now claim that associated to the sequence of part-frequency matrices of partitions is a class of natural combinatorial statistics which realizes not only every known congruence but indeed every possible congruence for the partition function. to construct this statistic we turn to a natural transformation on the part-frequency matrix: rotating the antidiagonals. to be precise, given modulus m and describing any part size as jmk with m j, define the map ρ on a partition which, operating on the part-frequency matrices (ai,`), yields (ρ(mj))i,` = ai−1,`+1 for i > 1 and (ρ(mj))1,` = a`,1. that is, we move each entry down and left 1 place, and the lowest entry in an antidiagonal is moved to the upper end. when this action is applied to partitions in which parts appear fewer than m times, i.e. nonzero entries only appear in the first column, all nonzero entries rotate to the first row. thus, this map still contains glaisher’s bijection as a special case. our interest now is in the size of the orbits among partitions produced by this action. for example, let m = 2 and consider the partition (20,5,5,4,2,2,1,1,1,1,1) illustrated in the introduction. we see that m1, m3 and mj for j ≥ 7 are unchanged by this operation. in the case of m1, all antidiagonals are “full” in that if any entry in the antidiagonal is 1, all are, while the others are empty. however, m5 is altered by this operation, with an orbit of length 6. its images are, in order: 5 0 1 0 0 0 0 1 0 0 5 0 0 1 1 0 0 0 0 0 5 0 1 0 0 1 0 0 0 0 5 0 0 0 1 0 0 1 0 0 5 0 1 1 0 0 0 0 0 0 5 0 0 0 1 1 0 0 0 0 thus, overall, the partition (20,5,5,4,2,2,1,1,1,1,1) is part of an orbit of size 6. one immediately observes that information about the sizes of the orbits of this action will give us information about p(n). in particular, denote by pρ,m(n,k) the number of partitions of n that lie in orbits of size k under the action of ρ with modulus m, and by oρ,m(n,k) the number of orbits of size k. we then have that theorem 2.1. given any partition congruence p(an + b) ≡ 0 (mod c), if b < a and a|m, then pρ,m(an + b,k) ≡ 0 (mod c) for all k. theorem 2.2. under the previous hypotheses, oρ,m(an + b,k) ≡ 0 (mod c) for all k also. thus, every linear congruence for the partition function can be realized by the orbit statistic for at least m = a. for example, the orbit sizes for modulus 5, 7, and 11 realize ramanujan’s congruences, that 179 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 p(mn + k) ≡ 0 (mod m) for (m,k) ∈ {(5,4),(7,5),(11,6)}. it seems interesting, and hopefully useful, that a map which is a small modification of glaisher’s old bijection should have such close connections to the overarching structure of partition congruences now known to exist. first we prove theorem 2.1. we will employ the standard notation ∞∏ i=1 ( 1−qi ) =: (q;q)∞. proof of theorem 2.1. observe partitions into parts not divisible by m, appearing fewer than m times. these are those partitions whose part-frequency matrices are nonzero in no entry other than the upper-leftmost corner. refer to such a partition as an upper-left filling. denote the number of these by pm,m(n) and their generating function by pm,m(q). this function is pm,m(q) := ∞∑ n=0 pm,m(n)q n := ∞∏ k=1 m-k 1 + qk + · · ·+ q(m−1)k = ∞∏ k=1 m-k 1−qmk 1−qk = ∞∏ k=1 m-k 1−qmk 1−qk ∞∏ j=1 1−qmj 1−qmj = (qm;qm)2∞ (q;q)∞(qm 2 ;qm 2 )∞ . (1) thus pm,m(q) = ∞∏ i=1 1 1−qi (qm;qn)2∞ (qm 2 ;qm 2 )∞ = ( ∞∑ n=0 p(n)qn )( ∞∑ i=0 c(i)qmi ) where the c(i) are the integral coefficients of the power-series expansion of the second factor in the middle term. note that the powers of q in this function with nonzero coefficients are all multiples of m. this gives us a recurrence for pm,m(n) in terms of values of the original partition function p(n−`m) for various nonnegative integral `: pm,m(n) = p(n)c(0) + p(n−m)c(1) + p(n−2m)c(2) + . . . . if all terms in the progression are divisible by c, then pm,m(n) will be as well. if n ≡ b (mod a), a|m, and p(ak + b) ≡ 0 (mod c) for all k, then it will hold that p(n−km) ≡ 0 (mod c). remark 2.3. the upper left fillings are precisely the fixed points of glaisher’s bijection. this makes them a very natural object of interest, and yet it appears that only one class of them has been well-studied: the m = 2 case of partitions of n into parts both odd and distinct, the number of which are well-known to be of equal parity with partitions of n, as they are in bijection with the self-conjugate partitions of n. the generating functions of upper-left fillings are simple η-quotients, which by work of treener [8] will possess their own congruences, but of perhaps greater interest is that many of these, and variants in which we permit the divisibility and frequency moduli to be different, appear in connection with mckay-thompson series [5]. investigation of these objects thus seems like it could be interesting in its own right. now observe that we may write partitions in which the orbit size is any stipulated value as an upper left filling plus the presence of various specified sets of fillings of other diagonals. for instance, suppose m = 5. then any partition consisting of an upper-left filling of some kind, 1s in the entries a0,1 and a1,0 for m1, and a 3 in the entry a0,1 for m4, representing 1s appearing 5 times (plus up to 4 more appearances), a single 5, and a 4 appearing 15 times (plus up to 4 more appearances), and no other entries, will have orbit size 2. the number of such partitions is just the number of upper-left filling partitions of n−70. this gives us a recurrence for pρ,m(n,k), the number of partitions of n of orbit size k, in terms of the partitions that are upper-left fillings. obviously this would be an exceedingly complicated recurrence, but 180 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 we only require that each of the terms in the recurrence is of the form pm,m(n−`m) for some nonnegative integral `. since in the arithmetic progressions an + b we have established that pm,m(an + b) is 0 mod c, the numbers pρ,m(an + b,k) will themselves be 0 mod c. � proof of theorem 2.2. each orbit is an equivalence class of antidiagonals under the action of rotation. for instance, one orbit of size 6 is the equivalence class represented by the six m5 matrices in the opening example partition, and the fixed matrices making up the rest of the sequence. for such an orbit, define the weight of the orbit as the sum of the entries in the antidiagonals outside the upper left corner, multiplied by the part-size powers of m and the frequency power of m associated to each element, i.e. the sum∑ i,j ai,jm ij. for the example partition, the weight is 52 – only the 1 in the upper left corner of m1 is part of the upper-left filling. for a given orbit, the weight in the antidiagonals outside of the upper left corner is some constant multiple of m, say km. then for each equivalence class (under rotation) of arrangements of entries in the antidiagonals of length greater than 1, there is one orbit per upper left filling of n − km. but if n ≡ b (mod a), then n−km ≡ b (mod a) since a|m, and so the upper left fillings of n−km, being defined by some recurrence in p(n−km− `m), also possess the property that their number is divisible by c. thus, the number of orbits is divisible by c. � remark 2.4. a reader may prudently object that if some sequence f(n), n ≥ 0 has the property f(ak + b) ≡ 0 (mod c), and a second power series f(q) has integral coefficients, and we consider the coefficients of the generating function ∞∑ n=0 φ(n)qn = ( ∞∑ n=0 f(n)qn ) f(qm) for some m such that a|m, the coefficients φ(n) will obviously share the congruence φ(ak + b) ≡ 0 (mod c) by a similar recurrence argument. of course this is true; we are mathematically interested in such a relation when, (a), the coefficients count an easily describable combinatorial object, and, (b), the function f(qm) arises in a natural way from a simple action on the elements counted by f(n), which finally (c) poses some hope of obtaining further research utility in the future. the first example of the theorems arises from the first ramanujan congruence p(5n+4) ≡ 0 (mod 5). modulo 5, the number of orbits of given sizes are: 5n + 4 1 2 3 6 4 5 0 0 0 9 20 5 0 0 14 75 30 0 0 19 220 135 0 0 24 605 485 0 0 29 1480 1535 5 0 34 3470 4375 20 5 39 7620 11580 75 30 (an orbit of size 6 can arise if there are diagonals with period 2 and 3; this will happen sooner than an orbit of size 4.) it would have been most interesting if the rank or the crank, restricted to classes of partitions with equal orbit sizes, also divided these classes into equinumerous sets; this is not the case, and so there arises the question question 1. does there exist a simple combinatorial statistic on upper left fillings, partitions of orbit size 1, or the orbits of size 1, that divides these sets into classes of equal size, or classes of size divisible by m? 181 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 if this property could be proved in an elementary fashion, we would have a completely elementary proof of the congruences the statistic supported – something which, to date, is lacking for ramanujan’s congruences. remark 2.5. if one calculates the andrews-garvan dyson crank restricted to partitions in which parts are not divisible by m and in which parts appear fewer than m times, one notices a significant deficiency in the number of partitions with crank 0 mod m. the reason for this is when there are no 1s in the partition, the crank is simply the size of the largest part, and under these restrictions this group never contributes a crank 0 mod m. a modification will have to figure out a way to work around this difficulty. question 2. recent work by breuer, eichhorn and kronholm [3] gives a geometric statistic they label a “supercrank” that realizes all congruences for partitions into three parts via the transformations of a cubic fundamental domain in the cone that describes these partitions (with work in progress on partitions into general fixed numbers of parts). it is probably unlikely that these statistics are intimately related, but any connection between the two families would be fascinating. 2.1. generating functions for orbit sizes a natural question that arises is the generating function for each orbit size. observe that an orbit of size 1 appears only when an antidiagonal, if occupied by anything other than 0, must be filled with copies of the same digit. suppose an antidiagonal in mj is of length k. then its entries are the elements ai,` with i + ` = k − 1, so each element represents a contribution of parts totaling jai,`mk−1 to the partition. thus, a fixed diagonal can be considered as contributing k copies of jmk−1 some number of times ai,` ranging over 0 ≤ ai,` < m. thus, if the matrices mj are modulus m, the orbits of size 1 can be described as partitions of n into parts k · j ·mk−1 appearing fewer than m times: p(1)m (q) = ∏ j:m-j k≥1 ( 1 + qjkm k−1 + q2jkm k−1 + · · ·+ q(m−1)jkm k−1 ) = ∏ j:m-j k≥1 1−qjkm k 1−qjkmk−1 = ∏ j:m-j k≥1 1−qjkm k 1−qjkmk−1 · ∏ b,k≥1 1−q(bm)km k−1 1−q(bm)kmk−1 = ∏ k≥1 ( qkm k ;qkm k )2 ∞( qkm k−1 ;qkm k−1 ) ∞ · 1( qkm k+1 ;qkm k+1 ) ∞ = 1 (q;q)∞ ∏ k≥1 ( qkm k ;qkm k )2 ∞( q(k+1)m k ;q(k+1)m k ) ∞ ( qkm k+1 ;qkm k+1 ) ∞ . question 3. this is a somewhat curious generating function. it is an infinite η-quotient, most of the factors of which do not cancel. finite η-quotients are weakly holomorphic modular forms, but does this fit in any of the usual classes of generating functions? since the number of partitions of orbit size 1 is of interest in possible elementary proofs for partition congruences, what can be said – whether combinatorially, analytically or asymptotically – about its coefficients? orbits of size 2 will will contain either a fixed point as in the orbits of size 1, or fillings of every other entry in an antidiagonal with the same entry. let the population of an antidiagonal denote the number of its nonzero entries. every odd population k thus arises in three ways: either as the filled antidiagonal of length k, or half of the antidiagonal of length 2k, interleaved with zeroes. those cases in which both halves appear cover the case of even population k filling the antidiagonal of length k, so we need add only 182 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 two more varieties of even population k to cover the case of the two interleaved halves of the antidiagonal of length 4k. finally, we subtract p(1)m to account for the count of partitions where any occupied halves happen to match. letting ak = 3 if k is odd, and ak = 2 if n is even, we have by logic similar to the previous function, p(2)m (q) = ∏ k≥1   ( qkm k ;qkm k )2 ∞( qkm k−1 ;qkm k−1 ) ∞ · 1( qkm k+1 ;qkm k+1 ) ∞   ak −p(1)m (q) the logic is easily extended, noting that for composite orbit sizes there will be an inclusion-exclusion alternating sum of orbit sizes dividing the desired orbit. 2.2. orbits of size not divisible by a given modulus one may wonder if this construction could yield any useful information on the parity or 3-arity of the partition numbers. unfortunately, it seems rather weak for that purpose. after all, it is already well known that the partition function p(n) is congruent modulo 2 to the number of partitions of n into distinct odd parts, which is just an upper left filling; analysis of additional orbit sizes could hardly be expected to yield better information. it is easy to observe that p(n) will be congruent modulo 2 or 3 to the number of partitions in orbits of size not divisible by 2 or 3 respectively. suppose that we set out to construct such partitions for modulus 2. consider the part-frequency matrices for m = 2. in each antidiagonal of length k, we desire that the period of the antidiagonal under the rotation action be odd. let k = t · 2r(k) where t is the largest odd divisor of k. then we allow orbits of size t or any divisor of t, so we build the antidiagonal of length k by choosing entries in periods of size t; there will be 2r(k) entries in each such period. we may choose these freely; different choices will yield different partitions. the 2r(k) entries in each periodic section of the antidiagonal will be either 0 or 1 and each entry represents parts totaling 2k−1, so if nonzero the amount contributed by those entries to the partition will be 2r(k)2k−1. the end result will be a partition with odd orbit size. we chose parts in the matrices mj, j = 2n−1. we can freely choose i of the t available periods, giving ( t i ) possible fillings for each antidiagonal. we end up with the generating function ∞∏ n=1 ∞∏ k=1 t∑ i=0 ( t i ) q(2n−1)2 k−12j(k)i = ∞∏ n=1 ∞∏ k=1 (1 + q(2n−1)2 k−12j(k))t ≡ ∞∏ n=1 ∞∏ k=1 (1 + q(2n−1)2 k−1 )k (mod 2). in the last line we used the fact that (1 − qk)2 ≡ (1 − q2k) (mod 2). alas, this is easily seen to be equivalent to partitions into distinct odd parts. if the experiment is run for general modulus, we can fairly easily obtain the fact that theorem 2.6. fix b > 1. letting km = r + 1 if br‖m, ∞∑ n=0 p(n)qn ≡ ∞∏ m=1 (1 + qm + q2m + · · ·+ q(b−1)m)km (mod b). that is, the number of partitions of n is equivalent modulo b to the number of partitions of n in which parts of size m appear at most b−1 times in each of km types. 183 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 remark 2.7. if we wish to consider parts appearing at most once, and exponents only 0, 1, or 2, we may consider the fact that 1 (q;q)∞ = ∏∞ n=1(1 + q n)an in which an is the power of 2 that divides n, plus 1. reducing products modulo 3 we obtain ∞∑ n=1 p(n)qn ≡ ∞∏ n=1 (1 + qn)an (mod 3) in which, if n = 2r3km, gcd(m,6) = 1, then an is the residue modulo 3 of the k-th iteration of the map starting from 0th iteration r+1 and, at each successive iteration, adds the floor of the previous exponent, divided by 3, to the residue of the previous exponent mod 3. 3. generalization of a theorem related to third-order mock theta functions we next illustrate the utility of part-frequency matrices for proving combinatorial identities of suitable type. in this section and the succeeding we employ the standard notation (a;q)n = (1−a)(1−aq) . . .(1−aqn−1) , (a;q)∞ = ∞∏ k=0 (1−aqk). a classical collection of objects in partition theory is ramanujan’s mock theta functions, which have spurred a phenomenal amount of work [4]. in [1] andrews, dixit and yee consider combinatorial interpretations of q-series identities related to several of the third order mock theta functions. most of these involve partitions into odd or distinct parts with restrictions on the sizes of odd or even parts appearing. one such identity is (theorem 3.4 in [1]) theorem 3.1. ∞∑ n=1 qn (qn+1;q)n(q2n+1;q2)∞ = −1 + (−q;q)∞ which, interpreted combinatorially, yields (theorem 3.5 in [1]). corollary 3.2. the number of partitions of a positive integer n with unique smallest part in which each even part does not exceed twice the smallest part equals the number of partitions of n into distinct parts. using part-frequency matrices, we can re-prove and generalize this to theorem 3.3. the number of partitions of n with smallest part appearing fewer than m times, in which each part divisible by m does not exceed m times the smallest part, equals the number of partitions of n into parts appearing fewer than m times. proof. let the smallest part of the partition be jmk, and consider the matrices mi with modulus m. for any other matrix mi with m i, if i < jmk, then there is exactly one value of k1 for which jmk < imk1 < jmk+1. if i > jmk, then only parts im0 are allowed under the conditions of the theorem. thus, in each matrix mi, there is one and exactly one part size imk1 permissible, with the exception of mj, in which jmk appears with nonzero entry a0,jmk and all other ab,jmk = 0, along with possible entries ab,jmk+1. now the map is simply a revised form of glaisher’s bijection: we transpose the portions of the matrices mi starting at row k1, i.e, we exchange the entries in row k1 for entries in the m0 column starting no lower than row k1. 184 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 the resulting matrix still has smallest part jmk, and any part larger than this may appear fewer than m times. observe that as long as jmk remains the smallest part, the allowable parts for a partition of the previous type are still identifiable. in particular, we can take a partition into parts appearing fewer than m times and exchange the m0 column of matrix mi for the k1 row which makes jmk < imk1 < jmk+1, or for the 0 row if i > jmk. example 3.4. let m = 2 for convenience and say λ = (15,15,15,10,10,8,6). write the matrices mj and mark the allowed row: 1 0 0 0 0 0 0 0 0 0 ∗ 1 0 0 3 0 0 0 1 0 0 ∗ 0 0 0 0 0 0 5 0 0 0 ∗ 0 1 0 0 0 0 0 0 0 . . . 15 ∗ 1 1 0 0 0 0 0 0 0 0 0 0 transpose the part of the matrices at and below the marked row. 1 0 0 0 0 0 0 0 0 0 ∗ 1 0 0 3 0 0 0 1 0 0 ∗ 0 0 0 0 0 0 5 0 0 0 ∗ 0 0 0 1 0 0 0 0 0 . . . 15 ∗ 1 0 0 1 0 0 0 0 0 0 0 0 we have the refined identity (qmn;qm)∞ (qn+1;q)(m−1)n(q mn;q)∞ = (qm(n+1);qm)∞ (qn+1;q)∞ which, multiplied on both sides by q n(1−q(m−1)n) 1−qn and summed over all n save the exceptional case n = 0, gives the additionally parametrized q-series identity corollary 3.5. ∞∑ n=1 qn(1−q(m−1)n)(qmn;qm)∞ (1−qn)(qn+1;q)(m−1)n(qmn;q)∞ = −1 + (qm;qm)∞ (q;q)∞ for which m = 2 is the original identity (3.1). using the part-frequency matrices we can play with this theorem in a variety of ways. for instance, suppose we relax the restriction that the smallest part be unique to require that it be not divisible by m. then we can pivot the row in matrix mj of smallest part jmk around the a1,jmk entry, preserving the smallest part, and any entries of size ai,jmk+1 are rotated to a0,jmk+1+i. now the resulting partition has parts which appear fewer than m times, except that multiples of the smallest part by mk may appear less than m2 times. other variations can be explored at leisure. question 4. can other known theorems related to third-order mock theta functions be generalized in this fashion, or the behavior of the relevant partitions related to glaisher-type maps? acknowledgment: this note is an expansion of a short talk given at “algebraic combinatorics and applications,” the first annual kliakhandler conference at michigan technological university. the author would like to thank igor kliakhandler for his generous support of mathematics at michigan tech and vladimir tonchev for organizing the 2015 conference. the anonymous referee is also appreciated for a sharp eye contributing clarity to the presentation in several places. 185 w. j. keith / j. algebra comb. discrete appl. 3(3) (2016) 177–186 references [1] g. e. andrews, a. dixit, a. j. yee, partitions associated with the ramanujan/watson mock theta functions ω(q), ν(q), and φ(q), res. number theory 1 (2015) 1–9. 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[8] s. treneer, congruences for the coefficients of weakly holomorphic modular forms, proc. london math. soc. 93(2) (2006) 304–324. 186 http://dx.doi.org/10.1007/s40993-015-0020-8 http://dx.doi.org/10.1007/s40993-015-0020-8 http://dx.doi.org/10.1090/s0273-0979-1988-15637-6 http://dx.doi.org/10.1090/s0273-0979-1988-15637-6 http://arxiv.org/abs/1508.00397 http://arxiv.org/abs/1508.00397 http://dx.doi.org/10.1007/s00222-005-0493-5 http://dx.doi.org/10.1007/s00222-005-0493-5 http://dx.doi.org/10.1080/00927879408825127 http://dx.doi.org/10.1080/00927879408825127 http://dx.doi.org/10.1007/bf01231493 http://dx.doi.org/10.1073/pnas.0506702102 http://dx.doi.org/10.1073/pnas.0506702102 http://dx.doi.org/10.1112/s0024611506015814 http://dx.doi.org/10.1112/s0024611506015814 introduction a congruence statistic: the orbit size generalization of a theorem related to third-order mock theta functions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.404964 j. algebra comb. discrete appl. 5(2) • 65–70 received: 9 november 2016 accepted: 22 november 2017 journal of algebra combinatorics discrete structures and applications some new binary codes with improved minimum distances research article eric zhi chen abstract: it has been well-known that the class of quasi-cyclic (qc) codes contain many good codes. in this paper, a method to conduct a computer search for binary 2-generator qc codes is presented, and a large number of good 2-generator qc codes have been obtained. 5 new binary qc codes that improve the lower bounds on minimum distance are presented. furthermore, with new 2-generator qc codes and construction x, 2 new improved binary linear codes are obtained. with the standard construction techniques, another 16 new binary linear codes that improve the lower bound on the minimum distance have also been obtained. 2010 msc: 94b05, 94b65 keywords: binary linear codes, quasi-cyclic codes, algorithms 1. introduction a binary linear [n, k, d] code is a k-dimensional subspace of gf(2 )n, where n is the block length, k the dimension of the code, and d is the minimum distance between any two codewords. the minimum distance determines the error-correcting or error-detecting capability. therefore, for a given block length n and dimension k, it is desired to have an [n, k, d] code with the minimum distance as large as possible. one of the most fundamental problems in coding theory is to construct codes with the best possible minimum distances. grassl [13] maintains online code tables of linear codes for small block length, code dimension over small finite fields. the code tables contain both the lower bounds and upper bounds on the minimum distance. a code with a minimum distance meeting the upper bound is said to be optimal, while a code with a minimum distance meeting the lower bound is called best-known. the problem to construct codes with the best possible minimum distances is shown to be very difficult. for small code dimension and block length, it is possible to do exhaustive computer search for optimal codes. but when both the eric zhi chen; department of computer science, kristianstad university, 291 88 kristianstad, sweden (email: eric.chen@hkr.se). 65 https://orcid.org/0000-0002-2492-7754 e. z. chen / j. algebra comb. discrete appl. 5(2) (2018) 65–70 code dimension and block length increase, it becomes intractable. the researchers turn to some promising subclasses of linear codes with rich mathematical structures to reduce the search time complexity. during the last decades, the class of quasi-cyclic (qc) codes and quasi-twisted codes has been shown to contain a large number of good codes. with the help of modern computers, a large number of record-breaking qc codes have been constructed [1, 2, 4–7, 10–12, 14–19]. the further improvements on [13] become difficult, and it is even difficult to improve the binary linear codes. in this paper, a method to construct better binary linear codes is presented. in the next section, a new weight matrix for constructing 2-generator qc codes is presented, and the iterative computer search algorithm is then conducted. in section 3, new binary qc codes that improve the minimum distance in [13] are given, and the well-known construction x is applied to produce 2 more binary linear codes. with the standard code constructions, 16 more codes that improve the minimum distance in [13] are obtained. 2. quasi-cyclic codes and their computer constructions a linear [n, k, d] code c is called cyclic if whenever a codeword (a0, a1, ..., an−1) is in c, then so is (an−1, a0, a1, ..., an−2). a code is said to be quasi-cyclic (qc) if a cyclic shift of any codeword by p positions is also a codeword. therefore, a cyclic code is a qc code with p = 1. the length n of a qc code is a multiple of p, i.e., n = pm. a cyclic matrix is also called a circulant matrix. the circulant matrices are basic components in the generator matrix for a qc code. an m × m cyclic matrix is defined as a =   c0 c1 c2 · · · cm−1 cm−1 c0 c1 · · · cm−2 cm−2 cm−1 c0 · · · cm−3 ... ... ... ... ... c1 c2 c3 · · · c0   (1) and the algebra of m × m cyclic matrices over gf(2) is isomorphic to the algebra in the ring gf(2)[x]/(xm − 1), if c is mapped onto the polynomial formed by the elements of its first row, c(x) = c0 + c1x + · · · + cm−1xm−1, with the least significant coefficient on the left. the polynomial c(x) is also called the defining polynomial of the matrix c and it is written in octal with least significant coefficients on the right in this paper. the generator matrix of a qc code can be transformed into rows of m × m circulant matrices by suitable permutation of columns. an h-generator qc code has a generator matrix of the following form: g =   g1,1 g1,2 g1,3 · · · g1,p g2,1 g2,2 g2,3 · · · g2,p g3,1 g3,2 g3,3 · · · g3,p ... ... ... ... ... gh,1 gh,2 gh,3 · · · gh,p   (2) where gij are m × m circulant matrices, for i = 1, 2, · · · , h, and j = 1, 2, · · · , p. let gij(x) be the defining polynomial of the matrix gij. then the defining polynomials for the h-generator qc code with generator matrix given in (2) can be written as (g11(x), g12(x), g13(x), · · · , g1p(x), · · · , gh1(x), gh2(x), gh3(x), · · · , ghp(x)). in magma [3], the parameter h is called the height. most quasi-cyclic codes studied in the literature are 1-generator qc codes (h = 1). very few studies on h-generator qc codes are found in the literature. in [17], 2 new rate 2/p qc codes were presented. in fact, they are 2-generator qc codes. in [6, 10], construction methods have been presented to obtain h-generator qc codes with improved minimum distances. in the computer search algorithms presented in [7, 14, 15], a weight matrix is used in the computation of the minimum distance of a 1-generator qc code. the general r × s weight matrix has the following 66 e. z. chen / j. algebra comb. discrete appl. 5(2) (2018) 65–70 form: w =   w0,0 w0,1 · · · w0,s−1 w1,0 w1,1 · · · w1,s−1 ... ... ... ... wr−1,0 wr−1,1 · · · wr−1,s−1   (3) where the entry wi,j is the hamming weight of ii(x)gj(x) mod xm 1, ii(x) is the i-th distinct information polynomial, and gj(x) is the j-th defining polynomial [14, 15]. as demonstrated in [6, 7, 10], it is often possible to extend the qc codes by adding one or more rows to the generator matrix of a 1-generator qc code. for example, with m = 57, a new binary 1-generator qc [228, 18, 96] code was constructed [7], with the defining polynomials g1(x)= 4524727255730403632, g2(x)= 5052140564035060426, g3(x)= 3041362270077724243, and g4(x)= 6624210767535636614. by extending one row, a 2-generator qc [228, 19, 95] code with the following generator matrix[ g1 g2 g3 g4 1 1 1 0 ] was obtained, where 1 is a vector of 57 1’s, and 0 is a vector of 57 0’s, and gi are the circulant matrix defined by the polynomial gi(x), i = 1, 2, 3, 4. with this motivation, the known good qc codes in [9, 13] have been investigated to obtain good augmented h-generator qc codes, and many good h-generator qc codes have constructed in [10]. the interesting questions to investigate in this paper are: how to do computer search for 2-generator qc codes and will new improved codes be constructed? the general 2-generator qc codes are quite complicated. so in this paper, we study a special form of 2-generator qc codes, as motivated in the last example: [ g11 g12 g13 · · · g1p g21 g22 g23 · · · g2p ] (4) where the first row of the defining polynomials are for 1-generator qc code, while the second row of defining polynomials are very special, g2j = 0 or 1(x) = 1 + x + x2 + · · · + xm−1, for i = 1, 2, 3, · · · , p. we start with derivation of distinct information polynomial ii(x)(i = 1, 2, 3, · · · , r), and distinct defining polynomials gj(x)(j = 1, 2, 3, · · · , s) as did in [14, 15] for finding 1-generator qc codes. then we try to extend the code with another row of defining polynomials. so for each possible defining polynomial gj(x), we have 2 possible combination in constructing 2-generator qc code:[ gj(x) 0 ] , [ gj(x) 1(x) ] for the sake of convenience, we write them as gi(x)/0 and gi(x)/1(x). we arrange all defining polynomials in the order of g0(x)/0, g1(x)/0, g2(x)/0, · · · , gs−1(x)/0, g0(x)/1(x), g1(x)/1(x), g2(x)/1(x), · · · , gs−1(x)/1(x). so we obtain the weight matrix as follows: w ′ =   w w0 · · · 0 m · · · m w m − w   (5) it has 2r + 1 rows, and 2s columns, where w is the weight matrix calculated as in (3), 0 · · · 0 is a vector of all zeros of length s, m · · · m is a vector of length s and each element has a value of m (generated by all 1’s vector), m is a r ×s matrix with each entry of value m. the first row of this new weight matrix is from the r distinct information polynomials by only considering the first row in (4); the middle row [0 · · · 0m · · · m] corresponds to the weights generated by 0 and 1(x) as the defining polynomials in (4); while the third row [wm − w ] is obtained by the considering the combined effect of both rows in (4). with such a block structure, it is only necessary to store an r × s weight matrix w, since all other 67 e. z. chen / j. algebra comb. discrete appl. 5(2) (2018) 65–70 elements of w ′ are 0, m, or can be calculated from w. with this weight matrix, we apply the iterative search algorithm presented in [7], and many new 2-generator qc codes have been found. next section will present the new codes with improved minimum distances. when the code dimension becomes large, this weight matrix becomes too large to be able to reside in a computer memory, and complete the search in reasonable time. as introduced in [8], we reduce the size of the weight matrix by selecting t distinct generator polynomials randomly and compute the (2r + 1) × (2t) weight matrix for constructions of 2-generator qc codes. 3. the new binary codes with the weight matrix (5) developed in the last section, the iterative search algorithm [7] has been used to conduct the search for good 2-generator qc codes. more than 150 good 2-generator qc codes have been obtained. they can be accessed via the on line database of quasi-twisted codes [9]. in this paper, only the codes that improve the minimum distances on [13] are presented. theorem 3.1. there exist binary qc [219, 18, 92], [225, 18, 96], [210, 20, 83], [81, 21, 25], [210, 24, 80] codes. proof. the 1-generator qc [219, 18, 92] code is constructed with m = 73, and the defining polynomials g1(x) = 3212271004340324237, g2(x) = 17721056076522411474157, and g3(x) = 3744160632054554 3443755. the 1-generator qc [225, 18, 96] code is constructed with m = 45, and the defining polynomials g1(x) = 30426152246431, g2(x) = 404750035361, g3(x) = 1342223621127, g4(x) = 1776673524175, and g5(x) = 36670644573317. the 2-generator qc [210, 20, 83] code is constructed with m = 35, and the defining polynomials g1(x) = 23477263277, g2(x) = 17461151113, g3(x) = 1631721217, g4(x) = 11576655613, g5(x) = 2267175171, g6(x) = 14354313511, g7(x) = g9(x) = g10(x) = g11(x) = g12(x) = 377777777777, and g8(x) = 0. the 2-generator qc [81, 21, 25] code is constructed with m = 27, and the defining polynomials g1(x) = 273277337, g2(x) = 14234775, g3(x) = 132552753, g4(x) = 0, g5(x) = 777777777, and g6(x) = 0. the 3-generator qc [210, 24, 80] code is constructed with m = 105, and the defining polynomials g1(x) = 6334264131043230150262137101, g2(x) = 4377421050451574564521102407255, g3(x) = g6(x) = 77777777777777777777777777777777777, and g4(x) = g5(x) = 0. the weight distributions of these codes can be found in on line database of quasi-twisted codes [9]. the well-known construction x is a simple and efficient method to construct new codes by combining 3 codes. theorem 3.2. (construction x) let c1 = [n, k1, d1] and c2 = [n, k2, d2] be a pair of nested codes, where c1 ⊂ c2. let c3 = [n3, k2 − k1, d3] be an auxiliary code. then there exists a c = [n + n3, k2, d] code with d ≥ min(d1, d2 + d3). theorem 3.3. there exists binary [86, 18, 30], and [107, 18, 40] codes. proof. let m = 21. with the search method given above, a best-known 2-generator qc [84, 18, 28] code was found. its defining polynomials are g1(x) = 54211, g2(x) = 26515, g3(x) = 321125, g4(x) = 244147, g5(x) = g8(x) = 7777777, and g6(x) = g7(x) = 0. let c1 be its sub-code of dimension 17. it is defined by the first row defining polynomials and is a 1-generator qc [84, 17, 30] code. let c2 be the 2-generator qc [84, 18, 28] code, and let c3 be a binary [2, 1, 2] code. by applying construction x, we obtain a new binary [86, 18, 30] code. 68 e. z. chen / j. algebra comb. discrete appl. 5(2) (2018) 65–70 let m = 21. with the search method given above, a best-known 2-generator qc [105, 18, 38] code was found. its defining polynomials are g1(x) = 77415, g2(x) = 1525677, g3(x) = 13427, g4(x) = 22137, g5(x) = 141531, and g6(x) = g7(x) = g10(x) = 0, and g8(x) = g9(x) = 7777777. let c1 be its sub-code of dimension 17. it is defined by the first row defining polynomials, and is a 1-generator qc [105, 17, 40] code. let c2 be the 2-generator qc [105, 18, 38] code and let c3 be a binary [2, 1, 2] code. by applying construction x, we obtain a new binary [107, 18, 30] code. it should be noted that all the codes given above improve the minimum distances in [13]. by applying standard construction methods, such as puncturing, shortening and extending, 16 more improvements on [13] are obtained. all the codes given in the paper have been checked with the magma algebraic system [3]. 4. conclusion in this paper, a construction method for binary 2-generator qc codes is presented and many good new qc codes are obtained. although it is quite difficult to improve the binary codes, we have made a total of 23 improvements on [13]. it should also be noted that these codes (and ones given in [10]) are only special cases of h-generator qc codes. further investigation on general h-generator qc codes is promising. acknowledgment: the author is grateful to the referees for their helpful comments and suggestions that improved the presentation of the results. references [1] n. aydin, i. siap, d. k. ray–chaudhuri, the structure of 1–generator quasi–twisted codes and new linear codes, des. codes cryptogr. 24(3) (2001) 313–326. 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[19] h. van tilborg, on quasi–cyclic codes with rate 1/m, ieee trans. inform. theory 24(5) (1978) 628–630. 70 http://www.codetables.de http://www.codetables.de https://doi.org/10.1109/18.79911 https://doi.org/10.1109/18.79911 https://doi.org/10.1109/18.144718 https://doi.org/10.1109/18.144718 https://doi.org/10.1109/18.259670 https://doi.org/10.1109/18.259670 https://doi.org/10.1109/18.333889 https://doi.org/10.1109/18.333889 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/tit.1978.1055929 https://doi.org/10.1109/tit.1978.1055929 introduction quasi-cyclic codes and their computer constructions the new binary codes conclusion references issn 2148-838xhttps://doi.org/10.13069/jacodesmath.1111733 j. algebra comb. discrete appl. 9(2) • 101–114 received: 3 february 2021 accepted: 22 october 2021 journal of algebra combinatorics discrete structures and applications on the generation of alpha graphs research article christian barrientos abstract: graceful labelings constitute one of the classical subjects in the area of graph labelings; among them, the most restrictive type are those called α-labelings. in this work, we explore new techniques to generate α-labeled graphs, such as vertex and edge duplications, replications of the entire graph, and k-vertex amalgamations. we prove that for some families of graphs, it is possible to duplicate several vertices or edges. using k-vertex amalgamations we obtain an α-labeling of a graph that can be decomposed into multiple copies of a given α-labeled graph as well as a robust family of irregular grids that can α-labeled. 2010 msc: 05c78, 05c51 keywords: α-labeling, graceful graph, amalgamation, duplication, replication 1. introduction the roots of difference vertex labelings can be found in the area of combinatorial design, in particular, this type of graph labeling was introduced as a mechanism to determine the veracity of two conjectures associated with the decomposition of complete graphs into copies of a given tree. the first of these conjectures, due to ringel [13], states that the complete graph k2n+1 can be decomposed into copies of any tree of size n. the second conjecture, posed by kotzig [10], goes even further by adding the condition that the decomposition is cyclic. let g be a graph of size n, a difference vertex labeling of g is a one-to-one function f : v (g) → s, where s is a set of non-negative integers, with the property that every uv ∈ e(g) has associated a weight determined by |f(u) − f(v)|. in [14], rosa defined four of these labelings by introducing some conditions to the set s and to the set of weights induced by the function. rosa said that the function f is a β-valuation if s = {0, 1, . . . ,n} and the set of weights is w = {1, 2, . . . ,n}. years later, golomb [7] used the term graceful labeling to refer to this valuation. graceful labelings are a special case of another labeling introduced by rosa. a ρ-valuation satisfies s = {0, 1, . . . , 2n} and the set of weights is w = {w1,w2, . . . ,wn}, where wi = i or wi = 2n + 1−i. rosa [14] proved that a cyclic decomposition of k2n+1 into copies of a given graph of size n exists if and only if christian barrientos; department of mathematics, valencia college, orlando, fl 32832, usa (email: chr_barrientos@yahoo.com). 101 https://orcid.org/0000-0003-2838-8687 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 there exists a ρ-labeling of g. therefore, in these three articles we can see the origin of the most famous conjecture in the area of graph labeling, known as the graceful tree conjecture that expresses that every tree admits a graceful labeling. rosa and širáň [16], introduced the concept of bipartite labeling of a tree. in [3], barrientos and minion extended their definition to include all bipartite graphs. let g be a bipartite graph and {a,b} be the natural bipartition of v (g); we refer to a and b as the stable sets of g and reserve the symbols a and b to represent the respective cardinalities of these sets. assuming that g is a bipartite graph, a difference vertex labeling of g is said to be bipartite if there exists an integer λ, called the boundary value of f, such that f(u) ≤ λ < f(v) for every (u,v) ∈ a × b. in [14], rosa defined an α-valuation (or α-labeling) as a bipartite graceful labeling. we say that an α-graph is any graph that admits an α-labeling. rosa [14] proved that if a graph g of size n admits an α-labeling, then there exists a cyclic decomposition of k2tn+1 into subgraphs isomorphic to g, where t is any natural number. this labeling is the most versatile of the labelings introduced by rosa; it can be transformed into several types of labelings such as harmonious, magic, and antimagic. a detailed description of its interaction with other labelings can be found in [11] and also in [19]. thus, the study of α-graphs has implications in other related research areas. this work is completely devoted to the study of this labeling. in section 2 we introduce some existing results that are used in the coming sections to build new families of α-graphs. in section 3 we work with the concept of vertex duplication, proving that any vertex of a caterpillar can be duplicated to produce a new α-graph; we go even further, proving that an α-graph is obtained when any number of pairwise non-adjacent vertices of a caterpillar are duplicated. in section 4 we study edge-duplications, proving that a graceful graph is obtained when an edge of a cycle is duplicated and when the size of the cycle is even the labeling obtained is in fact an α-labeling. moreover, we prove a similar result if two selected edges of a cycle are duplicated. we conclude that section by proving that any number of edges on the spine of a caterpillar can be duplicated and the outcome is an α-graph. in section 5 we extend the idea of vertex duplication; now we duplicate every vertex of a graph with an action that resembles the cartesian and the (weak) tensor product of a graph and a path. we show that when a graph admits an α-labeling and all the vertices are duplicated the same number of times, the final graph is also an α-graph. in section 6 we analyze the process of vertex amalgamation by studying some conditions that allow us to amalgamate multiple vertices of α-labeled graphs. this construction is used to generate new α-graphs, among the graphs obtained we have a supersubdivision of cycles of even size and a family of quadrangular cacti whose bases are some α-trees. we finish this section introducing a family of graphs whose vertices are points in the orthogonal integer lattice; we call these graphs irregular grids; an α-labeling for these grids is obtained using the tools presented in this section. the reader interested in this type of problems can find more information in the work of lópez and muntaner-batle [11] and in gallian’s survey [6]. all graphs considered here are finite with no loops nor multiple edges. the technical terminology not defined here is taken from [5] and/or [6]. 2. the essential results since the introduction of the concepts of graceful and α-labeling, many results have been obtained. in this section we provide, without proof, some of the results that are essential in the coming sections. suppose that g is a graph of size n and f is a difference vertex labeling of g. a shifting of f in c units is the labeling g of g defined for every v ∈ v (g) as g(v) = f(v) + c; clearly, the weight of any given edge of g is the same under both labelings. if f is a graceful labeling, its complementary labeling is the function f defined as f(v) = n − f(v) for every v ∈ v (g). if f is an α-labeling with boundary value λ, then f is an α-labeling with boundary value n − λ. note that if f assigns the label 0 on the stable set a, f does it on the stable set b; in other terms, if f assigns the smaller labels to the elements of a, f assigns the larger labels on this stable set. each α-labeling of g is also associated with other two important labelings of g; the first one is called the reverse of f, denoted by fr, and is defined as fr(v) = λ−f(v) if f(v) ≤ λ and fr(v) = n + λ + 1 −f(v) if f(v) > λ. take note of the fact that fr is 102 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 also an α-labeling; thus, in general terms, the number of graceful labelings of a graph is always divisible by 2 and the number of α-labelings is always divisible by 4. this labeling was also introduced by rosa [15] but under the name of inverse labeling. a graph g of size n is said to be arbitrarily graceful if for every positive integer d, there exists a difference vertex labeling that induces the weights d,d + 1, . . . ,d + n − 1. when d > 1, the function is called d-graceful labeling. for any positive integer d, an α-labeling f of g can be transformed into a d-graceful labeling, this transformation (also called amplification) can be done by adding the constant d − 1 to each vertex label larger than the boundary value of f. this property of the α-labelings was proven, independently, by maheo and thuillier [12] and slater [18]. in the coming sections, we transform some of the α-labelings into d-graceful labelings, to obtain, new and larger α-graphs. when g is an α-tree of size n and f is an α-labeling of g with boundary value λ, the associated d-graceful labeling shifted c units assigns the labels in the set {c,c + 1, . . . ,λ + c}∪{λ + c + d,λ + c + d + 1, . . . ,n + c + d−1} and induces the weights in the set {d,d + 1, . . . ,n + d− 1}. in [14], rosa proved, among other results, the existence of α-labelings for two types of graphs used in this work, caterpillars and cycles of size divisible by 4. in figure 1 we show an example of these labelings using a caterpillar and a cycle, both of order 12. the distribution of the labels can be extended to any other member of the respective family. 11 10 9 8 7 6 0 1 2 3 4 5 12 11 10 9 8 7 0 1 2 4 5 6 figure 1. a caterpillar and a cycle with their respective α-labeling let g1 and g2 be two graphs; a graph g is the result of a vertex amalgamation of g1 and g2 if a vertex of g1 is identified (merged) with a vertex of g2. let g1 and g2 be α-labeled graphs, it is well-known that the chain graph obtained amalgamating g1 and g2 is an α-graph; the amalgamation is done by identifying the vertex labeled 0 of g2 with the vertex of g1 which label is the boundary value of its α-labeling. furthermore, a wider category of graphs is obtained if g2 is just a graceful graph; in this case, we obtain a graceful chain graph. the k-vertex amalgamation of g1 and g2 is the graph obtained by identifying k independent vertices of g1 with k independent vertices of g2. in section 6 we discuss an alternative to use this multi-vertex amalgamation to generate new α-graphs. an edge amalgamation of g1 and g2 is the process of identifying an edge of g1 with an edge of g2. if both graphs have been α-labeled, there is an α-graph g that is the result of the amalgamation of the edge of weight n1 in g1 with the edge of weight 1 in g2, where n1 is the size of g1. this result was proven by barrientos and minion in [2]. observe that in both types of amalgamations, the second condition that asks for two α-graphs can be replaced by an α-graph and a graceful graph, but now the outcome is a graceful graph instead of an α-graph. the following definition was given by sethuraman and selvaraju [17]. let g be a graph of order m and size n; a graph h is called a supersubdivision of g if h is obtained from g by replacing every edge e of g by a complete bipartite graph k2,se in such a way that the end-vertices of e are amalgamated with the vertices of the 2-element stable set of k2,se and the edge e is deleted. note that the complete bipartite graphs used in these replacements do not need to be isomorphic. in [1], barrientos and barrientos proved that any graph of positive size has a supersubdivision that admits an α-labeling. 103 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 3. duplicating vertices the duplication of a vertex v of a graph g is the graph g′ obtained from g by adding a new vertex v′ to g and connecting v′ to all the neighbors of v. in [9], kaneria et al. proved that the duplication of a vertex of a cycle of even size is graceful. in the following proposition we prove that every vertex of a caterpillar can be duplicated to produce a new α-graph. recall that the spine of a caterpillar is the path containing all the vertices of degree larger than 1. proposition 3.1. every vertex of a caterpillar can be duplicated to produce a new α-graph. proof. suppose that g is a caterpillar of size n. let v ∈ v (g) be the vertex to be duplicated. if deg(v) = 1, then g′ is a caterpillar of size n + 1, which is an α-graph. suppose that deg(v) > 1. the size of g′ is n + deg(v) and its order is n + 2. since g is bipartite and v′ is only connected to the vertices adjacent to v, we get that g′ is also a bipartite graph. suppose that f is the α-labeling of g that follows the pattern in figure 1; we define an α-labeling g of g′ as follows: g(u) =   f(u) if f(u) ≤ f(v), f(u) + deg(v) if f(u) > f(v), f(v) + deg(v) if u = v′. let u1v,u2v, . . . ,ukv be the edges of g incident to v. the weights of these edges form a set of k consecutive integers. let wi = |f(ui) −f(v)|; without loss of generality we assume that w1 < w2 < · · · < wk. when v ∈ a, wi = f(ui) −f(v), we get g(ui) −g(v) = f(ui) + k −f(v) = f(ui) −f(v) + k = wi + k and g(ui) −g(v′) = f(ui) + k −f(v) −k = f(ui) −f(v) = wi. thus, the weights of the edges incident to v or v′ form the set {w1,w2, . . . ,wk,w1 +k,w2 +k,. . . ,wk +k} = {w1,w1 + 1, . . . ,w1 + 2k − 1} because wi+1 = wi + 1. let xy ∈ e(g) with x ∈ a. if f(y) −f(x) < w1, then g(y) − g(x) = f(y) − f(x) because both original labels are incremented k units. if f(y) − f(x) > wk = w1 + k − 1, then g(y) −g(x) = f(y) −f(x) + k because only the label of y increases k units. this implies that the new weight of xy is at least equal to w1 + 2k. consequently, the set of induced weights is {1, 2, . . . ,n + k}. when v ∈ b, wi = f(v)−f(ui) and g(v)−g(ui) = wi but g(v′)−g(ui) = f(v) + k−f(ui) = wi + k. thus, the set of weights induced on the edges incident to v or v′ is {w1,w1 + 1, . . . ,w1 + 2k − 1}. let xy ∈ e(g) with x ∈ a. if f(y)−f(x) < w1, then g(y)−g(x) = f(y)−f(x) because none of the original labels are incremented. if f(y)−f(x) > wk, then g(y)−g(x) = f(y)−f(x) + k because only the label of y increases k units. as in the previous case, these edges have weights in {w1 + 2k,w1 + 2k + 1, . . . ,n+k}. then, the set of induced weights is {1, 2, . . . ,n + k}. since the largest label used is n+k, and every label is used exactly once, this is a graceful labeling of g′; moreover, the labels assigned by g to the vertices of b are larger than those assigned to the elements of a, therefore, g is an α-labeling of g′. in figure 2 we show a sequence of examples where each internal vertex of a caterpillar has been duplicated. there are two features of the vertex duplication that we want to mention: • let v be the vertex of g, selected to be duplicated; if deg(v) = 2, then the graph g′, can be seen, as a supersubdivision of the caterpillar g. 104 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 0 1 2 3 4 5 6 7 8 910 1112 0 1 2 3 6 7 8 9 10 1112 1314 5 0 1 2 3 4 5 10 11 12 1314 1516 9 0 1 2 3 4 5 6 7 8 910 1112 16 0 1 2 3 4 5 6 7 8 910 1115 14 0 1 2 3 4 5 6 7 8 1213 1415 11 figure 2. α-labeled graphs obtained by duplicating every internal vertex of a caterpillar • if g is a tree, the subgraph of g induced by the edges incident to v is isomorphic to the complete bipartite graph k1,deg(v). in the case of a caterpillar with the α-labeling described before, the labels of the vertices adjacent to v form a set of consecutive integers. the labels of v and v′ are deg(v) units apart. thus, the subgraph formed by all the edges incident to v or v′ is isomorphic to k2,deg(v) and the labeling of this graph is a shifting of a d-graceful labeling. consequently, if instead of adding one new vertex we add any number of copies of v, the previous result still holds. in the next theorem we prove that for any subset of vertices of a caterpillar, such that the vertices in this subset are pairwise non-adjacent, we can duplicate all the elements of this set and the result is an α-graph. theorem 3.2. let g be a caterpillar and s be a subset of v (g). if the elements of s are pairwise non-adjacent, then an α-graph is obtained by duplicating all of them. proof. if s contains any vertex of degree one, its duplication results in another caterpillar, which admits an α-labeling. therefore we may assume that all the vertices in s are in the spine of the caterpillar. let s = {v1,v2, . . . ,vs} where dist(vi,vj) ≥ 2 and the path connecting vi and vi+1 does not contain any other member of s. the graph g can be decomposed into s subgraphs, being each of them a caterpillar containing exactly one element of s. thus, each of these caterpillars has only one vertex that has been duplicated; consequently, each of them is an α-graph. the α-labeling of each of these graphs can be amplified and shifted conveniently in such a way that there is no repetition of weights when they are taken collectively, the shiftings can be made in such a way that the vertices on the spine used in the decomposition are the only one where a label is repeated. the concatenation of these graphs is made using vertex amalgamation. thus, an α-labeling of g′ can be obtained. in figure 3 we show this procedure, where three vertices of the spine are duplicated, one of them more than once, and an end-vertex is also duplicated. the different colors are used to emphasize the subgraphs in the decomposition mentioned in the proof of this theorem. 105 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 30 33 4 9 10 13 16 19 20 2 27 23 21 0 1 28 29 3 26 25 24 22 17 18 figure 3. α-labeling of a graph obtained by duplicating multiple vertices of a caterpillar 4. duplicating edges the duplication of an edge e = uv of a graph g, is the process that consists of the introduction of an edge e′, with end-vertices u′ and v′, and the additional edges uu′ and vv′, that connect e′ to the graph g. kaneria et al. [9] proved that the duplication of an edge of an even cycle is graceful. in the following results we extend this result by showing, first, that this is also valid with an odd cycle, and second that more edges of a cycle can be duplicated with the same end result. proposition 4.1. if g is the result of an edge duplication of the cycle cn, then g admits a graceful labeling when n is odd and an α-labeling when n is even. proof. recall that cn is graceful if and only if n ≡ 0, 3(mod 4) and it is an α-graph if and only if n ≡ 0(mod 4). the graph g can be explained as the edge amalgamation of c4 and cn; based on the result in [2] and the fact that c4 is an α-graph, we know that g is graceful or an α-graph depending on the value of n. so, we need to analyze the cases where cn is not graceful. in either case we consider g as the outerplanar graph, obtained from cn+2, with a single chord connecting two vertices located at distance 3 from each other. note that the case n ≡ 2(mod 4) was solved in [9], i.e., we just need to study the case n ≡ 1(mod 4). we denote by v1,v2, . . . ,vn+2 the consecutive vertices of cn+2. for n ≡ 1(mod 4), we use the following labeling of the vertices of cn+2: f(vi) =   i−1 2 if i is odd, n + 4 − i 2 if 2 ≤ i ≤ n−1 2 is even, n + 3 − i 2 if i = n+3 2 , n + 2 − i 2 if n+7 2 ≤ i ≤ n + 1 is even. note that when i is odd, f is increasing with range {0, 1, . . . , n+1 2 }, when i is even, f is decreasing with range {n+3 2 , n+5 2 , . . . , 3n+1 4 }∪{3n+9 4 }∪{3n+17 4 , 3n+21 4 , . . . ,n + 3}. thus, the labels assigned by f are in the set {0, 1, . . . ,n + 3}, where n + 3 is the size of g. for each n+7 2 ≤ i ≤ n+1 even, the edge vi−1vi has weight f(vi)−f(vi−1) = n+2− i2− i−2 2 = n+3−i, while vi+1vi has weight f(vi) −f(vi+1) = n + 2 − i2 − i 2 = n + 2 − i. in other terms, the weights of the edges incident to vi are two consecutive integers. therefore, for these values of i, the weights on all these edges form the set {1, 2, . . . , n−1 2 }. the edge v1vn+2 has weight f(vn+2) − f(v1) = n+2−12 − 0 = n+1 2 . when i = n+3 2 , vivi−1 has weight f(vn+3 2 )−f(vn+1 2 ) = n+ 3− n+3 4 − n+1 2 −1 2 = n+5 2 and vivi+1 has weight f(vn+3 2 ) −f(vn+5 2 ) = n+3 2 . similarly, for each 2 ≤ i ≤ n−1 2 even, vi−1vi has weight n + 5 − i and vi+1vi has weight n + 4− i; thus, {n+9 2 , n+11 2 , . . . ,n + 3} is the set formed by the weights of these edges. hence, the weights on the edges of cn+2 form the set {1, 2, . . . ,n + 3}−{n+72 }. 106 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 since vn−1 2 and vn−1 2 +3 are three units apart, when they are connected, the cycles c4 and cn are formed, i.e., the graph g is obtained. the weight of the new edge is f(vn−1 2 ) − f(vn−1 2 +3) = n + 4 − n−1 4 − n+5 2 −1 2 = n+7 2 , that is, the weight that has not been obtained on cn+2. therefore, f is a graceful labeling of g. let e1,e2 ∈ e(g), we define the distance between e1 and e2, denoted by dist(e1,e2), as the length of the shortest path connecting an endvertex of e1 to an endvertex of e2. in the next theorem we prove that the duplication of two edges of the cycle cn is a graceful graph when the edges duplicated are at distance 1, 2, or 3. theorem 4.2. let e1,e2 be non-incident edges of cn and g be the graph obtained from cn by duplicating e1 and e2. if dist(e1,e2) ∈ {1, 2, 3}, then g is a graceful graph when n is odd or an α-graph when n is even. proof. the graph g has order n + 4 and size n + 6. let e1,e2 ∈ e(cn) such that σ = dist(e1,e2) ∈ {1, 2, 3}. the graph g can be formed by connecting four subgraphs, denoted by g1,g2,g3,g4, where each gi is vertex amalgamated to gi+1 and g4 is connected to g1 with an edge. on gi, we distinguish two vertices, denoted by ui and vi; thus, vi is amalgamated to ui+1 and an edge is used to connect v4 and u1. these subgraphs are chosen in such a way that g1 ∼= c4, c2 ∼= pr, and g3 ∼= ps, where s = n−r−σ and r =   n 2 if n ≡ 0(mod 4), n+1 2 if n ≡ 1(mod 4), n 2 if n ≡ 2(mod 4), n+3 2 if n ≡ 3(mod 4). the graph g4 varies with σ; in general, it can be described as the vertex amalgamation of c4 and a path of order σ. all these subgraphs admit an α-labeling. the labeling of g1 is (0, 4, 1, 2); the labeling of the paths g2 and g3 is the one given in section 2 or its complementary; the labeling of g4 is given in the following diagram, where each residue class of n is considered, the vertices u4 and v4 are highlighted. in g1, the vertices u1 and v1 are those labeled 1 and 2, respectively. in g2 and g3, the distinguished vertices are the corresponding end-vertices. the α-labeling of g1 is transformed into a (n + 3)-graceful labeling, thus u1 is labeled 1 and v1 is labeled n + 4. for i = 2, 3, the α-labeling of gi is chosen based on the new labeling of gi−1, then it is amplified and shifted in such a way that the label of ui matches with the label of vi−1, in this way there is no repetition of labels, except for the vertices that are going to be amalgamated. thus, assuming that f is the α-labeling of the path given in section 2, the α-labeling of g2 is f shifted 2 units and amplified to produce a largest weight equal to n + 2. the α-labeling of g3 is f only when n ≡ 0(mod 4), otherwise it is f, this is amplified to produce a largest weight equal to n+4 2 when n is even, n+3 2 when n ≡ 1(mod 4), or n+1 2 when n ≡ 3(mod 4). the α-labeling of g4, taken from the previous chart, is just shifted based on the final label of v3. then, the weights on the edges of the union of these four graphs form the set {1, 2, . . . ,n − 6} − {x}, where x = n+6 2 when n is even, x = n+5 2 when n ≡ 1(mod 4), or x = n+3 2 when n ≡ 3(mod 4). hence, the information of the following chart holds. final label of v3 final label of v4 weight of u1v4 n ≡ 0(mod 4) n/2 (n + 8)/2 (n + 6)/2 n ≡ 1(mod 4) (n + 11)/2 (n + 7)/2 (n + 5)/2 n ≡ 2(mod 4) (n + 2)/2 (n + 8)/2 (n + 6)/2 n ≡ 3(mod 4) (n + 11)/2 (n + 5)/2 (n + 3)/2 consequently, if we connect with an edge the vertices u1 and v4 we obtain an edge with a weight that complements the set of weights on the concatenation of these graphs. therefore, the graph g is 107 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 different α-labeled alternatives for the graph g4 σ 1 2 3 n ≡ 0(mod 4) n ≡ 1(mod 4) n ≡ 2(mod 4) n ≡ 3(mod 4) 4 0 1 2 v4 u4 2 4 3 0 3 0 2 4 1 4 3 0 4 1 5 2 0 v4 u4 3 4 0 2 5 3 2 5 4 0 2 3 0 1 5 5 3 4 0 1 6 v4 u4 3 1 2 6 5 0 4 3 5 0 2 6 2 3 1 6 4 0 graceful. furthermore, when n is even, g and its labeling are bipartite, which implies that g is indeed an α-graph. the boundary value of this α-labeling is λ = n+2 2 when n ≡ 0(mod 4) and λ = n+6 2 when n ≡ 2(mod 4). in the previous section we study how to duplicate any vertex on the spine of a caterpillar, now we analyze how to duplicate any edge on the spine. proposition 4.3. let g be a caterpillar of diameter at least three. an α-graph is obtained by duplicating any edge on the spine of g. proof. let e = v1v2 be any edge on the spine of g; thus, g−e has two components that are caterpillars on their own. we denote these components by g1 and g2 and assume that vi ∈ v (gi) and ni is the size of gi. we know that there exists an α-labeling fi of gi, with boundary value λi, such that f1(v1) = λ1 and f2(v2) = n2. if h is the graph that result of the duplication of e, then h can be seen as the chain graph obtained using g1, c4, and g2 in this specific order. to generate an α-labeling of h we proceed as follows. the function f1 is transformed into a (n2 + 5)-graceful labeling; this new labeling uses the labels in {0, 1, . . . ,λ1} ∪ {n2 + λ1 + 5,n2 + λ1 + 6, . . . ,n1 + n2 + 4} and induces the weights in {n2 + 5,n2 + 6, . . . ,n1 + n2 + 4}. the labeling f2 is shifted λ1 + 2 units; thus, the new labeling of g2 uses the labels in {λ1 + 2,λ1 + 3, . . . ,n2 + λ1 + 2} and induces the weights in {1, 2, . . . ,n2}. the α-labeling of c4 is the one that assigns the labels 0, 4, 1, 2 to the consecutive vertices; this labeling is transformed into a (n2 + 1)-graceful labeling shifted λ1 units. hence, the new labels are λ1,n2 +λ1 + 4,λ1 + 1 and n2 +λ1 + 2, respectively; the induced weights are {n2 + 1,n2 + 2,n2 + 3,n2 + 4}. note that the weights on the edges of the union of these three graphs form the set {1, 2, . . . ,n1 + n2 + 4}, where n1 + n2 + 4 is the size of h, and the labels are in {1, 2, . . . ,n1,n2 + 4}, being λ1 and n2 + λ1 + 2 the only labels used twice. so, to form the graph h we identify the vertices labeled λ1 of g1 and c4 and the vertices labeled n2 + λ1 + 2 of g2 and c4. since all the labelings used are bipartite, the final labeling of h is in fact an α-labeling. 108 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 an important characteristic of this labeling of h is that its boundary value is assigned either on a leaf maximum of eccentricity or on a vertex adjacent to a leaf of maximum eccentricity. therefore, we can create a chain graph using multiple graphs similar to h, that is, built in the same way, to produce an α-graph that is the result of duplicating multiple edges of the spine of a caterpillar. thus, we have the following corollary. corollary 4.4. let g be a caterpillar of diameter at least three. an α-graph is obtained by duplicating any number of edges on the spine of g. in figure 4 we show an example of this corollary, where the four edges on the spine of a caterpillar of size 12 have been duplicated. 0 1 2 3 4 5 6 7 8 9 10 11 24 22 21 19 17 15 16 14 12 figure 4. α-labeling of a graph obtained by duplicating some edges of a caterpillar 5. connecting multiple copies of an α-graph some authors used g�h to represent the traditional cartesian product of the graphs g and h, the reason for this notation is based on the fact that a given edge uv ∈ e(g) induces a square in g×h. in this section we present a graph obtained with r copies of a graph g, that could be written as g ./ pr. the r-tie of a graph g, is the graph h obtained with r copies of g, say g1,g2, . . . ,gr, where every vertex in gi is adjacent to all the vertices adjacent to its copy in gi+1. more formally, for each i ∈ {1, 2, . . . ,r}, let gi be a copy of a graph g of order m and size n, where v (gi) = {vi1,vi2, . . . ,vim}; assuming that vi+1k is the copy of v i k in gi+1, we say that a graph h is the r-tie of g if v (h) = ∪ r i=1v (gi) and e(h) = ∪ri=1e(gi) ∪{u i jv i+1 k : 1 ≤ i ≤ r − 1 ∧u i jv i k ∈ e(gi)}. thus, we have that h is a graph of order rm and size n(3r − 2) that can be decomposed into 3r − 2 copies of g. in the next theorem we prove that when g is an α-graph, any r-tie of g is also an α-graph. theorem 5.1. if g is an α-graph, then for each r ≥ 2 the r-tie of g is also an α-graph. proof. suppose that g is an α-graph of order m and size n with stable sets a and b. then, there exists an α-labeling f of g that assigns the label 0 to a vertex of a; this implies that the boundary value of f is at least λ = |a| − 1. let h be the r-tie of g, where r ≥ 2; if v (g) = {v1,v2, . . . ,vm}, then v (gi) = {vi1,vi2, . . . ,vim} where gi is the ith copy of g in h and 1 ≤ i ≤ r. consider the following labeling of the vertices of h: g(vij) = { (3r − 2)f(vj) + (i− 1) if f(vj) ≤ λ, (3r − 2)f(vj) + 2(i− 1) if f(vj) > λ. 109 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 note that the first part of this function is strictly increasing, while the second part is strictly decreasing. moreover, when f(vj) ≤ λ, we get that min{g(vij) : 1 ≤ i ≤ r} = 0 and max{g(v i j) : 1 ≤ i ≤ r} = (3r−2)λ+r−1. when f(vj) > λ, we get that min{g(vij) : 1 ≤ i ≤ r} = (3r−2)(λ+ 1)−2(r−1) = (3r − 2)λ + r and max{g(vij) : 1 ≤ i ≤ r} = (3r − 2)n− 2(1 − 1) = n(3r − 2). hence, g is an injective function with range in the interval [0,n(3r − 2)]. if a is the stable set of h that contains the vertex labeled 0 by g, then the largest label on a vertex of a is λ = (3r − 2)λ + r − 1. now we turn our attention to the weights induced by g on the edges of h. let vavb ∈ e(g) such that f(vb) − f(va) = w, where w ∈ {1, 2, . . . ,n}. then, g(vib) = (3r − 2)f(vb) − 2(i − 1), g(v i a) = (3r − 2)f(va) + (i− 1), g(vi+1b ) = (3r − 2)f(vb) − 2i, and g(v i+1 a ) = (3r − 2)f(va) + i. therefore, g(vib) −g(v i a) = (3r − 2)(f(vb) −f(va)) − 3(i− 1) = (3r − 2)w − 3i + 3, g(vib) −g(v i+1 a ) = (3r − 2)(f(vb) −f(va)) − 3i + 2 = (3r − 2)w − 3i + 2, g(vi+1b ) −g(v i a) = (3r − 2)(f(vb) −f(va)) − 3i + 1 = (3r − 2)w − 3i + 1, g(vi+1b ) −g(v i+1 a ) = (3r − 2)(f(vb) −f(va)) − 3i = (3r − 2)w − 3i. in other terms, these four weights form the set wi = {(3r−2)w−3i+k : 0 ≤ k ≤ 3}, where 1 ≤ i ≤ r−1 and ∪r−1i=1 wi = [(3r − 2)w − 3(r − 1), (3r − 2)w]. taking the union of these intervals over all the values of w, we get ∪nw=1[(3r − 2)w − 3(r − 1), (3r − 2)w] = [1,n(3r − 2)]. thus, the set of weights induced by g on the edges of h is {1, 2, . . . ,n(3r − 2)}. therefore, g is a bipartite graceful labeling, in other terms, g is an α-labeling of h with boundary value λ. in figure 5 we show an example of this labeling method for g ∼= c8 and r = 3. 24 26 28 31 33 35 45 47 49 52 54 56 0 1 2 7 8 9 14 15 16 21 22 23 figure 5. α-labeling of the 3-tie of c8 6. k-vertex amalgamation the k-vertex amalgamation is the process that allows us to construct a new graph starting with two graphs g1 and g2 each of order at least k, where k selected vertices from each graph are merged. note that if two adjacent vertices from g1 are amalgamated with two adjacent vertices from g2 we obtain a 110 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 multigraph. in the context of this work, the k vertices on each set are pairwise non-adjacent, in this way, we can be sure that the outcome of the amalgamation is indeed a graph, that is, there are no multiple edges present. lemma 6.1. for i = 1, 2, let γi be a tree with stable sets ai and bi. if γi admits an α-labeling, then there exists an α-graph obtained amalgamating k vertices of b1 with k vertices of b2, where 1 ≤ k ≤ min{|b1|, |b2|}. proof. for i = 1, 2, assume that γi is an α-tree of size ni with stable sets ai and bi, where |ai| = ai and |bi| = bi. any graph obtained via k-vertex amalgamation of γ1 and γ2 is bipartite of size n1+n2 and order n1 +n2 +2−k, where 1 ≤ k ≤ min{b1,b2} is the amount of pairs of vertices amalgamated. since γi is an αtree, there exists an α-labeling fi of γi that assigns the label 0 to a vertex of ai. the labeling f2 is shifted a1+k−1 units; thus, the set of new labels is l2 = {a1+k−1,a1+k,. . . ,a1+a2+k−2}and the set of induced weights is still w2 = {1, 2, . . . ,n2}. the labeling of γ1 is transformed into a (n2 + 1)-graceful labeling. hence, the set of labels on the vertices of γ1 is now l1 = {0, 1, . . . ,a1−1}∪{n2+a1,n2+a1+1, . . . ,n1+n2} and the set of induced weights is w1 = {n2 + 1,n2 + 2, . . . ,n1 + n2}. note that w1∪w2 = {1, 2, . . . ,n1 +n2} and that l1∩l2 = {n2 +a1,n2 +a1 + 1, . . . ,n2 +a1 +k−1}, i.e., a set of cardinality k formed by consecutive integers. so, identifying the vertices of b1 and b2 with the same label we obtain a graph with an α-labeling. in figure 6 we show an example of the construction presented in this lemma, where γ2 has size 9, a2 = 6, b2 = 4, γ1 has size 10, a1 = 5, b1 = 6, the graph on the left side corresponds to the amalgamation of k = 4 vertices while the graph on the right side is obtained with k = 3. γ1 : 0 1 2 3 4 5 9 8 7 6 γ2 : 10 9 8 7 6 5 0 1 2 3 4 k = 4 : 8 9 10 11 12 13 19 18 17 16 15 14 0 1 2 3 4 k = 3 : 7 8 9 10 11 12 1319 18 17 16 15 14 0 1 2 3 4 figure 6. α-labelings of some k-vertex amalgamations of two α-trees the core of the proof of this lemma can be found on the fact that the labels on a stable set of an α-labeled tree are consecutive integers; this characteristic is also present in the α-labelings of some other graphs like complete bipartite graphs and some cycles. in particular, the well-known α-labeling of the cycle c4n assigns consecutive integers to the elements of one stable set, therefore, it can be used in the context of this lemma. when γ1 = γ2 = c4n, the edge amalgamation of the 2n vertices of one stable set results in a graph h that can be understood as well as the supersubdivision of every edge of c2n where each edge of this last cycle is replaced by a copy of c4 ∼= k2,2. summarizing, using this lemma together with the α-labeling of c4n given by rosa in [14], we can prove the following theorem. theorem 6.2. let n ≥ 2, if every edge of the cycle c2n is supersubdivided, that is, replaced by k2,2, the resulting graph admits an α-labeling. 111 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 in figure 7 we show two examples of this construction for the cycles c4 and c6. 16 15 1413 0 8 1 9 311 4 12 24 23 22 212019 0 12 1 13 2 14 416 5 17 6 18 figure 7. α-labelings of a supersubdivision of c4 and c6 a quadrangular cactus is a connected graph all of whose blocks are squares (the cycle c4 ∼= k2,2) and the block-cutpoint graph is a tree. the lemma also allows us to prove the existence of an α-labeling for a family of quadrangular cacti. theorem 6.3. if t is an α-tree such that all the vertices of one stable set have degree two, then the quadrangular cactus obtained by duplicating all these vertices is an α-graph. proof. to obtain an α-graph from t we just need to apply the construction in lemma 6.1, where γ1 ∼= γ2 ∼= t and the α-labeling of t is one that assigns the largest label to the vertices of the stable set containing no interior vertices. the fact that each vertex in this stable set has degree two guarantees the obtainment of a square, for every subpath u1 −v −u2, where v is the vertex of degree two. we show an example of a quadrangular cactus in figure 8, which α-labeling was attained following the procedure of lemma 6.1. we must observe here that the condition that states that the degree must be two can be dropped, the resulting graph is still an α-graph but it is not a cactus. 9 10 11 12 13 14 15 16 25 2627 28 29 30 31 32 0 1 2 3 4 5 6 7 8 figure 8. α-labeling of a quadrangular cactus built on an α-tree we conclude this work with another application of lemma 6.1. the m × n grid is the cartesian product g = pm × pn. assuming that the vertices of this graph have integer coordinates, a cell of g is the subgraph induced by the vertices (i,j), (i + 1,j), (i + 1,j + 1), and (i,j + 1). this graph can be understood as a sequence of vertex amalgamations of paths. in particular, if t ≥ 0, the grid pm ×pm+t is obtained with the sequence of paths h1,h2, . . . ,h2m+t−2, where 112 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 hi ∼=   p2i+1 if 1 ≤ i ≤ m− 1, p2m if m ≤ i ≤ m + t− 1, h2m+t−1−i if m + t ≤ i ≤ 2m + t− 2, where all the vertices in the largest stable set of the ith term of the sequence are amalgamated with all the vertices in the smallest stable set of the (i + 1)th term. in other words, an α-labeling of g can be obtained applying the method of lemma 6.1 to this sequence of paths. this labeling does not correspond with the α-labeling of the grid given by jungreis and reid [8]. in [4], barrientos and minion introduced the concept of analogous caterpillars. the caterpillars g1 and g2 are said to be analogous if the stable sets of g1 have the same cardinality as the stable sets of g2. suppose that for each i ∈ {1, 2, . . . , 2m + t− 2}, gi is analogous to the path hi defined above. we say that g is an irregular grid of order m× (m + t) if g is obtained from pm ×pm+t by replacing the subgraph hi by an analogous gi. note that g is also a bipartite graph where all the cells are copies of c4 too. we claim that all irregular grids are α-graphs. theorem 6.4. all irregular grids are α-graphs. proof. suppose that g is an irregular grid of order m× (m + t) built with the sequence of subgraphs g1,g2, . . . ,g2m+t−2. we assume that the stable set bi of gi has the same cardinality that the stable set bi+1 of gi+1 and that all the vertices in bi and bi+1 are amalgamated. since each of these subgraphs is a caterpillar, there exists an α-labeling for each of them. for every 2 ≤ i ≤ 2m + t − 2, we apply the lemma with γ2 = gi and γ1 being the graph obtained with g1,g2, . . . ,gi−1. when this process is concluded we have an irregular graph with an α-labeling as claimed. in figure 9 we present an example of an irregular grid of order 6 × 6 which α-labeling is obtained using the procedure described in the last theorem. 0 2 3 4 8 9 10 11 12 18 19 20 21 22 26 27 28 30 60 59 56 55 54 53 48 47 46 45 44 43 38 37 36 35 32 31 figure 9. α-labeling of an irregular grid acknowledgment: i would like to thank the referee for his/her valuable comments and suggestions. 113 c. barrientos / j. algebra comb. discrete appl. 9(2) (2022) 101–114 references [1] c. barrientos, s. barrientos, on graceful supersubdivisions of graphs, bull. inst. combin. appl. 70 (2014) 77–85. [2] c. barrientos, s. minion, alpha labelings of snake polyominoes and hexagonal chains, bull. inst. combin. appl. 74 (2015) 73–83. [3] c. barrientos, s. minion, new attack on kotzig’s conjecture, electron. j. graph theory appl. 4(2) (2016) 119–131. [4] c. barrientos, s. minion, snakes and caterpillars in graceful graphs, j. algorithms and comput. 50(2) (2018) 37–47. [5] g. chartrand, l. lesniak, graphs & digraphs, 4th edition. crc press, boca raton (2005). [6] j. a. gallian, a dynamic survey of graph labeling, electron. j. combin. (2020) #ds6. [7] s. w. golomb, how to number a graph, r. c. read (editor), graph theory and computing, academic press, new york (1972) 23-37. [8] d. jungreis, m. reid, labeling grids, ars combin. 34 (1992) 167–182. [9] v. j. kaneria, s. k. vaidya, g. v. ghodasara, s. srivastav, some classes of disconnected graceful graphs, proc. first internat. conf. emerging technologies and appl. engin. tech. sci. (2008) 1050– 1056. [10] a. kotzig, on certain vertex valuations of finite graphs, util. math. 4 (1973) 67–73. [11] s. c. lópez, f. a. muntaner-batle, graceful, harmonious and magic type labelings: relations and techniques, springer (2017). [12] m. maheo, h. thuillier, on d-graceful graphs, ars combin. 13 (1982) 181–192. [13] g. ringel, problem 25, in: theory of graphs and its applications, proc. symposium smolenice 1963, czech. acad. sci., prague, czech. (1964) 162. on certain valuations of the vertices of a graph, theory of graphs (internat. symposium, rome, july 1966) [14] a. rosa, on certain valuations of the vertices of a graph, theory of graphs (internat. symposium, rome, july 1966), gordon and breach ny and dunod paris (1967) 349–355. [15] a. rosa, labelling snakes, ars combin. 3 (1977) 67–74. [16] a. rosa, j. širáň, bipartite labelings of trees and the gracesize, j. graph theory 19(2) (1995) 201–215. [17] g. sethuraman, p. selvaraju, gracefulness of arbitrary supersubdivisions of graphs, indian j. pure appl. math. 32 (2001) 1059–1064. [18] p. j. slater, on k-graceful graphs, proc. of the 13th s. e. conf. on combinatorics, graph theory and computing (1982) 53–57. [19] b. yao, x. liu, m. yao, connections between labellings of trees, bull. iranian math. soc. 43(2) (2017) 275–283. 114 https://mathscinet.ams.org/mathscinet-getitem?mr=3204924 https://mathscinet.ams.org/mathscinet-getitem?mr=3204924 https://mathscinet.ams.org/mathscinet-getitem?mr=3363625 https://mathscinet.ams.org/mathscinet-getitem?mr=3363625 http://dx.doi.org/10.5614/ejgta.2016.4.2.1 http://dx.doi.org/10.5614/ejgta.2016.4.2.1 http://dx.doi.org/10.22059/jac.2018.69503 http://dx.doi.org/10.22059/jac.2018.69503 https://www.worldcat.org/title/graphs-digraphs/oclc/55510638 https://doi.org/10.37236/27 https://doi.org/10.1016/b978-1-4832-3187-7.50008-8 https://doi.org/10.1016/b978-1-4832-3187-7.50008-8 https://mathscinet.ams.org/mathscinet-getitem?mr=1206560 https://mathscinet.ams.org/mathscinet-getitem?mr=384616 https://link.springer.com/book/10.1007/978-3-319-52657-7 https://link.springer.com/book/10.1007/978-3-319-52657-7 https://mathscinet.ams.org/mathscinet-getitem?mr=666937 https://doi.org/10.1002/jgt.3190190207 https://doi.org/10.1002/jgt.3190190207 https://mathscinet.ams.org/mathscinet-getitem?mr=1846112 https://mathscinet.ams.org/mathscinet-getitem?mr=1846112 http://bims.iranjournals.ir/article_930.html http://bims.iranjournals.ir/article_930.html introduction the essential results duplicating vertices duplicating edges connecting multiple copies of an -graph k-vertex amalgamation references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.47485 j. algebra comb. discrete appl. 3(2) • 45–59 received: 16 april 2015 accepted: 22 december 2015 journal of algebra combinatorics discrete structures and applications on the metric dimension of rotationally-symmetric convex polytopes∗ research article muhammad imran, syed ahtsham ul haq bokhary, a. q. baig abstract: metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). let f be a family of connected graphs gn : f = (gn)n ≥ 1 depending on n as follows: the order |v (g)| = ϕ(n) and lim n→∞ ϕ(n) = ∞. if there exists a constant c > 0 such that dim(gn) ≤ c for every n ≥ 1 then we shall say that f has bounded metric dimension, otherwise f has unbounded metric dimension. if all graphs in f have the same metric dimension, then f is called a family of graphs with constant metric dimension. in this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. it is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. it is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. 2010 msc: 05c12 keywords: metric dimension, basis, resolving set, prism, antiprism, convex polytopes 1. notation and preliminary results slater refereed to the metric dimension of a graph as its location number and motivated the study of this invariant by its application to the placement of a minimum number of sonar/loran detecting devices ∗ this research is supported by the grant of higher education commission of pakistan ref. no. 20367/nrpu/r&d/hec/12/831. muhammad imran (corresponding author); department of mathematics, school of natural sciences (sns), national university of sciences and technology (nust), sector h-12, islamabad, pakistan (email: imrandhab@gmail.com). syed ahtsham ul haq bokhary; centre for advanced studies in pure and applied mathematics, bahauddin zakariya university, multan, pakistan (email: sihtsham@gmail.com). a. q. baig; department of mathematics, comsats institute of information technology, attock, pakistan (email: aqbaig1@gmail.com). 45 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 in a network so that the position of every vertex in the network can be uniquely described in terms of its distances to the devices in the set ([18],[19]). these concepts have also some applications in chemistry for representing chemical compounds ([5],[12]) or to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures [16]. if g is a connected graph, the distance d(u, v) between two vertices u, v ∈ v (g) is the length of a shortest path between them. let w = {w1, w2, . . . , wk} be an ordered set of vertices of g and let v be a vertex of g. the representation r(v|w ) of v with respect to w is the k-tuple (d(v, w1), d(v, w2), d(v, w3), . . . , d(v, wk)). w is called a resolving set [5] or locating set [18] if every vertex of g is uniquely identified by its distances from the vertices of w , or equivalently, if distinct vertices of g have distinct representations with respect to w . a resolving set of minimum cardinality is called a basis for g and this cardinality is the metric dimension or location number of g, denoted by dim(g) [3]. the concepts of resolving set and metric basis have previously appeared in the literature (see [3-6, 8-12, 15-21]). for a given ordered set of vertices w = {w1, w2, . . . , wk} of a graph g, the ith component of r(v|w ) is 0 if and only if v = wi. thus, to show that w is a resolving set it suffices to verify that r(x|w ) 6= r(y|w ) for each pair of distinct vertices x, y ∈ v (g)\w . a useful property in finding dim(g) is the following lemma: lemma 1.1. [20] let w be a resolving set for a connected graph g and u, v ∈ v (g). if d(u, w) = d(v, w) for all vertices w ∈ v (g) \{u, v}, then {u, v}∩w 6= ∅. by denoting g + h the join of g and h a wheel wn is defined as wn = k1 + cn, for n ≥ 3, a fan is fn = k1 + pn for n ≥ 1 and jahangir graph j2n, (n ≥ 2) (also known as gear graph) is obtained from the wheel w2n by alternately deleting n spokes. buczkowski et al. [3] determined the dimension of wheel wn, caceres et al. [8] the dimension of fan fn and tomescu and javaid [21] the dimension of jahangir graph j2n. theorem 1.2. ([3], [8], [21]) let wn be a wheel of order n ≥ 3, fn be fan of order n ≥ 1 and j2n be a jahangir graph. then (i) for n ≥ 7, dim(wn) = b2n+25 c; (ii) for n ≥ 7, dim(fn) = b2n+25 c; (iii) for n ≥ 4, dim(j2n) = b2n3 c. the metric dimension of all these plane graphs depends upon the number of vertices in the graph. on the other hand, we say that a family g of connected graphs is a family with constant metric dimension if dim(g) is finite and does not depend upon the choice of g in g. in [5] it was shown that a graph has metric dimension 1 if and only if it is a path, hence paths on n vertices constitute a family of graphs with constant metric dimension. similarly, cycles with n(≥ 3) vertices also constitute such a family of graphs as their metric dimension is 2 and does not depend upon on the number of vertices n. javaid et al. proved in [11] that the plane graph antiprism an constitute a family of regular graphs with constant metric dimension as dim(an) = 3 for every n ≥ 5. the prism and the antiprism are archimedean convex polytopes defined e.g. in [13]. the metric dimension of cartesian product of graphs has been discussed in [4, 17]. the metric dimension of some classes of convex polytopes has been determined in [9] and [10] where it was shown that these classes of convex polytopes have constant metric dimension 3 and following open problems were raised in [9] and [10]. open problem [9]: is it the case that the graph of every convex polytope has constant metric dimension? open problem [10]: let g′ be the graph of a convex polytope obtained from the graph of convex polytope g by adding extra edges in g such that v (g′) = v (g). is it the case that g′ and g will always have the same metric dimension? note that the problem of determining whether dim(g) < k is an np-complete problem [6]. some bounds for this invariant, in terms of the diameter of the graph, are given in [15] and it was shown in [5, 15–17] that the metric dimension of trees can be determined efficiently. it appears unlikely that significant progress can be made in determining the dimension of a graph unless it belongs to a class for which the distances between vertices can be described in some systematic manner. 46 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 bača defined in [2] the graph of convex polytope rn which is obtained as a combination of the graph of a prism and the graph of an antiprism. the prism and antiprism have constant metric dimension [4, 11] and it was proved in [9] that the graph of convex polytope rn also has constant metric dimension. in this paper, we extend this study to some classes of convex polytopes which are obtained by combination of two different graph of convex polytopes. we prove that these classes of convex polytopes have constant metric dimension and only three vertices appropriately chosen suffice to resolve all the vertices of these classes of convex polytopes. in what follows all indices i which do not satisfy the given inequalities will be taken modolu n. 2. the graph of convex polytope bn the graph of convex polytope bn (fig. 1) consisting of 2n 4-sided faces, n 3-sided faces, n 5-sided faces and a pair of n-sided faces is obtained by the combination of the graph of convex polytope qn [2] and graph of a prism dn. we have v (bn) = {ai; bi; ci; di, ei : 1 ≤ i ≤ n} and e(bn) = {aiai+1; bibi+1; didi+1; eiei+1 : 1 ≤ i ≤ n} ∪{aibi; bici; bi+1ci; cidi; diei : 1 ≤ i ≤ n}. for our purpose, we call the cycle induced by {ai : 1 ≤ i ≤ n}, the inner cycle, cycle induced by an−2 an−1 an a1 a2 a3 a4 bn−2 bn−1 bn b1 b2 b3 b4 cn−2 cn−1 cnc1 c2 c3 dn−2 dn−1 dnd1 d2 d3 en−2 en−1 ene1 e2 e3 figure 1. the graph of convex polytope bn {bi : 1 ≤ i ≤ n}, the interior cycle, cycle induced by {di : 1 ≤ i ≤ n}, the exterior cycle, cycle induced by {ei : 1 ≤ i ≤ n}, the outer cycle and set of vertices {ci : 1 ≤ i ≤ n}, the set of interior vertices. the metric dimension of graph of convex polytope qn and graph of a prism dn have been studied in [9] and [4]. in the next theorem, we show that the metric dimension of the graph of convex polytope bn is 3. note that the choice of appropriate basis vertices (also referred to as landmarks in [14]) is core of the problem. theorem 2.1. for n ≥ 6, let the graph of convex polytopes be bn; then dim(bn) = 3. proof. we will prove the above equality by double inequalities. we consider the two cases. case(i) when n is even. in this case, we can write n = 2k, k ≥ 3, k ∈ z+. let w = {a1, a2, ak+1} ⊂ v (bn), we show that w is a resolving set for bn in this case. for this we give representations of any vertex of v (bn)\w with 47 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 respect to w . representations of the vertices on inner cycle are r(ai|w ) = { (i− 1, i− 2, k − i + 1), 3 ≤ i ≤ k; (2k − i + 1, 2k − i + 2, i−k − 1), k + 2 ≤ i ≤ 2k. representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k + 1), i = 1; (i, i− 1, k − i + 2), 2 ≤ i ≤ k + 1; (2k − i + 2, 2k − i + 3, i−k), k + 2 ≤ i ≤ 2k. representations of the set of interior vertices are r(ci|w ) =   (2, 2, k + 1), i = 1; (i + 1, i, k − i + 2), 2 ≤ i ≤ k; (k + 1, k + 1, 2), i = k + 1; (2k − i + 2, 2k − i + 3, i−k + 1), k + 2 ≤ i ≤ 2k. representations of the vertices on interior cycle are r(di|w ) =   (3, 3, k + 2), i = 1; (i + 2, i + 1, k − i + 3), 2 ≤ i ≤ k; (k + 2, k + 2, 3), i = k + 1; (2k − i + 3, 2k − i + 4, i−k + 2), k + 2 ≤ i ≤ 2k. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 3), i = 1; (i + 3, i + 2, k − i + 4), 2 ≤ i ≤ k; (k + 3, k + 3, 4), i = k + 1; (2k − i + 4, 2k − i + 5, i−k + 3), k + 2 ≤ i ≤ 2k. we note that there are no two vertices having the same representations implying that dim(bn) ≤ 3. on the other hand, we show that dim(bn) ≥ 3 by proving that there is no resolving set w such that |w | = 2. suppose on contrary that dim(bn) = 2, then there are following possibilities to be discussed. (1) both vertices are in the inner cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is at (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(an|{a1, at}) = r(b1|{a1, at}) = (1, t) and for t = k + 1, r(a2|{a1, ak+1}) = r(an|{a1, ak+1}) = (1, k−1), a contradiction. (2) both vertices are in the interior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is bt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(bn|{b1, bt}) = r(cn|{b1, bt}) = (1, t) and for t = k + 1, r(b2|{b1, bk+1}) = r(bn|{b1, bk+1}) = (1, k − 1), a contradiction. (3) both vertices are in the set of interior vertices. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is ct (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(b1|{c1, ct}) = r(d1|{c1, ct}) = (1, t) and for t = k + 1, r(d2|{c1, ck+1}) = r(dn|{c1, ck+1}) = (1, k − 1), a contradiction. (4) both vertices are in the exterior cycle. without loss of generality we suppose that one resolving vertex is d1. suppose that the second resolving vertex is dt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(c1|{d1, dt}) = r(dn|{d1, dt}) = (1, t) and for t = k + 1, r(d2|{d1, dk+1}) = r(dn|{d1, dk+1}) = (1, k − 1), a contradiction. (5) both vertices are in the outer cycle. without loss of generality we suppose that one resolving vertex is e1. suppose that the second resolving vertex is et (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(d1|{e1, et}) = r(en|{e1, et}) = (1, t) and for t = k + 1, r(e2|{e1, ek+1}) = r(en|{e1, ek+1}) = (1, k − 1), a contradiction. (6) one vertex is in the inner cycle and other in the interior cycle. without loss of generality we suppose 48 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 that one resolving vertex is a1. suppose that the second resolving vertex is bt (1 ≤ t ≤ k + 1). then for t = 1, we have r(a2|{a1, b1}) = r(an|{a1, b1}) = (1, 2) and when 2 ≤ t ≤ k + 1, r(a2|{a1, bk+1}) = r(b1|{a1, bk+1}) = (1, t− 1), a contradiction. (7) one vertex is in the inner cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(bn|{a1, ct}) = r(cn|{a1, ct}) = (2, t + 1) and when t = k + 1, r(a3|{a1, bk+1}) = r(cn|{a1, bk+1}) = (2, k), a contradiction. (8) one vertex is in the inner cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for t = 1, we have r(d2|{a1, d1}) = r(e1|{a1, d1}) = (4, 1). if 2 ≤ t ≤ k, r(a2|{a1, dt}) = r(b1|{a1, dt}) = (1, t + 1) and when t = k + 1, r(an|{a1, dk+1}) = r(b1|{a1, dk+1}) = (1, k + 1), a contradiction. (9) one vertex is in the inner cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for t = 1, we have r(b2|{a1, e1}) = r(cn|{a1, e1}) = (2, 3) and when 2 ≤ t ≤ k + 1, r(b2|{a1, et}) = r(c1|{a1, et}) = (2, t + 1), a contradiction. (10) one vertex is in the interior cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(bn|{b1, ct}) = r(cn|{b1, ct}) = (1, t + 1). for t = k, we have r(c1|{b1, ck}) = r(bn|{b1, ck}) = (1, k) and when t = k + 1, r(b2|{b1, ck+1}) = r(cn|{b1, ck+1}) = (1, k), a contradiction. (11) one vertex is in the interior cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, dt}) = r(bn|{b1, dt}) = (1, t + 2). for t = k, we have r(bn|{b1, dk}) = r(c1|{b1, dk}) = (1, k + 1) and when t = k + 1, r(b2|{b1, dk+1}) = r(cn|{b1, dk+1}) = (1, k + 1), a contradiction. (12) one vertex is in the interior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, et}) = r(bn|{b1, et}) = (1, t + 3). for t = k, we have r(bn|{b1, ek}) = r(c1|{b1, ek}) = (1, k + 2) and when t = k + 1, r(b2|{b1, ek+1}) = r(cn|{b1, ek+1}) = (1, k + 2), a contradiction. (13) one vertex is in the set of interior vertices and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, r(a1|{c1, dt}) = r(bn|{c1, dt}) = (2, t + 2) and when t = k + 1, we have r(d2|{c1, dk+1}) = r(dn|{c1, dk+1}) = (2, k), a contradiction. (14) one vertex is in the set of interior vertices and other in the outer cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(a1|{c1, et}) = r(bn|{c1, et}) = (2, t + 3) and when t = k + 1, r(e2|{c1, ek+1}) = r(en|{c1, ek+1}) = (3, k − 1), a contradiction. (15) one vertex is in the set of exterior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is d1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(c1|{d1, et}) = r(dn|{d1, et}) = (1, t + 1) and when t = k + 1, we have r(e2|{d1, ek+1}) = r(en|{d1, ek+1}) = (2, k), a contradiction. hence, from above it follows that there is no resolving set with two vertices for v (bn) implying that dim(bn) = 3 in this case. case(ii) when n is odd. in this case, we can write n = 2k + 1, k ≥ 3, k ∈ z+. again we show that w = {a1, a2, ak+1}⊂ v (bn) is a resolving set for bn in this case. for this we give representations of any vertex of v (bn)\w with respect to w . representations of the vertices on inner cycle are r(ai|w ) =   (i− 1, i− 2, k − i + 1), 3 ≤ i ≤ k; (k, k, 1), i = k + 2 (2k − i + 2, 2k − i + 3, i−k − 1), k + 3 ≤ i ≤ 2k + 1. 49 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k), i = 1; (i, i− 1, k − i + 2), 2 ≤ i ≤ k + 1; (k + 1, k + 1, 2), i = k + 2; (2k − i + 3, 2k − i + 4, i−k), k + 3 ≤ i ≤ 2k + 1. representations of set of interior vertices are r(ci|w ) =   (2, 2, k + 1), i = 1; (i + 1, i, k − i + 2), 2 ≤ i ≤ k; (k + 2, k + 1, 2), i = k + 1; (2k − i + 3, 2k − i + 4, i−k + 1), k + 2 ≤ i ≤ 2k + 1. representations of the vertices on exterior cycle are r(di|w ) =   (3, 3, k + 2), i = 1; (i + 2, i + 1, k − i + 3), 2 ≤ i ≤ k; (k + 3, k + 2, 3), i = k + 1; (2k − i + 4, 2k − i + 5, i−k + 2), k + 2 ≤ i ≤ 2k + 1. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 3), i = 1; (i + 3, i + 2, k − i + 4), 2 ≤ i ≤ k; (k + 4, k + 3, 4), i = k + 1; (2k − i + 5, 2k − i + 6, i−k + 3), k + 2 ≤ i ≤ 2k + 1. again we see that there are no two vertices having the same representations which implies that dim(bn) ≤ 3. on the other hand, suppose that dim(bn) = 2, then there are the same possibilities as in case (i) and contradiction can be deduced analogously. this implies that dim(bn) = 3 in this case, which completes the proof. 3. the graph of convex polytope cn the graph of convex polytope cn (fig. 2) consisting of 3n 3-sided faces, n 4-sided faces, n 5-sided faces and a pair of n-sided faces is obtained by the combination of the graph of convex polytope qn [2] and graph of an antiprism an [1]. we have v (cn) = {ai; bi; ci; di; ei : 1 ≤ i ≤ n} and e(cn) = {aiai+1; bibi+1; didi+1; eiei+1 : 1 ≤ i ≤ n} ∪{aibi; bici; cidi; diei; bi+1ci; di+1ei : 1 ≤ i ≤ n}. the graph of convex polytope cn can also be obtained from the graph of convex polytope bn by adding new edges di+1ei and having the same vertex set. i.e. v (cn) = v (bn) and e(cn) = e(bn) ∪{di+1ei : 1 ≤ i ≤ n}. for our purpose, we call the cycle induced by {ai : 1 ≤ i ≤ n}, the inner cycle, cycle induced by {bi : 1 ≤ i ≤ n}, the interior cycle, cycle induced by {di : 1 ≤ i ≤ n}, the exterior cycle, cycle induced by {ei : 1 ≤ i ≤ n}, the outer cycle and set of vertices {ci : 1 ≤ i ≤ n}, the set of interior vertices. the metric dimension of graph of convex polytope qn and graph of an antiprism an have been studied in [9] and [11]. in the next theorem, we show that the metric dimension of the graph of convex polytope cn is 3. again, choice of appropriate landmarks is crucial. 50 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 an−2 an−1 an a1 a2 a3 a4 bn−2 bn−1 bn b1 b2 b3 b4 cn−2 cn−1 cnc1 c2 c3 dn−2 dn−1 dnd1 d2 d3 en−1 en e1 e2 e3 figure 2. the graph of convex polytope cn theorem 3.1. let cn denotes the graph of convex polytope; then dim(cn) = 3 for every n ≥ 6. proof. we will prove the above equality by double inequalities. we consider the two cases. case(i) when n is even. in this case, we can write n = 2k, k ≥ 3, k ∈ z+. let w = {a1, a2, ak+1}⊂ v (cn), we show that w is a resolving set for cn in this case. for this we give representations of any vertex of v (cn)\w with respect to w . representations of the vertices on inner cycle are r(ai|w ) = { (i− 1, i− 2, k − i + 1), 3 ≤ i ≤ k; (2k − i + 1, 2k − i + 2, i−k − 1), k + 2 ≤ i ≤ 2k. representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k + 1), i = 1; (i, i− 1, k − i + 2), 2 ≤ i ≤ k + 1; (2k − i + 2, 2k − i + 3, i−k), k + 2 ≤ i ≤ 2k. representations of the set of interior vertices are r(ci|w ) =   (2, 2, k + 1), i = 1; (i + 1, i, k − i + 2), 2 ≤ i ≤ k; (k + 1, k + 1, 2), i = k + 1; (2k − i + 2, 2k − i + 3, i−k + 1), k + 2 ≤ i ≤ 2k. representations of the vertices on exterior cycle are r(di|w ) =   (3, 3, k + 2), i = 1; (i + 2, i + 1, k − i + 3), 2 ≤ i ≤ k; (k + 2, k + 2, 3), i = k + 1; (2k − i + 3, 2k − i + 4, i−k + 2), k + 2 ≤ i ≤ 2k. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 2), i = 1; (i + 3, i + 2, k − i + 3), 2 ≤ i ≤ k − 1; (k + 3, k + 2, 4), i = k; (2k − i + 3, 2k − i + 4, i−k + 3), k + 1 ≤ i ≤ 2k − 1; (4, 4, k + 3), i = 2k. 51 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 we note that there are no two vertices having the same representations implying that dim(cn) ≤ 3. on the other hand, we show that dim(cn) ≥ 3 by proving that there is no resolving set w such that |w | = 2. suppose on contrary that dim(cn) = 2, then there are following possibilities to be discussed. (1) both vertices are in the inner cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is at (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(an|{a1, at}) = r(b1|{a1, at}) = (1, t) and for t = k + 1, r(a2|{a1, ak+1}) = r(an|{a1, ak+1}) = (1, k−1), a contradiction. (2) both vertices are in the interior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is bt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(bn|{b1, bt}) = r(cn|{b1, bt}) = (1, t) and for t = k + 1, r(b2|{b1, bk+1}) = r(bn|{b1, bk+1}) = (1, k − 1), a contradiction. (3) both vertices are in the set of interior vertices. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is ct (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(b1|{c1, ct}) = r(d1|{c1, ct}) = (1, t) and for t = k + 1, r(d2|{c1, ck+1}) = r(dn|{c1, ck+1}) = (1, k − 1), a contradiction. (4) both vertices are in the exterior cycle. without loss of generality we suppose that one resolving vertex is d1. suppose that the second resolving vertex is dt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(c1|{d1, dt}) = r(dn|{d1, dt}) = (1, t) and for t = k + 1, r(d2|{d1, dk+1}) = r(dn|{d1, dk+1}) = (1, k − 1), a contradiction. (5) both vertices are in the outer cycle. without loss of generality we suppose that one resolving vertex is e1. suppose that the second resolving vertex is et (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(d1|{e1, et}) = r(en|{e1, et}) = (1, t) and for t = k + 1, r(e2|{e1, ek+1}) = r(en|{e1, ek+1}) = (1, k − 1), a contradiction. (6) one vertex is in the inner cycle and other in the interior cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is bt (1 ≤ t ≤ k + 1). then for t = 1, we have r(a2|{a1, b1}) = r(an|{a1, b1}) = (1, 2) and when 2 ≤ t ≤ k + 1, r(a2|{a1, bt}) = r(b1|{a1, bt}) = (1, t− 1), a contradiction. (7) one vertex is in the inner cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(bn|{a1, ct}) = r(cn|{a1, ct}) = (2, t + 1) and when t = k + 1, r(a3|{a1, bk+1}) = r(cn|{a1, bk+1}) = (2, k), a contradiction. (8) one vertex is in the inner cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for t = 1, we have r(d2|{a1, d1}) = r(e1|{a1, d1}) = (4, 1). if 2 ≤ t ≤ k, r(a2|{a1, dt}) = r(b1|{a1, dt}) = (1, t + 1) and when t = k + 1, r(an|{a1, dk+1}) = r(b1|{a1, dk+1}) = (1, k + 1), a contradiction. (9) one vertex is in the inner cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for t = 1, we have r(b2|{a1, et}) = r(cn|{a1, et}) = (2, 3) and when 2 ≤ t ≤ k + 1, r(b2|{a1, et}) = r(c1|{a1, et}) = (2, t + 1), a contradiction. (10) one vertex is in the interior cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(bn|{b1, ct}) = r(cn|{b1, ct}) = (1, t + 1). for t = k, we have r(c1|{b1, ck}) = r(bn|{b1, ck}) = (1, k) and when t = k + 1, r(b2|{b1, ck+1}) = r(cn|{b1, ck+1}) = (1, k), a contradiction. (11) one vertex is in the interior cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, dt}) = r(bn|{b1, dt}) = (1, t + 2). for t = k, we have r(bn|{b1, dk}) = r(c1|{b1, dk}) = (1, k + 1) and when t = k + 1, r(b2|{b1, dk+1}) = r(cn|{b1, dk+1}) = (1, k + 1), a contradiction. (12) one vertex is in the interior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, et}) = r(bn|{b1, et}) = (1, t + 3). for t = k, we have r(bn|{b1, ek}) = r(c1|{b1, ek}) = (1, k + 2) and when t = k + 1, r(b2|{b1, ek+1}) = r(cn|{b1, ek+1}) = (1, k + 1), a contradiction. 52 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 (13) one vertex is in the set of interior vertices and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(a1|{c1, dt}) = r(bn|{c1, dt}) = (2, t + 2), and when t = k + 1, r(d2|{c1, dk+1}) = r(dn|{c1, dk+1}) = (2, k), a contradiction. (14) one vertex is in the set of interior vertices and other in the outer cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(a1|{c1, et}) = r(bn|{c1, et}) = (2, t + 3) and when t = k + 1, r(cn|{c1, ek+1}) = r(e1|{c1, ek+1}) = (2, k), a contradiction. (15) one vertex is in the set of exterior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is d1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k, we have r(c1|{d1, et}) = r(dn|{d1, et}) = (1, t + 1) and when t = k + 1, r(dn|{d1, ek+1}) = r(en|{d1, ek+1}) = (1, k − 1), a contradiction. hence, from above it follows that there is no resolving set with two vertices for v (cn) implying that dim(cn) = 3 in this case. case(ii) when n is odd. in this case, we can write n = 2k + 1, k ≥ 3, k ∈ z+. let w = {a1, a2, ak+1} ⊂ v (cn), we show that w is a resolving set for cn in this case. for this we give representations of any vertex of v (cn)\w with respect to w . representations of the vertices on inner cycle are r(ai|w ) =   (i− 1, i− 2, k − i + 1), 3 ≤ i ≤ k; (k, k, 1), i = k + 2; (2k − i + 2, 2k − i + 3, i−k − 1), k + 3 ≤ i ≤ 2k + 1. representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k + 1), i = 1; (i, i− 1, k − i + 2), 2 ≤ i ≤ k + 1; (k + 1, k + 1, 2), i = k + 2; (2k − i + 3, 2k − i + 4, i−k), k + 3 ≤ i ≤ 2k + 1. representations of the set of interior vertices are r(ci|w ) =   (2, 2, k + 1), i = 1; (i + 1, i, k − i + 2), 2 ≤ i ≤ k; (k + 2, k + 1, 2), i = k + 1; (2k − i + 3, 2k − i + 4, i−k + 1), k + 2 ≤ i ≤ 2k + 1. representations of the vertices on exterior cycle are r(di|w ) =   (3, 3, k + 2), i = 1; (i + 2, i + 1, k − i + 3), 2 ≤ i ≤ k; (k + 3, k + 2, 3), i = k + 1; (2k − i + 4, 2k − i + 5, i−k + 2), k + 2 ≤ i ≤ 2k + 1. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 2), i = 1; (i + 3, i + 2, k − i + 3), 2 ≤ i ≤ k − 1; (k + 3, k + 2, 4), i = k; (k + 3, k + 3, 4), i = k + 1; (2k − i + 4, 2k − i + 5, i−k + 3), k + 2 ≤ i ≤ 2k; (4, 4, k + 3), i = 2k + 1. again we see that there are no two vertices having the same representations which implies that dim(cn) ≤ 3 in this case. on the other hand, suppose that dim(cn) = 2, then there are the same subcases as in case (i) and contradiction can be obtained analogously. this implies that dim(cn) = 3 in this case, which completes the proof. 53 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 note that the result in above theorem gives the positive answer to the open problem raised in [10] in this case. 4. the graph of convex polytope en the graph of convex polytope en is obtained as a combination of graph of convex polytope tn [10] and graph of an antiprism an [1]. the graph of convex polytope en can also be obtained from the graph an−1 an a1a2 a3 a4 bn−2 bn−1 bn b1 b2 b3 b4 cn−2 cn−1 cnc1 c2 c3 dn−2 dn−1 dnd1 d2 d3 en−2 en−1 en e1 e2 figure 3. the graph of convex polytope en of convex polytope cn by adding new edges ai+1bi and having the same vertex set. i.e. v (en) = v (cn) and e(en) = e(cn) ∪{ai+1bi : 1 ≤ i ≤ n}. for our purpose, we call the cycle induced by {ai : 1 ≤ i ≤ n}, the inner cycle, cycle induced by {bi : 1 ≤ i ≤ n}, the interior cycle, cycle induced by {di : 1 ≤ i ≤ n}, the exterior cycle, cycle induced by {ei : 1 ≤ i ≤ n}, the outer cycle and set of vertices {ci : 1 ≤ i ≤ n}, the set of interior vertices. the metric dimension of graph of convex polytope tn and graph of a prism dn have been studied in [10] and [4]. in the next theorem, we show that the metric dimension of the graph of convex polytope en is 3. once again, the choice of appropriate landmarks is crucial. theorem 4.1. let en denotes the graph of convex polytope; then dim(en) = 3 for every n ≥ 6. proof. we will prove the above equality by double inequalities. we consider the two cases. case(i) when n is even. in this case, we can write n = 2k, k ≥ 3, k ∈ z+. let w = {a1, a3, ak+1} ⊂ v (en), we show that w is a resolving set for en in this case. for this we give representations of any vertex of v (en)\w with respect to w . representations of the vertices on inner cycle are r(ai|w ) =   (1, 1, k − 1), i = 2; (i− 1, i− 3, k − i + 1), 4 ≤ i ≤ k; (k − 1, k − 1, 1), i = k + 2; (2k − i + 1, 2k − i + 3, i−k − 1), k + 3 ≤ i ≤ 2k. 54 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k), i = 1; (2, 1, k − 1), i = 2; (i, i− 2, k − i + 1), 3 ≤ i ≤ k; (k, k − 1, 1), i = k + 1; (k − 1, k, 2), i = k + 2; (2k − i + 1, 2k − i + 3, i−k), k + 3 ≤ i ≤ 2k. representations of the vertices on exterior cycle are r(ci|w ) =   (2, 2, k), i = 1; (3, 2, k − 1), i = 2; (i + 1, i− 1, k − i + 1), 3 ≤ i ≤ k − 1; (k + 1, k − 1, 2), i = k; (k, k, 2), i = k + 1; (2k − i + 1, 2k − i + 3, i−k + 1), k + 2 ≤ i ≤ 2k − 1; (2, 3, k + 1), i = 2k. and r(di|w ) =   (3, 3, k + 1), i = 1; (4, 3, k), i = 2; (i + 2, i, k − i + 1), 3 ≤ i ≤ k − 1; (k + 2, k, 3), i = k; (k + 1, k + 1, 3), i = k + 1; (2k − i + 2, 2k − i + 4, i−k + 2), k + 2 ≤ i ≤ 2k − 1; (3, 3, k + 2), i = 2k. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 1), i = 1; (5, 4, k), i = 2; (i + 3, i + 1, k − i + 2), 3 ≤ i ≤ k − 2; (k + 2, k, 4), i = k − 1; (k + 2, k + 1, 4), i = k; (k + 1, k + 2, 4), i = k + 1; (2k − i + 2, 2k − i + 4, i−k + 3), k + 2 ≤ i ≤ 2k − 2; (4, 5, k + 2), i = 2k − 1; (4, 4, k + 2), i = 2k. we note that there are no two vertices having the same representations implying that dim(en) ≤ 3. on the other hand, we show that dim(en) ≥ 3. suppose on contrary that dim(en) = 2, then there are following possibilities to be discussed. (1) both vertices are in the inner cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is at (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(an|{a1, at}) = r(bn|{a1, at}) = (1, t) and for t = k + 1, r(a2|{a1, ak+1}) = r(an|{a1, ak+1}) = (1, k−1), a contradiction. (2) both vertices are in the interior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is bt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(a1|{b1, bt}) = r(bn|{b1, bt}) = (1, t) and for t = k + 1, r(b2|{b1, bk+1}) = r(bn|{b1, bk+1}) = (1, k − 1), a contradiction. (3) both vertices are in the set of interior vertices. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is ct (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(an|{c1, ct}) = r(bn|{c1, ct}) = (2, t + 1) and for t = k + 1, r(a1|{c1, ck+1}) = r(b1|{c1, ck+1}) = (1, k), a contradiction. (4) both vertices are in the exterior cycle. without loss of generality we suppose that one resolving 55 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 vertex is d1. suppose that the second resolving vertex is dt (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(c1|{d1, dt}) = r(dn|{d1, dt}) = (1, t) and for t = k + 1, r(d2|{d1, dk+1}) = r(dn|{d1, dk+1}) = (1, k − 1), a contradiction. (5) both vertices are in the outer cycle. without loss of generality we suppose that one resolving vertex is e1. suppose that the second resolving vertex is et (2 ≤ t ≤ k + 1). then for 2 ≤ t ≤ k, we have r(d1|{e1, et}) = r(en|{e1, et}) = (1, t) and for t = k + 1, r(e2|{e1, ek+1}) = r(en|{e1, ek+1}) = (1, k − 1), a contradiction. (6) one vertex is in the inner cycle and other in the interior cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is bt (1 ≤ t ≤ k + 1). then for t = 1, we have r(a2|{a1, b1}) = r(bn|{a1, b1}) = (1, 1) and when 2 ≤ t ≤ k + 1, r(a2|{a1, bk+1}) = r(b1|{a1, bk+1}) = (1, t− 1), a contradiction. (7) one vertex is in the inner cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for t = 1, we have r(a2|{a1, c1}) = r(bn|{a1, c1}) = (1, 2) and when 2 ≤ t ≤ k+1, r(a2|{a1, bk+1}) = r(b1|{a1, bk+1}) = (1, t), a contradiction. (8) one vertex is in the inner cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for t = 1, we have r(d2|{a1, d1}) = r(e1|{a1, d1}) = (4, 1). if 2 ≤ t ≤ k, r(a2|{a1, dt}) = r(b1|{a1, dt}) = (1, t + 1) and when t = k + 1, r(an|{a1, dk+1}) = r(bn|{a1, dk+1}) = (1, k), a contradiction. (9) one vertex is in the inner cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is a1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for t = 1, we have r(a2|{a1, e1}) = r(bn|{a1, e1}) = (1, 4). for 2 ≤ t ≤ k, r(a2|{a1, et}) = r(b1|{a1, et}) = (1, t + 2) and when t = k + 1, r(d2|{a1, ek+1}) = r(e1|{a1, ek+1}) = (1, k), a contradiction. (10) one vertex is in the interior cycle and other in the set of interior vertices. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is ct (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, ct}) = r(bn|{b1, ct}) = (1, t + 1). for t = k, we have r(a2|{b1, ck}) = r(bn|{b1, ck}) = (1, k) and when t = k + 1, r(a1|{b1, ck+1}) = r(cn|{b1, ck+1}) = (1, k), a contradiction. (11) one vertex is in the interior cycle and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, dt}) = r(bn|{b1, dt}) = (1, t + 2). for t = k, we have r(a2|{b1, dk}) = r(bn|{b1, dk}) = (1, k + 1) and when t = k + 1, r(a1|{b1, dk+1}) = r(cn|{b1, dk+1}) = (1, k + 1), a contradiction. (12) one vertex is in the interior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is b1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{b1, et}) = r(bn|{b1, et}) = (1, t + 3). for t = k, we have r(an|{b1, ek}) = r(cn|{b1, ek}) = (1, k + 2) and when t = k + 1, r(a2|{b1, ek+1}) = r(c1|{b1, ek+1}) = (1, k + 2), a contradiction. (13) one vertex is in the set of interior vertices and other in the exterior cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is dt (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(a1|{c1, dt}) = r(bn|{c1, dt}) = (2, t + 2). for t = k, we have r(dn|{c1, dk}) = r(en|{c1, dk}) = (2, k) and when t = k + 1, r(d2|{c1, dk+1}) = r(dn|{c1, dk+1}) = (1, k − 1), a contradiction. (14) one vertex is in the set of interior vertices and other in the outer cycle. without loss of generality we suppose that one resolving vertex is c1. suppose that the second resolving vertex is et (1 ≤ t ≤ k−1). then for 1 ≤ t ≤ k − 1, we have r(a1|{c1, et}) = r(bn|{c1, et}) = (2, t + 3). for t = k, we have r(dn|{c1, ek}) = r(en|{c1, ek}) = (2, k) and when t = k+1, r(dn|{c1, ek+1}) = r(en|{c1, ek+1}) = (2, k−1), a contradiction. (15) one vertex is in the set of exterior cycle and other in the outer cycle. without loss of generality we suppose that one resolving vertex is d1. suppose that the second resolving vertex is et (1 ≤ t ≤ k + 1). then for 1 ≤ t ≤ k − 1, we have r(c1|{d1, et}) = r(dn|{d1, et}) = (1, t + 1). for t = k, we have r(dn|{d1, ek}) = r(en|{d1, ek}) = (1, k) and when t = k + 1, r(dn|{d1, ek+1}) = r(en|{d1, ek+1}) = (1, k − 1), a contradiction. hence, from above it follows that there is no resolving set with two vertices for v (en) implying that 56 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 dim(en) = 3 in this case. case(ii) when n is odd. in this case, we can write n = 2k + 1, k ≥ 3, k ∈ z+. let w = {a1, a3, ak+1} ⊂ v (en), again we show that w is a resolving set for en in this case. for this we give representations of any vertex of v (en)\w with respect to w . representations of the vertices on inner cycle are r(ai|w ) =   (1, 1, k − 1), i = 2; (i− 1, i− 3, k − i + 1), 4 ≤ i ≤ k; (k, k − 1, 1), i = k + 2; (k − 1, k, 2), i = k + 3; (2k − i + 2, 2k − i + 4, i−k − 1), k + 4 ≤ i ≤ 2k + 1. representations of the vertices on interior cycle are r(bi|w ) =   (1, 2, k), i = 1; (2, 1, k − 1), i = 2; (i, i− 2, k − i + 1), 3 ≤ i ≤ k; (k + 1, k − 1, 1), i = k + 1; (2k − i + 2, 2k − i + 4, i−k), k + 2 ≤ i ≤ 2k + 1. representations of the set of interior vertices are r(ci|w ) =   (2, 2, k), i = 1; (3, 2, k − 1), i = 2; (i + 1, i− 1, k − i + 1), 3 ≤ i ≤ k − 1; (k + 1, k − 1, 2), i = k; (k + 1, k, 2), i = k + 1; (k, k + 1, 3), i = k + 2; (2k − i + 2, 2k − i + 4, i−k + 1), k + 3 ≤ i ≤ 2k; (2, 3, k + 1), i = 2k + 1. representations of the vertices on exterior cycle are r(di|w ) =   (3, 3, k + 1), i = 1; (4, 3, k), i = 2; (i + 2, i, k − i + 2), 3 ≤ i ≤ k − 1; (k + 2, k, 3), i = k; (k + 2, k + 1, 3), i = k + 1; (k + 1, k + 2, 4), i = k + 2; (2k − i + 3, 2k − i + 5, i−k + 2), k + 3 ≤ i ≤ 2k; (3, 4, k + 2), i = 2k + 1. representations of the vertices on outer cycle are r(ei|w ) =   (4, 4, k + 1), i = 1; (5, 4, k), i = 2; (i + 3, i + 1, k − i + 2), 3 ≤ i ≤ k − 1; (k + 3, k + 1, 4), i = k; (k + 2, k + 2, 4), i = k + 1; (2k − i + 3, 2k − i + 5, i−k + 3), k + 2 ≤ i ≤ 2k − 1; (4, 5, k + 3), i = 2k; (4, 4, k + 2), i = 2k + 1. again we see that there are no two vertices having the same representations which implies that dim(en) ≤ 3 in this case. on the other hand, suppose that dim(en) = 2, then there are the same subcases as in case (i) and contradiction can be obtained analogously. this implies that dim(en) = 3 in this case, which completes the proof. this result also supports the open problem raised in [10] positively. 57 m. imran et al. / j. algebra comb. discrete appl. 3(2) (2016) 45–59 5. concluding remarks in this paper, we have studied the metric dimension of some classes of convex polytopes which are obtained by the combination of two different graph of convex polytopes. we see that the metric dimension of these classes of convex polytopes is finite and does not depend upon the number of vertices in these graphs and only three vertices appropriately chosen suffice to resolve all the vertices of these classes of convex polytopes. it is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. we also note that the results proved in theorem 3 and 4 give the answer to the open problem raised in [10] positively. note that in [16] melter and tomescu gave an example of infinite regular plane graph (namely the digital plane endowed with city-block distance) having no finite metric basis. we close this section by raising a question as an open problem that naturally arises from the text. open problem: let g be the graph of a convex polytope which is obtained by the combination of graph of two different convex polytopes g1 and g2 both having constant metric dimension. is it the case that g will always have the constant metric dimension? references [1] m. bača, labelings of two classes of convex polytopes, utilitas math. 34 (1988) 24–31. 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[21] i. tomescu, i. javaid, on the metric dimension of the jahangir graph, bull. math. soc. sci. math. roumanie. 50(98)(4) (2007) 371–376. 59 http://www.ams.org/mathscinet-getitem?mr=2573990 http://www.ams.org/mathscinet-getitem?mr=2573990 http://www.ams.org/mathscinet-getitem?mr=2370323 http://www.ams.org/mathscinet-getitem?mr=2370323 notation and preliminary results the graph of convex polytope bn the graph of convex polytope cn the graph of convex polytope en concluding remarks references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327377 j. algebra comb. discrete appl. 4(3) • 271–280 received: 11 october 2016 accepted: 17 april 2017 journal of algebra combinatorics discrete structures and applications twin bent functions, strongly regular cayley graphs, and hurwitz-radon theory research article paul leopardi abstract: the real monomial representations of clifford algebras give rise to two sequences of bent functions. for each of these sequences, the corresponding cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. the proof of this nonisomorphism is a simple consequence of a theorem of radon. 2010 msc: 05e30, 11t71, 15a24, 15b34 keywords: bent functions, strongly regular graphs, clifford algebras, hurwitz-radon 1. introduction two recent papers [10, 11] describe and investigate two infinite sequences of bent functions and their cayley graphs. the bent function σm on z2m2 is described in the first paper [10], on generalizations of williamson’s construction for hadamard matrices. the bent function τm on z2m2 is described in the second paper [11], which investigates some of the properties of the two sequences of bent functions. in this second paper it is shown that the bent functions σm and τm both correspond to hadamard difference sets with the same parameters (vm,km,λm,nm) = (4 m,22m−1 −2m−1,22m−2 −2m−1,22m−2), and that their corresponding cayley graphs are both strongly regular with the same parameters (vm,km,λm,λm). the main result of the current paper is the following. theorem 1.1. the cayley graphs of the bent functions σm and τm are isomorphic only when m = 1,2, or 3. paul leopardi; university of melbourne, australian government–bureau of meteorology (email: paul.leopardi@gmail.com). 271 http://orcid.org/0000-0003-2891-5969 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 the remainder of the paper is organized as follows. section 2 outlines some of the background of this investigation. section 3 includes further definitions used in the subsequent sections. section 4 proves the main result, and resolves the conjectures and the question raised by the previous papers. section 5 puts these results in context, and suggests future research. 2. background a recent paper of the author [10] describes a generalization of williamson’s construction for hadamard matrices [16] using the real monomial representation of the basis elements of the clifford algebras rm,m. briefly, the general construction uses some ak ∈{−1,0,1}n×n, bk ∈{−1,1}b×b, k ∈{1, . . . ,n}, where the ak are monomial matrices, and constructs h := n∑ k=1 ak ⊗bk, (h0) such that h ∈{−1,1}nb×nb and hht = nbi(nb), (h1) i.e. h is a hadamard matrix of order nb. the paper [10] focuses on a special case of the construction, satisfying the conditions aj ∗ak = 0 (j 6= k), n∑ k=1 ak ∈{−1,1}n×n, aka t k = i(n), aja t k + λj,kaka t j = 0 (j 6= k), (1) bjb t k −λj,kbkb t j = 0 (j 6= k), λj,k ∈{−1,1}, n∑ k=1 bkb t k = nbi(b), where ∗ is the hadamard matrix product. in section 3 of the paper [10], it is noted that the clifford algebra r2 m×2m has a canonical basis consisting of 4m real monomial matrices, corresponding to the basis of the algebra rm,m, with the following properties: pairs of basis matrices either commute or anticommute. basis matrices are either symmetric or skew, and so the basis matrices aj,ak satisfy aka t k = i(2m), aja t k + λj,kaka t j = 0 (j 6= k), λj,k ∈{−1,1}. (2) additionally, for n = 2m, we can choose a transversal of n canonical basis matrices that satisfies conditions (1) on the a matrices, aj ∗ak = 0 (j 6= k), n∑ k=1 ak ∈{−1,1}n×n. (3) 272 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 section 3 also contains the definition of ∆m, the restricted amicability / anti-amicability graph of rm,m, and the subgraphs ∆m[−1] and ∆m[1], as well as the term “transversal graph”. these definitions are repeated here since they are used in the conjectures and question below. definition 2.1. [10, p. 225] let ∆m be the graph whose vertices are the n2 = 4m positive signed basis matrices of the real representation of the clifford algebra rm,m, with each edge having one of two labels, −1 or 1: • matrices aj and ak are connected by an edge labelled by −1 (“red”) if they have disjoint support and are anti-amicable, that is, aja −1 k is skew. • matrices aj and ak are connected by an edge labelled by 1 (“blue”) if they have disjoint support and are amicable, that is, aja −1 k is symmetric. • otherwise there is no edge between aj and ak. the subgraph ∆m[−1] consists of the vertices of ∆m and all edges in ∆m labelled by −1. similarly, the subgraph ∆m[1] contains all of the edges of ∆m that are labelled by 1. a transversal graph for the clifford algebra rm,m is any induced subgraph of ∆m that is a complete graph on 2m vertices. that is, each pair of vertices in the transversal graph represents a pair of matrices, aj and ak with disjoint support. the following three conjectures appear in section 3 of the paper [10]: conjecture 2.2. for all m > 0 there is a permutation π of the set of 4m canonical basis matrices, that sends an amicable pair of basis matrices with disjoint support to an anti-amicable pair, and vice-versa. conjecture 2.3. for all m > 0, for the clifford algebra rm,m, the subset of transversal graphs that are not self-edge-colour complementary can be arranged into a set of pairs of graphs with each member of the pair being edge-colour complementary to the other member. conjecture 2.4. for all m > 0, for the clifford algebra rm,m, if a graph t exists amongst the transversal graphs, then so does at least one graph with edge colours complementary to those of t. note that conjecture 2.2 implies conjecture 2.3, which in turn implies conjecture 2.4. the significance of these conjectures can be seen in relation to the following result, which is part 1 of theorem 10 of the paper [10]. lemma 2.5. if b is a power of 2, b = 2m, m > 0, the amicability / anti-amicability graph pb of the matrices {−1,1}b×b contains a complete two-edge-coloured graph on 2b2 vertices with each vertex being a hadamard matrix. this graph is isomorphic to γm,m, the amicability / anti-amicability graph of the group gm,m. the definitions of γm,m and gm,m are given in section 3 of the paper [10], and the definition of gm,m is repeated below. for the current paper, it suffices to note that ∆m is a subgraph of γm,m, and so, therefore, are all of the transversal graphs. an n-tuple of a matrices of order n = 2m satisfying properties (2) and (3) yields a corresponding transversal graph t . as noted in section 5 of the paper [10], if conjecture 2.4 were true, this would guarantee the existence of an edge-colour complementary transversal graph t. in turn, because lemma 2.5 guarantees the existence of a complete two-edge-coloured graph isomorphic to γm,m within pb, and because ∆m is a subgraph of γm,m, the graph pb would have to contain a two-edge-coloured subgraph isomorphic to t . this would imply the existence of an n-tuple of b matrices of order n satisfying the condition (1) such that the construction (h0) would satisfy the hadamard condition (h1), with a matrix of order n2. the author’s subsequent paper on bent functions [11] refines conjecture 2.2 into the following question. 273 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 question 2.6. consider the sequence of edge-coloured graphs ∆m for m > 1, each with red subgraph ∆m[−1], and blue subgraph ∆m[1]. for which m > 1 is there an automorphism of ∆m that swaps the subgraphs ∆m[−1] and ∆m[1]? the main result of this paper, theorem 1.1, leads to the resolution of these conjectures and this question. 3. further definitions and properties this section sets out the remainder of the definitions and properties used in this paper. it is based on the previous papers [10, 11] with additions. 3.1. clifford algebras and their real monomial representations the following definitions and results appear in the paper on hadamard matrices and clifford algebras [10], and are presented here for completeness, since they are used below. further details and proofs can be found in that paper, and in the paper on bent functions [11], unless otherwise noted. an earlier paper on representations of clifford algebras [9] contains more background material. the signed group [4] gp,q of order 21+p+q is extension of z2 by z p+q 2 , defined by the signed group presentation gp,q := 〈 e{k} (k ∈ sp,q) | e2{k} = −1 (k < 0), e 2 {k} = 1 (k > 0), e{j}e{k} = −e{k}e{j} (j 6= k) 〉 , where sp,q := {−q, . . . ,−1,1, . . . ,p}. the 2×2 orthogonal matrices e1 := [ 0 −1 1 0 ] , e2 := [ 0 1 1 0 ] generate p(g1,1), the real monomial representation of group g1,1. the cosets of {±i} ≡ z2 in p(g1,1) are ordered using a pair of bits, as follows. 0 ↔ 00 ↔{±i}, 1 ↔ 01 ↔{±e1}, 2 ↔ 10 ↔{±e2}, 3 ↔ 11 ↔{±e1e2}. for m > 1, the real monomial representation p(gm,m) of the group gm,m consists of matrices of the form g1 ⊗gm−1 with g1 in p(g1,1) and gm−1 in p(gm−1,m−1). the cosets of {±i}≡ z2 in p(gm,m) are ordered by concatenation of pairs of bits, where each pair of bits uses the ordering as per p(g1,1), 274 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 and the pairs are ordered as follows. 0 ↔ 00 . . .00 ↔{±i}, 1 ↔ 00 . . .01 ↔{±i⊗(m−1) (2) ⊗e1}, 2 ↔ 00 . . .10 ↔{±i⊗(m−1) (2) ⊗e2}, . . . 22m −1 ↔ 11 . . .11 ↔{±(e1e2)⊗m}. this ordering is called the kronecker product ordering of the cosets of {±i} in p(gm,m). the group gm,m and its real monomial representation p(gm,m) satisfy the following properties. 1. pairs of elements of gm,m (and therefore p(gm,m)) either commute or anticommute: for g,h ∈ gm,m, either hg = gh or hg = −gh. 2. the matrices e ∈ p(gm,m) are orthogonal: eet = et e = i. 3. the matrices e ∈ p(gm,m) are either symmetric and square to give i or skew and square to give −i: either et = e and e2 = i or et = −e and e2 = −i. taking the positive signed element of each of the 22m cosets listed above defines a transversal of {±i} in p(gm,m) which is also a monomial basis for the real representation of the clifford algebra rm,m in kronecker product order, called this basis the positive signed basis of p(rm,m). the function γm : z22m → p(gm,m) chooses the corresponding basis matrix from the positive signed basis of p(rm,m), using the kronecker product ordering. this ordering also defines a corresponding function on z2m2 , also called γm. 3.2. hurwitz-radon theory the key concept used in the proof of lemma 4.1 below is that of a hurwitz-radon family of matrices. a set of real orthogonal matrices {a1,a2, . . . ,as} is called a hurwitz-radon family [6, 7, 13] if 1. atj = −aj for all j = 1, . . . ,s, and 2. ajak = −akaj for all j 6= k. the hurwitz-radon function ρ is defined by ρ(24d+c) := 2c + 8d, where 0 6 c < 4. as stated by geramita and pullman [6], radon [13] proved the following result, which is used as a lemma in this paper. lemma 3.1. [6, theorem a] any hurwitz-radon family of order n has at most ρ(n)−1 members. 3.3. the two sequences of bent functions the previous two papers [10, 11] define two binary functions on z2m2 , σm and τm, respectively. their key properties are repeated below. see the two papers for the proofs and for more details and references on bent functions. the function σm : z2m2 → z2 has the following properties. 275 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 1. for i ∈ z2m2 , σm(i) = 1 if and only if the number of digits equal to 1 in the base 4 representation of i is odd. 2. since each matrix γm(i) is orthogonal, σm(i) = 1 if and only if the matrix γm(i) is skew. 3. the function σm is bent. the function τm : z2m2 → z2 has the following properties. 1. for i ∈ z2m2 , τm(i) = 1 if and only if the number of digits equal to 1 or 2 in the base 4 representation of i is non zero, and the number of digits equal to 1 is even. 2. the value τm(i) = 1 if and only if the matrix γm(i) is symmetric but not diagonal. 3. the function τm is bent. 3.4. the relevant graphs for a binary function f : z2m2 → z2, with f(0) = 0 we consider the simple undirected cayley graph cay(f) [1, 3.1] where the vertex set v (cay(f)) = z2m2 and for i,j ∈ z2m2 , the edge (i,j) is in the edge set e(cay(f)) if and only if f(i + j) = 1. in the paper on hadamard matrices [10] it is shown that since σm(i) = 1 if and only if γm(i) is skew, the subgraph ∆m[−1] is isomorphic to the cayley graph cay(σm). the paper on bent functions [11] notes that since τm(i) = 1 if and only if γm(i) is symmetric but not diagonal, the subgraph ∆m[1] is isomorphic to the cayley graph cay(τm). in that paper, these isomorphisms and the characterization of cay(σm) and cay(τm) as cayley graphs of bent functions are used to prove the following theorem. theorem 3.2. [11, theorem 5.2] for all m > 1, both graphs ∆m[−1] and ∆m[1] are strongly regular, with parameters vm = 4m, km = 2 2m−1 −2m−1, λm = µm = 22m−2 −2m−1. 4. proof of theorem 1.1 and related results here we prove the main result, and examine its implications for conjectures 2.2 to 2.4 and question 2.6. the proof of theorem 1.1 follows from the following two lemmas. the first lemma puts an upper bound on the clique number of the graph cay(σm) ' ∆m[−1]. lemma 4.1. the clique number of the graph cay(σm) is at most ρ(2m), where ρ is the hurwitz-radon function. therefore ρ(2m) < 2m for m > 4. proof. if we label the vertices of the graph cay(σm) with the elements of z2m2 , then any clique in this graph is mapped to another clique if a constant is added to all of the vertices. thus without loss of generality we can assume that we have a clique of order s+ 1 with one of the vertices labelled by 0. if we then use γm to label the vertices with elements of rm,m to obtain the isomorphic graph ∆m[−1], we have one vertex of the clique labelled with the identity matrix i of order 2m. since the clique is in ∆m[−1], the other vertices a1 to as (say) must necessarily be skew matrices that are pairwise anti-amicable, aja t k = −aka t j for all j 6= k. 276 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 but then ajak = −akaj for all j 6= k, and therefore {a1, . . . ,as} is a hurwitz-radon family. by lemma 3.1, s is at most ρ(2m) − 1 and therefore the size of the clique is at most ρ(2m). the second lemma puts a lower bound on the clique number of the graph cay(τm) ' ∆m[1]. lemma 4.2. the clique number of the graph cay(τm) is at least 2m. proof. we construct a clique of order 2m in cay(τm) with the vertices labelled in z2m2 , using the following set of vertices, denoted in base 4: cm := {00 . . .00,00 . . .02,00 . . .20, . . . ,22 . . .22}. the set cm is closed under addition in z2m2 , and therefore forms a clique of order 2 m in cay(τm), since the sum of any two distinct elements of cm is in the support of τm. with these two lemmas in hand, the proof of theorem 1.1 follows easily. proof of theorem 1.1. the result is a direct consequence of lemmas 4.1 and 4.2. for m > 4, the clique numbers of the graphs cay(σm) and cay(τm) are different, and therefore these graphs cannot be isomorphic. lemmas 4.1 and 4.2, along with theorem 1.1 imply the failure of the conjectures 2.2 to 2.4, as well as the resolution of question 2.6, as follows. theorem 4.3. for m > 4 the following hold. 1. there exist transversal graphs that do not have an edge-colour complement, and therefore conjecture 2.4 does not hold. 2. as a consequence, conjectures 2.2 and 2.3 also do not hold. 3. question 2.6 is resolved. the only m > 1 for which there is an automorphism of ∆m that swaps the subgraphs ∆m[−1] and ∆m[1] are m = 1,2 and 3. proof. assume that m > 4. a transversal graph is a subgraph of ∆m which is a complete graph of order 2m. the edges of a transversal graph are labelled with the colour red (if the edge is contained in ∆m[−1]) or blue (if the edge is contained in ∆m[1]). by lemma 4.1, the largest clique of ∆m[−1] is of order ρ(2m) < 2m, and by lemma 4.2, the largest clique of ∆m[1] is of order 2m. if we take a blue clique of order 2m as a transversal graph, this cannot have an edge-colour complement in ∆m, because no red clique can be this large. more generally, we need only take a transversal graph containing a blue clique with order larger than ρ(2m) to have a clique with no edge-colour complement in ∆m. this falsifies conjecture 2.4. since conjecture 2.4 fails for m > 4, the pairing of graphs described in conjecture 2.3 is impossible for m > 4. thus conjecture 2.3 is also false. finally, conjecture 2.2 fails as a direct consequence of theorem 1.1 since, for m > 4, the subgraphs ∆m[−1] and ∆m[1]are not isomorphic. therefore, for m > 4, there can be no automorphism of ∆m that swaps these subgraphs. 277 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 5. discussion the result of lemma 4.1 is well known. for example, the graph ∆m[−1] is the complement of the graph v + of yiu [17], and the result for v + in his theorem 2 is equivalent to lemma 4.1. the main consequence of theorem 4.3 is that for m > 3 there is at least one n-tuple of a matrices, with n = 2m such that no n-tuple of b matrices of order n can be found to satisfy construction (h0) under condition (h1). the proof of theorem 5 of the hadamard construction paper [10] shows by construction that for any m, and any n-tuple of a matrices satisfying (1), there is an n-tuple of b matrices of order nc that satisfies construction (h0) under condition (h1), where c = m(n−1), with m(q) := { dq 2 e+ 1, if q ≡ 2,3,4 (mod 8), dq 2 e otherwise. (4) thus theorem 5 remains valid. the question remains as to whether the the order nc is tight or can be reduced. in the special case where the n-tuple of a matrices is mutually amicable, the answer is given by corollary 15 of the paper [10]: the set of {−1,1} matrices of order c contains an n-tuple of mutually anti-amicable hadamard matrices. so in this special case, the required order can be reduced from nc to c. this leads to the following question. question 5.1. in the general case, for any m > 1, n = 2m, for any n-tuple of a matrices satisfying (1), does there always exist an n-tuple of b matrices of order c that satisfies construction (h0) under condition (h1), where c = m(n−1), with m defined by (4)? as a result of theorems 3.2 and 4.3, we see that we have two sequences of strongly regular graphs, ∆m[−1] and ∆m[1] (m > 1), sharing the same parameters, vm = 4m, km = 22m−1 − 2m−1, λm = µm = 22m−2 − 2m−1, but the graphs are isomorphic only for m = 1,2,3. for these three values of m, the existence of automorphisms of ∆m that swap ∆m[−1] and ∆m[1] as subgraphs [10, table 1] is remarkable in the light of theorem 4.3. a paper of bernasconi and codenotti describes the relationship between bent functions and their cayley graphs, implying that a bent function corresponding to a (v,k,λ,n) hadamard difference set has a cayley graph that is strongly regular with parameters (v,k,λ,µ) where λ = µ [1, lemma 12]. the current paper notes that for two specific sequences of bent functions, σm and τm, the corresponding cayley graphs are not necessarily isomorphic. this raises the subject of classifying bent functions via their cayley graphs, raising the following questions. question 5.2. which strongly regular graphs with parameters (v,k,λ,λ) occur as cayley graphs of bent functions? question 5.3. what is the relationship between other classifications of bent functions and the classification via cayley graphs? this classification is the topic of a paper in preparation [8]. with respect to the specific bent functions σm and τm investigated here, one of the anonymous reviewers of an earlier draft of this paper has asked whether each of these functions are part of a larger class of bent functions. the function σm is a quadratic form, as can be seen from its definition and its recursive identity [10, lemma 7]. specifically, σm(0) = 0, and, in terms of algebraic normal form, using a particular convention for the mapping of bits to boolean variables, the identity is σ1(x0,x1) = x0x1 + x0, and σm+1(x0,x1, . . . ,x2m,x2m+1) = σm(x0,x1) + σm(x2,x3, . . . ,x2m,x2m+1) = x0x1 + x0 + x2x3 + x2 + . . . + x2mx2m+1 + x2m. 278 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 in a paper in preparation [8], it is proven that all quadratic bent functions with the same dimension and weight have isomorphic cayley graphs. as for τm, it is a bent iterative function [2, theorem v.4] [3, theorem 2] [15], as can be seen from its definition, and from the proof that it is a bent function [11, theorem 3.1]. since the ps(−) partial spread bent functions are formed using m-dimensional subspaces of z2m which are disjoint except for the 0 vector [5, p. 95], these bent functions also have cayley graphs whose clique number is at least 2m. it could therefore be speculated that τm is also a ps(−) bent function, but exhaustive search using sagemathcloud [14] shows that τ3 cannot be in ps(−). each clique of size 8 in cay(τ3) that contains the 0 vector intersects each other such clique at two vectors, only one of which is the 0 vector [12]. acknowledgment: thanks to christine leopardi for her hospitality at long beach. thanks to robert craigen, joanne hall, william martin, padraig ó catháin and judy-anne osborn for valuable discussions. this work was begun in 2014 while the author was a visiting fellow at the australian national university, continued while the author was a visiting fellow and a casual academic at the university of newcastle, australia, and concluded while the author was an employee of the bureau of meteorology of the australian government, and an honorary fellow of the university of melbourne. thanks also to the anonymous reviewers of previous drafts of this paper. references [1] a. bernasconi, b. codenotti, spectral analysis of boolean functions as a graph eigenvalue problem, ieee trans. comput. 48(3) (1999) 345–351. 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[15] n. tokareva, on the number of bent functions from iterative constructions: lower bounds and hypotheses, adv. math. commun. 5(4) (2011) 609–621. 279 https://doi.org/10.1109/12.755000 https://doi.org/10.1109/12.755000 https://doi.org/10.1109/18.923730 https://doi.org/10.1109/18.923730 https://doi.org/10.1109/tit.2003.814476 https://doi.org/10.1109/tit.2003.814476 https://doi.org/10.1016/0097-3165(95)90002-0 https://doi.org/10.1016/0097-3165(95)90002-0 https://doi.org/10.1090/s0002-9939-1974-0332764-4 https://doi.org/10.1090/s0002-9939-1974-0332764-4 http://doi.org/10.1007/bf01448439 https://arxiv.org/abs/1705.04507 http://www.ams.org/mathscinet-getitem?mr=2130632 http://www.ams.org/mathscinet-getitem?mr=2130632 http://www.ams.org/mathscinet-getitem?mr=3211780 http://www.ams.org/mathscinet-getitem?mr=3211780 http://doi.org/10.1007/978-3-319-17729-8_15 http://doi.org/10.1007/978-3-319-17729-8_15 http://doi.org/10.1007/978-3-319-17729-8_15 http://tinyurl.com/boolean-cayley-graphs http://tinyurl.com/boolean-cayley-graphs http://tinyurl.com/boolean-cayley-graphs http://doi.org/10.1007/bf02940576 http://doi.org/10.1007/bf02940576 https://cloud.sagemath.com/ http://dx.doi.org/10.3934/amc.2011.5.609 http://dx.doi.org/10.3934/amc.2011.5.609 p. leopardi / j. algebra comb. discrete appl. 4(3) (2017) 271–280 [16] j. williamson, hadamard’s determinant theorem and the sum of four squares, duke math. j. 11(1) (1944) 65–81. [17] p. y. yiu, strongly regular graphs and hurwitz–radon numbers, graphs and combin. 6(1) (1990) 61–69. 280 http://doi.org/10.1215/s0012-7094-44-01108-7 http://doi.org/10.1215/s0012-7094-44-01108-7 https://doi.org/10.1007/bf01787481 https://doi.org/10.1007/bf01787481 introduction background further definitions and properties proof of theorem 1.1 and related results discussion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.29560 j. algebra comb. discrete appl. 3(3) • 145–154 received: 19 september 2015 accepted: 06 december 2015 journal of algebra combinatorics discrete structures and applications enumeration of symmetric (45,12,3) designs with nontrivial automorphisms∗ research article dean crnković, doris dumičić danilović, sanja rukavina abstract: we show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. we describe the full automorphism groups of these designs and analyze their ternary codes. r. mathon and e. spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. further, we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs. we prove that k-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric (45, 12, 3) designs. 2010 msc: 05b05, 20d45, 94b05, 05c38 keywords: symmetric design, linear code, automorphism group, k-geodetic graph 1. introduction the terminology and notation in this paper for designs and codes are as in [2, 3, 6]. one of the main problems in design theory is that of classifying structures with given parameters. classification of designs has been considered in detail in the monograph [17]. complete classification of designs with certain parameters has been done just for some designs with relatively small number of points, and in the case of symmetric designs complete classification is done just for a few parameter triples (see [22]). the classification of projective planes of order 9 has been solved in 1991 (see [20]), and kaski and östergård classified all biplanes with k=11 in 2008 (see [18]). hence, the parameter triple (45,12,3) is the next for symmetric designs of order 9 to be classified. since the complete classification of symmetric (45,12,3) designs seems to be out of reach with the current techniques and computers, only ∗ this work has been fully supported by croatian science foundation under the project 1637. dean crnković (corresponding author), doris dumičić danilović, sanja rukavina; department of mathematics, university of rijeka, radmile matejčić 2, 51000 rijeka, croatia (email: deanc@math.uniri.hr, ddumicic@math.uniri.hr, sanjar@math.uniri.hr). 145 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 partial classification of such designs, with certain constrains, is possible. in this paper we manage to classify all symmetric (45,12,3) designs with nontrivial automorphisms. the first symmetric (45,12,3) design was constructed in [1], and further two symmetric (45,12,3) designs were constructed in [23] as (45,12,3) difference sets. later on, kölmel [19] and ćepulić [5] have independently constructed symmetric (45,12,3) designs having an automorphism of order 5. in his doctoral dissertation [19] kölmel also determined all (45,12,3) designs having a fixed-point-free automorphism of order 3. finally, mathon and spence [25] showed that there are at least 3752 symmetric (45,12,3) designs, 1136 of them having a trivial automorphism group. furthermore, coolsaet, de jager and spence, established in [7] that there are exactly 78 non-isomorphic strongly regular graphs with parameters (45,12,3,3), meaning that there are exactly 78 symmetric designs having symmetric incidence matrix with zero diagonal. ternary codes spanned by the adjacency matrices of these strongly regular graphs (i.e. incidence matrices of the corresponding symmetric designs) have been studied in [8]. the symmetric (45,12,3) design admitting a primitive action of the group psp(4, 3) is described in [10] and [12]. in this paper we give the classification of all symmetric (45,12,3) designs having a nontrivial automorphism group. we show that there exist exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms, which means that there are at least 5421 symmetric (45,12,3) designs. furthermore, we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs and prove that mutually non-isomorphic designs produce mutualy non-isomorphic k-geodetic graphs. the paper is organized as follows: after the brief introduction, in section 2 we give basic information concerning the construction method, in section 3 we describe the construction of symmetric (45,12,3) designs with nontrivial automorphisms and give a list of the designs and their full automorphism groups, section 4 gives information about the codes of the constructed designs, and in section 5 we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs. for the construction of designs we have used our own computer programs. for isomorphism testing, and to obtain and analyze the full automorphism groups of the designs we have used [14] and [30]. the codes have been analyzed using magma [4]. 2. outline of the construction an incidence structure d = (p,b,i), with point set p, block set b and incidence i is a t-(v,k,λ) design, if |p| = v, every block b ∈ b is incident with precisely k points, and every t distinct points are together incident with precisely λ blocks. a design is called symmetric if it has the same number of points and blocks. an automorphism of a design d is a permutation on p which sends blocks to blocks. the set of all automorphisms of d forms its full automorphism group denoted by aut(d). let d = (p,b,i) be a symmetric (v,k,λ) design and g ≤ aut(d). the group action of g produces the same number of point and block orbits (see [21, theorem 3.3]). we denote that number by t, the point orbits by p1, . . . ,pt, the block orbits by b1, . . . ,bt, and put |pr| = ωr and |bi| = ωi. an automorphism group g is said to be semi-standard if, after possibly renumbering orbits, we have ωi = ωi, for i = 1, . . . , t. we denote by γir the number of points of pr which are incident with a representative of the block orbit bi. for these numbers the following equalities hold (see [5, 9, 16]): t∑ r=1 γir = k , (1) t∑ r=1 ωj ωr γirγjr = λωj + δij · (k −λ) . (2) definition 2.1. a (t × t)-matrix (γir) with entries satisfying conditions (1) and (2) is called an orbit matrix for the parameters (v,k,λ) and orbit lengths distributions (ω1, . . . ,ωt), (ω1, . . . , ωt). 146 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 the construction of designs admitting an action of a presumed automorphism group, using orbit matrices, consists of the following two basic steps (see [5, 9, 16]): 1. construction of orbit matrices for the given automorphism group, 2. construction of block designs for the orbit matrices obtained in this way. this step is often called an indexing of orbit matrices. in order to construct the orbit matrices for an action of a presumed automorphism group we have to determine all possibilities for the orbit lengths distributions. the following facts, that one can use in that purpose, can be found in [21]. theorem 2.2. an automorphism ρ of a symmetric design fixes an equal number of points and blocks. moreover, ρ has the same cyclic structure, whether considered as a permutation on points or on blocks. theorem 2.3. suppose that a nonidentity automorphism ρ of a nontrivial symmetric (v,k,λ) design fixes f points. then f ≤ v − 2n and f ≤ λ k − √ n v, where n = k −λ is the order of the design. moreover, if equality holds in either inequality, ρ must be an involution and every non-fixed block contains exactly λ fixed points. theorem 2.4. suppose that d is a nontrivial symmetric (v,k,λ) design, with an involution ρ fixing f points and blocks. if f 6= 0, then f ≥ { 1 + k λ , if k and λ are both even, 1 + k−1 λ , otherwise. suppose that d is a symmetric (v,k,λ) design with an automorphism ρ of prime order p fixing f points. then f ≡ v (mod p), and 〈ρ〉 acts semi-standardly on d. in that case, since the action of g = 〈ρ〉 is semi-standard, it is sufficient to determine point orbit lengths distribution (ω1, . . . ,ωt). after determining the orbit lengths distributions we proceed with the construction of orbit matrices and corresponding designs, as described in [9]. 3. classification of symmetric (45,12,3) designs with nontrivial automorphisms in this section we give the classification of all symmetric (45,12,3) designs that admit nontrivial automorphisms. it is known that if ρ is a nonidentity automorphism of a symmetric (45,12,3) design, then |ρ| ∈ {2, 3, 5, 11} (see [25]). it has been shown in [5, 19, 25] that there are exactly 13 symmetric (45,12,3) designs with an automorphism of order 5, and exactly one symmetric (45,12,3) design with an automorphism of order 11. to complete the classification of symmetric (45,12,3) designs with nontrivial automorphisms, we have to classify all symmetric (45,12,3) designs that admit an automorphism of order 2 or 3. 3.1. symmetric (45,12,3) designs admitting z2 as an automorphism group let ρ be an involutory automorphism of a symmetric (45, 12, 3) design fixing f points. then 5 ≤ f ≤ 15 and f ≡ 1 (mod 2), hence f ∈ {5, 7, 9, 11, 13, 15}. up to isomorphism there are 682 orbit structures, that produce 2987 mutually non-isomorphic designs. information about the number of the orbit structures and the designs are given in table 1. 147 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 table 1. symmetric (45,12,3) designs having z2 as an automorphism group number of fixed points 5 7 9 11 13 15 number of orbit structures 233 397 32 4 11 5 number of orbit structures that produce designs 45 271 30 0 7 5 number of designs 603 1898 524 0 225 28 3.2. symmetric (45,12,3) designs admitting z3 as an automorphism group it was determined in [8] that there are exactly 591 orbit matrices for the group z3 acting on symmetric (45,12,3) designs. from these orbit matrices we have obtained up to isomorphism exactly 2108 symmetric (45,12,3) designs that admit an automorphism of order three. information about the number of the orbit matrices and the constructed designs are presented in table 2. table 2. symmetric (45,12,3) designs having z3 as an automorphism group number of fixed points 0 3 6 9 number of orbit structures 293 245 49 4 number of orbit structures that produce designs 19 25 24 4 number of designs 244 482 125 1775 3.3. all symmetric (45,12,3) designs admitting a nontrivial automorphism group comparing the designs described in subsections 3.1 and 3.2 we conclude that up to isomorphism there are exactly 4280 symmetric (45,12,3) designs that admit an automorphism of order 2 or 3. it is known from [5, 19, 25] that there are exactly 13 symmetric (45,12,3) designs with an automorphism of order 5, and only four of them have the full automorphism group whose order is not divisible by 2 or 3. further, there is exactly one symmetric (45,12,3) design with an automorphism of order 11, and the full automorphism group of that designs is z11. that shows that there exist exactly 4285 symmetric (45,12,3) designs with a nontrivial automorphism group. among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. information about these 4285 designs and their full automorphism groups are given in table 3. some od the automorphism groups have the same description of the structure, but they are not isomorphic. in that case, nonisomorphic groups with the same structure are listed in separate rows of table 3 (e.g. two groups of order 324 having the structure (e27 : z3) : e4). since mathon and spence have constructed 1136 symmetric (45,12,3) designs with a trivial automorphism group (see [25]), we conclude that up to isomorphism there are at least 5421 symmetric (45,12,3) designs. 4. ternary codes from symmetric (45,12,3) designs the code cf (d) of a design d = (p,b,i) over the finite field f is the space spanned by the incidence vectors of the blocks over f. if q is any subset of the point set p, then we will denote the incidence vector of q by vq. thus cf (d) = 〈vb |b ∈b〉, and is a subspace of fp, the full vector space of functions from p to f . the following theorem, that can be found in [2], shows that the code cf (d) over a field f of characteristic p is not interesting if p does not divide the order of d. in theorem 4.2 rankp(d) denotes the dimension of cf (d), and j denotes the all-one vector. 148 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 table 3. symmetric (45,12,3) designs with nontrivial automorphisms |aut(d)| structure of aut(d) no. designs |aut(d)| structure of aut(d) no. designs 51840 psp(4, 3) : z2 1 45 z15 ×z3 1 19440 (e81 : sl(2, 5)) : z2 1 36 s3 ×s3 4 1296 e27 : (s4 ×z2) 1 36 e9 : z4 1 486 e81 : z6 1 36 z2 × (e9 : z2) 1 486 e81 : s3 1 32 z4 : q8 1 432 ((s3 ×s3) : z2)×s3 2 30 z5 ×s3 1 360 (z15 ×z3) : z8 1 27 e27 11 324 (e27 : z3) : e4 1 27 z9 : z3 6 324 (e27 : z3) : e4 1 27 e9 : z3 4 216 (z3 ×s3 ×s3) : z2 3 24 z3 ×q8 1 216 (e9 : z4)×s3 2 20 z5 : z4 1 216 s3 ×s3 ×s3 2 20 z5 : z4 1 192 (e4 ×q8) : s3 2 18 z3 ×s3 87 162 e27 : z6 8 18 z6 ×z3 4 162 e27 : s3 5 18 e9 : z2 4 162 e27 : s3 1 16 qd16 7 162 s3 × (e9 : z3) 1 16 z2 ×d8 2 144 (e9 : z8) : z2 2 16 (z4 ×z2) : z2 1 108 s3 × (e9) : z2) 4 15 z15 2 108 e27 : z4 2 12 d12 65 108 e27 : e4 1 11 z11 1 108 z3 ×s3 ×s3 1 9 e9 213 81 e27 : z3 4 8 d8 12 81 e27 : z3 1 8 q8 7 64 (e4.(z4 ×z2)) : z2 1 8 z8 4 64 ((z2 ×q8) : z2) : z2 1 8 e8 2 54 e27 : z2 24 8 z4 ×z2 2 54 e27 : z2 9 6 s3 446 54 (z9 : z3) : z2 6 6 z6 104 54 e9 ×s3 6 5 z5 4 54 e9 : z6 3 4 e4 128 48 (z3 ×q8) : z2 3 4 z4 71 48 z2 ×s4 1 3 z3 1051 48 (z4 ×z4) : z3 1 2 z2 1931 theorem 4.1. let d = (p,b,i) be a nontrivial 2-(v,k,λ) design of order n. let p be a prime and let f be a field of characteristic p, where p does not divide n. then rankp(d) ≥ (v − 1) with equality if and only if p divides k; in the case of equality we have that cf (d) = 〈j〉⊥ and otherwise cf (d) = fp. since the order of a symmetric (45,12,3) design is 9, we consider only the ternary codes of the constructed designs, i.e. codes over the field of order 3. the ternary codes of the 4285 symmetric 149 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 (45,12,3) designs with nontrivial automorphisms are divided in 1005 equivalence classes. in table 4 we give information about code parameters and orders of automorphism groups of representatives of equivalence classes, where the definitions of automorphisms and equivalence of codes are the same as in magma [4]. the following theorem states that all the codes obtained are self-orthogonal. theorem 4.2. let d be a symmetric (45,12,3) design and c(d) be the ternary code of the design d. then the code c(d) is self-orthogonal, and j ∈ c(d)⊥. proof. the code c(d) is spanned by the rows of the row-point incidence matrix of d. since each row of d has 12 points, and any two blocks intersect in 3 points, the code c(d) is self-orthogonal. it is obvious that j ∈ c(d)⊥, because each row of the design d consist of 12 points. table 4. ternary codes of the symmetric (45,12,3) designs with nontrivial automorphisms parameters (|aut(c)|, no. of inequivalent codes) [45, 22, 9] (11,1) [45, 21, 6] (60466176,1), (5832,1), (2592,1), (1944,1), (216,1), (108,1), (24,3), (12,1), (9,8), (8,4), (6,3), (4,10), (3,4), (2,19) [45, 20, 12] (2,3) [45, 20, 9] (15,1), (9,8), (6,4), (3,7), (2,2) [45, 20, 6] (45349632,1), (31104,1), (23328,1), (11664,1), (10368,1), (5832,1), (3888,2), (1944,2), (1152,1),(972,1), (648,2), (324,3), (288,1), (216,1), (162,5), (108,2), (72,6), (54,4), (36,10), (27,3), (24,11), (18,1), (12,34), (9,5), (8,6), (6,7), (5,2), (4,59), (3,1), (2,139) [45, 19, 12] (64,1), (32,1), (4,1), (2,1) [45, 19, 9] (162,1), (81,4), (54,2), (27,2), (18,1), (9,6), (4,1), (3,4) [45, 19, 6] (226748160,1), (52488,1), (23328,1), (17496,1), (11664,1), (5832,3), (4608,1), (1944,1), (1296,1), (972,1), (648,1), (486,2), (324,1), (288,2), (216,2), (108,3), (72,7), (54,4), (36,20), (24,9), (20,1), (18,9), (16,3), (12,37), (9,2), (8,6), (6,32), (4,51), (3,16), (2,132) [45, 18, 12] (20,1) [45, 18, 9] (18,2), (9,1), (6,2), (2,2) [45, 18, 6] (209952,1), (52488,1), (23328,1), (8748,1), (7290,1), (5832,1), (1944,2), (1296,3), (432,1), (324,3), (216,2), (162,2), (108,6), (72,7), (54,4), (48,3), (36,14), (27,1), (24,1), (18,11), (16,1), (12,18), (8,5), (6,22), (4,16), (3,8), (2,34) [45, 17, 12] (192,2), (48,2) [45, 17, 9] (360,1), (81,1) [45, 17, 6] (69984,1), (3888,1), (2916,1), (1944,2), (486,1), (432,1), (324,6), (288,1), (216,1), (162,1), (144,1), (108,3), (54,1), (36,3), (18,1), (16,2), (12,4), (9,1), (8,1), (6,3), (4,1), (2,1) [45, 16, 9] (486,1), (324,1) [45, 16, 6] (972,1), (432,1), (216,1), (108,1) [45, 15, 12] (51840,1) [45, 15, 9] (19440,1) in table 5 we give information about the dual codes of the codes presented in table 4. according to [15] and [26], the [45,28,8] code has the greatest minimum distance among the known ternary [45,28] codes. further, the best known ternary [45,30] code has minimum distance 7, hence the [45,30,6] code has minimum distance one less than the best known code. a linear code whose dual code supports the blocks of a t-design admits one of the simplest decoding algorithms, majority logic decoding (see [28]). if a codeword x = (x1, . . . ,xn) ∈ c is sent over a communication channel, and a vector y = (y1, . . . ,yn) is received, for each symbol yi a set of values 150 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 y (1) i , . . . ,y (ri) i of ri linear functions defined by the blocks of the design are computed, and yi is decoded as the most frequent among the values y(1)i , . . . ,y (ri) i . the following result have been obtained by rudolph [28]. theorem 4.3. if c is a linear [n,k] code such that c⊥ contains a set s of vectors of weight w whose supports are the blocks of a 2-(n,w,λ) design, the code c can correct up to e = ⌊ r + λ− 1 2λ ⌋ errors by majority logic decoding, where r = λn−1 w−1. consequently, the codes listed in table 5 can correct up to two errors by majority logic decoding. table 5. dual codes of ternary codes of the symmetric (45,12,3) designs with nontrivial automorphisms parameters (|aut(c)|, no. of inequivalent codes) [45, 30, 6] (51840,1), (19440,1) [45, 29, 6] (972,1), (486,1), (432,1), (324,1), (216,1), (108,1) [45, 28, 8] (48,1) [45, 28, 6] (69984,1), (3888,1), (2916,1), (1944,2), (486,1), (432,1), (360,1), (324,6), (288,1), (216,1), (192,2), (162,1), (144,1), (108,3), (81,1), (54,1), (48,1), (36,3), (18,1), (16,2), (12,4), (9,1), (8,1), (6,3), (4,1), (2,1) [45, 27, 6] (209952,1), (52488,1), (23328,1), (8748,1), (7290,1), (5832,1), (1944,1), (1296,3), (432,1), (324,3), (216,2), (162,2), (108,6), (72,7), (54,4), (48,3), (36,14), (27,1), (24,1), (20,1), (18,13), (16,1), (12,18), (9,1), (8,5), (6,24), (4,16), (3,8), (2,36) [45, 26, 8] (3,2) [45, 26, 6] (226748160,1), (52488,1), (23328,1), (17496,1), (11664,1), (5832,1), (4608,1), (1944,1), (1296,1), (972,1), (648,1),(486,2), (324,1), (288,2), (216,2), (162,1), (108,3), (81,4), (72,7), (64,1), (54,6), (36,20), (32,1), (27,2), (24,9), (20,1), (18,10), (16,3), (12,37), (9,8), (8,6), (6,32), (4,53), (3,18), (2,133) [45, 25, 8] (15,1), (2,2), (3,1) [45, 25, 6] (45349632,1), (31104,1), (23328,1), (11664,1), (10368,1), (5832,1), (3888,2), (1944,2), (1152,1), (972,1), (648,2), (324,3), (288,1), (216,1), (162,5), (108,2), (72,6), (54,4), (36,10), (27,3), (24,11), (18,1), (12,34), (9,13), (8,6), (6,11), (5,2), (4,59), (3,7), (2,142) [45, 24, 6] (60466176,1), (5832,1), (2592,1), (1944,1), (216,1), (108,1), (24,3), (12,1), (9,8), (8,4), (6,3), (4,10), (3,4), (2,19) [45, 23, 9] (11,1) 5. on k-geodetic graphs from symmetric (45, 12, 3) designs in this section we present results concerning 3-geodetic (trigeodetic) graphs. we prove that kgeodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic. for applications of k-geodetic graphs in the topological design of computer networks the reader may consult [13]. for further reading on k-geodetic graphs we refer the reader to [27] and [29]. for every 2-(v,k,λ) design d with replication number r and b blocks it is possible to construct k−connected biregular block k∗v (r,k,λ) (a block is a graph with vertex connectivity > 1) of diameter 4 151 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 or 5 with vertex degrees r and k, in which there are at most µ paths of minimum length between any pair of vertices, where µ = max{max{|bi ∩bj| : i,j = 1, 2, . . . ,b, i 6= j} , λ} , b1,b2, . . . ,bb being blocks of the design (see [29]). k∗v (r,k,λ) has v(r + 1) vertices and vr(k+1) 2 edges. if d is a symmetric design then k∗v (r,k,λ) is k−regular graph in which there are at most λ paths of minimum length between each pair of vertices. graphs in which every pair of nonadjacent vertices has a unique path of minimum length between them are called geodetic graph, bigeodetic graphs are graphs in which each pair of nonadjacent vertices has at most two paths of minimum length between them and graphs in which each pair of nonadjacent vertices has at most k paths of minimum length between them are called k-geodetic graphs (see [13], [29]). we follow the construction of k∗v (r,k,λ) from a 2-(v,k,λ) design given in [29]. if bi = {pi1, . . . pik}, 1 ≤ i ≤ b, is a block of a design d, then {xi1i, . . . ,xiki} are vertices of the complete graph (kk)i. graphs (kk)i , 1 ≤ i ≤ b, together with v vertices xi0, 1 ≤ i ≤ v, xi0 and xst being adjacent if i = s, form the graph k∗v (r,k,λ) for the design d. the adjacency matrix of a graph k∗v (r,k,λ) is given as follows a =   (jk − ik) 0k . . . 0k m1 0k (jk − ik) · · · 0k m2 ... ... ... ... ... 0k 0k · · · (jk − ik) mb mt1 m t 2 . . . m t b 0v   , where mi = [mr,s], 1 ≤ i ≤ b, are k × v matrices with m1,i1 = m2,i2 = ... = mk,ik = 1 for bi = {pi1, . . . pik}, 1 ≤ i1 ≤ i2 ≤ ··· ≤ ik ≤ v and mr,s = 0 otherwise, 0k is the k ×k zero-matrix, jk is the k ×k all-one matrix, and ik is the k ×k identity matrix. rows of mi, 1 ≤ i ≤ b, are labeled with xi1i, ...,xiki, and columns of mi, 1 ≤ i ≤ b, are labeled with x10, ...,xv0. the number of columns labeled with xs0, 1 ≤ s ≤ v in which matrices mi and mj both have an entry 1 is equal to |bi ∩bj|, since in the column xs0 there is an entry 1 in both matrices if and only if ps ∈ bi ∩bj. moreover, the matrix mi is determined by the ith row of the incidence matrix im = [di,s] of the design d. vice versa, the ith row of the incidence matrix im = [di,s] is determined by the matrix mi, putting di,s = 1 if there exists a row of mi having 1 on the position xs0. theorem 5.1. let d1 and d2 be 2-(v,k,λ) designs. then the corresponding graphs k∗v (r,k,λ)1 and k∗v (r,k,λ) 2 are isomorphic if and only if the designs d1 and d2 are isomorphic. proof. let d1 = (p1,b1,i1) and d2 = (p2,b2,i2) be 2-(v,k,λ) designs and α be an isomorphism from d1 onto d2. then there exists unique isomorphism β between the corresponding graphs k∗v (r,k,λ)1 and k∗v (r,k,λ) 2 that satisfy( p1s α = p 2 t ∧b 1 i α = b 2 j ) ⇒ ( (kk) 1 i β = (kk) 2 j ∧x 1 s0β = x 2 t0 ) , where 1 ≤ s ≤ v, 1 ≤ i ≤ b. conversely, each isomorphism from the graph k∗v (r,k,λ) 1 onto k∗v (r,k,λ) 2 induces unique isomorphism from the design d1 onto d2. to prove this statement it is crusial to show that an isomorphism from k∗v (r,k,λ) 1 onto k∗v (r,k,λ) 2 maps vertices {x110, . . . ,x1v0} of k∗v (r,k,λ)1 onto vertices {x210, . . . ,x2v0} of k∗v (r,k,λ) 2. if the designs d1 and d2 are not symmetric, then r 6= k and since the vertices x110, . . . ,x1v0 and x210, . . . ,x 2 v0 have degree r and the other vertices of k ∗ v (r,k,λ) 1 and k∗v (r,k,λ) 2 have degree k, it is clear that an isomorphism from k∗v (r,k,λ) 1 onto k∗v (r,k,λ) 2 maps the set {x110, . . . ,x1v0} onto {x210, . . . ,x2v0}. 152 d. crnković et al. / j. algebra comb. discrete appl. 3(3) (2016) 145–154 if d1 and d2 are symmetric designs then r = k. a vertex x1i0 and a vertex adjacent to x 1 i0 have no common neighbour, while a vertex that do not belong to {x110, . . . ,x1v0} has k − 2 commmon neighbours with any of its neighbour. similarly, a vertex x2i0 and a vertex adjacent to him have no common neighbour, while a vertex that do not belong to {x210, . . . ,x2v0} has k − 2 commmon neighbours with any of its neighbour. hence, we conclude that {x110, . . . ,x1v0} is mapped onto {x210, . . . ,x2v0}. so, an isomorphism from k∗v (r,k,λ) 1 onto k∗v (r,k,λ) 2 maps (kk) 1 i onto (kk) 2 j, and m 1 i onto m 2 j, and it induces unique isomorphism from the design d1 onto d2. graphs k∗45(12, 12, 3) constructed from symmetric (45,12,3) designs are 12-connected and 12-regular graphs of diameter 4 with 585 vertices and 3510 edges. for each pair of nonadjacent vertices there are at most three paths of minimum length between them. from all known triplanes of order nine one can obtain 5421 non-isomorphic graphs k∗45(12, 12, 3), since non-isomorphic designs produce non-isomorphic trigeodetic graphs. the following theorem, which is proved in [11], shows that table 3 gives information on automorphism groups of all trigeodetic graphs constructed from the symmetric (45,12,3) designs with nontrivial automorphisms, and that there are at least 1136 trigeodetic graphs k∗45(12, 12, 3) having trivial group as the full automorphism group. theorem 5.2. let d be a 2-(v,k,λ) design. then the full automorphism group of d is isomorphic to the full automorphism group of the corresponding k-geodetic graph k∗v (r,k,λ). all symmetric (45,12,3) designs admitting nontrivial automorphisms can be found at http://www.math.uniri.hr/∼sanjar/structures/. references [1] r. w. ahrens, g. szekeres, on a combinatorial generalization of the 27 lines associated with a cubic surface, j. austral. math. soc. 10 (1969) 485–492. 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[26] mint, dept. of mathematics, university of salzburg, the online database for optimal parameters of (t,m,s)-nets, (t,s)-sequences, orthogonal arrays, linear codes, and ooas, online available at http://mint.sbg.ac.at/index.php, accessed on 2014-03-11. [27] r. m. ramos, j. sicilia, m. t. ramos, a generalization of geodetic graphs: k-geodetic graphs, investigacion operativa 6 (1998) 85–101. [28] l. rudolph, a class of majority logic decodable codes, ieee trans. inform. theory 13(2) (1967) 305–307. [29] n. srinivasan, j. opatrný, v. s. alagar, construction of geodetic and bigeodetic blocks of connectivity k ≥ 3 and their relation to block designs, ars combin. 24 (1987) 101–114. [30] l.h. soicher, design a gap package, version 1.6, 23/11/2011. (http://www.gapsystem.org/packages/design.html) 154 http://dx.doi.org/10.1023/a:1008373207617 http://dx.doi.org/10.1023/a:1008373207617 http://www.gap-system.org http://www.gap-system.org http://www.codetables.de http://www.codetables.de http://dx.doi.org/10.1016/s0167-5060(08)70919-1 http://dx.doi.org/10.1016/s0167-5060(08)70919-1 http://dx.doi.org/10.1002/jcd.20145 http://dx.doi.org/10.1002/jcd.20145 http://dx.doi.org/10.1016/0012-365x(91)90280-f http://dx.doi.org/10.1016/0012-365x(91)90280-f http://dx.doi.org/10.1016/0097-3165(73)90031-9 http://dx.doi.org/10.1016/0097-3165(73)90031-9 http://www.ams.org/mathscinet-getitem?mr=1884442 http://www.ams.org/mathscinet-getitem?mr=1884442 http://dx.doi.org/10.1002/(sici)1520-6610(1996)4:3<155::aid-jcd1>3.0.co;2-e http://mint.sbg.ac.at/index.php http://mint.sbg.ac.at/index.php http://mint.sbg.ac.at/index.php http://dx.doi.org/10.1109/tit.1967.1053994 http://dx.doi.org/10.1109/tit.1967.1053994 http://www.ams.org/mathscinet-getitem?mr=0917965 http://www.ams.org/mathscinet-getitem?mr=0917965 http://www.gap-system.org/packages/design.html http://www.gap-system.org/packages/design.html introduction outline of the construction classification of symmetric (45,12,3) designs with nontrivial automorphisms ternary codes from symmetric (45,12,3) designs on k-geodetic graphs from symmetric (45,12,3) designs references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.42848 j. algebra comb. discrete appl. 3(1) • 31–36 received: 31 october 2015 accepted: 30 november 2015 journal of algebra combinatorics discrete structures and applications an active attack on a multiparty key exchange protocol research article reto schnyder, juan antonio lópez-ramos, joachim rosenthal, davide schipani abstract: the multiparty key exchange introduced in steiner et al. and presented in more general form by the authors is known to be secure against passive attacks. in this paper, an active attack is presented assuming malicious control of the communications of the last two users for the duration of only the key exchange. 2010 msc: 94a60 keywords: multiparty key exchange, active attacks, group actions 1. introduction the increased use of light and mobile devices has led to the study of the so called mobile ad hoc networks. these are created, operated and managed by the nodes themselves and therefore are solely dependent upon the cooperative and trusting nature of the nodes. the ad hoc property of these mobile networks implies that the network is formed in an unplanned manner to meet an immediate demand and specific goal, and that the nodes are continuously joining or leaving the network. thus, key management in this type of networks is a very important issue and has been the aim of numerous works since then (see [1] or [5] and their references). one of the most widely known such schemes is due to steiner et al. and is known as cliques (cf. [4]). cliques is a multiparty key exchange protocol generalizing the diffie-hellman key exchange based on the discrete logarithm problem. it is composed of an initial key agreement (ika) to set up a first common key and an auxiliary key agreement (aka) in order to refresh the key at any later stage. the research was supported in part by the swiss national science foundation under grant no. 149716. juan antonio lópez-ramos is partially suppported by ministerio de educacion, cultura y deporte grant “salvador de madariaga” prx14/00121, ministerio de economia y competitividad grant mtm2014-54439 and junta de andalucia (fqm0211). reto schnyder is supported by armasuisse. reto schnyder, joachim rosenthal, davide schipani (corresponding author); institute of mathematics, university of zurich, switzerland (email: reto.schnyder@math.uzh.ch, rosenthal@math.uzh.ch, davide.schipani@math.uzh.ch). juan antonio lópez-ramos; department of mathematics, university of almeria, spain (email: jlopez@ual.es). 31 r. schnyder et. al. / j. algebra comb. discrete appl. 3(1) (2016) 31–36 in [3], the authors propose a systematic way for analyzing protocol suites which extend the diffiehellman key-exchange scheme to a group setting. they find interesting attacks which exploit algebraic properties of diffie-hellman exponentiation. however, our attack uses a different approach that exploits a weakness of a specific protocol and allows for prolonged eavesdropping. we will consider in particular one of the proposed initial key agreements referred to (in [4]) as ika.2. the authors generalize these schemes in [2], considering a general action on a semigroup, and this is how ika.2 is presented below. we will then show an active attack on this protocol that requires control of the communications of two particular parties for only the duration of the key exchange. that is, unlike in a regular man-in-themiddle attack, it is not necessary for the attacker to control the communications after the key exchange in order to translate messages, since all users are made to agree on the same key. although it is not possible for the attacker to keep a copy of the key after the users initiate aka operations, we will show how she can avoid being noticed at that point. 2. an initial key agreement protocol the protocol below gives n users the possibility to share an initial common key built using their private keys. a proof of its correctness and security against passive attacks can be found in [2, 4], assuming the diffie-hellman problem is hard for the given group action. suppose we have n users u1, . . . ,un who wish to agree upon a common key. let g be an abelian group, written multiplicatively. let s be a set, and suppose we have a group action g×s → s (g, s) 7→ g ·s. the users publicly agree on a common element c0 = s ∈ s, and for each i = 1, . . . , n, the user ui selects a secret group element gi ∈ g. the protocol proceeds as follows: 1. for i = 1, . . . , n− 2, ui sends to ui+1 the message ci = gi ·ci−1. 2. un−1 broadcasts cn−1 = gn−1 ·cn−2 to the other users u1, . . . ,un−2,un. 3. un computes the shared key k = gn ·cn−1. 4. for i = 1, . . . , n− 1, ui sends di = g−1i ·cn−1 to un. 5. un broadcasts {gn ·d1, gn ·d2, . . . , gn ·dn−1, cn−1} to ui, i = 1, . . . , n− 1. 6. for i = 1, . . . , n− 1, ui computes the shared key k = gi · (gn ·di). it is easy to see that for i = 1, . . . , n− 1, we have that ci = ( i∏ j=1 gj ) ·s, di = (n−1∏ j=1 j 6=i gj ) ·s, 32 r. schnyder et. al. / j. algebra comb. discrete appl. 3(1) (2016) 31–36 and finally k = cn = ( n∏ j=1 gj ) ·s. from the above, we can also observe that cn−1 is not needed by any user to recover the session key k. however, this information is disclosed for future rekeying purposes, as we will see later. example 2.1. let fq be a finite field. let us consider an element g of prime order p, generating the subgroup s ⊂ f∗q. then the action φ : z∗p × s → s defined by φ(x, h) = hx provides the initial key agreement protocol introduced in [4, section 4.2] as ika.2. example 2.2. let us denote by ε the group of points of an elliptic curve of prime order p. then the action φ : z∗p ×ε → ε defined by φ(x, p ) = xp gives an elliptic curve version of ika.2 cited above. 3. an active attack on the initial key agreement we describe an active attack on the protocol of the preceding section. suppose that the attacker m wants the users u1, . . . ,un to agree on a shared key as usual, except that she is in possession of the key as well. in order to carry out our attack, m needs to have full control over the communication of the users un−1 and un for the duration of the key exchange. however, unlike in a regular man-in-the-middle attack, she does not need to maintain this control after the key exchange is completed. in the beginning, m chooses her own secret group element ĝ ∈ g. she then proceeds as follows: 1. step (1) is carried out as usual. 2. m intercepts the broadcast of un−1 during step (2) and remembers the value cn−1. at this point, all users except for un−1 are sitting in step (2), waiting for the broadcast that was halted. 3. un−1 proceeds to step (4), where he sends g−1n−1 · cn−1 = cn−2 to un. this is also intercepted by m. un−1 is now waiting in step (5). 4. m now makes un believe that he received the broadcast of step (2), but actually sends him ĝ·cn−1. at this point, un computes the shared key k = gnĝ ·cn−1 and waits in step (4). 5. m now sends to un the values {m1, . . . , mn−3, cn−2, cn−1}, pretending that they were sent by the other users in step (4). the mi are random elements of the orbit g ·s. 6. in step (5), un sends back, among others, the values gn ·cn−2 and gn ·cn−1, which m intercepts. the user un is now finished, and m can compute the shared key k = ĝgn ·cn−1. 7. until now, u1, . . . ,un−2 have been waiting for the broadcast in step (2), which m now provides in the form of gn ·cn−1. 8. ui, i = 1 . . . , n− 2, go to step (4) and send back g−1i gn ·cn−1, which m intercepts. 9. in step (5), m broadcasts to ui, i = 1 . . . , n− 2, the message {ĝg−11 gn ·cn−1, ĝg −1 2 gn ·cn−1, . . . , ĝg −1 n−1gn ·cn−1, gn ·cn−1} user un−1 is sent the same message, but the last element, gn ·cn−1 is substituted by cn−1. 10. the users u1, . . . ,un−2 now all compute the shared secret k = giĝg−1i gn ·cn−1. 33 r. schnyder et. al. / j. algebra comb. discrete appl. 3(1) (2016) 31–36 let us make some comments on the attack introduced above. first, we can observe that at the end of this procedure, all users as well as the attacker share the same key k = ( n∏ j=1 gj ) · (ĝ ·s). any passive observer will still be unable to determine the key, for the same reason that the original protocol is secure against passive attacks, cf. [4, theorem 2.1], whose proof also applies to the general setting given in section 2 whenever the action is transitive and the diffie-hellman problem is hard. the attacker’s secret ĝ is not strictly required for the attack to work, but without it, the users may notice that something is amiss. namely, in step (e), if we leave out ĝ, the user un may notice that m sent the same value cn−1 as in step (d). similarly, in step (i), the other users could notice that the attacker just returned their transmission from (h). using ĝ, however, the users should be unable to tell the difference between a regular execution of the protocol and the attack, again as a consequence of [4, theorem 2.1]. as in the initial key agreement (ika) protocol introduced in section 2, the broadcast element gn ·cn−1 is added at the end of the message in (i) in view of future rekeying operations and is not needed by any of the users u1, . . . ,un−2 to recover the shared key. note that users ui, i = 1, . . . , n − 2, expect that the last element of the message sent in step (i) is the one broadcast in step (2) of the protocol, which the attacker substitutes precisely by gn · cn−1. in the case of user un−1, who is also expecting the element sent in step (2) of the protocol, the element that m sends in step (b) is cn−1. if this is not satisfied, the users might notice that something is wrong. 4. an exit strategy after the attack of section 3, the attacker m shares the key with the users u1, . . . ,un and can listen in on their conversation without any further active measures. however, at some point after that, the users may wish to execute an aka operation, which is to say a key refreshment, the addition of a new member to the group, etc. as described in [4, section 5]. after this point, the attacker can certainly no longer listen to the conversation. even worse, the values the users remember from step (5) of the protocol are substantially different from normal, and any key refresh operation will thus fail completely, alerting the users about the attack. in what follows, we will describe how the attacker can avoid being noticed by forging key refresh operations herself, assuming that any user may initiate a key refreshment at any time. first, we recall the key refresh operation after a regular execution of ika.2, adapted from [4, section 5.6]. suppose user uc wishes to initiate a key refreshment. he remembers from step (5) of the key agreement protocol the values {e1, . . . , en}, where ek = (∏n j=1,j 6=k gj ) ·s, k = 1, . . . , n. he picks a new secret g′c ∈ g and broadcasts {g′c ·e1, . . . , g ′ c ·ec−1, ec, g ′ c ·ec+1, . . . , g ′ c ·en}. now, all users can compute the new key g′c ·cn = g′c · (∏n j=1 gj ) ·s. user uc also replaces his own secret with g′cgc, and everyone replaces the information remembered from step (5) with this new broadcast. remark 4.1. one important detail to note is that when uc initiates the key refreshment, the value ec he sends in position c is unchanged and already known to the other users. hence, if m wishes to forge a key refreshment coming from uc, she has to make sure that each user receives in position c the value he previously held there. otherwise, the attack could be discovered. suppose now that the attacker m has just executed the attack from section 3. instead of {e1, . . . , en}, the users now remember the following values: 34 r. schnyder et. al. / j. algebra comb. discrete appl. 3(1) (2016) 31–36 • for i = 1, . . . , n− 2, ui remembers {ĝ ·e1, . . . , ĝ ·en−1, cn}. • un−1 remembers {ĝ ·e1, . . . , ĝ ·en−1, en}. • un remembers {gn ·m1, . . . , gn ·mn−3, en−1, cn, ĝ ·en}. evidently, if some user tries to initiate a key refreshment with these values, the operation will fail. however, m can bring the users into a consistent state by forging two key refresh operations herself. for this, she needs to still have control over the communications of un−1 and un, as in the original attack. first, m picks two new random values f̂ and ĥ ∈ g. then, she forges a key refresh operation by sending the following values to the different users: • to ui, i = 1, . . . , n− 2, she sends {ĥĝ ·e1, ĥĝ ·e2, . . . , ĥĝ ·en−2, ĝ ·en−1, f̂ĥĝ ·en}, pretending it came from un−1. • to un−1, she sends {f̂ĥĝ ·e1, ĥĝ ·e2, . . . , ĥĝ ·en−2, ĥĝ ·en−1, en}, pretending it came from un. • to un, she sends {f̂ĥĝ ·e1, ĥĝ ·e2, . . . , ĥĝ ·en−2, cn, ĥĝen}, pretending it came from un−1. after this, the users will agree on the shared key ĥĝ ·cn, which is also known to m. as remarked above, if a user is made to believe that he received a key refreshment from uc, he must receive in position c the value he already held there. now, the values held by the users are still inconsistent, so m has to forge a second key refreshment: • to ui, i = 1, . . . , n− 2, she sends {f̂ĥĝ ·e1, . . . , f̂ĥĝ ·en}, pretending it came from un. • to un−1 and un, she sends {f̂ĥĝ ·e1, . . . , f̂ĥĝ ·en}, pretending it came from u1. now, all users and the attacker agree on the shared key f̂ĥĝ · cn. furthermore, all users remember the same consistent values for key refreshment. if in the future any user initiates a key refreshment or other aka operation, the attacker will lose access to the key, but the operation itself will work out without problem and without the users noticing anything wrong. remark 4.2. an alternative course of action for m is to convert the attack into a regular man-in-themiddle attack on un at the time of the first key refreshment. for this, note that given the values each user remembers, a key refreshment initiated by uc, c ≤ n − 2, works well for all users but un. the attacker can then intercept the broadcast arriving at un and replace it with random values, except that at position n she sends ĥ · en for some random ĥ ∈ g, and at position c she sends gn · mc, which she knows from step (f) of the attack. then, m will have the key ĝg′c ·cn in common with ui, i ≤ n−1, as well as ĥ ·cn with un. from then on, she can run a regular man-in-the-middle attack. a similar attack can be carried out if un initiates a key refreshment, but not if un−1 does so. in this case, the attacker can intercept and apply ĝ to the message for un so that all users agree on a common key without noticing the previous attack. 35 r. schnyder et. al. / j. algebra comb. discrete appl. 3(1) (2016) 31–36 5. conclusions we have presented an active attack against cliques, one of the most popular multiparty key exchange protocols currently known. by assuming control of the communications of the last two users in the initial key agreement process, an attacker can eavesdrop into the subsequent communications between legitimate users without the need of a regular man-in-the-middle attack. the attacker is also able to leave the group unnoticed when later key renewals do not allow further eavesdropping. references [1] p. p. c. lee, j. c. s. lui, d. k. y. yau, distributed collaborative key agreement and authentication protocols for dynamic peer groups, ieee/acm trans. netw. 14(2) (2006) 263–276. [2] j. a. lópez-ramos, j. rosenthal, d. schipani, r. schnyder, group key management based on semigroup actions, arxiv.org/pdf/1509.01075v1.pdf. [3] o. pereira, j. -j. quisquater, generic insecurity of cliques-type authenticated group key agreement protocols, proc. 17th ieee computer security foundations workshop, 16–29, 2004. [4] m. steiner, g. tsudik, m. waidner, key agreement in dynamic peer groups, ieee trans. parallel distrib. systems 11(8) (2000) 769–780. [5] j. van der merwe, d. dawoud, s. mcdonald, a survey on peer-to-peer key management for mobile ad hoc networks, acm computing surveys 39(1) (2007) 1–45. 36 http://dx.doi.org/10.1109/tnet.2006.872575 http://dx.doi.org/10.1109/tnet.2006.872575 http://arxiv.org/pdf/1509.01075v1.pdf http://arxiv.org/pdf/1509.01075v1.pdf http://dx.doi.org/10.1109/csfw.2004.1310729 http://dx.doi.org/10.1109/csfw.2004.1310729 http://dx.doi.org/10.1109/71.877936 http://dx.doi.org/10.1109/71.877936 http://dx.doi.org/10.1145/1216370.1216371 http://dx.doi.org/10.1145/1216370.1216371 introduction an initial key agreement protocol an active attack on the initial key agreement an exit strategy conclusions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284943 j. algebra comb. discrete appl. 4(2) • 123–129 received: 17 june 2015 accepted: 5 february 2016 journal of algebra combinatorics discrete structures and applications on some radicals and proper classes associated to simple modules research article septimiu crivei, derya keskin tütüncü abstract: for a unitary right module m, there are two known partitions of simple modules in the category σ[m]: the first one divides them into m-injective modules and m-small modules, while the second one divides them into m-projective modules and m-singular modules. we study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes. 2010 msc: 16s90, 16d60, 16d50, 16d40, 16d90 keywords: radical, proper class, simple module, relative supplement submodule, m-injective module, m-projective module, m-small module, m-singular module 1. introduction for an associative ring r with identity, let m be a unitary right r-module. then the category σ[m] is a grothendieck category which consists of all right r-modules subgenerated by m, that is, submodules of m-generated right r-modules. an object x of σ[m] is called: 1. m-injective if the functor homr(−,x) preserves exactness of all short exact sequences 0 → k → m → n → 0. 2. m-projective if the functor homr(x,−) preserves exactness of all short exact sequences 0 → k → m → n → 0. 3. m-small if x is superfluous in some module y ∈ σ[m]. 4. m-singular if x ∼= y/z for some essential submodule z of a module y ∈ σ[m]. septimiu crivei; faculty of mathematics and computer science, babeş-bolyai university, 400084 cluj-napoca, romania (email: crivei@math.ubbcluj.ro). derya keskin tütüncü (corresponding author); department of mathematics, hacettepe university, 06800, ankara, turkey (email: keskin@hacettepe.edu.tr). 123 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 there are two classical dichotomies concerning the simple modules in σ[m]. the first one partitions them into the pair of classes (s(i ), s(m )), where s(i ) is the class of simple m-injective modules in σ[m] and s(m ) is the class of simple m-small modules in σ[m] [4, 4.2]. the second one partitions them into the pair of classes (s(p), s(s )), where s(p) is the class of simple m-projective modules in σ[m] and s(s ) is the class of simple m-singular modules in σ[m] [3, 8.2]. in the present note we give equivalent conditions under which the above two pairs of classes of simple modules may be linked in terms of some associated radicals and proper classes. more precisely, we are interested in such characterizations for possible inclusions between the four classes of simple modules s(i ), s(m ), s(p) and s(s ). they are inspired by and complete results by preisser montaño [5]. 2. preliminaries in this paper r will be an associative ring with identity and any module m will be a unitary right r-module. 2.1. radicals let a be an abelian category. we first recall the definitions of preradicals and radicals from [6, chapter 6, §1]. a preradical τ of a is a subfunctor of the identity functor on a, that is, a functor τ : a−→a such that: 1. τ(a) ⊆ a for every object a of a. 2. τ(f) = f |τ(a): τ(a) −→ τ(b) for every morphism f : a −→ b in a. a preradical τ of a is idempotent if ττ = τ. preradicals of a are partially ordered by the relation defined by τ ≤ τ′ if and only if τ(a) ⊆ τ′(a) for every object a of a. if the abelian category a is complete and cocomplete, then τ(⊕i∈iai) = ⊕i∈iτ(ai) and τ( ∏ i∈i ai) ⊆ ∏ i∈i τ(ai) for any family {ai}i∈i of objects of a and any preradical τ of a. note that τ(p) = pτ(rr) for all projective modules p in the category mod-r of right r-modules and any preradical τ of mod-r. a preradical τ of a is called radical if τ(a/τ(a)) = 0, for all objects a of a. in particular, one has the jacobson radical rad on a grothendieck category, where rad(x) is the intersection of maximal subobjects of x for every object x. let τ be a radical of σ[m]. let l be a submodule of a module n ∈ σ[m]. then l is called a τsupplement in n if there exists a submodule x of n such that n = l+x and l∩x ⊆ τ(l), equivalently, n = l + x and l∩x = τ(l) (see [1, 1.11] or [5, 4.9]). a similar concept may also be given with respect to a class c of modules in σ[m]. a submodule k of a module n ∈ σ[m] is called c-small in n if for every submodule x ≤ n, the equality k + x = n and n/x ∈ c imply x = n. we denote this by k �c n (see [5, 8.1]). for the class s of m-singular modules, note that s -small submodules are the δ-small submodules in σ[m] as defined by zhou in [7] (see [5, 8.2(i)]). a submodule l of a module n ∈ σ[m] is called a c-supplement in n if there exists a submodule l′ of n such that l + l′ = n and l∩l′ �c l (see [5, 9.1]). denote by s(c) the class of simple modules in c, and let n ∈ σ[m]. then denote rads(c)(n) = rej(n,s(c)) = ⋂ {ker(f) | f : n −→ s,s ∈s(c)} (see [5, 10.1]). note that rads(c) is a radical, rad ≤ rads(c) and rads(c)(n) = n if and only if n has no nonzero simple factor modules in c (see [5, 10.2]). on the other hand, if c is closed under submodules and factor modules, then we also have rads(c)(n) = ∑ {l ≤ n | l �c n} 124 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 (see [5, 10.3]). then rads(c)(n) = n if and only if every finitely generated submodule of n is c-small in n (see [5, 10.2 and 10.3]). 2.2. proper classes let a be an abelian category. we now recall the definition of proper classes from [2] (also see [5, chapter 2, section 3]). let p be a class of short exact sequences in a. if 0 −→ a f−→ b g−→ c −→ 0 belongs to p, then f is called a p-monomorphism and g a p-epimorphism. a class p of short exact sequences in a is called a proper class if it satisfies the following axioms: (p1) p is closed under isomorphisms. (p2) p contains all splitting short exact sequences of a. (p3) if f and f′ are p-monomorphisms and the composition f′f is defined, then f′f is a pmonomorphism. (p4) if f and f′ are monomorphisms such that the composition f′f is defined and is a p-monomorphism, then f is a p-monomorphism. (p5) if g and g′ are p-epimorphisms and the composition gg′ is defined, then gg′ is a p-epimorphism. (p6) if g and g′ are epimorphisms such that the composition gg′ is defined and is a p-epimorphism, then g is a p-epimorphism. proper classes in a are partially ordered by the relation defined by p ⊆p′ if and only if every short exact sequence in p belongs to p′. the class of all short exact sequences in σ[m] 0 −→ a −→ b −→ c −→ 0 such that a is a c-supplement in b, is a proper class and it is denoted by c-suppl (see [5, 9.2]). the class of all short exact sequences in σ[m] 0 −→ a −→ b −→ c −→ 0 such that a is a rads(c)-supplement in b, is a proper class and it is denoted by rads(c)-suppl (see [5, 10.4]). if c = σ[m], then rads(c) = rad. and if c is a class in σ[m] closed under submodules and factor modules, then c-suppl ⊆ rads(c)-suppl (see [5, 10.7]). 3. class inclusions following [5, 10.20] and keeping the notation from the introduction, we consider the radicals α = rads(i ), β = rads(p), γ = rads(m ), 125 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 δ = rads(s ) of σ[m] associated to the corresponding classes of simple modules which are m-injective, m-projective, m-small and m-singular respectively. note that rad = α ∩ γ = β ∩ δ and α is an idempotent radical (see [5, 10.22]). then we also have the proper classes α-suppl, β-suppl, γ-suppl and δ-suppl associated to the radicals α, β, γ and δ. we begin with four class inclusions for which we have equivalent conditions in terms of both associated radicals and associated proper classes. we first recall the following result, which motivated our study. proposition 3.1. [5, proposition 10.26] let m be a module. the following conditions are equivalent: 1. s(s ) ⊆s(i ), i.e. every m-singular simple module is m-injective. 2. α ≤ δ. 3. α-suppl ⊆ δ-suppl. proposition 3.2. let m be a module. the following are equivalent: 1. s(m ) ⊆s(p), i.e. every m-small simple module is m-projective. 2. β ≤ γ. 3. β-suppl ⊆ γ-suppl. proof. we only need to prove (3) ⇒ (1). let s be an m-small simple module. then γ(s) = 0. since s is simple, we have β(s) = 0 or β(s) = s. assume that β(s) = s. then the short exact sequence 0 −→ s −→ em(s) −→ em(s)/s −→ 0 belongs to β-suppl, and hence it belongs to γ-suppl by hypothesis, where em(s) is the m-injective hull of s in σ[m]. so, there exists a submodule t of em(s) such that em(s) = s +t and s∩t = γ(s) = 0. this means that s is m-injective, a contradiction. therefore β(s) = 0. now there exists a nonzero homomorphism f : s −→ s ′ with s ′ m-projective simple. thus s is m-projective. one obtains the following two propositions in a similar way. proposition 3.3. let m be a module. the following are equivalent: 1. s(p) ⊆s(i ), i.e. every m-projective simple module is m-injective. 2. α ≤ β. 3. α-suppl ⊆ β-suppl. proposition 3.4. let m be a module. the following are equivalent: 1. s(m ) ⊆s(s ), i.e. every m-small simple module is m-singular. 2. δ ≤ γ. 3. δ-suppl ⊆ γ-suppl. the remaining four possible class inclusions may only be partially given in general as above. proposition 3.5. let m be a module. consider the following conditions: 126 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 1. s(i ) ⊆s(s ), i.e. every m-injective simple module is m-singular. 2. δ ≤ α. 3. δ-suppl ⊆ α-suppl. then (1) ⇔ (2) and (2) ⇒ (3). proof. for (1) ⇒ (2) ⇒ (3) see [5, proposition 10.27]. (2) ⇒ (1) let s be an m-injective simple module. then α(s) = 0. since δ ≤ α, we have δ(s) = 0. now there exists a nonzero homomorphism f : s −→ s ′ with s ′ simple m-singular, because δ(s) 6= s. therefore s is m-singular. one obtains the following three propositions in a similar way. proposition 3.6. let m be a module. consider the following conditions: 1. s(p) ⊆s(m ), i.e. every m-projective simple module is m-small. 2. γ ≤ β. 3. γ-suppl ⊆ β-suppl. then (1) ⇔ (2) and (2) ⇒ (3). proposition 3.7. let m be a module. consider the following conditions: 1. s(i ) ⊆s(p), i.e. every m-injective simple module is m-projective. 2. β ≤ α. 3. β-suppl ⊆ α-suppl. then (1) ⇔ (2) and (2) ⇒ (3). proposition 3.8. let m be a module. consider the following conditions: 1. s(s ) ⊆s(m ), i.e. every m-singular simple module is m-small. 2. γ ≤ δ. 3. γ-suppl ⊆ δ-suppl. then (1) ⇔ (2) and (2) ⇒ (3). obviously, the equalities s(i ) = s(p) and s(m ) = s(s ) hold if and only if the associated radicals are equal if and only if the associated proper classes coincide. in what follows we are interested in conditions under which we have equivalences in propositions 3.5, 3.6, 3.7 and 3.8. let τ be a preradical of σ[m]. recall that an epimorphism f : p −→ l is called a projective τ-cover of l in σ[m] if ker(f) ⊆ τ(p) and p is projective in σ[m] (see [1, 2.11]). proposition 3.9. let m be a module. assume that the class tγ = {x ∈ σ[m] | γ(x) = x} is closed under submodules and δγ(p) = 0 for every projective module in σ[m]. if every simple module has a projective γ-cover in σ[m], then the implication (3) ⇒ (1) holds in proposition 3.8. 127 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 proof. assume that γ-suppl ⊆ δ-suppl. let s be an m-singular simple module. so, we have δ(s) = 0. if γ(s) = 0, then s will be m-small. suppose the contrary. then γ(s) = s, because s is simple. since every simple module has a projective γ-cover in σ[m], there exists an epimorphism f : p −→ s such that ker(f) ⊆ γ(p) and p is projective in σ[m]. by [1, lemma 2.12], f(γ(p)) = γ(s) = s. now we have p = γ(p) + ker(f) = γ(p). since tγ is closed under submodules, we have γ(ker(f)) = ker(f), and hence the short exact sequence 0 −→ ker(f) −→ p −→ s −→ 0 belongs to γ-suppl. by hypothesis, there exists a submodule l of p such that p = ker(f) + l and l ∩ ker(f) = δ(ker(f)). on the other hand, δ(p) = δγ(p) = 0 and hence δ(ker(f)) = 0. therefore p = ker(f)⊕l. this implies that s is m-projective, a contradiction. therefore γ(s) = 0, and hence s is m-small. in a similar manner one obtains the following three propositions. proposition 3.10. let m be a module. assume that the class tβ = {x ∈ σ[m] | β(x) = x} is closed under submodules and αβ(p) = 0 for every projective module in σ[m]. if every simple module has a projective β-cover in σ[m], then the implication (3) ⇒ (1) holds in proposition 3.7. proposition 3.11. let m be a module. assume that the class tγ = {x ∈ σ[m] | γ(x) = x} is closed under submodules and βγ(p) = 0 for every projective module in σ[m]. if every simple module has a projective γ-cover in σ[m], then the implication (3) ⇒ (1) holds in proposition 3.6. proposition 3.12. let m be a module. assume that the class tδ = {x ∈ σ[m] | δ(x) = x} is closed under submodules and αδ(p) = 0 for every projective module in σ[m]. if every simple module has a projective δ-cover in σ[m], then the implication (3) ⇒ (1) holds in proposition 3.5. example 3.13. (see [7, example 4.1]) consider the category mod-r. let q = ∏∞ i=1 fi, where each fi = z2. let r be the subring of q generated by ⊕∞i=1fi and 1q. by [7, example 4.1], every simple r-module has a projective δ-cover, while f0 = r/(⊕∞i=1fi),f1,f2, . . . are the only simple r-modules and f0 is the only singular simple r-module. therefore f1,f2, . . . are the only projective simple r-modules. this means that every simple r-module has a projective β-cover except for f0. for, assume that f0 has a projective β-cover f : p −→ f0. since f0 is simple singular, β(f0) = f0. therefore β(p) = p = pβ(rr). note that β(qr) = β ( ∞∏ i=1 fi ) ⊆ ∞∏ i=1 β(fi) = 0 implies that β(rr) = 0. so p = 0, which is a contradiction. therefore f0 does not have a projective β-cover. 128 s. crivei, d. keskin tütüncü / j. algebra comb. discrete appl. 4(2) (2017) 123–129 references [1] k. al–takhman, c. lomp, r. wisbauer, τ−complemented and τ−supplemented modules, algebra discrete math. 3 (2006) 1–16. [2] d. buchsbaum, a note on homology in categories, ann. math. 69(1) (1959) 66–74. [3] j. clark, c. lomp, n. vanaja, r. wisbauer, lifting modules, frontiers in mathematics, birkhäuser, basel, 2006. [4] n. v. dung, d. v. huynh, p. smith, r. wisbauer, extending modules, pitman research notes in mathematics, harlow, longman, 1994. [5] c. f. preisser montaño, proper classes of short exact sequences and structure theory of modules, ph.d. thesis, düsseldorf, 2010. [6] b. stenström, rings of quotients, springer, berlin, heidelberg, new york, 1975. [7] y. zhou, generalizations of perfect, semiperfect and semiregular rings, algebra colloq. 7(3) (2000) 305–318. 129 http://mi.mathnet.ru/eng/adm266 http://mi.mathnet.ru/eng/adm266 http://dx.doi.org/10.2307/1970093 http://dx.doi.org/10.1007/3-7643-7573-6 http://dx.doi.org/10.1007/3-7643-7573-6 http://dx.doi.org/10.1007/978-3-642-66066-5 http://dx.doi.org/10.1007/s10011-000-0305-9 http://dx.doi.org/10.1007/s10011-000-0305-9 introduction preliminaries class inclusions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284970 j. algebra comb. discrete appl. 4(2) • 207–217 received: 12 june 2015 accepted: 8 june 2016 journal of algebra combinatorics discrete structures and applications the extension problem for lee and euclidean weights research article philippe langevin, jay a. wood abstract: the extension problem is solved for the lee and euclidean weights over three families of rings of the form z/nz: n = 2`+1, n = 3`+1, or n = p = 2q + 1 with p and q prime. the extension problem is solved for the euclidean psk weight over z/nz for all n. 2010 msc: 94b05 keywords: extension problem, lee weight, euclidean weight, egalitarian weight 1. introduction one of the themes of the mini-cours at the lens conference was the extension theorem of macwilliams. in its original form, this theorem says that any linear isomorphism between linear codes defined over a finite field that preserves the hamming weight must extend to a monomial transformation. this result has been generalized in several directions, including: for linear codes defined over finite frobenius rings with respect to the hamming weight [12] or the homogeneous weight [7]; for linear codes defined over finite frobenius rings with respect to symmetrized weight compositions [11] (with an improved proof in [2]); for linear codes defined over finite commutative chain rings with respect to any weight function satisfying certain conditions [13, 14]; for linear codes equipped with a weight function having maximal symmetry and satisfying certain conditions, defined over products of finite chain rings [6] or over a finite principal ideal ring [5]. despite all this progress, there are glaring gaps in our knowledge: does the extension theorem hold for linear codes defined over z/nz with respect to the lee weight or the euclidean weight? in general, we do not know. the purpose of this paper is to describe what we do know. in [15, examples 3.7 and 3.9], it was claimed that the extension theorem holds for the lee and euclidean weights over the rings z/2kz, z/3kz, philippe langevin; laboratoire imath, université de toulon, 83957 la garde cedex, france (email: langevin@univ-tln.fr). jay a. wood (corresponding author); department of mathematics, western michigan university, 1903 w. michigan ave., kalamazoo, mi 49008–5248 usa (email: jay.wood@wmich.edu). 207 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 and z/pz for a prime p of the form p = 2q + 1 with q prime. proofs of those claims will constitute the bulk of this paper. barra [1] has proved that the extension theorem holds for the lee weight over z/pz for a prime p of the form p = 4q + 1 with q prime. in [15, example 3.8], it was claimed that the extension theorem holds for euclidean psk weight over the rings z/3kz and z/pz for a prime p of the form p = 2q + 1 with q prime. for this weight, we give a simple proof that works for all z/nz. this proof is similar to the proof in [2, theorem 6.6] for the homogeneous weight over any finite frobenius ring. 2. background let r = z/nz be the ring of integers modulo n. every element x of r is represented uniquely by an integer a satisfying −n/2 < a ≤ n/2 with x ≡ a mod n. the lee weight wl(x) (resp., euclidean weight we(x)) of x ∈ r is the ordinary absolute value |a| (resp., the square |a|2) of the representative a. alternatively, if one represents x ∈ r by a unique integer representative b satisfying 0 ≤ b < n with x ≡ b mod n, then the lee weight and the euclidean weight have the form wl(x) = { b, 0 ≤ b ≤ n/2, n − b, n/2 ≤ b < n; we(x) = { b2, 0 ≤ b ≤ n/2, (n − b)2, n/2 ≤ b < n. there is another euclidean weight, the euclidean psk weight wpsk, which is used in phase-shift key modulation. it is defined using the squared euclidean distance in the complex numbers; for x ∈ r, wpsk(x) = |exp(2πix/n) − 1|2 = 2 − 2 cos(2πx/n). note that all three weights satisfy w(−x) = w(x), for x ∈ r. the lee and the euclidean weights can be extended to real-valued functions (still denoted by wl, we, and wpsk) on the product rn: wl(x) = n∑ i=1 wl(xi), we(x) = n∑ i=1 we(xi), wpsk(x) = n∑ i=1 wpsk(xi), for x = (x1,x2, . . . ,xn) ∈ rn. a monomial transformation t on rn is an r-module isomorphism t : rn → rn of the form t(x1,x2, . . . ,xn) = (u1xσ(1),u2xσ(2), . . . ,unxσ(n)), for (x1,x2, . . . ,xn) ∈ rn, where u1,u2, . . . ,un are units of r and σ is a permutation of the set {1, 2, . . . ,n}. if g is a subgroup of the group of units of r and each ui ∈ g, we say that t is a g-monomial transformation. in the case where g = {±1}, i.e., each ui = ±1, we say that t is a signed permutation. the following proposition is now immediate. proposition 2.1. suppose t : rn → rn is a signed permutation. then t preserves the lee and euclidean weights. that is, for every x ∈ rn, wl(t(x)) = wl(x), we(t(x)) = we(x), and wpsk(t(x)) = wpsk(x). 208 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 conversely, the signed permutations are precisely the linear isometries of rn with respect to any of the three weights. proposition 2.2. let w be one of the weights wl,we,wpsk, and suppose f : rn → rn is an r-module homomorphism that preserves w: w(f(x)) = w(x), for all x ∈ rn. then f is a signed permutation. proof. observe that f is injective. indeed, only the zero vector has weight zero, so if f(x) = 0, then 0 = w(f(x)) = w(x) implies x = 0. because rn is a finite set, f is also surjective, hence an r-module isomorphism. let ei = (0, . . . , 0, 1, 0, . . . , 0), with the 1 in position i, 1 ≤ i ≤ n. then w(1) = w(ei) = w(f(ei)). but w(1) = w(−1) is the smallest positive value of w on r, so it follows that f(ei) = ±ej for some j, 1 ≤ j ≤ n. because f is known to be an isomorphism, it follows that f is a signed permutation. again, let w be one of the weights wl,we,wpsk. we say that r = z/nz has the extension property with respect to w if the following property is satisfied for every r-submodule c ⊆ rn and every rmodule homomorphism f : c → rn: if f preserves w, w(f(x)) = w(x), x ∈ c, then f extends to a signed permutation. that is, there exists a signed permutation t defined on all of rn such that t(x) = f(x) for x ∈ c. another way to view the extension property is that every linear w-isometry on c extends to a linear w-isometry on rn. the main results of this paper are the following theorems. theorem 2.3. the rings in the following list have the extension property with respect to the lee weight wl and the euclidean weight we: 1. r = z/2`+1z, l ≥ 0; 2. r = z/3`+1z, l ≥ 0; 3. r = z/pz, with prime p of the form p = 2q + 1, q prime. remark 2.4. barra proved that r = z/pz, with prime p of the form p = 4q + 1, q prime, has the extension property with respect to the lee weight wl [1]. remark 2.5. (added in proof.) after the submission of this paper, the authors, together with serhii dyshko, have extended theorem 2.3, first to the cases r = z/pz for all primes p [3], and then to the cases r = z/p`+1z for all primes p and ` ≥ 0 [p. langevin, j. a. wood, the extension theorem for the lee and euclidean weights over z/pkz, in preparation, 2016]. the methods used for the new results are very different from those of sections 4 and 5 of this paper. theorem 2.6. for any positive integer n, the ring r = z/nz has the extension property with respect to the euclidean psk weight wpsk. after an explanation of the method of attack in section 3 and an analysis of cases in section 4, the proof of theorem 2.3 appears in section 5. the proof of theorem 2.6 appears in section 6. 3. method of attack for theorem 2.3 in this section we will describe how the proof of theorem 2.3 reduces to showing that certain fourier coefficients are nonzero. in both this section and the next, we will use some basic facts from number theory, such as the structure of the group of units for z/nz and the form of cyclotomic polynomials, which may be found in textbooks such as [10]. there is an extension theorem for general weight functions over a finite frobenius ring, [13, theorem 3.1]. the theorem below is the form that this theorem takes in the context of the lee or the euclidean weights on the frobenius ring r = z/nz. 209 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 to establish notation, let r = z/nz and let w be one of wl, we, or wpsk on r. let m = bn/2c be the largest integer less than or equal to n/2. form an m×m real matrix aw as follows. the entry of aw in position (i,j), 1 ≤ i,j ≤ m, equals w(ij), the value of the weight function on the product of i and j in the ring r. theorem 3.1 ([13, theorem 3.1]). if the matrix aw is nonsingular over c, then r has the extension property with respect to w. remark 3.2. in fact, theorem 3.1 applies to any complex-valued weight w on r that satisfies two properties: w(0) = 0 and sym(w) = {±1}. here, sym(w) denotes the symmetry group of the weight w: sym(w) = {u ∈ r : u is a unit, and w(ux) = w(x) for all x ∈ r}. example 3.3. for r = z/9z, the determinants of the matrices for wl and we are detal = det   1 2 3 4 2 4 3 1 3 3 0 3 4 1 3 2   = 189, detae = det   1 4 9 16 4 16 9 1 9 9 0 9 16 1 9 4   = 45 927. remark 3.4. maple has been used to verify that r = z/nz has the extension property for wl and we for n ≤ 2048. the smallest singular values of the matrices al and ae were calculated and seen to be nonzero. barra has verified the extension property for wl over r = z/pz for the first 2012 primes p [1, p. 44]. now assume that r = z/nz is a local ring, so that n equals a prime power; r is then a finite commutative chain ring. we continue to assume that w is one of wl,we,wpsk. in this chain ring context, the determinant detaw can be factored into linear expressions that are fourier transforms of values of the weight function w, [14, theorem 7]. we explain this next. suppose r = z/p`+1z, with p prime. let u be the group of units of r. then |u| = φ(p`+1) = p`(p− 1), where φ is euler’s totient function. the group u is cyclic when p is an odd prime. for powers of 2, u is cyclic for z/2z and z/4z; for z/2`+1z, l ≥ 2, u is isomorphic to the product of a cyclic group of order 2 times a cyclic group of order 2`−1, with generators for the two cyclic factors being the residue classes of −1 and 5, respectively. the group u acts on the ring r by ring multiplication. the orbits have the form oi = {upi : u ∈u}, i = 0, 1, . . . ,` + 1. the orbit oi equals the set-theoretic difference of ideals (pi)\(pi+1), where (a) = ra denotes the principal ideal generated by a ∈ r. because u is abelian, the stabilizer subgroup of a point in an orbit oi, i = 0, 1, . . . ,`+ 1, is independent of the point chosen; the stabilizer subgroup depends only on the orbit. denote the stabilizer subgroup of a point of oi by ui. then ui = {u ∈ u : upi = pi}, and oi can be identified with the coset space u/ui. the ideal (pi) has order |(pi)| = p`+1−i, i = 0, 1, . . .` + 1, so that |oi| = p`+1−i −p`−i = p`−i(p− 1) and |ui| = pi. the group of complex characters (group homomorphisms from u to the multiplicative group of nonzero complex numbers) of the group u will be denoted û. a pair (π,oi) consisting of a character π ∈ û and a u-orbit oi is admissible if ui ⊆ ker π. for π ∈ û, let iπ be the largest integer j ≤ ` such that (π,oj) is admissible. because u0 = {1}, every pair (π,o0) is admissible, so that iπ ≥ 0 for all π ∈ û. given a subgroup h ⊆ u, the annihilator (û : h) of h in û is defined by (û : h) = {π ∈ û : π(h) = 1}. let u = {±1} ⊆ u. suppose a character π satisfies (1) π ∈ (û : u) and (2) (π,oi) is admissible. for w equal to wl,we,wpsk, we define w̌(π,i) = ∑ u∈u/uui w(upi)π(u). (1) 210 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 observe that this character sum is well-defined. here is the factorization of detaw, for w equal to wl,we,wpsk. theorem 3.5 ([14, theorem 7]). let r = z/p`+1z and u = {±1}⊆u. for w equal to wl,we,wpsk, there exists a nonzero integer constant c such that detaw = c ∏ π∈(û:u) w̌(π,iπ) 1+iπ. remark 3.6. theorem 3.5 also holds for any weight w satisfying the conditions in remark 3.2. theorem 3.5 can be viewed as a generalization of the factorization of the group determinant given by dedekind and frobenius in 1896 [4]. example 3.7. for r = z/9z, the factorizations have the form detaw = 3w(3)2[w(1) + w(2)ζ + w(4)ζ2][w(1) + w(2)ζ2 + w(4)ζ] = 3w(3)2[w(1)2 −w(1)w(2) −w(1)w(4) + w(2)2 −w(2)w(4) + w(4)2], where ζ is a primitive third root of unity in c. using wl,we, we recover the values of detaw in example 3.3. for additional details of this computation, see [14, example 12]. remark 3.8. in order to prove theorem 2.3, we will show that w̌(π,iπ) 6= 0 for all π ∈ (û : u) and then appeal to theorems 3.1 and 3.5. to illustrate the type of argument that will be used in the next two sections, consider the factorization of detaw in example 3.7. under what conditions could one of the factors vanish? in the example, there are three factors. one possibility is w(3) = 0. for the other two factors, make use of the minimal polynomial for ζ over q, i.e., ζ2 + ζ + 1 = 0. then w(1) + w(2)ζ + w(4)ζ2 = (w(1) −w(4)) + (w(2) −w(4))ζ, w(1) + w(4)ζ + w(2)ζ2 = (w(1) −w(2)) + (w(4) −w(2))ζ. both these factors would vanish when w(1) = w(2) = w(4), and, if the values of w are rational numbers, this is the only way in which these factors could vanish. for wl and we, it is easy to see that the factors do not vanish. 4. analysis of cases in this section we will analyze the structure of the orbits oi and the stabilizer subgroups ui that occur in the character sum (1). the case where n = 2`+1 let r = z/2`+1z. when r = z/2z or r = z/4z, the extension property for wl or we follows easily from theorem 3.1 (also, see remark 3.4). for the rest of the discussion, we assume ` ≥ 2. then the group of units u of r is isomorphic to the product of a cyclic group b of order 2 and a cyclic group c of order 2`−1, with b and c generated by the residue classes of −1 and 5, respectively. in particular, b = u = {±1}. in theorem 3.5, it is the characters π ∈ (û : u), i.e., characters such that π(−1) = 1, that contribute to the factorization of detaw. since (û : u) ∼= (u/u)̂ and u ∼= u ×c, we see that u/u ∼= c is a cyclic group of order 2`−1, and (û : u) ∼= ĉ. 211 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 the orbits of u on r have the form oi = {u2i : u ∈u}, i = 0, 1, . . . ,` + 1, and the stabilizer subgroup of the point 2i ∈oi is ui = {1 + m2`+1−i : 0 ≤ m < 2i}, i = 0, 1, . . . ,`. when i = ` + 1, u`+1 = u. the stabilizer subgroups satisfy {1} = u0 ⊂u1 ⊂ ···⊂ul = u`+1 = u, and |ui| = 2i (for 0 ≤ i ≤ `). observe that −1 ∈ui only for i = ` or ` + 1. thus, for i < `, the image of ui under the projection u ∼= u ×c → c ∼= u/u is the unique subgroup in c of order 2i. in particular, ui is cyclic, for i < `. proposition 4.1. let r = z/2`+1z, ` ≥ 2, and u = {±1}. if π ∈ (û : u), then ker π = uuiπ. proof. take any π ∈ (û : u). under the isomorphism (û : u) ∼= ĉ, we can view π as a character on c. from that perspective, ker π is a subgroup e of c of order 2k, for some k ≤ ` − 1 (since c is a group of order 2`−1). note that e is the image of uk under the projection u → c. when we view π as a character on u, we see that uk ⊂ ker π, that ker π ∼= u ×e (since π ∈ (û : u)), and that ker π = uuk. if k < `− 1, we claim that k = iπ, i.e., that k is the largest integer j such that uj ⊂ ker π. on the one hand, ker π ∼= u ×e has order 2k+1 but is not cyclic. if uk+1 ⊂ ker π, then uk+1 = ker π, because both groups have order 2k+1. but uk+1 is cyclic, forcing ker π to be cyclic as well; contradiction. thus, if k < `− 1, we have iπ = k and ker π = uuiπ. if k = `− 1, then ker π ∼= u ×e has order 2`, so that ker π = u and π is the trivial character. in that case, iπ = `, so that ker π = u = u` = uu`. the case where n = 3`+1 we first assume that n = p`+1, with p an odd prime. we will specialize to p = 3 later. set r = z/p`+1z, and let u be its group of units. the group u is cyclic of order p`(p− 1). let u = {±1}; then u/u is cyclic of order p`(p− 1)/2. the orbits of u on r have the form oi = {upi : u ∈u}, i = 0, 1, . . . ,` + 1, and the stabilizer subgroup of the point pi ∈oi is ui = {1 + mp`+1−i : 0 ≤ m < pi}, i = 0, 1, . . . ,`. when i = ` + 1, u`+1 = u. the stabilizer subgroups satisfy {1} = u0 ⊂u1 ⊂ ···⊂u` ⊂u`+1 = u, and |ui| = pi (for 0 ≤ i ≤ `). observe that −1 ∈ui only for i = ` + 1. proposition 4.2. let r = z/3`+1z and u = {±1}. if π ∈ (û : u), then ker π = uuiπ. proof. take any π ∈ (û : u). viewing π as a character on u/u, ker π is a subgroup of the cyclic group u/u, which, for a general odd prime p, has order p`(p − 1)/2. for a general odd prime p, there is not much control on the order of ker π. but this proposition assumes that p = 3, so that the cyclic group u/u has order 3`. thus ker π is the unique subgroup of order 3k for some k ≤ l. under the projection u → u/u, ui projects to a subgroup of order 3i (for i ≤ `) because −1 6∈ ui. thus, when we view π as a character on u, we see that uk ⊂ ker π, and ker π = uuk has order 2 · 3k. note that uk+1 6⊆ ker π by size considerations, so that iπ = k and ker π = uuiπ. for i = ` + 1, ker π = u, so that iπ = ` and ker π = uuiπ. 212 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 the case where n = p = 2q + 1, p, q prime let r = z/pz, where p is prime and p = 2q + 1, with q prime. then the group of units u is cyclic of order p− 1 = 2q. set u = {±1}; then u/u is cyclic with prime order q. because r is a field, the orbit structure of u acting on r is very simple. there are only two orbits: o0 = u = r \ (0) and o1 = (0). the stabilizer subgroups of a point are u0 = {1} and u1 = u, respectively. proposition 4.3. let r = z/pz, with prime p satisfying p = 2q + 1, q prime. set u = {±1}. if π ∈ (û : u), then ker π = uuiπ. proof. take any π ∈ (û : u). because u/u has prime order, ker π (viewed as a subgroup of u/u) is either trivial or all of u/u. when viewing π as a character on u, ker π = u or ker π = u. in the first case, iπ = 0, so that ker π = u = uu0; in the second case, iπ = 1, and ker π = u = uu1. a common corollary propositions 4.1, 4.2, and 4.3 all lead to a common corollary. corollary 4.4. let r = z/nz, with, respectively, n = 2`+1, n = 3`+1, or n = p = 2q + 1, with p and q prime. let u = {±1}. let w be wl or we on r. for any π ∈ (û : u), • the quotient groups u/uuj are cyclic groups of prime power order; • the character π, viewed as a homomorphism π : u/uuiπ → c, is injective; and • the function f : u/uuiπ → q, f(u) = w(upiπ ), is well-defined and injective. (here, p is 2, 3, or p, respectively.) proof. the group u/uuj is a quotient of the group u/u, which is cyclic of prime power order (under the hypotheses on n). by propositions 4.1, 4.2, and 4.3, ker π = uuiπ, so that the character π viewed as π : u/uuiπ → c is injective. that the function f is well-defined and injective follows from the definition of uiπ and the ±-symmetry of w (i.e., w(−x) = w(x), for x ∈ r). 5. proof of theorem 2.3 the proof of theorem 2.3 will depend upon understanding the fourier transform of a function defined on a cyclic group of prime power order. in this section we isolate the necessary lemma that will be used and then prove theorem 2.3. let p be a prime, and suppose g is a cyclic group of order pk. let γ ∈ g be a generator of g. given any function f : g → c, the fourier transform of f is a function f̂ : ĝ → c, given by f̂(π) = ∑ g∈g π(g)f(g), π ∈ ĝ. lemma 5.1. assume g is a cyclic group of order pk, p prime. if π ∈ ĝ is injective as a function π : g → c and f : g → q is any injective function with rational values, then f̂(π) 6= 0. proof. the image ξ = π(γ) under π of the generator γ of g is a pkth root of unity. because π is assumed to be injective, ξ is a primitive pkth root of unity. as g = γj varies over g, the images π(g) = ξj 213 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 vary over all the pkth roots of unity. the fourier transform of f has the form f̂(π) = pk−1∑ j=0 f(γj)ξj. (2) the summation in (2), with 0 ≤ j < pk, can be written as a double summation by making use of the residue class m of j mod pk−1. write j = ipk−1 + m, with 0 ≤ i < p and 0 ≤ m < pk−1. as a double summation, the fourier transform is f̂(π) = p−1∑ i=0 pk−1−1∑ m=0 f(γip k−1+m)ξip k−1+m. (3) to simplify (3), we make use of the minimal polynomial of ξ over q. the minimal polynomial of ξ over q has degree φ(pk) = pk−1(p− 1), and the minimal polynomial itself is the cyclotomic polynomial φpk(x) = x (p−1)pk−1 + x(p−2)p k−1 + · · · + xp k−1 + 1. (4) in (3), those terms where i = p − 1 will be replaced by the following expression derived from the minimal polynomial (4): ξ(p−1)p k−1+m = − p−2∑ i=0 ξip k−1+m. (5) after substituting the expressions from (5) into (3), the form of f̂(π) becomes f̂(π) = p−2∑ i=0 pk−1−1∑ m=0 {f(γip k−1+m) −f(γ(p−1)p k−1+m)}ξip k−1+m. (6) observe from (6) that f̂(π), as a polynomial in ξ, has degree < (p − 1)pk−1, the latter being the degree of the minimal polynomial of ξ over q. if f̂(π) = 0, then all the (rational) coefficients of the powers of ξ must vanish. this contradicts f being injective. proof of theorem 2.3. when n = 2`+1, n = 3`+1, or n = p = 2q + 1, p, q prime, corollary 4.4 shows that the groups u/uuj are cyclic of prime power order. it also shows that a character π ∈ (û : u) is injective on u/uuj when j = iπ. set g = u/uuiπ, and define f : g → q by f(u) = w(upiπ ), where w is wl or we. this is a well-defined injective function on g. under the assumptions above, lemma 5.1 implies that w̌(π,iπ) = ∑ u∈u/uuiπ w(upiπ )π(u) 6= 0 (see (1)). because the w̌(π,iπ) are the factors of detaw in theorem 3.5, we conclude that detaw 6= 0. by theorem 3.1, the extension property holds in the cases claimed in the statement of the theorem. the same arguments show that theorem 2.3 holds for any rational-valued weight satisfying the conditions of remark 3.2 such that w(x) = w(y) implies y = ±x. rational weights yield rational coefficients in (6), which are needed to apply the minimal polynomial argument. 214 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 remark 5.2. consider the case where r = z/pz, with p = 4q + 1 and p,q prime, as in remark 2.4. the group u/u is cyclic of order 2q. for any π ∈ (û : u), ker π (viewed as a subgroup of u/u) will be cyclic of order 1, 2, q, or 2q. barra provides separate arguments for each of the four cases. refer to [1, theorem 5.28] for the details. for other values of n, the type of analysis given above quickly becomes very complicated. the group u/u need not be cyclic, and even in cases (such as n prime) where u/u is cyclic, it need not have prime power order. the degree and form of the cyclotomic polynomial becomes complicated, and the number of different cases to consider quickly gets out of hand. 6. an egalitarian weight and the proof of theorem 2.6 theorem 2.6 will be a special case of a general result valid over any finite module with a cyclic socle, in particular, over any finite frobenius ring. the treatment here generalizes that for frobenius bimodules in [18, section 2.3] and will be rather brisk. readers wanting more background information are also referred to [16]. we work in the following context. let r be a finite ring with 1, and let a be a finite unitary left r-module. suppose χ is an generating character for a, i.e., χ : a → c× is a homomorphism from the additive group of a to the multiplicative group of nonzero complex numbers that has the property that ker χ contains no nonzero left r-submodules. such a generating character exists if and only if the socle of a is cyclic [17, proposition 14] (and in the case where a = r, a generating character exists if and only if the ring r is frobenius [12, theorem 3.10]). it follows that any character π on a is a right scalar multiple of χ; i.e., π = χr, for some r ∈ r, where χr(a) = χ(ra) for a ∈ a. let g be a subgroup of the automorphism group glr(a). we will write elements g ∈ glr(a) as acting on the right of a, so that the preservation of scalar multiplication takes the form (rx)g = r(xg), for r ∈ r, x ∈ a, and g ∈ glr(a). define a weight wg : a → c on a by wg(x) = 1 − 1 |g| ∑ g∈g χ(xg), x ∈ a. a weight w on a is egalitarian if there exists a nonzero constant γ such that ∑ x∈b w(b) = γ|b| for any nonzero left r-submodule b ⊆ a. proposition 6.1. the weight wg has the following properties. 1. the value wg(0) = 0. 2. the weight wg is right g-invariant; i.e., wg(xg) = wg(x), for all x ∈ a and g ∈ g. 3. the weight wg is egalitarian with γ = 1. moreover, wg is egalitarian on cosets with γ = 1; i.e.,∑ b∈b wg(x0 + b) = |b| for any nonzero left r-submodule b ⊆ a and element x0 ∈ a. proof. because χ(0) = 1, the first result is immediate. the right invariance follows immediately by a re-indexing argument. to prove that wg is egalitarian on cosets, let x0 ∈ a and b ⊆ a be a nonzero submodule. we calculate: ∑ b∈b wg(x0 + b) = ∑ b∈b  1 − 1 |g| ∑ g∈g χ((x0 + b)g)   = |b|− 1 |g| ∑ g∈g χ(x0g) ∑ b∈b χ(bg) = |b|, 215 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 where we have used the fact that ∑ b∈b χ(bg) = 0 for any nonzero left submodule b. indeed, because ker χ contains no nonzero left submodules, there exists some b0 ∈ b with χ(b0g) 6= 1. by re-indexing (b = b0 + c), we see that ∑ b∈b χ(bg) = χ(b0g) ∑ c∈b χ(cg). thus, ∑ b∈b χ(bg) = 0. remark 6.2. when a = r and g = u(r), the group of units of r, the resulting weight wu(r) is the homogeneous weight on r [9]. the egalitarian property for the homogeneous weight was proved in [8, theorem 2]. the proof of the next theorem generalizes that for the homogeneous weight found in [2, theorem 6.6]. theorem 6.3. the weight wg has the extension property. that is, if c ⊆ an is a left r-linear code in an and f : c → an is an injective homomorphism of r-modules that preserves wg, wg(xf) = wg(x) for all x ∈ c, then f extends to a g-monomial transformation of an. proof. write the components of f as f = (f1,f2, . . . ,fn); each fi : c → a is a homomorphism of left r-modules. inputs to fi are written on the left. weight preservation and the definition of wg yield, for all x ∈ c, n∑ i=1  1 − 1 |g| ∑ g∈g χ((xfi)g)   = n∑ i=1  1 − 1 |g| ∑ g∈g χ(xig)   . this simplifies to n∑ i=1 ∑ g∈g χ((xfi)g) = n∑ i=1 ∑ g∈g χ(xig), x ∈ c. (7) this is an equation of functions on c, specifically, linear combinations of characters on c. because characters are linearly independent over the complex numbers, if we set i = 1 and g = idg on the left side of (7), there exist i = σ(1) and g1 ∈ g on the right side so that χ(xf1) = χ(xσ(1)g1) for all x ∈ c. this implies that the image of the module homomorphism c → a, x 7→ xf1 − xσ(1)g1, is contained in ker χ. but ker χ contains no nonzero left submodules, so xf1 = xσ(1)g1 for all x ∈ c. it now follows, by re-indexing, that∑ g∈g χ(xf1g) = ∑ g∈g χ(xσ(1)g1g) = ∑ g∈g χ(xσ(1)g) for all x ∈ c. this allows us to reduce by one the size of the outer summation in (7) and to proceed by induction to produce a permutation σ of {1, 2, . . . ,n} and elements gi ∈ g with xfi = xσ(i)gi for all x ∈ c and i = 1, 2, . . . ,n. proof of theorem 2.6. let r = z/nz, a = r, and g = {±1}. a generating character for z/nz is χ(x) = exp(2πix/n), x ∈ z/nz, where exp is the usual complex exponential function. then χ(x) + χ(−x) = 2 cos(2πx/n), so that wg(x) = 1 − cos(2πx/n), x ∈ z/nz. up to a factor of 2, wg(x) equals wpsk. acknowledgment: we thank aleams barra for renewing our interest in this problem. the second author thanks the université de toulon for its support for research visits in 2000 and 2015, when the ideas of this paper were developed and finalized. 216 p. langevin, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 207–217 references [1] a. barra, equivalence theorems and the local-global property, proquest llc, phd thesis university of kentucky, ann arbor, mi, usa, 2012. 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[18] j. a. wood, relative one-weight linear codes, des. codes cryptogr. 72(2) (2014) 331–344. 217 http://dx.doi.org/10.1016/j.jpaa.2014.04.026 http://dx.doi.org/10.1016/j.jpaa.2014.04.026 http://dx.doi.org/10.1016/j.crma.2016.05.004 http://dx.doi.org/10.1016/j.crma.2016.05.004 http://dx.doi.org/10.1016/j.jcta.2014.03.005 http://dx.doi.org/10.1016/j.jcta.2014.03.005 http://dx.doi.org/10.1007/s10623-012-9671-9 http://dx.doi.org/10.1007/s10623-012-9671-9 http://dx.doi.org/10.1006/jcta.1999.3033 http://dx.doi.org/10.1006/jcta.1999.3033 http://dx.doi.org/10.1007/pl00000451 http://dx.doi.org/10.1007/3-540-63163-1_26 http://dx.doi.org/10.1007/3-540-63163-1_26 http://dx.doi.org/10.1007/3-540-63163-1_26 http://www.ams.org/mathscinet-getitem?mr=1738408 http://www.ams.org/mathscinet-getitem?mr=1738408 http://www.ams.org/mathscinet-getitem?mr=1650644 http://www.ams.org/mathscinet-getitem?mr=1650644 http://www.ams.org/mathscinet-getitem?mr=1650644 http://dx.doi.org/10.1007/978-3-642-57189-3_23 http://dx.doi.org/10.1007/978-3-642-57189-3_23 http://dx.doi.org/10.1090/s0002-9947-01-02905-1 http://dx.doi.org/10.1090/s0002-9947-01-02905-1 http://www.ams.org/mathscinet-getitem?mr=2850303 http://www.ams.org/mathscinet-getitem?mr=2850303 http://www.ams.org/mathscinet-getitem?mr=2850303 http://www.ams.org/mathscinet-getitem?mr=2975625 http://www.ams.org/mathscinet-getitem?mr=2975625 http://www.ams.org/mathscinet-getitem?mr=2975625 http://dx.doi.org/10.1007/s10623-012-9769-0 introduction background method of attack for theorem ?? analysis of cases proof of theorem ?? an egalitarian weight and the proof of theorem ?? references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.349383 j. algebra comb. discrete appl. 5(1) • 19–27 received: 23 january 2017 accepted: 26 june 2017 journal of algebra combinatorics discrete structures and applications coloring sums of extensions of certain graphs research article johan kok, saptarshi bej abstract: we recall that the minimum number of colors that allow a proper coloring of graph g is called the chromatic number of g and denoted χ(g). motivated by the introduction of the concept of the b-chromatic sum of a graph the concept of χ′-chromatic sum and χ+-chromatic sum are introduced in this paper. the extended graph gx of a graph g was recently introduced for certain regular graphs. this paper furthers the concepts of χ′-chromatic sum and χ+-chromatic sum to extended paths and cycles. bipartite graphs also receive some attention. the paper concludes with patterned structured graphs. these last said graphs are typically found in chemical and biological structures. 2010 msc: 05c15, 05c20, 05c38, 05c62 keywords: chromatic number, χ′-chromatic sum, χ+-chromatic sum, extended path, extended cycle 1. introduction for general notation and concepts in graph and digraph theory, we refer to [2, 4, 5]. unless mentioned otherwise, all graphs mentioned in this paper are simple, connected, finite and undirected graphs of order n ≥ 2. we recall that the minimum number of colors that allow a proper coloring of graph g is called the chromatic number of g and denoted χ(g). since coloring may be effected from any alphabet like in an application of cryptographical coding of vertices, we use a preferred definition slightly different from the contemporary definition found in the literature. consider a proper k-coloring of a graph g and denote the set of k colors, c = {c1,c2,c3, . . . ,ck}. also consider the disjoint subsets of v (g) i.e. vci = {vj : vj → ci,vj ∈ v (g),ci ∈c}, 1 ≤ i ≤ k. clearly, v (g) = k⋃ i=1 vci. if for largest k ∈ n, a proper k-coloring is found such that, for all color class vci; 1 ≤ i ≤ k, there exists at least one vs ∈ vci such that vsvt ∈ e(g), for at least one vt ∈ vcj ; 1 ≤ j ≤ k, i 6= j, then the b-chromatic number of g is defined to be ϕ(g) = k. such a coloring is called a b-coloring of g. johan kok (corresponding author); centre for studies in discrete mathematics, vidya academy of science & technology, thrissur, india (email: kokkiek2@tshwane.gov.za). saptarshi bej; department of mathematics & statistics, indian institute of science education, kolkata, india (email: saptarshibej24@gmail.com). 19 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 the concept of extending regular graphs has been introduced in [1]. we shall use the idea of extending a graph (also called an extended graph) in general and study how certain interesting invariants change in the extended graph. 2. on extensions of graphs we formally define the notion of a degree-extension of a graph g. definition 2.1. (i) let g be a graph of even order n ≥ 2. if the complement graph gc has a perfect matching say m, then gxc = g + m is called a complete degree-extension of g, also called an c-extended g. (ii) let g be a graph of odd order n ≥ 3 and let u ∈ v (g) such that dg(u) = ∆(g). if the complement graph (g−u)c has a perfect matching say m, then gxic = g + m is called an incomplete degree-extension of g, also called an ic-extended g. the choices of vertex u, dg(u) = ∆(g) in definition 2.1(ii) is by default. other criteria can apply of which u, dg(u) = δ(g) is an obvious other criteria. an immediate and important result following from definition 2.1 is given by the next corollary. corollary 2.2. (i) in gxc, dgxc (v) = dg(v) + 1, ∀v ∈ v (g). (ii) in gxic, dgxic (v) = dg(v) + 1, ∀v ∈ v (g), v 6= u and ∆(g x ic) ∈{∆(g), ∆(g) + 1}. in [1] it is shown that an r-regular graph g of even order n with r < n 2 always has a degree-extension gxc . it means that g can always be extended to an (r + 1)-regular graph g x c by adding edges to g. first, we generalise the result to any simple connected graph g for which ∆(g) < n 2 . theorem 2.3. let g be a graph of order n, n even with ∆(g) < n 2 , then graph g has a complete degree-extension gxc. also, if n is odd and a unique vertex u exists such that dg(u) = ∆(g) then g has a complete degree-extension. proof. following from definition 2.1 we consider two cases. case (i): see [1]. assume that n is even. since ∆(g) < n 2 the complement graph gc has δ(gc) ≥ n 2 . recalling dirac’s theorem it follows that gc is hamiltonian. so for each hamilton cycle in gc exactly two perfect matchings exist in gc. therefore, g has at least two complete degree-extensions, gxc . case (ii): assume that n is odd and let here exists a unique vertex u ∈ v (g) such that dg(u) = ∆(g). consider the graph g−u then certainly ∆(g−u) < n−1 2 . following from case (i) we have that g−u has a degree extension (g − u)xc . from definition 2.1(ii) it follows that (g − u)x + u is an incomplete degree-extension gxic of g, with ∆(g x ic) ∈{∆(g), ∆(g) + 1}. it is observed that bn 2 c edges must be added to obtain an incomplete degree-extension gxic. henceforth we will only consider a graph g of even order n. if n is even and if k distinct hamilton cycles exist in gc then 2k complete extended graphs exist. for graphs on even number of vertices we have the next result. theorem 2.4. let g be a graph of even order n. if the complement graph gc has a path of length n−1 (spanning path), then the graph g has a complete degree-extension gxc. proof. if the complement graph gc has a spanning path of length n− 1 exactly one perfect matching corresponding to that path in gc exists. the result follows immediately. corollary 2.5. let g be a graph of even order n. if v (g) can be partitioned into a number of subsets v1(g),v2(g), . . . ,v`(g), each containing even number of vertices such that the complement of each induced subgraph 〈vi(g)〉c, 1 ≤ i ≤ ` has a spanning path, then g has a complete degree-extension gxc. 20 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 proof. denote the spanning path in 〈vi(g)〉c byp(i), 1 ≤ i ≤ `. from each p(i) we select the unique perfect matching s(i) in 〈vi(g)〉c. clearly, ⋃̀ i=1 s(i) is a perfect matching in gc. clearly, the smallest path for which theorem 2.2 finds application is p4. let the vertices of p4 consecutively be labelled v1,v2,v3,v4. then, the unique px4,c = p4 + (v1v3,v2v4). for p6 we find the application of theorem 2.2 allows the complete degree-extensions, px16,c = p6 + (v1v6,v2v4,v3v5) or p x2 6,c = p6 + (v1v3,v2v5,v4v6) or p x3 6,c = p6 + (v1v4,v2v5,v3v6) or p x4 6,c = p6 + (v1v5,v2v4,v3v6) or p x5 6,c = p6 + (v1v4,v2v6,v3v5). 3. χ′-chromatic sum and χ+-chromatic sum of certain graphs we recall that a vertex coloring of a graph g such that adjacent vertices are not allocated the same color is called a proper coloring of g. the minimum number of colors in a proper coloring of g is called the chromatic number, χ(g). 3.1. the χ′-chromatic sum and χ+-chromatic sum of extended paths and extended cycles a new concept called the b-chromatic sum of a graph was introduced in [10]. analogous to this the notion of general color sum of graphs has been defined in [7] as: let c = {c1,c2,c3, . . . ,ck} allow a b-coloring s of g. as stated in [7] there are k! ways of allocating the colors to the vertices of g. let the color weight θ(ci) be the number of times a color ci is allocated to vertices. in general we refer to the color sum of a coloring s and define it as, ω(s) = k∑ i=1 i ·θ(ci). the b-chromatic sum is given by ϕ′(g) = min{ k∑ i=1 i ·θ(ci) : ∀ b-colorings of g}. the general color sums of certain cycle related graphs has been studied in [11]. this interesting new invariant motivated similar concepts in graph coloring. definition 3.1. [7] for a graph g the χ′-chromatic sum is defined to be: χ′(g) = min{ k∑ i=1 i ·θ(ci) : ∀ minimum proper colorings of g}. definition 3.2. citeks1 for a graph g the χ+-chromatic sum is defined to be: χ+(g) = max{ k∑ i=1 i ·θ(ci) : ∀ minimum proper colorings of g}. further motivation for these new invariants is as follows. if the colors represent different technology types and the configuration requirement is that at least one unit per technology type must be placed at a point in a network without similar technology types being adjacent, two further considerations come into play. firstly, if the higher indexed colors represent technology types with higher failure rate (risk) then the placement of the maximal number of higher indexed units is the solution to ensure a functional network. on the other hand, if the lower indexed colors represent a less costly (procurement, installation, commissioning and maintenance) technology type, and minimising total cost is the priority, then the placement of maximal number of lower indexed units is the desired solution. we recall two important results from [7]. theorem 3.3. [7] for a path pn, n ≥ 1 the χ′-chromatic sum and χ+-chromatic sum are given by: 21 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 (i) χ′(pn)) =   1, if n = 1, 3n 2 , if n is even, 3 · bn 2 c + 1, if n is odd. (ii) χ+(pn)) =   1, if n = 1, 3n 2 , if n is even, 3 · bn 2 c + 2, if n is odd. theorem 3.4. [7] for a cycle cn the χ′-chromatic sum and χ+-chromatic sum are given by: (i) χ′(cn)) = 3 · d n 2 e. (ii) χ+(cn)) = { 3n 2 , if n is even, 5 · bn 2 c + 1, if n is odd. from pxi6,c, 1 ≤ i ≤ 5 we note that p x3 6,c allows a minimum proper coloring v1 → c1,v2 → c2,v3 → c1,v4 → c2,v5 → c1,v6 → c2, such that χ′(px36,c) = 9 = min{χ ′(pxi6 ) : 1 ≤ i ≤ 5}. also we note that px46 , (by symmetry p x5 6 as well) allows a minimum proper coloring v1 → c3,v2 → c2,v3 → c1,v4 → c3,v5 → c2,v6 → c3, such that χ+(px46,c) = 14 = max{χ +(pxi6,c) : 1 ≤ i ≤ 5}. up to isomorphism p6 has 4 distinct complete extensions. these observations lead to the next results which indeed follow directly from definitions 3.1 and 3.2. proposition 3.5. (i) for a graph g of even order n, such that the complement graph gc has at least one perfect matching then, if up to isomorphism the graph g has gxic , 1 ≤ i ≤ ` complete degree-extensions then, χ′(gxc ) = min{χ′(gxic ) : 1 ≤ i ≤ `}. (ii) for a graph g of even order n, such that the complement graph gc has at least one perfect matching then, if up to isomorphism the graph g has gxic , 1 ≤ i ≤ ` complete degree-extensions then, χ+(gxc ) = max{χ+(gxic ) : 1 ≤ i ≤ `}. theorem 3.6. for a path pn, n ≥ 4 and even, we have (i) χ′(px4,c) = 7 and χ +(px4,c) = 9. (ii) χ′(pxn,c) = 3n 2 and χ+(pxn,c) = 5n 2 − 1, for n ≥ 6. proof. case (i): consider the unique px4,c = p4 +(v1v3,v2v4) and allocate the minimum proper coloring v1 → c1,v2 → c2,v3 → c3,v4 → c1. it follows that, χ′(px4,c) = 7. now interchange the colors c1 and c3. it follows that, χ+(px4,c) = 9. case (ii)(a): consider pn, n ≥ 6. subcase (ii)(a)(1): if n 2 is odd, consider pxn,c = pn + (v1vn2 +1,vn2 vn,v1+ivn−i), 1 ≤ i ≤ n 2 − 1. allocate the minimum proper coloring v1 → c1,v2 → c2,v3 → c1, . . . ,vn → c2. so, θ(c1) = n2 and θ(c2) = n 2 . hence, χ′(pxn,c) = 1 ·θ(c1) + 2 ·θ(c2) = n 2 + 2 · n 2 = 3n 2 = min{χ′(pxin,c) : ∀pxin,c}. subcase (ii)(a)(2): if n 2 is even, consider pxn,c = pn + (v1vn2 ,vn2 +1vn,v1+ivn−i), 1 ≤ i ≤ n 2 −1. the result now follows like in subcase (ii)(a)(1). 22 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 case (ii)(b): consider pn, n ≥ 6. subcase (ii)(b)(1): if n 2 is odd let, t = n 2 . construct the graph p∗n = pn + (v1vt+1,vt−1vt+3,vtvt+2). now, allocate the proper coloring v1 → c3,v2 → c2,v3 → c3, . . . ,vt−1 → c2,vt → c3,vt+1 → c1,vt+2 → c2,vt+3 → c3, . . . ,vn−1 → c2,vn → c3. since equal number of vertices are left with degree 2 and colored c3, c2 respectively, it is always possible to complete the extension to obtain pxn,c having the allocated minimum proper coloring. clearly, θ(c1) = 1, θ(c2) = t−1 and θ(c3) = t. hence, χ+(pxn,c) = 3 · n 2 + 2 ·(n 2 −1) + 1 = 5n 2 − 1 = max{χ+(pxin,c) : ∀pxin,c}. subcase (ii)(b)(2): if n 2 is even let, t = n 2 . construct the graph p∗n = pn + (v1vt+1,vt−1vt+3,vtvt+2). now, allocate the proper coloring v1 → c3,v2 → c2,v3 → c3, . . . ,vt−1 → c2,vt → c3,vt+1 → c1,vt+2 → c2,vt+3 → c3, . . . ,vn−1 → c2,vn → c3. since equal number of vertices are left with degree 2 and colored c3, c2 respectively, it is always possible to complete the extension to obtain pxn,c having the allocated minimum proper coloring. clearly, θ(c1) = 1, θ(c2) = t−1 and θ(c3) = t. hence, χ+(pxn,c) = 3 · n 2 + 2 ·(n 2 −1) + 1 = 5n 2 − 1 = max{χ+(pxin,c) : ∀pxin,c}. theorem 3.7. for a cycle cn, n ≥ 4, we have: (i) χ′(cx4,c) = χ +(cx4,c) = 10. (ii) χ′(cxn,c) = 3n 2 and: χ+(cxn,c)) = { 5n 2 − 1, if n 2 is even, n ≥ 6, 5n 2 − 3, if n 2 is odd, n ≥ 6. proof. case (i): consider the unique cx4,c = c4 + (v1v3,v2v4) = k4. hence, χ +(cx4,c) = 10. case (ii)(a): consider cn, n ≥ 6. subcase (ii)(a)(1): if n 2 is odd, consider cxn,c = cn + (v1vn2 +1,vn2 vn,v1+ivn−i), 1 ≤ i ≤ n 2 − 1. allocate the minimum proper coloring v1 → c1,v2 → c2,v3 → c1, . . . ,vn → c2. so, θ(c1) = n2 and θ(c2) = n 2 . hence, χ′(cxn,c) = 1 ·θ(c1) + 2 ·θ(c2) = n 2 + 2 · n 2 = 3n 2 = min{χ′(cxin,c) : ∀cxin,c}. subcase (ii)(a)(2): if n 2 is even, consider cxn,c = cn + (v1vn2 ,vn2 +1vn,v1+ivn−i), 1 ≤ i ≤ n 2 −1. the result now follows like in subcase (ii)(a)(1). case (ii)(b): consider cn, n ≥ 6. subcase (ii)(b)(1): if n 2 is even let t = n 2 . construct the graph c∗n = cn + (v1vt+1,vt−1vt+3,vtvt+2). now, allocate the proper coloring v1 → c3,v2 → c2,v3 → c3, . . . ,vt−1 → c2,vt → c3,vt+1 → c1,vt+2 → c2,vt+3 → c3, . . . ,vn−1 → c2,vn → c1. since equal number of non-adjacent vertices are left with degree 2 and half colored c2 and other half colored c3 respectively, it is always possible to complete the extension to obtain cxn,c having the allocated minimum proper coloring. clearly, θ(c1) = 1, θ(c2) = t − 1 and θ(c3) = t. hence, χ+(cxn,c) = 3 · n 2 + 2 · (n 2 − 1) + 1 = 5n 2 − 1 = max{χ+(cxin,c) : ∀cxin,c}. subcase (ii)(b)(2): if n 2 is odd let t = n 2 . construct the graph c∗n = cn + (v1vt+1,vt−1vt+3,vtvt+2). now, allocate the proper coloring v1 → c3,v2 → c2,v3 → c3, . . . ,vt−1 → c2,vt → c3,vt+1 → c1,vt+2 → c2,vt+3 → c3, . . . ,vn−1 → c1,vn → c3. since equal number of non-adjacent vertices are left with degree 2 and half colored c2 and other half colored c3 respectively, it is always possible to complete the extension to obtain cxn,c having the allocated minimum proper coloring. clearly, θ(c1) = 2, θ(c2) = t − 1 and θ(c3) = t− 1. hence, χ+(cxn,c) = 3 · ( n 2 − 1) + 2 · (n 2 − 1) + 2 = 5n 2 − 3 = max{χ+(cxin,c) : ∀cxin,c}. 3.2. on bipartite graphs it is well known that a graph g is bipartite if and only g contains no odd cycle. up to isomorphism a graph (connected) which contains no odd cycle has a unique vertex-set partition say, x,y such that if v,u ∈ x then vu /∈ e(g) and similarly for vertices in y . if |x| and |y | are even we say that the partition 23 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 is even balanced, else if |x| and |y | are odd it is odd balanced. we also say that the ordered pairs (a,b) and (c,d) are bi-distinct if and only if a 6= c and b 6= d. theorem 3.8. if g is isomorphic to a balanced bipartite graph bn,m with n ≥ m then g has a complete extended graph gxc. proof. case 1: consider the even balanced bipartite graph bn,m which is isomorphic to g. without loss of generality assume that |x| = n ≥ m = |y |. label the vertices in x to be, v1,v2,v3 . . . ,vn and those in y to be, u1,u2,u3, . . . ,um. identify the maximum number of bi-distinct non-adjacent vertex pairs (vi,uj) in g. assume that there are ` such bi-distinct pairs. subcase 1(i): if ` is even, add the edges viuj for all ` bi-distinct pairs. clearly, n−` vertices in x can be paired into bi-distinct vertex pairs. add an edge for each bi-distinct vertex pair. similarly, m−` vertices in y can be paired into bi-distinct vertex pairs. add those corresponding edges as well. the new graph obtained through this construction is a complete degree-extension gxc of g. subcase 1(ii): if ` is odd select any ` − 1 bi-distinct vertex pairs. construct a new graph similar to subcase 1(i) which is clearly gxc . case 2: consider the odd balanced bipartite graph bn,m which is isomorphic to g. as before, assume that |x| = n ≥ m = |y | and label the vertices similarly. subcase 2(i): if ` is even select any ` − 1 bi-distinct vertex pairs. construct a new graph similar to subcase 1(i) which is clearly, gxc . subcase 2(ii): if ` is odd, construct a new graph similar to subcase 1(ii) which is clearly, gxc . we note that in all cases above the subgraph, g − {vi,uj : ∀` or ` − 1, (vi,uj) is a bi-distinct non-adjacent pair of vertices}, is a complete bipartite subgraph. theorem 3.9. if g is isomorphic to a balanced bipartite graph bn,m with |x| = n ≥ m = |y | and g has a maximum ` bi-distinct pairs of vertices (vi,uj), vi ∈ x, uj ∈ y , then, case 1: if both n, m are even: (i) if ` is even, χ′(gxc ) = 3(n−`) 2 + 7(m−`) 2 + 3`. (ii) if ` is odd, χ′(gxc ) = b n−` 2 c + 2 · dn−` 2 e + 7(m−`+1) 2 + 3`. case 2: if both n, m are odd: (i) if ` is odd, χ′(gxc ) = 3(n−`) 2 + 7(m−`) 2 + 3`. (ii) if ` is even, χ′(gxc ) = b n−` 2 c + 2 · dn−` 2 e + 7(m−`+1) 2 + 3`. proof. case 1: assume that both |x| and |y | are even. label the vertices in x to be, v1,v2,v3 . . . ,vn and those in y to be, u1,u2,u3, . . . ,um. subcase 1(i): let ` be even. without loss of generality label the vertices of x,y which belong to the ` bi-distinct non-adjacent pairs as v1,v2,v3, . . . ,v` and u1,u2,u3, . . . ,u`, respectively such that (vi,uj), 1 ≤ i,j ≤ ` are non-adjacent. add edges viui, 1 ≤ i ≤ ` and assign the colors vi → c1,ui → c2, 1 ≤ i ≤ `. label the rest of the vertices of x and y to be v`+1,v`+2 . . . ,vn and u`+1,u`+2, . . . ,um, respectively. now, add the edges v`+1v`+2,u`+1u`+2 and assign the colors v`+1 → c1,v`+2 → c3,u`+1 → c2,u`+2 → c4. note that, v`+1,v`+2,u`+1,u`+1 forms a complete graph of order 4 (k4) in gxc since we have chosen ` to be maximum. recursively proceed with the edge-adding and coloring protocols until only the (n − m) vertices vm+1,vm+2, . . . ,vn are left. add the edges vm+1vm+2,vm+3vm+4, . . .vn−1vn, and assign the colors, alternating vm+1 → c1,vm+2 → c2, . . .vn−1 → c1,vn → c2. clearly, the assignment of colors is a minimum proper coloring of gxc . it follows that, θ(c1) = n−`2 + `, θ(c2) = n−` 2 + `, θ(c3) = m−` 2 , θ(c4) = m−` 2 . therefore, χ′(gxc ) = min{ k∑ i=1 i ·θ(ci) : ∀ minimum proper colorings of gx} = θ(c1) + 2 ·θ(c2) + 3 ·θ(c3) + 4 ·θ(c4) = 3(n−`) 2 + 7(m−`) 2 + 3`. 24 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 subcase 1(ii): follows similar to case (i) by considering only ` − 1 of the maximum bi-distinct nonadjacents pairs of vertices. case 2: assume that both |x| and |y | are odd. subcase 2(i): let ` be odd and label the vertices as in subcase 1(i). without loss of generality label the vertices of x,y which belong to the ` bi-distinct non-adjacent pairs as v1,v2,v3, . . . ,v` and u1,u2,u3, . . . ,u`, respectively such that (vi,uj), 1 ≤ i,j ≤ ` are non-adjacent. add edges viui, 1 ≤ i ≤ ` and assign the colors vi → c1,ui → c2, 1 ≤ i ≤ `. label the rest of the vertices of x and y to be v`+1,v`+2 . . . ,vn and u`+1,u`+2, . . . ,um, respectively. now, add the edges v`+1v`+2,u`+1u`+2 and assign the colors v`+1 → c1,v`+2 → c3,u`+1 → c2,u`+2 → c4. note that, v`+1,v`+2,u`+1,u`+1 forms a complete graph of order 4 (k4) in gxc since we have chosen ` to be maximum. recursively proceed with the edgeadding and coloring protocols until only the (n−m) vertices vm+1,vm+2, . . . ,vn are left. add the edges vm+1vm+2,vm+3vm+4, . . .vn−1vn, and assign the colors, alternating vm+1 → c1,vm+2 → c2, . . .vn−1 → c1,vn → c2. clearly, the assignment of colors is a minimum proper coloring of gxc . it follows that, θ(c1) = n−`2 + `, θ(c2) = n−` 2 + `, θ(c3) = m−` 2 , θ(c4) = m−` 2 . therefore, χ′(gxc ) = min{ k∑ i=1 i · θ(ci) : ∀ minimum proper colorings of gx} = θ(c1) + 2 · θ(c2) + 3 · θ(c3) + 4 ·θ(c4) = 3(n−`) 2 + 7(m−`) 2 + 3`. subcase 2(ii): let ` be even and label the vertices as before. the proof follows from similar reasoning found in subcase 1(i). note that the maximum matching in a bipartite graph hence, the maximum bi-distinct non-adjacent pairs of vertices and the value ` can be determined by amongst others, the n 5 2 -algorithm described in [6]. we now discuss a special case of this theorem where |x| = |y |, which means both the partitions of the bipartite graph have equal cardinality. this class of graph includes all regular bipartite graphs. for this case we calculate both χ′(gx) and χ+(gxc ). definition 3.10. let g be a bipartite graph, with vertex partitions x and y . let s be a subset of x. for, v ∈ s we define nc(v) to be the set of vertices in y that are not adjacent to v. we define nc(s) to be ⋃ v∈s n c(v). corollary 3.11. let g be a bipartite graph of even order n, with vertex partitions x and y , such that |x| = |y |. if for every s ⊆ x, |nc(s)| ≥ |s| then, (i) χ′(gxc ) = 3n 2 . (ii) χ+(gxc ) = 5n 2 − 3. proof. case (i): note that in g, the sets x and y are independent sets. hence, in gc vertices in 〈x〉 and 〈y 〉 will be cliques of size |x| and |y |, respectively. let us denote these cliques as k|x| and k|y | respectively. also, note that the graph gc−e(k|x|)−e(k|y |) is bipartite. by hall’s theorem (see [2]) the graph gc−e(k|x|)−e(k|y |) has a perfect matching m, since in g, for every s ⊆ x, |nc(s)| ≥ |s|. thus, g can always be extended to a bipartite graph. hence, χ′(gxc ) = 3n 2 . case (ii): now, let us assign a different extension of g. choose two arbitrary edges from m, x1y1 and x2y2, such that x1,x2 ∈ x and y1,y2 ∈ y . now, for extending g, use the edges m−x1y1−x2y2 +x1x2 +y1y2. this can always be done since the graph is bipartite. now, we can color x1 and y1 using c1. vertices in x −x1 and y −y1 can be colored using c2 and c3, respectively. this obviously shows that, χ+(gxc ) = 5n 2 − 3. 4. on some specific classes of graphs lemma 4.1. let g be a graph of order n, such that δ(g) > n 2 , then diam(g) ≤ 2. 25 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 proof. let a be the adjacency matrix of g. every entry ai,j of a2 is the scalar product of the i-th row and the j-th column of a. since every row and column of a has at least n 2 entries equal to 1, every ai,j > 0, ai,j an entry of a2. hence, for any two distinct vertices v,u ∈ v (g) we have dg(v,u) ≤ 2. therefore, diam(g) ≤ 2. theorem 4.2. let g be a graph of order n, such that δ(g) > n 2 . then g is bipartite if and only it is triangle free and kr,r is the only such graph. proof. necessary condition: it follows from lemma 4 that diam(g) ≤ 2. therefore, the largest holes g can have are of the kind, c4. if g is triangle free it has no odd cycle hence, g is bipartite. sufficient condition: if g is bipartite it contains no odd cycle hence, g is triangle free. finally kr,r is the only r-regular bipartite graph on exactly (at most) 2r vertices. lemma 4.3. let graphs g and h have minimum proper coloring sets cg = {c1,c2,c3, . . . ,cχ(g)} and ch = {c1,c2,c3, . . . ,cχ(h)}, respectively. assume χ(g) ≤ χ(h). let vi → ck, vi ∈ v (g) and uj → ct, uj ∈ v (h) and both ck,ct ∈cg hence, ct ∈ch, as well. (i) if g and h are joined by merging vertices viinv (g) and uj ∈ v (h) as a common vertex which is now assigned the color ck; an equivalent minimum proper coloring is possible by re-assigning color ck to all vertices in h which were assigned ct; and assigning color ct to all vertices in h which were assigned color ck. (ii) if graphs g and h are allowed to join by merging an edge each into a common edge, result (i) holds by applying it consecutively to the pairs of vertices (vi,uj) and (v`,um), viv` ∈ e(g) and ujum ∈ e(h). proof. the results are trivial. now we introduce a family of pattern structured graphs. when a cluster of two copies of a graph h are allowed to merge at least one edge (not necessarily structurally equivalent edges) to share at least one common edge, the new graph is called a h-gridlike cluster. two or more h-gridlike clusters are allowed to merge similarly to get an expanded h-gridlike cluster. when a cluster of two or more copies of a graph h are all allowed to merge a vertex (not necessarily structurally equivalent vertices) to share a common vertex, the new graph is called a h-cloverlike cluster. when a cluster of two or more copies of a graph h are all allowed to merge an edge (not necessarily structurally equivalent edges) to all share a common edge, the new graph is called a h-booklike cluster. when two copies h1,h2 are joined by a path ve0w1e1w2e2w3 . . .em−1wmemu, v ∈ v (h1),u ∈ v (h2) we say they are adjacent. if a graph g∗ is the composition of the aforesaid then; a vertex, a h-gridlike cluster, a h-cloverlike cluster, a h-booklike cluster are called h-elements of g∗. if g∗ has a cycle between at least two h-elements g∗ is called cyclic, else it is called acyclic or h-treelike. theorem 4.4. for a h-treelike graph g∗ with χ(h) ≥ 2 we have, χ(g∗) = χ(h). proof. consider any h-treelike graph g∗. case (i): consider a h-gridlike element and without loss of generality consider any pair h1,h2 sharing a common edge say, uv. if in a minimum proper coloring of h1 the vertex coloring is v → ci,u → cj then, after applying lemma 4.3 the χ-number of the partial h-gridlike element remains the same. case (ii): consider a h-cloverlike element and without loss of generality assume the common vertex is v. if in a minimum proper coloring of h1 the vertex coloring is v → ci then after applying lemma 4.3(i) the χ-number in the partial h-cloverlike element remains the same. by iteratively applying lemma 4.3(i) to all pairs, the χ-number of the whole h-cloverlike element remains the same. case (iii): consider a h-booklike element. similar reasoning as in case (i) follows. 26 http://orcid.org/0000-0003-0106-1676 j. kok, s. bej / j. algebra comb. discrete appl. 5(1) (2018) 19–27 case (iv): first consider adjacent h-elements h1 and h2 as the vertex join between h1 and an end vertex of a path and apply lemma 4.3(i). thereafter, consider the vertex join between the other end vertex of the path and h2 and apply lemma 4.3(i) again. clearly, the χ-number remains the same. invoking cases (i) to (iv) throughout a h-treelike graph g∗ settles the result in general. 5. conclusion we recall that an almost regular graph g is such that, ∆(g) − δ(g) = 1. a partial extension of an almost regular graph g is defined to be the graph gpx obtained by adding edges to g such that, gpx is ∆(g)-regular. a path is such an almost regular graph so it is easy to see that, ppxn = cn therefore, χ′(ppxn ) = χ ′(cn) and χ+(ppxn ) = χ +(cn). there is scope to research χ′(gpxn ) and χ +(gpxn ) for other classes of almost regular graphs. for an r-regular graph g on at least r4 vertices the b-chromatic number is ϕ(g) = r + 1 (see [8]). this result has been improved in [3] by bounding the result to graphs having at least 2r3 vertices. so it is easy to see that for cn, n ≥ 54, we have, ϕ(cn) = 3 and ϕ(cxn,c) = 4. in general, it will be worthy to research the relationship between ϕ′(cn) and ϕ′(cxn,c) as well as that, between ϕ +(cn) and ϕ+(cxn,c) and perhaps for other classes of regular graphs, if such exist. acknowledgment: the authors express their sincere gratitude to professor anirban banerjee, department of mathematics and statistics, indian institute of science education, kolkata, india, for his constructive and insightful comments. the authors also wish professor banerjee the complete return of great health. we need his amazing intellectual capacity to guide many of us for many years to come. the authors are also grateful to the anonymous referees who guided this paper to technical closure. references [1] a. banerjee, s. bej, on extension of regular graphs, arxiv:1509.05476v1 [math.co]. [2] j. a. bondy, u. s. r. murty, graph theory, springer, 2008. [3] s. cabello, m. jacovac, on the b–chromatic number of regular graphs, discrete appl. math. 159 (2011) 1303–1310. [4] g. chartrand, l. lesniak, graphs and digraphs, crc press, 2000. [5] j. t. gross, j. yellen, graph theory and its applications, crc press, 2006. [6] j. e. hopcroft, r. m. karp, an n 5 2 algorithm for maximum matchings in bipartite graphs, siam j. comput. 2(4) (1973) 225–231. [7] j. kok, n. k. sudev, k. p. chithra, generalised colouring sums of graphs, cogent math. 3(1) (2016) 1–11. [8] m. kouider, a. el sahili, about b–coloring of regular graphs, rapport de recherche, no. 1432, cnrs–universite paris sud–lri. [9] e. kubicka, a. j. schwenk, an introduction to chromatic sums, proc. acm computer sci. conf. (louisville) (1989) 39–45. [10] p. c. lisna, m. s. sunitha, b–chromatic sum of a graph, discrete math. algorithm. appl. 7(4) (2015) 1–15. [11] n. k. sudev, k. p. chithra, j. kok, certain chromatic sums of some cycle-related graph classes, discrete math. algorithm. appl. 8(3) (2016) 1–25. 27 http://orcid.org/0000-0003-0106-1676 https://arxiv.org/pdf/1509.05476v1.pdf https://doi.org/10.1016/j.dam.2011.04.028 https://doi.org/10.1016/j.dam.2011.04.028 https://doi.org/10.1137/0202019 https://doi.org/10.1137/0202019 http://dx.doi.org/10.1080/23311835.2016.1140002 http://dx.doi.org/10.1080/23311835.2016.1140002 https://www.lri.fr/~bibli/rapports-internes/2006/rr1432.pdf https://www.lri.fr/~bibli/rapports-internes/2006/rr1432.pdf https://doi.org/10.1145/75427.75430 https://doi.org/10.1145/75427.75430 https://doi.org/10.1142/s1793830915500408 https://doi.org/10.1142/s1793830915500408 https://doi.org/10.1142/s1793830916500506 https://doi.org/10.1142/s1793830916500506 introduction on extensions of graphs '-chromatic sum and +-chromatic sum of certain graphs on some specific classes of graphs conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.37019 j. algebra comb. discrete appl. 4(1) • 75–91 received: 6 october 2015 accepted: 7 june 2016 journal of algebra combinatorics discrete structures and applications multivariate asymptotic analysis of set partitions: focus on blocks of fixed size research article guy louchard abstract: using the saddle point method and multiseries expansions, we obtain from the exponential formula and cauchy’s integral formula, asymptotic results for the number t(n,m,k) of partitions of n labeled objects with m blocks of fixed size k. we analyze the central and non-central region. in the region m = n/k−nα, 1 > α > 1/2, we analyze the dependence of t(n,m,k) on α. this paper fits within the framework of analytic combinatorics. 2010 msc: 05a16, 60c05, 60f05 keywords: set partitions, bell numbers, asymptotics, saddle point method, multiseries expansions, analytic combinatorics 1. introduction set partitions parameters have long been a topic of investigation. see, for example, graham et al. [7], knuth [9], mansour [13], stanley [15], for other investigations of set partitions. moreover, set partitions continue to be of interest recently; see chern et al. [1, 2]. during a talk given by p. diaconis at the conference in honour of svante janson’s 60th birthday, our attention was attracted by a classical parameter of set partitions: the number t(n,m,k) of partitions of n labeled objects with m blocks of fixed size k. the value of k will be fixed in this paper, so we suppress the k and simply write this number as t(m,n). our goal is to analyze the asymptotic growth of t(m,n) in several regimes. let π(n) be the set of partitions of n labeled objects, with bn := |π(n)| denoting the nth bell number. we define the random variable jn as the number of blocks of size k in a set partition chosen (uniformly) at random from the class of all set partitions of n labeled objects. then the distribution of jn guy louchard; université libre de bruxelles, département d’informatique, cp 212, boulevard du triomphe, b-1050 bruxelles, belgium (email: louchard@ulb.ac.be). 75 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 is p(jn = m) = t(m,n) bn . fristedt [5] proved that this distribution is asymptotically gaussian. most of our techniques for studying t(m,n) and the distribution of jn rely on generating functions treated as analytic (complex-valued) functions. thus, we now introduce some of these functions. we note that exp(ez − 1) is the exponential generating function (gf) of the class of set partitions (see [4, pp. 106–111]). therefore the bn’s are exactly the coefficients of this gf: ∞∑ n=0 bn zn n! = exp(ez − 1). now we decompose the number of such set partitions. the celebrated exponential formula (or polymer expansion) is given as follows. let g(z) := ∞∑ k=1 ak zk k! . then exp(g(z)) = ∞∑ n=0 bn(a1, . . . ,an) zn n! , where the nth bell polynomial reads as bn(a1, . . . ,an) = ∑ λ∈π(n) πnk=1a xk(λ) k , and xk(λ) is the number of blocks in λ of fixed size k, λ denoting a set partition. hence, we use f2(z,y) := ∞∑ n=0 ∞∑ m=0 t(m,n)ym zn n! to denote the analogous bivariate gf (which is exponential in z and ordinary in y); the variable y is used here to “mark” the blocks of size k. see [4, p. 156] for an introduction to the marking technique. it follows immediately that f2(z,y) = exp ( ez − 1 + (y − 1) zk k! ) . (1) now we define f3(z) := [y m]f2(z,y) = ∞∑ n=0 t(m,n) zn n! . (2) to find f3(z), we take the mth derivative of f2(z) with respect to y, then divide by m! and evaluate at y = 1, so that f3(z) = exp ( ez − 1 − zk k! ) (zk/k!)m m! . (3) 76 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 in this paper, we will use multiseries expansions: multiseries are in effect power series (in which the powers may be non-integral but must tend to infinity) and the variables are elements of a scale. the scale is a set of variables of increasing order. the series is computed in terms of the variable of maximum order, the coefficients of which are given in terms of the next-to-maximum order, etc. this is more precise than mixing different terms. this technique was used in the analysis of stirling numbers (of the first and second kind) and eulerian numbers in louchard [10–12]. our paper is organized as follows: in section 2, we consider the central region, where we re-derive, with more precision, the asymptotic gaussian property of jn. in section 3, we analyze the large deviation m = n/k − nα, with 1 > α > 1/2. the appendix provides a brief justification of some integration procedures. 2. central region we use several saddle points. to ease this paper’s reading, we summarize the different values we need. we consistently use “root” to be interpreted as “root of smallest modulus.” in section 2, ρ is the root of ρeρ = n, ρ̃k is the root of ρ̃k exp(ρ̃k) = n−k, ρk is the root of ρkeρk − k(ρk) k k! + km−n = 0, in section 3, ρ∗ is the root of ρ∗eρ ∗ − k(ρ∗)k k! −knα = 0, ρ is the root of ρeρ = knα. 2.1. the moments in this section, we first derive asymptotics for the bell numbers and then proceed to the analysis of the moments of jn. for the bell numbers, we could use salvy and shackell [14] or chern et al. [1]. but the first paper uses n/ρ in the scale and the second paper uses the solution of ueu = n + 1. we prefer to use n and ρ: the root (of smallest modulus) of ρeρ = n. the scale of the multiseries is n � ρk � ρ (we assume here k ≥ 2). we define m` := ∏`−1 j=0(m− j) = (m)(m− 1)(m− 2) · · ·(m− ` + 1) as the `th falling factorial of m. now we use this notation to study the `th falling factorial of jn. we have e((jn)`) = ∞∑ m=0 m`p(jn = m) = ∞∑ m=0 m` t(m,n) bn = ∑∞ m=0 m ` t(m,n) bn . from (1), we have ∞∑ m=0 m` t(m,n) = n! [zn] ∂` ∂y` f2(z,y) ∣∣∣∣ y=1 = n! [zn](zk/k!)` exp(ez − 1) = n! (k!)` [zn−k`] exp(ez − 1) = n! (k!)` bn−k` (n−k`)! 77 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 = ( n k,k, . . . ,k︸ ︷︷ ︸ ` ,n−k` ) bn−k`. in particular, for ` = 1 and ` = 2, respectively, we have ∞∑ m=0 mt(m,n) = ( n k ) bn−k and ∞∑ m=0 (m)(m− 1)t(m,n) = ( n k,k,n− 2k ) bn−2k. therefore, the first and second falling moments of jn are e(jn) = ( n k ) bn−k bn and e((jn)(jn − 1)) = ( n k,k,n− 2k ) bn−2k bn . now we need an asymptotic expansion of bn. set f4(z) := exp(e z). let the saddle point be given by ρ and let ω denote the circle ρeiθ. we compute (bn/n!)(e) = ( [zn] exp(ez − 1) ) (e) = [zn] exp(ez) = [zn]f4(z). by cauchy’s theorem, it follows that (bn/n!)(e) = 1 2πi ∫ ω f4(z) zn+1 dz = 1 ρn 1 2π ∫ π −π f4(ρe iθ)e−niθ dθ using z = ρeiθ = 1 ρn 1 2π ∫ π −π exp ( ln(f4(ρe iθ)) −niθ ) dθ = 1 ρn f4(ρ) 2π ∫ π −π exp [ − 1 2 κ2θ 2 − i 6 κ3θ 3 + · · · ] dθ, (4) where κi(ρ) := ( ∂ ∂u )i ln(f4(ρe u))|u=0 . see good [6] for a neat description of this technique. we have ρ = w(n), where w is lambert’s w function (see corless et al. [3]). we use the principal branch, which is analytic at 0. let us set l := ln(n). we have the well-known asymptotic expressions ρ = l− ln l + ln l l + o ( (ln l)2 l2 ) , ln(ρ) = ln (l) − ln (l) l + o ( (ln l)2 l2 ) . all expressions involving ρ in the sequel can of course be expanded into powers of l, but this would lead to huge formulae. 78 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 the dominant part of (4) gives f4(ρ) ρn = exp(n/ρ−n ln(ρ)). now we turn to the integral. we have for instance κ2 = n(1 + ρ), κ3 = n(1 + 3ρ + ρ 2). we now proceed as in flajolet and sedgewick [4, ch. viii]. let us choose a splitting value θ0 such that κ2θ20 → ∞, and κ3θ30 → 0, as n → ∞. for instance, we can use θ0 = n−5/12. we must prove that the integral kn = ∫ 2π−θ0 θ0 exp ( ln(f4(ρe iθ)) −niθ ) dθ is such that |kn| is exponentially small. this is done in appendix 3. now we use the classical trick of setting −κ2θ2/2! + ∞∑ `=3 κ`(iθ) `/`! = −u2/2. computing θ as a series in u, this gives, by lagrange’s inversion, θ = 1 √ n ∞∑ i=1 aiu i, with, for instance, a1 = 1 √ 1 + ρ . this expansion is valid in the dominant integration domain |u| ≤ √ nθ0 a1 = √ 1 + ρn1/12. setting dθ = dθ du du, we integrate on u = [−∞,∞]. this extension of the range is justified as in flajolet and sedgewick [4, ch. viii]. (from now on, we only provide a few terms in our expansions, but of course we use more terms in our computations. also, all o terms in the sequel may depend on k,ρ.) this integration gives bne n! = f4(ρ) ρn h1 = 1 √ 2π √ 1 + 1/ρ exp(e1)h2, where e1 := n/ρ−n ln(ρ) −l/2 − ln(ρ)/2; (5) h1 := h2√ 2πn(1 + ρ) ; h2 := 1 − 1 24 2ρ4 + 9ρ3 + 16ρ2 + 6ρ + 2 (1 + ρ) 3 n + o ( 1 n2 ) . (6) 79 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 now we turn to bn−`k. we compute ln(n− `k) = l− `k n − 1 2 (`k) 2 n2 + o ( 1 n3 ) . we will use ρ̃`,k as the root of ρ̃`,k exp(ρ̃`,k) = n− `k, i.e., ρ̃`,k = w(n− `k). this gives ρ̃`,k = ρ− ρ`k n (1 + ρ) − 1 2 ρ2 (2 + ρ) (`k) 2 (1 + ρ) 3 n2 + o ( 1 n3 ) , with ln(ρ̃`,k) = ln (ρ) − `k n (1 + ρ) − 1 2 (`k) 2 ( 1 + 3ρ + ρ2 ) (1 + ρ) 3 n2 + o ( 1 n3 ) . now we specialize to the case ` = 1 to get the mean. plugging into e1, we obtain bn−ke (n−k)! = exp(e1k)h2k√ 2π √ 1 + 1/ρ̃k , where e1k := k ln (ρ) − 1 2 l− 1 2 ln (ρ) − 1 2 k (−ρ + k − 2) (1 + ρ)n + o ( 1 n2 ) , and h2k := 1 − 1 24 2ρ4 + 9ρ3 + 16ρ2 + 6ρ + 2 (1 + ρ) 3 n + o ( 1 n2 ) . we need bn−k bn . this leads to conclude bn−k bn = (n−k)! n! h6ρ k, with h6 := h3h4h5 = 1 − 1 2 k ( −ρ2 + k(1 + ρ) − 3ρ− 1 ) (1 + ρ) 2 n + o ( 1 n2 ) , where h3 := √ 1 + 1/ρ√ 1 + 1/ρ̃k = 1 − 1 2 k (1 + ρ) 2 n − 1 8 k2 ( 8ρ + 2ρ2 + 1 ) (1 + ρ) 4 n2 + o ( 1 n3 ) , and h4 := h2k h2 = 1 − 1 24 k ( 11ρ5 + 28ρ4 + 36ρ3 + 10ρ2 + 10ρ + 2ρ6 + 2 ) (1 + ρ) 5 n2 + o ( 1 n3 ) , and where h5 is computed as follows: e1k −e1 = k ln(ρ) − 1 2 k (−ρ + k − 2) (1 + ρ)n + o ( 1 n2 ) , 80 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 and exp(e1k −e1) = ρkh5, so h5 = 1 − 1 2 k (−ρ + k − 2) n (1 + ρ) + 1 24 ( 9ρ3 + 39ρ2 − 10ρ2k + 60ρ + 3k2(1 + ρ) − 30ρk + 24 − 16k ) k2 (1 + ρ) 3 n2 + o ( 1 n3 ) . the mean m of jn is given by m = bn−kn! bn(n−k)!k! = ρk k! h6. similarly (we omit the details) e(jn(jn − 1)) = ρ2k (k!)2 h7, where h7 := 1 − k ( −ρ2 + 2k(1 + ρ) − 3ρ− 1 ) (1 + ρ) 2 n + o ( 1 n2 ) . hence the variance σ2 of jn is given by σ2 = e(jn(jn − 1)) + m −m2, and σ = √ ρk k! h8, with h8 := 1 − 2k2 (ρ + 2) ρk + kk! ( −3ρ + k(1 + ρ) −ρ2 − 1 ) 4k! (1 + ρ) 2 n + o ( 1 n2 ) . more generally, the `th falling moment is given by n! (n−k`)!(k!)` bn−k` bn = ρk` (k!)` h7,`, h7,` = 1 − 1 2 k` ( −ρ2 + k`(1 + ρ) − 3ρ− 1 ) (1 + ρ) 2 n + o ( 1 n2 ) . 2.2. distribution of jn fristedt [5] proved that the distribution of jn is asymptotically gaussian. this can also be obtained with hwang’s techniques: see [8]. we want here to re-derive this property with more precision. the corresponding gf f5(z) is derived from f3(z): f5(z) = exp ( ez − zk k! + km ln(z) − ln(m!) −m ln (k!) ) , 81 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 with m = m + xσ, x = θ(1). the saddle point equation is now ρke ρk − ρk kk k! + km−n = 0. (7) it is easily seen that we have ρk ∼ ρ + δ, with δ = ∞∑ j=1 αj nj . when we write ρk ∼ ρ + δ, this is simply a statement that ρk −ρ can be written as a taylor series in terms of powers of n−1, and then α1 is just the first coefficient in this taylor series. solving (7) gives, for instance, α1 := −kx √ ρk k! ρ (1 + ρ) −1 . also, we have the classical result that ln(m!) = −m + m ln (m) + 1 2 ln (2πm) + 1 12 m−1 + o ( 1 m2 ) . we must analyze p(jn = m) = n! ebn [zn]f5(z). first of all, we must compute the dominant term of n!f5(ρk) ebnρ n k . we have n!f5(ρk) ebnρ n k = exp(t1) 1 h2 with h2 given by (6) and, with (5), t1 = neδ ρ − ρk k k! + km ln (ρk ) −n ln (ρk ) − ln(m!) −m ln (k!) − ( n ρ − 1 2 l−n ln (ρ) − 1 2 ln (ρ) − 1 2 ln ( 1 + ρ−1 ) − 1 2 ln (2π) ) . we compute t1 = t0 + t4 n + o ( 1 n2 ) . the dominant term of t0 is computed as t2 := 1 2 [ l + ln (1 + ρ) −x2 −k ln (ρ) + ln (k!) ] . 82 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 set now the next term of t0 as t3 := t0 −t2. we have n!f5(ρk) ebnρ n k = exp(t1) 1 h2 = e−x 2/2 √ k!√ ρk √ 1 + ρ √ n exp ( t3 + t4/n + o ( 1 n2 ))( 1 + t5 n + o ( 1 n2 )) , (8) and t3 := 1/6x ( x2 − 3 )√ k! ρk/2 − 1 12 k! ( −3x2 + x4 + 1 ) ρk + 1 60 x ( 3x4 + 5 − 10x2 ) (k!) 3/2 ρ3k/2 + o ( 1 ρ2k ) . later on, we will need t6 := exp(t3) = 1 + 1/6x ( x2 − 3 )√ k! ρk/2 + 1 72 ( 27x2 − 12x4 − 6 + x6 ) k! ρk + o ( 1 ρ3k/2 ) . also t5 := 1 24 2ρ4 + 9ρ3 + 16ρ2 + 6ρ + 2 (1 + ρ) 3 . we now turn to the coefficient of 1/n in t1. t4 = 1 2 k3x √ (k!) −1 ln (ρ) k! (1 + ρ) ρ3k/2 + 1 2 k2 ln(ρ) ( −ρ2 + x2ρ ln (ρ) − 3ρ + x2 ln (ρ) + k(1 + ρ) − 1 ) ρk (k!) −1 (1 + ρ) −2 and exp ( t4 n ) = 1 + t7 n + o ( 1 n2 ) , and t7 ≡ t4. to compute the integral, we obtain, for instance, κ2 = (1 + ρ)n +  −k2ρk k! − x √ (k!) −1 ( 1 + 3ρ + ρ2 ) kρk/2 (1 + ρ) + o ( 1 ρk/2 ) + o( 1 n ) , and the integral leads to i = 1 √ n √ 2π √ 1 + ρ ( 1 + t8 n + o ( 1 n2 )) , with t8 := 1 24 ( 24k2ρ2 + 12k2ρ3 + 12k2ρ ) ρk (1 + ρ) 3 ρk! + o ( ρk/2 ) . we compute now t9 := t5 + t7 + t8 = t91ρ 3k/2 + t92ρ k + o ( ρk/2 ) , 83 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 with t91 := 1 2 k3x ln (ρ) (k!) 3/2 (1 + ρ) , and t92 := 1 2 k2 ( ρ + x2(1 + ρ) + 1 ) (1 + ρ) 2 k! + 1 2 k2 ( −ρ2 + k(1 + ρ) − 3ρ− 1 ) ln (ρ) (1 + ρ) 2 k! . combining the integral i with (8) gives the local limit theorem: theorem 2.1. the asymptotic distribution of jn is given by the local limit theorem: p(jn = m) = e−x 2/2 √ k!√ ρk √ 2π t6 ( 1 + t9 n + o ( 1 n2 )) . (9) of course more terms can be mechanically computed, but the expressions become much more intricate. to check the quality of our asymptotics, we have chosen k = 2,n = 1000, the range of interest for m is given by m ∈ ( ρk k! + 2 √ ρk k! , ρ k k! − 2 √ ρk k! ) = (6, 21). to numerically compute t(m,n), we use (3), which gives t(m,n) n! = [ zn−km ] exp ( ez − 1 − zk k! ) 1 m!(k!)m . figure 1 shows t(m,n) (circle), the asymptotics e−x 2/2 √ k!√ ρk √ 2π (line, of course we use here ρ k k! as mean and variance) and equ. (9) (box). figure 2 gives the quotient of equ. (9) and t(m,n). figure 3 gives the quotient of equ. (9) and t(m,n) (box) and the quotient of e−x 2/2 √ k!√ ρk √ 2π and t(m,n) (line). note that the same technique would lead to the joint distribution of jn(k1), jn(k2) for two (or more) different values of k. 3. large deviation m = n/k −nα, 1 > α > 1/2 we have a maximum of n/k blocks of size k in a partition. so we use m = n k −nα. this can be written as m = n k (1 −ε), with ε := knα−1. in this section, we choose α ≥ 1 2 , but the other case is similarly analyzed. the saddle point equation, from (7) becomes ρ∗eρ ∗ − ρ∗kk k! + km−n = ρ∗eρ ∗ − ρ∗kk k! − ñ = 0 84 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 0 0.02 0.04 0.06 0.08 0.1 5 10 15 20 25 j figure 1. t(m,n) (circle), the asymptotics e−x 2/2 √ k!√ ρk √ 2π (line) and equ. (9) (box) 0.9 0.95 1 1.05 1.1 1.15 5 10 15 20 25 j figure 2. quotient of equ. (9) and t(m,n) 85 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 0.6 0.8 1 1.2 1.4 5 10 15 20 25 j figure 3. quotient of equ. (9) and t(m,n) (box), quotient of e−x 2/2 √ k!√ ρk √ 2π and t(m,n) (line) with ñ := knα. the multiseries’ scale is here n � ñ � 1 ε � l. the solution of ρeρ = ñ is asymptotically given by ρ = αl + ln (k) − ln (α) − ln (l) + ( − ln (k) α + ln (α) + ln (l) α ) l−1 + o ( 1 l2 ) . as previously, we have ρ∗ = ρ + δ, with, here, δ = ρkkρ k! (1 + ρ) ñ + 1 2 ( ρk )2 k2 ( −2ρ−ρ2 + 2k + 2ρk ) ρ (k!) 2 (1 + ρ) 3 ñ2 + o ( 1 ñ3 ) . first of all, we have n!f5(ρ ∗) eρ∗n = n! e exp(r0), with r0 = eδñ ρ − ρ∗k k! + km ln (ρ∗) −n ln (ρ∗) − ln (m!) −m ln (k!) . we have, successively, r0 = nr1 + r2 + o ( 1 n ) , 86 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 r1 := r10 + r11 ñ + r12 ñ2 + o ( 1 ñ3 ) , r2 := ñ ρ + r20 + r21 ñ + o ( 1 ñ2 ) , with r10 := 1 2 2 − 2 ln (k!) + 2 ln (k) − 2l k + 1 2 −2 ln (k) + 2l− 2k ln (ρ) + 2 ln (k!) k ε− 1 2 k−1ε2 + o(ε3), r11 := − ρkk k! (1 + ρ) ε, r12 := − 1 2 ρ2k ( −ρ2 + 2k(1 + ρ) − 3ρ− 1 ) k2 (k!) 2 (1 + ρ) 3 ε, and r20 = − ρk k! − 1 2 l + ρkk k! (1 + ρ) − 1 2 ln ( 2 π k ) + 1 2 ε + o(ε2), and r21 = 1 2 ρ2k ( −2 − 5ρ− 2ρ2 + 2k(1 + ρ) ) (1 + ρ) 3 (k − 1)!2 . computing the integral, we have, for instance, κ2 := (1 + ρ) ñ − ρkk ( −ρ2 + k(1 + ρ) − 3ρ− 1 ) k! (1 + ρ) + o ( 1 ñ ) , and the integral leads to i = 1 √ 1 + ρ √ 2π 1 √ ñ ( 1 + r5 ñ + o ( 1 ñ2 )) , and r5 := r3 r4 , with r3 := 12k 2ρk+1(1 + ρ) − 36ρk+1k(1 + ρ) + 12ρkk2(1 + ρ) − 12ρk+2k(1 + ρ) − 12ρkk(1 + ρ) − 2k!ρ4 − 9k!ρ3 − 16k!ρ2 − 6k!ρ− 2k!, where r4 := 24 (1 + ρ) 3 k!. finally, we obtain the following asymptotic result 87 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 theorem 3.1. the asymptotic expression of the t(m,n) for large deviation is given by t(m,n) = n! e exp(r0) 1 √ 1 + ρ √ 2π 1 √ ñ ( 1 + r5 ñ + o ( 1 ñ2 )) . (10) let us analyze the importance of our terms. we have two sets: the set a of dominant terms, which stay in the exponent and the set b of small terms, leading to a coefficient of type (1 + ∆), with ∆ small. the property of each term may depend on α. for instance, in r10, the first term leads to an o(nl) term, the ε term leads to an nα term, the ε2 term leads to an n2α−1 term, which are all ∈ a. the ε3 term leads to an n3α−2 term which is ∈ a if α ≥ 2/3 and ∈ b otherwize. in r11 the ε term leads to a term ∈ a. in r12 all terms are ∈ b. in r2, the ñ term is ∈ a , in r20 the first term is ∈ a, all other terms are ∈ b, in r21, all terms are ∈ b. we finally mention that our non-central range is not sacred: other types of ranges can be analyzed with similar methods. to check the quality of our asymptotics, we have first chosen k = 2,α = .52, a range n ∈ [10000, 70000] and m = bn k − nαc. figure 4 shows the quotient equ. (10)/t(m,n). the fluctuations are due to the fact that m is integer, so the value of α we need is actually the root of m− (n k −nα) = 0. 1.005 1.006 1.007 1.008 1.009 1.01 1.011 10000 20000 30000 40000 50000 60000 70000 figure 4. quotient equ. (10)/t(m,n), α = .52,n ∈ [10000,70000] for α = .65, we choose n ∈ [100, 2300], this leads to figure 5. the quality of the asymptotics decreases for α ≥ .7, more terms would be necessary. another way is to fix n. we have chosen k = 2,n = 10000 and a range α ∈ (.52, .65) hence m ∈ [4600, 4900]. again α is chosen as the root of m − (n k − nα) = 0. figure 6 shows the quotient equ. (10)/t(m,n). acknowledgment: we would like to thank two referees for many useful suggestions that improved the paper. 88 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 1.7 1.8 1.9 2 500 1000 1500 2000 figure 5. quotient equ. (10)/t(m,n), α = .65,n ∈ [100,2300] 1 1.1 1.2 1.3 1.4 1.5 4600 4650 4700 4750 4800 4850 4900 figure 6. quotient equ. (10)/t(m,n), n = 10000,α ∈ (.52, .65),m ∈ [4600,4900] 89 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 references [1] b. chern, p. diaconis, d. m. kane, r. c. rhoades, closed expressions for set partition statistics, res. math. sci. 1(2) (2014) 1–32. [2] b. chern, p. diaconis, d. m. kane, r. c. rhoades, central limit theorems for some set partitions, adv. appl. math. 70 (2015) 92–105. [3] r. m. corless, g. h. gonnet, d. e. g. hare, d. j. jeffrey, d. e. knuth, on the lambertw function, adv. comput. math. 5 (1996) 329–359. [4] p. flajolet, r. sedgewick, analytic combinatorics, cambridge university press, 2009. [5] b. fristedt, the structure of random partitions of large sets, technical report, university of minnesota, 1987. [6] i. j. good, saddle–point methods for the multinomial distribution, ann. math. statist. 28(4) (1957) 861–881. [7] r. l. graham, d. e. knuth, o. patashnik, concrete mathematics, second edition, addison wesley, 1994. [8] h. k. hwang, on convergence rates in the central limit theorems for combinatorial structures, european j. combin. 19(3) (1998) 329–343. [9] d. e. knuth, the art of computer programming, vol. 4a: combinatorial algorithms. part i, addison–wesley, upper saddle river, new jersey, 2011. [10] g. louchard, asymptotics of the stirling numbers of the first kind revisited: a saddle point approach, discrete math. theor. comput. sci. 12(2) (2010) 167–184. [11] g. louchard, asymptotics of the stirling numbers of the second kind revisited: a saddle point approach, appl. anal. discrete math. 7(2) (2013) 193–210. [12] g. louchard, asymptotics of the eulerian numbers revisited: a large deviation analysis, online j. anal. comb. 10 (2015) 1–11. [13] t. mansour, combinatorics of set partitions, discrete mathematics and its applications series, crc press, boca raton, fl, 2013. [14] b. salvy, j. shackell, symbolic asymptotics: multiseries of inverse functions, j. symbolic comput. 20(6) (1999) 543–563. [15] r. p. stanley, enumerative combinatorics, volume 1, 2nd edn, cambridge studies in advanced mathematics, vol. 49. cambridge university press, cambridge, 2012. 90 http://dx.doi.org/10.1186/2197-9847-1-2 http://dx.doi.org/10.1186/2197-9847-1-2 http://dx.doi.org/10.1016/j.aam.2015.06.008 http://dx.doi.org/10.1016/j.aam.2015.06.008 http://dx.doi.org/10.1007/bf02124750 http://dx.doi.org/10.1007/bf02124750 http://dx.doi.org/10.1214/aoms/1177706790 http://dx.doi.org/10.1214/aoms/1177706790 http://dx.doi.org/10.1006/eujc.1997.0179 http://dx.doi.org/10.1006/eujc.1997.0179 http://www.ams.org/mathscinet-getitem?mr=2676669 http://www.ams.org/mathscinet-getitem?mr=2676669 http://dx.doi.org/10.2298/aadm130612011l http://dx.doi.org/10.2298/aadm130612011l http://web.math.rochester.edu/ojac/vol10/117.pdf http://web.math.rochester.edu/ojac/vol10/117.pdf http://dx.doi.org/10.1006/jsco.1999.0281 http://dx.doi.org/10.1006/jsco.1999.0281 g. louchard / j. algebra comb. discrete appl. 4(1) (2017) 75–91 appendix: justification of the integration procedure for the bn case, we must analyze <(ln(f4(ρeiθ) −niθ). let us first notice that niθ does not contribute to the analysis. next, we have < ( eρe iθ ) = eρ cos(θ) cos(ρ sin(θ)) which has a dominant peak at 0. for the gaussian case, we use f5(z). we must analyze < ( eρe iθ − (ρeiθ)k k! ) = eρ cos(θ) cos(ρ sin(θ)) − ρk k! cos(kθ) which has a dominant peak at 0. the non-central region leads to the same analysis. 91 introduction central region large deviation m=n/k-n, 1>>1/2 references appendix: justification of the integration procedure issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.01767 j. algebra comb. discrete appl. 3(2) • 81–90 received: 22 july 2015 accepted: 21 january 2016 journal of algebra combinatorics discrete structures and applications on the spectral characterization of kite graphs∗ research article sezer sorgun, hatice topcu abstract: the kite graph, denoted by kitep,q is obtained by appending a complete graph kp to a pendant vertex of a path pq. in this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. let g be a graph which is cospectral with kitep,q and let w(g) be the clique number of g. then, it is shown that w(g) ≥ p − 2q + 1. also, we prove that kitep,2 graphs are determined by their adjacency spectrum. 2010 msc: 05c50, 05c75 keywords: kite graph, cospectral graphs, clique number, determined by adjacency spectrum 1. introduction all of the graphs considered here are simple and undirected. let g = (v,e) be a graph with vertex set v (g) = {v1,v2, . . . ,vn} and edge set e(g). for a given graph f, if g does not contain f as an induced subgraph, then g is called f − free. a complete subgraph of g is a clique of g. the clique number of g is the number of the vertices in the largest clique of g and it is denoted by w(g). let a(g) be the (0,1)-adjacency matrix of g and dk denotes the degree of the vertex vk. the polynomial pa(g)(λ) = det(λi − a(g)) is the adjacency characteristic polynomial of g, where i is the identity matrix. eigenvalues of the matrix a(g) are adjacency eigenvalues. since a(g) is real and symmetric matrix, adjacency eigenvalues are all real numbers and could be ordered as λ1(a(g)) ≥ λ2(a(g)) ≥ . . . ≥ λn(a(g)). adjacency spectrum of the graph g consists of the adjacency eigenvalues with their multiplicities. the largest absolute value of the adjacency eigenvalues of a graph is known as its adjacency spectral radius. two graphs g and h are said to be cospectral if they have same spectrum (i.e., same characteristic polynomial). a graph g is determined by its adjacency spectrum, shortly das, if every graph cospectral with g w.r.t the adjacency matrix, is isomorphic to g. it is conjectured in [5] that almost all graphs are determined by their spectrum, ds for short. but it is difficult to show that a given ∗ this work was supported by the nevsehir haci bektas veli univesity coordinatorship of scientific research projects (no. neulup15f17). sezer sorgun (corresponding author), hatice topcu; department of mathematics, nevşehir hacı bektaş veli university, nevşehir 50300, turkey (email: srgnrzs@gmail.com, haticekamittopcu@gmail.com). 81 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 graph is ds. up to now, some graphs are proved to be ds [1, 2, 4–11, 13, 15]. recently, some papers have appeared in the literature that researchers focus on some special graphs (oftenly under some conditions) and prove that these special graphs are ds or non-ds [1, 2, 6–11, 13, 15]. for a recent survey, one can see [5]. the kite graph, denoted by kitep,q, is obtained by appending a complete graph with p vertices kp to a pendant vertex of a path graph with q vertices pq. if q = 1, it is called short kite graph. in this paper, firstly we obtain the characteristic polynomial of kite graphs and show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. then for a given graph g which is cospectral with kitep,q, the clique number of g is w(g) ≥ p−2q + 1. also we prove that kitep,2 graphs are das for all p. 2. preliminaries first, we give some lemmas that will be used in the next sections of this paper. lemma 2.1. [8] let x1 be a pendant vertex of a graph g and x2 be the vertex which is adjacent to x1. let g1 be the induced subgraph obtained from g by deleting the vertex x1. if x1 and x2 are deleted, the induced subgraph g2 is obtained. then, pa(g)(λ) = λpa(g1)(λ)−pa(g2)(λ) lemma 2.2. [4] for nxn matrices a and b, followings are equivalent : (i) a and b are cospectral (ii) a and b have the same characteristic polynomial (iii) tr(ai) = tr(bi) for i = 1,2, ...,n lemma 2.3. [4] for the adjacency matrix of a graph g, following parameters can be deduced from the spectrum; (i) the number of vertices (ii) the number of edges (iii) the number of closed walks of any fixed length. theorem 2.4. [14] if a given connected graph g has the same order, same clique number and same spectral radius with kitep,q then g is isomorphic to kitep,q. in the rest of the paper, we denote the number of subgraphs of a graph g which are isomorphic to complete graph k3 by t(g). theorem 2.5. [14] for any integers p ≥ 3 and q ≥ 1, if we denote the spectral radius of a(kitep,q) with ρ(kitep,q) then p−1 + 1 p2 + 1 p3 < ρ(kitep,q) < p−1 + 1 4p + 1 p2 −2p theorem 2.6. [12] let g be a graph with n vertices, m edges and spectral radius µ. if g is kr+1−free, then µ ≤ √ 2m( r −1 r ) lemma 2.7. [3](interlacing lemma) if g is a graph on n vertices with eigenvalues λ1(g) ≥ . . . ≥ λn(g) and h is an induced subgraph on m vertices with eigenvalues λ1(h) ≥ . . . ≥ λm(h), then for i = 1, . . . ,m λi(g) ≥ λi(h) ≥ λn−m+i(g) 82 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 lemma 2.8. [3] a connected graph with the largest adjacency eigenvalue less than 2 are precisely induced subgraphs of the smith graphs shown in figure 1. figure 1. smith graphs 3. characteristic polynomial of kite graphs we use the method similar to that given in [8] to obtain the general form of characteristic polynomials of kitep,q graphs. obviously, if we delete the vertex with one degree from short kite graph, the induced subgraph will be the complete graph kp. then, by deleting the vertex with one degree and its adjacent vertex, we obtain the complete graph kp−1 with p−1 vertices. from lemma 2.1, we get pa(kitep,1)(λ) = λpa(kp)(λ)−pa(kp−1)(λ) = λ(λ−p + 1)(λ + 1)p−1 − [(λ−p + 2)(λ + 1)p−2] = (λ + 1)p−2[(λ2 −λp + λ)(λ + 1)−λ + p−2] = (λ + 1)p−2[λ3 − (p−2)λ2 −λp + p−2]. similarly for kitep,2, induced subgraphs will be kitep,1 and kp respectively. by lemma 2.1, we get pa(kitep,2)(λ) = λpa(kitep,1)(λ)−pa(kp))(λ) = λ(λpa(kp)(λ)−pa(kp−1)(λ))−pa(kp))(λ) = (λ2 −1)pa(kp)(λ)−λpa(kp−1)(λ). by using these polynomials, we calculate the characteristic polynomial of kitep,q where n = p + q. again, by lemma 2.1 we have pa(kitep,1)(λ) = λpa(kp)(λ)−pa(kp−1)(λ). 83 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 coefficients of above equation are b1 = −1, a1 = λ. simultaneously, we get pa(kitep,2)(λ) = (λ 2 −1)pa(kp)(λ)−λpa(kp−1)(λ). and coefficients of above equation are b2 = −a1 = −λ, a2 = λa1 −1 = λ2 −1. then for kitep,3, we have pa(kitep,3)(λ) = λpa(kitep,2)(λ)−pa(kitep,1))(λ) = (λ(λ2 −1)−λ)pa(kp)(λ)− ((λ 2 −1)pa(kp−1)(λ)) and coefficients of above equation are b3 = −a2 = −(λ2 − 1),a3 = λa2 − a1 = λ(λ2 − 1) − λ. in the following steps, for n ≥ 3, an = λan−1 −an−2. from this difference equation, we get an = n∑ k=0 ( λ + √ λ2 −4 2 )k( λ− √ λ2 −4 2 )n−k now, let λ = 2cosθ and u = eiθ. then, we have an = n∑ k=0 u2k−n = u−n(1−u2n+2) 1−u2 and by calculation the characteristic polynomial of any kite graph kitep,q where n = p + q is pa(kitep,q)(u + u −1) = an−ppa(kp)(u + u −1)−an−p−1pa(kp−1)(u + u −1) = u−n+p(1−u2n−2p+2) 1−u2 .((u + u−1 −p + 1).(u + u−1 + 1)p−1) − u−n+p+1(1−u2n−2p+4) 1−u2 .((u + u−1 −p + 2).(u + u−1 + 1)p−2) = u−n+p(1 + u−u−1)p−2 1−u2 [(2−p).(1 + u−1 −u2n−2p+2 −u2n−2p+3) +(u−2 −u2n−2p+4)] = u−q(1 + u−u−1)p−2 1−u2 [(2−p).(1 + u−1 −u2q+2 −u2q+3) +(u−2 −u2q+4)]. theorem 3.1. no two non-isomorphic kite graphs have the same adjacency spectrum. proof. assume that there are two cospectral kite graphs with number of vertices respectively, p1 + q1 and p2 + q2. since they are cospectral, they must have same number of vertices and same characteristic polynomials. hence, n = p1 + q1 = p2 + q2 and we get pa(kitep1,q1 )(u + u −1) = pa(kitep2,q2 )(u + u −1) i.e., u−n+p1(1 + u−u−1)p1−2 1−u2 [(2−p1).(1 + u−1 −u2n−2p1+2 −u2n−2p1+3) +(u−2 −u2n−2p1+4)] = u−n+p2(1 + u−u−1)p2−2 1−u2 [(2−p2).(1 + u−1 −u2n−2p2+2 −u2n−2p2+3) +(u−2 −u2n−2p2+4]) 84 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 i.e., up1.(1 + u−u−1) p1 .[(2−p1).(1 + u−1 −u2n−2p1+2 −u2n−2p1+3) +(u−2 −u2n−2p1+4)] = up2.(1 + u−u−1) p2 .[(2−p2).(1 + u−1 −u2n−2p2+2 −u2n−2p2+3) +(u−2 −u2n−2p2+4)] let p1 > p2. it follows that n−p2 > n−p1. then, we have up1−p2.(1 + u−u−1) p1−p2{[(2−p1).(1 + u−1 −u2n−2p1+2 −u2n−2p1+3) +(u−2 −u2n−2p1+4)]− [(2−p2).(1 + u−1 −u2n−2p2+2 −u2n−2p2+3) +(u−2 −u2n−2p2+4)]} = 0 by using the fact that u 6= 0 and 1 + u + u−1 6= 0, we get f(u) = [(2−p1).(1 + u−1 −u2n−2p1+2 −u2n−2p1+3) + (u−2 −u2n−2p1+4)] −[(2−p2).(1 + u−1 −u2n−2p2+2 −u2n−2p2+3) + (u−2 −u2n−2p2+4)] = 0 since f(u) = 0, the derivation of (2n−2p2 + 5)th of f equals to zero again. thus, we have [(p1 −2)(2n−2p2 + 4)!(u−2n+2p2−6)]− [(p2 −2).(2n−2p2 + 4)!(u−2n+2p2−6)] = 0 i.e., [(p1 −2)− (p2 −2)].(u−2n+2p2−6) = 0 i.e., p1 = p2 since u 6= 0. this is a contradiction with our assumption that is p1 > p2. for p2 > p1, we get the similar contradiction. so p1 must be equal to p2. hence q1 = q2 and these graphs are isomorphic. 4. spectral characterization of kitep,2 graphs lemma 4.1. let g be a graph which is cospectral with kitep,q. then we get w(g) ≥ p−2q + 1. proof. since g is cospectral with kitep,q, from lemma 2.3, g has the same number of vertices, same number of edges and same spectrum with kitep,q. so, if g has n vertices and m edges, n = p + q and m = ( p 2 ) + q = p 2−p+2q 2 . also, ρ(g) = ρ(kitep,q). from theorem 2.6, we say that if µ > √ 2m(r−1 r ) then g isn’t kr+1 −free. it means that, g contains kr+1 as an induced subgraph. now, we claim that for r < p− 2q, √ 2m(r−1 r ) < ρ(g). by theorem 2.5, we’ve already known that p− 1 + 1 p2 + 1 p3 < ρ(g). hence, we need to show that √ 2m(r−1 r ) < p−1 + 1 p2 + 1 p3 , when r < p−2q . indeed, 85 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 ( √ 2m( r −1 r ))2 − (p−1 + 1 p2 + 1 p3 )2 = (p2 −p + 2q)(r −1)−r(p−1 + 1 p2 + 1 p3 )2 = (p2 −p + 2q)(r −1)− ( r(p2 + p3) p5 )(2(p−1) + (p2 + p3) p5 ) = (pr −p2 + p + (2q −1)r −2q)− ( r(p2 + p3) p5 )(2(p−1) + (p2 + p3) p5 ) by the help of mathematica, for r < p−2q we can see (pr −p2 + p + (2q −1)r −2q)− ( r(p2 + p3) p5 )(2(p−1) + (p2 + p3) p5 ) < 0 i.e., ( √ 2m( r −1 r ))2 − (p−1 + 1 p2 + 1 p3 )2 < 0 i.e., ( √ 2m( r −1 r ))2 < (p−1 + 1 p2 + 1 p3 )2 since √ 2m(r−1 r ) > 0 and p−1 + 1 p2 + 1 p3 > 0, we get √ 2m( r −1 r ) < p−1 + 1 p2 + 1 p3 < ρ(g). thus, we proved our claim and so g contains kr+1 as an induced subgraph such that r < p− 2q. consequently, w(g) ≥ p−2q + 1. theorem 4.2. kitep,2 graphs are determined by their adjacency spectrum for all p. proof. if p = 1 or p = 2, kitep,2 graphs are actually the path graphs p3 or p4. also if p = 3, then we obtain the lollipop graph h5,3. as is known, these graphs are already das [8]. hence we will continue our proof for p ≥ 4. adjacency characteristic polynomial of kitep,2 is as below, pa(kitep,2)(λ) = (λ + 1) p−2[λ4 + (2−p)λ3 − (p + 1)λ2 + (2p−4)λ + p−1] by calculation, for the adjacency eigenvalues of kitep,2, we obtain the following facts; p − 1 < λ1(a(kitep,2)) < p, 0 < λ2(a(kitep,2)) < 2, λ3(a(kitep,2)) < 0, λ4(a(kitep,2)) = . . . = λp+1(a(kitep,2)) = −1 and λp−1(a(kitep,2)) < −1. for a given graph g with n vertices and m edges, assume that g is cospectral with kitep,2. then by lemma 2.3, n = p + 2, m = ( p 2 ) + 2 = p 2−p+4 2 and t(g) = t(kitep,2) = ( p 3 ) = p 3−3p2+2p 6 . from 86 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 lemma 4.1, w(g) ≥ p−2q + 1. when q = 2, w(g) ≥ p−3 = n−5. it’s well-known that complete graph kn is ds. so w(g) 6= n. if w(g) = n−1 = p + 1, then g contains at least one clique with size p−1. it means that the edge number of g is greater than or equal to ( p + 1 2 ) . but it is a contradiction since( p + 1 2 ) > ( p 2 ) + 2 = m. hence, w(g) 6= n − 1. because of these facts, we get p − 3 ≤ w(g) ≤ p. from interlacing lemma, g can not contain the graphs in the following figure as an induced subgraph because λ3(g1) = λ3(g2) = 0. figure 2. graphs g1 and g2 if g is disconnected, from lemma 2.8, components of g except one of them must be induced subgraphs of smith graphs. clearly, this is impossible because g1 is forbidden and any path graph (since they have symmetric eigenvalues) can not be a component of g. hence g must be a connected graph. if w(g) = p, then by theorem 2.4., g ∼= kitep,2. so we continue for w(g) < p. since w(g) ≥ p − 3, g contains at least one clique with size at least p− 3. this clique is denoted by kw(g). there may be at most five vertices out of the clique kw(g). let us label these five vertices respectively with 1,2,3,4,5 and call the set of these five vertices with a. so, we get |a| ≤ 5. moreover, ∀i,j ∈ a we get i ∼ j since g1, g2 are not induced subgraphs of g and there is no isolated vertex in g. then, we can say that p ≥ 6 since w(g) ≥ p−3. for i ∈ a, xi denotes the number of adjacent vertices of i in kw(g). by the fact that p−1 ≥ w(g) ≥ p−3, for all i ∈ a we say xi ≤ w(g)−|a|+ 1 (1) also, xi∧j denotes the number of common adjacent vertices in kw(g) of i and j such that i,j ∈ a and i < j. similarly, if i ∼ j then xi∧j ≤ w(g)−|a| (2) let d denotes the number of edges between the vertices of a and kw(g), also α denotes the number of cliques with size 3 which are not contained by a or kw(g). then, we get m = ( p 2 ) + 2 = ( w(g) 2 ) + ( |a| 2 ) + d. (3) similarly, we get t(g) = ( p 3 ) = ( w(g) 3 ) + ( |a| 3 ) + α. (4) on the other hand for α and d, we have d = |a|∑ i=1 xi (5) 87 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 and α = |a|∑ i=1 ( xi 2 ) + ∑ i∼j xi∧j. (6) if w(g) = p−3 then |a| = 5 and so p ≥ 8. thus we have d = 3p−14 (7) and α = ( p 3 ) − ( p−3 3 ) −10 = 3p2 2 − 15p 2 . (8) from (1),(2),(5),(6) and (7) we have α = 5∑ i=1 ( xi 2 ) + ∑ i∼j xi∧j ≤ 3 ( p−7 2 ) + ( 7 2 ) + 2 5∑ i=1 xi = 3 ( p−7 2 ) + ( 7 2 ) + 6p−28 = 3p2 −33p 2 + 77. but obviously for p = 8 this result gives contradiction. also for p > 8, 3p2 −33p 2 + 77 < 3p2 −15p 2 = α. so this is again a contradiction. if w(g) = p−2 then |a| = 4 and so p ≥ 7. thus we have d = 2p−7 and α = ( p 3 ) − ( p−2 3 ) −4 = p2 −4p. on the other hand we have α = 4∑ i=1 ( xi 2 ) + ∑ i∼j xi∧j ≤ 2 ( p−5 2 ) + ( 3 2 ) + 2 4∑ i=1 xi = p2 −7p + 19. clearly for p ≥ 7, p2 −7p + 19 < p2 −4p = α. so this is a contradiction. similarly, if w(g) = p−1 then |a| = 3 and so p ≥ 6. hence we have d = p−2 and α = p2 −3p 2 . 88 s. sorgun, h. topcu / j. algebra comb. discrete appl. 3(2) (2016) 81–90 also we have α = 3∑ i=1 ( xi 2 ) + ∑ i∼j xi∧j ≤ ( p−3 2 ) + p−2 = p2 −5p 2 + 4. clearly for p ≥ 6, p2 −5p 2 + 4 < p2 −3p 2 = α. again we obtain a contradiction. by all of these facts, we can conclude that our assumption is actually false, then w(g) 6< p. hence w(g) = p and so that by theorem 2.4., g ∼= kitep,2. in the final of the paper, we give a conjecture below. conjecture 4.3. for q > 2, kitep,q graphs are das. acknowledgment: the authors are grateful to the referees for many suggestions which led to an improved version of this paper. references [1] r. boulet, b. jouve, the lollipop graph is determined by its spectrum, electron. j. combin. 15(1) (2008) research paper 74, 43 pp. [2] m. camara, w. h. haemers, spectral characterizations of almost complete graphs, discrete appl. math. 176 (2014) 19–23. 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[15] x. zhang, h. zhang, some graphs determined by their spectra, linear algebra appl. 431(9) (2009) 1443–1454. 90 http://dx.doi.org/10.13001/1081-3810.1253 http://dx.doi.org/10.13001/1081-3810.1253 http://dx.doi.org/10.1016/j.laa.2009.05.018 http://dx.doi.org/10.1016/j.laa.2009.05.018 introduction preliminaries characteristic polynomial of kite graphs spectral characterization of kitep,2 graphs references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.458240 j. algebra comb. discrete appl. 5(3) • 129–136 received: 17 february 2017 accepted: 10 april 2018 journal of algebra combinatorics discrete structures and applications game chromatic number of cartesian and corona product graphs research article syed ahtsham ul haq bokhary, tanveer iqbal, usman ali abstract: the game chromatic number χg is investigated for cartesian product g�h and corona product g◦h of two graphs g and h. the exact values for the game chromatic number of cartesian product graph of s3�sn is found, where sn is a star graph of order n+1. this extends previous results of bartnicki et al. [1] and sia [5] on the game chromatic number of cartesian product graphs. let pm be the path graph on m vertices and cn be the cycle graph on n vertices. we have determined the exact values for the game chromatic number of corona product graphs pm ◦ k1 and pm ◦ cn. 2010 msc: 05c15, 05c65 keywords: game chromatic number, cartesian product, corona product 1. introduction we consider the following well-known graph coloring game, played on a simple graph g with a color set c of cardinality k. two players, alice and bob, alternately color an uncolored vertex of g with a color from c such that no adjacent vertices receive the same color (such a coloring of a graph g is known as proper coloring). alice has the first move and the game ends when no move is possible any more. if all the vertices are properly colored, alice wins, otherwise bob wins. the game chromatic number of g, denoted by χg(g), is the least cardinality k of the set c for which alice has a winning strategy. in other words, necessary and sufficient conditions for k to be the game chromatic number of a graph g are: i) bob has winning strategy for k − 1 colors or less and ii) alice has winning strategy for k colors. this parameter is well defined, since it is easy to see that alice always wins if the number of colors is larger than the maximum degree of g. clearly, χg(g) is at least as large as the ordinary chromatic number χ(g), but it can be considerably more. for example, let g be a complete bipartite graph kn,n syed ahtsham ul haq bokhary (corresponding author), tanveer iqbal, usman ali; centre for advanced studies in pure and applied mathematics, bahauddin zakariya university, multan, pakistan (email: sihtsham@gmail.com, tanveerm8@yahoo.com, uali@bzu.edu.pk). 129 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 minus a perfect matching m and consider the following strategy for bob. if alice colors vertex v with color c then bob responds by coloring the vertex u matched with v in the matching m with the same color c. note that now c cannot be used on any other vertex in the graph. therefore, if the number of colors is less than n, bob wins the game. this shows that there are bipartite graphs with arbitrarily large game chromatic number and thus there is no upper bound on χg(g) as a function of χ(g). the obvious bounds for the game chromatic number are: χ(g) ≤ χg(g) ≤ ∆(g) + 1 (1) where χ(g) is the chromatic number and ∆(g) is the maximum degree of the graph g. a lot of attempts have been made to determine the game chromatic number for several classes of graphs. this work was first initiated by faigle et al. [3]. it was proved by kierstead and trotter [4] that the maximum of the game chromatic number of a forest is 4, also that 33 is an upper bound for the game chromatic number of planar graphs. bodlaender [2], found that the game chromatic number of cartesian product is bounded above by constant in the family of planar graphs. later, bartnicki et al. [1] determine the exact values of χg(g�h) when g and h belong to certain classes of graphs, and showed that, in general, the game chromatic number χg(g�h) is not bounded from above by a function of game chromatic numbers of the graphs g and h. after that, zhu [6] established a bound for game coloring number and acyclic chromatic number for cartesian product of two graphs h and s. sia [5] determined the exact values for the cartesian product of different families of graph like sm�pn, sm�cn, p2�wn , where pn, cn, denotes the path graph and cycle graph with n vertices, respectively. while sm is a star graph with m + 1 vertices and wn is a wheel graph with n + 1 vertices. in this work, we have extended the study of game chromatic number and found the exact values of the game chromatic number of s3�sn, pm ◦k1 and pm ◦cn, where ◦ denote corona product of graphs. 2. exact value of χg(s3�sn) a cartesian product of two graphs g and h, denoted by g�h, is the graph with vertex set v (g)× v (h), where two vertices (u,u′) and (v,v′) are adjacent if and only if u = v and u′v′ ∈ e(h) or u′ = v′ and uv ∈ e(g). note that the cartesian product operation is both commutative and associative up to isomorphism. before stating our results, we introduce some definitions and notational conventions. suppose that alice and bob plays the coloring game with k colors. we say that there is a threat to an uncolored vertex v if there are k − 1 colors in the neighborhood of v, and it is possible to color a vertex adjacent to v with the last color, so that all k colors would then appear in the neighborhood of v. the threat to the vertex v is said to be blocked if v is subsequently assigned a color, or it is no longer possible for v to have all k colors in its neighborhood. we shall also use the convention that color numbers are only used to differentiate distinct colors, and should not be regarded as ascribed to particular colors. for example, if only colors 1 and 2 have been used so far and we introduce a new color, color 3, then color 3 can refer to any color that is not the same as color 1 or color 2. finally, we label figures in the following manner: vertices are labeled in the form “color(player)turn”, with the information in parentheses being omitted if the same configuration can be attained in multiple ways. in the following result, we have found the exact value of χg(s3�sn). theorem 2.1. for integer n ≥ 3, χg(s3�sn) = 4. proof. first, we show that bob has a winning strategy with three or fewer colors. we give bob’s winning strategy when there are exactly three colors, it will be easy to see that bob can win using the same strategy when there are fewer than three colors. denote the vertices of s3�sn by a1,a2, ...,an+1,b1, b2, ...,bn+1,c1,c2, ...,cn+1,d1,d2, ....dn+1. the vertex a1 has degree n+ 3, three vertices b1,c1 and d1 have degree n + 1. the vertices a′is for i ≥ 2 has degree 4. while all the remaining vertices have degree two. 130 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 the graph shown in figure 1 is isomorphic to s3�sn. bob’s winning strategy with three colors: case 1: let d = {1, 2, 3} be set of colors. if alice begins by playing color 1 in the vertex a1 of highest degree. bob should plays color 2 in any of the vertex dk of degree two (say d2), see figure 1. alice can not block the threats to both the vertices d1 and a2, so bob wins. a1 1a1 2b1 d2 d3d4dndn+1 c2c3c4cncn+1 b2b3b4bnbn+1 b1c1d1 a2a3 a4an an+1 figure 1. bob’s winning strategy in case 1 for the graph s3�sn case 2: if alice start the game by playing color 1 in any of the vertex bi, cj or dk 2 ≤ i,j,k ≤ n (say b2), bob respond with color 2 in the vertex a1, see figure 2. now there are two vertices a2 and b1, which are under threat. alice can save only one of these vertices and bob wins the game. a1 1a12b1 d2 d3d4dndn+1 c2c3c4cncn+1 b2b3b4bnbn+1 b1c1d1 a2a3 a4an an+1 figure 2. bob’s winning strategy in case 2 for the graph s3�sn case 3: if alice begins the game by assigning color to any of the vertices al, 2 ≤ l ≤ n. suppose, she plays a color 1 in the vertex a2. in response, bob plays color 2 in the vertex ai, where i 6= 2 and i > 2 say a3. this forces alice to assign color 3 to the vertex a1, because this is the only way, she can block the threat to this vertex. after that bob plays color 1 in the neighbor of a′ls (l ≥ 4) and wins the games. case 4: if alice start the game by playing color 1 in vertex b1, c1 or d1 (say b1). bob should plays color 2 in any of the vertices ai, where 2 ≤ i ≤ n, for simplicity, say a2. there are two distinct colors in the neighbor of vertex a1, so alice must block the threat to this vertex by assigning it the color 3. in the next move bob plays color 1 in dk (k 6= 2) say d3. now alice can’t block the threat of d1 and a3 and bob wins the game. thus χg(s3�sn) ≥ 4. next, alice’s winning strategy is given for four colors, which proves that χg(s3�sn) ≤ 4. alice’s winning strategy with four colors: alice in her strategy, first assigns colors to all the vertices of degree n + 1, because in this way, she can prevent these vertices from threats. now al, where l ≥ 1 are 131 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 the only vertices which can come under threat. since al ∼ bi (l = i), al ∼ cj (l = j) and al ∼ dk (l = k). so, whenever, bob plays a color in a vertex adjacent to ai, say cj, alice will play the color of vertex a1 or color of the vertex cj in bi (adjacent to al). in this way, alice can prevent these vertices from threats, since every vertex has two same colors in its neighbor. so, in order to win the game, she need only to assign colors in the vertices of degree greater than 4. below, we have given the alice strategy to color these vertices with four given colors. alice makes her first move by playing color 1 in any of three vertices having degree n + 1, say b1. we, now consider all the cases for bob’s first move. case 1: if bob plays color 2 in any of the vertex having degree four, say a2. alice makes her second move by playing color 2 in the vertex c1. now bob has two possibilities for his second move. case 1.1: if bob wants that the vertex a1 be under threat in his second move and assigns color 3 to the vertex a3. alice, then has to play color 4 in a1, in order to prevent this vertex from threat. alice is safe to assign suitable color in the vertex d1 in her fourth move. now all the vertices of degree greater than four have assigned color. after that whenever bob plays a color in a vertex bi or cj say cj (adjacent to al), alice will play the color of cj in bi where l = i = j and win the game. (strategy is shown in figure 3) a1 d2d3d4dndn+1 c2c3c4cncn+1 b2b3b4bnbn+1 b1c1d1 a2a3 a4an an+1 1a11a22b13b2 4a3 figure 3. alice’s winning strategy in case 1 for the graph s3�sn case 1.2: bob plays color 1 in d′ks, say d3 which is adjacent to d1. in respond, alice assigns color 2 in a1. in his third move bob plays according to the following strategy. i: if bob plays color 2 in the vertex b′is or c ′ js, alice then plays color of the vertex cj in bi (i = j). now vertex al (i = l = j) has two same colors in its neighbors. in this way alice protects all the vertices of degree four from threat. alice will play only color in a1, whenever bob plays color in a′is. case 2: in this case, bob plays his first move in the vertices of degree two and assigns color 1 in dk or cj say d2. alice then plays color 1 in c1. now we consider all the cases for bob’s second move. case 2.1: if bob plays again in the neighbor of d1 say d3 by assigning it color 2. alice, in respond plays color 3 in d1. now there are following possibilities of bob’s third move. we distinguish 3 subcases. i: if bob wants that the vertex a1 be under threat and plays color 2 in al say a2. alice then should play color 4 in a1. now all the vertices of degree greater than four have assigned color. after that alice follow the same strategy as in case 1. ii: if bob plays in the neighbor of that al which has already two distinct colors say a2 and assigns color 2 in bi or cj say c2. alice can prevent this vertex from threat by playing color of vertex c2 in vertex b2. after that alice follows the same strategy as in case 1. iii: if bob wants to plays in the neighbor of that al which has has no color assigned say a4 and assigns color 1 in d4 (d4 is adjacent to a4), then alice plays color 2 or color 4 in the vertex a1. in his next move if, bob again plays in the neighbor of a4 and assigns color 3 in c4. alice then should play color of c4 in b4. now a4 has two same colors in its neighbor. whenever, bob plays color 1 in any one of the remaining uncolored dk′s say d5, alice then plays color of vertex a1 in c5. now there are two same colors in the 132 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 a1 d2d3d4dndn+1 c2c3c4cncn+1 b2b3b4bnbn+1 b1c1d1 a2a3 a4an an+1 1a11a2 2b3 3a3 4a4 1b1 2b2 figure 4. alice’s winning strategy in case 2 for the graph s3�sn neighbor of a5. after that alice follow the same strategy as in case 1. case 2.2: now vertices d1 and a2 have one color in its neighbor. if bob make his second move by playing the color 2 in bi or cj say c2, then alice assigns color of vertex c2 in b2. in this way, alice can block the threat of vertex a2. after that alice follows the same strategies as in case 1 and case 2.1. case 2.3: if bob plays suitable color in cj or bi say cj (k 6= j), then alice plays color of that cj in bi (i = j). after that alice follows the same strategies as in above cases. case 3: if alice assigns color 1 in b1 in her first move and bob plays color 2 in c1. alice then plays color in d1. now all the vertices of degree n + 1 have colored. after that alice will follow the strategy of case 1 and case 2 in her next moves to win the game. case 4: bob plays color 2 in the vertex a1 in his first move. alice then plays color 1 in c1. now, irrespective of bob’s next move. alice plays color 3 in d1. in this way, alice can succeed to assign color to all the vertices of degree greater than n + 1. now bob will in bi or cj or dk to create threat in al. alice will respond by following the strategies of case 1 and case 2 and she wins the game. 3. exact values of χg(pm ◦k1) and χg(cm ◦pn) let g and h be two graphs of order n1 and n2, respectively. the corona product g◦h is defined as the graph obtained from g and h by taking one copy of g and n1 copies of h and joining by an edge each vertex from the ith−copy of h with the ith−vertex of g. we will denote by v = {v1,v2, . . . ,vn} the set of vertices of g and by hi = (vi,ei) the copy of h such that vi ∼ v for every v ∈ vi. it is important to note that the corona product operation is not commutative. in this section we prove the exact values of χg(pm ◦k1) and χg(cm ◦pn). theorem 3.1. the game chromatic number of pm ◦k1 = 3, where m ≥ 2. proof. it is obvious that χg(pm ◦ k1) ≥ 3. in order to show that χg(pm ◦ k1) ≤ 3, alice winning strategy is given. alice’s winning strategy with 3 colors: alice starts the game by playing color 1 in the vertex w1. suppose, in his first move, bob plays one of the available color in vertex vi, then alice responds by playing the same color in wi+1 or wi−1 . in this way she can block the threat to the vertex vi, because it has two same colors in its neighbor. similarly if bob plays some color in vertex wi, alice responds by playing the same color in vi+1 or vi−1. if during the game the following two cases arises, then alice will play according to the following strategy. case 1: suppose the vertex vj is already colored and bob plays a color in the vertex wi. in this 133 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 situation, if d(vi, vm) is odd then alice assigns the same color in vi−1 (if i > j) or vi+1 (if i < j), otherwise she plays any available color in the vertex vi. case 2: if there are two vertices say vm and vn that are already colored and bob plays a color 1 in the vertex wi where m < i < n. in response, alice plays the same color 1 in vi−1, (if d(vi, vm) is even) or in vi+1 (if d(vi, vn) is even). in case when vi is at odd distance to both vm and vn, she will play any available color in the vertex vi. similarly if bob plays some color in vi then alice plays the same color in wi−1 or wi+1 according to the above two cases. if during the game, the following situations arise then alice modifies her strategy according to the situation. situation 1: a part of graph pm ◦k1 is shown in figure 5. if bob plays the color 3 on wi+2. suppose, bob already assigned the color 3 in the vertex vi+4. then vi vi+1 vi+2 vi+3 wi+1 wi+2 wi+3 wi+4 vi+4 vi+5 3b 3bi 2ai+1 3a wi 2a 2b figure 5. p3 ◦ k1 according to her strategy alice can not assign the same color in the vertex vi+3. in this case alice plays color 2 in vi+3, because this is the only way she can block the threat to the vertex vi+2. situation 2: if bob plays a color (say 3) in the vertex wi+2 and the vertices vi+4 and vi−1 are already colored with colors 3 and 2 respectively. in this situation alice plays the same color as of the vertex vi−1 in wi. vi vi+1 vi+2 vi+3 wi+1 wi+2 wi+3 wi+4 vi+4 vi+5 2a 3b 3bi2ai+1 3a2b wiwi-1wi-2 vi-1 1bi+1 1ai+2 figure 6. p3 ◦ k1 theorem 3.2. for any integer n ≥ 2 and m ≥ 3 we have that χg(cm ◦pn) = 4. 134 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 proof. the graph c8 ◦pn is shown in the figure 7. bob’s winning strategy for 3 colors: a1 a2 a3 a4 a5 a6 a7 a8 b1,1 b1,n b8,1 b8,n 1a1 2b1 3a2 3a3 1b2 3a4 2b3 3b4 figure 7. c8 ◦ pn case 1: alice makes her first move by playing color 1 in any of the vertex from the inner cycle, say a1. in response bob plays the color 2 in the vertex a3. in her second move alice will play color 3 in the vertex a2 because this is the only way she can block the threat to this vertex. in his next moves bob plays suitable color in the alternative vertices a5,a7,a9, . . . , respectively, forcing alice to play suitable color in the vertices a4,a6,a8, . . . . continuing in this way, bob plays his last move according to the following two situations: i: if the number of vertices in inner cycle are even then bob plays any suitable color in the vertex an−1. alice cannot block the threat to both the vertices an−2 and an, and bob wins. ii: if the number of vertices in inner cycle are odd then bob in his second last move plays a color in any of the vertex bi,n−1, which is different to that of a1. in response alice is forced to play color in the vertex an−1 because this is the only way to block the threat to this vertex. by playing a color different to both a1 and an−1 in any of the vertex bi,n−1, bob can succeed to assign three distinct colors in the neighbor of the vertex an. case 2: alice makes her first move by playing color 1 in any of the outer vertex bi,n, say b1,1. in response bob plays color 2 in the vertex b2,1. in her second move alice has to play color 3 in the vertex a1 because this is the only way she can block the threat to this vertex. in his next moves, bob plays in the alternative vertices a3,a5,a7, . . . , respectively, forcing alice to play in the vertices a2,a4,a6, . . . continuing in this way, bob plays his last move according to case 1. hence χg(cm ◦pn) ≥ 4. alice’s winning strategy with 4 colors: the graph cm ◦pn have m + mn vertices. the vertices of degree greater than four are only the vertices of the inner cycle, as shown in the above figure. alice, in her strategy assigns color to the vertices a1,a2, ...,am and consequently blocks the threat to these vertices. since alice has to start the game, 135 s. a. h. bokhary et al. / j. algebra comb. discrete appl. 5(3) (2018) 129–136 suppose she makes her first move by playing color 1 in the vertex a1. in respond, bob can play the game in the following two ways. case 1: bob plays in the vertices of inner cycle say ai, for some 2 ≤ i ≤ m. alice responds by playing any color from the set of available colors in the uncolored vertices of inner cycle. case 2: bob plays in the vertices bi,j, then alice assigns a suitable color in the corresponding ai. bob can only win if he succeeded to assign 4 distinct colors in the neighborhood of any vertex. but whenever he plays in the outer vertices, bi,j, alice assigns a suitable color in the corresponding ai. in this way alice can color all the vertices in the inner cycle and she wins the game. from both these strategies we conclude that χg(cm ◦pn) = 4. acknowledgment: the authors are very thankful to a referee for careful reading with corrections and useful comments. references 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[6] x. zhu, game coloring the cartesian product of graphs, j. graph theory 59(4) (2008) 261–278. 136 https://mathscinet.ams.org/mathscinet-getitem?mr=2411449 https://mathscinet.ams.org/mathscinet-getitem?mr=2411449 https://doi.org/10.1142/s0129054191000091 https://doi.org/10.1142/s0129054191000091 https://mathscinet.ams.org/mathscinet-getitem?mr=1220515 https://mathscinet.ams.org/mathscinet-getitem?mr=1220515 https://mathscinet.ams.org/mathscinet-getitem?mr=1292976 https://mathscinet.ams.org/mathscinet-getitem?mr=1292976 https://mathscinet.ams.org/mathscinet-getitem?mr=2549583 https://mathscinet.ams.org/mathscinet-getitem?mr=2549583 https://mathscinet.ams.org/mathscinet-getitem?mr=2463181 introduction exact value of g(s3sn) exact values of g(pmk1) and g(cmpn) references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(3) • 169-190 received: 17 march 2015; accepted: 17 july 2015 doi 10.13069/jacodesmath.79416 journal of algebra combinatorics discrete structures and applications a further study for the upper bound of the cardinality of farey vertices and application in discrete geometry∗ research article daniel khoshnoudirad∗∗ abstract: the aim of the paper is to bring new combinatorial analytical properties of the farey diagrams of order (m, n), which are associated to the (m, n)-cubes. the latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. we give a new upper bound for the number of farey vertices fv (m, n) obtained as intersections points of farey lines ([14]): ∃c > 0,∀(m, n) ∈ n∗2, ∣∣∣fv (m, n)∣∣∣ ≤ cm2n2(m + n) ln2(mn) using it, in particular, we show that the number of (m, n)-cubes um,n verifies: ∃c > 0,∀(m, n) ∈ n∗2, ∣∣∣um,n∣∣∣ ≤ cm3n3(m + n) ln2(mn) which is an important improvement of the result previously obtained in [6], which was a polynomial of degree 8. this work uses combinatorics, graph theory, and elementary and analytical number theory. 2010 msc: 05a15, 05a16, 05a19, 05c30, 68r01, 68r05, 68r10 keywords: combinatorial number theory, farey diagrams, theoretical computer sciences, discrete planes, diophantine equations, arithmetical geometry, combinatorial geometry, discrete geometry, graph theory in computer sciences 1. introduction discrete geometry is the meeting point between several domains of mathematics and computer sciences: combinatorics, graph theory and number theory. to understand better the images, one studies ∗ the subject of this research was initiated at the university of strasbourg under the contract conv 10, blanc-205 of the a.n.r. the research was precised, pursued, and accomplished after the end of the contract by my own means. ∗∗ e-mail: daniel.khoshnoudirad@hotmail.com 169 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes very advanced theoretical problems coming from pure mathematics. in 2d, many progresses have been done. in 3d, there is a very active research in understanding the discretization of planes. this article brings progress to the combinatorial studies of 3d-patterns. paul erdös obtained many results in the field of combinatorial geometry for some particular configurations (see [9] for example). one particular instance of configurations are the farey diagrams. these diagrams and farey sequences have many applications. in particular, farey sequences and farey diagrams are directly involved in medicine. for instance, in cardiology to modelize optimized systems for pacemakers, or in research against cancer, in imagery and surgery [28]. a very active field of research is the tomography and reconstruction, in which farey sequences and farey diagrams also apply [12]. another example of application to vision is given in [20]. they can also be used for the detection of pieces of discrete planes in 3d-image. for example in [28]. tomás proves in [27] and [26] that there is an important link between accelerator physics and farey diagrams. we notice that the farey diagram of order (n,n) has the same degree as the resonance diagram of order n. the asymptotic behaviour of the two different structures only differs by a factor. there are some similarities between (m,n)-cubes, that we redefine below, and threshold functions on a two-dimensional rectangular grid, for which an asymptotic value for the cardinality of these functions has been derived in [11]. and farey diagrams are used, since long time in computer science: for example, they are also used when we study the preimage of a discrete piece of plane in discrete mathematics, and the farey diagram for discrete segments were studied by mcilroy in [19]. some problems related to farey diagrams remain unsolved. we are going to focus on this field to study the farey diagrams from the point of view of combinatorics and number theory. in [6], one of the strategies for the enumeration of pieces of discrete planes, was to estimate the number of vertices in a farey diagram. this work, combined with a basic property of graph theory, yields an upper bound. this upper bound is an homogeneous polynomial of degree 8: m3n3(m + n)2. in her thesis [7], debled-rennesson also studied this problem. another step forward has been taken by domenjoud, jamet, vergnaud, and vuillon in [8] where an exact formula (from combinatorial number theory) for the cardinality of the (2,n)-cubes has been derived. in [14], i found that the number of straight farey lines is asymptotically mn(m + n) ζ(3) when m and n go to infinity. henceforth, the strategy consisting in focusing on farey lines to study farey vertices combinatorics is not sufficient if we want to have a deeper understanding of the combinatorics of the (m,n)-cubes, and we can directly focus on the farey vertices [14] with some tools of number theory. in the following, i derive an upper bound of degree strictly lower than 6, and not 6, as it was the case in [6]. 2. preliminaries let j−m,mk denote the set {−m,.. . ,−1,0,1, . . . ,m} of consecutive integers between −m and m. definition 2.1. [14](farey lines of order (m,n)) a farey line of order (m,n) is a line whose equation is uα + vβ + w = 0 with (u,v,w) ∈ j−m,mk × j−n,nk × z, and which has at least 2 intersection points with the frontier of [0,1]2. (u,v,w) are the coefficients. (α,β) are the variables. let denote the set of farey lines of order (m,n) by fl(m,n). definition 2.2. [14](farey vertex) a farey vertex of order (m,n) is the intersection of two farey lines. we will denote the set of farey vertices of order (m,n), obtained as intersection points of farey lines of order (m,n), by fv (m,n). definition 2.3. [14](farey diagrams for the pieces of discrete planes of order (m,n) (or (m,n)-cubes)) the farey diagram for the (m,n)-cubes of order (m,n) is the diagram defined by the passage of farey lines in [0,1]2. we recall that b c denotes the integer part, and 〈 〉 denotes the fractional part. if a and b are two integers, a∧ b denotes the greatest common divisor of a and b, and a∨ b denotes the least common multiple. 170 daniel khoshnoudirad figure 1. farey lines of order (3, 3) ϕ denotes the euler’s totient function. card(a) denotes the cardinality of the set a. definition 2.4. [10](farey sequences of order n) the farey sequence of order n is the set fn = { 0 }⋃{p q , ∣∣∣1 ≤ p ≤ q ≤ n,p∧q = 1} . we mention [10] as a forthcoming modern reference work on the farey sequences. several standard variants of the notion of farey diagram are mentioned there. definition 2.5. [14](farey edge) a farey edge of order (m,n) is an edge of the farey diagram of order (m,n). we denote the set of farey edges by fe(m,n). definition 2.6. [14](farey graph) the farey graph of order (m,n) is the graph gf(m,n) = (fv (m,n),fe(m,n)). definition 2.7. (farey facet) a farey facet of order (m,n) is a facet of the farey graph of order (m,n). we will denote the set of farey facets of order (m,n) by ff(m,n). let m and n be two positive integers. we let fm,n denote the set = j0,m− 1k × j0,n− 1k. um,n denotes the set of all (m,n)-cubes. furthermore, the proposition 3 of [6] shows that the set of (m,n)cubes of the discrete planes pα,β,γ only depends of (α,β), and is denoted by cm,n,α,β. 171 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes definition 2.8. [6]((m,n)-pattern) let m and n be two positive integers. a (m,n)-pattern is a map w:fm,n −→ z. m × n is called the size of the (m,n)-pattern w. the set of the (m,n)-patterns will be denoted by mm,n. definition 2.9. [6]((m,n)-cube, see figure 2) the (m,n)-cube wi,j(α,β,γ) at the position (i,j) of a discrete plane pα,β,γ is the (m,n)-pattern w defined by: w(i′,j′) = pα,β,γ(i + i ′,j + j′)−pα,β,γ(i,j) for all (i′,j′) ∈fm,n = bα(i + i′) + β(j + j′) + γc−bαi + βj + γc for all (i′,j′) ∈fm,n where pα,β,γ(i,j) = bαi + βj + γc and { (i,j,pα,β,γ(i,j)), ∣∣∣(i,j) ∈ z2} defines the discrete plane pα,β,γ. this definition shows that: ∀ (i,j) ∈ z2,∀(α,β,γ) ∈ r3, wi,j(α,β,γ) = w0,0(α,β,αi + βj + γ). now, we recall some results obtained in [6], and some direct consequences of this result. proposition 2.10. (recall [6]) 1. the (k,l)-th point of the (m,n)-cube at the position (i,j) of the discrete plane pα,β,γ can be computed by the formula : wi,j (α,β,γ) (k,l) = { bαk + βlc if 〈αi + βj + γ〉 < cα,βk,l bαk + βlc+ 1 else where cα,βk,l = 1−〈αk + βl〉. 2. the (m,n)-cube wi,j(α,β,γ) only depends on the interval [b α,β h ,b α,β h+1[ containing 〈αi + βj + γ〉 where the bα,βh are the number c α,β k,l ordered by ascending order. 3. for all h ∈ j0,mn−1k, if [bα,βh ,b α,β h+1[ is non-empty, then there exists i,j such that 〈αi + βj + γ〉 ∈ [b α,β h ,b α,β h+1[. such a way, the number of (m,n)-cubes in the discrete plane pα,β,γ is equal to card ({ c α,β k,l ∣∣∣(k,l) ∈fm,n}) ≤ mn. corollary 2.11. ([6]) 1. ∀(α,β,γ) ∈ [0,1]2 ×r,w0,0(α,β,γ) = w0,0(α,β,〈γ〉). 2. ∀(α,β,γ) ∈ [0,1]2 ×r,∀(i,j) ∈ z2, wi,j(α,β,γ) = w0,0(α,β,αi + βj + γ) = w0,0(α,β,〈αi + βj + γ〉). 3. by the proposition 2.10, the set of (m,n)-cubes of the discrete planes pα,β,γ only depends of (α,β) and is denoted by cm,n,α,β. corollary 2.12. ([6]) let o be a farey connected component, then o is a convex polygon and if p1,p2,p3 are distinct vertices of the polygon o, then : • for any point p ∈o, cm,n,p = cm,n,p1 ∪cm,n,p2 ∪cm,n,p3. 172 daniel khoshnoudirad figure 2. example of two (4, 3)-cubes (red and green) • for any point p ∈o in the interior of the segment of vertices p1 and p2, cm,n,p = cm,n,p1 ∪cm,n,p2. by this corollary, all the (m,n)-cubes are associated to farey vertices. and according to the proposition 2.10, there are at most mn (m,n)-cubes associated to a farey vertex, therefore∣∣∣um,n∣∣∣ ≤ mn∣∣∣fv (m,n)∣∣∣. 173 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes 3. fundamental properties lemma 3.1. (reminder of graph theory) let us consider n straight lines. the number of vertices constructed from these n lines is at most n(n−1) 2 . we know by [14], that the number of farey lines, is equivalent to a polynomial of degree 3 in m and n, when m and n go to infinity. according to lemma 3.1, these lines form a number of vertices, given at most by a polynomial of order 6 ([6]). but this method is far from giving an optimal upper bound for the cardinality of the farey vertices. in order to obtain a new and more powerful result of combinatorics on this set of vertices, we are going to study the properties of the farey lines passing through a farey vertex. our idea is to use the theorem: proposition 3.2. (reminder of graph theory) in a simple graph g = (v,e), we have:∑ x∈v deg(x) = 2 |e| where v is the set of vertices, and e is the set of edges, and deg(x) is the degree of the vertex x, that is the number of edges which are adjacent to the vertex x. theorem 3.3. (gauss theorem) if (a,b,c) ∈ z3, such that a | bc, and a∧ b = 1. then, a | c. 4. modeling of the problem of farey vertices of order (m, n) to have a precise estimate on the number of edges in the farey graph of order (m,n), it is sufficient to compute the degree of any farey vertex. henceforth, below we to study in more detail the mapping defined on fv (m,n): x 7→ deg(x). proposition 4.1. (upper bound for the number of farey lines of order (m,n) passing through a farey vertex of order (m,n)) let p = ( p q , p′ q′ ) be a farey vertex of order (m,n). let us define r,r′,s,s′,d and d′ as follows:  p = (p∧p′)r, q = (q ∧q′)s p′ = (p∧p′)r′, q′ = (q ∧q′)s′ d = p∧p′, d′ = q ∧q′. (1) • if (p,p′) ∈ n∗2, then we have deg(p) ≤ 2 [ 2 ⌊ 1 dr ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) + 2 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋] . • if p = 0 then we have deg ( 0, p′ q′ ) ≤ ( 1 + ⌊ n q′ ⌋) (2m + 1). the vertices such that p = 0, are the vertices of the set{( 0, p′ q′ ) with p′ q′ ∈ fn } . 174 daniel khoshnoudirad • if p′ = 0, then we have deg ( p q ,0 ) ≤ ( 1 + 2 ⌊ m q ⌋) (n + 1). the vertices such that p′ = 0 are the vertices of the set{( p q ,0 ) with p q ∈ fm } . proof. we can always suppose that in the equation of a farey line, (of the type: uα + vβ + w = 0, with (u,v,w) ∈ j−m,mk × j−n,nk ×z), we have v ≥ 0. because if v < 0, it is sufficient to multiply the equation by −1. and we obtain the same line, but (−u,−v,−w) ∈ j−m,mk × j0,nk ×z. first, we handle the case where p = 0 or p′ = 0. p = 0 ⇒ p′v + q′w = 0 ⇒ { v = q′k w = −p′k ⇒ 0 ≤ k ≤ n q′ (because of the preliminary). there are at most 1 + ⌊ n q′ ⌋ such integers. and there are 2m + 1 integers in the interval j−m,mk. the vertices such that p = 0, are the vertices of the set{( 0, p′ q′ ) with p′ q′ ∈ fn } . p′ = 0 ⇒ pu + qw = 0 ⇒ { u = qk w = −pk ⇒ 0 ≤ |k| ≤ m q . there are at most 1 + 2 ⌊ m q ⌋ such integers. the vertices such that p′ = 0, are the vertices of the set {( p q ,0 ) with p q ∈ fm } . now, we can handle the general case remaining: (p,p′) ∈ n∗2 in gf(m,n), because a farey line generates at most 2 edges passing through the farey vertex p , we have: deg(p) ≤ 2×card ({ farey lines passing through p }) (2) so, we are looking for an optimal bound for the cardinality of (u,v,w) ∈ j−m,mk × j1,nk ×z such that u p q + v p′ q′ = −w ⇔ upq′ + vp′q qq′ = −w (with the condition u∧v ∧w = 1), that is u(p∧p′)r(q ∧q′)s′ + v(p∧p′)r′(q ∧q′)s qq′ = −w. 175 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes (p∧p′)(q ∧q′) urs′ + vr′s (q ∧q′)2ss′ = −w. after simplification: (p∧p′) urs′ + vr′s (q ∧q′)ss′ = −w. (p∧p′)(urs′ + vr′s) = −w(q ∧q′)ss′ (p∧p′)urs′ = −w(q ∧q′)ss′ − (p∧p′)vr′s ⇒ s | (p∧p′)urs′ as s∧ [(p∧p′)rs′] = 1, the gauss theorem (theorem 3.3) implies that s | u. so,{ ∃u′ ∈ z such that u = su′ ∃v′ ∈ z such that v = s′v′. so, 0 ≤ |u′| ≤ m s and 1 ≤ v′ ≤ n s′ . (p∧p′) su′rs′ + s′v′r′s (q ∧q′)ss′ = −w ⇒ (p∧p′) u′r + v′r′ (q ∧q′) = −w (theorem of gauss) ⇒ (p∧p′) | w and (q ∧q′) | u′r + v′r′. when w is fixed, the consequence of the hypothesis of primality enables to solve this diophantine equation: let us fix w,   u′ = u0 ( − w p∧p′ (q ∧q′) ) + r′k v′ = v0 ( − w p∧p′ (q ∧q′) ) −rk for k ∈ z (3) where (u0,v0) is a particular solution of the diophantine equation in (x,y): rx + r′y = 1. the determinant of this system in ( w p∧p′ ,k ) is: u0q ∧q′r + v0(q ∧q′)r′ = (q ∧q′)[u0r + v0r′] = q ∧q′. so, one can determine ( w p∧p′ ,k ) by the cramer formulas: ( w p∧p′ k ) =  − rq ∧q′ − r ′ q ∧q′ v0 −u0  ( u′ v′ ) . as in [14], we can always suppose that v ≥ 0 (else we mutiply uα + vβ + w = 0, by −1). moreover, we have seen that as we have: |w| ≤ m p q + n p′ q′ , 176 daniel khoshnoudirad and as p∧p′ | w, we can deduce that there exists w′ such that w = w′(p∧p′). so, 0 ≤ |w′| ≤ ⌊ 1 p∧p′ ( m p q + n p′ q′ )⌋ . as for a farey line, (u,v) 6= (0,0), we deduce that (u′,v′) 6= (0,0). first we handle the case where v = 0 ⇒ v′ = 0 ⇒ v0 ( w p∧p′ ) q∧q′ +rk = 0 ⇒ p | w. as ru0 +r′v0 = 1, we deduce by the bezout theorem that r ∧ v0 = 1. in addition to it, we have r ∧ (q ∧ q′) = 1. so, r ∧ (v0(q ∧ q′)) = 1. and by the gauss theorem, we have that r | w d . in this case, it is impossible that w = 0, else k = 0 and u = v = w = 0. so, there are at most 2 ⌊ 1 p ( m p q + n p′ q′ )⌋ . farey lines passing through the farey vertex. in the second case, v > 0, so we have  − ⌊m s ⌋ ≤ u′ ≤ ⌊m s ⌋ , 1 ≤ v′ ≤ ⌊n s′ ⌋ . when w is fixed,   r′k = u′ + u0 ( w p∧p′ ) q ∧q′, rk = v0 ( − w p∧p′ ) q ∧q′ −v′ for k ∈ z. so,   − ⌊m s ⌋ + u0 ( w p∧p′ ) q ∧q′ ≤ r′k ≤ ⌊m s ⌋ + u0 ( w p∧p′ ) q ∧q′, − ⌊n s′ ⌋ + v0 ( − w p∧p′ ) q ∧q′ ≤ rk ≤−1−v0 ( w p∧p′ ) q ∧q′ for k ∈ z. now, we distinguish 2 cases: • if w = 0, the number of suitable integers k is bounded by min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) . • w 6= 0 – the case where r′k = u0 ( w p∧p′ ) q ∧q′ ⇒ r′ | ( w p∧p′ ) . then, the number of suitable w dr′ is at most 2 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ . – else r′k 6= u0 ( w p∧p′ ) q ∧q′, so the number of suitable integers k is bounded by min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) 177 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes and the number of ( k, w d ) is bounded by: min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋ . and, of course, the existence of such couples implies that: min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋ ≥ 1. in particular,   ⌊ m sr′ ⌋ ≥ 1⌊ n s′r ⌋ ≥ 1. (4) so card ({ farey lines passing through p }) ≤ 2 ⌊ 1 dr ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) + 2 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋ and by (2), we have deg(p) ≤ 2 [ 2 ⌊ 1 dr ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) + 2 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ + min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ×2 ⌊ 1 d ( m p q + n p′ q′ )⌋] which proves the claim. 5. bound for |fv (m, n)| the equations in (1) are:   r = p p∧p′ ,s = q q ∧q′ , r′ = p′ p∧p′ ,s′ = q′ q ∧q′ , d = p∧p′,d′ = q ∧q′. 178 daniel khoshnoudirad proposition 5.1. let  a(m,n,p,q,p′,q′) = 2 min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) b(m,n,p,q,p′,q′) = 4 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ c(m,n,p,q,p′,q′) = 4 ⌊ 1 dr ( m p q + n p′ q′ )⌋ d(m,n,p,q,p′,q′) = 2 ⌊ 1 d ( m p q + n p′ q′ )⌋ e(m,n,p,q,p′,q′) = 2a(m,n,r,r′,s,s′,d′)×d(m,n,r′,p,q,p′,q′) a′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 a(m,n,p,q,p′,q′) b′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 b(m,n,p,q,p′,q′) c′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 c(m,n,p,q,p′,q′) d′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 d(m,n,p,q,p′,q′) e′(m,n) = ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 e(m,n,p,q,p′,q′) we have: 2 |fe(m,n)| ≤ a′(m,n) + b′(m,n) + c′(m,n) + e′(m,n) + ∑ p′ q′ ∈fn deg ( 0, p′ q′ ) + ∑ p q ∈fm deg ( p q ,0 ) . proof. by the proposition 3.2, we have: 2 |fe(m,n)| ≤ ∑ x∈fv (m,n) deg(x) p(α,β) ∈ fv (m,n) ⇒∃ ( p q , p′ q′ ) such that   α = p q with p∧q = 1,p ≤ q ≤ 2mn β = p′ q′ with p′ ∧q′ = 1,p′ ≤ q′ ≤ 2mn. 2 |fe(m,n)| ≤ ∑ 0≤p<q≤2mn p∧q=1 ∑ 0≤p′<q′≤2mn p′∧q′=1 deg ( p q , p′ q′ ) to conclude,we use the result of the proposition 4.1, which proves the claim. 179 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes lemma 5.2. ( p q , p′ q′ ) ∈ fv (m,n) ⇒ 1 ≤ q ∨q′ ≤ 2mn. proof. ( p q , p′ q′ ) ∈ fv (m,n) ⇒ { ∃((u,u′),(v,v′),(w,w′)) ∈ j−m,mk2 × j−n,nk2 ×z2 u′v −uv′ 6= 0 such that: ( p q , p′ q′ ) = ( |w′v −wv′| |uv′ −u′v| , |wu′ −w′u| |uv′ −u′v| ) . so, q ∨q′ | |uv′ −u′v|. in particular, q ∨q′ ≤ 2mn which proves the claim. therefore, as we remember that ∀(a,b) ∈ n∗2,(a∧b)×(a∨b) = ab we can rewrite q∨q′ = qq′ q ∧q′ = ss′d′ ≤ 2mn. we need to recall another theorem: theorem 5.3. [2](asymptotic development of the harmonic series) if x ≤ 1, then∑ n≤x 1 n = log x + c +o( 1 x ) where c is euler’s constant. we can apply this theorem and we are able to say in particular: corollary 5.4. there exists k > 0 such that, ∀n ∈ n\{0,1}, we have n∑ i=1 1 i ≤ k log n. proposition 5.5. ∃k > 0,∀(m,n) ∈ n∗2, a′(m,n) ≤ km2n2(m + n) ln2(mn). proof. a′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 a(m,n,p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 2 min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋) ≤ 2 ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 ⌊ n s′r ⌋ ≤ 2 2mn∑ q=1 q∑ p=0 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 ⌊ n s′r ⌋ . 180 daniel khoshnoudirad i point out that i chose ⌊ n s′r ⌋ , and after ⌊ m sr′ ⌋ , in order to obtain a symmetric upper bound. in the following, we use the boundaries for r,r′,s,s′ given by:  q = d′s so 1 ≤ s ≤ 2mn d′ q′ = d′s′ so 1 ≤ s′ ≤ 2mn d′ p = dr so 1 ≤ r ≤ d′s d p′ = dr′ so 1 ≤ r′ ≤ d′s′ d . now, we can continue the calculation of a′(m,n). for this purpose, let us permute the sums and let us change the variables by using, as before,   r = p p∧p′ ,s = q q ∧q′ r′ = p′ p∧p′ ,s′ = q′ q ∧q′ d = p∧p′,d′ = q ∧q′. in particular,   r ≤ d′s d r′ ≤ d′s′ d ⇒ d ≤ ⌊ min ( d′s r , d′s′ r′ )⌋ . a′(m,n) ≤ 2 2mn∑ q=1 q∑ p=0 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 ⌊ n s′r ⌋ ≤ m+n∑ d=1 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s d∑ r=1 d′s′ d∑ r′=1 ⌊ n s′r ⌋ ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ n s′r ⌋⌊min( d′sr , d′s′r′ )⌋∑ d=1 1. now, we have to distinguish the case where d′s = 1 and the case d′s > 1 in the sums in order to use the corollary 5.4 (and the same for d′s′). a′ ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s>1 d′s′∑ r′=1 d′s′>1 d′ n rr′ + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 2mn d′s + km2n3 ln2(mn) 181 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes ≤ k′n ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 2mn s + km2n3 ln2(mn) ≤ k′n ln2(mn) 2mn∑ s=1 b2mns c∑ d′=1 2mn s + km2n3 ln2(mn) ≤ k′m2n3 ln2(mn) 2mn∑ s=1 1 s2 + km2n3 ln2(mn) ≤ k”m2n3 ln2(mn) a′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 a(p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 4 ⌊ m sr′ ⌋ ≤ 4 ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 ⌊ m sr′ ⌋ ≤ 4 2mn∑ q=1 q∑ p=1 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 ⌊ m sr′ ⌋ ≤ m+n∑ d=1 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s d∑ r=1 d′s′ d∑ r′=1 ⌊ m sr′ ⌋ ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ m sr′ ⌋⌊min( d′sr , d′s′r′ )⌋∑ d=1 1 ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 m sr′ d′s r ≤ 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s>1 d′s′∑ r′=1 d′s′>1 d′ m rr′ + kn2m3 ln2(mn) ≤ k′m ln2(mn) 2mn∑ d′=1 d′ b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 + kn2m3 ln2(mn) ≤ k′m ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 2mn d′s + kn2m3 ln2(mn) ≤ k′m2n ln2(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 1 s + kn2m3 ln2(mn) 182 daniel khoshnoudirad ≤ k”n2m3 ln2(mn). proposition 5.6. ∃k > 0,∀(m,n) ∈ n∗2, b′(m,n) ≤ km2n2(m + n) ln2(mn). proof. b′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 b(p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 4 ⌊ 1 dr′ ( m p q + n p′ q′ )⌋ ≤ k[b ′ 1(m,n) + b ′ 2(m,n)] where b ′ 1(m,n) = m ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 r r′sd′ b ′ 2(m,n) = n ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 d′s′ b ′ 1(m,n) ≤ m 2mn∑ q=1 q∑ p=1 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 r r′sd′ ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 r r′sd′ ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 1 r′ now, we have to distinguish the case where d′s′ = 1 and the case d′s′ > 1 in the sum in order to use the corollary 5.4. b ′ 1(m,n) ≤ k ′m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s ln(d′s′) + km3n2 ln2(mn) ≤ k′m ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 d′s 2mn d′s + km3n2 ln2(mn) 183 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes ≤ k′m2n ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 1 + km3n2 ln2(mn) ≤ k′m2n ln(2mn) 2mn∑ d′=1 2mn d′ + km3n2 ln2(mn) ≤ k”m3n2 ln2(mn). b ′ 2(m,n) ≤ n 2mn∑ q=1 q∑ p=1 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 1 d′s′ ≤ n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 d′s′ ≤ n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 1 r′ ≤ k′n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s ln(d′s′) + kn3m2 ln2(mn) ≤ k′n ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 d′s 2mn d′s + kn3m2 ln2(mn) ≤ k′n2m ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 1 + kn3m2 ln2(mn) ≤ k′n2m ln(2mn) 2mn∑ d′=1 2mn d′ + kn3m2 ln2(mn) ≤ k”n3m2 ln2(mn). proposition 5.7. ∃k > 0,∀(m,n) ∈ n∗2, c′(m,n) ≤ km2n2(m + n) ln2(mn). proof. c′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 c(p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 4 ⌊ 1 dr ( m p q + n p′ q′ )⌋ ≤ k[c ′ 1(m,n) + c ′ 2(m,n)] 184 daniel khoshnoudirad where c ′ 1(m,n) = m ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 sd′ c ′ 2(m,n) = n ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 r′ rd′s′ . c ′ 1(m,n) ≤ m 2mn∑ q=1 q∑ p=1 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 1 sd′ ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 sd′ ≤ m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 1 r ≤ k′m 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s′ ln(d′s) + km3n2 ln2(mn) ≤ k′m ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s′=1 d′s′ 2mn d′s′ + km3n2 ln2(mn) ≤ k′m2n ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s′=1 1 + km3n2 ln2(mn) ≤ k′m2n ln(2mn) 2mn∑ d′=1 2mn d′ + km3n2 ln2(mn) ≤ k”m3n2 ln2(mn). c ′ 2(m,n) ≤ n 2mn∑ q=1 q∑ p=1 p∧q=1 2mn∑ q′=1 q′∑ p′=1 p′∧q′=1 r′ rd′s′ ≤ n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 r′ rd′s′ ≤ n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 d′s′∑ r′=1 r′ rd′s′ d′s′ r′ 185 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes ≤ k′n 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s′ ln(d′s) + kn3m2 ln2(mn) ≤ k′n ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s′=1 d′s′ 2mn d′s′ + kn3m2 ln2(mn) ≤ k′n2m ln(2mn) 2mn∑ d′=1 b2mn d′ c∑ s′=1 1 + kn3m2 ln2(mn) ≤ k′n2m ln(2mn) 2mn∑ d′=1 2mn d′ + kn3m2 ln2(mn) ≤ k”n3m2 ln2(mn). proposition 5.8. ∃k > 0, ∀(m,n) ∈ n∗2,e′(m,n) ≤ km2n2(m + n) ln(mn). proof. e′(m,n) = ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 e(p,q,p′,q′) ≤ ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 2a(m,n,r,r′,s,s′,d′)×d(m,n,r′,p,q,p′,q′) ≤ k ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 min ( 2 ⌊ m sr′ ⌋ , ⌊ n s′r ⌋)⌊1 d ( m p q + n p′ q′ )⌋ ≤ k[e ′ 1(m,n) + e ′ 2(m,n)] with the condition (4). e ′ 1(m,n) = mn ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 s′r 1 d p q . e ′ 2(m,n) = mn ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 sr′ 1 d p′ q′ . let us study further e ′ 1(m,n), then the results for e ′ 2(m,n) are computed in a similar manner. e ′ 1(m,n) ≤ mn ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 s′ 1 q ≤ mn ∑ 1≤p<q≤2mn p∧q=1 ∑ 1≤p′<q′≤2mn p′∧q′=1 1 ss′d′ 186 daniel khoshnoudirad ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′rc≥1 d′s′∑ r′=1 b m sr′c≥1 ⌊ min( d ′s r , d ′s′ r′ ) ⌋∑ d=1 1 ss′d′ ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′rc≥1 d′s′∑ r′=1 b m sr′c≥1 d′s′ r′ 1 ss′d′ ≤ mn 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 d′s∑ r=1 b n s′rc≥1 d′s′∑ r′=1 b m sr′c≥1 d′s′>1 1 r′s + km2n3 ln(mn) ≤ k′mn2 ln(mn) 2mn∑ d′=1 b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 ss′ + km2n3 ln(mn) ≤ k′m2n3 ln(mn) b2mn d′ c∑ s=1 b2mn d′s c∑ s′=1 1 s2s′2 + km2n3 ln(mn) ≤ k”m2n3 ln(mn). the computation is exactly the same for e ′ 2(m,n). it remains to treat the two simple cases where p = 0 or p′ = 0 of the proposition 4.1: proposition 5.9. ∃k > 0,∀(m,n) ∈ n∗2, ∑ p′ q′ ∈fn deg ( 0, p′ q′ ) + ∑ p q ∈fm deg ( p q ,0 ) ≤ kmn(m2 + n2). proof. • [( 1 + ⌊ n q′ ⌋) (2m + 1) ] ≤ 2m + 1 + 2nm + n ≤ 5mn + 1 ∑ p′ q′ ∈fn deg ( 0, p′ q′ ) ≤ ∑ p′ q′ ∈fn [( 1 + ⌊ n q′ ⌋) (2m + 1) ] ≤ ∑ p′ q′ ∈fn (5mn + 1) ≤ (5mn + 1) |fn| • [( 1 + 2 ⌊ m q ⌋) (n + 1) ] ≤ n + 1 + 2mn + 2m ≤ 5mn + 1 ∑ p q ∈fm deg ( p q ,0 ) ≤ ∑ p q ∈fm [( 1 + 2 ⌊ m q ⌋) (n + 1) ] ≤ ∑ p q ∈fm (5mn + 1) ≤ (5mn + 1) |fm| we know [29] that |fn| = 1 + n∑ k=1 ϕ(k) ∼ +∞ n2 2ζ(2) . 187 a new upper bound for farey vertices in farey diagrams and application to (m,n)-cubes so there exists k > 0 such that∑ p′ q′ ∈fn deg ( 0, p′ q′ ) + ∑ p q ∈fm deg ( p q ,0 ) ≤ kmn(m2 + n2). hence, we have: ∃k > 0,∀(m,n) ∈ n∗2, 2|fe(m,n)| = ∑ x∈fv (m,n) deg(x) ≤ km2n2(m + n) ln2(mn). so, we can say that: ∃k ≥ 0, |fe(m,n)| ≤ km2n2(m + n) ln2(mn). hence, in the farey graph of order (m,n), we have: proposition 5.10. ∃k > 0,∀(m,n) ∈ n∗2, |fe(m,n)| ≤ km2n2(m + n) ln2(mn). moreover, as the farey graph of order (m,n) is planar, we can apply to it the euler’s formula: theorem 5.11. (recall)(euler’s formula for the connex planar graphs) in a connex planar multi-graph, having v vertices, e edges, and f facets, we have: v −e + f = 2. in particular, this involves that we have v ≤ e + 2. ∃k > 0,∀(m,n) ∈ n∗2, |fv (m,n)| ≤ km2n2(m + n) ln2(mn). this greatly improves the upper bound previously found in [6]. and we can add: ∃k > 0,∀(m,n) ∈ n∗2, |ff(m,n)| ≤ km2n2(m + n) ln2(mn). corollary 5.12. (upper bound for the (m,n)-cubes) ∃k > 0,∀(m,n) ∈ n∗2, |umn| ≤ km3n3(m + n) ln2(mn). 6. summary-conclusion in order to improve the upper bound for ∣∣∣um,n∣∣∣, an interesting work using combinatorics, graph theory, and number theory, has been to focus on the diophantine aspects of farey diagrams, combined with some other arguments of graph theory to estimate better the cardinality of farey vertices. and in this work, we obtained two important results: • ∃c > 0,∀(m,n) ∈ n∗2, ∣∣∣fv (m,n)∣∣∣ ≤ cm2n2(m + n) ln2(mn) whereas the previous published result was a polynomial of degree 6 [6]. 188 daniel khoshnoudirad • and we obtained: ∃c > 0,∀(m,n) ∈ n∗2, ∣∣∣um,n∣∣∣ ≤ cm3n3(m + n) ln2(mn) whereas the previous published result existing was a polynomial of degree 8 [6]. acknowledgment: i dedicate this work to the memory of my beloved father, who did his best for me and for the science. i also want to thank my brother(and his wife) and my sister(and her husband) for their support during the preparation of this article. i am thankful to grégory apou for his help. i also would like to thank the professor andre leroy for his meaningful help, his expertise in algebra, and for having given to me an invitation to give a successful seminar on combinatorics and number theory, and their applications to computer sciences, in the l.m.l. of the university of artois, on january, 2015. i also thank the director of the i.m.b., the professor alain bachelot for his help, and two other persons: professor hugues talbot, and professor pierre villon for their great kindness. and i also thank the experts of number theory: henri cohen and yuri bilu for their advises. my gratitude also goes to david, and jacques(and his wife) the friends of my father, for their encouragement to publish this article. this work could not take his final version without the meaningful help of the anonymous referees to whom i express my sincere acknowledgements for their insightful remarks, as they carefully reviewed my work and gave generously of their time and their expertise. references [1] d. m. acketa and j. d. žunić, on the number of linear partitions of the (m, n)-grid, inform. process. lett., 38(3), 163-168, 1991. 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[22] i. svalbe and a. kingston, on correcting the unevenness of angle distributions arising from integer ratios lying in restricted portions of the farey plane, combinatorial image analysis, springer, 110121, 2005. [23] g. tenenbaum, introduction to analytic and probabilistic number theory, 46, cambridge university press, 1995. [24] e. thiel, les distances de chenfrein en analyse d’images : fondements et applications, thélse de doctorat, ujf grenoble, 1994. [25] e. thiel and a. montanvert, chamfer masks: discrete distance functions, geometrical properties and optimization, pattern recognition, vol. iii. conference c: image, speech and signal analysis, proceedings., 11th iapr international conference on, ieee, 244-247, 1992. [26] r. tomás, asymptotic behavior of a series of euler’s totient function ϕ(k) times the index of 1/k in a farey sequence, arxiv:1406.6991, 2014. 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[29] g. h. hardy and e. m. wright, introduction à la théorie des nombres, traduction de françois sauvageot, springer. 190 introduction preliminaries fundamental properties modeling of the problem of farey vertices of order (m,n) bound for |fv(m,n)| summary-conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.09585 j. algebra comb. discrete appl. 4(1) • 49–60 received: 9 october 2015 accepted: 29 may 2016 journal of algebra combinatorics discrete structures and applications one–generator quasi–abelian codes revisited∗ research article somphong jitman, patanee udomkavanich abstract: the class of 1-generator quasi-abelian codes over finite fields is revisited. alternative and explicit characterization and enumeration of such codes are given. an algorithm to find all 1-generator quasi-abelian codes is provided. two 1-generator quasi-abelian codes whose minimum distances are improved from grassl’s online table are presented. 2010 msc: 94b15, 94b60, 16a26 keywords: group algebras, quasi-abelian codes, minimum distances, 1-generator 1. introduction as a family of codes with good parameters, rich algebraic structures, and wide ranges of applications (see [8], [9], [11], [10], [13], [14] , and references therein), quasi-cyclic codes have been studied for a halfcentury. quasi-abelian codes, a generalization of quasi-cyclic codes, have been introduced in [15] and extensively studied in [7]. given finite abelian groups h ≤ g and a finite field fq, an h-quasi-abelian code is defined to be an fq[h]-submodule of fq[g]. note that h-quasi-abelian codes are not only a generalization of quasi-cyclic codes (see [7], [8], [9], and [15]) if h is cyclic but also of abelian codes (see [1] and [2]) if g = h, and of cyclic codes (see [12]) if g = h is cyclic. the characterization and enumeration of quasi-abelian codes have been established in [7]. an h-quasi-abelian code c is said to be of 1-generator if c is a cyclic fq[h]-module. such a code can be viewed as a generalization of 1-generator quasi-cyclic codes which are more frequently studied and applied (see [11], [13], and [14]). analogous to the case of 1-generator quasi-cyclic codes, the number of 1-generator quasi-abelian codes has been determined in [7]. however, an explicit construction and an algorithm to determine all 1-generator quasi-abelian codes have not been well studied. ∗ this research is supported by the dpst research grant 005/2557 and the thailand research fund under research grant trg5780065. somphong jitman (corresponding author); department of mathematics, faculty of science, silpakorn university, nakhon pathom 73000, thailand (email: sjitman@gmail.com). patanee udomkavanich; department of mathematics and computer science, faculty of science, chulalongkorn university, bangkok 10330, thailand (email: pattanee.u@chula.ac.th). 49 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 in this paper, we give an alternative discussion on the algebraic structure of 1-generator quasi-abelian codes and an algorithm to find all 1-generator quasi-abelian codes. examples of new codes derived from 1-generator quasi-abelian codes are presented. the paper is organized as follows. in section 2, we recall some notations and basic results. an alternative discussion on the algebraic structure of 1-generator quasi-abelian codes is given in section 3 together with an algorithm to find all 1-generator quasi-abelian codes and the number of such codes. examples of new codes derived from 1-generator quasi-abelian codes are presented in section 4. 2. preliminaries let fq denote a finite field of order q and let g be a finite abelian group of order n, written additively. denote by fq[g] the group ring of g over fq. the elements in fq[g] will be written as ∑ g∈g αgy g, where αg ∈ fq. the addition and the multiplication in fq[g] are given as in the usual polynomial rings over fq with the indeterminate y , where the indices are computed additively in g. we note that y 0 = 1 is the identity of fq[g], where 1 is the identity in fq and 0 is the identity of g. given a ring r, a linear code of length n over r refers to a submodule of the r-module rn. a linear code in fq[g] refers to an fq-subspace c of fq[g]. this can be viewed as a linear code of length n over fq by indexing the n-tuples by the elements in g. the hamming weight wt(u) of u = ∑ g∈g ugy g ∈ fq[g] is defined to be the number of nonzero term ug’s in u. the minimum hamming distance a code c is defined by d(c) := min{wt(u) | u ∈ c,u 6= 0}. a linear code c in fq[g] is referred to as an [n,k,d]q code if c has fq-dimension k and minimum hamming distance d. given a subgroup h of g, a code c in fq[g] is called an h-quasi-abelian code if c is an fq[h]module, i.e., c is closed under the multiplication by the elements in fq[h]. such a code will be called a quasi-abelian code if h is not specified or where it is clear in the context. an h-quasi-abelian code c is said to be of 1-generator if c is a cyclic fq[h]-module. since every h-quasi-abelian code c in fq[g] is an fq[h]-module, it is also an fq[a]-module for all cyclic subgroups of h. it follows that c is quasi-cyclic of index |g|/|a|. however, being 1-generator h-quasi-abelian does not imply that c is 1-generator quasi-cyclic. therefore, it makes sense to study 1-generator h-quasi-abelian codes. assume that h ≤ g such that |h| = m and the index [g : h] = n m = l. let {g1,g2, . . . ,gl} be a fixed set of representatives of the cosets of h in g. let r := fq[h]. define φ : fq[g] → rl by φ (∑ h∈h l∑ i=1 αh+giy h+gi ) = (α1(y ),α2(y ), . . . ,αl(y )) , (1) where αi(y ) = ∑ h∈h αh+giy h ∈ r, for all i ∈ {1, 2, . . . , l}. it is not difficult to see that φ is an r-module isomorphism, and hence, the next lemma follows. lemma 2.1. the map φ induces a one-to-one correspondence between h-quasi-abelian codes in fq[g] and linear codes of length l over r. throughout, assume that gcd(q, |h|) = 1, or equivalently, fq[h] is semisimple. following [7, section 3], the group ring r = fq[h] is decomposed as follows. for each h ∈ h, denote by ord(h) the order of h in h. the q-cyclotomic class of h containing h ∈ h, denoted by sq(h), is defined to be the set sq(h) := {qi ·h | i = 0, 1, . . .} = {qi ·h | 0 ≤ i ≤ νh}, where qi ·h := ∑qi j=1 h in h and νh is the multiplicative order of q in zord(h). an idempotent in a ring r is a non-zero element e such that e2 = e. an idempotent e is said to be primitive if for every other idempotent f, either ef = e or ef = 0. the primitive idempotents in r 50 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 are induced by the q-cyclotomic classes of h (see [4, proposition ii.4]). every idempotent e in r can be viewed as a unique sum of primitive idempotents in r. the fq-dimension of an idempotent e ∈ r is defined to be the fq-dimension of re. from [7, subsection 3.2], r := fq[h] can be decomposed as r = re1 + re2 + · · · + res, where e1,e2, . . . ,es are the primitive idempotents in r. moreover, every ideal in r is of the form re, where e is an idempotent in r. 3. 1-generator quasi-abelian codes in [7], characterization and enumeration of 1-generator h-quasi-abelian codes in fq[g] have been given. in this section, we give alternative characterization and enumeration of such codes. the characterization in subsection 3.1 allows us to express an algorithm to find all 1-generator h-quasi-abelian codes in fq[g] in subsection 3.2. using the r-module isomorphism φ defined in (1), to study 1-generator h-quasi-abelian codes in fq[g], it suffices to consider cyclic r-submodules ra, where a = (a1,a2, . . . ,al) ∈ rl. for each a = (a1,a2, . . . ,al) ∈ rl, there exists a unique idempotent e ∈ r such that re = ra1 + ra2 + · · ·+ ral. the element e is called the idempotent generator element for ra. an idempotent f ∈ r of largest fq-dimension such that fa = 0 is called the idempotent check element for ra. let s = fql [h], where fql is an extension field of fq of degree l. let {α1,α2, . . . ,αl} be a fixed basis of fql over fq. let ϕ : rl → s be an r-module isomorphism defined by a = (a1,a2, . . . ,al) 7→ a = l∑ i=1 αiai. (2) using the map ϕ, the code ra can be regarded as an r-module ra in s. lemma 3.1 ([7, lemma 6.1]). let a ∈ rl and let e and f be the idempotent generator and idempotent check elements of ra, respectively, then e + f = 1 and dimfq (ra) = dimfq (re) = m− dimfq (rf). for a ring r, denote by r∗ and r× the set of non-zero elements and the group of units of r, respectively. in order to enumerate and determine all 1-generator h-quasi-abelian codes in fq[g], we need the following result. lemma 3.2. let a,b ∈ rl and let e be the idempotent generator of ra. let a = ϕ(a) and b = ϕ(b), where ϕ is defined in (2). then ra = rb if and only if there exists u ∈ (re)× such that b = ua. equivalently, ra = rb if and only if there exists u ∈ (re)× such that b = ua. 51 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 proof. write a = (a1,a2, . . . ,al) and b = (b1,b2, . . . ,bl), where ai,bi ∈ r for all i ∈{1, 2, . . . , l}. assume that ra = rb. then b = va for some v ∈ r. let u = ve ∈ re. note that, for each i ∈ {1, 2, . . . , l}, we have ai = rie for some ri ∈ r. then uai = (ve)(rie) = vrie2 = v(rie) = vai = bi for all i ∈{1, 2, . . . , l}. hence, b = ua and re = ra = rb = r(ua) = ura = ure. since u ∈ re and re = ure, we have u ∈ (re)×. conversely, assume that there exists u ∈ (re)× such that b = ua. then rb = rua ⊆ ra. we need to show that dimfq (ra) = dimfq (rb). let e ′ be an idempotent generator of rb. we have re′ = rb = r(ub) = u(rb) = u(re) = re since u ∈ (re)×. hence, by lemma 3.1, we have dimfq (ra) = dimfq (re) = dimfq (re ′) = dimfq (rb). therefore, rb = ra as desired. 3.1. the enumeration of 1-generator quasi-abelian codes first, we focus on the number of 1-generator h-quasi-abelian codes of a given idempotent generator in fq[h]. using the fact that the idempotents in fq[h] are known, the number of 1-generator h-quasiabelian codes in fq[g] can be concluded. proposition 3.3. let {e1,e2, . . . ,er} be a set of primitive idempotents of r and e = e1 + e2 + · · · + er. then the following statements hold. i) e1,e2, . . . ,er are pairwise orthogonal (non-zero) idempotents of se. ii) ej is the identity of sej for all j ∈{1, 2, . . . ,r}. iii) e is the identity of se. iv) se = se1 ⊕se2 ⊕···⊕ser. proof. for i), it is clear that e1,e2, . . . ,er are pairwise orthogonal (non-zero) idempotents in s. they are in se since ej = eje ∈ se for all j ∈{1, 2, . . . ,r}. the statements ii) and iii) follow since sej = se2j = (sej)ej for all sej ∈ sej and se = se2 = (se)e for all se ∈ se. the last statement can be verified using i). corollary 3.4. let {e1,e2, . . . ,er} be a set of primitive idempotents of r and e = e1 + e2 + · · · + er. then the following statements hold. i) e1,e2, . . . ,er are pairwise orthogonal (non-zero) idempotents of re. ii) ej is the identity of rej for all j ∈{1, 2, . . . ,r}. iii) e is the identity of re. iv) re = re1⊕re2⊕···⊕rer, where rej is isomorphic to an extension field of fq for all j ∈{1, 2, . . . ,r}. let ω = {∑r j=1 aj ∣∣∣aj ∈ (sej)∗} ⊂ se. then we have the following results. lemma 3.5. let a = ∑l i=1 αiai ∈ s, where ai ∈ r, and let b ∈ r. then ra ⊆ sb if and only if ra1 + ra2 + · · · + ral ⊆ rb. 52 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 proof. assume that ra ⊆ sb. then a = bb for some b ∈ s. write b = ∑l i=1 αibi, where bi ∈ r. then ai = bbi for all i ∈{1, 2, . . . , l}. hence, we have l∑ i=1 riai = l∑ i=1 ribbi = ( l∑ i=1 ribi ) b ∈ rb for all ∑l i=1 riai ∈ ra1 + ra2 + · · · + ral. conversely, it suffices to show that a ∈ sb. since ra1 + ra2 + · · · + ral ⊆ rb, we have ai ∈ rb for all i ∈{1, 2, . . . , l}. then, for each i ∈{1, 2, . . . , l}, there exists ri ∈ r such that ai = rib. hence, a = l∑ i=1 αiai = l∑ i=1 αirib = ( l∑ i=1 αiri ) b ∈ sb as desired. lemma 3.6. let a = ∑l i=1 αiai ∈ se, where ai ∈ r. then a ∈ ω if and only if re = ra1 + ra2 + · · · + ral. proof. first, we note that ra ⊆ se since a ∈ se. then ra1 + ra2 + · · · + ral ⊆ re by lemma 3.5. assume that a ∈ ω. then a = a1 + a2 + · · · + ar, where aj ∈ (sej)∗. we have aej = aj 6= 0 for all j ∈ {1, 2, . . . ,r}. suppose that ra1 + ra2 + · · · + ral ( re. by corollary 3.4, we have re = re1 ⊕re2 ⊕···⊕rer. then ra1 + ra2 + · · · + ral ⊆ r̂ej = r(e−ej) for some j ∈{1, 2, . . . ,r}, where r̂ej := re1 ⊕···⊕rej−1 ⊕rej+1 ⊕···⊕rer. by lemma 3.5, we have 0 6= aj = aej ∈ ra ⊆ s(e−ej), a contradiction. therefore, ra1 + ra2 + · · · + ral = re. conversely, assume that re = ra1 + ra2 + · · ·+ ral. then ra ⊆ se by lemma 3.5. since a ∈ se, by theorem 3.3, we have a = a1 + a2 + · · · + ar, where aj ∈ sej for all j ∈{1, 2, . . . ,r}. suppose that aj = 0 for some j ∈{1, 2, . . . ,r}. then ra = r̂aj ⊆ ŝej = s(e−ej). by lemma 3.5, we have re = ra1 + ra2 + · · · + ral ⊆ r(e−ej) which is a contradiction. hence, aj ∈ (sej)∗ for all j ∈{1, 2, . . . ,r}. corollary 3.7. let a = ∑l i=1 αiai ∈ sej, where ai ∈ r. then a ∈ (sej) ∗ if and only if rej = ra1 + ra2 + · · · + ral. let j ∈ {1, 2, . . . ,r} and let kj denote the fq-dimension of ej. then rej is isomorphic to a finite field of qkj elements. define an equivalence relation on (sej)∗ by a ∼ b ⇐⇒ ∃u ∈ (rej)× such that a = ub. for a ∈ (sej)∗, denote by [a] the equivalence class of a and let [(sej)∗] = {[a] | a ∈ (sej)∗}. lemma 3.8. let j ∈{1, 2, . . . ,r}. then |[a]| = qkj − 1 for all a ∈ (sej)∗. 53 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 proof. let a ∈ (sej)∗ and define ρ : (rej)× → [a], u 7→ ua. from the definition of ∼, ρ is a well-defined surjective map. for each u1,u2 ∈ (rej)×, if u1a = u2a, then (u1 −u2)a = 0. write a = ∑l i=1 αiai, where ai ∈ r. then ai(u1 −u2) = 0 for all i ∈{1, 2, . . . , l}. since a ∈ (sej)∗, by corollary 3.7, we can write ej = ∑i i=1 riai, where ri ∈ r. hence, ej(u1 −u2) = ( i∑ i=1 riai ) (u1 −u2) = i∑ i=1 riai(u1 −u2) = 0 ∈ rej. since ej is the identity of rej, it follows that u1 = u2 ∈ (rej)×. hence, ρ is a bijection. therefore, |[a]| = |(rej)×| = |f∗ q kj | = qkj − 1. corollary 3.9. for each i ∈{1, 2, . . . ,r}, we have |[(sej)∗]| = |(sej)∗| |[a]| = qlkj − 1 qkj − 1 . let [ω] = r∏ j=1 [(sej) ∗]. then |[ω]| = r∏ j=1 qlkj − 1 qkj − 1 . the number of 1-generator quasi-abelian codes sharing a idempotent has been determined in [7, corollary 6.1]. here, an alternative proof using a different technique is provided. theorem 3.10. let c denote the set of all 1-generator h-quasi-abelian codes in fq[g] with idempotent generator e. then there exists a one-to-one correspondence between [ω] and c. hence, the number of 1-generator quasi-abelian codes having e as their idempotent generator is r∏ j=1 qlkj − 1 qkj − 1 . proof. define σ : [ω] → c, ([a1], [a2], . . . , [ar]) 7→ ra, where a := a1 + a2 + · · · + ar ∈ se is viewed as a = ∑l i=1 αiai and a := (a1,a2, . . . ,al). since aj ∈ (sej)∗ for all j ∈ {1, 2, . . . ,r}, we have a ∈ ω. then re = ra1 + ra2 + · · · + ral by lemma 3.6, and hence, ra is a 1-generator quasi-abelian code with idempotent generator e, i.e., ra ∈ c. for ([a1], [a2], . . . , [ar]) = ([b1], [b2], . . . , [br]) ∈ [ω], there exists uj ∈ (rej)× such that aj = ujbj for all j ∈{1, 2, . . . ,r}. let u := u1 + u2 + · · · + ur. then u ( u−11 + u −1 2 + · · · + u −1 r ) = e1 + e2 + · · · + er = e is the identity of re (see corollary 3.4), where u−1j refers to the inverse of uj in rej. hence, u is a unit in (re)×. let b := ∑r j=1 bj. then a = r∑ j=1 aj = r∑ j=1 ujbj = ub. hence, ra = rb by lemma 3.2. therefore, σ is a well-defined map. 54 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 for ([a1], [a2], . . . , [ar]), ([b1], [b2], . . . , [br]) ∈ [ω], if ra = rb, then, by lemma 3.2, there exists u ∈ (re)× such that a = ub. then aj = ubj = uejbj since ej is the identity of sej by proposition 3.3. since aj ∈ (sej)∗, uej is a non-zero in rej which is a finite field. thus uej is a unit in (rej)×. hence, ([a1], [a2], . . . , [ar]) = ([b1], [b2], . . . , [br]) which implies that σ is an injective map. to verify that σ is surjective, let ra ∈ c, where a = (a1,a2, . . . ,al) ∈ rl. then re = ra1 + ra2 + · · · + ral. hence, by lemma 3.6, we conclude that a := l∑ i=1 αiai ∈ ω. write a = ∑r j=1 aj, where aj ∈ (sej) ∗. then ([a1], [a2], . . . , [ar]) ∈ [ω], and hence, σ(([a1], [a2], . . . , [ar])) = ra. 3.2. the generators for 1-generator quasi-abelian codes in this subsection, we establish an algorithm to find all 1-generator h-quasi-abelian codes in fq[g]. note that every idempotent in r := fq[h] can be written as a unique sum of primitive idempotents in r. hence, it is sufficient to study h-quasi-abelian codes of a given idempotent generator. let e = e1 + e2 + · · ·+ er be an idempotent in r, where, for each j ∈{1, 2, . . . ,r}, ej is the primitive idempotent in r induced by a q-cyclotomic class sq(hj) for some hj ∈ h. for each j ∈{1, 2, . . . ,r}, assume that ej is decomposed as ej = ej1 + ej2 + · · · + ejsj, where, for each i ∈ {1, 2, . . . ,sj}, eji is the primitive idempotent in s defined corresponding to a qlcyclotomic class sql (hji) for some hji ∈ sq(hj). note that all the elements in sq(hj) have the same order. hence, the ql-cyclotomic classes sql (hji) have the same size for all 1 ≤ i ≤ sj. without loss of generality, we assume that ej1 is defined corresponding to sql (hj). for each j ∈ {1, 2, . . . ,r}, let kj and dj denote the fq-dimension of ej and the fql-dimension of ej1, respectively. then kj and dj are the smallest positive integers such that qkj ·hj = hj and qldj ·hj = hj. then kj|ldj which implies that kj gcd(l,kj) |dj. since q l kj gcd(l,kj ) · hj = q kj l gcd(l,kj ) · hj = hj, we have dj| kj gcd(l,kj) . it follows that dj = kj gcd(l,kj) . hence, eji’s have the same ql-size dj = kj gcd(l,kj) and sj = gcd(l,kj). using arguments similar to those in the proof of proposition 3.3, we conclude the following result. proposition 3.11. let {e1,e2, . . . ,er} be a set of primitive idempotents of r. assume that ej = ej1 + ej2 + · · · + ejsj , where eji is a primitive idempotent in s for all i ∈ {1, 2, . . . ,sj}. then the following statements hold. i) for j ∈ {1, 2, . . . ,r}, the elements ej1,ej2, . . . ,ejsj are pairwise orthogonal (non-zero) idempotents of sej. ii) eji is the identity of seji for all j ∈{1, 2, . . . ,r} and i ∈{1, 2, . . . ,sj}. 55 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 iii) ej = ej1 + ej2 + · · · + ejsj is the identity of sej for all j ∈{1, 2, . . . ,r}. iv) for j ∈{1, 2, . . . ,r}, we have sej = sej1 ⊕sej2 ⊕···⊕sejsj , where seji is an extension field of fq of order qldj for all i ∈{1, 2, . . . ,sj}. theorem 3.12. let j ∈{1, 2, . . . ,r} be fixed. for i ∈{1, 2, . . . ,sj}, let πi be a primitive element of seji, a finite field of qldj elements. let lj = q ldj−1 q kj−1 and tj = {∞, 0, 1, 2, . . . ,qldj − 2}. then the elements πνtt + π νt+1 t+1 + · · · + π νsj sj , (3) for all 1 ≤ t ≤ sj, 0 ≤ νt ≤ lj − 1, and νt+1,νt+2, . . . ,νsj ∈ tj, are a complete set of representatives of [(sej) ∗]. (by convention, π∞i = 0.) proof. note that the number of elements in (3) is ljq ldj(sj−1) + ljq ldj(sj−2) + · · · + lj = qlkj − 1 qkj − 1 = |[(sej)∗]|. hence, it suffices to show that the elements in (3) are in different equivalence classes. let a = πνtt + π νt+1 t+1 + · · · + π νsj sj and b = π µx x + π µx+1 x+1 + · · · + π µsj sj , where 0 ≤ νt,µx ≤ lj−1, νt+1, νt+2, . . . ,νsj ∈ tj, and µx+1, µx+2, . . . ,µsj ∈ tj. assume that [a] = [b]. then there exists u ∈ (rej)× such that πνtt + π νt+1 t+1 + · · · + π νsj sj = a = ub = uπ µx x + uπ µx+1 x+1 + · · · + uπ µsj sj . since πνtt ∈ (sejt)× and uπµxx ∈ (sejx)×, by the decomposition in proposition 3.11, t = x and π νt t = uπ µt t ∈ sejt. then uejt = π νt−µt t . since u ∈ (rej)×, we have uq kj−1 = ej, and hence, ejt = ejtej = π (νt−µt)(q kj−1) t . since 0 ≤ νt,µt ≤ lj − 1 and πt has order qldj − 1, we conclude that νt = µt. hence, uejt = ejt = ejejt which implies (u−ej)ejt = 0 in sejt. it follows that s(u−ej) ⊆ s(ej1 + · · · + ej,t−1 + ej,t+1 + · · · + ejsj ) ( sej. since u,ej ∈ rej, we have u− ej ∈ rej and r(u− ej) ( rej. hence, r(u− ej) is the zero ideal, i.e., u = ej. therefore, a = ub = ejb = b since ej is the identity of sej. the following corollary now follows from theorem 3.10 and theorem 3.12. corollary 3.13. let {e1,e2, . . . ,er} be a set of primitive idempotents of r and e = e1 + e2 + · · · + er. then all 1-generator quasi-abelian codes having e as their idempotent generator are of the form a1 + a2 + · · · + ar, where aj ∈ (sej)∗ is as defined in (3). combining the results above, we summarize the steps of finding all 1-generator h-quasi-abelian codes in fq[g] as in algorithm 1. we note that the 1-generator h-quasi-abelian codes in fq[g] are possible to determined using [7, theorem 6.1] which depend on linear codes of dimension 1 over various extension fields of fq. using this concept, the algorithm might look more tedious and complicated. an illustrative example for algorithm 1 is given as follows. example 3.14. let q = 2, g = z3 × z6 and h = z3 × 2z6. denote by a0 := (0, 0), a1 := (1, 0), a2 := (2, 0), a3 := (0, 2), a4 := (1, 2), a5 := (2, 2), a6 := (0, 4), a7 := (1, 4), and a8 := (2, 4), the elements in h. then l = [g : h] = 2 and the elements in h can be partitioned into the following 2-cyclotomic 56 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 for abelian groups h ≤ g and a finite field fq with gcd(q, |h|) = 1 and [g : h] = l, do the following steps. 1. compute the q-cyclotomic classes of h in g. 2. compute the set {e1, e2, . . . , er} of primitive idempotents of r = fq[h] (see [4, proposition ii.4]). 3. for each 1 ≤ j ≤ r, compute a set bj of a complete set of representatives of [(sej)∗] (see theorem 3.12). 4. compute the idempotents of r, i.e., the set t = { t∑ j=1 eij ∣∣∣∣∣1 ≤ t ≤ r and 1 ≤ i1 < i2 < · · · < it ≤ r } . 5. for each e = ∑t j=1 eij ∈ t , compute the 1-generator quasi-abelian codes having e as their idempotent generator of the form a1 + a2 + · · ·+ at, where aj ∈ bij (see corollary 3.13). 6. run e over all elements of t , the 1-generator h-quasi-abelian codes in fq[g] are obtained. algorithm 1. steps for determining all 1-generator h-quasi-abelian codes in fq[g] classes s2(a0) = {a0}, s2(a1) = {a1,a2}, s2(a3) = {a3,a6}, s2(a4) = {a4,a8}, and s2(a5) = {a7,a5}. from [4, proposition ii.4], we note that e1 =y a0 + y a1 + y a2 + y a3 + y a4 + y a5 + y a6 + y a7 + y a8, e2 =y a1 + y a2 + y a4 + y a5 + y a7 + y a8, e3 =y a3 + y a4 + y a5 + y a6 + y a7 + y a8, e4 =y a1 + y a2 + y a3 + y a4 + y a6 + y a8, e5 =y a1 + y a2 + y a3 + y a5 + y a6 + y a7 are primitive idempotents of r := f2[h] induced by s2(a0), s2(a1), s2(a3), s2(a4), and s2(a5), respectively. let e := e1 + e2 + e3. from theorem 3.10, it follows that the number of 1-generator h-quasi abelian codes in f2[g] with idempotent generator e is 3 · 5 · 5 = 75. let s := f4[h], where f4 = {0, 1,α,α2 = 1 + α}. then e2 = e21 + e22 and e3 = e31 + e32, where e21 =y a0 + α2y a1 + αy a2 + y a3 + α2y a4 + αy a5 + y a6 + α2y a7 + αy a8, e22 =y a0 + αy a1 + α2y a2 + y a3 + αy a4 + α2y a5 + 1y a6 + αy a7 + α2y a8, e31 =y a0 + y a1 + y a2 + α2y a3 + α2y a4 + α2y a5 + αy a6 + αy a7 + αy a8, e32 =y a0 + y a1 + y a2 + αy a3 + αy a4 + αy a5 + α2y a6 + α2y a7 + α2y a8 are primitive idempotents in s induced by 4-cyclotomic classes {a1}, {a2}, {a3} and {a6}, respectively. now, we have k1 = 1, k2 = k3 = 2, d1 = d2 = d3 = 1, s1 = 1, and s2 = s3 = 2. it follows that l1 = 22−1 2−1 = 3, l2 = l3 = 22−1 22−1 = 1, and t1 = t2 = t3 = {∞, 0, 1, 2}. then αe1, αe21, αe22, αe31, and αe32 are primitive elements of se1, se21, se22, se31, and se32, 57 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 respectively. therefore, we have that b1 = {e1,αe1,α2e1}, b2 = {e21,e21 + e22,e21 + αe22,e21 + α2e22, e22}, and b2 = {e31,e31 + e32,e31 + αe32,e31 + α2e32, e32} are complete sets of representatives of [(se1)∗], [(se2)∗], and [(se3)∗], respectively. hence, all the generators of the 75 1-generator h-quasi abelian codes in f2[g] with idempotent generator e are of the form a1 + a2 + a3, where ai ∈ bi for all i ∈{1, 2, 3}. in order to find permutation inequivalent 1-generator h-quasi abelian codes, the following theorem is useful. theorem 3.15. let h ≤ g be finite abelian groups of index [g : h] = l and let {αq i | 1 ≤ i ≤ l} be a fixed basis of fql over fq. if a = ∑l i=1 aiα qi ∈ se, then a and aq = ∑l i=1 a q iα qi+1 generate permutation equivalent h-quasi abelian codes (viewed in fq[g]) with the same idempotent generator. proof. let e be the idempotent generator of a quasi-abelian code ra. then ra q 1 + ra q 2 + · · · + ra q l ⊆ ra1 + ra2 + · · · + ral = re assume that e = ∑l i=1 riai, where ri ∈ r. it follows that e = eq = l∑ i=1 r q i a q i ∈ ra q 1 + ra q 2 + · · · + ra q l . hence, we have re = raq1 + ra q 2 + · · ·+ ra q l . therefore, a and a q generate 1-generator h-quasi-abelian codes with the same idempotent generator e. let ψ : r → r be a ring homomorphism defined by γ 7→ γq. let γ = ∑ h∈h γhy h and β = ∑ h∈h βhy h be elements in r, where γh and βh are elements in fq. if ψ(γ) = ψ(β), then 0 = γq −βq = (γ −β)q = ∑ h∈h (γh −βh)y q·h. by comparing the coefficients, we have γh = βh for all h ∈ h, i.e., γ = β. hence, ψ is a ring automorphism and r(a q l ,a q 1, . . . ,a q l−1) = r(ψ(al),ψ(a1), . . . ,ψ(al−1)) = ψ(r(al,a1, . . . ,al−1)), (4) where ψ is a natural extension of ψ to rl. since ψ(γ) = ∑ h∈h γhy q·h, ψ(γ) is just a permutation on the coefficients of γ. hence, by (4), ψ ◦ φ is a permutation on fq[g] such that φ−1 ( r(a q l ,a q 1, . . . ,a q l−1) ) is permutation equivalent to φ−1 (r(al,a1, . . . ,al−1)) in f[g], where φ is the r-module isomorphism defined in (1). therefore, the result follows since r(al,a1, . . . ,al−1) is permutation equivalent to r(a1,a2, . . . ,al). 58 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 4. computational results it has been shown in [6] and [7] that a family of quasi-abelian codes contains various new and optimal codes. here, we present other 2 new codes from 1-generator quasi-abelian codes together with 1 new code obtained by shortening of one of these codes. given an abelian group h = zn1 ×zn2 of order n = n1n2, denote by u = (u0,u1,u2, . . . ,un−1) ∈ fnq the vector representation of u = n2−1∑ j=0 n1−1∑ i=0 ujn1+iy (i,j) in fq[h]. let c(a,b) := {(fa,fb) | f ∈ fq[h]}, (5) where a and b are elements in fq[h]. using (5), 2 quasi-abelian codes whose minimum distance improves on grassl’s online table [5] can be found. the codes c1 and c2 are presented in table 1 and the generator matrices of c1 and c2 are g1 =   1 3 0 3 4 1 3 2 0 4 1 2 1 4 0 4 1 0 4 3 0 4 1 3 4 4 3 1 4 0 2 4 1 3 0 2 2 4 3 1 1 3 4 0 1 4 4 3 4 0 4 0 0 1 0 3 1 2 0 1 0 3 2 4 4 4 4 4 3 3 4 2 3 3 1 3 4 0 3 3 2 1 1 1 1 0 3 0 4 3 3 4 3 2 4 2 3 2 3 2 2 3 0 3 2 1 0 1 4 3 4 4 2 4 4 1 4 1 2 4 2 1 4 0 0 1 1 2 0 4 0 4 0 2 1 1 3 1 4 1 1 2 1 0 1 1 4 2 0 0 1 3 2 3 i14 0 1 2 1 4 3 1 2 1 1 1 1 0 2 1 4 1 0 0 3 3 2 0 1 1 2 1 4 3 1 2 1 0 1 1 4 2 1 0 1 0 2 3 3 1 2 2 2 3 4 4 4 4 1 3 1 4 4 3 3 1 0 1 2 2 4 1 2 3 1 4 0 2 2 4 3 4 0 4 1 2 2 0 1 1 3 3 2 1 1 3 2 2 1 3 4 2 3 4 1 3 0 4 1 0 0 2 1 4 3 4 0 4 1 0 3 2 4 0 1 0 3 2 2 2 1 1 0 4 1 4 0 4 1 4 0 2 3 0 0 4 1 2 3 0 3 4 3 0 1 4 1 0 4   and g2 =   0 1 0 4 4 0 0 1 4 4 0 4 1 3 2 3 3 1 1 3 3 2 0 1 4 4 4 1 1 2 1 2 4 1 4 3 2 1 4 4 3 2 4 2 0 1 1 0 1 2 1 0 4 0 0 0 4 4 4 1 4 1 0 2 3 3 1 1 3 3 2 3 1 4 0 0 1 0 0 4 0 4 1 0 3 1 3 0 3 1 4 1 3 4 1 4 3 3 4 1 4 4 0 0 0 0 1 1 4 3 3 4 1 4 3 1 4 1 3 0 3 1 3 0 1 i11 1 0 0 0 0 4 4 0 3 1 3 0 1 1 4 3 3 4 1 4 3 1 4 1 3 1 1 4 0 4 0 4 3 2 1 0 0 4 1 3 1 2 3 3 2 3 4 2 4 2 4 0 0 4 0 0 1 4 1 0 2 3 3 1 1 3 3 2 3 1 4 0 4 4 1 0 4 1 1 2 1 1 2 1 3 2 1 2 4 2 2 4 4 3 1 2 0 0 3 3 1 1 0 0 4 4 4 2 2 2 2 2 2 0 0 0 0 0 0 3 3 3 3 3 3 0 0 1 1 1 1 1 2 2 2 4 4 4 1 1 1 1 1 1 1 1 1 4 4 4   , respectively. by puncturing c2 at the first coordinate, a [35, 11, 17]5 code can be obtained with minimum distance improved by 1 from grassl’s online table [5]. all the computations are done using magma [3]. acknowledgment: the authors thank to san ling for useful discussions and to the anonymous referees for their helpful comments. 59 s. jitman, p. udomkavanich / j. algebra comb. discrete appl. 4(1) (2017) 49–60 table 1. new codes from quasi-abelian codes name c(a,b) h a and b c1 [36, 14, 15]5 z3 ×z6 a = (3, 3, 3, 0, 0, 1, 4, 3, 4, 0, 4, 4, 4, 4, 3, 0, 1, 0) b = (2, 4, 1, 1, 3, 3, 0, 0, 4, 4, 1, 0, 0, 1, 4, 2, 2, 4) c2 [36, 11, 18]5 z3 ×z6 a = (2, 4, 4, 3, 4, 4, 3, 2, 4, 3, 4, 4, 3, 4, 2, 3, 4, 4) b = (3, 0, 0, 0, 3, 3, 3, 0, 3, 0, 3, 0, 1, 1, 1, 1, 1, 1) references [1] s. d. berman, semi–simple cyclic and abelian codes. ii, kibernetika 3(3) (1967) 21–30. [2] s. d. berman, on the theory of group codes, kibernetika 3(1) (1967) 31–39. [3] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symbolic comput. 24(3–4) (1997) 235–265. [4] c. ding, d. r. kohel, s. ling, split group codes, ieee trans. inform. theory 46(2) (2000) 485–495. [5] m. grassl, bounds on the minimum distance of linear codes and quantum codes, online available at http://www.codetables.de, accessed on 2015-10-09. [6] s. jitman, generator matrices for new quasi–abelian codes, online available at https://sites.google.com/site/quasiabeliancodes, accessed on 2015-10-09. [7] s. jitman, s. ling, quasi–abelian codes, des. codes cryptogr. 74(3) (2015) 511–531. [8] k. lally, p. fitzpatrick, algebraic structure of quasicyclic codes, discrete appl. math. 111(1–2) (2001) 157–175. [9] s. ling, p. solé, on the algebraic structure of quasi–cyclic codes i: finite fields, ieee trans. inform. theory 47(7) (2001) 2751–2760. [10] s. ling, p. solé, good self–dual quasi–cyclic codes exist, ieee trans. inform. theory 49(4) (2003) 1052–1053. [11] s. ling, p. solé, on the algebraic structure of quasi–cyclic codes iii: generator theory, ieee trans. inform. theory 51(7) (2005) 2692–2700. [12] f. j. macwilliams, n. j. a. sloane, the theory of error–correcting codes, amsterdam, the netherlands: north–holland, 1977. [13] j. pei, x. zhang, 1−generator quasi–cyclic codes, j. syst. sci. complex. 20(4) (2007) 554–561. [14] g. e. seguin, a class of 1−generator quasi–cyclic codes, ieee trans. inform. theory 50(8) (2004) 1745–1753. [15] s. k. wasan, quasi abelian codes, pub. inst. math. 21(35) (1977) 201–206. 60 http://dx.doi.org/10.1007/bf01119999 http://dx.doi.org/10.1007/bf01072842 http://dx.doi.org/10.1006/jsco.1996.0125 http://dx.doi.org/10.1006/jsco.1996.0125 http://dx.doi.org/10.1109/18.825811 http://www.codetables.de http://www.codetables.de https://sites.google.com/site/quasiabeliancodes https://sites.google.com/site/quasiabeliancodes http://dx.doi.org/10.1007/s10623-013-9878-4 http://dx.doi.org/10.1016/s0166-218x(00)00350-4 http://dx.doi.org/10.1016/s0166-218x(00)00350-4 http://dx.doi.org/10.1109/18.959257 http://dx.doi.org/10.1109/18.959257 http://dx.doi.org/10.1109/tit.2003.809501 http://dx.doi.org/10.1109/tit.2003.809501 http://dx.doi.org/10.1109/tit.2005.850142 http://dx.doi.org/10.1109/tit.2005.850142 http://dx.doi.org/10.1007/s11424-007-9053-y http://dx.doi.org/10.1109/tit.2004.831861 http://dx.doi.org/10.1109/tit.2004.831861 http://www.ams.org/mathscinet-getitem?mr=469498 introduction preliminaries 1-generator quasi-abelian codes computational results references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327373 j. algebra comb. discrete appl. 4(3) • 247–260 received: 23 january 2016 accepted: 23 january 2017 journal of algebra combinatorics discrete structures and applications gaussian elimination in split unitary groups with an application to public-key cryptography∗ research article ayan mahalanobis, anupam singh abstract: gaussian elimination is used in special linear groups to solve the word problem. in this paper, we extend gaussian elimination to split unitary groups. these algorithms have an application in building a public-key cryptosystem, we demonstrate that. 2010 msc: 20h30, 94a60 keywords: unitary groups, gaussian elimination, row-column operations 1. introduction gaussian elimination is a very old theme in computational mathematics. it was developed to solve linear simultaneous equations. the modern day matrix theoretic approach was developed by john von neumann and the popular textbook version by alan turing. gaussian elimination has many applications and is a very well known mathematical method. we will not elaborate on it any further, but will refer an interested reader to a nice article by grcar [10]. the way we look at gaussian elimination is: it gives us an algorithm to write any matrix of the general linear group, gl(d,k), of size d over a field k as the product of elementary matrices and a diagonal matrix with all ones except one entry, using elementary operations. that entry in the diagonal is the determinant of the matrix. there are many ways to look at this phenomena. one simple way is: one can write the matrix as a word in generators. so in the language of computational group theory the word problem in gl(d,k) has an efficient algorithm – gaussian elimination. we write this paper to say that one can have a very similar result with split unitary groups as well. it is well known that unitary groups over a finite field are split. so we completely solve the problem for unitary groups over finite fields for most characteristics. however, over infinite fields, our algorithm works only for the split case. split unitary groups are defined by the hermitian form with maximal witt ∗ this work was supported by a serb research grant ms: 831/13. ayan mahalanobis (corresponding author), anupam singh; iiser pune, dr. homi bhabha road, pashan, pune 411008, india (email: ayan.mahalanobis@gmail.com, anupamk18@gmail.com). 247 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 index. from now on, by a unitary group we mean a split unitary group. we define elementary matrices and elementary operations for unitary groups. these matrices and operations are similar to that of elementary transvections and elementary row-column operations for special linear groups. using these elementary matrices and elementary operations, we solve the word problem in unitary groups in a way that is very similar to the general linear groups. similar algorithms are being developed for other classical groups and will be presented elsewhere. unitary groups are of interest in computational group theory, in the matrix group recognition project. in this paper, we work with a different set of generators than that is usual in computational group theory. the usual generators are called the standard generators [14, tables 1 & 2]. our generators, we call them elementary matrices and are defined later, have their root in the root spaces in lie theory [6, sections 11.3, 14.5] and have the disadvantage of being a larger set compared to that of the standard generators. however, standard generators being “multiplicative” in nature, depends on the primitive element of a finite field, works only for finite fields. on the other hand, our generators, work for arbitrary fields. using standard generators, one needs to solve the discrete logarithm problem often. no such need arises in our case. in the current literature, the best row-column operations in unitary groups is by costi [8] and implemented in magma [3] by costi and c. schneider. using their magma function classicalrewritenatural, we show that our algorithm is much faster, see figure 1. in costi’s algorithm one needs to compute various powers of ω a primitive element of the finite field. this makes his algorithm slower. a need for row-column operations in classical groups was articulated by seress [20, page 677] in 1997. computational group theory and in particular constructive recognition of classical groups have come a long way till then. we will not give a historical overview of this, an interested reader can find such an overview in the works of brooksbank [5, section 1.1], leedham-green and o’brien [14, section 1.3] and o’brien [18]. two recent works that are relevant to our work are costi [8] and ambrose et. al. [1]. brooksbank [4, section 5] deals with a similar algorithm which only works for finite fields. in coming years, public key cryptography will go through a major change because of quantum computers. the ubiquitous public key cryptosystems like the elgamal cryptosystem over elliptic curves and rsa will become obsolete. the need of the day are new public key cryptosystems whose security does not rely on the discrete logarithm problem or factoring integers. we study mor cryptosystem on various groups with the hope to discover new quantum-secure cryptographic primitives. in this paper, we only deal with unitary groups defined by the hermitian form β defined later. the hermitian form for the even-order case works for all characteristic. however, in the odd-order case the 2 in the upper-left makes it useless in the even characteristic. one can change this 2 to a 1 in β, however, then one needs to compensate that by putting 1 2 in the generators. we tried, but were unable to extend our algorithm for the odd-order unitary group to even characteristic. for even-order unitary groups, the algorithm developed in this paper works for all characteristic. however, for the odd-order case only odd characteristic will be considered. 1.1. notations for the rest of the paper, let k be the quadratic extension of a field k with an automorphism σ : x 7→ x̄ of order two that fixes k elementwise. in the case of c : r, σ is the complex conjugation. in the case of a finite field fq2 : fq, σ is the map x 7→ xq. we define ko = {x ∈ k | x̄ = −x}. we also denote k1 = {x ∈k | xx̄ = 1}. a d×d matrix x is called hermitian (skew-hermitian) if tx̄ = x (tx̄ = −x). two important examples of k : k pairs that we have in mind for this work are c : r and fq2 : fq. the main result that we prove in this paper follows. the result is well known, however the algorithmic proof of the result is original. moreover, this algorithm is of independent interest in other areas, for example, constructive recognition of classical groups. for a definition of elementary matrices and elementary operations, see section 3. theorem a. for d ≥ 4, using elementary operations, one can write any matrix a in u(d,k), the unitary group of size d over k, as product of elementary matrices and a diagonal matrix. the diagonal 248 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 matrix is of the following form: •   1 ... 1 λ 1 ... 1 λ̄−1   where λλ̄−1 = deta and d = 2l. •   α 1 ... 1 λ 1 ... 1 λ̄−1   where αᾱ = 1 and αλλ̄−1 = deta and d = 2l + 1. here λ̄ is the image of λ under the automorphism σ. a trivial corollary (theorem 6.1) of our algorithm is very similar to a result by steinberg [21, §6.2], where he describes the generators of a projective-unitary group over odd characteristic. our work is somewhat similar in nature to the work of cohen et. al. [7], where the authors study generalized rowcolumn operations in chevalley groups. they did not study twisted groups. we use the algorithm developed to construct a mor cryptosystem in unitary groups and study its security. 2. unitary groups let k be a field with a non-trivial field automorphism σ of order 2 with fixed field k. let v be a vector space of dimension d over k. we denote the image of α under σ by ᾱ. let β : v ×v → k be a non-degenerate hermitian form, i.e., bar-linear in the first coordinate and linear in the second coordinate satisfying β(x,y) = β(y,x). we fix a basis for v and slightly abuse the notation to denote the matrix of β by β. thus β is a non-singular matrix satisfying β = tβ̄. definition 2.1 (unitary group). the unitary group is: u(d,k) = {x ∈ gl(d,k) | tx̄βx = β}. the special unitary group su(d,k) consists of matrices of u(d,k) of determinant 1. note that the unitary group depends on the hermitian form β. it is known that corresponding to equivalent hermitian forms, corresponding unitary groups are conjugate in gl(d,k). however over a infinite field there could be more than one non-equivalent nondegenerate hermitian form giving rise to more than one non-isomorphic unitary groups. in this article, we deal with a specific form β and the corresponding split unitary group. recall, we assumed that 249 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 characteristic of k is odd whenever d is odd. for the convenience of computations we index the basis of the vector space by 1, . . . , l,−1, . . . ,−l when d = 2l and by 0, 1, . . . , l,−1, . . . ,−l when d = 2l + 1; where l > 1. we also fix the matrix β as follows: • d = 2l fix β = ( il il ) . • d = 2l + 1 fix β =  2 il il  . there are two important examples of fields: complex numbers c over reals r with σ the complex conjugation and the other, finite field fq2 over fq with σ : α 7→ αq. in the case of c : r, hermitian forms are classified by signatures and unitary groups denoted by u(p,q) where p + q = d (see [12] discussion following theorem 6.19). the form corresponding to p being maximum is the split hermitian form and is of interest in this paper. however there is only one non-degenerate hermitian form up to equivalence [11, corollary 10.4] over finite fields. in this case a unitary group will be denoted by u(d,q2) and special unitary group as su(d,q2). a word of caution: in the literature u(d,q2), u(d,fq) and u(d,q) are used interchangeably. 3. elementary matrices and elementary operations in unitary groups solving the word problem in any group is of interest in computational group theory. in a special linear group, it can be easily solved using gaussian elimination. however, for many groups, it is a very hard problem. in this paper we present a fast, cubic-time solution to the word problem in unitary groups. gaussian elimination in sl(d,k) uses elementary transvections as the elementary matrices and row-column operations as elementary operations. these elementary operations are multiplication by elementary matrices. the elementary matrices are of the form i + tei,j (t ∈k), where ei,j is the matrix unit with 1 in the (i,j)th position and zero elsewhere. in the same spirit, one can define chevalley-steinberg generators for the unitary group [6, section 14.5] as follows: 3.1. elementary matrices for u(2l,k) in what follows, l ≥ 2. for 1 ≤ i,j ≤ l, t ∈k and s ∈ko: xi,j(t) = i + tei,j − t̄e−j,−i for i 6= j, xi,−j(t) = i + tei,−j − t̄ej,−i for i < j, x−i,j(t) = i + te−i,j − t̄e−j,i for i < j, xi,−i(s) = i + sei,−i, x−i,i(s) = i + se−i,i, 3.2. row-column operations for u(2l,k) rephrasing the earlier definition in matrix format, we have three kinds of elementary matrices. 250 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 e1: ( r tr̄−1 ) where r = i + tei,j; i 6= j. e2: ( i r i ) where r is either tei,j − t̄ej,i; i < j or sei,i. e3: ( i r i ) where r is either tei,j − t̄ej,i; i < j or sei,i. let g = ( a b c d ) be a 2l× 2l matrix written in block form of size l× l. note the effect of multiplying g by matrices from above. er1 : ( r tr̄−1 )( a b c d ) = ( ra rb tr̄−1c tr̄−1d ) . ec1 : ( a b c d )( r tr̄−1 ) = ( ar btr̄−1 cr dtr̄−1 ) . er2 : ( i r i )( a b c d ) = ( a + rc b + rd c d ) . ec2 : ( a b c d )( i r i ) = ( a ar + b c cr + d ) . er3 : ( i r i )( a b c d ) = ( a b ra + c rb + d ) . ec3 : ( a b c d )( i r i ) = ( a + br b c + dr d ) . 3.3. elementary matrices for u(2l + 1,k) for l ≥ 2, 1 ≤ i,j ≤ l, t ∈k, s ∈ko and characteristic of k odd xi,j(t) = i + tei,j − t̄e−j,−i for i 6= j, xi,−j(t) = i + tei,−j − t̄ej,−i for i < j, x−i,j(t) = i + te−i,j − t̄e−j,i for i < j, xi,−i(s) = i + sei,−i, x−i,i(s) = i + se−i,i, xi,0(t) = i − 2t̄ei,0 + te0,−i − tt̄ei,−i, x0,i(t) = i + te0,i − 2t̄e−i,0 − tt̄e−i,i, 3.4. row-column operations for u(2l + 1,k) rephrasing in matrix format: 251 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 e1:  1 r tr̄−1   where r = i + tei,j; i 6= j. e2:  1 i r i   where r is either tei,j − t̄ej,i; i < j or sei,i. e3:  1 i r i   where r is either tei,j − t̄ej,i; i < j or sei,i. e4:     1 r−2r̄ i −trr̄ i   where r = tei   1 ri −2r̄ −trr̄ i   where r = tei here ei is the row vector with 1 at ith place and zero elsewhere. let g =  α x ye a b f c d   be a (2l+1)×(2l+1) matrix where a,b,c,d are l × l matrices. the matrices x = (x1,x2, . . . ,xl), y = (y1,y2, . . . ,yl), e = t(e1,e2, . . . ,el) and f = t(f1,f2, . . . ,fl) are rows of length l. furthermore α ∈k. note the effect of multiplication by elementary matrices from above is as follows: er1 :  1 r tr̄−1    α x ye a b f c d   =   α x yre ra rb tr̄−1f tr̄−1c tr̄−1d   . ec1 :  α x ye a b f c d    1 r tr̄−1   =  α xr y tr̄−1 e ar btr̄−1 f cr dtr̄−1   . er2 :  1 i r i    α x ye a b f c d   =   α x ye + rf a + rc b + rd f c d   . ec2 :  α x ye a b f c d    1 i r i   =  α x xr + ye a ar + b f c cr + d   . er3 :  1 i r i    α x ye a b f c d   =   α x ye a b re + f ra + c rb + d   . ec3 :  α x ye a b f c d    1 i r i   =  α x + y r ye a + br b f c + dr d   . for e4 we only write the equations that we need later. 252 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 • let the matrix g has c = diag(d1, . . . ,dl). er4 : [(i + te0,−i − 2t̄ei,0 − tt̄ei,−i)g]0,i = xi + tdi ec4 : [g(i + te0,−i − 2t̄ei,0 − tt̄ei,−i)]−i,0 = fi − 2t̄di. • let the matrix g has a = diag(d1, . . . ,dl). er4 : [(i + te0,i − 2t̄e−i,0 − tt̄e−i,i)g]0,i = xi + tdi ec4 : [g(i + te0,−i − 2t̄ei,0 − tt̄ei,−i)]i,0 = ei − 2t̄di. 3.5. row-interchange matrices we need certain row interchange matrices, multiplication with these matrices from left, interchanges ith row with −ith row for 1 ≤ i ≤ l. these are certain weyl group elements. these matrices can be produced as follows: for s ∈ko, wi,−i(s) = xi,−i(s)x−i,i(−s−1)xi,−i(s) = i + sei,−i −ei,i −s−1e−i,i −e−i,−i. note that our row interchange multiplies one row by s and the other by −s−1 and then swaps them. this scalar multiplication of rows produce no problem for our cause. 4. gaussian elimination in unitary group now we present the main result of this paper, two algorithms, one for even-order unitary groups and other for the odd-order unitary groups. 4.1. the algorithm for even-order unitary groups let g = ( a b c d ) be an element of the unitary group u (2l,k). one principal reason our algorithm works is that we are able to exploit a symmetry that comes out of the use of the hermitian form β described earlier. notice that, tḡβg = β implies after straightforward computations that tc̄a + tāc = 0 and td̄b + tb̄d = 0. this implies that tāc and tb̄d are skew-hermitian matrices. we now describe the algorithm. step 1 using er1 and ec1 make a into a diagonal matrix. this is the usual gaussian elimination algorithm. this new diagonal matrix will be referred to a as well. there are two possibilities. a the diagonal matrix has full rank and is of the form diag(λ1,λ2, . . . ,λl), where each λi are non-zero. b the diagonal matrix a do not have a full rank and is of rank r less than l and is of the form diag(λ1,λ2, . . . ,λr, 0, 0 . . . , 0). step 2 in case a, use er3 to make c into a zero matrix. in case b, bottom l − r rows in a are zero. make the top r rows of c zero using er3. we now interchange bottom l − r rows of a with the corresponding l−r rows of c. this makes c a zero matrix. we claim that a must be of full rank. we know that tc̄b + tād = i. since, c = 0, a must be of full rank. we now go back to step 1 and make a a diagonal matrix. 253 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 step 3 note that td̄a + tb̄c = i and since c is zero and a is diagonal makes d a diagonal matrix of full rank l. using this diagonal matrix, we can make b a zero matrix using er2. so now we have a diagonal matrix instead of g. step 4 in this step we only work with a. using er1 and ec1, we reduce all diagonal elements of a to 1 except the last. 4.2. the algorithm for odd-order unitary groups recall that for odd-order unitary groups, we assumed the characteristic of k to be odd. let g = α x ye a b f c d   be an element of the unitary group u(2l + 1,k). as with even-order case, our algorithm exploits the symmetry that come out of the chosen bilinear form β. from the equation tḡβg = β we get two useful equations: 2tx̄x + tc̄a + tāc = 0 (1) 2tȳ y + td̄b + tb̄d = 0 (2) now if x = 0, tāc is skew-hermitian and similarly for the other case tb̄d is skew-hermitian. the reader will notice our insistence in making x zero as quickly as possible in the algorithm. once we do that rest of the algorithm is very similar to that of the even-order algorithm. the algorithm is as follows: step 1 using er1 and ec1, make a a diagonal matrix. this new diagonal matrix will be referred to a as well. two things can happen a the diagonal matrix a has full rank and is of the form diag(λ1,λ2, . . . ,λl) where λi are non-zero for 1 ≤ i ≤ l. b the matrix a is not of full rank, but of rank r and is diag(λ1,λ2, . . . ,λr, 0, . . . , 0) where λi are non-zero for 1 ≤ i ≤ r. step 2 the purpose of this step is to make x and e zero using a. in the case a above, this can be easily done using er4 and ec4 respectively. in the case b above, from lemma 5.1, if x1 = x2 = . . . = xr = 0 then x = 0. however x1,x2, . . . ,xr can be made zero as above and that will make whole of x zero. similarly, use the non-zero diagonals of a to make the corresponding entries of e zero. step 3 the purpose of this step is to make c zero matrix. we first use the non-zero diagonal entries in a to make the corresponding rows in c zero. if there are any zero rows in a, i.e., we are in case b of step 1, we use the row interchange operation to exchange the zero rows of a with the corresponding ones in c. then c is a zero matrix and that makes a of full rank. diagonalize a using er1 and ec1 and make e a zero matrix. this makes d a full-rank diagonal matrix. step 4 using er2 and d we make b a zero matrix. step 5 we now have a diagonal matrix, using er1 and ec1 we can make the diagonal entries of a except the last one 1. 5. some lemmas lemma 5.1. let g be an element of u(2l + 1,k) as described earlier. furthermore assume that a is of the form diag(λ1,λ2, . . . ,λr, 0, . . . , 0) where λi are non-zero and the row vector x has first r entries 0. then x is zero. 254 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 proof. notice that from the equation tḡβg = β, it follows that 2tx̄x + tc̄a + tāc = 0. then tx̄x is a matrix with the lower right (l−r) × (l−r) block possibly non-zero. however since, a is a diagonal matrix with lower l−r entries zero, it is clear from the equation that the possible non-zero block is zero and this proves that x = 0. lemma 5.2. once e, c and x are zero in g defined earlier, it follows that f and y are zero. proof. the important equation from tḡβg = β for this lemma is 2tx̄α + tc̄e + tāf = 0 from which it follows that f = 0. then y = 0 follows from 2ᾱy + tf̄b + ēd = 0 a cautious reader must have noticed that in the definition of the generators we define xi,−i(s) = i + sei,−i and x−i,i(s) = i + se−i,i where s ∈ ko. we need these generators to clear diagonal elements from c and b respectively. we only talk about clearing diagonal elements from c, b follows similarly. now if a is diagonal and λi is a non-zero element in the diagonal, then using ra + c we will clear ci,i in c. notice that from earlier discussion it follows λ̄ici,i = −c̄i,iλi. then take s = − ci,i λ and it is easy to check that it belongs to ko. 5.1. proof of theorem a proof. let g ∈ u(d,k). using the gaussian elimination above we can reduce g to a matrix of the form diag(1, . . . , 1,λ, 1, . . . , 1, λ̄−1) when d = 2l and diag(α, 1 . . . , 1,λ, 1, . . . , 1, λ̄−1) when d = 2l + 1. we further note that row-column operations are multiplication by elementary matrices from left or right and each of these elementary matrices have determinant one. thus we get the required result. 5.2. asymptotic complexity is o(l3) in this section, we show that the asymptotic complexity of the algorithm that we developed is o(l3). we count the number of field multiplications. we can break the algorithm into three parts. one, reduce a to a diagonal, then deal with c and then with d. it is easy to see that reducing a to the diagonal has complexity o(l3) and dealing with c and d has complexity o(l3). row interchange has complexity o(l2). in the odd-order case there is a complexity of o(l) to deal with x,y,e and f. in all, the worst case complexity is o(l3). 6. finite unitary groups in the next section, we talk about cryptography. in cryptography, we need to deal explicitly with finite fields. in this context, when k = fq2, we prove a theorem similar in spirit to steinberg [21, section 6.2]. the proof is an obvious corollary to our algorithm. theorem 6.1. fix an element ζ which generates the cyclic group f× q2 , the subgroup f1 q2 is generated by ζ1 = ζ q−1. we add following matrices to the respective set of elementary matrices: • h (ζ) = diag(1, . . . ,ζ, 1, . . . , ζ̄−1) whenever d = 2l. • { h (ζ) = diag(1, . . . ,ζ, 1, . . . , ζ̄−1) h (ζ1) = diag(ζ1, 1, . . . , 1) whenever d = 2l + 1. then the group u(d,q2) is generated by elementary matrices and the matrices defined above. 255 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 6.1. special unitary group su(d, q2) in the case of su(2l,q2) a simple and straightforward enhancement of our algorithm reduces a matrix g ∈ su(2l,q2) to the identity matrix. thus the word problem in su(2l,q2) is completely solved as with sl(d,q) using only elementary matrices; this is particularly useful for the mor cryptosystem. an analysis of a mor cryptosystem similar to the mor cryptosystem over sl(d,q) [15] will be done in the next section. for the reduction to identity, note that theorem 1.1 reduces g to diag(1, . . . , 1,λ, 1 . . . , 1, λ̄−1). however, since det(g) = λλ̄−1 = 1, we have λ = λ̄ and λ ∈ f×q . thus the word problem is completely solved for even characteristics. for odd characteristics, let s = ελ where ε ∈ko, then wl,−l(s)wl,−l(−ε) = diag(1, . . . , 1,λ, 1, . . . ,λ−1). so if we add wl,−l(s)wl,−l(−ε) to the output of theorem 1.1, we have the identity matrix. in the case of su(2l + 1,q2) we need to add an extra generator h(ζ1) = diag(ζ1, 1, . . . , 1) where ζ1 is a generator of f1q2. now we can reduce an element of the form diag(α, 1 . . . ,λ, 1, . . . , λ̄ −1) to diag(1, 1 . . . ,λ, 1, . . . , λ̄−1) by multiplying with the suitable power of h(ζ1). note that finding the suitable power involves solving a discrete logarithm problem. then we use similar computations for even-order case to reduce diag(1, 1 . . . ,λ, 1, . . . , λ̄−1) to identity. 7. the mor cryptosystem on unitary groups in this section, we will work with the mor cryptosystem over u(2l,q2) most of time. however, we will occasionally refer to the odd-order unitary group as well. briefly speaking, the mor cryptosystem is a simple and straightforward generalization of the classic elgamal cryptosystem and was put forward by paeng et. al. [19]. in a mor cryptosystem one works with the automorphism group rather than the group itself. it provides an interesting change in perspective in public-key cryptography – from finite cyclic groups to finite non-abelian groups. the mor cryptosystem was studied for the special linear group in details by mahalanobis [15]. for many other classical groups, except the orthogonal groups, the analysis of a mor cryptosystem remains almost the same. so we will remain brief in this paper and refer an interested reader to [15] (see also [16]). the description of the mor cryptosystem is as follows: let g = 〈g1,g2, . . . ,gs〉 be a finite group. let φ be a non-identity automorphism. • public-key: let {φ(gi)}si=1 and {φ m(gi)}si=1 is public. • private-key: the integer m is private. encryption: to encrypt a plaintext m ∈ g, get an arbitrary integer r ∈ [1, |φ|] compute φr and φrm. the ciphertext is (φr,φrm (m)). decryption: after receiving the ciphertext (φr,φrm (m)), the user knows the private key m. so she computes φmr from φr and then computes m. to develop a mor cryptosystem we need a thorough understanding of the automorphisms group of the group involved. the automorphisms of unitary groups are well described in the literature. we mention them briefly to facilitate further discussion. 7.1. automorphism group of unitary groups first we define the similitude group. we need these groups to define diagonal automorphisms. 256 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 definition 7.1 (unitary similitude group). the unitary similitude group is defined as: gu(d,q2) = {x ∈ gl(d,q2) | tx̄βx = µβ, for some µ ∈ f×q }. note that the multiplier µ defines a group homomorphism from gu(d,q2) to f×q with kernel the unitary group. conjugation automorphisms: when the conjugation map g 7→ ngn−1 for n ∈ gu(d,q2) is an automorphism, we call it a conjugation automorphism. they are a composition of two types of automorphisms – inner automorphisms given as conjugation by elements of u(d,q2) and diagonal automorphisms given as conjugation by diagonals of gu(d,q2). central automorphisms: let χ: u(d,q2) → f1 q2 be a group homomorphism. then the central automorphism cχ is given by g 7→ χ(g)g. since [u(d,q2),u(d,q2)] = [su(d,q2),su(d,q2)] = su(d,q2) [11, theorem 11.22], any χ is equivalent to a group homomorphism from u(d,q2)/su(d,q2) to f1 q2 . there are at most q + 1 such maps. field automorphisms: for any automorphism σ of the field fq2, replacing all entries of a matrix by their image under σ give us a field automorphism. the following theorem, due to dieudonné [9, theorem 25], describes all automorphisms: theorem 7.2. let q be odd and d ≥ 4. then any automorphism φ of the unitary group u(d,q2) is written as cχιδσ where cχ is a central automorphism, ι is an inner automorphism, δ is a diagonal automorphism and σ is a field automorphism. as we saw above there are three kind of automorphisms in an unitary group. one is conjugation automorphism, the others are central and field automorphisms. a central automorphism being multiplication by an element of the center which is a field element. exponentiation of a central automorphism will give rise to a discrete logarithm problem in fq2. similar is the case with a field automorphism. so the only choice for a better mor cryptosystem is a conjugation automorphism. once, we have decided that the automorphism that we are going to use in the mor cryptosystem will act by conjugation. further analysis is straightforward and follows [15, section 7]. recall that we insisted that automorphisms in the mor cryptosystem are presented as action on generators. in this case, the generators are elementary matrices and the group is a special unitary group of even-order. other groups can be used and analyzed similarly. note that two things can happen: one can find the conjugator element for the automorphism in use, finding the conjugator up to a scalar multiple is enough or one cannot find the conjugator in the automorphism. in the first case, the discrete logarithm problem in the automorphism becomes a discrete logarithm problem in a matrix group. assume that we found the conjugating matrix a up to a scalar multiple, where a ∈ gu(d,q2). now the discrete logarithm problem in φ becomes a discrete logarithm problem in a. one can show that by suitably choosing a, the discrete logarithm in a is embedded in the field fq2d. this argument is presented in details [15, section 7.1]. we will not repeat it here. in the next section (reduction of security), we show that one can find this conjugating element for unitary groups. the success of any cryptosystem comes from a balance between speed and security. in this paper, we deal with both speed and security of the mor cryptosystem briefly. for an implementation of the mor cryptosystem, we need to compute power of an automorphism. the algorithm of our choice is the famous square-and-multiply algorithm. since we do not use any special algorithm for squaring, squaring and multiplying is the same for us. so we talk about multiplying two automorphisms. we present automorphisms as action on generators, i.e., φ(gi) is a matrix for i = 1, 2, . . . ,s. the first step of the algorithm is to find the word in generators from the matrix1. so now the automorphism is φ(gi) = wi where each wi is a word in generators. once that is done then composing with an automorphism is 1 one can also present the automorphisms as word in generators, we choose matrices. 257 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 substituting each generator in the word by another word. this can be done fast. the challenging thing is to find the matrix corresponding to the word thus formed. this is not a hard problem, but can be both time and memory intensive. what is the best way to do it is still an open question! however, there are many shortcuts available. one being an obvious time-memory trade off, like storing matrices corresponding to a word in generators. the other being there are many trivial and non-trivial relations among these generators and moreover these generators are sparse matrices. one can use these properties in the implementation. this problem, which is of independent interest in computational group theory and is a reason that we insist on automorphisms being presented as generators for the mor cryptosystem. for more information, see [15, section 8]. 7.2. reduction of security in this subsection, we show that for unitary groups, the security of the mor cryptosystem reduces to the discrete logarithm problem in fq2d. this is the same as saying that we can find the conjugating matrix up to a scalar multiple. let φ be an automorphism that works by conjugation, i.e., φ = ιg for some g and we try to determine g. step 1: the automorphism φ is presented as action on generators. thus φ(xi,−i(s)) = g(i + sei,−i)g −1 = i + sgei,−ig −1. this implies that we know εgei,−ig−1 and similarly εge−i,ig−1 for a fixed ε ∈ko. we first claim that we can determine n := gd where d is diagonal. when d = 2l, write g in the column form [g1, . . .gl,g−1, . . . ,g−l]. now, 1. [g1, . . .gl,g−1, . . . ,g−l] εei,−i = [0, . . . , 0,εgi, 0, . . . , 0] where gi is at −ith place. multiplying this with g−1 gives us scalar multiple of gi, say di. 2. [g1, . . .gl,g−1, . . . ,g−l] εe−i,i = [0, . . . , 0,εg−i, 0, . . . , 0] where g−i is at ith place. multiplying this with g−1 gives us scalar multiple of g−i, say d−i. thus we get n = gd where d is a diagonal matrix diag(d1, . . . ,dl,d−1, . . . ,d−l). in the case when d = 2l + 1 we write g = [g0,g1, . . . ,gl,g−1, . . . ,g−l] and get scalar multiple of columns gi and g−i. we now use xi,0(t) and x0,i(t) to get linear combination of g0 with gi or g−i, say we get αg0 + βg−1. in this case we get n = gd where d is of the form  α d1 ... dl β d−1 ... d−l   . step 2: now we compute n−1φ(xr(t))n = d−1g−1(gxr(t)g−1)gd = d−1xr(t)d. substituting various xr(t) it amounts to computing d−1erd. when d = 2l, we first compute d−1(ei,j −e−j,−i)d and get d−1i dj, d −1 −id−j for i 6= j. then we compute d −1ei,−id,d −1e−i,id and get did −1 −i ,d−id −1 i . we form a matrix diag(1,d−12 d1, . . . ,d −1 l d1,d −1 −1d1, . . . ,d −1 −l d1) and multiply it to n = gd to get d1g. thus we can determine g up to a scalar multiple and the attack follows [15, section 7.1.1]. 258 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 figure 1. some simulations comparing our algorithm with the one inbuilt in magma. in the case d = 2l+1, the matrix d is almost a diagonal matrix except the first column. however while computing d−1(e12 −e−2,−1)d we also get d−1−2β and by computing d −1(e0,1 − 2e−1,0 −e−1,1)d we get α−1d. thus we can multiply αg0 +βg−1 by β−1d1 = β−1d−2d −1 −2d−1d −1 −1d1 and get αβ −1d1g0 +d1g−1. with the computation in even case we can determine d1g−1 and hence can determine αg0. furthermore, since we know α−1d1 we can determine d1g0 thus in this case as well we can determine d1g, i.e., g up to a scalar multiple. 8. conclusion for us, this paper is an interplay of finite (non-abelian) groups and public key cryptography. computational group theory, in particular computations with quasi-simple groups have a long and distinguished history [2, 7, 13, 14, 17]. the interesting thing to us is, some of the questions that arise naturally when dealing with the mor cryptosystem are interesting in its own right in computational group theory and are actively studied. the row-column operations that we developed is one example of that. in the rowcolumn operations we developed, we used a different set of generators. these generators have a long history starting with chevalley. in our knowledge, we are the first to use them in row-column operations in unitary groups. earlier works were mostly done using the standard generators. it seems that chevalley generators might offer a paradigm shift in algorithms with quasi-simple groups. in magma, there is an implementation of row-column operations in unitary groups in a function classicalrewritenatural. we compared that function with our algorithm in an actual implementation on even order unitary groups using identical parameters. to select parameters for our simulation, we followed costi’s work [8, table 6.2]. in one case, the characteristic of the field was fixed at 7 and the size of the matrix at 20, we varied the degree of the extension of the field from 4 to 34. we then picked at random elements from the generalunitarygroup and timed our algorithm. we did the same with the magma function using special unitary group. the final time was the average over one thousand such repetitions. times of both these algorithms were tabulated and is presented in figure 1. in another case, we kept the field fixed at 710 and changed the size of the matrix. in all cases, the final time was the average of one thousand random repetitions. the timing was tabulated and presented in figure 1. it seems the our algorithm is much better than that of costi’s from all aspects. acknowledgment: authors owe a debt of gratitude to the referee and the handling editor for careful reading and wonderful comments which has improved the paper substantially. 259 http://orcid.org/0000-0001-9822-3672 http://orcid.org/0000-0002-6162-4605 a. mahalanobis, a. singh / j. algebra comb. discrete appl. 4(3) (2017) 247–260 references [1] s. ambrose, s. murray, c. e. praeger, c. schneider, constructive membership testing in black–box classical groups, proceedings of the third international congress on mathematical software, lncs 6327 (2011) 54–57. 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http://www.ams.org/mathscinet-getitem?mr=1859189 http://www.ams.org/mathscinet-getitem?mr=780184 http://www.ams.org/mathscinet-getitem?mr=1804385 http://www.ams.org/mathscinet-getitem?mr=1804385 http://dx.doi.org/10.1016/j.jalgebra.2009.04.028 http://dx.doi.org/10.1016/j.jalgebra.2009.04.028 http://dx.doi.org/10.1080/00927872.2011.602998 http://dx.doi.org/10.1080/00927872.2011.602998 http://dx.doi.org/10.1080/00927872.2014.974254 https://doi.org/10.1112/s0024611598000422 https://doi.org/10.1112/s0024611598000422 http://www.ams.org/mathscinet-getitem?mr=2858866 http://www.ams.org/mathscinet-getitem?mr=2858866 http://dx.doi.org/10.1007/3-540-44647-8_28 http://dx.doi.org/10.1007/3-540-44647-8_28 http://www.ams.org/mathscinet-getitem?mr=1452069 http://www.ams.org/mathscinet-getitem?mr=1452069 http://www.ams.org/mathscinet-getitem?mr=109191 introduction unitary groups elementary matrices and elementary operations in unitary groups gaussian elimination in unitary group some lemmas finite unitary groups the mor cryptosystem on unitary groups conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.458601 j. algebra comb. discrete appl. 5(3) • 143–151 received: 21 august 2017 accepted: 13 june 2018 journal of algebra combinatorics discrete structures and applications new extremal singly even self-dual codes of lengths 64 and 66∗ research article damyan anev, masaaki harada, nikolay yankov abstract: for lengths 64 and 66, we construct six and seven extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist, respectively. we also construct new 40 inequivalent extremal doubly even self-dual [64, 32, 12] codes with covering radius 12 meeting the delsarte bound. these new codes are constructed by considering four-circulant codes along with their neighbors and shadows. 2010 msc: 94b05 keywords: self-dual code, weight enumerator 1. introduction a (binary) [n,k] code c is a k-dimensional vector subspace of fn2 , where f2 denotes the finite field of order 2. all codes in this note are binary. the parameter n is called the length of c. the weight wt(x) of a vector x is the number of non-zero components of x. a vector of c is a codeword of c. the minimum non-zero weight of all codewords in c is called the minimum weight of c. an [n,k] code with minimum weight d is called an [n,k,d] code. the dual code c⊥ of a code c of length n is defined as c⊥ = {x ∈ fn2 | x · y = 0 for all y ∈ c}, where x · y is the standard inner product. a code c is called self-dual if c = c⊥. a self-dual code c is doubly even if all codewords of c have weight divisible by four, and singly even if there is at least one codeword x with wt(x) ≡ 2 (mod 4). it is known that a self-dual code of length n exists if and only if n is even, and a doubly even self-dual code of length n exists if and only if n is divisible by 8. let c be a singly even self-dual code. let c0 denote the subcode of c consisting of codewords x with wt(x) ≡ 0 (mod 4). the shadow s of c is defined to be c⊥0 \ c. shadows for self-dual codes ∗ this work was supported by jsps kakenhi grant number 15h03633. damyan anev, nikolay yankov; faculty of mathematics and informatics, konstantin preslavski university of shumen, shumen, 9712, bulgaria (email: damian_anev@mail.bg, jankov_niki@yahoo.com). masaaki harada (corresponding author); research center for pure and applied mathematics, graduate school of information sciences, tohoku university, sendai 980–8579, japan (email: mharada@tohoku.ac.jp). 143 https://orcid.org/0000-0002-3175-0168 https://orcid.org/0000-0002-2748-6456 https://orcid.org/0000-0003-3703-5867 d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 were introduced by conway and sloane [6] in order to give the largest possible minimum weight among singly even self-dual codes, and to provide restrictions on the weight enumerators of singly even self-dual codes. the largest possible minimum weights among singly even self-dual codes of length n were given for n ≤ 72 in [6]. the possible weight enumerators of singly even self-dual codes with the largest possible minimum weights were given in [6] and [7] for n ≤ 72. it is a fundamental problem to find which weight enumerators actually occur for the possible weight enumerators (see [6]). by considering the shadows, rains [13] showed that the minimum weight d of a self-dual code of length n is bounded by d ≤ 4b n 24 c+6 if n ≡ 22 (mod 24), d ≤ 4b n 24 c+ 4 otherwise. a self-dual code meeting the bound is called extremal. the aim of this note is to construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. more precisely, we construct extremal singly even self-dual [64,32,12] codes with weight enumerators w64,1 for β = 35, and w64,2 for β ∈ {19,34,42,45,50} (see section 2 for w64,1 and w64,2). these codes are constructed as self-dual neighbors of extremal four-circulant singly even self-dual codes. we construct extremal singly even self-dual [66,33,12] codes with weight enumerators w66,1 for β ∈ {7,58,70,91,93}, and w66,3 for β ∈{22,23} (see section 2 for w66,1 and w66,3). these codes are constructed from extremal singly even self-dual [64,32,12] codes by the method given in [14]. we also demonstrate that there are at least 44 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the delsarte bound. all computer calculations in this note were done with the help of the algebra software magma [1] and the computer system q-extensions [2]. 2. weight enumerators of extremal singly even self-dual codes of lengths 64 and 66 the possible weight enumerators w64,i and s64,i of extremal singly even self-dual [64,32,12] codes and their shadows are given in [6]:{ w64,1 = 1 + (1312 + 16β)y 12 + (22016−64β)y14 + · · · , s64,1 = y 4 + (β −14)y8 + (3419−12β)y12 + · · · ,{ w64,2 = 1 + (1312 + 16β)y 12 + (23040−64β)y14 + · · · , s64,2 = βy 8 + (3328−12β)y12 + · · · , where β are integers with 14 ≤ β ≤ 104 for w64,1 and 0 ≤ β ≤ 277 for w64,2. extremal singly even self-dual codes with weight enumerator w64,1 are known for β ∈ { 14,16,18,20,22,24,25,26,28,29,30,32, 34,36,38,39,44,46,53,59,60,64,74 } (see [4], [10], [11] and [16]). extremal singly even self-dual codes with weight enumerator w64,2 are known for β ∈ { 0,1, . . . ,41,44,48,51,52,56,58,64,65,72, 80,88,96,104,108,112,114,118,120,184 } \{19,31,34,39} (see [4], [10], [16] and [18]). the possible weight enumerators w66,i and s66,i of extremal singly even self-dual [66,33,12] codes 144 d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 and their shadows are given in [7]:{ w66,1 = 1 + (858 + 8β)y 12 + (18678−24β)y14 + · · · , s66,1 = βy 9 + (10032−12β)y13 + · · · ,{ w66,2 = 1 + 1690y 12 + 7990y14 + · · · , s66,2 = y + 9680y 13 + · · · ,{ w66,3 = 1 + (858 + 8β)y 12 + (18166−24β)y14 + · · · , s66,3 = y 5 + (β −14)y9 + (10123−12β)y13 + · · · , where β are integers with 0 ≤ β ≤ 778 for w66,1 and 14 ≤ β ≤ 756 for w66,3. extremal singly even self-dual codes with weight enumerator w66,1 are known for β ∈{0,1, . . . ,92,94,100,101,115}\{4,7,58,70,91} (see [5], [8], [10], [17] and [18]). extremal singly even self-dual codes with weight enumerator w66,2 are known (see [8] and [15]). extremal singly even self-dual codes with weight enumerator w66,3 are known for β ∈{24,25, . . . ,92}\{65,68,69,72,89,91} (see [9], [10], [11] and [12]). 3. extremal four-circulant singly even self-dual [64, 32, 12] codes an n×n circulant matrix has the following form:  r0 r1 r2 · · · rn−1 rn−1 r0 r1 · · · rn−2 ... ... ... ... r1 r2 r3 · · · r0   , so that each successive row is a cyclic shift of the previous one. let a and b be n×n circulant matrices. let c be a [4n,2n] code with generator matrix of the following form:( i2n a b bt at ) , (1) where in denotes the identity matrix of order n and at denotes the transpose of a. it is easy to see that c is self-dual if aat + bbt = in. the codes with generator matrices of the form (1) are called four-circulant. two codes are equivalent if one can be obtained from the other by a permutation of coordinates. in this section, we give a classification of extremal four-circulant singly even self-dual [64,32,12] codes. our exhaustive search found all distinct extremal four-circulant singly even self-dual [64,32,12] codes, which must be checked further for equivalence to complete the classification. this was done by considering all pairs of 16 × 16 circulant matrices a and b satisfying the condition that aat + bbt = i16, the sum of the weights of the first rows of a and b is congruent to 1 (mod 4) and the sum of the weights is greater than or equal to 13. since a cyclic shift of the first rows gives an equivalent code, we may assume without loss of generality that the last entry of the first row of b is 1. then our computer search shows that the above distinct extremal four-circulant singly even self-dual [64,32,12] codes are divided into 67 inequivalent codes. proposition 3.1. up to equivalence, there are 67 extremal four-circulant singly even self-dual [64,32,12] codes. we denote the 67 codes by c64,i (i = 1,2, . . . ,67). for the 67 codes c64,i, the first rows ra (resp. rb) of the circulant matrices a (resp. b) in generator matrices (1) are listed in table 1. we verified that the codes c64,i have weight enumerator w64,2, where β are also listed in table 1. 145 d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 table 1. extremal four-circulant singly even self-dual [64, 32, 12] codes codes ra rb β c64,1 (0000001100111111) (0001011010101111) 0 c64,2 (0000010101111101) (0010011010111011) 0 c64,3 (0000011001101111) (0010110101011011) 0 c64,4 (0000000001011111) (0001001100101011) 8 c64,5 (0000000010101111) (0011011011110111) 8 c64,6 (0000000011010111) (0000100110011011) 8 c64,7 (0000000011010111) (0000101100010111) 8 c64,8 (0000000011010111) (0011101110101111) 8 c64,9 (0000000110111111) (0101101111111111) 8 c64,10 (0000001001011101) (0001000101011011) 8 c64,11 (0000001100011111) (0010101011011111) 8 c64,12 (0000001100011111) (0010111011011011) 8 c64,13 (0000001100111011) (0001101011101111) 8 c64,14 (0000001101111111) (0011101111011111) 8 c64,15 (0000010000111101) (0010111011011111) 8 c64,16 (0000010001011111) (0001110101101111) 8 c64,17 (0000010110111011) (0001101110001111) 8 c64,18 (0000000100011111) (0010111111110011) 16 c64,19 (0000000100111101) (0000101011000111) 16 c64,20 (0000000110010111) (0001001111111111) 16 c64,21 (0000000111001111) (0010101110111101) 16 c64,22 (0000000111001111) (0010110110111011) 16 c64,23 (0000001000101111) (0011101011110111) 16 c64,24 (0000001011100011) (0010101111110111) 16 c64,25 (0000001011100011) (0011011011111011) 16 c64,26 (0000010010011111) (0010110011101111) 16 c64,27 (0000011001101111) (0001001011011111) 16 c64,28 (0000011011011111) (0010010101011101) 16 c64,29 (0000011011100111) (0001011111001011) 16 c64,30 (0000011101111111) (0101101110110111) 16 c64,31 (0000101110111111) (0011101011110111) 16 c64,32 (0000000000100111) (0001011101101011) 24 c64,33 (0000000001011011) (0010010101101011) 24 c64,34 (0000000100111111) (0001001000101011) 24 c64,35 (0000000101001011) (0010010110011011) 24 c64,36 (0000000101001011) (0010011001011011) 24 c64,37 (0000000110111111) (0000001000100111) 24 c64,38 (0000001001111111) (0010101111001011) 24 c64,39 (0000001100011111) (0001010011111111) 24 c64,40 (0000001100011111) (0001110011110111) 24 c64,41 (0000010001011111) (0010101111001111) 24 c64,42 (0000010001101111) (0011001110101111) 24 c64,43 (0000010011101111) (0001011101100111) 24 c64,44 (0000010101010111) (0001010111101111) 24 c64,45 (0000010101010111) (0010110011111011) 24 c64,46 (0000010101110111) (0000101111110011) 24 c64,47 (0000010101110111) (0001011101101011) 24 c64,48 (0000011011110111) (0101101110111111) 24 c64,49 (0000000001001011) (0000111010110111) 32 c64,50 (0000000001100111) (0001001111100011) 32 146 d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 table 1. extremal four-circulant singly even self-dual [64, 32, 12] codes (continued) codes ra rb β c64,51 (0000001010111011) (0001011111100111) 32 c64,52 (0000010101011111) (0001101111000111) 32 c64,53 (0000010101111101) (0010110010110111) 32 c64,54 (0000011010111111) (0000101110011101) 32 c64,55 (0000101011101011) (0001011111001011) 32 c64,56 (0000000000100111) (0001011010111011) 40 c64,57 (0000000010101101) (0001001011011011) 40 c64,58 (0000001000011101) (0000100101111011) 40 c64,59 (0000001110011111) (0001010111101101) 40 c64,60 (0000011000111111) (0001010111101101) 40 c64,61 (0000011011001111) (0000101010111111) 40 c64,62 (0000100111011111) (0001010101011011) 40 c64,63 (0000001001101011) (0001010011001101) 48 c64,64 (0000000001011011) (0001011000101111) 56 c64,65 (0000010111011111) (0010100101011011) 56 c64,66 (0000101110011101) (0001000101111111) 64 c64,67 (0000000001011111) (0001011111110111) 72 4. extremal self-dual [64, 32, 12] neighbors of c64,i two self-dual codes c and c′ of length n are said to be neighbors if dim(c∩c′) = n/2−1. any selfdual code of length n can be reached from any other by taking successive neighbors (see [6]). since every self-dual code c of length n contains the all-one vector 1, c has 2n/2−1 − 1 subcodes d of codimension 1 containing 1. since dim(d⊥/d) = 2, there are two self-dual codes rather than c lying between d⊥ and d. if c is a singly even self-dual code of length divisible by 8, then c has two doubly even selfdual neighbors (see [3]). in this section, we construct extremal self-dual [64,32,12] codes by considering self-dual neighbors. for i = 1,2, . . . ,67, we found all distinct extremal singly even self-dual neighbors of c64,i, which are equivalent to none of the 67 codes. then we verified that these codes are divided into 385 inequivalent codes d64,i (i = 1,2, . . . ,385). these codes d64,i are constructed as 〈(c64,j ∩〈x〉⊥),x〉. to save space, the values j, the supports supp(x) of x, the values (k,β) in the weight enumerators w64,k are listed in “http://www.math.is.tohoku.ac.jp/~mharada/paper/64-se-d12.txt” for the 385 codes. for extremal singly even self-dual [64,32,12] codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist, j, supp(x) and (k,β) are list in table 2. hence, we have the following: proposition 4.1. there is an extremal singly even self-dual [64,32,12] code with weight enumerator w64,1 for β = 35, and w64,2 for β ∈{19,34,42,45,50}. now we consider the extremal doubly even self-dual neighbors of c64,i (i = 1,2,3). since the shadow has minimum weight 12, the two doubly even self-dual neighbors c164,i and c 2 64,i are extremal doubly even self-dual [64,32,12] codes with covering radius 12 (see [4]). thus, six extremal doubly even selfdual [64,32,12] codes with covering radius 12 are constructed. in addition, among the 385 codes d64,i (i = 1,2, . . . ,385), the 19 extremal singly even self-dual codes d64,j have shadow of minimum weight 12, where j ∈{1,2,12,19,22,33,44,58,66,68,84,95,108,115,136,143,191,240,254}. 147 http://www.math.is.tohoku.ac.jp/~mharada/paper/64-se-d12.txt d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 table 2. extremal singly even self-dual [64, 32, 12] neighbors codes j supp(x) (k,β) d64,138 24 {1,2,3,38,42,43,45,46,48,54,56,57} (2,19) d64,270 49 {1,2,8,32,38,41,48,49,50,53,55,61} (1,35) d64,283 52 {1,2,4,33,36,37,41,43,46,51,61,64} (2,42) d64,293 56 {3,7,9,10,11,37,43,53,57,58,62,64} (2,34) d64,314 64 {6,8,26,37,38,40,43,46,48,59,61,63} (2,50) d64,329 65 {1,6,8,9,37,47,50,52,57,60,63,64} (2,45) d64,1 1 {4,7,9,34,38,40,45,46,47,50,51,53} (2,0) d64,2 1 {3,37,38,47,48,50,52,53,54,59,60,63} (2,0) d64,12 4 {2,4,5,16,17,38,40,46,56,57,60,62} (2,0) d64,19 4 {2,3,6,7,9,35,41,49,55,56,57,61} (2,0) d64,22 4 {2,33,34,35,38,39,42,45,48,52,61,62} (2,0) d64,33 6 {8,9,10,16,17,33,44,45,54,55,59,61} (2,0) d64,44 6 {1,3,6,33,36,38,39,45,47,55,57,59} (2,0) d64,58 8 {1,3,5,16,17,35,36,38,42,44,54,59} (2,0) d64,66 8 {4,6,9,34,36,39,41,42,48,51,57,63} (2,0) d64,68 8 {3,6,9,33,36,37,38,49,56,57,60,62} (2,0) d64,84 13 {1,4,5,35,37,38,41,44,53,60,61,62} (2,0) d64,95 13 {2,4,9,34,35,40,42,47,49,52,59,64} (2,0) d64,108 15 {2,16,17,37,43,48,49,52,54,57,58,64} (2,0) d64,115 16 {1,3,6,7,8,41,45,46,49,50,57,60} (2,0) d64,136 21 {3,16,17,33,34,37,42,44,47,51,52,56} (2,0) d64,143 26 {1,2,9,34,37,38,41,48,57,58,59,64} (2,0) d64,191 35 {1,2,6,8,10,33,37,46,54,59,60,63} (2,0) d64,240 47 {2,4,7,9,13,16,17,44,56,59,62,64} (2,0) d64,254 48 {1,2,5,7,8,35,36,37,45,47,49,63} (2,0) d64,14 4 {1,7,8,35,36,37,41,43,46,49,51,53} (1,14) d64,383 67 {1,33,34,36,37,38,40,41,47,49,50,53,55,59,61,63} (2,40) the constructions of the 19 codes d64,j are listed in table 2. their two doubly even self-dual neighbors d164,j and d 2 64,j are extremal doubly even self-dual [64,32,12] codes with covering radius 12. we verified that there are the following equivalent codes among the four codes in [4], the six codes c164,i, c 2 64,i and the 38 codes d164,j, d 2 64,j, where d264,22 ∼= d 2 64,68,d 2 64,33 ∼= d264,84,d 2 64,44 ∼= d264,95,d 2 64,136 ∼= d264,143, where c ∼= d means that c and d are equivalent, and there is no other pair of equivalent codes. therefore, we have the following proposition. proposition 4.2. there are at least 44 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the delsarte bound. in order to distinguish two doubly even neighbors d164,i and d 2 64,i (i = 68,84,95,143), we list in table 3 the supports supp(x) for the 8 codes, where d164,i and d 2 64,i are constructed as 〈(d64,i∩〈x〉 ⊥),x〉. 148 d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 table 3. extremal doubly even self-dual [64, 32, 12] neighbors codes supp(x) d164,68 {1,4,7,34,35,36,47,54,55,58,60,63} d264,68 {1,4,5,6,30,42,45,47,54,56,58,64} d164,84 {16,17,33,39,43,46,48,49,51,54,58,64} d264,84 {1,2,6,33,35,38,40,42,52,57,59,60} d164,95 {1,2,6,33,35,38,40,42,52,57,59,60} d264,95 {3,33,38,41,45,47,51,53,58,60,62,64} d164,143 {1,4,10,40,43,46,52,54,58,61,62,63} d264,143 {1,31,34,42,44,45,46,50,51,52,54,62} 5. four-circulant singly even self-dual [64, 32, 10] codes and selfdual neighbors using an approach similar to that given in section 3, our exhaustive search found all distinct fourcirculant singly even self-dual [64,32,10] codes. then our computer search shows that the distinct four-circulant singly even self-dual [64,32,10] codes are divided into 224 inequivalent codes. proposition 5.1. up to equivalence, there are 224 four-circulant singly even self-dual [64,32,10] codes. we denote the 224 codes by e64,i (i = 1,2, . . . ,224). for the codes, the first rows ra (resp. rb) of the circulant matrices a (resp. b) in generator matrices (1) can be obtained from “http://www.math.is.tohoku.ac.jp/~mharada/paper/64-4cir-d10.txt”. the following method for constructing self-dual neighbors was given in [4]. for c = e64,i (i = 1,2, . . . ,224), let m be a matrix whose rows are the codewords of weight 10 in c. suppose that there is a vector x of even weight such that mxt = 1t . (2) then c0 = 〈x〉⊥ ∩c is a subcode of index 2 in c. we have self-dual neighbors 〈c0,x〉 and 〈c0,x + y〉 of c for some vector y ∈ c \ c0, which have no codeword of weight 10 in c. when c has a self-dual neighbor c′ with minimum weight 12, there is a vector x satisfying (2) and we can obtain c′ in this way. for i = 1,2, . . . ,224, we verified that there is a unique vector satisfying (2) and c has two self-dual neighbors, where c0 is a doubly even [64,31,12] code. in this case, the two neighbors are automatically doubly even. hence, we have the following: proposition 5.2. there is no extremal singly even self-dual [64,32,12] neighbor of e64,i for i = 1,2, . . . ,224. 6. extremal singly even self-dual [66, 33, 12] codes the following method for constructing singly even self-dual codes was given in [14]. let c be a self-dual code of length n. let x be a vector of odd weight. let c0 denote the subcode of c consisting of all codewords which are orthogonal to x. then there are cosets c1,c2,c3 of c0 such that c0 ⊥ = c0 ∪c1 ∪c2 ∪c3, where c = c0 ∪c2 and x + c = c1 ∪c3. it was shown in [14] that c(x) = (0,0,c0)∪ (1,1,c2)∪ (1,0,c1)∪ (0,1,c3) (3) 149 http://www.math.is.tohoku.ac.jp/~mharada/paper/64-4cir-d10.txt d. anev et al. / j. algebra comb. discrete appl. 5(3) (2018) 143–151 is a self-dual code of length n + 2. in this section, we construct new extremal singly even self-dual codes of length 66 using this construction from the extremal singly even self-dual [64,32,12] codes obtained in sections 3 and 4. our exhaustive search shows that there are 1166 inequivalent extremal singly even self-dual [66,33,12] codes constructed as the codes c(x) in (3) from the codes c64,i (i = 1,2, . . . ,67). 1157 codes of the 1166 codes have weight enumerator w66,1 for β ∈ {7,8, . . . ,92} \ {9,11}, 3 of them have weight enumerator w66,3 for β ∈ {30,49,54}, and 6 of them have weight enumerator w66,2. extremal singly even self-dual [66,33,12] codes with weight enumerator w66,1 for β ∈ {7,58,70,91} are constructed for the first time. for the four weight enumerators w , as an example, codes c66,i with weight enumerators w are given (i = 1,2,3,4). we list in table 4 the values β in w , the codes c and the vectors x = (x1,x2, . . . ,x32) of c(x) in (3), where xj = 1 (j = 33, . . . ,64). table 4. extremal singly even self-dual [66, 33, 12] codes codes β w c (x1, . . . ,x32) c66,1 7 w66,1 c64,1 (01101101101010010111111010101100) c66,2 58 w66,1 c64,56 (00001101100000011000110000011100) c66,3 70 w66,1 c64,66 (00100110011011001001011100000010) c66,4 91 w66,1 c64,67 (00001110110111110000011101000010) d66,1 22 w66,3 d64,14 (10100011100100110111101010011111) d66,2 23 w66,3 d64,14 (10111100111100000100101000100011) d66,3 93 w66,1 d64,383 (10100101011110010011001101001101) by applying the construction given in (3) to d64,i, we found more extremal singly even self-dual [66,33,12] codes d66,j with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. for the codes d66,j, we list in table 4 the values β in the weight enumerators w , the codes c and the vectors x = (x1,x2, . . . ,x32) of c(x) in (3), where xi = 1 (i = 33, . . . ,64). hence, we have the following: proposition 6.1. there is an extremal singly even self-dual [66,33,12] code with weight enumerator w66,1 for β ∈{7,58,70,91,93}, and weight enumerator w66,3 for β ∈{22,23}. remark 6.2. the code d66,1 has the smallest value β among known extremal singly even self-dual [66,33,12] codes with weight enumerator w66,3. references [1] w. bosma, j. cannon, c. playoust, the magma algebra system i: the user language, j. symb. comput. 24(3–4) (1997) 235–265. 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[18] n. yankov, m. ivanova, m. h. lee, self–dual codes with an automorphism of order 7 and s–extremal codes of length 68, finite fields appl. 51 (2018) 17–30. 151 https://doi.org/10.1109/18.641574 https://doi.org/10.1109/18.641574 https://mathscinet.ams.org/mathscinet-getitem?mr=2350723 https://mathscinet.ams.org/mathscinet-getitem?mr=2350723 https://doi.org/10.1016/j.jfranklin.2013.05.015 https://doi.org/10.1016/j.jfranklin.2013.05.015 https://doi.org/10.1016/j.ffa.2017.04.003 https://doi.org/10.1016/j.ffa.2017.04.003 https://doi.org/10.1016/j.disc.2015.10.041 https://doi.org/10.1016/j.disc.2015.10.041 https://doi.org/10.1016/j.ffa.2015.05.004 https://doi.org/10.1016/j.ffa.2015.05.004 https://doi.org/10.1109/18.651000 https://doi.org/10.1109/18.119711 https://doi.org/10.1109/18.119711 https://doi.org/10.1109/18.782156 https://doi.org/10.1109/18.782156 http://dx.doi.org/10.3934/amc.2014.8.73 http://dx.doi.org/10.3934/amc.2014.8.73 https://doi.org/10.1109/tit.2015.2396915 https://doi.org/10.1109/tit.2015.2396915 https://doi.org/10.1016/j.ffa.2017.12.001 https://doi.org/10.1016/j.ffa.2017.12.001 introduction weight enumerators of extremal singly even self-dual codes of lengths 64 and 66 extremal four-circulant singly even self-dual [64,32,12] codes extremal self-dual [64,32,12] neighbors of c64,i four-circulant singly even self-dual [64,32,10] codes and self-dual neighbors extremal singly even self-dual [66, 33, 12] codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284934 j. algebra comb. discrete appl. 4(2) • 105–113 received: 12 june 2015 accepted: 8 february 2016 journal of algebra combinatorics discrete structures and applications properties of dual codes defined by nondegenerate forms research article steve szabo, jay a. wood abstract: dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite frobenius ring. these dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the macwilliams identities for the hamming weight. 2010 msc: 94b05, 15a63 keywords: frobenius ring, sesquilinear form, bilinear form, dual code, generating character, macwilliams identities 1. introduction and overview one of the topics discussed in the mini-cours at the lens conference in 2015 was dual codes and the macwilliams identities for linear codes defined over finite rings. in those lectures two notions of duality were used: (1) left and right annihilators defined by the standard dot product and (2) the charactertheoretic annihilator. a key idea for proving useful properties of duality, including the cardinality of dual codes and the macwilliams identities, is to establish an identification between these two notions of annihilators. this can be accomplished over finite frobenius rings. as researchers expand their studies of linear codes over finite rings, questions naturally arise concerning the behavior of dual codes defined using more general inner products than just the standard dot product. this paper begins to address these questions. our main goal is to prove a “model theorem” that captures the main features of a dual code. such a model theorem is proved in [7, sections 10–12], first in the context of character-theoretic annihilators over any ring, and then in the context of the standard dot product over a finite frobenius ring. the dual code is proved to have a module structure if the code does, to satisfy a double-dual property, to steve szabo; department of mathematics and statistics, eastern kentucky university, richmond, ky 40475 usa (email: steve.szabo@eku.edu). jay a. wood (corresponding author); department of mathematics, western michigan university, kalamazoo, mi 49008 usa (email: jay.wood@wmich.edu). 105 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 have cardinality complementary to that of the code, and to satisfy the macwilliams identities for the hamming weight enumerator. in this paper, we show the same results over a finite frobenius ring for nondegenerate sesquilinear forms (with respect to an anti-automorphism of the ring) and for non-degenerate bilinear forms. 2. preliminaries in this section we collect several definitions that will be used throughout the paper. for a ring a, a subset c of an is a code of length n over a. if c is an additive subgroup of an, then c is an additive code over a, and if c is a left (right) a-submodule of an, then c is a left (right) linear code over a. we will usually state results only for left linear codes, but there are always versions of those results that apply to right linear codes. if g is a finite abelian group, a character of g is a group homomorphism π : g → c×, the multiplicative group of nonzero complex numbers. the set of all characters of g is denoted ĝ, and ĝ is itself a finite abelian group under pointwise multiplication of functions: (π1π2)(g) = π1(g)π2(g), for g ∈ g. the groups g and ĝ are isomorphic; in particular, |g| = |ĝ|. in particular, the definition of characters applies to the additive groups of finite rings and finite modules. we will always assume that our finite rings have a multiplicative identity 1 and that all modules are unitary. if m is a finite left module over a finite ring a, then m̂ is a right a-module with right scalar multiplication of π ∈ m̂ by a ∈ a denoted by πa; πa(m) = π(am) for m ∈ m. we use the exponential notation so that the distributive law for the right a-module m̂ (i.e., right scalar multiplication distributing over the “addition” given by pointwise multiplication) reads (π1π2)a = πa1π a 2. similarly, if n is a finite right a-module, then n̂ is a left a-module under aπ(n) = π(na) for a ∈ a, n ∈ n, and π ∈ n̂. if b is a finite bimodule over a, then b̂ is also a bimodule over a. in particular, â is a bimodule over a. suppose m is a finite left a-module and π ∈ m̂ is a character of m. the right scalar multiplication defines a homomorphism of right a-modules a → m̂ by a 7→ πa. we say that π is a generating character of m if this homomorphism is surjective (so that π generates m̂ as a right a-module). a character π ∈ m̂ of m is a generating character if and only if kerπ ⊆ m contains no nonzero left a-submodules [9, dual of proposition 12]. we will often assume that a finite ring is frobenius. there are several equivalent definitions of frobenius rings [4, theorem 16.14], and we will say that a finite ring a is frobenius if there exists a character ρ ∈ â of a that is a generating character of a, both as a left a-module and as a right a-module. in fact, for a finite ring a, a character ρ ∈ â is a left generating character of a if and only if ρ is a right generating character of a. see [6, sections 3 and 4] for more details. for later use, we state two lemmas, the first of which dates from [1]. lemma 2.1 ([1, corollary 3.6]). let a be a finite ring. a character ρ of a is a generating character if and only if kerρ contains no nonzero one-sided ideals. lemma 2.2. let ρ ∈ â be a generating character of a. for every a1 ∈ a, there exists a2 ∈ a such that a1ρ = ρa2. likewise, for every a3 ∈ a, there exists a4 ∈ a such that ρa3 = a4ρ. proof. for any a1 ∈ a, a1ρ belongs to â. but the homomorphism of right a-modules a → â, a 7→ ρa, is surjective. thus there exists a2 ∈ a with ρa2 = a1ρ, as desired. the proof of the other case is similar, using the homomorphism a 7→ aρ of left a-modules. 106 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 3. sesquilinear forms for a ring a, an anti-automorphism σ on a and a left a-module m, a σ-sesquilinear form on m is a map 〈·, ·〉 : m × m → a such that if x,y,z ∈ m and a ∈ a, then 〈x + z,y〉 = 〈x,y〉 + 〈z,y〉, 〈x,y + z〉 = 〈x,y〉 + 〈x,z〉, 〈ax,y〉 = a〈x,y〉 and 〈x,ay〉 = 〈x,y〉σ(a). in addition, if σ(〈x,y〉) = 〈y,x〉, then the form is called a σ-hermitian form. a σ-sesquilinear form with the property that 〈x,y〉 = 0 ⇐⇒ 〈y,x〉 = 0 is called reflexive. clearly a σ-hermitian form is reflexive. a σ-sesquilinear form is called non-degenerate if 〈x,y〉 = 0 for all y ∈ m implies x = 0 and 〈y,x〉 = 0 for all y ∈ m implies x = 0. proposition 3.1. let a be a ring and σ be an anti-automorphism on a. define the map 〈·, ·〉 : ak ×ak → a, 〈x,y〉 = k∑ i=1 xiσ(yi), where x = (x1,x2, . . . ,xk), y = (y1,y2, . . . ,yk) ∈ ak. then 〈·, ·〉 is a σ-sesquilinear form. furthermore, 〈·, ·〉 is a σ-hermitian form if and only if σ is involutory. moreover, 〈xa,y〉 = 〈 x,yσ−1(a) 〉 , for all x,y ∈ ak and a ∈ a. proof. clearly, 〈·, ·〉 is a σ-sesquilinear form. assume 〈·, ·〉 is hermitian. for a ∈ a, a = a〈(1,0, . . . ,0),(1,0, . . . ,0)〉 = 〈(a,0, . . . ,0),(1,0, . . . ,0)〉 . since 〈·, ·〉 is hermitian, σ2(a) = σ2(〈(a,0, . . . ,0),(1,0, . . . ,0)〉) = a, showing σ is involutory. now, assume σ is involutory. for x,y ∈ ak, σ(〈x,y〉) = σ( k∑ i=1 xiσ(yi)) = k∑ i=1 σ(xiσ(yi)) = k∑ i=1 yiσ(xi) = 〈y,x〉 , showing 〈·, ·〉 is σ-hermitian. the final identity is a straight-forward verification left for the reader. for the rest of the section, let n be a natural number, a be a finite frobenius ring, ρ be a generating character of a, σ be an anti-automorphism on a, and 〈·, ·〉 be a non-degenerate σ-sesquilinear form on an. for any code (linear or not) c ⊆ an, the left and right dual codes of c, denoted by l(c) and r(c), are l(c) = {v ∈ an : 〈v,c〉 = 0} and r(c) = {v ∈ an : 〈c,v〉 = 0}, and the left and right ρ-character dual codes of c, denoted by lρ(c) and rρ(c), are lρ(c) = {v ∈ an : ρ(〈v,c〉) = 1}, rρ(c) = {v ∈ an : ρ(〈c,v〉) = 1}. it is immediate that l(c) and r(c) are left a-submodules of an, and lρ(c) and rρ(c) are additive subgroups of an. in fact, l(c) = l(ac) and r(c) = r(ac), where ac is the left a-module generated by c, while lρ(c) = lρ(zc) and rρ(c) = rρ(zc), where zc is the additive group generated by c. (it need not be true that lρ(ac) = lρ(zc). let a = f4 and c = f2 ⊂ f4; c = zc is an additive subgroup but not an ideal. of course, ac = f4, and lρ(c) = f2, while lρ(ac) = {0}.) as noted earlier, every hermitian form is reflexive; thus l(c) = r(c) when the form is hermitian. when c is a left a-linear code, the following lemma shows that lρ(c) and rρ(c) are left a-modules as well. 107 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 lemma 3.2. let c ⊆ an be a left a-linear code over a finite frobenius ring a. then lρ(c) and rρ(c) are left a-modules. proof. we prove the lρ(c) case; the rρ(c) case is similar. let x ∈ lρ(c) and r ∈ a. by lemma 2.2, there exists s ∈ a with ρr = sρ. then ρ(〈rx,c〉) = ρ(r〈x,c〉) = ρr(〈x,c〉) = sρ(〈x,c〉) = ρ(〈x,c〉s) = ρ( 〈 x,σ−1(s)c 〉 ) = 1. lemma 3.3. let c ⊆ an be a left a-linear code over a finite frobenius ring a. then l(c) = lρ(c) and r(c) = rρ(c). proof. clearly, l(c) ⊆ lρ(c). let x ∈ lρ(c), so that 〈x,c〉 ⊆ kerρ. for c ∈ c and a ∈ a, 〈x,c〉a =〈 x,σ−1(a)c 〉 ∈ 〈x,c〉, which implies that 〈x,c〉 is a right ideal. since ρ is a generating character, lemma 2.1 implies 〈x,c〉 = 0, showing x ∈ l(c). thus, lρ(c) = l(c). a similar argument, using 〈c,x〉 being a left ideal, shows rρ(c) = r(c). define two maps as follows. α : an → ân, x 7→ αx, αx(y) = ρ(〈x,y〉),y ∈ an; β : an → ân, x 7→ βx, βx(y) = ρ(〈y,x〉),y ∈ an. lemma 3.4. over a finite frobenius ring a, the maps α and β are group isomorphisms. proof. we prove the result for α; the proof for β is similar. then αx(z)αy(z) = ρ(〈x,z〉)ρ(〈y,z〉) = ρ(〈x,z〉+ 〈y,z〉) = ρ(〈x + y,z〉) = αx+y(z), for all x,y,z ∈ an. so, α is a group homomorphism. assume αx = αy. then for any w ∈ an, 1 = αx(w)α−y(w) = ρ(〈x−y,w〉). since 〈x−y,an〉 is a right ideal and ρ is a generating character, by lemma 2.1, 〈x−y,an〉 = 0. this implies x = y since 〈·, ·〉 is non-degenerate. so, α is injective and hence bijective. for an additive subgroup h of an, define (ân : h) = {π ∈ ân : π(h) = 1}. clearly, (ân : h) is a subgroup of ân. if c ⊆ an is a left (right) linear code, then (ân : c) is a right (left) a-submodule of ân. the next lemma follows directly from lemma 3.4. lemma 3.5. over a finite frobenius ring a, there are group isomorphisms lρ(c) ∼= (ân : c) and rρ(c) ∼= (ân : c). proof. the character dual lρ(c) corresponds to (ân : c) under the isomorphism α, while rρ(c) corresponds to (ân : c) under β. theorem 3.6. for a left a-linear code c ⊆ an over a finite frobenius ring a, |c| · |l(c)| = |c| · |r(c)| = |an|. 108 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 proof. the annihilator (ân : c) is isomorphic to the character group (an/c)̂ of the quotient group an/c. thus |c| · |(ân : c)| = |an|. by lemmas 3.3 and 3.5, the result follows. for any code (linear or not) c ⊆ an, we have c ⊆ r(l(c)), c ⊆ l(r(c)), c ⊆ rρ(lρ(c)), and c ⊆ lρ(rρ(c)). when c is a left linear code, all these containments are equalities. proposition 3.7. let c ⊆ an be a left a-linear code over a finite frobenius ring a. then c = l(r(c)), c = lρ(rρ(c)), c = r(l(c)), c = rρ(lρ(c)). proof. by using theorem 3.6, one sees that c and the double annihilators all have the same cardinalities. thus the containments described above are equalities. 4. bilinear forms this section discusses bilinear forms. most of the results are similar to those in section 3, and their proofs are essentially the same. similar results in the more general setting of bilinear forms with values in a bimodule may be found in [7, section 12.5]. a very general approach using biadditive forms was initiated in [5]; additional information about that approach appears in [7] and [8]. for a ring a and a two-sided module m over a, a bilinear form on m is a map 〈·, ·〉 : m ×m → a such that if x,y,z ∈ m and a ∈ a, then 〈x + z,y〉 = 〈x,y〉 + 〈z,y〉, 〈x,y + z〉 = 〈x,y〉 + 〈x,z〉, 〈ax,y〉 = a〈x,y〉, and 〈x,ya〉 = 〈x,y〉a. a bilinear form is called non-degenerate if 〈x,y〉 = 0 for all y ∈ m implies x = 0 and 〈y,x〉 = 0 for all y ∈ m implies x = 0. proposition 4.1. let a be a ring. define the map 〈·, ·〉 : ak ×ak → a; 〈x,y〉 = k∑ i=1 xiyi. then 〈·, ·〉 is a bilinear form. moreover, 〈xa,y〉 = 〈x,ay〉 for all x,y ∈ ak and a ∈ a. for the rest of the section, let n be a natural number, a be a finite frobenius ring, ρ be a generating character for a, and 〈·, ·〉 be a non-degenerate bilinear form on an. for any code (linear or not) c ⊆ an, the left and right dual codes of c, denoted by l(c) and r(c), are l(c) = {v ∈ an : 〈v,c〉 = 0} and r(c) = {v ∈ an : 〈c,v〉 = 0}, and the left and right ρ-character dual codes of c, denoted by lρ(c) and rρ(c), are lρ(c) = {v ∈ an : ρ(〈v,c〉) = 1}, rρ(c) = {v ∈ an : ρ(〈c,v〉) = 1}. then l(c) is a left a-submodule and r(c) is a right a-submodule of an, while lρ(c) and rρ(c) are additive subgroups of an. in fact, l(c) = l(ca), r(c) = r(ac), lρ(c) = lρ(zc), and rρ(c) = rρ(zc), where ca is the right a-submodule of an generated by c. lemma 4.2. suppose a is a finite frobenius ring. if c ⊆ an is a left a-linear code, then rρ(c) is a right a-module. if c ⊆ an is a right a-linear code, then lρ(c) is a left a-module. proof. see lemma 3.2. 109 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 lemma 4.3. suppose a is a finite frobenius ring. if c ⊆ an is a left a-linear code, then r(c) = rρ(c). if c ⊆ an is a right a-linear code, then l(c) = lρ(c). proof. see lemma 3.3. define two maps as follows. α : an → ân, x 7→ αx, αx(y) = ρ(〈x,y〉),y ∈ an; β : an → ân, x 7→ βx, βx(y) = ρ(〈y,x〉),y ∈ an. lemma 4.4. over a finite frobenius ring a, the maps α and β are group isomorphisms. proof. see lemma 3.4. the following lemma follows directly from lemma 4.4. lemma 4.5. suppose a is a finite frobenius ring. let c ⊆ an be any code. then the map α restricts to a group isomorphism lρ(c) ∼= (ân : c), while β restricts to a group isomorphism rρ(c) ∼= (ân : c). theorem 4.6. let a be a finite frobenius ring. if c ⊆ an is a left a-linear code, then |c|·|r(c)| = |an|. if c ⊆ an is a right a-linear code, then |c| · |l(c)| = |an|. proof. see theorem 3.6 and lemma 4.3. just as for sesquilinear forms, for any code (linear or not) c ⊆ an, we have c ⊆ r(l(c)), c ⊆ l(r(c)), c ⊆ rρ(lρ(c)), and c ⊆ lρ(rρ(c)). when c is a linear code, appropriate containments are equalities. proposition 4.7. let a be a finite frobenius ring. if c ⊆ an is a left a-linear code, then c = l(r(c)), c = lρ(rρ(c)). if c ⊆ an is a right a-linear code, then c = r(l(c)), c = rρ(lρ(c)). proof. see proposition 3.7 and theorem 4.6. remark 4.8. the double-annihilator property of proposition 4.7 holds in more general settings. suppose a is any finite ring with 1, and consider the standard dot product on an of proposition 4.1. then the results of proposition 4.7 hold if and only if a is a quasi-frobenius ring [3]. in fact, the double-annihilator property for ideals (i.e., n = 1) is often taken as the definition of a quasi-frobenus ring. remark 4.9. we thank the referee for the observation that the results in the sesquilinear case are special cases of those in the bilinear case. suppose a is a finite ring with 1 and σ is an anti-automorphism on a. suppose m is a left a-module equipped with a σ-sesquilinear form 〈·, ·〉 : m ×m → a. define a right a-module structure on m by ma = σ−1(a)m, for m ∈ m and a ∈ a, where the left scalar multiplication is used for σ−1(a)m. this makes m a two-sided module over a. moreover, 〈·, ·〉 is now a bilinear form: 〈x,ya〉 = 〈 x,σ−1(a)y 〉 = 〈x,y〉σ(σ−1(a)) = 〈x,y〉a, for x,y ∈ m and a ∈ a. 110 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 5. macwilliams identities in this section we will prove the macwilliams identities relating the hamming weight enumerators of a linear code over a finite frobenius ring and its dual code with respect to a non-degenerate sesquilinear form or a non-degenerate bilinear form. given a finite ring a, the hamming weight of an element a ∈ a is wt(a) = { 0, a = 0, 1, a 6= 0. we extend wt to apply to vectors x = (x1, . . . ,xn) ∈ an by wt(x) = ∑n i=1 wt(xi), where the sum takes place in z. thus wt(x) counts the number of nonzero entries of the vector x. we will also need to define the hamming weight on â: wt(π) = 0 when π = 1 (the trivial character with π(a) = 1 for all a ∈ a) and wt(π) = 1 for π 6= 1. then extend to ân as above. given an additive code c ⊆ an, the hamming weight enumerator wc(x,y ) of c is the polynomial in c[x,y ] defined by wc(x,y ) = ∑ x∈c xn−wt(x)y wt(x). we quote the macwilliams identities for additive codes from [7], but the result in this generality goes back to delsarte [2]. theorem 5.1 ([7, theorem 11.3]). let c ⊆ an be an additive code over a. then the macwilliams identities hold: w (ân:c) (x,y ) = 1 |c| wc(x + (|a|−1)y,x −y ). we now state the macwilliams identities with respect to non-degenerate forms. theorem 5.2 (macwilliams identities). suppose a is a finite frobenius ring. then wd(x,y ) = 1 |c| wc(x + (|a|−1)y,x −y ) holds in the following cases: • 〈·, ·〉 is a non-degenerate sesquilinear form on an, c ⊆ an is a left a-linear code, and d = l(c) or r(c); • 〈·, ·〉 is a non-degenerate bilinear form on an, c ⊆ an is a left a-linear code, and d = r(c); • 〈·, ·〉 is a non-degenerate bilinear form on an, c ⊆ an is a right a-linear code, and d = l(c). theorem 5.2 will follow from theorem 5.1 once we prove the next lemma. lemma 5.3. the group isomorphisms α and β of lemmas 3.4 and 4.4 preserve the hamming weight. proof. we will prove the result for α; the proof for β is similar. express x = (x1, . . . ,xn) ∈ an as the linear combination x = ∑n i=1 xiei, where the ei are the standard basis of a n. i.e., ei is the n-tuple with 1 in position i and 0s elsewhere. then αx(y) = ρ(〈x,y〉) = ρ( n∑ i=1 xi 〈ei,y〉) = n∏ i=1 ρ(xi 〈ei,y〉) = n∏ i=1 ρxi(〈ei,y〉) 111 s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 the lemma is equivalent to showing that y 7→ ρxi(〈ei,y〉) is the trivial character if and only if xi = 0. if xi = 0, then certainly y 7→ ρxi(〈ei,y〉) is trivial. for the converse, suppose y 7→ ρxi(〈ei,y〉) is trivial. then xi 〈ei,an〉 ⊆ kerρ. but xi 〈ei,an〉 is a right ideal of a, so xi 〈ei,an〉 = 0 by lemma 2.2. this means 〈xiei,an〉 = 0, so that xiei = 0 by the non-degeneracy of the form 〈·, ·〉. thus xi = 0. proof of theorem 5.2. by lemma 5.3, wd(x,y ) = w(ân:c)(x,y ) via the weight-preserving isomorphisms α or β. the result then follows from theorem 5.1. 6. summary in this final section, we collect in one place the main results of the paper, arranged in the same manner as the “model theorem” in [7]. theorem 6.1. let c ⊆ an be a left a-linear code over a finite frobenius ring a. suppose an is equipped with a non-degenerate sesquilinear form. then • l(c) and r(c) are left a-linear codes in an; • l(r(c)) = c and r(l(c)) = c; • |c| · |l(c)| = |an| and |c| · |r(c)| = |an|; • the macwilliams identities hold, for d equaling l(c) or r(c): wd(x,y ) = 1 |c| wc(x + (|a|−1)y,x −y ). theorem 6.2. let c ⊆ an be a left a-linear code and c′ ⊆ an be a right a-linear code over a finite frobenius ring a. suppose an is equipped with a non-degenerate bilinear form. then • in an, r(c) is a right a-linear code, and l(c′) is a left a-linear code; • l(r(c)) = c and r(l(c′)) = c′; • |c| · |r(c)| = |an| and |c′| · |l(c′)| = |an|; • the macwilliams identities hold: wr(c)(x,y ) = 1 |c| wc(x + (|a|−1)y,x −y ), wl(c′)(x,y ) = 1 |c′| wc′(x + (|a|−1)y,x −y ). acknowledgment: we thank the referee for valuable comments which helped improve the clarity of the paper and which reminded us that we initially forgot to include proposition 4.7. references [1] h. l. claasen, r. w. goldbach, a field–like property of finite rings, indag. math. (n.s.) 3(1) (1992) 11–26. 112 http://dx.doi.org/10.1016/0019-3577(92)90024-f http://dx.doi.org/10.1016/0019-3577(92)90024-f s. szabo, j. a. wood / j. algebra comb. discrete appl. 4(2) (2017) 105–113 [2] p. delsarte, bounds for unrestricted codes, by linear programming, philips res. rep. 27 (1972) 272– 289. [3] m. hall, a type of algebraic closure, ann. of math. 40(2) (1939) 360–369. [4] t. y. lam, lectures on modules and rings, graduate texts in mathematics, vol. 189, springer– verlag, new york, 1999. [5] g. nebe, e. m. rains, n. j. a. sloane, self–dual codes and invariant theory, algorithms and computation in mathematics, vol. 17, springer–verlag, berlin, 2006. [6] j. a. wood, duality for modules over finite rings and applications to coding theory, amer. j. math. 121(3) (1999) 555–575. [7] j. a. wood, foundations of linear codes defined over finite modules: the extension theorem and the macwilliams identities. codes over rings, 124–190, ser. coding theory cryptol., 6, world sci. publ., hackensack, nj, 2009. [8] j. a. wood, anti–isomorphisms, character modules and self–dual codes over non-commutative rings, int. j. inf. coding theory 1(4) (2010) 429–444. [9] j. a. wood, applications of finite frobenius rings to the foundations of algebraic coding theory. proceedings of the 44th symposium on ring theory and representation theory, 223–245, symp. ring theory represent. theory organ. comm., nagoya, 2012. 113 http://www.ams.org/mathscinet-getitem?mr=314545 http://www.ams.org/mathscinet-getitem?mr=314545 http://dx.doi.org/10.2307/1968924 http://dx.doi.org/10.1007/978-1-4612-0525-8 http://dx.doi.org/10.1007/978-1-4612-0525-8 http://dx.doi.org/10.1007/3-540-30731-1 http://dx.doi.org/10.1007/3-540-30731-1 http://www.ams.org/mathscinet-getitem?mr=1738408 http://www.ams.org/mathscinet-getitem?mr=1738408 http://www.ams.org/mathscinet-getitem?mr=2850303 http://www.ams.org/mathscinet-getitem?mr=2850303 http://www.ams.org/mathscinet-getitem?mr=2850303 http://dx.doi.org/10.1504/ijicot.2010.032867 http://dx.doi.org/10.1504/ijicot.2010.032867 http://www.ams.org/mathscinet-getitem?mr=2975625 http://www.ams.org/mathscinet-getitem?mr=2975625 http://www.ams.org/mathscinet-getitem?mr=2975625 introduction and overview preliminaries sesquilinear forms bilinear forms macwilliams identities summary references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284962 j. algebra comb. discrete appl. 4(2) • 189–196 received: 12 june 2015 accepted: 25 february 2016 journal of algebra combinatorics discrete structures and applications commuting probability for subrings and quotient rings research article stephen m. buckley, desmond machale abstract: we prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison. 2010 msc: 05e15 keywords: commuting probability, subring, quotient ring 1. introduction suppose r is a finite (possibly nonunital) ring. the commuting probability of r is pr(r) := |{(x,y) ∈ r×r : xy = yx}| |r|2 , where | · | denotes cardinality. there has been much written on the commuting probability of a finite group: see for instance [5], [7], [8], [10], [4], and [6]. the commuting probability of a ring has been discussed in [9], [3], [2], and [1]. work on the commuting probability of rings r has so far mainly concentrated on the possible values of pr(r). however, it was shown in [9] that pr(r) is no larger than pr(s) whenever s is a subring of r. our first result gives a new proof of this result, one that allows us to characterize when equality occurs. theorem 1.1. suppose s is a subring of a finite ring r. then pr(r) ≤ pr(s). equality holds if and only if [x,s] = [x,r] for all x ∈ r. our second result is similar, but involves a comparison with quotient rings. stephen m. buckley (corresponding author); department of mathematics and statistics, maynooth university, maynooth, co. kildare, ireland (email: stephen.buckley@maths.nuim.ie). desmond machale; school of mathematical sciences, university college cork, cork, ireland (email: d.machale@ucc.ie). 189 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 proposition 1.2. suppose i is an ideal in a finite ring r. then pr(r) ≤ pr(r/i). equality holds if and only if [x,r]∩ i = {0} for all x ∈ r. in the above results, [x,s] = {[x,s] : s ∈ s}, and [x,s] = xs−sx is the commutator of x and s. after some preliminaries in section 2, we prove generalizations of the above results in section 3. in section 4, we give various counterexamples which rule out seemingly plausible variants of the above conditions for equality. 2. preliminaries given a set s of finite cardinality, and a function f : s → r, we write |s| for the cardinality of s, and define the arithmetic mean ——— ∑ x∈s f(x) = 1 |s| ∑ x∈s f(x) . in this paper, a ring is not necessarily unital. our results do not use associativity either and, to emphasize this, we sometimes talk of pn rings (where “pn” stands for “possibly nonassociative”). in the absence of the “pn” qualifier, rings and algebras are assumed to be associative. however, an ideal in a pn ring is not assumed to be associative. suppose r is a pn ring and x ∈ r. the annihilator ann(r), center z(r), and centralizer cr(x) are defined by ann(r) := {u ∈ r : uv = vu = 0 for all v ∈ r} , z(r) := {u ∈ r : [u,v] = 0 for all v ∈ r} , cr(x) := {u ∈ r : [u,x] = 0} . if a and b are finite subsets of a pn ring r, then we define the commuting probability for the triple (a,b;r) to be prr(a,b) := |{(x,y) ∈ a×b : xy = yx}| |a| · |b| , where juxtaposition indicates multiplication in r. we also write prr(a) := prr(a,a) and pr(r) := prr(r). if x,y are elements of a pn ring r, and s is an additive subgroup of r, then we define the commutator [x,y] := xy −yx, and we write [x,s] := {[x,s] : s ∈ s}. note that [x,s] is always an additive subgroup of r. if t is another additive subgroup of r, we define [s,t] to be the additive subgroup of r given by the set of finite sums of commutators [s,t], s ∈ s, t ∈ t . a + b denotes the additive subgroup {a+b : a ∈ a, b ∈ b} whenever a,b are additive subgroups of a pn ring r, and spans is the subspace of finite linear combinations of elements of a subset s of an algebra r. if a pn ring r is the direct sum of pn rings r1 and r2, it follows easily that pr(r) = pr(r1) pr(r2). 3. proofs theorem 1.1 follows immediately from the following more general result. theorem 3.1. suppose a pn ring r has finite additive subgroups a1,a2,b1,b2 satisfying a1 ⊆ a2 and b1 ⊆ b2. then prr(a2,b2) ≤ prr(a1,b1). furthermore, the following conditions are equivalent: 190 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 (ab1) prr(a1,b1) = prr(a2,b2). (ab2) [x,a1] = [x,a2] and [y,b1] = [y,b2], for all x ∈ b1, y ∈ a2. (ab3) [x,a1] = [x,a2] and [y,b1] = [y,b2], for all x ∈ b2, y ∈ a2. proof. note that for any finite subsets a,b of r, we have prr(a,b) = ——— ∑ x∈b ——— ∑ y∈a f(y,x) , where f : r×r →{0,1} is the function defined by f(y,x) = 1 if xy = yx, and f(x,y) = 0 otherwise. we first prove the result in the special case b1 = b2. for each x ∈ b2, define a surjective homomorphism of additive groups, φx : a2 → [x,a2], by φx(y) = [x,y] , y ∈ a2 . for x ∈ b2, y ∈ a2, and f as in the previous paragraph, we have f(x,y) = 1 if and only if y ∈ kerφx. by the first isomorphism theorem, it follows that ——— ∑ y∈a2 f(y,x) = |kerφx| |a2| = 1 |[x,a2]| . and so prr(a2,b2) = ——— ∑ x∈b2 1 |[x,a2]| . by the same argument, we have prr(a1,b2) = ——— ∑ x∈b2 1 |[x,a1]| . it follows readily that prr(a2,b2) ≤ prr(a1,b2), with equality if and only if [x,a1] = [x,a2] for all x ∈ b2. this proves the equivalence of (ab1)–(ab3) in the special case b1 = b2. we wish to employ symmetry between the aand b-subgroups. for this, we note that (ab2) can be written in a simpler form in our special case b1 = b2: (ab2′) [x,a1] = [x,a2], for all x ∈ b2. moreover, let us say that (ab2′) has data (a1,a2;b2). by symmetry, we can now handle the special case a1 = a2. in fact, we have prr(a2,b2) ≤ prr(a2,b1), with equality if and only if [y,b1] = [y,b2] for all y ∈ a2, and we deduce the equivalence of (ab1)–(ab3) as before. for the special case a1 = a2, (ab2) can be written in the simpler form (ab2′′) [y,b1] = [y,b2], for all y ∈ a2. moreover, let us say that (ab2′′) has data (b1,b2;a2). we now consider the general case. by the two special cases considered above, we have prr(a2,b2) ≤ prr(a2,b1) ≤ prr(a1,b1) , (1) as required. moreover, prr(a1,b1) = prr(a2,b2) if and only if both of inequalities in (1) are equalities, which is equivalent to the conjunction of (ab2′) with data (a1,a2;b1), and (ab2′′) with data (b1,b2;a2). this conjunction is just the required general form of (ab2). thus, (ab1) is equivalent to (ab2). because of the symmetry between the aand b-subgroups in (ab1) that is lacking in (ab2), we get a version of (ab2) where the equations are instead true for all y ∈ a1 and all x ∈ b2. putting this together with the original form of (ab2), we derive the formally stronger (ab3). thus, all three conditions (ab1)–(ab3) are mutually equivalent. 191 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 next, we tackle proposition 1.2. in fact we prove a slight generalization of it in the context of pn rings. proposition 3.2. suppose i is an ideal in a finite pn ring r. then pr(r) ≤ pr(r/i), with equality if and only if [x,r]∩ i = {0} for all x ∈ r. proof. take x + i,y + i ∈ r/i, where x,y ∈ r. since [x + i,y + i] = [x,y] + i is independent of the representatives x,y of these elements in r/i, it follows that pr(r/i) = |{(x,y) ∈ r×r : xy −yx ∈ i}| |r|2 . it is now clear that pr(r) ≤ pr(r/i). furthermore, we have equality if and only if the condition xy −yx ∈ i is equivalent to xy = yx for all x,y ∈ r. this is equivalent to the desired condition. 4. counterexamples here we pose three questions and give a negative answer in each case. together, these answers show that our results cannot be simplified or improved in any obvious way. question 4.1. can we improve theorem 3.1 by dropping one equation from (ab2) or (ab3), or by significantly restricting the set of elements for which the equations hold, and still obtaining a condition equivalent to (ab1)? question 4.2. can we strengthen the first conclusion of theorem 3.1 by dropping one of the assumptions that ai,bi are additive subgroups of r? question 4.3. can we strengthen the statement of either theorem 3.1 or proposition 1.2 by replacing the necessary and sufficient condition for equality of commuting probabilities by a simpler quantifier-free commutator subgroup property? we will see that the first two questions are easily answered, but that the third one is rather more interesting (although we will need to make clearer what we have in mind by this question separately for each of the two results to which it refers). our first proposition gives a negative answer to question 4.1. proposition 4.4. neither of the following conditions are equivalent to conditions (ab1)–(ab3) in theorem 3.1. (ab4) [x,a1] = [x,a2] and [y,b1] = [y,b2], for all x ∈ b1, y ∈ a1. (ab5) [x,a1] = [x,a2] for all x ∈ b2. proof. we get counterexamples in an arbitrary finite noncommutative ring r. in (ab4), let a1 = b1 = z(r) and a2 = b2 = r. then [x,a1] = [x,a2] = [y,b1] = [y,b2] = {0} for all x ∈ b1, y ∈ a1. however, prr(a1,b1) = 1 > prr(a2,b2). for (ab5), let b1 = z(r) and a1 = a2 = b2 = r. trivially, [x,a1] = [x,a2] for all x ∈ b2. however, prr(a1,b1) = 1 > prr(a2,b2). the following proposition gives a negative answer to question 4.2. proposition 4.5. if we drop any one of the assumptions that ai,bi are additive subgroups of r in theorem 3.1, then the main inequality prr(a2,b2) ≤ prr(a1,b1) may fail. 192 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 proof. by symmetry, it suffices to show that the inequality may fail if either b1 or b2 is not an additive subgroup. below, p is any prime number. let r be the zp-algebra with basis {u,v}, where u2 = vu = u and v2 = uv = v. let b2 = a1 = a2 = r, and let b1 = {u}. it is clear that prr(a1,b1) = 1/p, whereas it is well known and straightforward to verify (see [3, theorem 5.1]) that prr(a2,b2) = pr(r) = p2 + p−1 p3 > 1 p . next, let s be the ring of order p3 given by an internal direct sum of zp and the ring r of the previous paragraph. let a1 = a2 = b1 = r, and let b2 be any subset of s such that b2 = r ∪{z} where z ∈ z(s) \ {0}; note that |z(s) \ {0}| = p − 1 and z(s) \ {0} does not intersect r. then prr(a1,b1) = pr(r) = (p 2 + p−1)/p3 as before, but prr(a2,b2) = (p3 + p2 −p) + p2 p2(p2 + 1) = p2 + 2p−1 p3 + p > p2 + p−1 p3 . since (p2 + 2p−1)/(p3 + p) is a weighted mean of pr(r) and 1. remark 4.6. the counterexamples in proposition 4.5 do not immediately imply that the assumption that s is a subring in theorem 1.1 is essential. however, this is easily shown. for instance, if r is any finite non-commutative ring and s = {a,b}, where a,b ∈ r do not commute, then pr(s) = 0 < pr(r). we next address question 4.3 in relation to theorem 3.1. assume that the hypotheses of theorem 3.1 are in effect and that (ab1)–(ab3) hold. suppose u ∈ [a2,b2]. by definition, u is a finite sum of terms of the form [y,x], y ∈ a2, x ∈ b2. by (ab2), we may assume that x ∈ b1 for each such term, and so [a2,b2] = [a2,b1]. by symmetry, it is also true that [a2,b2] = [a1,b2]. if we restrict y to a1 then, by essentially the same argument, it follows that [a1,b1] = [a1,b2]. thus, (ab1)–(ab3) imply the following quantifier-free condition: (ab6) [a1,b1] = [a1,b2] = [a2,b1] = [a2,b2]. if (ab6) were equivalent to (ab1)–(ab3), then we could weaken the condition for equality in theorem 1.1 to [s,s] = [r,r]. however, we will see that this is false. in fact, we can say more. let us consider the following four conditions for a subring s of a ring r. (s1) s + z(r) = r. (s2) pr(r) = pr(s). (s3) [s,s] = [r,r]. (s4) [r,s] = [r,r]. if (s1) holds, then r is a disjoint union of cosets of the form z+s, z ∈ z(r). since [z1+s1,z2+s2] = [s1,s2] for s1,s2 ∈ s, z1,z2 ∈ z(r), it follows readily that (s2) holds. since (ab1) implies (ab6), it follows in particular that (s2) implies (s3), and trivially (s3) implies (s4). the above implications cannot be reversed. first, it is easy to see that (s4) does not imply (s3): just take r to be a two-dimensional non-commutative zp-algebra (as in the proof of proposition 4.5), where p is a prime, and let s be a one-dimensional subalgebra. then [r,s] = [r,r] has order p, but [s,s] = {0}. the following pair of results show that the other two reverse implications also fail. proposition 4.7. for each prime p, there exists a 5-dimensional zp-algebra r with a subalgebra s of codimension 1 such that [s,s] = [r,r] and pr(r) < pr(s). theorem 4.8. there exists a 7-dimensional z2-algebra r with a subalgebra s of codimension 1 such that pr(s) = pr(r) and s + z(r) 6= r. 193 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 in the proofs of the above pair of results and the proof of one subsequent result, r will in each case be a finite nilpotent zp-algebra. in each proof, we list a basis b of r and define xy for all x,y ∈ b. by distributivity, this defines r uniquely as a pn zp-algebra. however the products of basis elements will be of a special type that in each case allows us to drop the “pn” qualifier: for x,y ∈b, we will either have xy = 0 or xy = z for some z ∈ b ∩ ann(r). by distributivity, it follows that (uv)w = u(vw) = 0 for all u,v,w ∈ r, and so in particular r is associative. we will in each proof denote elements of b∩ ann(r) as zi (or simply z if there is only one such element). proof of proposition 4.7. let r be the zp-algebra with basis{x1,x2,y1,y2,z}, where the only nonzero products of basis elements are x1y1 = x2y2 = z. letting s be the 4-dimensional subalgebra of r with basis {x1,x2,y1,z}, it is clear that [s,s] = [r,r] = span{z}. next, let t be the 3-dimensional subalgebra of s with basis {x1,y1,z}. thus, s = t ⊕ span{x2} is isomorphic to a direct sum of t and zp, and so it is readily verified that pr(s) = pr(t) pr(zp) = p2 + p−1 p3 ·1 = p2 + p−1 p3 . the ring r is what we call an augmentation of t in [3, section 4], so it follows from that paper, or by direct calculation, that pr(r) = p4 + p−1 p5 < pr(s) . proof of theorem 4.8. let r be the algebra with basis b = {u1,u2,v1,v2,w,z1,z2} , where the only nonzero products of basis elements are v1v2 = z1 and uivi = uiw = z2 for i = 1,2. let s be the codimension 1 subalgebra with basis b′ := b\{w}. let ai := spanbi for i = 1,2, where b1 := {u1,u2,v1,v2,w} and b2 := {z1,z2}. it is clear that (r,+) is a direct sum of a1 and a2. we claim that z(r) = a2. clearly a2 ⊆ ann(r) ⊆ z(r), so we need only show that z(r) ⊆ a2. first, note that dim[v1,r] = dima2 = 2, so cr(v1) has codimension 2, and we easily deduce that cr(v1) = span{u2,v1,w,z1,z2} . by symmetry, cr(v2) = span{u1,v2,w,z1,z2}. now a2 ⊆ z(r) ⊆ cr(v1)∩cr(v2) = span{w,z1,z2} . moreover, w is not central, so we deduce that z(r) = a2, as claimed. since a2 = z(r) ⊂ s ⊂ r, we have also proved that s + z(r) 6= r. the fact that pr(s) = pr(r) is a routine exercise, but we indicate how to carry out the required work efficiently. we need to show that [x,s] = [x,r] for all x ∈ r. since a2 = z(r), it suffices to examine x ∈ a1. let us write z2 := span{z2}. since r = span(s ∪{w}) and [w,r] = z2, we have [x,s] ⊆ [x,r] ⊆ [x,s] + z2 , x ∈ r. consequently, [x,s] = [x,r] if either z2 ∈ [x,s] or x ∈ cr(w). we claim that one of these two conditions holds for all x ∈ a1. let x = a1u1 + a2u2 + b1v1 + b2v2 + cw, where a1,a2,b2,b2,c ∈ z2 . now, [x,u1] = (b1 +c)z2, so z2 ∈ [x,s] if b1 +c = 1. thus, without loss of generality, it suffices to consider the case b1 = c and, by symmetry, we may also assume that b2 = c. it follows that [x,v1+v2] = (a1+a2)z2 so, again without loss of generality, it suffices to consider the case a1 = a2. since u1 + u2, v1, v2, and w all lie in cr(w), our claim is proved, and we have shown that [x,s] = [x,r]. 194 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 finally, we address question 4.3 in relation to proposition 1.2. the quantifier-free condition [r,r]∩ i = {0} certainly implies that [x,r] ∩ i = {0} for x ∈ r. however, the next result shows that this quantifier-free condition is not necessary for pr(r) = pr(r/i). theorem 4.9. there exists a 15-dimensional z2-algebra r containing a nontrivial ideal i such that pr(r) = pr(r/i) and i ⊂ [r,r]. proof. let r be the z2-algebra with basis b = {xi,yi | i = 1,2,3}∪{zi,j : (i,j) ∈ s} , where s = {(i,j) | 1 ≤ i,j ≤ 3} , and the only nonzero products of basis elements are xiyj = zi,j , (i,j) ∈ s . it is readily verified that ann(r) = r2 = span{zi,j : (i,j) ∈ s} . let i = span{s}, where s := z1,1 + z2,2 + z3,3. since i ⊂ ann(r), i is an ideal. we claim that s is not a commutator in r. suppose that c := [u,u′] is a sum of the form ∑3 i,j=1 ci,jzi,j for some u,u′ ∈ r, where ci,j ∈ z2 and ci,i = 1 for 1 ≤ i ≤ 3. we claim at least one of the coefficients ci,j, i 6= j, equals 1, regardless of the choice of u,u′; note that it follows from this claim that s is not a commutator. it suffices to assume that both u and u′ are linear combinations of the six basis elements that lie outside ann(r): u := 3∑ i=1 (aixi + biyi) u′ := 3∑ i=1 (a′ixi + b ′ iyi)   , ai,bi,a ′ i,b ′ i ∈ z2 , i = 1,2,3 . note that ci,j = aib′j + a ′ ibj. since zi,i occurs as a term in c, it follows that exactly one of aib′i and a ′ ibi is nonzero for each i ∈ {1,2,3}. by swapping u and u′ if necessary, we may assume that aib′i = 1 and a ′ ibi = 0 for at least two indices i. in fact, by symmetry of the indices, we may assume that these two equations hold for i ∈{1,2} and, in particular, a1 = a2 = b′1 = b′2 = 1. because 0 = (a′1b1)(a ′ 2b2) = (a ′ 2b1)(a ′ 1b2) , it also follows that either a′1b2 = 0 or a ′ 2b1 = 0. thus, either a1b ′ 2 + a ′ 1b2 = 1 or a2b ′ 1 + a ′ 2b1 = 1, and so either c1,2 = 1 or c2,1 = 1, as claimed. we have shown that [x,r] ∩ i = {0} for all x ∈ r and so, by proposition 1.2, pr(r) = pr(r/i). however, s ∈ [r,r] because s = [x1,y1] + [x2,y2] + [x3,y3], and so i ⊂ [r,r]. references [1] s. m. buckley, distributive algebras, isoclinism, and invariant probabilities, contemp. math. 634 (2015) 31–52. 195 http://dx.doi.org/10.1090/conm/634/12689 http://dx.doi.org/10.1090/conm/634/12689 s. m. buckley, d. machale / j. algebra comb. discrete appl. 4(2) (2017) 189–196 [2] s. m. buckley, d. machale, commuting probabilities of groups and rings, preprint. 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[10] d. rusin, what is the probability that two elements of a finite group commute?, pacific j. math. 82(1) (1979) 237–247. 196 http://www.maths.nuim.ie/staff/sbuckley/papers/bm_g-vs-r.pdf http://www.maths.nuim.ie/staff/sbuckley/papers/bms.pdf http://www.ams.org/mathscinet-getitem?mr=1882359 http://dx.doi.org/10.1007/bf01894517 http://dx.doi.org/10.1007/bf01894517 http://dx.doi.org/10.1016/j.jalgebra.2005.09.044 http://dx.doi.org/10.1016/j.jalgebra.2005.09.044 http://dx.doi.org/10.2307/3615961 http://dx.doi.org/10.2307/3615961 http://dx.doi.org/10.2307/2318829 http://dx.doi.org/10.2140/pjm.1979.82.237 http://dx.doi.org/10.2140/pjm.1979.82.237 introduction preliminaries proofs counterexamples references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.560404 j. algebra comb. discrete appl. 6(2) • 53–62 received: 10 july 2018 accepted: 12 march 2019 journal of algebra combinatorics discrete structures and applications some explicit expressions for the structure coefficients of the center of the symmetric group algebra involving cycles of length three research article omar tout abstract: we use the combinatorial way to give an explicit expression for the product of the class of cycles of length three with an arbitrary class of cycles. in addition, an explicit formula for the coefficient of an arbitrary class in the expansion of the product of an arbitrary class by the class of cycles of length three is given. 2010 msc: 05e15, 20c08 keywords: the center of the symmetric group algebra, structure coefficients, product of conjugacy classes of the symmetric group, representation theory of the symmetric group 1. introduction if n is a positive integer, we denote by sn the symmetric group of permutations on the set [n] := {1,2, · · · ,n}. the center of the symmetric group algebra, denoted by z(c[sn]), is linearly generated by elements cλ, indexed by partitions of n, where cλ is the sum of permutations of [n] with cycle-type λ. the structure coefficients cρλδ describe the product in this algebra, they are defined by the equation: cλcδ = ∑ ρ`n c ρ λδcρ. in other words, cρλδ counts the number of pairs of permutations (x,y) with cycle-type λ and δ such that x · y = z for a fixed permutation z with cycle-type ρ. there is no general formula to compute the coefficients cρλδ. omar tout; department of mathematics, faculty of sciences iii, lebanese university, tripoli, lebanon (email: omar-tout@outlook.fr). 53 https://orcid.org/0000-0003-0963-9639 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 the main two tools to compute the structure coefficients of the center of the symmetric group algebra are the combinatorial way and the character theory. the first one is about studying the cycletype of the product of two permutations which can be a difficult task, see for example [1], [2], [3] and [9]. however, this way had led to a complete formula for the structure coefficients when δ is the partition coding transpositions, see [8]. by using the representation theory of the symmetric group, the structure coefficients can be expressed in terms of irreducible characters of the symmetric group due to the frobenius theorem, see [7, lemma 3.3]. this theorem was used by goupil and schaeffer to obtain an explicit formula for the structure coefficients cρλδ when one of the partitions codes the cycles of length n, see [6]. in 1958, farahat and higman proved in [5, theorem 2.2] that the coefficients cρλδ are polynomials in n when λ, δ and ρ are fixed partitions, completed with parts equal to 1 to get partitions of n. in 1975, cori showed in his thesis [4] that these coefficients count the number of embeddings of certain graphs into orientable surfaces. the goal of this paper is to give new explicit formulas for the structure coefficients of the center of the symmetric group algebra using the combinatorial method. we will be interested in computing the product of an arbitrary class of the symmetric group by another coding cycles, especially of lengh three. we hope that these results will be useful to obtain an explicit expression for the product of the class of cycles of length three with an arbitrary class of the symmetric group. 2. definitions and the combinatorial way to compute the structure coefficients a partition λ is a list of integers (λ1, . . . ,λl) where λ1 ≥ λ2 ≥ . . .λl ≥ 1. the λi are called the parts of λ; the size of λ, denoted by |λ|, is the sum of all of its parts. if |λ| = n, we say that λ is a partition of n and we write λ ` n. the number of parts of λ is denoted by l(λ). we will especially use the exponential notation λ = (1m1(λ),2m2(λ),3m3(λ), . . .), where mi(λ) is the number of parts equal to i in the partition λ. in case there is no confusion, we will omit λ from mi(λ) to simplify our notation. if λ = (1m1(λ),2m2(λ),3m3(λ), . . . ,nmn(λ)) is a partition of n then ∑n i=1 imi(λ) = n. we will dismiss imi(λ) from λ when mi(λ) = 0. for example, we will write λ = (12,3,62) instead of λ = (12,20,3,40,50,62,70). if λ and δ are two partitions we define the union λ∪δ and subtraction λ\δ (if exists) as the following partitions: λ∪ δ = (1m1(λ)+m1(δ),2m2(λ)+m2(δ),3m3(λ)+m3(δ), . . .). λ\ δ = (1m1(λ)−m1(δ),2m2(λ)−m2(δ),3m3(λ)−m3(δ), . . .) if mi(λ) ≥ mi(δ) for any i. the cycle-type of a permutation of sn is the partition of n obtained from the lengthes of the cycles that appear in its decomposition into product of disjoint cycles. for example, the permutation (2,4,1,6)(3,8,10,12)(5)(7,9,11) of s12 has cycle-type (1,3,42). in this paper we will denote the cycle-type of a permutation ω by ct(ω). it is well known that two permutations of sn belong to the same conjugacy class if and only if they have the same cycle-type. thus the conjugacy classes of the symmetric group sn can be indexed by partitions of n. if λ = (1m1(λ),2m2(λ),3m3(λ), . . . ,nmn(λ)) is a partition of n, we will denote by cλ the conjugacy class of sn associated to λ : cλ := {σ ∈sn | ct(σ) = λ}. the cardinal of cλ is given by: |cλ| = n! zλ , 54 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 where zλ = 1 m1(λ)m1(λ)!2 m2(λ)m2(λ)! · · ·nmn(λ)mn(λ)!. the family (cρ)ρ`n, indexed by partitions of n where cλ is the sum of permutations of sn with cycle-type ρ cρ = ∑ σ∈cρ σ forms a basis for the center of the symmetric group algebra. if λ and δ are two partitions of n, the product cλcδ can be written as a linear combination of the elements (cρ)ρ`n as follows: cλcδ = ∑ ρ`n c ρ λδcρ. the coefficients cρλδ are called the structure coefficients of the center of the symmetric group algebra. there is a combinatorial way to find these structure coefficients. to compute cρλδ, we fix a permutation γ ∈ cρ, that means γ is a permutation of sn with ct(γ) = ρ, then c ρ λδ is the cardinal of the followin set: c ρ λδ =| {(α,β) ∈ cλ ×cδ such that α◦β = γ} | . (1) alternatively, c ρ λδ =| {β ∈ cδ such that ct(γ ◦β −1) = λ} | . (2) we will use the above two formulas (1) and (2) to compute the coefficients cρλδ in this paper. since there is no general formula for the structure coefficients of z(c[sn]), the combinatorial way is useful to investigate a formula in case of special partitions. for example, to compute the product c(1n−2,2)c(1n−2,2), we need to know how a transposition acts on another transposition. the composite of two transpositions of sn gives: • the identity if both transpositions are equals, • a cycle of length three if they share only one element in their support, • a product of two transpositions if their support are disjoint. by using the combinatorial way, see [12, example 6.5],we deduce that: c(1n−2,2)c(1n−2,2) = n(n−1) 2 c(1n) + 3c(1n−3,3) + 2c(1n−4,22). in fact, if λ is a partition of n, there exists an explicit formula for the product cλc(1n−2,2) of the class corresponding to λ with the class of transpositions. it appears for the first time in [8] and as stated here it can be found in [11, proposition 2.8]. proposition 2.1. let λ = (1m1,2m2, . . . ,nmn) be any partition of n then: cλ.c(1n−2,2) = n∑ i=2,mi 6=0 ∑ k≤e(i/2) φkicλki + n∑ i=1,mi≥2 φ ′i i cλ′i i + n∑ i=1,mi 6=0 n−i∑ k>i,mk 6=0 φ ′k i cλ′k i 55 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 where λki = { λ∪ (k,i−k)\ (i) if i−k 6= k λ∪ (k2)\ (i) if i−k = k λ ′k i = { λ\ (i,k)∪ (i + k) if k 6= i λ\ (i2)∪ (2i) if k = i φki = { k(mk + 1)(i−k)(mi−k + 1) if i−k 6= k k2(mk+1)(mk+2) 2 if i−k = k φ ′k i = { (i + k)(mi+k + 1) if k 6= i i(m2i + 1) if k = i example 2.2. 1. when λ = (1n−2,2), we get: c(1n−2,2).c(1n−2,2) = φ12cλ12 + φ ′1 1 cλ′11 + φ ′2 1 cλ′21 = n(n−1) 2 c(1n) + 2c(1n−4,22) + 3c(1n−3,3). 2. if r ≥ 2 and n ≥ 2r + 2, we have: c(1n−2r,2r).c(1n−2,2) = ∑ 1≤k≤r φk2rcλk2r + φ ′1 1 cλ′11 + φ ′2r 1 cλ′2r1 = (n−2r + 1)(2r −1)c(1n−2r+1,2r−1) + ∑ 1<k<r k(2r −k)c(1n−2r,k,2r−k) + r2c(1n−2r,r2) +c(1n−2r−2,2,2r) + (1 + 2r)c(1n−2r−1,2r+1), 3. if r ≥ 1 and n ≥ 2r + 3, we have: c(1n−2r−1,2r+1).c(1n−2,2) = ∑ 1≤k≤r φk2r+1cλk2r+1 + φ ′1 1 cλ′11 + φ ′2r+1 1 cλ′2r+11 = (n−2r)(2r)c(1n−2r,2r) + ∑ 1<k<r k(2r + 1−k)c(1n−2r−1,k,2r+1−k) + c(1n−2r−3,2,2r+1) + (2 + 2r)c(1n−2r−2,2r+2), 4. the above proposition along with the derived formulas in items (1), (2) and (3) give the following expressions in case n = 6 : c(14,2).c(14,2) = 15c(1n) + 2c(12,22) + 3c(13,3) c(13,3).c(14,2) = 8c(14,2) +c(1,2,3) + 4c(12,4) c(1,5).c(14,2) = 8c(12,4) + 6c(6) c(12,4).c(14,2) = 9c(13,3) + 4c(12,22) +c(2,4) + 5c(1,5) c(6).c(14,2) = 5c(1,5) + 8c(2,4) + 9c(32) c(23).c(14,2) = φ12cλ12 + φ ′2 2 cλ′22 = c(12,22) + 2c(2,4) c(32).c(14,2) = φ13cλ13 + φ ′3 3 cλ′33 = 2c(1,2,3) + 3c(6) c(12,22).c(14,2) = φ12cλ12 + φ ′1 1 cλ′11 + φ ′2 2 cλ′22 + φ ′2 1 cλ′21 = 6c(14,2) + 3c(23) + 2c(12,4) + 3c(1,2,3) c(1,2,3).c(14,2) = φ12cλ12 + φ 1 3cλ13 + φ ′2 1 cλ′21 + φ ′3 1 cλ′31 + φ ′3 2 cλ′32 = 3c(13,3) + 8c(12,22) + 6c(32) + 4c(2,4) + 5c(1,5) c(2,4).c(14,2) = φ12cλ12 + φ 1 4cλ14 + φ 2 4cλ24 + φ ′4 2 cλ′42 = c(12,4) + 3c(1,2,3) + 12c(23) + 6c(6) 56 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 3. some new explicit formulas for the structure coefficients in this section we give new explicit formulas for the structure coefficients of the center of the symmetric group algebra involving cycles of length three. specifically, in proposition 3.4 we give an explicit formula for the coefficients cλ λ,(1n−3,3) and in proposition 3.10 the complete expression of the the product c(1n−k,k)c(1n−3,3) is given. we hope that these two results will help find an explicit expression for the product of an arbitrary class of permutations with the class of cycles of length three similar to that given in proposition 2.1. a cycle of length k will be sometimes called a k-cycle. if f and g are two permutations we will write fg from time to time to denote the composite f ◦ g. occasionally and to simplify our notation, we will dismiss the fixed elements when writting a permutation as a product of disjoint cycles. let λ = (1m1,2m2, . . . ,nmn) be a partition of n. the following permutation of n pλ := (1,2)(3,4) · · ·(2m2 −1,2m2)︸ ︷︷ ︸ m2(λ) transposition (2m2 + 1,2m2 + 2,2m2 + 3) · · ·(2m2 + 1,2m2 + 2,2m2 + 3)︸ ︷︷ ︸ m3(λ) cycles of length 3 · · · will be called the canonical permutation of n associated to λ. for example: p(22,3) = (1,2)(3,4)(5,6,7), p(13,42,5) = (1,2,3,4)(5,6,7,8)(9,10,11,12,13)(14)(15)(16). lemma 3.1. let n and k be integers such that n ≥ k ≥ 3, then card ( {(α,β) ∈ c(1n−k,k) ×c(1n−3,3) such that α◦β = p(1n−k,k)} ) = ( k 3 ) . lemma 3.2. let n and k be integers such that n > k ≥ 2, then p(1n−k,k) ◦ (k k −1 k + 1) = (1 2 3 · · ·k −1 k + 1)(k)(k + 2) · · ·(n). lemma 3.3. let (i1, i2, · · · , ik) and (r1,r2, · · · ,rt) be two disjoint cycles of size k and t respectively with t > k ≥ 2, then: (i1, i2, · · · , ik)(r1,r2, · · · ,rt)(ik,rt,rt−k) = (i1, i2, · · · , ik,r1,r2, · · · ,rt−k)(rt−k+1, · · · ,rt). proposition 3.4. let λ = (1m1,2m2, . . . ,nmn) be any partition of n ≥ 3 then: cλλ(1n−3,3) = ∑ i≥3, mi≥1 mi ( i 3 ) + ∑ i≥2, mi≥1 m1.i.mi + ∑ i≥2, mi≥1 ∑ j>i, mj≥1 i.j.mimj. (3) proof. to compute cλ λ(1n−3,3) by the combinatorial way, we should find the number of couples (α,β) such that ct(α) = λ, ct(β) = (1n−3,3) and αβ = pλ. we have to consider the following three cases: 1. there exists i ≥ 3 such that mi 6= 0, say c is a cycle of length i in pλ. we will choose the three elements of the 3-cycle in β among the elements of the cycle c and thus α will act as pλ outside the support of c. by lemma 3.1, and after renormalisation we can find ( i 3 ) such couples (α,β). we can repeat the same process as many times as we have cycles of length i in λ which justify the first coefficient ∑ i≥3, mi≥1 mi ( i 3 ) in the formula (3). 57 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 2. there exists i ≥ 2 such that mi 6= 0 and m1 6= 0. consider (a1,a2, · · · ,ai) to be a cycle of length i in pλ. choose α to be the same as pλ except that the cycle (a1,a2, · · · ,ai) is replaced by (a1,a2, · · · ,ai−1,b) where b is a chosen element among the m1 unchaged integer by pλ. then by lemme 3.2, α ◦ β = pλ, where β = (ai−1,ai,b). since there exists i way to write the cycle (a1,a2, · · · ,ai−1,b) and for each way the corresponding β is different, we conclude that for this case there are ∑ i≥2, mi≥1 m1.i.mi ways to choose the couple (α,β). 3. there exists i ≥ 2 and j > i such that mi 6= 0 and mj 6= 0. suppose (a1,a2, · · · ,ai) and (b1,b2, · · · ,bj) are two cycles of length i and j in pλ. then by lemma 3.3, if we take α to be the same as pλ except that the cycles (a1,a2, · · · ,ai) and (b1,b2, · · · ,bj) are replaced by (bi+1,bi+2, · · · ,bj,a1,a2, · · · ,ai) and (b1, · · · ,bi). then it is clear that ct(α) = λ and αβ = pλ where β = (bi,ai,bj). since there are j ways to write the cycle (bi+1,bi+2, · · · ,bj,a1,a2, · · · ,ai) and i ways to write (b1, · · · ,bi), we will get in total ij couples (α,β) satisfying the desired conditions. we can repeat the same process as many times as we have cycles of lengthes i and j in pλ which gives reason to the coefficient ∑ i≥2, mi≥1 ∑ j>i, mj≥1 i.j.mimj that appears in formula (3). example 3.5. we illustrate the above proof by an example. suppose λ = (12,2,4) is a partition of 8. we will list below all the possible permutations α and β of 8 such that ct(α) = λ, ct(β) = (15,3) and α◦β = pλ. there are three cases to distinguish as in the above proof: a. the support of β is taken from the cycle of length 4 of pλ : 1. α = (1,2)(3,5,4,6) and β = (3,5,4) 2. α = (1,2)(3,6,4,5) and β = (3,6,5) 3. α = (1,2)(4,6,5,3) and β = (4,6,5) 4. α = (1,2)(5,6,4,3) and β = (3,6,4) b. two elements in the support of β are taken either from the cycle of length 2 or the cycle of lentgh 4 of pλ and the third one is taken among the invariant elements of pλ : 1. α = (1,7)(3,4,5,6) and β = (1,2,7) 2. α = (1,8)(3,4,5,6) and β = (1,2,8) 3. α = (2,7)(3,4,5,6) and β = (2,1,7) 4. α = (2,8)(3,4,5,6) and β = (2,1,8) 5. α = (1,2)(3,4,5,7) and β = (7,5,6) 6. α = (1,2)(3,4,5,8) and β = (8,5,6) 7. α = (1,2)(3,4,7,6) and β = (4,5,7) 8. α = (1,2)(3,4,8,6) and β = (4,5,8) 9. α = (1,2)(3,7,5,6) and β = (3,4,7) 10. α = (1,2)(3,8,5,6) and β = (3,4,8) 58 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 11. α = (1,2)(7,4,5,6) and β = (3,7,6) 12. α = (1,2)(8,4,5,6) and β = (3,8,6) b. two elements in the support of β are taken from the cycle of length 4 of pλ and the third one is taken from the cycle of length 2 : 1. α = (3,4)(5,6,1,2) and β = (2,6,4) 2. α = (3,4)(5,6,2,1) and β = (1,6,4) 3. α = (6,3)(4,5,1,2) and β = (2,5,3) 4. α = (6,3)(4,5,2,1) and β = (1,5,3) 5. α = (5,6)(3,4,1,2) and β = (2,4,6) 6. α = (5,6)(3,4,2,1) and β = (1,4,6) 7. α = (4,5)(6,3,1,2) and β = (2,3,5) 8. α = (4,5)(6,3,2,1) and β = (1,3,5) thus [c(12,2,4)]c(12,2,4)c(15,3) = 4 + 12 + 8 = 24. example 3.6. by formula (3), we get: c (2,6) (2,6),(15,3) = 32, c (1,42) (1,42),(16,3) = 16, c (2,7) (2,7),(16,3) = 49, c (3,6) (3,6),(16,3) = 39, c (14,5) (14,5),(16,3) = 30, c (12,2,5) (12,2,5),(16,3) = 34, c (2,3,4) (2,3,4),(16,3) = 31, c (1,2,32) (1,2,32),(16,3) = 22, c (13,2,4) (13,2,4),(16,3) = 30, c (12,2,32) (12,2,32),(17,3) = 30, c (12,2,6) (12,2,6),(17,3) = 48, c (3,7) (3,7),(17,3) = 57. we used the sagemath software [10] to confirm these results. we turn now to compute remarkable coefficients in the product cλc(1n−3,3), the coefficients c (1n−3,3) λ,(1n−3,3) , c λ\(13)∪(3) λ,(1n−3,3) and cλ\(3)∪(1 3) λ,(1n−3,3) . proposition 3.7. let λ = (1m1,2m2, . . . ,nmn) be any partition of n ≥ 6 then: c (1n−3,3) λ(1n−3,3) =   2 ( n−3 3 ) if λ = (1n−6,32), 3m1 + 1 if λ = (1 n−3,3), 0 otherwise. proof. first if λ = (1n−3,3), use formula (3) to get the result. suppose now that λ = (1n−6,32), then c(1 n−3,3) λ(1n−3,3) is the number of couples (α,β) such that ct(α) = (1n−6,32), ct(β) = (1n−3,3) and αβ = p(1n−3,3). to choose α, pick the first 3-cycle of α to be (1,2,3) and form the second 3-cycle c by choosing arbitrary 3 integers of the remaining n−3 integers, there are 2 ( n−3 3 ) such way to do that. take β = c−1, then αβ = p(1n−3,3) and c (1n−3,3) (1n−6,32),(1n−3,3) = 2 ( n−3 3 ) . proposition 3.8. let λ = (1m1,2m2, . . . ,nmn) be any partition of n ≥ 3 such that m1 ≥ 3 then: c λ\(13)∪(3) λ(1n−3,3) = m3 + 1. 59 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 proof. any permutation with cycle-type λ \ (13) ∪ (3) has m3 + 1 cycle of length 3. let γ be such a permutation. the coefficient cλ\(1 3)∪(3) λ,(1n−3,3) is the number of couples (α,β) such that ct(α) = λ, ct(β) = (1n−3,3) and αβ = γ.we can choose β to be any 3 cycle among the m3 + 1 3-cycles of γ and α is the same as γ except that the cycle β is replaced by the identity on its support. proposition 3.9. let λ = (1m1,2m2, . . . ,nmn) be any partition of n ≥ 3 such that m3 ≥ 1 then: c λ\(3)∪(13) λ(1n−3,3) = 2 ( m1 + 3 3 ) . proof. let ρ be a permutation such that ct(ρ) = λ\(3)∪(13). the coefficient cλ\(3)∪(1 3) λ,(1n−3,3) is the number of couples (α,β) such that ct(α) = λ, ct(β) = (1n−3,3) and αβ = ρ. the permutation ρ has m1 + 3 fixed element. let β be any 3-cycle formed with these m1 + 3 fixed elements. there are 2 ( m1+3 3 ) way to make β. since β is a 3-cycle, its inverse β−1 is a 3-cycle also and ρβ−1 has cycle-type λ. the next proposition gives an explicit expression for the product of the class of 3-cycles by the class of cycles of length k for an arbitrary k. proposition 3.10. let n ≥ k > 3 then: c(1n−k,k)c(1n−3,3) = ∑ i≤j≤r, i+j+r=k 2ijr ( m1((1 n−k)∪ (i,j,r)) n−k ) c(1n−k)∪(i,j,r) + [(k+1)/2]∑ i=2 2i(k + 1− i)c(1n−k−1)∪(i,k+1−i)︸ ︷︷ ︸ if n≥k+1 + (k + 2)c(1n−k−2)∪(k+2)︸ ︷︷ ︸ if n≥k+2 + c(1n−k−3,k)∪(3)︸ ︷︷ ︸ if n≥k+3 + [(k 3 ) + (n−k)k ] c(1n−k,k) proof. we use formula (2) in this proof. let i,j and r be three integers such that i ≤ j ≤ r and i + j + r = k. consider the permutation p(1n−k)∪(i,j,r), the coefficient c (1n−k)∪(i,j,r) (1n−k,k),(1n−3,3) is the number of permutations β such that ct(β) = (1n−3,3) and ct(p(1n−k)∪(i,j,r)β−1) = (1n−k,k). choose β = (x,y,z) to be any cycle formed with an element x of [i], an element y of [i + j]\ [i] and an element z of [k]\ [i + j]. there are thus 2ijr ways to form β. then p(1n−k)∪(i,j,r)β−1 is the following cycle of length k (1,2, · · · ,x,z + 1, · · · ,k,i + j + 1, · · · ,z,y + 1, · · · , i + j,i + 1, · · · ,y,x + 1, · · · , i). to obtain the coefficient 2ijr ( m1((1 n−k)∪(i,j,r)) n−k ) that appears in formula (4), one should remark that when i = 1, x can be chosen among the m1 + 1 fixed elements of p(1n−k)∪(i,j,r), and when i = 1 and j = 1, x and y can be chosen among the m1 + 2 fixed elements of p(1n−k)∪(i,j,r). now fix an integer 2 ≤ i ≤ [(k + 1)/2]. to compute c(1 n−k−1)∪(i,k+1−i) (1n−k,k),(1n−3,3) , we need to determine the number of permutations β such that ct(β) = (1n−3,3) and ct(p(1n−k−1)∪(i,k+1−i)β−1) = (1n−k,k). we can choose β to be: • any 3-cycle (a,k,k + 1) with a ∈ [i], then p(1n−k−1)∪(i,k+1−i)β −1 = (1,2, · · · , i)(i + 1, · · · ,k + 1)(k + 1,k,a)(k + 2) · · ·(n) = (1,2, · · · ,a,i + 1, · · · ,k,a + 1, · · · , i)(k + 1)(k + 2) · · ·(n). since there are k − i + 1 ways to write the cycle (i + 1, · · · ,k + 1), in total there are i.(k − i + 1) possibilities to choose β in this case. 60 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 • any 3-cycle (b,i−1, i) with b ∈ [k]\ [i], then p(1n−k−1)∪(i,k+1−i)β −1 = (1,2, · · · , i)(i + 1, · · · ,k + 1)(i, i−1,b)(k + 2) · · ·(n) = (1,2, · · · , i−1,b + 1, · · · ,k + 1, i + 1, · · · ,b)(k + 2) · · ·(n). since there are i ways to write the cycle (1, · · · , i), in total there are i.(k − i + 1) possibilities to choose β in here. thus in total, we obtain the coefficient 2i(k + 1− i). we turn now to calculate the number of permutations β such that ct(β) = (1n−3,3) and ct(p(1n−k−2)∪(k+2)β −1) = (1n−k,k). there are k + 2 ways to choose β, it can be taken to be any cycle (i, i + 1, i + 2) with [i] ∈ [k + 2] ( k + 3 ≡ 1 and k + 4 ≡ 2 in here). then p(1n−k−2)∪(k+2)β −1 = (1,2, · · · ,k + 2)(i + 2, i + 1, i)(k + 3) · · ·(n) = (1,2 · · · , i, i + 3, · · · ,k + 2)(i + 1)(i + 2)(k + 3) · · ·(n) is a k-cycle. it remains to remark that the last two coefficients in formula (4) can be derived from proposition 3.8 and proposition 3.4 respectively. example 3.11. 1. we list below all the cases for n = 9 : c(9)c(16,3) = 14c(12,7) + 24c(1,2,6) + 30c(1,3,5) + 40c(22,5) + 32c(1,42) +48c(2,3,4) + 54c(33) + 84c(9). c(1,8)c(16,3) = 36c(13,6) + 40c(12,2,5) + 48c(12,3,4) + 32c(1,22,4) + 36c(1,2,32) +40c(4,5) + 36c(3,6) + 28c(2,7) + 64c(1,8). c(12,7)c(16,3) = 60c(14,5) + 48c(13,2,4) + 24c(12,22,3) + 54c(13,32) + 32c(1,42) +30c(1,3,5) + 24c(1,2,6) + 49c(12,7) + 9c(9). c(13,6)c(16,3) = 80c(15,4) + 48c(14,2,3) + 16c(13,23) + 20c(12,2,5) + 24c(12,3,4) +38c(13,6) +c(3,6) + 8c(1,8). c(14,5)c(16,3) = 90c(16,3) + 40c(15,22) + 7c(12,7) + 16c(13,2,4) + 18c(13,32) +c(1,3,5) + 30c(14,5). c(15,4)c(16,3) = 84c(17,2) + 6c(13,6) + 12c(14,2,3) +c(12,3,4) + 24c(15,4). c(16,3)c(16,3) = 168c(19) + 5c(14,5) + 8c(15,22) + 2c(13,32) + 19c(16,3). 2. c(13,7)c(17,3) = 100c(15,5) + 64c(14,2,4) + 24c(13,22,3) + 72c(14,32) + 32c(12,42) + 30c(12,3,5) + 24c(12,2,6) +c(3,7) + 9c(1,9) + 56c(13,7). the coefficients cρλδ can be expressed in terms of irreducible characters of the symmetric group: c ρ λδ = |cλ||cδ| n! ∑ µ`n χ µ λχ µ δχ µ ρ dimvµ , (4) where χµλ is the irreducible character of the symmetric group associated to the partition µ evaluated on an element of the conjugacy class cλ and dimvµ is the dimension of the irreducible representation associated to the partition µ. the formula (4) is due to frobenius. it was used along with the murnaghan-nakayama rule to compute the structure coefficients of the center of the symmetric group algebra, see [6] for the computation of the coefficients c(n)λµ . it will be interesting to investigate how the explicit formulas for the structure coefficients given in this paper could be found by the frobenius formula. 61 o. tout / j. algebra comb. discrete appl. 6(2) (2019) 53–62 references [1] z. arad, m. herzog (eds.), products of conjugacy classes in groups, lecture notes in math., vol. 1112, springer-verlag, berlin, 1985. 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[8] j. katriel, j. paldus, explicit expression for the product of the class of transpositions with an arbitrary class of the symmetric group, r gilmore (ed.), group theoretical methods in physics, world scientific, singapore (1987) 503–506. [9] r. p. stanley, factorization of permutations into n-cycles, discrete math. 37(2-3) (1981) 255–262. [10] w. a. stein et al., sage mathematics software (version 4.8), the sage development team, http://www.sagemath.org. [11] o. tout, polynomialité des coefficients de structure des algèbres de doubles-classes. ph.d thesis, université de bordeaux, 2014. [12] o. tout, a general framework for the polynomiality property of the structure coefficients of double– class algebras, j. algebr. comb. 45(4) (2017) 1111–1152. 62 https://doi.org/10.1007/bfb0072284 https://doi.org/10.1007/bfb0072284 https://doi.org/10.1137/0601050 https://doi.org/10.1137/0601050 https://doi.org/10.1016/0012-365x(80)90001-1 https://doi.org/10.1016/0012-365x(80)90001-1 https://mathscinet.ams.org/mathscinet-getitem?mr=404045 https://doi.org/10.1098/rspa.1959.0060 https://doi.org/10.1098/rspa.1959.0060 https://doi.org/10.1006/eujc.1998.0215 https://doi.org/10.1006/eujc.1998.0215 https://doi.org/10.1090/s0002-9947-1990-1012516-6 https://doi.org/10.1090/s0002-9947-1990-1012516-6 https://mathscinet.ams.org/mathscinet-getitem?mr=1026700 https://mathscinet.ams.org/mathscinet-getitem?mr=1026700 https://mathscinet.ams.org/mathscinet-getitem?mr=1026700 https://doi.org/10.1016/0012-365x(81)90224-7 http://www.sagemath.org http://www.sagemath.org https://tel.archives-ouvertes.fr/tel-01152767 https://tel.archives-ouvertes.fr/tel-01152767 https://doi.org/10.1007/s10801-017-0736-8 https://doi.org/10.1007/s10801-017-0736-8 introduction definitions and the combinatorial way to compute the structure coefficients some new explicit formulas for the structure coefficients references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284950 j. algebra comb. discrete appl. 4(2) • 141–154 received: 16 june 2015 accepted: 18 february 2016 journal of algebra combinatorics discrete structures and applications codes and the steenrod algebra research article steven t. dougherty, tane vergili abstract: we study codes over the finite sub hopf algebras of the steenrod algebra. we define three dualities for codes over these rings, namely the eulidean duality, the hermitian duality and a duality based on the underlying additive group structure. we study self-dual codes, namely codes equal to their orthogonal, with respect to all three dualities. 2010 msc: 11t71, 94b05 keywords: self-dual codes, non-commutative rings, steenrod algebra 1. introduction codes over commutative rings have received a great deal of attention since the discovery in the early 1990s that certain non-linear binary codes were in fact the images under a gray map of codes over z4. very little work has been done yet on codes over non-commutative rings. in [11], j. wood gave foundational results for codes over commutative and non-commutative rings. specifically, he showed that frobenius rings were the class of rings for which it was natural to study codes since both macwilliams theorems hold in this case. in [4], dougherty and leroy described some general theorems about self-dual codes over non-commutative rings with respect to the euclidean inner-product. in this work, we shall study codes over a family of non-commutative frobenius rings that are of great importance in the study of algebraic topology. namely, we study codes over the finite sub hopf algebras of the steenrod algebra. we take a broader approach to duality in that we consider both the euclidean and hermitian innerproducts as well as duality based on the underlying additive group structure. we consider linear codes as well as additive codes. namely, linear codes are when the code is a submodule of the ambient space and additive codes are when they are simply a subgroup of the ambient space in terms of the additive operation. steven t. dougherty (corresponding author); department of mathematics, university of scranton, scranton, pa 18510, usa (email: prof.steven.dougherty@gmail.com). tane vergili; department of mathematics, ege university, 35100 izmir, turkey (email: tanevergili@gmail.com). 141 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 2. definitions and notations 2.1. steenrod algebra in this paper, we shall use as our coding alphabet, the steenrod algebra at the prime 2. we assume throughout the paper that all computations in the steenrod algebra and in the sub hopf algebra are done with the assumption that the prime is 2. for a complete topological discussion about the steenrod algebra see [10]. we shall now give an algebraic description of these algebras. we start our description by defining the steenrod squaring operations sqk. by convention we have that sq0 = 1 and sqk is assigned grading k. the steenrod algebra a is the free associative graded algebra generated by sqk over the field f2 subject to the following relations: sqksqj = ∑ 0≤i≤bk 2 c ( j − i− 1 k − 2i ) sqj+k−isqi (1) for 0 < k < 2j. these relations are known as the adem relations. the steenrod squares sqk are group homomorphisms sqk : hi(x; z2) −→ hi+k(x; z2) between the cohomology groups of a topological space x, for k,i ≥ 0 satisfying the following (see [10] for a complete description): 1. the square sq0 is an identity and if i < k, then sqk = 0. 2. if k = i, then sqkx = x2 for all x ∈ hi(x; z2). 3. the square sqk(x ∪ y) = ∑ k=k1+k2 sqk1 (x) ∪ sqk2 (y), where the operation ∪ is the cup product of the cohomology ring h∗(x; z2) := ⊕ n≥0 hn(x; z2) and x,y ∈ h∗(x; z2). by utilizing z2 as the coefficient group of the cohomology group, no sign problems occur. the grading of the steenrod square sqk is k and for the monomial formed as the composition of the steenrod squares, sqk1sqk2 · · ·sqki, is k1 + k2 + . . . + ki. formally, the steenrod algebra a is the graded associative algebra generated over the finite field f2 by the steenrod squares subject to the adem relations and the identity homomorphism sq0. the operations sq0 and sq2 k , k ≥ 0, constitute a system of multiplicative generators for a, see [9] for a complete description. the steenrod algebra has a hopf algebraic structure (see [7]) and is the union of the finite sub hopf algebras a(n), for n ≥ 0, where a(n) is generated by the squares sq2 j for 0 ≤ j ≤ n and sq0. note that a(n) ⊆ a(n + 1) for all n ≥ 0. r. wood [13] has defined the atomic squares which are of the form sq2 s(2t−1) where s ≥ 0, t > 0 are integers such that s + t ≤ n + 1, to form a z base system for the a(n) which can be extended to the whole algebra. the z base system for a(n) is constructed as follows: let xn = sq 1·2nsq3·2 n−1 sq7·2 n−2 · · ·sq2 n+1−1 and define zn := xnxn−1 · · ·x1x0. for instance, 142 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 z1 = sq 2sq3sq1, z2 = sq 4sq6sq7sq2sq3sq1, and z3 = sq8sq12sq14sq15sq4sq6sq7sq2sq3sq1. the element zn is the top element for a(n) in terms of the grading. the set of 2(n+1)(n+2)/2 monomials obtained by selecting all subsets of atomic factors in zn, in the given order, is an additive basis for a(n) (see [13]). we note that the product of the top element in a(i) and the top element in a(j) is 0 in the steenrod algebra, that is zizj = 0, for all i,j, since it exceeds the maximum grading in a(n) where n = max{i,j}. see [t. vergili, i. karaca, a note on the new basis in the mod 2 steenrod algebra, in preparation, 2016] for a description of the basis of the steenrod algebra. all computations done in this paper for the steenrod algebra where performed using the computational tool given in [6]. throughout the paper, we shall denote sqasqb by sqa,b for convenience. 2.2. codes and rings a code c of length m over a ring r is a subset of rm. if the code is a left module then we say that c is left linear and if c is a right module then we say that c is right linear. suppose r is a finite ring. let m̂ denote the character module homz(m,c) where m is a module. the following are equivalent for finite rings, see [11]: • r is a frobenius ring. • as a left module, r̂ ∼=r r. • as a right module r̂ ∼= rr. it is well known that a(n) is a frobenius ring for all n, see [11] for example. next, we shall define an involution of the steenrod algebra which also applies to the sub hopf algebras. define the map τ : a → a by τ(sq0) = sq0, and τ(sqk) = k∑ i=1 sqiτ(sqk−i). (2) the map τ can be restricted to a(n) in a natural way as long as sqk ∈ a(n). it is well known that τ is an anti-isomorphism, that is, τ is additive and τ(ab) = τ(b)τ(a) and that τ2 is the identity map. we note that τ is often written as χ in the literature, see [7], [14] and [10] for example, but we shall use χ as a generating character of the character module as is standard and use τ for this anti-isomorphism. 2.3. orthogonals we shall now describe inner-products which can be used in a(n)m. in classical coding theory, the euclidean inner-product is the standard inner-product. however, it is often the case that the hermitian inner-product is used for specific applications. for example, it is used when self-dual codes over finite rings are used to construct complex and quaternionic lattices, see [3] and [2] for example. define the two following inner-products. the euclidean inner-product is defined as [v,w] = ∑ viwi. (3) the hermitian inner-product is defined as [v,w]h = ∑ viτ(wi). (4) 143 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 let c be a code then the left euclidean orthogonal is l(c) = {v | [v,w] = 0, ∀w ∈ c} (5) and the right euclidean orthogonal is r(c) = {v | [w,v] = 0, ∀w ∈ c}. (6) let c be a code then left hermitian orthogonal is lh (c) = {v | [v,w]h = 0, ∀w ∈ c} (7) and the right hermitian orthogonal is rh (c) = {v | [w,v]h = 0, ∀w ∈ c}. (8) in [4], is shown that l(c) is a left linear code and r(c) is a right linear code. of course, left linearity does not imply right linearity nor does right linearity imply left linearity. we note that the notion of hermitian and euclidean duality are not identical. for example, let a1 = a2 = sq 2,1. then a1a2 = sq2,3,1 but a1τ(a2) = sq2,1sq3 = 0. so a2 ∈ rh (a(1)[a1]) but a2 6∈r(a(1)[a1]). theorem 2.1. let c be a code over a(n) then lh (c) is a left linear code. proof. let c be a code over a(n). let v,w ∈lh (c). then [av + cw,u]h = ∑ (avi + cwi)τ(ui) = a ∑ viτ(ui) + c ∑ wiτ(ui) = 0 + 0 = 0. hence av + cw ∈lh (c) and it is a left linear code. unlike in the euclidean case, r(c) is not necessarily right linear, since [u,va + wc]h = ∑ ui(τ(via + wic)) = ∑ uiτ(a)τ(vi) + ∑ uiτ(c)τ(wi) which may or may not be 0. however, we do have the following theorem, which again is unlike the euclidean case. theorem 2.2. let c be a code over a(n). then lh (c) = rh (c). proof. let w ∈lh (c). then [w,v]h = 0 for all v ∈ c. this implies that ∑ wiτ(vi) = 0 which gives τ( ∑ wiτ(vi)) = τ(0) = 0. then, we have ∑ τ(τ(vi))τ(wi) = 0 and finally ∑ viτ(wi) = 0. this gives that w ∈rh (c). let w ∈ rh (c). then [v,w]h = 0 for all v ∈ c. this implies that ∑ viτ(wi) = 0 which gives τ( ∑ viτ(wi)) = τ(0) = 0. then we have ∑ τ(τ(wi))τ(vi) = 0 and finally ∑ wiτ(vi) = 0. this gives that w ∈lh (c). example 2.3. consider the two sided ideal a(1)[sq3,1] = 〈sq3,1,sq2,3,1〉 in a(1). then we have that rh (a(1)[sq3,1]) = 〈sq1,sq3,sq2,1,sq2,3,sq3,1,sq2,3,1〉 = lh (a(1)[sq3,1]) is also a two sided ideal in a(1). since the left and right hermitian orthogonals are equal this gives that rh (c) is left linear but it may not be right linear. example 2.4. let c be the code of length 1 over a(1) defined by a = a(1)[sq2 + sq3,1]. then c = 〈sq2+sq3,1,sq3,sq2,3,sq3,1+sq2,3,1,sq2,3,1〉. then rh (c) = 〈sq3,sq2,3,sq2,3,1〉. we have that rh (c) is not right linear since sq3sq1 6∈rh (c). 144 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 for any ring r let j(r) denote the jacobson radical of r, which is defined as the intersection of all maximal left ideals in r. theorem 2.5. let a(n) be the sub hopf algebra and let b1,b2, . . . ,bt be the basis elements with b1 = sq0 and bt = zn. the ring a(n) is a left and right local ring with unique two sided maximal ideal m(n) = a(n)[b2,b3, . . . ,bt] = j(a(n)). we have that l(m(n)) = r(m(n)) = soc(a(n)) = a(n)[bt] = {0,zn}. proof. we know that btbi = bibt = 0 for all i, 1 < i ≤ t. therefore a(n)[bt] = [bt]a(n) = {0,zn} and bt ∈ l(i) for all left and right ideals i ⊂ a(n),i 6= a(n). therefore r(l(i)) = i ⊆ r(a(n)[bt]) and l(r(i)) = i ⊆l(a(n)[bt]). then we have that r(a(n)[bt]) = l(a(n)[bt]) = a(n)[b2,b3, . . . ,bt] = [b2,b3, . . . ,bt]a(n) = m(n). hence a(n)[bt] is the unique minimal ideal and its left and right dual is the unique maximal ideal. therefore soc(a(n)) = a(n)[bt] = {0,zn} and j(a(n)) = a(n)[b2,b3, . . . ,bt]. this leads naturally to the following corollary. corollary 2.6. the two sided ideal {0,zn} is contained in all non-trivial ideals of a(n). 3. macwilliams relations the macwilliams relations are one of the foundational results of algebraic coding theory. they relate the weight enumerator of a linear code with the weight enumerator of its dual. the critical part of finding specific macwilliams relations for a code over a ring r is to find a generating character for r̂. namely, if φ : r → r̂ is a right r-module isomorphism then the generating character is φ(1). a generating character was given for a(1) in [12]. theorem 3.1. let b1, . . . ,bt be a basis for a(n) with bt = zn. define χ : a(n) → c∗ by χ( t∑ i=1 aibi) = (−1)at, (9) where the ai ∈ f2. then χ is a generating character of â(n). proof. it is immediate that χ is a homomorphism and hence a character of a(n). we know from corollary 2.6 that bt ∈ i for all non-zero left ideals i in a(n). also we have that χ(bt) = −1 so χ contains no non-zero ideals in its kernel. by lemma 3.1 in [11], which states that a character is both a left generating and right generating character if it contains no non-trivial ideals in its kernel, we have that χ is a generating character for â(n). notice that the generating character for a(n + 1) is not an extension of the generating character for a(n). we are not claiming that there is a unique generating character. to the contrary, any character whose kernel contains no non-trivial ideal is a generating character. we are simply identifying a useful generating character. we know that a(n) and â(n) are isomorphic (although not canonically). let χa be the character associated with the element a, then we have that χa(c) = χ(ac), where χ is the generating character for â(n). definition 3.2. for a code over an alphabet a = {a0,a1, . . . ,as−1}, the complete weight enumerator is defined as: cwec (xa0,xa1, . . . ,xas−1 ) = ∑ c∈c s−1∏ i=0 xni(c)ai (10) 145 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 where there are ni(c) occurrences of ai in the vector c. let t be the |a(n)| by |a(n)| matrix defined by ta,c = χ(ac). for a matrix m and vector v we let m · v = (mvt)t so that the result is a row vector. in [11], wood establishes the macwilliams relations for codes over frobenius rings. theorem 3.3. if c is a left submodule of a(n)m, then cwec (x0,x1, . . . ,xk) = 1 |r(c)| cwer(c)(t t · (x0,x1, . . . ,xk)). if c is a right submodule of a(n)m, then cwec (x0,x1, . . . ,xk) = 1 |l(c)| cwel(c)(t · (x0,x1, . . . ,xk)). let th be the |a(n)| by |a(n)| matrix with (th )a,c = χ(aτ(c)). we notice that (th )a,c is not identical to ta,c. let a = sq3, c = sq3. then ac = sq2,3,1 = z1 and aτ(c) = 0. thus χ(ac) = −1 but χ(aτ(c)) = 1. while t 6= tt in general, we do have the following for th. theorem 3.4. let (th )a,c = χ(aτ(c)) where χ is the generating character for â(n). then th = tth. proof. we note that the anti-isomorphism τ preserves the grading of a(n), so χ defined as (−1)at for the element ∑ aibi with bi the basis of a(n), satisfies χ(a) = χ(τ(a)). then (th )a,c = χ(aτ(c)) = χ(τ(aτ(c)) = χ(cτ(a)) = (th )c,a. a similar proof to theorem 3.3 applies to the hermitian dual although it is not stated in [11]. namely we have the following. theorem 3.5. if c is a left submodule of a(n)m, then cwec (x0,x1, . . . ,xk) = 1 |rh(c)| cwer(c)(t t h · (x0,x1, . . . ,xk)). if c is a right submodule of a(n)m, then cwec (x0,x1, . . . ,xk) = 1 |lh(c)| cwel(c)(th · (x0,x1, . . . ,xk)). the standard proof, setting xi = 1, gives the following corollary. corollary 3.6. if c is a left linear code over a(n) then |c||rh (c)| = |a(n)|m and if c is a right linear code over a(n) then |c||lh (c)| = |a(n)|m. example 3.7. we continue with example 2.4. let c be the code of length 1 over a(1) defined by a = a(1)[sq2 + sq3,1]. then c = 〈sq2 + sq3,1,sq3,sq2,3,sq3,1 + sq2,3,1,sq2,3,1〉. then rh (c) = 〈sq3,sq2,3,sq2,3,1〉. then |c| = 25 and |rh (c)| = 23 and 2523 = |a(1)|. 146 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 4. self-dual and hermitian self-dual codes self-dual codes are one of the most widely studied families of codes, for both codes over rings and fields. they have interesting applications to designs, lattices and invariant theory. in a recent text [8] a very broad view of self-dual codes have been given with respect to various dualities and interesting connections to invariant theory have been given. in this section, we shall study self-dual codes over the finite ring a(n). we begin with the definition for a self-dual code over a non-commutative ring. definition 4.1. a linear code c is said to be euclidean self-dual if c = l(c). it is shown in [4] that a code c that is equal to l(c) must also be equal to r(c). this implies that c is both left linear and right linear when it is self-dual. this implies that a self-dual code must be a bimodule. definition 4.2. a linear code c is said to be hermitian self-dual if c = lh (c). we know that lh (c) = rh (c) so a hermitian self-dual code satisfies c = lh (c) = rh (c). we now investigate some results about self-orthogonality and self-duality. in [4] it is shown that if v1,v2, . . . ,vs are vectors over a(n) such that [vi,vj] = 0 for all i and j, then [v1,v2, . . . ,vs]a(n) ⊆r(a(n)[v1,v2, . . . ,vs]). (11) notice that we do not necessarily have that a(n)[v1,v2, . . . ,vs] ⊆ r(a(n)[v1,v2, . . . ,vs]) as we would have for commutative rings. for example, if a = sq1 and c = sq2 then a2 = 0 but (ca)2 = (ca)(ca) = sq2,3,1 6= 0. so the code a(1)[a] 6⊆r(a(1)[a]). this means that more must be considered when generating a self-orthogonal code. specifically, it is shown in [4] that if v1,v2, . . . ,vs are vectors in rn, where r is any frobenius ring, then [vi,αvj] = 0 for all i,j and α ∈ r if and only if 〈v1,v2, . . . ,vs〉l is a self-orthogonal code. theorem 4.3. there exists euclidean and hermitian self-dual codes of length 2 over a(n) for all n. proof. consider the code c = a(n)[(sq0,sq0)]. then v ∈ c implies that v = (a,a) which gives [(a,a), (c,c)] = ac+ac = 0. hence it is both left and right self-orthogonal. then |c| = |a(n)| = √ |a(n)|2 and so the code is euclidean self-dual. for the hermitian dual of c, we have [(a,a), (c,c)]h = aτ(c) + aτ(c) = 0 and the remainder of the proof is identical. using the standard techniques we have the following corollary. corollary 4.4. there exist euclidean and hermitian self-dual codes for all even lengths over a(n) for all n. proof. if c and d are self-dual codes (euclidean or hermitian) of length m and m′ respectively then c ×d is a self-dual code of length m + m′. this gives the result. theorem 4.5. let c be a binary self-dual code of length m, then reading 1 as sq0, we have a(n)[c] is a euclidean and hermitian self-dual code. proof. the code c has a basis of vectors vi over f2. we note that m must be even for a binary self-dual code to exist. then |a(n)[v1,v2, . . . ,v m 2 ]| = |a(n)| m 2 . then [ ∑ aivi, ∑ cjvj] = [ ∑ aivi, ∑ vjcj] = ∑ i,j ai[vi,vj]cj = 0, (12) since the elements in the coordinates of vi commute with all of the elements of a(n). therefore the code is euclidean self-dual. 147 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 next consider [ ∑ aivi, ∑ cjvj]h = [ ∑ aivi, ∑ vjcj]h ∑ i,j ai[vi,vj]hcj = 0, (13) since the hermitian inner-product and the euclidean inner-product are identical for vectors with coordinates containing only 0 and sq0. the key to this result was that 0 and sq0 are in the center of the ring. if we take a self-orthogonal code over a subring which is not in the center the proof would not apply and the code over the larger ring generated by the code over the subring may not be self-orthogonal. theorem 4.6. let c be a non-trivial code of length 1 over a(n). then zn ∈ r(c),zn ∈ l(c) and zn ∈rh (c) = lh (c), proof. the codes l(c), r(c), and lh (c) = rh (c) are left linear or right linear regardless if c is linear. therefore, as non-trivial ideals, {0,zn} is a subset of all of them by corollary 2.6. theorem 4.7. 1. let c = a(n)[v1,v2, . . . ,vs] and c′ = a(n + t)[v1,v2, . . . ,vs], t > 0. then r(c) ⊆r(c′) and rh (c) ⊆rh (c′). 2. let c = [v1,v2, . . . ,vs]a(n) and c′ = [v1,v2, . . . ,vs]a(n + t), t > 0. then l(c) ⊆ l(c′) and lh (c) ⊆lh (c′). proof. we prove only the first item, the second follows similarly. let w ∈r(c). consider the following inner-product: [a1v1 + a2v2 + · · · + asvs,w] = a1[v1,w] + a2[v2,w] + · · · + as[vs,w] = 0. (14) therefore w ∈r(c′). for the second part of the statement, let w ∈rh (c). consider the following inner-product: [a1v1 + a2v2 + · · · + asvs,w]h = a1[v1,w]h + a2[v2,w]h + · · · + as[vs,w]h = 0. (15) therefore w ∈rh (c′). 5. code over a(n) and binary codes recall that a(n) has a canonical basis with 2 (n+1)(n+2) 2 elements. then |a(n)| = 22 ( (n+1)(n+2) 2 ) . for example, a(1) has 8 basis elements and 28 elements. for a(2), the algebra has 26 basis elements and 264 elements. we now fix a basis b1,b2, . . . ,bt for a(n) with t = 2( (n+1)(n+2) 2 ). let a ∈ a(n) = ∑ aibi. define the map ψ : a(n) → ft2 by ψ(a) = ψ( ∑ aibi) = (a1,a2, . . . ,at). (16) note that the map ψ is dependent on the basis b1,b2, . . . ,bt for a(n) and so we keep this ordering of the basis elements throughout the remainder of the paper. by the definition of addition in a(n) we have that ψ is an additive map. we extend c to a(n)m by allowing it to act on each coordinate. definition 5.1. a code c over a(n) is an additive code if for all v,w ∈ c, v + w ∈ c. 148 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 we see that an additive code is a subgroup of (a(n))m but it may not be a submodule. that is, a linear code is necessarily additive, but an additive code may not be linear. theorem 5.2. let c be an additive code over a(n) of length m. then ψ(c) is a linear binary code of length 2( (n+1)(n+2) 2 )m. proof. we already have that ψ is an additive map. then the theorem follows by noting that an additive code over f2 is linear over f2. example 5.3. let c be the code of length 1 over a(1) defined as c = a(1)[sq2,1 + sq3]. then ψ(c) is the linear binary code generated by   0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1   . the following theorem has a proof similar to the one of theorem 5.2. theorem 5.4. let c be a linear code over f2 of length 2( (n+1)(n+2) 2 )m. then ψ−1(c) is an additive code of length m over a(n). example 5.5. let c be the binary hamming code of length 8. then ψ−1(c) is a subgroup of a(1) but not a submodule. for example, the elements sq0 + sq1 + sq2 + sq2,3,1 and sq2 + sq3 + sq2,3 + sq2,3,1 are both elements of ψ−1(c) but their product sq2 + sq3,1 is not. it is easy to see that this element is not in the code since its corresponding vector in f82 would have hamming weight 2 whereas the minimum hamming weight of the length 8 hamming code is 4. we can define an orthogonality relation for additive codes that will correspond to the orthogonality for binary codes. let g(n) be the additive group of a(n). order the elements of g(n) by g1,g2, . . . ,gs, where s = 2(2 (n+1)(n+2) 2 ). fix a character table tg for ĝ defined by tgg1,g2 = −1 [ψ(g1),ψ(g2)], (17) where [ψ(g1), ψ(g2)] indicates the usual binary inner-product. then χgi corresponds to the row of tg given by χ(gigj) where j goes from 1 to s. definition 5.6. let v,w ∈ a(n)m. define [v,w]g = ∏ χvi (wi). we note that the result of this inner-product is either 1 or −1. if c is an additive code over a(n) define the orthogonal to be c∗ = {(c1,c2, . . . ,cm) | ∏ χci (vi) = 1, for all (v1,v2, . . . ,vm) ∈ c}. (18) theorem 5.7. if c is an additive code in a(n)m, then cwec (x0,x1, . . . ,xk) = 1 |c∗| cwec∗ (tg · (x0,x1, . . . ,xk)). (19) proof. it follows from the standard macwilliams relations for codes over groups. namely, the matrix tg serves as a duality for the underlying additive group of the ring. theorem 5.8. let c be a code over a(n) then ψ(c∗) = ψ(c)⊥. proof. let v,w ∈ a(n)m. the following are equivalent statements: 149 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 1. [v,w]g = 1 2. ∏ χvi (wi) = 1 3. ∏ (−1)[ψ(vi),ψ(wi)] = 1 4. [ψ(v), ψ(w)] = 0. this gives that [v,w]g = 1 if and only if [ψ(v), ψ(w)] = 0. the next corollary follows immediately from theorem 5.8. corollary 5.9. a code c in (a(n))m is self-dual with respect to the duality tg if and only if ψ(c) is a binary self-dual code. corollary 5.10. self-dual codes exists for all lengths over a(n) with respect to the duality tg. proof. since 2 (n+1)(n+2) 2 is even, for all n ≥ 0, there exists binary self-dual codes of all lengths 2 (n+1)(n+2) 2 m. then apply corollary 5.9. note that we can replace the duality tg with a different duality for the group, which may or may not correspond directly to the binary orthogonality. 6. codes over the steenrod algebra just as we described codes over a(n) we can extend these ideas to the infinite ring a. this was done in a similar way for codes over the p-adics in [1] and [5]. a code here is a subset of am and it is left linear or right linear if it is a left submodule or right submodule of am. similarly, we can define l(c), r(c), lh (c) and rh (c) as in the finite case. we cannot define the group orthogonality since the underlying additive group is infinite and the technique no longer applies. notice that in this infinite case, we have that lh (c) = rh (c) as it is in the finite case. we can now define a projection to a(n). let c be a code over a, then let cn = c ∩ (a(n))m. (20) theorem 6.1. let c be a left (right) linear code over a then cn is a left (right) linear code for all n. proof. assume c is left linear. let v,w ∈ c and a,c ∈ a(n). then av + cw ∈ c since c is left linear. each coordinate of v and w is an element of a(n) so av + cw ∈ (a(n))m since the ring a(n) is closed under addition and multiplication. then av + cw ∈ c ∩ (a(n))m = cn and cn is left linear. the proof in the right linear case is similar. in general we have c0 ⊆ c1 ⊆ ···⊆ c. theorem 6.2. let c be a code over a. • if c ⊆l(c) then cn ⊆l(cn). • if c ⊆r(c) then cn ⊆r(cn). • if c ⊆lh (c) = rh (c) then cn ⊆lh (cn) = rh (c). proof. we prove the first case and the rest are similar. if c ⊆ l(c) then cn ⊆ c ⊆ l(c) and cn ⊆l(c) ∩ (a(n))m ⊆l(cn). 150 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 in this sense self-orthogonality projects down but self-duality may not. for example, let c1 = a(1)(sq2,1 + sq3), which is self-dual. but c0 ∩a(0) = {0} is not self-dual. lemma 6.3. let g be a binary matrix in standard form (i |m) and let c = a[g]. then cn = a(n)[g]. proof. if v ∈ c ∩ (a(n))m then the coefficients of the rows in the linear combination resulting in v must all be from a(n) since the first part of the matrix is the identity. theorem 6.4. let g be a matrix in standard form that generates a self-dual binary code, then a[g] is a self-dual code over a. proof. the proof of theorem 4.5 shows that the code must be self-orthogonal. however, it does not show self-duality since it uses a cardinality argument. assume there exists v ∈l(c) with v 6∈ c. then for some n, we have that v ∈ a(n)m. this implies that cn = a(n)[g], by lemma 6.3, has an element v ∈ l(cn),v 6∈ cn which contradicts theorem 4.5. therefore the code is self-dual. let g generate a binary self-dual code of length m and let c = a[g]. then c is self-dual and cn is self-dual for all n. this gives infinite families of self-dual codes for all even lengths. we shall now investigate some codes over a which we can then project down. lemma 6.5. for all α ∈ a we have that z1αz1 = 0. proof. any α in a can be written as a sum of atomic squares so it is sufficient to prove the result for atomic squares. • if α is one of the atomic squares in a(1) then the claim is true since z1 is the top element of a(1). • next, we consider the case for atomic squares with an odd power. let α = sq2 t−1 for t ≥ 3 then sq1sq2 t−1 = 0 (21) since 2t − 1 is an odd number and the result follows from the adem relations. hence z1αz1 = 0. • next, we consider the case for atomic squares where the power of the atomic square is a power of 2. let α = sq2 s for s ≥ 2. first we multiply z1 and α. note that if k is an even integer then from the adem relations we have that sq1sqk = sqk+1. (22) then we have z1α = sq 2sq3sq1sq2 s = sq2sq3sq2 s+1. (23) for s ≥ 2, 3 < 2(2s + 1), we then apply the adem relations to sq3sq2 s+1 which gives sq2sq3sq2 s+1 = sq2[ 1∑ k=0 ( 2s −k 3 − 2k ) sq2 s+4−ksqk] = sq2[ ( 2s 3 ) sq2 s+4 + ( 2s − 1 1 ) sq2 s+3sq1]. 151 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 the first term will be 0 since the binomial coefficient is an even number. hence z1α = sq2sq3sq2 s+1 = sq2sq2 s+3sq1. since 2 < 2(2s + 3) for s ≥ 2, we can apply the adem relations to sq2sq2 s+3, which gives sq2sq2 s+3sq1 = [ 1∑ k=0 ( 2s + 2 −k 2 − 2k ) sq2 s+5−ksqk]sq1 = [ ( 2s + 2 2 ) sq2 s+5 + ( 2s + 1 0 ) sq2 s+4sq1]sq1 = sq2 s+5sq1 + sq2 s+4sq1sq1 = sq2 s+5sq1. now we have z1α = sq2 s+5sq1. if we multiply these with z1 from the right we have z1αz1 = sq 2s+5sq1sq2sq3sq1 = sq2 s+5(sq3sq3sq1) = sq2 s+50 = 0. (24) • next we consider the remaining two cases of atomic squares. let α = sq2(2 t−1) where t ≥ 2. from the adem relations we have that sq2sq3sq1 = sq5sq1. then z1α = sq 2sq3sq1sq2(2 t−1) = sq5sq1sq2(2 t−1) = sq5sq2 t+1−1. (25) since 5 < 2(2t+1 − 1) for t ≥ 2, we can apply the adem relations to sq5sq2 t+1−1 which gives sq5sq2 t+1−1 = 2∑ k=0 ( 2t+1 −k − 2 5 − 2k ) sq2 t+1+4−ksqk = ( 2t+1 − 2 5 ) sq2 t+1+4 + ( 2t+1 − 3 3 ) sq2 t+1+3sq1 + ( 2t+1 − 4 1 ) sq2 t+1+2sq2 = 0, since the binomial coefficients are always even numbers. hence we get z1α = 0 so z1αz1 = 0. • we now consider the final case. let α = sq2 s(2t−1) where s,t ≥ 2. we have that z1α = sq 2sq3sq1sq2 s(2t−1) = sq5sq1sq2 s(2t−1) = sq5sq2 s(2t−1)+1. (26) since 5 < 2(2s(2t − 1) + 1) for s,t ≥ 0, we can apply the adem relations to sq5sq2 s(2t−1)+1 which gives sq5sq2 s(2t−1)+1 = 2∑ k=0 ( 2s(2t − 1) −k 5 − 2k ) sq2 s(2t−1)+6−ksqk = ( 2s(2t − 1) 5 ) sq2 s(2t−1)+6 + ( 2s(2t − 1) − 1 3 ) sq2 s(2t−1)+5sq1 + ( 2s(2t − 1) − 2 1 ) sq2 s(2t−1)+4sq2. the first and the last term are zero since the binomial coefficients are even. for the second term if the binomial coefficient is 0 then z1α = 0 and z1αz1 = 0. otherwise it will be 1 and then z1α = sq 2s(2t−1)+5sq1 and then z1αz1 will again be 0 as in the third case when the powers of the atomic squares were a power of 2. 152 s.t. dougherty, t. vergili / j. algebra comb. discrete appl. 4(2) (2017) 141–154 this does not say that αzi is necessarily 0 for all α. for example, z1sq4 = sq9,1 and sq4z1 = sq7,2,1 + sq9,1 but z1sq4z1 = 0. theorem 6.6. let c = a[z1,z2, . . . ] and d = [z1,z2, . . . ]a. then c and d are hermitian selforthogonal codes. proof. for the left linear code c, we need to show that [azi,czj]h = 0 for all integer i,j ≥ 1. we have that [azi,czj]h = aziτ(czj) = aziτ(zj)τ(c) = azizjτ(c) = a(0)τ(c) = 0. this gives that c is hermitian self-orthogonal. we note that zi = γz1 and zj = δz1 for some γ,δ ∈ a(n), n = max{i,j}. for the right linear code d, we need to show that [zia,zjc]h = 0 for all integers i,j ≥ 1. we have that [zia,zjc]h = ziaτ(zjc) = zi(aτ(c))zj = γ(z1(aτ(c)δ)z1) = 0 by lemma 6.5. this gives that d is hermitian self-orthogonal. similarly, we have the following theorem. theorem 6.7. let c = a[z1,z2, . . . ] and d = [z1,z2, . . . ]a. then c and d are euclidean selforthogonal codes. proof. as in the previous proof, we let zi = γz1 and zj = δz1 for some γ,δ ∈ a(n), n = max{i,j}. for the left linear code c we need to show that [azi,czj] = 0 for all integers i,j ≥ 1. we have that [azi,czj] = azi(czj) = aγz1βδz1 = aγ(z1(βδ)z1) = 0 by lemma 6.5. this gives that c is euclidean self-orthogonal. for the right linear code d, we need to show that [zia,zjc] = 0 for all integers i,j ≥ 1. we have that [zia,zjc] = γz1aδz1c = γ(z1(aδ)z1)c = 0 by lemma 6.5. this gives that d is euclidean self-orthogonal. this leads naturally to the following corollary. corollary 6.8. let c = a[z1,z2, . . . ], d = [z1,z2, . . . ]a, cn = c ∩ (a(n))m and dn = d ∩ (a(n))m then cn and dn are both euclidean and hermitian self-orthogonal codes for all n. proof. the results follow directly from theorem 6.6, theorem 6.7 and theorem 6.2. the code c = a[z1,z2, . . . ] is not self-dual. if it were then l(c) would be equal to r(c). however, sq1sq6z1 = sq 9,3,1 6= 0 but zn = γz1 = γsq2sq3sq1 is znsq1 = 0 and sq1 6∈ l(c) but sq1 ∈ r(c). in terms of the hermitian inner-product, sq1 is in both duals but sq1 is not in c, hence the code is not hermitian self-dual. similar results hold for d = [z1,z2, . . . ]a. references [1] a. r. calderbank, n. j. a. sloane, modular and p−adic cyclic codes, des. codes cryptog. 6(1) (1995) 21–35. 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[14] r. m. w. wood, problems in the steenrod algebra, bull. lond. math. soc. 30(5) (1998) 449–517. 154 https://math.berkeley.edu/~kruckman/adem/ http://dx.doi.org/10.2307/1969932 http://dx.doi.org/10.1007/3-540-30731-1 http://dx.doi.org/10.1007/3-540-30731-1 http://dx.doi.org/10.1007/bf02564562 http://dx.doi.org/10.1007/bf02564562 http://doi.org/10.1353/ajm.1999.0024 http://doi.org/10.1353/ajm.1999.0024 http://dx.doi.org/10.1504/ijicot.2010.032867 http://dx.doi.org/10.1504/ijicot.2010.032867 http://dx.doi.org/10.1112/blms/27.4.380 http://dx.doi.org/10.1112/blms/27.4.380 http://dx.doi.org/10.1112/s002460939800486x introduction definitions and notations macwilliams relations self-dual and hermitian self-dual codes code over a(n) and binary codes codes over the steenrod algebra references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.505364 j. algebra comb. discrete appl. 6(1) • 1–11 received: 21 august 2017 accepted: 2 december 2018 journal of algebra combinatorics discrete structures and applications weight distribution of a class of cyclic codes of length 2n research article manjit singh, sudhir batra abstract: let fq be a finite field with q elements and n be a positive integer. in this paper, we determine the weight distribution of a class cyclic codes of length 2n over fq whose parity check polynomials are either binomials or trinomials with 2l zeros over fq, where integer l ≥ 1. in addition, constant weight and two-weight linear codes are constructed when q ≡ 3 (mod 4). 2010 msc: 12e05, 12e20, 94b05, 11t71 keywords: linear codes, reversible codes, weight distributions, constant weight codes 1. introduction a linear [m,k]q code c over fq is a k-dimensional subspace of fmq . let λ be a nonzero element in fq. a linear code c of length m over fq is called a λ-constacyclic code if (c1,c2, . . . ,cm−1,c0λ) ∈ c, ∀ (c0,c1,c2, . . . ,cm−1) ∈ c. a λ-constacyclic code of length m over fq is called a simple-root code if gcd(m,q) = 1; otherwise it is called repeated-root code. if we identify a vector (a0,a1, . . . ,am−1) ∈ fmq with a polynomial a0xm−1 + a1x m−2 +· · ·+am−1 modulo (xm−λ) in fq[x], then a simple-root λ-constacyclic code c can be interpreted as an ideal of the quotient ring r = fq[x]/〈xm −λ〉. it is well known that each ideal in r is of the form 〈g(x)〉, where g(x) is a monic divisor of xm −λ in fq[x]. the polynomial g(x) is known as the generator polynomial of the code c and the corresponding factor h(x) = (xm −λ)/g(x) of xm −λ is referred to as the parity check polynomial of the code c. if h(x) is a product of t monic irreducible factors over fq, then we say c with t zeros over fq. a constacyclic code is called an irreducible code over fq if t = 1, manjit singh (corresponding author), sudhir batra; department of mathematics, deenbandhu chhotu ram university of science and technology, murthal-131039, sonepat, india (email: manjitsingh.math@gmail.com, batrasudhir@rediffmail.com). 1 https://orcid.org/0000-0003-3351-7287 https://orcid.org/0000-0003-4139-0589 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 and a reducible code over fq if t ≥ 2. a λ−constacyclic code c is called cyclic if λ = 1 and negacyclic if λ = −1. let ai be the number of codewords of a linear code c of length m over fq with hamming weight i, where 0 ≤ i ≤ m. note that a0 = 1 and ai = 0 for all 1 ≤ i < d, where d is the minimum hamming distance of the code. the sequence (1,a1,a2, . . . ,am) is called the weight distribution of the code c. cyclic codes are the most important class of linear codes for a wide variety of applications. in the last few decades, the weight distribution of irreducible cyclic codes have been studied extensively (see [3, 9, 12]). however, not much is known about the weight distribution of reducible codes except in very specific cases. vega [11] presented a new family of two-weight reducible cyclic codes that can be constructed as the direct sum of two one-weight irreducible codes. for any q = pm, where p is an odd prime, m ≥ 3, k ≥ 1 and gcd(k,m) = 1, feng and luo [4] obtained weight distribution of cyclic codes c1 of dimension 2m and c2 of dimension 3m over fp of length n = q − 1 with parity-check polynomial h2(x)h3(x) and h(x) = h1(x)h2(x)h3(x) respectively, where h1(x), h2(x) and h3(x) are the minimal polynomials of π−1, π−2 and π−(p k+1) over fp, respectively, for a primitive element π of fq with deghi(x) = m for i = 1, 2, 3. for doing this, they computed the value distribution of multi-sets of exponential sums using quadratic form over fp. yang, xiong, ding and luo [14] proposed a class of cyclic codes c of length n over fq with paritycheck polynomial h(x) = ha1 (x)ha2 (x) · · ·hat (x), where hai (x) is the minimal polynomial of γ−ai over fq, deghai (x) = m, r = q m, γ is a generator of f∗r and n = (r−1)/δ, δ = gcd(r−1,a1,a2, . . . ,at) and integer e ≥ t ≥ 2 with e|(q − 1). they remarked that it may be very difficult to find the weight distribution of this class of codes if the integers ai are not chosen in the right way or the gaussian periods have many different values. the values of the gaussian periods are in general very hard to determine. hence they obtained the weight distribution of this class of codes when t = e and the gaussian periods of order n are known, including the cases that n = 1, 2, 3, semi-primitive case and a special index 2 case. recently, assuming `v||(q − 1), where ` is a prime and v is an integer, and q ≡ 1 (mod 4) if ` = 2, zhu, yue and hu [15] have applied a combinatorial method to obtain not only the weight distribution of all cyclic codes of length `m with two zeros over fq, but also the weight distribution of a special cyclic code of length `m with three zeros over fq. in this paper, we present a class of cyclic codes of length 2n with 2l zeros over fq, where q is an odd prime power and n > l ≥ 1. further, using the explicit forms of codewords, the weight distribution of these codes is determined explicitly. we make use of diophantine equation and its solutions to obtain the explicit form of weights of codewords and the number of codewords of the given weight of these cyclic codes. it is observed that, when q ≡ 3 (mod 4), the problem of finding the weight distribution is transferred into a problem of determining the weight distribution of a two-weight negacyclic code, which turns out to be associated with counting the number of constant weight linear codes. these codes are known for applicability of various association schemes and traceability schemes, which justify their practical applications in engineering perspective (see [1, 2, 5–7]). in particular, these codes are of special interest in authentication codes [2] and traceability schemes [6]. the paper is organized as follows: the necessary notations and some auxiliary results are provided in section 2. in section 3, we describe a class of negacyclic and hence cyclic codes of length 2n with 2l zeros over fq. it is observed that these reducible codes are reversible when q ≡ 3 (mod 4). in section 4, constant weight linear codes and two-weight negacyclic codes are constructed and the weight distribution of cyclic codes of length 2n with 2l zeros over fq are determined. in the end of this section, we give an example illustrating the results. in section 5, we conclude the paper. 2. preliminaries the paper follows the standard notation of finite fields. the multiplicative group of fq is denoted by f∗q. it is well known that f ∗ q is a cyclic group of order q− 1. for any integer m ≥ 2, let ν2(m) denote 2 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 the highest power of 2 dividing m. let q be an odd prime power, s = ν2(q−1) and u = ν2(q2 −1). then u−s = ν2(q + 1) ≥ 1. readily note that (i) u−s = 1 if and only if q ≡ 1 (mod 4), and (ii) s = 1 if and only if q ≡ 3 (mod 4). let αk be a primitive 2kth root of unity in f∗q. also, let βk be a primitive 2 kth root of unity in f∗ q2 when s + 1 ≤ k ≤ u. notice that β2s+1 = αs. since βk ∈ fq2 \ fq, the minimal polynomial of βk over fq is x2 − (βk + β q k)x + β q+1 k . if f(x) is an irreducible polynomial over fq and f(0) 6= 0, then, for any integer k ≥ 2, the following result is useful to check the irreducibility of the polynomial of the type of f(xk) over fq. lemma 2.1. [13, theorem 10.15] let f(x) be any irreducible polynomial over fq of degree l ≥ 1. suppose that f(0) 6= 0 and f(x) is of order e which is equal to the order of any root of f(x). let k be a positive integer, then the polynomial f(xk) is irreducible over fq of order ke if and only if the following three conditions are satisfied: (i) every prime divisor of k divides e; (ii) gcd(k, q l−1 e ) = 1; (iii) if 4|k, then 4|(ql − 1). we end this section with the following lemma. lemma 2.2. let q be an odd prime power and r ≥ 1 be an integer. then (i) for any 2 ≤ k ≤ s, x2 r − αk is a product of 2l monic irreducible binomial factors of degree 2r−l over fq, where l = min{r,s−k}. (ii) for any s + 2 ≤ k ≤ u, x2 r+1 − (βk + β q k)x 2r + β q+1 k is a product of 2 l monic irreducible trinomial factors of degree 2r−l over fq, where l = min{r,u−k}. proof. (i) for any fixed 2 ≤ k ≤ s, let l = min{r,s − k}. then 2k+l|(q − 1), αk+l ∈ f∗q such that αk = α 2l k+l. the set sk = {α 2ki+1 k+l : 1 ≤ i ≤ 2 l} contains 2l distinct elements of order 2k+l in f∗q. for any α ∈ sk, x2 r−l −α is a factor of x2 r −αk. since gcd(2r−l, q−12k+l ) = 1, by lemma 2.1, x 2r−l −α is irreducible over fq for every α ∈ sk. (ii) for any fixed s + 2 ≤ k ≤ u, let l = min{r,u − k}. then 2k+l|(q2 − 1), βk+l ∈ f∗q2 such that βk = β 2l k+l. the set tk = {β 2ki+1 k+l : 1 ≤ i ≤ 2 l} contains 2l distinct elements of order 2k+l in f∗ q2 . in view of the part (i), x2 r−l − η is an irreducible factor of x2 r − βk over fq2 for every η ∈ tk. for any η ∈ tk, we notice that βk = η2 l and βqk = (η q)2 l . further, x2 r−l − η is a factor of x2 r − βk over fq2 if and only if x2 r−l −ηq is a factor of x2 r −βqk over fq2. this bijection of factors of x 2r −βk and x2 r −βqk over fq2 generates a unique factor (x2 r−l − η)(x2 r−l − ηq) = x2 r−l+1 − (η + ηq)x2 r−l + ηq+1 ∈ fq[x] of (x2 r −βk)(x2 r −βqk) = x 2r+1 − (βk + β q k)x 2r + β q+1 k ∈ fq[x] for every η ∈ tk. since gcd(2 r−l, q 2−1 2k+l ) = 1, hence by lemma 2.1, x2 r−l+1 − (η + ηq)x2 r−l + ηq+1 is irreducible over fq for every η ∈ tk. 3. negacyclic and cyclic codes for any 1 ≤ k ≤ s and integer r ≥ 0, we define a negacyclic [2k+r−1, 2r]q code and a cyclic [2k+r, 2r]q code over fq by n (k) r = 〈n (k) r (x)〉 and c (k) r = 〈c (k) r (x)〉, respectively, with the generator polynomials n(k)r (x) = x2 r+k−1 + 1 x2 r −αk and c(k)r (x) = x2 r+k − 1 x2 r −αk . 3 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 note that n(1)r is the whole space f2 r q . by lemma 2.2, for each 2 ≤ k ≤ s, n (k) r and c (k) r are the codes with 2l zeros over fq, where l = min{r,s−k}. lemma 3.1. for any 1 ≤ k ≤ s and integer r ≥ 0, n(k)r = {(b,bαk, . . . ,bα 2k−1−1 k ) : b ∈ f 2r q } and c(k)r = {(b,bαk, . . . ,bα 2k−1−1 k ,−b,−bαk, . . . ,−bα 2k−1−1 k ) : b ∈ f 2r q }. proof. for 1 ≤ k ≤ s, observe that α2 k−1 k = −1. it follows that x2 k−1 + 1 = (x−αk) 2k−1−1∑ i=0 αikx 2k−1−i−1 and for any r ≥ 0, x2 r+k−1 + 1 = (x2 r −αk) 2k−1−1∑ i=0 αikx 2r(2k−1−i−1) = (x2 r −αk)n(k)r (x). therefore the generator polynomial of n(k)r is n(k)r (x) = 2k−1−1∑ i=0 αikx 2r(2k−1−i−1). let b = (b0,b1, . . . ,b2r−1) ∈ f2 r q be a message word and the corresponding message polynomial be b(x) = ∑2r−1 j=0 bjx 2r−j−1 ∈ fq[x]. then the code polynomial of n (k) r is given by: b(x) ( x2 r+k−1 + 1 x2 r −αk ) = 2k−1−1∑ i=0 2r−1∑ j=0 bjα i kx 2r+k−1−2ri−j−1. the polynomial on the right hand side can be expressed as a vector of the form (b,bαk, . . . ,bα 2k−1−1 k ) by substituting i = 0, 1, . . . , 2k−1 − 1. hence, we have n(k)r = {(b,bαk, . . . ,bα 2k−1−1 k )}. since the generator polynomial of c(k)r is c(k)r (x) = n (k) r (x) ( x2 r+k−1 − 1 ) , so we obtain c(k)r = {(c,−c) : c = (b,bαk, . . . ,bα 2k−1−1 k ) ∈ n (k) r }. lemma 3.1 is valid for every odd q. consider the case q ≡ 3 (mod 4), i.e., s = 1, then we obtain the following trivial codes: n (1) r = f2 r q and c (1) r = {(c,−c) : c ∈ f2 r q }. 4 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 in the rest of this section, we assume q ≡ 3 (mod 4). in order to define a class of cyclic codes with parity check polynomial of the type x2 r+1 − (βk + β q k)x 2r + β q+1 k with 2 l zeros, where l = min{r,u−k}, we need a bit more notations of [10]. for any x ∈ fq2 \ fq, let t(x) = x + xq and n(x) = xq+1. it follows that the minimal polynomial of β2 over fq is x2 + 1, and the minimal polynomial of βk over fq is x2 − t(βk)x + n(βk) for every 3 ≤ k ≤ u. for every 1 ≤ i ≤ 2k−2 and 3 ≤ k ≤ u, there are exactly 2k−2 elements of the form t(β2i−1k ) and n(β2i−1k ) = β q+1 k = ±1. in order to avoid cumbersome notations, for a fixed k, we denote ξ = t(βk), � = β q+1 k . note that ξ = 0 if k = 2; � = 1 for 2 ≤ k ≤ u − 1 and � = −1 for k = u. further, for any integer i ≥ 0 and 3 ≤ k ≤ u, we define the followings: (i) δi = βik −β qi k βk −β q k with δ0 = 0, δ1 = 1 and δ2k−1−i = �iδi for every 0 ≤ i ≤ 2k−1 − 1. in view of [10, lemma 3.1], δi = ξδi−1 − �δi−2 for 2 ≤ i ≤ 2k−1 − 1, with δ0 = 0 and δ1 = 1. (ii) γi(a,b) = δi+1a + δib, where a,b ∈ fq. lemma 3.2. let 3 ≤ k ≤ u be a fixed integer and a,b ∈ fq. then, the sequence (γi(a,b))i≥0 satisfies the recursive relation γi(a,b) = ξγi−1(a,b) − �γi−2(a,b), for i ≥ 2 with γ0(a,b) = a, γ1(a,b) = ξa + �b. for any given (a,b) ∈ fq × fq, the sequence γi = γi(a,b) contains 2k terms satisfying γ2k−1+i = −γi for every 0 ≤ i ≤ 2k−1 − 1. proof. by our definition γi(a,b) = δi+1a+δib, where 0 ≤ i ≤ 2k−1−1 and a,b ∈ fq, it is immediate. for any fixed 3 ≤ k ≤ u, observe that x2 − ξx + � is a divisor of x2 k−1 + 1, but not a divisor of x2 k−1 − 1. let nk(x) = x2 k−1 + 1 x2 − ξx + � . (if k = 2, then ξ = 0, � = 1 and hence nk(x) = 1.) further, for any integer r ≥ 0, x2 r+1 − ξx2 r + � is a divisor of x2 r+k−1 + 1. let nr,k(x) = x2 r+k−1 + 1 x2 r+1 − ξx2r + � and cr,k(x) = nr,k(x)(x 2r+k−1 − 1). then nr,k(x) = nk(x2 r ). for any integer r ≥ 0, we define a negacyclic [2r+k−1, 2r+1]q code and a cyclic [2r+k, 2r+1]q code over fq by nr,k = 〈nr,k(x)〉, cr,k = 〈cr,k(x)〉 with nk = n0,k, a 2-dimensional negacyclic code of length 2k−1 over fq. for 3 ≤ k ≤ u, by lemma 2.2, nr,k and cr,k are the codes with 2l zeros over fq. remark 3.3. let f(x) = a0xm + a1xm−1 + · · ·+ am ∈ fq[x] be a polynomial of degree m. the reciprocal polynomial of f(x) is the polynomial f∗(x) = xmf(x−1) = amxm + am−1xm−1 + · · · + a0. further, f(x) is said to be reversible provided f∗(x) = f(x). a reversible code is a code such that reversing the order of the components of a codeword gives always a codeword. massey [8] showed that reversible cyclic codes are those which have self-reciprocal generator polynomials. for any integer r ≥ 0, the polynomial x2 r+1 − ξx2 r + 1 is reversible. by lemma 2.2, x2 r+1 − ξx2 r + 1 is reducible over fq and hence cr,k is a reducible and reversible cyclic code. cyclic codes can be decoded by sequential circuits, and hence the invariance of these codes under the reversing transformation is of special interest [8]. theorem 3.4. for any fixed k in 3 ≤ k ≤ u, the negacyclic [2k−1, 2]q-code is given by nk = {(γ0(a,b),γ1(a,b), . . . ,γ2k−1−1(a,b) : a,b ∈ fq}. further, for any integer r ≥ 0, the negacyclic [2r+k−1, 2r+1]q code nr,k is nr,k = nk ×nk ×···×nk︸ ︷︷ ︸ 2rcopies . 5 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 proof. for any fixed 3 ≤ k ≤ u, observe that nk(x) = x2 k−1 + 1 (x−βk)(x−β q k) = 2k−1−1∑ i=1 ( βik −β iq k βk −β q k ) x2 k−1−i−1 = 2k−1−1∑ i=1 δix 2k−1−i−1. the code polynomial of nk is (ax + b)nk(x) = 2k−1−1∑ i=1 δi(ax + b)x 2k−1−i−1 = 2k−1−1∑ i=0 (aδi+1 + bδi) x 2k−1−i−1. in view of lemma 3.2, we obtain the desired form of nk. now, for any integer r ≥ 0, we have nr,k(x) = nk(x 2r ) = 2k−1−1∑ i=1 δix 2r(2k−1−i−1). let a(x) = ∑2r+1−1 j=0 ajx 2r+1−j−1 be the polynomial representation of the vector a = (a0,a1, . . . ,a2r+1−1) ∈ f2 r+1 q . if we denote bj = aj+2r for 0 ≤ j ≤ 2r − 1, then a = (a0,a1, . . . ,a2r−1,b0,b1, . . . ,b2r−1) and a(x) = 2r−1∑ j=0 (ajx 2r + bj)x 2r−j−1. the code polynomial of nr,k is a(x)nr,k(x) = 2r−1∑ j=0 2k−1−1∑ i=1 δi(ajx 2r + bj)x 2r(2k−1−i)−j−1 = 2r−1∑ j=0  2k−1−2∑ i=0 ajδi+1x 2r(2k−1−i)−j−1 + 2k−1−1∑ i=1 bjδix 2r(2k−1−i)−j−1   = 2r−1∑ j=0 2k−1−1∑ i=0 (ajδi+1 + bjδi) x 2r+k−1−2ri−j−1. let γi,j = γi(aj,bj) = ajδi+1 + bjδi, where 0 ≤ i ≤ 2k−1 − 1 and 0 ≤ j ≤ 2r − 1. note that any codeword in nk is of the form (c0,c1, . . . ,c2k−1−1) with ci = γi(a,b) for 0 ≤ i ≤ 2k−1 − 1. similarly, let us denote c2k−1j+i = γi,j = γi(aj,bj) for 0 ≤ j ≤ 2r − 1. then any codeword of nr,k can be expressed as (c0,c1, . . . ,c2r+k−1−1) in which for every 0 ≤ i ≤ 2k−1 − 1, there exists 0 ≤ l ≤ 2k−1 − 1 such that c2k−1j+i = cl. let ej = {(c2k−1j+0,c2k−1j+1, . . . ,c2k−1j+2k−1−1)}. then ej is a block subcode of nr,k such that ej = nk. since nr,k = {(c̄0, c̄1, . . . , c̄2r−1) : c̄j ∈ej}, we have nr,k = {(c̄0, c̄1, . . . , c̄2r−1) : c̄j ∈ nk for 0 ≤ j ≤ 2r − 1}. this completes the proof. 6 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 4. weight distributions in this section, we determine the weight distribution of codes defined in the last section. the following theorem gives the weight distribution of c(k)r for 2 ≤ k ≤ s. theorem 4.1. let q ≡ 1 (mod 4) and 2 ≤ k ≤ s. then, for any integer r ≥ 0, the weight distribution of c(k)r is given by: a2kj = ( 2r j ) (q − 1)j ; where 0 ≤ j ≤ 2r. proof. let b = (b0,b1, . . . ,b2r−1) be a message word. further, let j denote the number of nonzero symbols in b. since each nonzero symbol has q − 1 choices and there are exactly ( 2r j ) message word with j nonzero symbols, the number of message word having j nonzero symbols is ( 2r j ) (q−1)j, where 0 ≤ j ≤ 2r. by lemma 3.1, any codeword of c(k)r is of the type (c,−c), where c ∈ n (k) r such that c = (b,bαk, . . . ,bα 2k−1−1 k ) with b = (b0,b1, . . . ,b2r−1). using the form of codeword c, the weight of c is 2k−1j and the number of codewords of weight 2k−1j are also ( 2r j ) (q−1)j. it follows that, the weight of any codeword of c(k)r is 2kj and the number of codewords of weight 2kj is same as the number of codewords of weight 2k−1j. therefore a2kj = ( 2r j ) (q−1)j. remark 4.2. for any integer r ≥ 0, c(1)r = {(c,−c) : c = (b0,b1, . . . ,b2r−1)}, the weight distribution of c (1) 1 is a2j = ( 2r j ) (q − 1)j. in rest of the section, we determine the weight distribution of cyclic codes cr,k for 3 ≤ k ≤ u and integer r ≥ 0. at first, we determine the weight distribution of nr,k. in view of theorem 3.4, the weight distribution of nr,k can be derived from the weight distribution of nk. fundamentally, we need to compute the weight distribution of nk. in this connection, some additional conventions are explained here. let θ : f2q → nk be a mapping, defined as θ((a,b)) = (c0,c1, . . . ,c2k−1−1), where ci = γi(a,b) for any a,b ∈ fq. then θ is an fq-isomorphism. further, for any fixed 0 ≤ i ≤ 2k−1 − 1, let ki = {(a,b) ∈ f2q : ci = 0}. observe that ki is an 1-dimensional subspace of f2q with k0 = {(0,b) : b ∈ fq} = fq and k2k−1−1 = {(a, 0) : a ∈ fq} = fq. denote ni = θ(ki), a copy of ki in nk. obviously, ni is also an 1-dimensional subspace of nk. a code c with the same hamming distance between every pair of its codewords is called an equidistant code. if all the codewords of a code c carry the same weight, then c is called constant weight code. a code c with both of these properties is known as equidistant constant weight code. a linear constant weight code is always an equidistant code. the following result presents a class of constant weight codes, which are applied in many areas [2, 5, 6]. lemma 4.3. for each 0 ≤ i ≤ 2k−1 − 1 and 3 ≤ k ≤ u, ni is a constant weight linear code such that n0 = {(0,δ1a,δ2a,. . . ,δ2k−1−1a) : a ∈ fq}, n2k−1−1 = {(δ1a,δ2a,. . . ,δ2k−1−1a, 0) : a ∈ fq} and for any 1 ≤ i ≤ 2k−1 − 2, 7 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 ni = {(a,δ1(λ1 −λi)a,. . . ,δj(λj −λi)a,. . . ,−�λia) : a ∈ fq} with ni ∩nj = {(0, 0, . . . , 0)} for 1 ≤ i 6= j ≤ 2k−1 − 1, where λi = δi+1/δi for 1 ≤ i ≤ 2k−1 − 2. proof. for any (a,b) ∈ ki, there exists a unique (c0,c1, . . . ,c2k−1−1) ∈ ni such that ci = γi(a,b) = δi+1a + δib = 0. if c0 = 0, then a = 0 and cj = δjb for 1 ≤ j ≤ 2k−1 − 1. it follows that n0 = {(0,δ1b, . . . ,δ2k−1−1b) : b ∈ fq}. also if c2k−1−1 = 0, then b = 0 and cj = δj+1a for 0 ≤ j ≤ 2k−1 − 2 and hence n2k−1−1 = {(δ1a,δ2a,. . . ,δ2k−1−1a, 0) : a ∈ fq}. further, let 1 ≤ i ≤ 2k−1 − 2 and (a,b) ∈ ki i.e., ci = γi(a,b) = 0. then b = −λia, where λi = δi+1/δi for 1 ≤ i ≤ 2k−1 − 2. it follows that ni = {(a,δ1(λ1 −λi)a,. . . ,δj(λj −λi)a,. . . ,−�λia) : a ∈ fq}. next we shall show that ni is constant weight code. this assertion obviously holds for i = 0 and i = 2k−1 − 1. further, let 1 ≤ i ≤ 2k−1 − 2. for this, we must show that ci is the only zero in (c0,c1, . . . ,c2k−1−1). assume ci = cj = 0 for j 6= i. then λi = λj for i 6= j. this implies βk ∈ fq for 3 ≤ k ≤ u, which is a contradiction as 2k (q − 1) for k ≥ 2 when q ≡ 3 (mod 4). so cj 6= 0 for any j 6= i. therefore the weight of any nonzero codeword of ni is 2k−1 − 1 for every 0 ≤ i ≤ 2k−1 − 1 and ni ∩nj = {(0, 0, . . . , 0)} for every 1 ≤ i 6= j ≤ 2k−1 − 1. this proves the result. we now move to determine the weight distribution of nk for 3 ≤ k ≤ u. theorem 4.4. the negacyclic code nk is a two-weight code for 3 ≤ k ≤ u. further, if ` (k) 0 and ` (k) 1 are non-zero weights of nk and if a (0) ` (k) i is the number of words of weight `(k)i in nk, then ` (k) 0 = 2 k−1, `(k)1 = ` (k) 0 − 1, a (0) ` (k) 0 = (q − 2k−1 + 1)(q − 1) and a(0) ` (k) 1 = 2k−1(q − 1). proof. let c = (c0,c1, . . . ,c2k−1−1) ∈ nk be a codeword with ci = γi(a,b), where a,b ∈ fq. for a moment, if ci = 0 and cj = 0 for some i 6= j, then a = b = 0. in this case c becomes a zero codeword. thus no two symbols of a nonzero codeword c can be simultaneously zero. therefore the possible weight of nonzero c is `(k)i = 2 k−1 − i for i = 0, 1. for any fixed 0 ≤ i ≤ 2k−1 − 1, by lemma 4.3, ni has q − 1 nonzero codewords of weight 2k−1 − 1 and one codeword of weight zero. let n = ⋃2k−1−1 i=0 ni and m = {c ∈ nk : c /∈ n}. then nk = m∪n and m∩n = ∅. observe that n has (q − 1)2k−1 codewords of weight 2k−1 − 1. therefore, m has (q2 − 1) − (q − 1)2k−1 = (q − 1)(q − 2k−1 + 1) nonzero codewords of weight 2k−1. remark 4.5. take q + 1 = 2k−1 for some 3 ≤ k ≤ u, then by lemma 4.3, m = ∅ and n = nk. equivalently, nk is a linear constant weight code with constant weight 2k−1 − 1. for example, n6 is a constant weight code of length 32 with the constant weight 31 over f31. further, if q + 1 6= 2k−1, but q + 1 = 2k for some 2 ≤ k ≤ u− 1, then nk is a linear 2-weight code of the same frequency 2k−1(q − 1), i.e., a(0) ` (k) i = 2k−1(q − 1) for i = 0, 1 (see example 4.8). the following result determines the weight distribution of nr,k recursively using the weight distribution of nk. theorem 4.6. for any r ≥ 0 and 3 ≤ k ≤ u, the weight distribution of negacyclic codes nr,k over fq is a`(k) = ∑ (x,y) ( 2r x )( 2r −x y )( (q − 2k−1 + 1)(q − 1) )x( 2k−1(q − 1) )y , where (x,y) varies over all possible solutions of the diophantine equation 2k−1x+ (2k−1−1)y = `(k) such that 0 ≤ x ≤ 2r, 0 ≤ y ≤ 2r −x. 8 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 proof. in view of theorem 3.4, nr,k = nk ×nk ×···×nk︸ ︷︷ ︸ 2r copies . if c ∈ nr,k is any codeword, then there are 2r vectors c̄1, c̄2, . . . , c̄2r (not necessarily distinct) in nk such that c = (c̄1, c̄2, . . . , c̄2r ). by theorem 4.4, the possible weights of codewords c̄j ∈ nk, where 1 ≤ j ≤ 2r, are `(k)i = 2 k−1 − i for i = 0, 1, 2k−1. let xi, for i = 0, 1, 2k−1, be the number of c̄j’s in a codeword c of weight `(k)i . clearly x1 + x2 + x2k−1 = 2 r. further, if `(k) denotes the weight of c, then `(k) = ` (k) 0 x0 + ` (k) 1 x1. for the given 2r copies of nk, we first can choose the x0 copies of weight 2k−1 in ( 2r x0 ) ways. from 2r − x0 copies of nk, we can select the x1 copies of them, having weight 2k−1 − 1, in ( 2r −x0 x1 ) ways. now we can select the x2k−1 copies from the remaining 2r −x0 −x1 copies, having weight zero, in( 2r −x0 −x1 x2k−1 ) ways. this can be done in only one way because 2r −x0 −x1 = x2k−1. by theorem 4.4, a2k−1 = (q − 1)(q − 2 k−1 + 1) and a2k−1−1 = (q − 1)2 k−1. therefore a2k−1x0 = ( 2r x0 )( (q − 2k−1 + 1)(q − 1) )x0 and a(2k−1−1)x1 = ( 2r −x0 x1 )( 2k−1(q − 1) )x1. since the diophantine equation `(k) = `(k)0 x + ` (k) 1 y admits finite solutions and (x,y) varies over the set of all possible solutions, the number of codewords of weight `(k) = 2k−1x + (2k−1 − 1)y is given by a`(k) = ∑ (x,y) 0≤x≤2r 0≤y≤2r−x a2k−1xa(2k−1−1)y. this completes the proof. theorem 4.7. let q ≡ 3 (mod 4) and 3 ≤ k ≤ u. then, the weight distribution of cyclic code cr,k is given by a` = ∑ (x,y) `=2kx+(2k−2)y 0≤x≤2r,0≤y≤2r−x ( 2r x )( 2r −x y )( 2k−1(q − 1) )x+y(q − 2k−1 + 1 2k−1 )x . proof. if c is any codeword of cr,k, there exists a codeword c0 ∈ nr,k such that c = (c0,−c0). observe that, for any 3 ≤ k ≤ u and q ≡ 3 (mod 4), 2k+1|(q + 1). now applying theorem 4.6, we obtain the weight distribution of cyclic code cr,k in the desired form. note that the main result in [9] is a special case of theorem 4.1 and theorem 4.7. example 4.8. let q = 7 and k = 3. by theorem 4.4, the weight distribution of a negacyclic [4, 2, 3]7 code n3 is a0 = 1, a3 = a4 = 24. further, by theorem 4.7, the weight distribution of a reducible cyclic [16, 4, 6]7 code is given by a` = ∑ (x,y) 0≤x≤2,0≤y≤2−x ( 2 x )( 2 −x y ) 24x+y with 8x + 6y = `. 9 m. singh, s. batra / j. algebra comb. discrete appl. 6(1) (2019) 1–11 table 4.8 characterizes the type of possible weights ` = 8x + 6y of a reducible cyclic [16, 4, 6]7 code with the number of codewords a` of a given weight `. table 4.8 ` | 0 6 8 12 14 16 (x,y) | (0, 0) (0, 1) (1, 0) (0, 2) (1, 1) (2, 0) a` | 1 48 48 576 1152 576 5. concluding remarks the main contributions of this paper are the followings: • the construction of a class of linear codes of length 2n with 2l zeros over fq and their weight distribution. these codes are reversible when l ≥ 1 and q ≡ 3 (mod 4). • the explicit form of the weight distribution in which the weights of codewords and the number of codewords of a given weight of these codes can be computed easily using a linear diophantine equation and its solutions (see example 4.8). • the construction of constant weight linear codes and two-weight negacyclic codes of length 2n, where 2n divides q + 1 and integer n ≥ 2. a class of linear codes with few weights are of special interest in authentication codes [2] and traceability schemes [6]. many authors have worked on the problem of determining the weight distribution of reducible cyclic codes using mathematical tools, such as gaussian periods and exponential sums. the values of the gaussian periods, exponential sums are in general very hard to compute. it would be interesting to use the combinatorics approach of this paper for obtaining the weight distribution of cyclic codes of length m over fq whose parity check polynomials are binomials or trinomials over fq for the case m ∈{2nd,dn} for some odd integer d such that d|(q − 1) or d|(q + 1). acknowledgment: the authors wish to thank the anonymous referees for their valuable comments that really helped us to improve the quality of this paper. references [1] e. r. berlekamp, algebraic coding theory, revised edition, world scientific publishing co. pte. ltd., 2015. 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[15] x. zhu, q. yue, l. hu, weight distributions of cyclic codes of length lm, finite fields appl. 31 (2015) 241–257. 11 https://doi.org/10.1016/s0019-9958(64)90438-3 https://doi.org/10.1016/j.ffa.2007.07.004 https://doi.org/10.1016/j.ffa.2007.07.004 https://doi.org/10.1142/s0219498817500025 https://doi.org/10.1109/tit.2012.2193376 https://doi.org/10.1109/tit.2012.2193376 https://doi.org/10.1006/jnth.1995.1133 https://doi.org/10.1006/jnth.1995.1133 https://doi.org/10.1142/5350 https://doi.org/10.1109/tit.2013.2266731 https://doi.org/10.1109/tit.2013.2266731 https://doi.org/10.1016/j.ffa.2014.07.005 https://doi.org/10.1016/j.ffa.2014.07.005 introduction preliminaries negacyclic and cyclic codes weight distributions concluding remarks references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 1(1) • 41-51 received: 26 june 2014; accepted: 3 september 2014 journal of algebra combinatorics discrete structures and applications some properties of topological pressure on cellular automata∗ research article chih-hung chang ∗∗ department of applied mathematics, national university of kaohsiung, kaohsiung 700, taiwan, roc abstract: this paper investigates the ergodicity and the power rule of the topological pressure of a cellular automaton. if a cellular automaton is either leftmost or rightmost permutive (due to the terminology given by hedlund [math. syst. theor. 3, 320-375, 1969]), then it is ergodic with respect to the uniform bernoulli measure. more than that, the relation of topological pressure between the original cellular automaton and its power rule is expressed in a closed form. as an application, the topological pressure of a linear cellular automaton can be computed explicitly. 2010 msc: 37b10 keywords: cellular automata, permutive, ergodicity, topological pressure 1. introduction let a = {0, 1, . . . ,υ − 1} for some υ ≥ 2, and let x = az be the space of infinite sequences x = (xi)i∈z. a continuous map f : x → x satisfying f ◦σ = σ◦f is called a cellular automaton (ca), where σ is the shift map defined by σ(x)i = xi+1, i ∈ z. ca is a particular class of dynamical systems introduced by s. ulam [21] and j. von neumann [23] as a model for self-production. it is widely studied in a variety of contexts in physics, biology and computer science [6, 7, 10, 12, 16, 20, 22, 26, 27]. greenberg and hastings [13] transfer a reaction-diffusion equation into a ca and demonstrate that such a simplified model can still generate spatial-temporal structures similar to the original system [11– 13]. in 1976, hardy et al. [14] proposed the so-called hpp model for the study of hydrodynamics. notably, this model is a ca. since ca is widely applied in physics, it is also interesting to consider the role thermodynamics plays in ca. the second law of thermodynamics says that the entropy of a system and its surroundings (universe) do not decrease. however, this encounters with a paradox to poincaré recurrence theorem. ∗ this work is partially supported by the ministry of science and technology, roc (contract no most 1032115-m-390-004-). ∗∗ e-mail: chchang@nuk.edu.tw 41 some properties of topological pressure on cellular automata to overcome this difficulty, l. boltzmann introduces “ergodic hypothesis". reader may refer to [17]. this leads to the development of ergodic theory. this elucidation studies two properties of ca. the first part investigates the ergodic property of ca. then the second part gives an application for ergodic ca by demonstrating the power rule of topological pressure of ca. one-dimensional ca consists of infinite lattice with finite states and an associated mapping, say local rule. a systematic study of ca from purely mathematical point of view is discussed by hedlund [15]. he treats ca as a particular class of symbolic dynamics and discovers that a permutive ca (defined later) implements many phenomena such as surjection. from the viewpoint of ergodic theory, it is interesting to investigate whether a ca is surjective or not since a surjective ca preserves the uniform bernoulli measure [9]. shereshevsky [18] demonstrates more ergodic properties for permutive ca, such as mixing, bernoulli automorphism, kolmogorov automorphism, and so on. he also defines the power rule of a ca and figures out many properties are preserved. read is referred to akın’s works [1, 2] for more results. shereshevsky and rogers [19] conjecture that every surjective ca, except those of the form fxi = kxi, where g.c.d.(k,υ) = 1, is ergodic with respect to the uniform bernoulli measure. this conjecture does not hold in general. a counterexample is discussed in section 3. the first result of this paper is, if a ca is permutive, then it is ergodic. besides, the permutivity of ca is necessary for the ergodicity for a particular class of ca. using this property, the power rule of topological pressure of ca can be expressed in a closed form. for t : ω → ω a continuous transformation defined on a compact metric space, it is well-known that the power rule of topological entropy of t satisfying htop(tn) = nhtop(t) for all n ∈ n. if, moreover, (ω,b,µ) is a probability space and t : ω → ω is also a measure-preserving transformation. then hµ(t n) = nhµ(t) for all n ∈ n, where hµ(t) is the measure-theoretic entropy of t. also, for the topological pressure of t, p(t, ·) : c(ω,r) → r, there is a similar expression p(tn,sng) = np(t,g), where sng = n−1∑ i=0 g ◦ti. in [5], the authors investigated the topological pressure of permutive ca. theorem 4.4 indicates that, if f is a permutive ca, then the topological pressure of f and its nth power rule can be presented in a closed form: p(fn,g) = np(f,g) − (n− 1) ∫ g dµ, (1) for g continuous function, n ∈ n. notably, this expression does not hold in general. a counterexample is studied in section 4. as an application, the topological pressure of linear ca can be formulated rigorously (cf. [3]), which extends our previous result [4]. the materials of this elucidation are organized as follows. section 2 states some definitions and notations. section 3 investigates the sufficient condition for the ergodic ca, and figures out a necessary condition for the ergodicity for a particular class of ca. section 4 studies the power rule of topological pressure of a ca. a closed expression is given via a strong generator. 2. preliminaries let a = {0, 1, . . . ,υ − 1} be a finite alphabet and let x = az be the space of infinite sequence x = (xn) ∞ −∞. whenever a local rule f : a2k+1 →a is given, there associates unique ca, say f : x → x, defined by f(x)i = f(xi−k, . . . ,xi+k), 42 c.-h. chang where an = {a1a2 · · ·an : ai ∈a, 1 ≤ i ≤ n}, n ∈ n. for any m ≥ 1, f can be extended to the mapping fm : a2k+m →am by fm(x−k, . . . ,xk+m−1) = (f(x−k, . . . ,xk), . . . ,f(x−k+m−1, . . . ,xk+m−1)), where f1 = f. in addition, the nth power rule of f, denote by fn : a2nk+1 →a, is defined as fn(x−nk, . . . ,xnk) = f(f n−1(x−nk, . . . ,x(n−1)k), . . . ,f n−1(x−(n−2)k, . . . ,xnk)), for n ≥ 1. hedlund [15] shows a necessary and sufficient condition for the surjection of a ca. proposition 2.1 ([15]). consider f a ca with local rule f : a2k+1 → a. then f is surjective if and only if card f−1m (y1, . . . ,ym) = (card a)2k for all m ∈ n and all (y1, . . . ,ym) ∈am, where card s denotes the cardinality of s. the study of the local rule of a ca is essential for the understanding of this system. a particular class of local rules, say permutive, is initially introduced by hedlund [15]. definition 2.2. the local rule f : a2k+1 → a for a given ca is said to be leftmost (respectively rightmost) permutive in xi if there exists an integer i, −k ≤ i ≤−1 (respectively 1 ≤ i ≤ k), such that (i) for (y1, . . . ,y2k) ∈a2k, g(xi) ≡ f(y1, . . . ,y2k; xi) : a→a is a permutation; (ii) f does not depend on xj for j < i (respectively j > i), where x = (xj)kj=−k ∈a 2k+1. definition 2.3. the local rule f is called bipermutive if it is not only rightmost permutive but also leftmost permutive. f is called permutive if f is one of these three cases, i.e., f is either leftmost or rightmost or bipermutive. hedlund demonstrates a permutive ca is also surjective. proposition 2.4 ([15]). if f is permutive, then f is surjective. it is easily verified that the permutivity is preserved by power rule. lemma 2.5. if f is permutive, then so is fn for all n ∈ n. more precisely, fn preserves the type of permutivity of f for all n ∈ n. proof. this follows immediately from the definition of fn. example 2.6. consider the alphabet a = {0, 1, 2, 3} and local rule f : a3 →a defined by f(x−1,x0,x1) = 2x−1 + x1 mod 4. then f is rightmost permutive. observe that the second and the third power rule of f are f2 : a5 →a, f2(x−2, . . . ,x2) = x2, and f3 : a7 →a, f3(x−3, . . . ,x3) = 2x1 + x3, respectively. it comes by induction that fn : a2n+1 →a, fn(x−n, . . . ,xn) = { xn, n is even; 2xn−2 + xn, n is odd. (2) hence fn is rightmost permutive for all n ∈ n. 43 some properties of topological pressure on cellular automata let (x,b,µ) be a probability space and let t : x → x be a measure-preserving transformation. t is called ergodic if the only elements a ∈b with t−1a = a satisfy µ(a) = 0 or µ(a) = 1. the following theorem gives some equivalent conditions for the ergodicity of t . theorem 2.7 ([24]). if t : x → x is a measure-preserving transformation of the probability space (x,b,µ), then the following statements are equivalent: (i) t is ergodic. (ii) if g is measurable and g ◦t = g a.e., then g is constant a.e. (iii) for every a,b ∈b with µ(a) > 0,µ(b) > 0, there exists n ∈ n such that µ(t−na∩b) > 0. define d : x ×x → r by d(x,y) = ∞∑ i=−∞ |xi −yi| υ|i| , x,y ∈ x. (3) it is easy to verify that d is a metric and (x,d) is a compact metric space. consider µ an invariant probability measure on (x,f). let α and β be two finite measurable partitions of x, define α∨β = {a∩b : a ∈ α,b ∈ β}. it is easily seen that α∨β is a refinement of α and β. we define the entropy of the partition α by hµ(α) = − ∑ a∈α µ(a) log µ(a). the entropy of the measure preserving transformation f with respect to the partition α is given by hµ(f,α) = lim n→∞ 1 n hµ( n−1∨ i=0 f−iα), (4) reader may refer to [24] for the existence of the limit. and the measure-theoretic entropy of f is defined by hµ(f) = sup hµ(f,α), (5) where the supremum is taken over all finite measurable partitions of x. let (x,f) be an endomorphism and let p be an open cover of x. set h(p) = inf{log cardp̂}, where the infimum is taken over the set of finite subcovers p̂ of p and carda is the cardinality of a. the topological entropy of the measure preserving transformation f with respect to the open cover p is defined by h(f,p) = lim n→∞ 1 n h( n−1∨ i=0 f−ip). (6) for the existence of the limit, we refer reader to [24]. the topological entropy of f is defined by htop(f) = sup h(f,p), (7) where the supremum is taken over all finite open covers of x. 44 c.-h. chang in addition, for α an open cover of x and φ ∈ c(x,r) a continuous function from x to r, denote by pn(f,φ,α) = inf{ ∑ b∈β sup x∈b e(snφ)(x) : β is a finite subcover of n−1∨ i=0 f−iα}, where n ∈ n and snφ = ∑n−1 i=0 φ◦f i. then limn→∞ 1n log pn(f,φ,α) exists [24]. for each δ > 0, define p(f,φ,δ) = sup{ lim n→∞ 1 n log pn(f,φ,α) : diam(α) ≤ δ}, (8) and p(f,φ) = lim δ→0 p(f,φ,δ). (9) the map p(f, ·) : c(x,r) → r∪{∞} is called the topological pressure of f. it comes immediately that p(f, 0) = htop(f). 3. ergodicity on permutive cellular automata this section studies the ergodicity of permutive ca. for simplicity, a local rule f : a2k+1 → a is presumed that f only depends on xi, r ≤ i ≤ s, for some r ≤ s, r,s ∈ z. in the rest of this elucidation, we refer µ to the uniform bernoulli measure unless otherwise stated. theorem 3.1. if f is permutive, then f is ergodic. proof. it suffices to show that f is ergodic if f is rightmost permutive. the case that f is leftmost permutive can be demonstrated via the same argument, thus is omitted. denote by c(i,j) = i[ai, . . . ,aj]j a cylinder of x, i.e., for all x = (xn) ∈ x, xn = an, where i ≤ n ≤ j and i ≤ j. first we show that, for any two cylinder c(i,j) = i[ci, . . . ,cj]j,d(m,l) = m[dm, . . . ,dl]l, there exists n ∈ n such that µ(f−nc(i,j) ∩d(m,l)) > 0, where i ≤ j,m ≤ l. without loss of generality, we may assume that r = 0. the case that r 6= 0 can be done analogously. we construct a scheme to demonstrate the ergodicity of f. since f is rightmost permutive, it comes immediately that f−nc(i,j) is a collection of cylinders e(i,j + ns) with cardinality card f−nc(i,j) = νns, for n ∈ n. therefore, if l < i or j < m−s, then the permutivity of f demonstrates that µ(f−1c(i,j) ∩d(m,l)) > 0. if m ≤ i ≤ l ≤ j. let α ∈ z+ be the smallest number such that l + α − i is a multiple of s, say l + α − i = τs for some τ ∈ n. pick d′(m,l + α) = m[dm, . . . ,dl, 0, . . . , 0]l+α ≡ [dm, . . . ,dl+α] a subcylinder of d(m,l). define κ1,1, . . . ,κ1,s,κ2,1, . . . by κ1,` = f τ−1(di+`−1, . . . ,di+`+(τ−1)s−1), 1 ≤ ` ≤ s, κ2,` = f τ−2(di+`−1, . . . ,di+`+(τ−2)s−1), 1 ≤ ` ≤ 2s, ... κτ−1,` = f(di+`−1, . . . ,di+`+s−1), 1 ≤ ` ≤ (τ − 1)s. since f is rightmost permutive, there exist unique β1,1, . . . ,β1,j−i+1 such that f([κ1,1, . . . ,κ1,s,β1,1, . . . ,β1,j−i+1]) = c(i,j), i.e., e1 ≡ i[κ1,1, . . . ,κ1,s,β1,1, . . . ,β1,j−i+1]j+s ⊂ f−1c(i,j). 45 some properties of topological pressure on cellular automata similarly, there exist unique β2,1, . . . ,β2,j−i+1 such that f([κ2,1, . . . ,κ2,2s,β2,1, . . . ,β2,j−i+1]) = e1, i.e., e2 ≡ i[κ2,1, . . . ,κ2,2s,β2,1, . . . ,β2,j−i+1]j+2s ⊂ f−1e1 ⊂ f−2c(i,j). inductively we can conclude that d′(m,l + α) ⊂ f−τc(i,j). this implies µ(f−τc(i,j) ∩d(m,l)) > 0. the other cases, such as m ≤ i ≤ j ≤ l, i ≤ m ≤ j ≤ l, and so on, can be done analogously. hence, for any two cylinders c(i,j) and d(m,l), there exists n ∈ n such that µ(f−1c(i,j) ∩d(m,l)) > 0. for any two measurable sets a,b with positive measure and � > 0, there exist two collections of pairwise disjoint cylinders a1, . . . ,am, b1, . . . ,bl such that (i) ∪ai ⊆ a and ∪bj ⊆ b; (ii) µ(a\∪ai) < � and µ(b \∪bj) < �. it can be easily verified that the permutivity of f asserts f−1ai∩f−1aj = ∅ and f−1bi∩f−1bj = ∅ for all i 6= j. moreover, f−na∩b ⊇ (f−n(∪ai)) ∩ (∪bj) = ∪i,j(f−nai ∩bj), for all n ∈ n. therefore, there exists n ∈ n such that µ(f−na∩b) > 0. the proof is completed. example 3.2. for a = {0, 1, . . . ,ν−1}, ν ≥ 2. let σ : az →az be the shift map. then σ is rightmost permutive. it is easy to see that σ is ergodic. for any two distinct cylinders c(i,j) and d(m,l), where i ≤ j,m ≤ l. there exists n ∈ n such that i + n > l. thus σ−nc(i,j) ∩d(m,l) = c(i + n,j + n) ∩d(m,l) is of positive measure. that is, σ is ergodic. example 3.3 (continued). in example 2.6, we consider the alphabet a = {0, 1, 2, 3} and local rule f : a3 →a defined by f(x−1,x0,x1) = 2x−1 + x1 mod 4. moreover, fn : a2n+1 →a, fn(x−n, . . . ,xn) = { xn, n is even; 2xn−2 + xn, n is odd. (10) it is easily seen that if fn is ergodic for some n ∈ n, then so is f. for all k ∈ n, f2k(x−2k, . . . ,x2k) = x2k. this indicates f2k is kind of a “multiple-shift map", thus is ergodic. therefore, f is ergodic. if the cardinality of states a is a prime, then every additive ca is permutive except for those local rules of the form f : a→a,f(x) = αx for some 0 ≤ α ≤ p− 1, where card a = p. there is an example that, f is permutive in x0 but not ergodic since f is not rightmost permutive. proposition 3.4. consider a = {0, 1} and local rule f : a3 →a defined by f(x0,x1,x2) = x0 + x1(x2 + 1) mod 2. then f is not ergodic. 46 c.-h. chang proof. it comes from [8] that f is not topologically transitive. therefore, f is not ergodic. however, the assumption of permutivity is the necessary condition for the ergodicity in a particular class of ca. proposition 3.5. consider a = {0, 1, . . . ,ν − 1} and f a ca satisfying the following conditions: (i) f is additive, i.e., f : a2k+1 →a can be represented as f(x−k, . . . ,xk) = ∑k i=−k aixi. (ii) there exists unique `,−k ≤ ` ≤ k, such that a` = 1. (iii) if f only depends on xi, r ≤ i ≤ s, for some −k ≤ r ≤ s ≤ k, then g.c.d.(am,ν) > 1 for r < m < s,m 6= `. then f is ergodic if and only if f is either leftmost or rightmost permutive. proof. it suffices to show that, if f is neither leftmost nor rightmost permutive, then f is not ergodic. write f(xr, . . . ,xs) = ∑s i=r aixi. there exists κ ∈ n,κ 6= 0 mod ν, such that κai = 0 mod ν for all i 6= `. then κf(x)j = κ s∑ i=r aixi+j = κxj+`. define g : az → r by g(x) = ∑ κxi, it is easily seen that g is measurable and g ◦f = g on az. the fact that g is not constant demonstrates f is not ergodic. this completes the proof. example 3.6. consider a = {0, 1, 2, 3} and local rule f : a3 →a defined by f(x−1,x0,x1) = 2x−1 + x0 + 2x1 mod 4. then f is neither rightmost nor leftmost permutive. consider g : x → r defined by g(x) = 2x0. then g is measurable and nonconstant, however, g ◦f = g for all x ∈ x. this demonstrates f is not ergodic. 4. the topological pressure of power rule of ca this section investigates the topological pressure via a strong generator. consider (x,f) an endomorphism and ξ a finite cover of x, we call ξ a strong generator if, for any δ > 0, there exists n ∈ n such that ‖ ∨m−1 i=0 f −iξ ‖< δ for all m ≥ n, where ‖ a ‖ denotes the diameter of a. in other words, ξ is a strong generator if and only if ‖ m−1∨ i=0 f−iξ ‖→ 0, as n →∞. for a,b ∈ z, a ≤ b, denote by a[sa, . . . ,sb]b = {x ∈ x : xa = sa, . . . ,xb = sb} a cylinder in x and ξ(a,b) = {a[xa, . . . ,xb]b : xa, . . . ,xb ∈ s} a partition of x. 47 some properties of topological pressure on cellular automata lemma 4.1. let f be permutive and let ξ ≡ ξ(a,b) be a partition of x for a ≤ b, a,b ∈ z. then ξ(a,b) is a strong generator. proof. we show the case that f is rightmost permutive, the other cases can be done similarly. assume that f is rightmost permutive in xs and does not depend on xj for all j < r, where r < s. denote by c(i,j) = i[ai, . . . ,aj]j a cylinder of x, i.e., for all x = (xm) ∈ x, xm = am, where i ≤ m ≤ j and i ≤ j. it is easy to verify that f−1a ∩ f−1b = ∅ since f is rightmost permutive. moreover, for any cylinder c(i,j), f−1c(i,j) ⊂ ξ(i + r,j + s) with cardinality υs−r. in other words, f−1ξ(a,b) = ξ(a + r,b + s). denote ξ ≡ ξ(a,b) for simplicity. it follows from the discussion above that ξ ∨f−1ξ = { ξ(a,b + s), 0 ≤ r; ξ(a + r,b + s), 0 ≥ r. (11) inductively, ξ ∨f−1ξ ∨·· ·∨f−(m−1)ξ = { ξ(a,b + (m− 1)s), 0 ≤ r; ξ(a + (m− 1)r,b + (m− 1)s), 0 ≥ r. (12) this elucidates that ‖ ξ ∨f−1ξ ∨·· ·∨f−(m−1)ξ ‖→ 0, as m →∞, i.e., ξ is a strong generator. the proof is completed. example 4.2 (continued). consider a and f same as given in example 2.6. pick ξ(0) = {[0]0, [1]0, [2]0, [3]0} the standard partition of x = az. then f−1[0]0 = {−1[0, ·, 0]1,−1[1, ·, 2]1,−1[2, ·, 0]1,−1[3, ·, 2]1}, f−1[1]0 = {−1[0, ·, 1]1,−1[1, ·, 3]1,−1[2, ·, 1]1,−1[3, ·, 3]1}, f−1[2]0 = {−1[0, ·, 2]1,−1[1, ·, 0]1,−1[2, ·, 2]1,−1[3, ·, 0]1}, f−1[3]0 = {−1[0, ·, 3]1,−1[1, ·, 1]1,−1[2, ·, 3]1,−1[3, ·, 1]1}. it follows that (i) f−1[i]0 ∩f−1[j]0 = ∅ for i 6= j; (ii) for 0 ≤ i ≤ 3, a∩b = ∅ for a 6= b,a,b ∈ f−1[i]0; (iii) f−1ξ(0) = ξ(−1, 1). it is easily verified that n−1∨ i=0 f−iξ(0) = ξ(−(n− 1),n− 1), and ‖ n−1∨ i=0 f−iξ(0) ‖→ 0, as n → ∞. in other words, ξ(0) is a strong generator. in addition, it can be checked without difficulty that ξ(a,b) is a strong generator for a ≤ b. it is well-known that htop(fn) = nhtop(f) and p(fn,sng) = np(f,g) for all n ∈ n, where sng =∑n−1 i=0 g ◦f i and g ∈ c(x,r) [24]. there is an intuitive connection between fn and the nth power rule of local rule f. 48 c.-h. chang lemma 4.3. consider a ca f with local rule f. then the nth power rule of f, fn, is a ca with local rule fn for n ∈ n. proof. this comes from the definition of fn and fn, thus is omitted. the main theorem of this section then follows. theorem 4.4. consider f a ca with permutive local rule f. the topological pressure of fn can be expressed as p(fn,g) = np(f,g) − (n− 1) ∫ x g dµ, (13) for all n ∈ n, g ∈ c(x,r). theorem 3.1 shows that, for a ca f with the local rule f, if f is permutive, then f is ergodic. therefore, there is an immediate consequence comes from birkhoff ergodic theorem. lemma 4.5. consider g ∈ c(x,r). for any � > 0, there exists n ∈ n such that for ` ≥ n, | sup x∈a s`g(x) − sup x∈b s`g(x)| < `�, for any two disjoint measurable set a,b, where s`g = ∑`−1 i=0 g ◦f ni. proof. the permutivity of f implies fn is permutive, thus fn is ergodic. birkhoff ergodic theorem demonstrates that 1 ` s`g converges a.e. to a function ḡ and ḡ = ∫ x g dµ is constant a.e. the proof is completed. proof of theorem 4.4. ξ is a strong generator implies that p(fn,g) = lim m→∞ 1 m log pm(f n,g,ξ), for g ∈ c(x,r). same discussion as in the proof of lemma 4.1 indicates that m−1∨ i=0 f−niξ = { ξ(a,b + (m− 1)ns), 0 ≤ r; ξ(a + (m− 1)nr,b + (m− 1)ns), 0 ≥ r. and card m−1∨ i=0 f−niξ = { υb−a+1+(m−1)ns, 0 ≤ r; υb−a+1+(m−1)n(s−r), 0 ≥ r. hence p(fn,g) = lim m→∞ 1 m log pm(f n,φ,ξ) = lim m→∞ 1 m log inf{ ∑ b sup x∈b e(smg)(x) : b ∈ m−1∨ i=0 f−niξ} = lim m→∞ 1 m log { υb−a+1+(m−1)nse(smg), 0 ≤ r; υb−a+1+(m−1)n(s−r)e(smg), 0 ≥ r. = { ns log υ + ∫ x g dµ, 0 ≤ r; n(s−r) log υ + ∫ x g dµ, 0 ≥ r. by lemma 4.5. it is easily seen that p(fn,g) = np(f,g) − (n− 1) ∫ x g dµ. this completes the proof. 49 some properties of topological pressure on cellular automata example 4.6 (continued). let a and f same as in example 2.6. replace n by 1 in the proof above, we have p(f,g) = 2 log 4 + ∫ x g dµ, and p(fn,g) = 2n log 4 + ∫ x g dµ. hence p(fn,g) = np(f,g) − (n− 1) ∫ x g dµ for all n ∈ n. furthermore, ward [25] shows that htop(f) = 2 log 4. this implies p(fn,g) = nhtop(f) + ∫ x g dµ = htop(f n) + ∫ x g dµ. notably, theorem 4.4 fails if the permutivity of f is omitted. proposition 4.7. consider f a ca with local rule f : a2k+1 →a, defined by f(x−k, . . . ,fk) = κx0, where 0 ≤ κ ≤ υ − 1. then p(fn,g) = 0 for all n ∈ n,g ∈ c(x,r). proof. it can be easily verified that f is not permutive. if κ = 0, it comes immediately that p(fn,g) = 0 for all n ∈ n,g ∈ c(x,r). it remains to show the cases κ 6= 0. for n ∈ n and ξ(a,b) a partition of x, where a ≤ b. since f(x−k, . . . ,xk) = κx0, we have∨m−1 i=0 f −niξ(a,b) = ξ(a,b). this implies p(fn,g) = 0 for any g ∈ c(x,r). the proof is completed. acknowledgment the author wishes to express his gratitude to the anonymous referees. their comments make a significant improvement to this paper. references [1] h. akin, on the ergodic properties of certain additive cellular automata over zm, appl. math. comput., 168, 192-197, 2005. 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[27] s. wolfram, statistical mechanics of cellular automata, rev. modern physics, 55, 601-644, 1983. 51 introduction preliminaries ergodicity on permutive cellular automata the topological pressure of power rule of ca acknowledgment references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.451218 j. algebra comb. discrete appl. 5(3) • 137–142 received: 11 april 2018 accepted: 16 july 2018 journal of algebra combinatorics discrete structures and applications finite rogers–ramanujan type continued fractions∗ research article helmut prodinger abstract: new finite continued fractions related to bressoud and santos polynomials are established. 2010 msc: 05a30, 11a55 keywords: bressoud polynomials, santos polynomials, rogers–ramanujan identities 1. introduction define, as it is common today, (x; q)n := (1 − x)(1 − xq) . . . (1 − xqn−1), where we assume that |q| < 1, and we allow n also to be 0 and infinity. we also need the coefficients [ n k ] := (q;q)n (q;q)k(q;q)n−k . these standard notations can be found e. g. in the classic book [1]. the two rogers-ramanujan identities [1, 6] ∑ n≥0 qn 2 (q; q)n = 1 (q; q5)∞(q4; q5)∞ , ∑ n≥0 qn 2+n (q; q)n = 1 (q2; q5)∞(q3; q5)∞ are very popular, influential, useful and historically interesting. let f(z) = ∑ n≥0 qn 2 zn (q; q)n and g(z) = ∑ n≥0 qn 2+nzn (q; q)n ; ∗ dedicated to peter paule on the occasion of his 60th birthday. helmut prodinger; department of mathematics, university of stellenbosch 7602, stellenbosch, south africa (email: hproding@sun.ac.za). 137 https://orcid.org/0000-0002-0009-8015 h. prodinger / j. algebra comb. discrete appl. 5(3) (2018) 137–142 the continued fraction (due to ramanujan) zg(z) f(z) = z 1 + zq 1 + zq2 1 + ... is also very well known, see [5, entry 5] and [6, (6.1)]. there are several families of polynomials that approximate f(z) and g(z). probably the the most well known are fn(z) = ∑ j≥0 qj 2 [ n + 1− j j ] zj → f(z) and gn(z) = ∑ j≥0 qj 2+j [ n− j j ] zj → g(z) for n →∞, because of their link to the schur polynomials, see [1]. the finite continued fraction zgn(z) fn(z) = z 1 + zq 1 + zq2 ... 1 + zqn 1 is also known [5, entry 16]. the polynomials sn(z) = ∑ j≥0 qj 2 [ n j ] zj → f(z) and tn(z) = ∑ j≥0 qj 2+j [ n j ] zj → g(z) for n →∞, due to bressoud [7], are less well known; see also [8]. 2. bressoud polynomials and continued fractions in this section, we will establish the following attractive finite continued fraction: theorem 2.1. ztn(z) sn(z) = z 1 + zq(1−qn) 1 + zq2 1 + zq3(1−qn−1) 1 + zq4 1 + ... proof. to prove this statement by induction, define the righthand side by tn(z). it is plain to see that t0(z) = z, and tn(z) = z 1 + zq(1−qn) 1 + tn−1(zq2) . 138 h. prodinger / j. algebra comb. discrete appl. 5(3) (2018) 137–142 we are left to prove that ztn(z) sn(z) = z 1 + zq(1−qn) 1 + ztn−1(zq 2) sn−1(zq2) = z 1 + zq(1−qn)sn−1(zq2) sn−1(zq2) + ztn−1(zq2) = z(sn−1(zq 2) + ztn−1(zq 2)) sn−1(zq2) + ztn−1(zq2) + zq(1−qn)sn−1(zq2) , which amounts to prove that tn(z) = sn−1(zq 2) + ztn−1(zq 2), sn(z) = sn−1(zq 2) + ztn−1(zq 2) + zq(1−qn)sn−1(zq2). we will show that the coefficients of zj coincide, which is trivial for j = 0, so we assume j ≥ 1: qj 2+j [ n j ] = qj 2+2j [ n−1 j ] + q(j−1) 2+3(j−1)+2 [ n−1 j −1 ] , which is equivalent to [ n j ] = qj [ n−1 j ] + [ n−1 j −1 ] and therefore true. the second one goes like this: qj 2 [ n j ] = qj 2+2j [ n−1 j ] + q(j−1) 2+3(j−1)+2 [ n−1 j −1 ] + q(1−qn)q(j−1) 2+2(j−1) [ n−1 j −1 ] , which is equivalent to [ n j ] = q2j [ n−1 j ] + qj [ n−1 j −1 ] + (1−qj) [ n j ] , and further to [ n j ] = qj [ n−1 j ] + [ n−1 j −1 ] , which finishes the proof. 3. identities 39 and 38 from slater’s list slater [11] produced a list of rogers-ramanujan type series/product identities; sills [10] in an amazing effort reworked and annotated this list, providing, in particular, finite versions of all of them. arguably the second most popular identities in the rogers-ramanujan world are slater’s [11] identities (39) and (38) ∑ n≥0 q2n 2 (q; q)2n = ∏ k≥1, k≡±2,±3,±4,±5 (mod 16) 1 1−qk , ∑ n≥0 q2n 2+2n (q; q)2n+1 = ∏ k≥1, k≡±1,±4,±6,±7 (mod 16) 1 1−qk . 139 h. prodinger / j. algebra comb. discrete appl. 5(3) (2018) 137–142 let sn(z) = ∑ 0≤2h≤n q2h 2 [ n 2h ] zh and tn(z) = ∑ 0≤2h≤n q2h 2+2h [ n 2h + 1 ] zh; these polynomials are called santos polynomials [2–4]. in order to describe the finite continued fraction expansion of ztn(z)/sn(z), we define the following numbers and polynomials (power series) which were originally found by guessing: a2k := (1−q4k+1)(qn+1−2k; q2)2k q2k(qn−2k; q2)2k+1 , a2k+1 := (1−q4k+3)(qn−2k; q2)2k+1 q2k+2(qn−2k−1; q2)2k+2 ; s2i := ∑ j≥0 q2(i+j)(i+j+1)(qn−2i−2j; q)2j(q n−2i; q2)2i+1 (q; q)2j+1(q2j+3; q2)2i zj, s2i+1 := ∑ j≥0 q2(i+j+1) 2 (qn−1−2i−2j; q)2j(q n−1−2i; q2)2i+2 (q; q)2j+1(q2j+3; q2)2i+1 zj. theorem 3.1. the polynomials si satisfy the second order recurrence zsi+1 = si−1−aisi, s−1 = sn(z), s0 = tn(z). consequently, we get the finite continued fraction expansion zt(n) s(n) = z a0 + z a1 + z a2 + z ... , or, more elegantly: zt(n) s(n) = zb0 1 + zb1 1 + zb2 1 + . . . with b0 = 1 a0 = 1−qn 1−q and bi = 1 ai−1ai = q2i(1−qn−i)(1−qn+i) (1−q2i−1)(1−q2i+1) for i ≥ 1. 140 h. prodinger / j. algebra comb. discrete appl. 5(3) (2018) 137–142 proof. the recursion will be shown for even i, the other instance being very similar: s2i−1 −a2is2i = ∑ j≥0 q2(i+j) 2 (qn+1−2i−2j; q)2j(q n+1−2i; q2)2i (q; q)2j+1(q2j+3; q2)2i−1 zj − (1−q4i+1)(qn+1−2i; q2)2i q2i(qn−2i; q2)2i+1 ∑ j≥0 q2(i+j)(i+j+1)(qn−2i−2j; q)2j(q n−2i; q2)2i+1 (q; q)2j+1(q2j+3; q2)2i zj = (qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2 (qn+1−2i−2j; q)2j (q; q)2j+1(q2j+3; q2)2i−1 zj − (1−q4i+1)(qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2+2j(qn−2i−2j; q)2j (q; q)2j+1(q2j+3; q2)2i zj = (qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2 (qn+1−2i−2j; q)2j−1(1−qn−2i)(1−q2j+4i+1) (q; q)2j+1(q2j+3; q2)2i zj − (1−q4i+1)(qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2+2j(qn+1−2i−2j; q)2j−1(1−qn−2i−2j) (q; q)2j+1(q2j+3; q2)2i zj = (qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2 (qn+1−2i−2j; q)2j−1 (q; q)2j+1(q2j+3; q2)2i zj × [ (1−qn−2i)(1−q2j+4i+1)−q2j(1−q4i+1)(1−qn−2i−2j) ] = (qn+1−2i; q2)2i ∑ j≥0 q2(i+j) 2 (qn+1−2i−2j; q)2j−1 (q; q)2j+1(q2j+3; q2)2i zj(1−q2j)(1−qn+1+2i) = (qn+1−2i; q2)2i+1 ∑ j≥1 q2(i+j) 2 (qn+1−2i−2j; q)2j−1 (q; q)2j+1(q2j+3; q2)2i zj(1−q2j) = z(qn−1−2i; q2)2i+2 ∑ j≥0 q2(i+j+1) 2 (qn−1−2i−2j; q)2j (q; q)2j+1(q2j+3; q2)2i+1 zj = zs2i+1. further, s−1 = ∑ j≥0 q2j 2 (qn+1−2j; q)2j (q; q)2j+1(q2j+3; q2)−1 zj = ∑ j≥0 q2j 2 (qn+1−2j; q)2j (q; q)2j zj = s0(z) and s0 = ∑ j≥0 q2j(j+1)(qn−2j; q)2j(1−qn) (q; q)2j+1 zj = ∑ j≥0 q2j(j+1)(qn−2j; q)2j+1 (q; q)2j+1 zj = t0(z). now we can iterate this relation in the following form: zt(n) s(n) = zs0 s−1 = z a0 + zs1 s0 = z a0 + z a1 + zs2 s1 = . . . this is the desired (finite) continued fraction expansion. we remark that for n →∞, the quantities ai and si appear already in [9]. 141 h. prodinger / j. algebra comb. discrete appl. 5(3) (2018) 137–142 4. conclusion we would like to mention that it is more challenging to find the continued fractions and the relevant quantities, as the present proofs (and possibly other ones) consist of routine manipulations. since there are many rogers-ramanujan type identities and polynomials approximating them are not even unique, there might be additional additional results; compare our previous effort [9] for infinite versions. most polynomials from sill’s list [10] are, however, not expressable in terms of one summation and thus not candidates for the present approach. references [1] g. e. andrews, the theory of partitions, encyclopedia of mathematics and its applications, vol. 2, addison–wesley publishing co., reading, mass.–london–amsterdam, 1976. [2] g. e. andrews, a. knopfmacher, p. paule, h. prodinger, q–engel series expansions and slater’s identities, quaest. math. 24(3) (2001) 403–416. [3] g. e. andrews, a. knopfmacher, p. paule, an infinite family of engel expansions of rogers– ramanujan type, adv. appl. math. 25(1) (2000) 2–11. [4] g. e. andrews, j. p. o. santos, rogers–ramanujan type identities for partitions with attached odd parts, ramanujan j. 1(1) (1997) 91–99. [5] b. c. berndt, ramanujan’s notebooks, part iii, springer–verlag, new york, 1991. [6] b. c. berndt, s. s. huang, j. sohn, s. h. son, some theorems on the rogers–ramanujan continued fraction in ramanujan’s lost notebook, trans. amer. math. soc. 352 (2000) 2157–2177. [7] d. m. bressoud, some identities for terminating q-series, math. proc. cambridge philos. soc. 89(2) (1981) 211–223. [8] r. chapman, a new proof of some identities of bressoud, int. j. math. math. sci. 32(10) (2002) 627–633. [9] n. s. s. gu, h. prodinger, on some continued fraction expansions of the rogers–ramanujan type, ramanujan j. 26(3) (2011) 323–367. [10] a. v. sills, finite rogers–ramanujan type identities, electron. j. combin. 10 (2003) research paper 13, 122 pp. [11] l. j. slater, further identities of the rogers–ramanujan type, proc. london math. soc. s2–54(1) (1952) 147–167. 142 https://mathscinet.ams.org/mathscinet-getitem?mr=557013 https://mathscinet.ams.org/mathscinet-getitem?mr=557013 https://doi.org/10.1080/16073606.2001.9639228 https://doi.org/10.1080/16073606.2001.9639228 https://doi.org/10.1006/aama.2000.0686 https://doi.org/10.1006/aama.2000.0686 https://doi.org/10.1023/a:1009719322309 https://doi.org/10.1023/a:1009719322309 https://doi.org/10.1007/978-1-4612-0965-2 https://doi.org/10.1090/s0002-9947-00-02337-0 https://doi.org/10.1090/s0002-9947-00-02337-0 https://doi.org/10.1017/s0305004100058114 https://doi.org/10.1017/s0305004100058114 http://dx.doi.org/10.1155/s0161171202110155 http://dx.doi.org/10.1155/s0161171202110155 https://doi.org/10.1007/s11139-011-9329-7 https://doi.org/10.1007/s11139-011-9329-7 https://mathscinet.ams.org/mathscinet-getitem?mr=1975763 https://mathscinet.ams.org/mathscinet-getitem?mr=1975763 https://doi.org/10.1112/plms/s2-54.2.147 https://doi.org/10.1112/plms/s2-54.2.147 introduction bressoud polynomials and continued fractions identities 39 and 38 from slater's list conclusion references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(1) • 1-16 received: 23 july 2014; accepted: 11 september 2014 doi 10.13069/jacodesmath.36947 journal of algebra combinatorics discrete structures and applications a database of linear codes over f13 with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm research article eric z. chen1∗, nuh aydin2∗∗ 1. department of computer science, kristianstad university 2. department of mathematics and statistics, kenyon college abstract: error control codes have been widely used in data communications and storage systems. one central problem in coding theory is to optimize the parameters of a linear code and construct codes with best possible parameters. there are tables of best-known linear codes over finite fields of sizes up to 9. recently, there has been a growing interest in codes over f13 and other fields of size greater than 9. the main purpose of this work is to present a database of best-known linear codes over the field f13 together with upper bounds on the minimum distances. to find good linear codes to establish lower bounds on minimum distances, an iterative heuristic computer search algorithm is employed to construct quasi-twisted (qt) codes over the field f13 with high minimum distances. a large number of new linear codes have been found, improving previously best-known results. tables of [pm, m] qt codes over f13 with best-known minimum distances as well as a table of lower and upper bounds on the minimum distances for linear codes of length up to 150 and dimension up to 6 are presented. 2010 msc: 94b05, 94b65 keywords: database of linear codes, quasi-twisted codes, heuristic search algorithm, iterative search 1. introduction and motivation let [n,k,d]q denote a linear code of length n, dimension k and minimum distance (weight) d over the finite field fq. a central and fundamental problem in coding theory is to find the optimal values of the parameters of a linear code and construct codes with these parameters. the problem can be formulated in a few different ways. for example, we may wish to maximize the minimum distance d for the given block length n and dimension k; or minimize the block length n for the given dimension k and minimum ∗ e-mail: eric.chen@hkr.se ∗∗ e-mail: aydinn@kenyon.edu 1 a database of linear codes over f13 distance. let dq(n,k) denote the largest value of d for which there exists an [n,k,d] code over fq, and nq(k,d) the smallest value of n for which there exists an [n,k,d] code over fq. an [n,k,d] code is called optimal (or length-optimal) if its block length n equals nq(k,d), or if its minimum distance d equals dq(n,k) (also called distance-optimal). this optimization problem is very difficult. in general, it is only solved for the cases where either k or n − k is small. computers are often used in searching for codes with best parameters but there is an inherent difficulty: computing the minimum distance of a linear code is computationally intractable [19]. since it is not possible to conduct exhaustive searches for linear codes if the dimension is large, researchers often focus on promising subclasses of linear codes with rich mathematical structures. as a generalization to cyclic and consta-cyclic codes, quasi-cyclic (qc) and quasi-twisted (qt) codes are known to have this characteristic. they have been shown to contain many good linear codes. with the help of modern computers, many record-breaking qc and qt codes have been constructed [2]-[13]. however, the problem still becomes intractable as the dimension and the block length of the code get large. the records of best-known linear codes are available. for example, the online database [21] is one that is commonly referred to. it contains records of best-known codes over fq for q ≤ 9 together with upper bounds on dq(n,k). the magma software [20] also contains a similar database. the online database of qt codes contains best-known qc and qt codes [24]. these databases are updated as new codes are discovered. there has been a growing interest in codes over f13 in recent years. several papers in the literature deal with self-dual or maximum distance separable (mds) codes over f13 . for example, betsumiya [25] et al studied mds self-dual codes over f13 of lengths up to 24 and determined largest minimum weights of such codes for lengths up to 20. de boer [16] constructed a self-dual [18, 9, 9] code and optimal codes with parameters [23, 3, 20] and [23, 17, 6] over f13. newhart [26] studied the extended quadratic residue (qr) codes [18, 9, 9], [24, 12, 10] and [30, 15, 12] over f13. grassl and gulliver [28] showed non-existence of a self-dual mds code over f13 with parameters [12,6,7]. in [29] the authors constructed a euclidean self-dual near-mds code over f13. kotsireas et al. constructed many mds and near-mds self-dual codes over f13 [27]. another reason for the interest in codes over f13 is the connection between linear codes and finite geometries. codes of dimension 3 are closely related to arcs in a projective geometry, and a lot of research has been carried out on projective codes of dimension 3 over finite fields of size up to 19 [4]. finally, venkaiah and gulliver [13] used the tabu search to construct quasi-cyclic codes over f13 of dimensions up to 6 and lengths less than 150. they constructed many qc codes of the form [pk,k], for over f13, and presented their results in several tables (one for each value of k). these tables constitute the most comprehensive set of best-known linear codes over f13 to date. in this paper, we present a database of linear codes over f13 for lengths ≤ 150 and dimensions 3 ≤ k ≤ 6. we employed an iterative, heuristic algorithm [15] to conduct a computer search to produce new codes. with this algorithm, a large number of new qc and qt codes have been constructed many of which improve the previous results. we achieve improvements on the parameters of the codes presented in [13] in many cases. combining the results presented in [13] with the new codes we have found, we create a comprehensive database of best-known linear codes over f13. to the best of our knowledge, this is the first time such a database appears in the literature. the remainder of the paper is organized as follows. in section 2, some basic definitions and facts on qt codes are presented. in section 3, the iterative heuristic algorithm that is used to find good qt codes is described. next, a database of linear codes over f13 with minimum distance bounds is presented. the paper contains several tables: tables of new, improved qc and qt codes, maximum known minimum distances for qt [pm,m] codes, optimal qt codes, as well as a comprehensive table of lower and upper bounds on linear codes over f13 that covers the range n ≤ 150 and 3 ≤ k ≤ 6. with these concrete results, this work can serve as a foundation for future research on linear codes over f13 (e.g. a more comprehensive database). 2 e. z. chen, n. aydin 2. quasi-twisted codes a linear q-ary [n,k,d] code is said to be α-consta-cyclic if there is a non-zero element α of fq such that for any codeword (a0,a1, . . . ,an−1), a consta-cyclic shift by one position, that is (αan−1,a0, . . . ,an−2), is also a codeword [14]. therefore, consta-cyclic codes are a generalization of cyclic codes, or a cyclic code is an α-consta-cyclic code with α = 1. a consta-cyclic code can be defined by a single generator polynomial. a code is said to be quasi-twisted (qt) if a consta-cyclic shift of any codeword by p positions is still a codeword. thus a consta-cyclic code is a qt code with p = 1, and a quasi-cyclic (qc) code is a qt code with α = 1. the length n of a qt code is a multiple of p, i.e., n = pm for some positive integer m. an α-consta-cyclic matrix of order n, also called a twistulant matrix, is defined as c =   c0 c1 c2 · · · cn−1 αcn−1 c0 c1 · · · cn−2 αcn−2 αcn−1 c0 · · · cn−3 ... ... ... ... αc1 αc2 αc3 · · · c0   (1) twistulant matrices are basic components in the generator matrix for a qt code. the algebra of n × n consta-cyclic matrices over fq is isomorphic to the algebra of the quotient ring fq[x]/(xn − α) if c is mapped onto the polynomial formed by the elements of its first row, c(x) = c0 + c1x + · · · + cn−1xn−1, with the least significant coefficient on the left. the polynomial c(x) is also called the defining polynomial of the matrix c. a twistulant matrix is called a circulant matrix if α = 1. the generator matrix of a qt code can be transformed into rows of twistulant matrices by a suitable permutation of columns. most research has been focused on 1-generator and 2-generator qt codes. the generator matrices for 1-generator and 2-generator qt codes consist of one row of twistulant matrices and two rows of twistulant matrices, respectively, g = [ g0 g1 · · · gp−1 ] and g = [ g1,0 g1,1 · · · g1,p g1,0 g2,1 . . . g2,p ] (2) where gj and gij are twistulant matrices, for j = 0,1,2, . . . ,p1 and i = 1,2. let gi,j(x) and gi,j(x) be the defining polynomials for the corresponding twistulant matrices gj and gij. then, the defining polynomials (g0(x),g1(x),g2(x), · · · ,gp−1(x)) and (g1,0(x),g1,1(x),g1,2(x), . . . ,g1,p−1(x);g2,0(x),g2,1(x), g2,2(x), . . . ,g2,p−1(x)) define a 1-generator qt [pm,k,d] code and 2-generator qt [pm,k,d] code, where k, the dimension of the code, is the rank of the generator matrix g. in magma algebra system [20], the number of generators is called the height. the parameters of all the codes presented in this paper have been verified by magma. 3. the search algorithm and new qt codes over f13 as a generalization to cyclic codes and consta-cyclic codes, quasi-cyclic (qc) codes and quasi-twisted (qt) codes have been known to contain many good codes. in fact, many record-breaking linear codes have been obtained from these classes [2]-[13]. gulliver et al. [4, 5, 9, 13] have done much work on the computer searches for good qc and qt codes. by eliminating the equivalent generator polynomials, and eliminating all redundant information polynomials, an r×s weight matrix w is used in the constructions, as given below, where ck(x) is the kth generator polynomial, ij(x) is the jth information polynomial, wjk is the hamming weight of ij(x)ck(x) 3 a database of linear codes over f13 w = c1(x) c2(x) · · · ck(x) · · · cs(x) i1(x) w11 w12 · · · w1k · · · w1s i2(x) w21 w22 · · · w2k · · · w2s ... ... ... · · · ... ... ij(x) wj1 wj2 · · · wjk · · · wjs ... ... ... · · · ... ... ir(x) wr1 wr2 · · · wrk · · · wrs mod (xm − α), m is the size of the twistulant matrix and α is the shift constant. to construct a good qt [pm,k] code, their algorithm selects a set of p columns among s columns such that the set of columns maximizes the smallest row sum of the corresponding p columns. when p and s are large, it is not possible to examine all (s,p) combinations. gulliver’s search is initialized with an arbitrary [pm,k] code (usually a good one) with p columns (or generator polynomials). to improve the code, a new column is found to replace one presently in the code so that the minimum distance is increased. later on, a stochastic optimization called tabu search has been used to construct good qc or qt codes by gulliver and östergård [9], and daskalov et al. [10]. in a recent paper, venkaiah and gulliver [13] used the tabu search to find good qc codes over f13. on the other hand, a method to obtain a weight matrix from a consta-cyclic simplex code of composite length was recently presented in [15]. the resulting weight matrix is cyclic, and therefore only one row is required to be in the memory during the search. a new iterative heuristic search is also presented, and many good qt codes have been constructed [15]. in this work, the algorithm from [15] is applied to both the weight matrix defined by gulliver’s method and the weight matrix derived from the consta-cyclic simplex code as given in [15]. as a result, many good qt codes have been obtained, allowing us to establish a database of linear codes over f13 with the range of parameters described above. given an r×s weight matrix w = (wij). the iterative algorithm tries to find a sequence of good qt [im,k]q codes, i = 1,2, . . . , t, where t < s. the basic idea of the algorithm is to extend a qt [(i−1)m,k]q code by one more column to obtain a good qt [im,k]q code, for i = 2,3, . . . , t. the algorithm is executed for a specified number of iterations. the algorithm records the best codes found so far, and stores them in files. when the algorithm stops, a summary of the codes found is presented. in the execution of the algorithm, the selection of columns is important as it determines if good codes can be found quickly. in order to avoid exhaustive search, we use a heuristic method to implement the selection. at each iteration, to obtain the best possible minimum distance for a qt [im,k]q code, we select a column that results in the largest minimum row sum (it is also the minimum distance of the constructed code). if there is more than one column that gives the same best minimum distance, we count how many such rows that result in the minimum row sum. we choose the column that will have the smallest number of such rows, since it is expected that such a selection will provide a better chance to get a good qt [(i + 1)m,k]q code in the next extension. in this way, the algorithm is greedy and heuristic. if there is more than one choice, a column is selected at random among suitable choices. so the algorithm contains some randomization. the effectiveness of this iterative heuristic search algorithm is evident from the fact that a large number of new qt codes over f13 for k = 3,4,5, and 6 have been obtained as a result of the application of the algorithm. the new codes improve the previously known results. table 1 lists the new qt codes over f13 that have larger minimum distances than the corresponding codes given in [13]. the defining polynomials are listed with the lowest degree coefficient on the left, and the finite field f13 elements 10, 11, 12 are denoted by a,b and c (as commonly used in a hexa-decimal system). for example, c024a9 corresponds to the polynomial 12 + 2x2 + 4x3 + 10x4 + 9x5. table 2 summarizes the maximum known minimum distances for qt [pm,m] codes over f13 for p up to 25. the authors can provide all best known qt codes for n up to 255, upon request. most entries in the table are from the results in [13], and the entries labeled with superscript “e” are new codes found with the algorithm in this paper. all codes with k = 6 are constructed from the weight matrix derived 4 e. z. chen, n. aydin from the consta-cyclic simplex [402234,6,371293] code. since the weight matrix is cyclic, only one row of 402234/6 = 67039 elements is required to be stored in memory. this makes it easier to search for good qt codes with k = 6 (otherwise, the weight matrix is too big to fit in the memory). 4. a database of linear codes over f13 with minimum distance bounds 4.1. lower bounds on minimum distance since there are no good, general analytical lower bounds available for the parameters of a linear code, the lower bounds on minimum distances have been established by explicitly constructing the codes [1]. as commented earlier, constructing good linear codes is a difficult task because finding the minimum distance of a linear code is computationally expensive [19]. therefore, researchers focus on certain promising classes of codes with rich mathematical structure. the class of qt codes has been an excellent source for producing best-known codes [2]-[13]. constacyclic codes are a special case of qt codes. following the approach given in [22], we have been able to compute all constacyclic codes exhaustively for most lengths since the dimension is restricted to 3 ≤ k ≤ 6. some of the best-known (or optimal) codes are constacyclic. another tool that can be used to obtain more new codes from existing codes in a computationally efficient way is to apply standard construction methods to derive codes from known codes, such as puncturing, shortening and extending [1]. with the codes constructed in [13], the new qt codes over f13 presented in the previous section, as well as the standard construction methods to derive new codes from existing codes, we are able to create a comprehensive table of lower bounds on the minimum distances for linear codes over f13 with dimensions between 3 and 6 and block length n up to 255. table 3 includes the lower bounds for block lengths up to 150. there is a connection between best-known linear codes and projective geometry. an (n,r)-arc in pg(k − 1,q) is a set of n points k with the property that every hyperplane is incident with at most r points of k and there is some hyperplane incident with exactly r points of k. it is known that there exists a projective [n,3,d]q code if and only if there exists an (n,n − d)-arc in pg(2,q) [13]. ball [17] maintains an online table of bounds on the sizes of (n,r)-arcs in pg(2,q) for q ≤ 19 . from that table, one can obtain lower bounds on the minimum distances of linear codes of dimension 3. some of the entries in table 3 for k = 3 can be derived from [17]. table 4 lists the defining polynomials for the new codes found in this paper and that are used to establish the lower bounds in table 3. there are 7 new 2-generator qt codes with k = 6 and m = 3 that are used to derive the lower bounds in table 3. 5. upper bounds on minimum distance we also determined upper bounds on the minimum distances by applying the standard bounds (such as griesmer, elias, sphere packing etc.) [1] and taking the best result for each parameter set. in the range of parameters considered here, griesmer bound turned out to be the best for most of the cases except that in some cases the levenshtein bound performed better. when a code whose minimum distance equals to the upper bound, an optimal code is constructed and there is no room for improvement in the table. when there is a gap between the minimum distance of a best-known code and the upper bound on the minimum distance, this is indicated in the table by listing the both values. for example, for a [51,4]-code, the minimum distance of a best-known code is 43 whereas the theoretical upper bound is 45. it is worth noting that the theoretical upper bound may be unattainable. to save the space, only entries for the block length n up to 150 are given below (table 3). interested readers can obtain the full table 5 a database of linear codes over f13 from the authors. 5.1. linear codes with dimension 3 suppose d ≤ qk−1 and that c is an [n,k,d] code over fq which attains the griesmer bound. then c is projective [13]. therefore, from the ball’s table, we conclude that there do not exist codes with the following parameters over f13: [15, 3, 13], [24, 3, 21], [25, 3, 22], [26, 3, 23], [27, 3, 24], [28, 3, 25], [29, 3, 26], [41, 3, 37], [42, 3, 38], [43, 3, 39], [54, 3, 49], [55, 3, 50], [56, 3, 51], [57, 3, 52], [70, 3, 64], [71, 3, 65], [80, 3, 73], [81, 3, 74], [82, 3, 75], [83, 3, 76], [84, 3, 77], [85, 3, 78], [93, 3, 85], [94, 3, 86], [95, 3, 87], [96, 3, 88], [97, 3, 89], [98, 3, 90], [99, 3, 91], [106, 3, 97], [107, 3, 98], [108, 3, 99], [109, 3, 100], [110, 3, 101], [111, 3, 102], [112, 3, 103], [113, 3, 104], [120, 3, 110], [121, 3, 111], [122, 3, 112], [123, 3, 113], [124, 3, 114], [125, 3, 115], [126, 3, 116], [127, 3, 117], [134, 3, 123], [135, 3, 124], [136, 3, 125], [137, 3, 126], [138,3 , 127], [139, 3, 128], [140, 3, 129], [141, 3, 130], [148, 3, 136], [149, 3, 137], and [150, 3, 138]. 5.2. some optimal codes over f13 table 3 presents the lower and upper bounds on d13(n,k) for k up to 6. many bounds are attained. it is possible that some of the current upper bounds may be improved and more codes may turn out to be optimal. in the rest of this section, we give more details on the optimal codes in table 3. with the algorithm given in the last section, many qc codes with k = 3 have been constructed whose minimum distances meet the griesmer bounds, and thus are optimal. table 5 lists those optimal qc [pm,3] codes that do not appear in [13]. it should be noted that codes with these parameters were not constructed in the qc form [17, 23]. codes constructed in qc or qt form have advantages in practical implementation. table 6 lists optimal qt [pm,k] codes for k = 4,5 and 6, over f13, and their defining polynomials. with the upper bounds given in table 3, we now know that the qc [20, 4, 16] and [28, 4, 23] codes constructed in [13] are optimal, since they reach the upper bounds. the optimal [153, 4, 139] code is included here, since two other optimal codes are obtained from it by puncturing: [150, 4, 136] and [149, 4, 135] codes. the optimal [15, 6, 9] code given in the table is a 2-generator qt code with shift constant 6, and is constructed with the method given in [15]. with these codes, and results on (n,r)-arcs, the exact values on d13(n,k) in table 3 are established. 6. conclusion in this paper, we present the construction of a large number of new qt codes over f13 obtained by an iterative heuristic search algorithm recently introduced. the results are presented in several tables. combining the new results with earlier work on linear codes over f13, a database of linear codes over f13 with both lower and upper bounds on the minimum distances is presented for the first time. we hope that the results presented in this paper serve as a basis for future study on codes over f13. 6 e. z. chen, n. aydin table 1 new qc and qt codes over f13 code m α defining polynomials [63, 3, 57] 3 1 531, 51, 61, c11, b31, 21, a31, 321, 211, 341, 641, c31, b11, 611, 91, 921, c1, 261, 241, 311, 651 [40, 4, 34] 5 1 c1, 7b71, 7b611, 2911, a9511, 3b921, bc21, 69731 [48, 4, 41] 4 6 c55b, 529b, b301, 0ac5, a418, 4ca2, 1a21, 0995, 1625, 1c21, 93b1, 7a9c [60, 4, 52] 4 6 c55b, 9578, 9997, 2586, a9c3, 4254, 6a96, a3a3, b0b4, b501, a61b, 45b3, 7255, 5c97, 2c3c [68, 4, 60] 17 6 c9566572b03055915, 663cc4022720508c5, 8680977c590521b4a, 972a15a2473369c09 [68, 4, 59] 4 1 38a, 1b8, 191, b873, 6aa1, 6c4, 103, ba11, 417, 468b, 6521, 315, 6712, 7133, 6691, 422a, 9631 [72, 4, 63] 4 1 [68, 4,59] code, 6171 [76, 4, 67] 4 2 9012, bb74, 3631, 849, a98, 7c26, c8ca, 6c74, 7ba1, 661, c219, 4148, 1c37, bb21, 7a, 2489, 5797, c668, a751 [80, 4, 71] 4 2 [76, 4, 67] code, 28 [88, 4, 77] 4 1 681, 6b21, 2c21, 2711, a51, 3421, 4b41, a621, 2851, 6a71, 4a11, c431, 2b21, a91, 361, 451, 6211, 3b41, 51, c11, 6b1, 7121 [92, 4, 81] 4 6 c55b, 732a, 614, b965, 290c, ba84, 9113, 8251, 42c3, c71a, 7b64, c3a4, c867,2a73, c081, 1c88, aacb, 95a8, abba, a61b, 0549, 0837, 887b [100, 4, 88] 4 1 691, 211, 581, 5281, a81, a11, 3231, 231, 8b31, 8531, 4941, 6531, 4621, a831, 4961, c411, 4a11, 2711, 5831, c111, 5721, 8321, 8911, c431, a51 [40, 5, 32] 5 1 8351, 6721, c1511, 83731, 5191, ca821, b3c31, 7a411 [75, 5, 63] 5 1 c841, 8611, a7521, 93211, ac81, 74b1, 2c411, 4b571, ca831, 48161, 9a721, b451, 4a131, 69a1, 38711 [85, 5, 72] 5 1 81c21, 41931, 47521, 98711, bab41, 54721, 71611, a621, 471, c6a1, 69a21, b7c21, 7cc1, 3c81, a8111, bc821, 56131 [95, 5, 81] 5 1 b9261, 61811, 9751, c9c11, a3c21, 5811, c2641, 64c11, c251, 93c1, 89c1, c1a11, b3761, 61831, 1231, 601, 28b41, 8611, bcb21 [100, 5, 85] 5 1 b191, a291, b631, a8c1, 41611, 81711, 27b1, 7801, 3331, b361, cb521, 43261, bc921, 53641, cbb1, 69611, 32311, 35731, 65261, 3201 [105, 5, 90] 5 1 48911, a1211, 6a71, 24621, 17a1, 63921, cac21, 6a651, 3b241, ca21, 37511, 46941, 1b91, 9c121, c2741, aba1, b4821, 4481, 39a1, ac911, 89531 [110, 5, 94] 5 1 b191, a291, b631, a8c1, 41611, 81711, 27b1, 7801, 3331, b361, cb521, 43261, bc921, 53641, cbb1, 69611, 32311, 35731, 65261, 2411, 37c1, 1 [115, 5, 99] 5 1 53641, c4c21, 95511, 45861, 9401, a9511, bab31, 5141, b3a1, 2211, 89641, 93b1, 66a1, 94321, 85c1, a161, 6a391, 7161, bb61, 3ab1, 58511, 64b21, 68111 [120, 5, 103] 5 1 b191, a291, b631, a8c1, 41611, 81711, 27b1, 7801, 3331, b361, cb521, 43261, bc921, 53641, cbb1, 69611, 32311, 35731, 65261, 2411, 37c1, 52411, 67a11, 6b1 [18, 6, 12] 6 6 c024c9, 16589b, ab836 [28, 6, 20] 28 1 83470747880b081737a7331 [36, 6, 27] 6 6 c024c9, 9064c3, a6666a, 980855, bcc956, 259089 [66, 6, 53] 6 6 c024c9, 422448, 5b6a6c, 918c06, 6016a2, 8111b4, 3c0676, 7c4a08, 1b18b, 32c246, b9c5a3 [72, 6, 58] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, 2c8c13 [84, 6, 69] 28 1 28bb602605a2731cb0b90b65031,9779c314425896634952a6b4541, 6aa2c9836262784120c570c3321 [90, 6, 74] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 01c3c [96, 6, 79] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, bc1263 [102, 6, 84] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, bb1818 [108, 6, 90] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8 [112, 6, 94] 28 1 693580a3c5b4b6b4114264b25b1, 4b3388ac355242875b3105a841, 498a29a8a8489b2497587593661, 354b13a9088905c58328b301941 [114, 6, 95] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, b283 [120, 6, 100] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, 0ab0ab [126, 6, 106] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, a67a0b,9c512a [132, 6, 111] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, a67a0b, 115a6, 348297 [138, 6, 116] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, a67a0b, 115a6, c8833a, bb343 [140, 6, 119] 28 1 a351438b0147a3abcb9a6bc5681, 15894745b677671461888533801, 2a2121c7a84423995189ab26401, 5396c6558b1b083bc216427981, cab6c982774602546921bbb6241 [144, 6, 122] 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, a67a0b, 115a6, c8833a, 6a3704, aa727 7 a database of linear codes over f13 1 table 2 maximum known minimum distances for qt [pk, k] codes over f13 k\p 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 3 4o 7o 10o 12 15o 18o 20 23 26o 29o 32o 34 37 40o 43o 45 48 51 54o 57oe 59 62 65o 68o 4 5o 9o 12 16o 19 23 26 30 33 37 41e 44 48 52e 55 59e 63e 67e 71e 73 77e 81e 84 88e 5 6 10 150 19 23 27 32e 36 40 45 49 54 58 63e 67 72e 76 81e 85e 90e 94e 99e 103e 108e 6 7 12 16 21 27e 32 37 42 47 53e 58e 63 68 74e 79e 84e 90e 95e 100e 106e 111e 116e 122e 127e n0 an optimal code ne new code found in this paper, and exceeds the best minimum distance in [13] 8 e. z. chen, n. aydin table 3 lower and upper bounds on minimum distances for linear codes over f13 n k = 3 4 5 6 n k = 3 4 5 6 n k = 3 4 5 6 1 51 45-46 43-45 41-44 39-43 101 92 89-91 86-90 83-89 2 52 46-47 44-46 42-45 40-44 102 93 90-92 87-91 84-90 3 1 53 47-48 45-47 43-46 41-45 103 94 91-93 88-92 85-91 4 2 1 54 48 46-48 44-47 42-46 104 95 92-94 ca 89-93 86-92 5 3 2 1 55 49 47-48 45-48 43-47 105 96 ba 92-95 90-93 ca 87-93 6 4 3 2 1 56 50 48-49 46-48 ca 44-48 106 96 93-96 90-94 88-93 7 5 4 3 2 57 51 49-50 46-49 45-48 ca 107 97 94-96 91-95 89-94 8 6 5 4 3 58 52 50-51 47-50 45-49 108 98 95-97 92-96 90-95 9 7 6 5 4 59 53 51-52 48-51 46-50 109 99 96-98 93-97 91-96 10 8 7 6 5 60 54 52-53 49-52 47-51 110 100 97-99 94 -98 92-97 11 9 8 7 6 61 55 53-54 50-53 48-52 111 101 98-100 95-99 93-98 12 10 9 8 7 62 56 54-55 51-54 49-53 112 102 99-101 96-100 94-99 ca 13 11 10 9 8 63 57 55-56 52-55 50-54 113 103 100-102 97-101 94-100 14 12 be 11 be 10 be 9 be 64 58 ba 56-57 53-56 51-55 114 104 101-103 98-102 95-101 15 12 11 10 9 65 58-59 57-58 54-57 52-56 115 105 102-104 99-103 96-102 16 13 12 11 10 66 59-60 58-59 55-58 53-57 ca 116 106 103-105 100-103 97-103 17 14 13 12 11 67 60 59 56-58 53-57 117 107 104-106 101-104 98-103 ca 18 15 14 13 12 ca 68 61 60 ca 57-59 ca 54-58 118 108 ba 105-107 102-105 98-104 19 16 15 14 12-13 69 62 60-61 57-60 55-59 119 108-109 106-108 103-106 ca 99-105 20 17 16 vg 15 vg 13-14 70 63 61-62 58-61 56-60 120 109 107-109 103-107 100-106 21 18 16-17 15-16 14-15 ca 71 64 62-63 59-62 57-61 121 110 108-109 104-108 101-107 22 19 17-18 16-17 14-16 72 65 63-64 60-63 58-62 ca 122 111 109-110 104-109 102-108 23 20 db 18-19 17-18 15-17 73 66 64-65 61-64 58-63 123 112 110-111 106-110 103-109 24 20 19-20 18-19 16-18 74 67 65-66 62-65 59-64 124 113 111-112 107-111 104-110 25 21 20 19-20 vg 17-19 75 68 66-67 63-66 ca 60-65 125 114 112-113 108-112 105-111 26 22 21 19-20 18-20 76 69 67-68 63-67 61-66 126 115 113-114 109-113 ca 106-112 ca 27 23 22 20-21 19-20 77 70 68-69 64-68 62-67 127 116 114-115 109-114 106-113 9 a database of linear codes over f13 table 3 lower and upper bounds on minimum distances for linear codes over f13 n k = 3 4 5 6 n k = 3 4 5 6 n k = 3 4 5 6 28 24 23 vg 21-22 20-21 ca 78 71 69-70 65-69 63-68 128 117 115-116 110-114 107-114 29 25 23-24 22-23 20-22 79 72 ba 70-71 66-70 64-69 129 118 116-117 111-115 108-114 30 26 24-25 23-24 21-23 80 72 71-72 ca 67-71 65-70 130 119 117-118 112-116 109-115 31 27 25-26 24-25 22-24 81 73 71-72 68-72 66-71 131 120 118-119 113-117 110-116 32 28 26-27 25-26 ca 23-25 82 74 72-73 69-72 67-72 132 121 ba 119-120 114-118 111-117 33 29 27-28 25-27 24-26 83 75 73-74 70-73 68-72 133 121-122 120-121 115-119 112-118 34 30 28-29 26-28 25-27 84 76 74-75 71-74 69-73 ca 134 122 121-122 116-120 113-119 35 31 29-30 27-29 26-28 85 77 75-76 ca 72-75 ca 69-74 135 123 122-122 117-121 114-120 36 32 30-31 28-30 27-29 ca 86 78 75-77 72-76 70-75 136 124 123-123 ca 118-122 ca 115-121 37 33 31-32 29-31 27-30 87 79 76-78 73-77 71-76 137 125 123-124 118-123 116-122 38 34 ba 32-33 30-32 28-31 88 80 77-79 74-78 72-77 138 126 124-125 119-124 117-123 39 34-35 33-34 31-33 29-32 89 81 78-80 75-79 73-78 139 127 125-126 120-125 118-124 40 35-36 34-35 ca 32-34 30-33 90 82 79-81 76-80 74-79 ca 140 128 126-127 121-125 119-125 ca 41 36 34-36 33-35 31-34 91 83 80-82 77-81 74-80 141 129 127-128 122-126 119-125 42 37 35-36 34-36 ca 32-35 vg 92 84 ba 81-83 ca 78-82 75-81 142 130 128-129 123-127 120-126 43 38 36-37 34-36 32-36 93 84 81-84 79-82 76-82 143 131 129-130 124-128 121-127 44 39 37-38 35-37 33-36 94 85 82-84 80-83 77-82 144 132 130-131 125-129 122-128 45 40 38-39 36-38 34-37 95 86 83-85 81-84 78-83 145 133 ba 131-132 126-130 123-129 46 41 39-40 37-39 35-38 96 87 84-86 82-85 ca 79-84 146 133-134 132-133 127-131 124-130 47 42 40-41 38-40 36-39 97 88 85-87 82-86 80-85 147 134-135 133-134 128-132 ca 125-131 ca 48 43 41-42 39-41 37-40 98 89 86-88 83-87 81-86 148 135 134-135 128-133 125-132 49 44 ba 42-43 40-42 ca 38-41ca 99 90 87-89 84-88 82-97 ca 149 136 135 129-134 126-133 50 44-45 43-44 40-43 38-42 100 91 88-90 ca 85-89 ca 82-88 150 137 136 130-135 127-134 ca ba – simeon ball [17 ] vg – quasi-cyclic code in [13 ] be—mds code for n < 15 [14] db – de boer code [16] ca – new codes presented in this paper unmarked entries can be obtained by puncturing technique on longer codes or [23] if k = 3 10 e. z. chen, n. aydin table 4 other new qt codes that are found in this work and used to derive the lower bounds in table 3 n k d m α defining polynomials 85 4 75 17 6 c9566572b03055915, 3b90bc822420ab4, 175010787b272ac12, a4586b7a4a1555aaa, 6824a1c67b8ba3b68 104 4 92 8 1 a9c77511, 5a613a31, c9695821, 9a4ac421, b8952121, 2ac2c01, 645b4621, a5b14521, 5661611, 74b99b1, 7809c21, b92b6b11, 43532621 136 4 123 34 6 c89658606957772cb509300552519b145a, 9a340568467b478a34aa1135a5253aaa9a, 33b2950bb9c98b2c26482101a4b841080a, 269882b42a512ca6b74b883b0a732b3638 153 4 139 17 6 c9566572b03055915, 3b90bc822420ab4, 175010787b272ac12, a4586b7a4a1555aaa, 6824a1c67b8ba3b68, 930647483a13a23a9, 298b252ab48307233, 8680977c590521b4a, 325b99bc68114818a 32 5 25 8 1 965c81, 117a5c1, 6b66521, 39612911 42 5 34 14 1 c364c197a1, c1406b7668b51, 273b691926081 49 5 40 7 1 5274a11, 8a21511, 2696211, 32991, cac131, 373491, 9716131 56 5 46 7 1 6b65b1, c65c81, 4b4961, 569781, 4a16121, 84c1711, 9283c1, 87c661 68 5 57 17 1 26778c10b7b73731, 42c4c825639842a1, 196c27211272c691, 6a30ba1c25566381 96 5 82 8 1 3ba79211, c57b3311, 25b43731, c647121, a981611, 16715b1, 98c1c811, 2472c951, 8470c41, 29327121, 7a1a781, c85b6b1 119 5 103 17 1 158bcb3bcb851, 43227b2cb6976681, 1939c57875c9391, 550102645a546201, a463983a2316c151, 3839b779c2980661, 315895a570c81241 126 5 109 14 1 c364c197a1, 913937ab67cc1, 21472632237c1, 8986292bb3ab1, 3062ca237511, 76b23513759411, 68b307a1cc521, 35096ba095421, 963ca473580a1 136 5 118 8 1 5abab321, a7916721, 3462211, 286b9621, 979baa1, b761ac1, 68c5971, 47681341, 79883511, a1a86a31, 95643721, b9643b51, b59b7321, 312b8c11, c7ac5a11, b78b7a1, 781b7b1 147 5 128 21 1 cbc6c37bc7b51a88c1111, c5843c3aa792418224c1, c46981c81a47c179b1c11, 2487c583226b53651b11, b1a625c6b3c6b5054c91, 64cb15b59406b66416a1, a34881b17c650445c121 150 5 130 5 1 b191, a291, b631, a8c1, 41611, 81711, 27b1, 7801, 3331, b361, cb521, 43261, bc921, 53641, cbb1, 69611, 32311, 35731, 65261, 2411, 37c1, 52411, 67a11, 4751, c4531, a6311, 67161, cba31, 82911, 81 21 6 14 3 6 c2c, aa8, 9bb, 88c, a81, 71a, 042; 0c2, 3a9, 9c5, 38, 5c4, 3a4, 054 27 6 19 3 6 c2c, aa8, 9bb, 88c, a81, 71a, 5bb, a87, 608; 0c2, 3a9, 9c5, 38, 5c4, 3a4, 803, 939, a35 49 6 38 7 1 7b6a311, 3cc651, 5b39641, a9b171, c89ca1, 893c421, 953b911 57 6 45 3 6 c2c, 015, 1a, 95, 1a2, 10c, 435, 204, 169, 0a5, 676, 769, 82c, b8c, 524, 4ba, 52c, 963, 249; 0c2, 605, 916, cc4, 689, 6a4, 5aa, 361, 88c, 099, 80b, cb5, 902, b66, 7b6, 576, 9c3, b34, 49b 81 6 66 3 6 c2c, aa8, 9bb, 88c, a81, 71a, 5bb, a87, 288, 8b3, 391, cab, 866, 371, 621, 453, 2a8, 9cb, 491, 345, 818, ba5, 26b, 248, 795, c6, 7b6; 0c2, 3a9, 9c5, 38, 5c4, 3a4, 803, 939, ca2, 479, 8a1, cca, 205, 14b, 805, 181, 416, 7aa, 30b, ac, 489, 418, 482, 80a, c35, 511, 2c9 99 6 82 3 6 c2c, 043, 991, a4a, 73, 342, 103, b55, c51, 054, 962, 052, b18, 9c9, a35, 95a, 7cc, 042, 788, 3c6, 45b, ca3, 377, 43, 025, 9c3, 7b9, 049, a82, b75, 628, a19, 09c; 0c2, 11a, b96, 577, 743, 1b7, 867, 915, 68, 6b7, 4b5, 977, b09, 681, 7ac, a51, 91, 8bb, 879, 8c3, 839, 5ac, 414, 165, b7b, 304, 8bc, 36a, 003, 368, cb8, 313, 38 117 6 98 3 6 c2c, b51, 80b, b18, 525, 38b, b36, a76, 68b, c61, 24c, 865, 6a4, 82c, 331, 879, b6a, 04c, 391, a0c, 217, a7, 711, cca, 35c, c84, a24, 0c1, 216, 993, c35, 8b8, a88, 747, a66, 3ab, 361, 044, 6a7; 0c2, c72, 4c3, 1a6, 901, 363, 4c8, 472, 5a6, c09, 2b9, b74, 729, b44, 2a4, 2b, 346, a04, c19, 304, 0c4, 6bc, 5ca, b6b, 6, 61c, c24, 7c8, 1b4, 282, 587, c87, 196, 03c, 68a, 346, 89b, c82, a98 135 6 114 3 6 c2c, 532, 556, ccb, 7b2, 59a, 26, bb4, c22, 53a, 968, 3aa, b3c, 37, 4c8, 905, 82b, 119, 271, 112, 565, a8, 9c5, b7b, 5ab, c22, 077, 216, 8b1, c14, 9cc, 06, 3c8, baa, 745, 501, 295, 0ca, b9c, 404, 23, 16c, 5c2, 031, 1a4; 0c2, 81c, a1, 038, c86, 5c3, b7a, 31a, aba, 54b, 591, bb4, 2a7, 096, 243, a64, 5b9, 37, 61b, c76, 1c9, 40a, 3a5, a42, c4b, 552, 252, 65, a68, 975, 96a, 989, 2b4, 383, 902, 94c, 626, 878, b45, 7a8, ccb, c48, 13b, 526, 374 147 6 125 21 1 15927a7c452b83136931, c8905a324208569c6611, 850175159485bb722991, 5796b3114c22c917c231, 32aa65b413508b2ba141, 46854b05322a8664661, b773147b873a94886041 150 6 127 6 6 c024c9, 015c, 73ca6a, 0073a6, 7742c9, 4c3651, 641374, 42bb6, 22133, 56723, 2cba85, cb02a6, bc404b, 571aa2, 8227c, 834c31, 93857, 2130b8, 7c85c6, a67a0b, 115a6, c8833a, 6a3704, 25574c, bb1818 11 a database of linear codes over f13 table 5 other new optimal qt [pm, 3] codes found in this work n p m d α defining polynomials 8 2 4 6 1 11, 2321 14 2 7 12 1 18481, a8b8a11 20 5 4 17 1 11, 4961, 6831, 671, 231 32 8 4 28 1 11, 4961, 6831, 671, 4521, 3311, ab21, 8921 35 5 7 31 1 3c72b1, 19cc91, 6a6011, 3494311, b323b11 77 11 7 70 1 a8b8a11, 18481, 19cc91, 8535811, 3494311, 2437a1, 64ba931, 2959211, 75a821, b786451, c64971 78 26 3 71 1 211, 261, 531, 341, 1, b1, 6a1, 491, a31, 851, 21, 581, 241, 451, 811, 11, 351, 621, 31, b41, 671, 421, 911, 81, 321, b51 87 29 3 79 1 [78, 3, 71] code, 411, c71, 51 88 22 4 80 1 11, 4961, 6831, 671, 4521, 3311, ab21, 8921, 341, 5731, 561, 2321, 4851, 6611, bc1, 3531, 121, 8a31, 9a1, 2211, 4741, 6941 90 30 3 82 1 [87, 3, 79] code, 311 91 13 7 83 1 [77, 3, 70] code, 47bac1, a39651 102 34 3 93 1 511, 321, c21, b51, a11, 821, 431, 231, 521, 241, 31, 611, 361, 911, 1, 541, 6a1, 91, 671, 581, b21, 921, 341, 471, 851, 311, c11, b11, 831, b41, 421, 11, 641, 491 105 35 3 96 1 421, 851, 411, c31, 21, 491, 11, 231, 61, 211, 511, 361, c91, 1, 341, 581, 921, 651, 6a1, 241, b1, a31, 51, 321, 471, 831, a1, 641, 531, 621, c21, 261, b51, 911, 541 116 29 4 106 1 11, 2321, 4961, 6941, 4521, 891, 2211, 341, 6831, 671, 9a1, 3a81, b361, 3531, c131, 4411, ab1, 8b41, 9a21, 8a31, 231, b141, 5621, 2651, 2431, 781, 5511, 4851, 6611 117 39 3 107 1 231, 51, 71, 431, 531, 261, b21, c91, 911, 811, c71, 1, 11, a11, 361, c31, 61, 541, 641, 671, 91, 611, 341, 621, c11, 471, 321, 21, 81, 491, b51, 831, 511, 451, 821, a31, 211, 921, 521 129 43 3 118 1 a31, 641, b31, c21, 921, 41, 321, 6a1, 531, a1, c11, 61, b51, 51, 511, 421, 11, 821, 541, b41, c71, 491, c31, 91, b1, 81, 471, 361, 241, 211, 711, 571, c1, 261, 341, 621, 71, 611, 431, 411, 581, 851, 671 132 44 3 121 1 341, c91, 261, 671, b1, 831, c1, b41, 921, 311, c31, 51, 821, 231, 91, 711, 811, 511, a1, 1, 431, 581, 471, 411, a31, 11, 491, 351, c21, 611, 6a1, 81, 641, 241, 31, 321, b31, a11, 911, 71, b11, 851, 451, c71 144 48 3 132 1 431, 581, 911, 711, 231, 471, 341, a11, 51, 321, 851, 361, 421, 821, b41, 261, c21, 81, b31, 511, 241, c91, 921, b51, 31, 211, 811, 611, a31, 411, 671, c11, c31, 521, 351, 21, 641, 451, 11, 491, 531, 311, 6a1, 651, 621, b1, c71, 1 159 53 3 146 1 211, 261, 531, 341, 1, b1, 6a1, 491, a31, 851, 21, 581, 241, 451, 811, 11, 351, 621, 31, b41, 671, 421, 911, 81, 321, b51, 411, c71, 311, 51, c31, 71, 361, 431, 651, 711, c21, b11, 511, a21, 571, c91, 611, 231, 921, 61, a11, c1, 471, 521, b31, c11, 41 160 40 4 147 1 11, 4961, 6831, 671, 4521, 3311, ab21, 8921, 341, 5731, 561, 2321, 4851, 6611, bc1, 3531, 121, 8a31, 9a1, 2211, 4741, 6941, 8b41, 231, 5621, 5511, 3a81, b361, c131, 781, 2651, 3421, ab1, b141, 2431, 4411, 3641, 6721, 451, 891 161 23 7 148 1 18481, a8b8a11, 19cc91, 8535811, 47bac1, 75a821, b323b11, 2959211, cb6bc11, 985b41, 86b9321, 2437a1, 64ba931, 3494311, b786451, 7aca711, 3548b21, 5747511, 6a6011, a39651, 522501, a96c521, 3c72b1 162 54 3 149 1 [159, 3, 146] + 821 164 41 4 151 1 11, 4961, 6831, 671, 4521, 3311, ab21, 8921, 341, 5731, 561, 2321, 4851, 6611, bc1, 3531, 121, 8a31, 9a1, 2211, 4741, 6941, 8b41, 231, 5621, 5511, 3a81, b361, c131, 781, 2651, 3421, ab1, b141, 2431, 4411, 3641, 6721, 451, c241, 891 12 e. z. chen, n. aydin table 5 other new optimal qt [pm, 3] codes found in this work n p m d α defining polynomials 165 55 3 152 1 [162, 3, 149], 641 168 56 3 155 1 [165,3, 155], 91 171 57 3 157 1 c91, 361, a31, 921, b11, 1, 51, a11, 211, 851, 311, 241, c21, b1, 31, b51, 491, 621, 11, 451, 71, 651, 671, b41, 421, 511, 81, 541, c11, 531, 341, 611, 521, 431, c71, 21, 581, 471, 831, 811, 911, 61, a1, 641, 411, b31, a21, c31, 41, c1, 571, 231, 261, 6a1, 321, 351, 91 174 58 3 160 1 [171, 3, 157], 711 175 25 7 161 1 18481, a8b8a11, 19cc91, 8535811, a39651, 2437a1, 47bac1, 75a821, cb6bc11, 6282611, b4a7621, 5747511, 6a6011, 7aca711, 3548b21, b323b11, 3c72b1, 985b41, 522501, 2959211, 3494311, b786451, 64ba931, a96c521, c64971 177 59 3 163 1 [174, 3, 160] code, 821 180 60 3 166 1 [177, 3, 163] code, b21 182 26 7 168 1 [175, 3, 161] code, 86b9321 183 61 3 169 2 431, 521, 721, a1, 81, 651, 471, 511, a21, 811, 91, b51, 711, 121, a41, 11, 31, a11, 411, 531, 21, 451, 341, 41, 321, c91, 641, 311, 831, 611, 71, 731, b71, b21, c1, 821, 621, 351, 891, b11, c11, c71, a81, 921, 1, b31, 941, 851, 61, 961, 911, 51, 111, b41, 951, 861, 671, a31, c31, c21, b1 186 62 3 170 1 c11, 811, 71, 471, 21, 521, 211, 491, c21, b11, 31, 11, 261, c91, 621, 921, 361, 51, 81, 641, a11, c71, b51, 611, b1, 571, a31, 671, 91, 411, 431, 41, 541, b21, a1, 341, 241, 581, 511, 421, 61, 6a1, 321, 711, 911, 531, 351, b41, 311, b31, b31, 651, c1, 451, 821, 1, c31, 851, 231, 831, 831, a21 188 47 4 172 1 11, 3a81, 2431, 231, ab1, 6941, 4521, 8b41, ab21, 9a21, b141, 3531, 2321, 4961, 2211, 6611, 3641, 8a31, 3421, 3311, 6721, 4851, 451, 341, c241, 9a1, 5511, bc1, 781, 121, 121, 561, c131, 891, 5621, 4411, 671, 4741, 5731, 2651, b361, 8921, 6831, c01, bc21, 6c71, ac31 192 48 4 176 1 c131, 5511, b361, 8921, ab1, 5621, bc1, 8921, 451, ab21, 671, 2211, 6941, 121, 6611, 6831, c01, 231, c241, 3531, 781, 5731, 2431, 341, 8a31, 3311, 4521, 9a21, 561, 3641, 6721, 11, 2651, 3421, 4741, b141, 891, 4851, 2321, 4961, 4411, 8b41, 8b41, 9a1, 3a81, cc11, 4c91, bc21 189 27 7 173 1 cb6bc11, 5747511, 522501, 18481, 2959211, 985b41, 3c72b1, 64ba931, b4a7621, c64971, 75a821, 86b9321, 8535811, 8535811, 6a6011, b323b11, b323b11, 19cc91, a8b8a11, 3494311, 2437a1, 47bac1, 6282611, 7aca711, a96c521, b786451, 3548b21 204 17 12 187 1 4ba782ca2831, 5a96a793501, a2085669411, 274a0532321, c4c133a1141, c4c133a1141, 5262343c8511, 55428ca6c1, 55428ca6c1, 6b299cbca81, b36b4b85991, 246a8a98c621, 246a8a98c621, 91756140c61, 2c1662ab6711, 57b65c291421, b837a821c111 13 a database of linear codes over f13 table 6 optimal qt [pm, 4] and [pm, 5] codes n k p m d α defining polynomials 10 4 2 5 7 1 c01, b2b11 14 4 2 7 11 1 c851, b636b11 15 4 3 5 11 1 c01, 96911, a111 16 4 4 4 12 1 4121, 7211, a431, 41 18 4 3 6 14 1 c71, 91a81, 9a6111 20 4 5 4 16 vg 1 116, 1b, 1186, 142, 134a 25 4 5 5 20 1 c01, 96911, 28111, 95b1, 5611 28 4 7 4 23 vg 1 14, 13, 1159, 163b, 1252, 112c, 1294 68 4 4 17 60 6 c9566572b03055915, 663cc4022720508c5, 8680977c590521b4a, 972a15a2473369c09 153 4 9 17 139 6 c9566572b03055915, 3b90bc822420ab4, 175010787b272ac12, a4586b7a4a1555aaa, 6824a1c67b8ba3b68, 930647483a13a23a9, 298b252ab48307233, 8680977c590521b4a, 325b99bc68114818a 10 5 2 5 6 vg 1 13a, 10aa 12 5 2 6 8 1 11, 512721 14 5 2 7 10 1 6b65b1, c65c81 15 5 3 5 10 1 b191, a291, 721 18 5 3 6 13 1 32b131, 8c4121, 51271 20 5 4 5 15 vg 1 18, 14ac4, 1c8b, 12b3c 15 6 5 3 9 6 c2c, aa8, 9bb, 88c, 2a2; 0c2, 3a9, 9c5, 38, a04 18 6 3 6 12 6 c024c9, 16589b, ab836 14 e. z. chen, n. aydin references [1] f. j. macwilliams f. j., n. j. a. sloane, the theory of error-correcting codes, north-holland, 1977. 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[17] s. ball, table of bounds on three dimensional linear codes or (n,r) arcs in pg(2,q), available at http://www-ma4.upc.es/∼ simeon/codebounds.html, accessed july 22, 2014 [18] a. r. caldebank, w. m. kantor, the geometry of two-weight codes, bull. london math. soc., 18, 97-122, 1986. [19] a. vardy, the intractability of computing the minimum distance of a code, ieee trans. inform. theory, 43, 1757-1766, 1997. [20] w. bosma, l. cannon, c. playoust, the magma algebra system i, the user language, j. symbolic comput., 24, 235-265, 1997. [21] m. grassl, bounds on the minimum distances of linear codes, available at http://www.codetables.de, accessed july 22, 2014. [22] n. aydin, j. murphree, new linear codes from constacyclic codes, j. franklin inst., 351, 1691-1699, 2014. [23] r. hill, optimal linear codes, in cryptography and coding ii, edited by c. mitchell, oxford university press, 74-104, 1992. 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[28] m. grassl, t. a. gulliver, on self-dual mds codes, proc. ieee int. symp. inform. theory, 19541957, 2008. 15 a database of linear codes over f13 [29] t. a. gulliver, j-l. kim, y. lee, new mds or near-mds self-dual codes, ieee trans. inform. theory, 54, 4354-4360, 2008. 16 introduction and motivation quasi-twisted codes the search algorithm and new qt codes over f13 a database of linear codes over f13 with minimum distance bounds upper bounds on minimum distance conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327385 j. algebra comb. discrete appl. 4(3) • 281–290 received: 12 august 2016 accepted: 17 april 2017 journal of algebra combinatorics discrete structures and applications a class of cyclic codes constructed via semiprimitive two-weight irreducible cyclic codes∗ research article jesús e. cuén-ramos, gerardo vega abstract: we present a family of reducible cyclic codes constructed as a direct sum (as vector spaces) of two different semiprimitive two-weight irreducible cyclic codes. this family generalizes the class of reducible cyclic codes that was reported in the main result of [10]. moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way. 2010 msc: 94b15, 11t71 keywords: weight distribution, reducible cyclic codes, semiprimitive cyclic codes, cyclotomic numbers 1. introduction it is said that a cyclic code is reducible if its parity-check polynomial is factorizable in two or more irreducible factors. each one of these irreducible factors can be seen as the parity-check polynomial of an irreducible cyclic code. therefore, a reducible cyclic code is, basically, a direct sum (as vector spaces) of these irreducible cyclic codes. reducible cyclic codes whose parity-check polynomials are factorizable in exactly two different irreducible factors have been extensively studied (see, for example, [10], [3], [11], [6], [5], [2], [12] and [8]). a very interesting problem regarding this kind of reducible cyclic codes is to obtain their full weight distributions. in particular, [10] employed an elaborate procedure that uses some sort of elliptic curves in order to obtain the weight distribution of a class of reducible cyclic codes. we present here a family of reducible cyclic codes constructed as a direct sum of two different semiprimitive two-weight irreducible cyclic codes, that generalizes such class of reducible cyclic codes. moreover, we show that, contrary to what was stated in [10, p. 7254], it is still possible, at least for the codes in this ∗ this work was partially supported by papiit-unam in107515. jesús e. cuén-ramos (corresponding author); posgrado en ciencias matemáticas, universidad nacional autónoma de méxico, 04530 ciudad de méxico, mexico (email: elisandro@ciencias.unam.mx). gerardo vega; dirección general de cómputo y de tecnologías de información y comunicación, universidad nacional autónoma de méxico, 04510 ciudad de méxico, mexico (email: gerardov@unam.mx). 281 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 table 1. weight distribution of c(ai), i = 1,2. weight frequency 0 1 (q−1) d (qk−1 −q(k−2)/2) (q k−1) 2 (q−1) d (qk−1 + q(k−2)/2) (qk−1) 2 table 2. weight distribution of c(a1,a2). weight frequency 0 1 2(q−1) 3d (qk−1 −q(k−2)/2) 3(q k−1) 2 2(q−1) 3d (qk−1 + q(k−2)/2) 3(qk−1) 2 q−1 d (qk−1 −q(k−2)/2) (q k−1)(qk−5) 8 q−1 3d (3qk−1 −q(k−2)/2) 3(q k−1)2 8 q−1 3d (3qk−1 + q(k−2)/2) 3(qk−1)2 8 q−1 d (qk−1 + q(k−2)/2) (qk−1)(qk−5) 8 family, to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way. in order to give a detailed explanation of what is the main result of this work, let p, t, q, k and ∆ be integers, such that p is a prime, q = pt and ∆ = (qk − 1)/(q − 1). in addition, let γ be a fixed primitive element of fqk and, for any integer a, denote by ha(x) ∈ fq[x] the minimal polynomial of γ−a. also, for any integers a1,a2,a3, . . . ,at, let c(a1,a2,a3,...,at) be the cyclic code with parity-check polynomial∏t i=1 hai (x). with this notation, the following result gives a description for the weight distribution of a family of reducible cyclic codes: theorem 1.1. suppose that 3|(qk − 1). let a1, a2, d and n be any integers such that a1 −a2 = ±q k−1 3 , d = gcd(qk − 1,a1,a2) and n = q k−1 d . if gcd(∆, 3a1) = 2 then (a) c(a1) and c(a2) are two different semiprimitive two-weight irreducible cyclic codes of length n and dimension k over fq. in addition, these codes have the same weight distribution which is given in table 1. (b) c(a1,a2) is an [n, 2k] cyclic code over fq, with the weight distribution given in table 2. recently, in [8] was given a unified explanation for the weight distribution of several families of codes whose parity-check polynomials are given by the products of the form ha(x)h a±q k−1 2 (x), where ha(x) 6= h a±q k−1 2 (x). from this perspective, therefore, it is important to keep in mind that the paritycheck polynomials of the kind of codes studied in [10], and those studied by theorem 1.1, are now given by the products of the form ha(x)h a±q k−1 3 (x). this work is organized as follows: in section 2 we establish some notations, recall some definitions and establish our main assumption. section 3 is devoted to presenting some preliminaries and general results. in section 4 we use these results in order to present a formal proof of theorem 1.1. in section 5 we show some applications of theorem 1.1. finally, section 6 is devoted to conclusions. 282 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 2. definitions, notations, preliminaries and main assumption first of all, we set for the rest of this work the following: notation. by using p, t, q, k and ∆, we will denote five positive integers such that p is a prime number, q = pt and ∆ = (qk − 1)/(q − 1). from now on, γ will denote a fixed primitive element of fqk. for any integer a, the polynomial ha(x) ∈ fq[x] will denote the minimal polynomial of γ−a. furthermore, we will denote by “tr", the absolute trace mapping from fqk to the prime field fp, and by “trf qk /fq " the trace mapping from fqk to fq. for any positive divisor m of qk −1 and for any 0 ≤ i ≤ m−1, we define d(m)i := γ i〈γm〉, where 〈γm〉 denotes the subgroup of f∗ qk generated by γm. the cosets d(m)i are called cyclotomic classes of order m in fqk. in connection with these cyclotomic classes, we have the cyclotomic numbers of order m: (i,j)(m,q k) := |(d(m)i + 1) ∩d (m) j | , where (d(m)i + 1) = {x + 1 |x ∈ d (m) i }, and 0 ≤ i,j ≤ m− 1. finally, the canonical additive character χ, of fqk, is: χ(y) = ζtr(y)p , for all y ∈ fqk , where ζp = exp( 2π √ −1 p ). the following definition is a proper generalization of the idea of a semiprimitive irreducible cyclic code that was introduced recently in [9]. definition 2.1. let p, q, k and ∆ be as before, and for any integer a let u = gcd(∆,a). then an irreducible cyclic code with parity-check polynomial ha(x), of degree k, is called a semiprimitive code if u ≥ 2, and if −1 is a power of p modulo u (that is, if the prime p is semiprimitive modulo u). now, we also set for the rest of this work the following: main assumption. from now on, we are going to suppose that 3|(qk−1), 2|∆ and 3 ∆. also, in what follows, we will reserve the greek letter τ in order to fix τ = γ qk−1 3 . remark 2.2. as a consequence of our main assumption, note that k should be an even integer, whereas q must be an odd integer greater than 5, and necessarily 3|(q−1) and 4|(qk−1). in addition, observe that f∗q ⊂ d (2) 0 , τ ∈ d (2) 0 , and also that the finite field element τ is a primitive three-root of unity satisfying τ2 + τ + 1 = 0. the following is a well known result ([5, lemma 4]): lemma 2.3. define ηi = ∑ x∈d(2) i χ(x) , i = 0, 1 . then η1 = −1 −η0, and η0 = { −1+(−1)tk−1qk/2 2 if p ≡ 1 (mod 4), −1+(−1)tk−1( √ −1)tkqk/2 2 if p ≡ 3 (mod 4). the exponential sums η0 and η1 are known as the gaussian periods of order 2. since we will be dealing with the gaussian period of order 2, we also need the cyclotomic numbers of order 2. the following result is on that direction ([5]). lemma 2.4. suppose that 4|(qk − 1), then (0, 0)(2,q k) = q k−5 4 , (0, 1)(2,q k) = (1, 0)(2,q k) = (1, 1)(2,q k) = q k−1 4 . 283 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 3. some preliminaries and general results with our current notation and main assumption in mind we present the following results. lemma 3.1. let a1, a2 and d be integers such that a2 = a1 ± q k−1 3 and d = gcd(qk − 1,a1,a2) 6= 0. then 3|q k−1 d and d = gcd(qk − 1,a1) or d = gcd(qk − 1,a2). proof. since d divides both a1 and a2, we have that d|q k−1 3 . therefore the first assertion is true. let di = gcd(qk − 1,ai) for i = 1, 2. without loss of generality, suppose that d1 ≤ d2. observe that d = gcd(q k−1 3 ,ai) for i = 1, 2, and therefore either di is d or 3d. now, if d1 = 3d, then d2 = 3d, which implies that 3d|qk − 1,a1,a2. but this last condition is impossible, thus d1 = d. lemma 3.2. if 3d is a divisor of qk − 1 and gcd(∆, 3d) = 2, then for any integer i {xy | x ∈d(3d)i and y ∈ f ∗ q} = 2(q − 1) 3d ∗d(2)i , where 2(q−1) 3d ∗d(2)i is the multiset in which each element of d (2) i appears with multiplicity 2(q−1) 3d . proof. since 2|∆ and 2|3d, f∗q ⊂ d (2) 0 and d (3d) 0 ⊂ d (2) 0 . but gcd(∆, 3d) = 2, therefore the result comes from the fact that |d(3d)0 | |f ∗ q|/|d (2) 0 | = 2(q − 1)/3d, and d (2) i = γ id(2)0 for any integer i. by using τ, and the cyclotomic classes of order 2, we define the following sets: g := {(α,−β) ∈ f2qk | (α−βτ i) 6= 0, 0 ≤ i < 3 }, and ei,j := {(α,−ατi) ∈ f2qk | (α−ατ) ∈d (2) j } for i = 0, 1, 2, and j = 0, 1. remark 3.3. through a direct inspection it is easy to see that the above seven sets are pairwise disjoint and their union is equal to f2 qk \(0, 0). in addition, clearly |ei,j| = |d (2) 0 | = qk−1 2 , |g| = q2k−1−6|e0,0| = (qk −1)(qk −2) and, due to remark 2.2, we have that if (α−ατ) ∈d(2)j , for some integer j = 0, 1, then necessarily (α−ατ2) = −τ2(α−ατ) ∈d(2)j . now, for each (α,−β) ∈ g, we define the function fα,β : {0, 1, 2} → {0, 1}, given by the rule fα,β(i) = j if and only if (α−βτi) ∈d (2) j . with the help of these functions we induce a partition of the set g into the following disjoint subsets: sl := {(α,−β) ∈g | wh(fα,β(0),fα,β(1),fα,β(2)) = l} , for l = 0, 1, 2, 3, where wh(·) stands for the usual hamming weight function. remark 3.4. for any α,β ∈ fqk, we define ui = (α − βτi), for i = 0, 1, 2. it is not difficult to see that these u values satisfy u0 + u1τ + u2τ2 = 0. furthermore, observe that if we arbitrarily choose the values of, say, u0 and u2 then there must exist a unique vector (α,β) ∈ f2qk, such that u0 = (α − β), u2 = (α−βτ2) and u1 = −τ−1(u0 + u2τ2). therefore, if we want to calculate, for example, |s0| then we can assume, without loss of generality, that u2 can take any value in d (2) 0 . this leads us to qk−1 2 possible choices for u2. but u1 = −u2τ(u0u2 τ −2 + 1) and −1,τ ∈ d(2)0 (see remark 2.2), thus, in order that u1 and u0 also belong to d (2) 0 it is necessary that ( u0 u2 τ−2 + 1) ∈d(2)0 . hence, the number of such instances is given by the cyclotomic number (0, 0)(2,q k). consequently, we have |s0| = q k−1 2 (0, 0)(2,q k). in a quite similar way, one can obtain |s1|, |s2| and |s3|. 284 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 table 3. value distribution of ∑ z∈d(2) 0 2∑ i=0 χ(z(α + βτ i )). value frequency 3(qk−1) 2 1 qk−1 2 + 2η0 3(qk−1) 2 qk−1 2 + 2η1 3(qk−1) 2 3η0 (qk−1)(qk−5) 8 −1 + η0 3(q k−1)2 8 −1 + η1 3(q k−1)2 8 3η1 (qk−1)(qk−5) 8 keeping in mind the previous definitions and observations we now present the following result, which will be important in order to determine the weight distribution of the class of reducible cyclic codes that we are interested in. lemma 3.5. with our notation and main assumption, we have |s0| = qk − 1 2 (0, 0)(2,q k), |s1| = 3(qk − 1) 2 (0, 1)(2,q k), |s2| = 3(qk − 1) 2 (1, 1)(2,q k), |s3| = (qk − 1)(qk − 2) − (|s0| + |s1| + |s2|). furthermore, if χ denotes the canonical additive character of fqk, and if η0 and η1 are as in lemma 2.3, then, for any α,β ∈ fqk, we also have ∑ z∈d(2)0 2∑ i=0 χ(z(α + βτi)) =   3(qk−1) 2 if (α,β) = (0, 0), qk−1 2 + 2η0 if (α,β) ∈∪2i=0ei,0, qk−1 2 + 2η1 if (α,β) ∈∪2i=0ei,1, 3η0 if (α,β) ∈s0, −1 + η0 if (α,β) ∈s1, −1 + η1 if (α,β) ∈s2, 3η1 if (α,β) ∈s3. proof. the first assertion comes from remark 3.4. since ∑ z∈d(2)0 χ(0) = |d(2)0 | = qk−1 2 , the second assertion comes from lemma 2.3, remark 3.3, and from the definitions of the sets ei,j and sl, with i = 0, 1, 2, j = 0, 1, and l = 0, 1, 2, 3. considering the actual values of the cyclotomic numbers in lemma 2.4, the following result is an important consequence. corollary 3.6. consider the same hypotheses as in the previous lemma. then the value distribution of the character sum ∑ z∈d(2)0 ∑2 i=0 χ(z(α + βτ i)) is given in table 3. 285 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 4. proof of theorem 1.1 proof. part (a): first note that gcd(∆, 3a1) = gcd(∆, 3a2) = 2, in consequence gcd(∆,ai) = 2 for i=1,2. let vi be an integer such that 1 ≤ vi < k and aiqvi ≡ ai (mod qk − 1). then q k−1 q−1 |ai qvi−1 q−1 , but gcd(∆,ai) = 2, thus (qk −1)|2(qvi −1) for i=1,2. however, this last condition is impossible if 1 ≤ vi < k and q ≥ 7 (recall remark 2.2). hence deg(hai (x)) = k. now, suppose that ha1 (x) = ha2 (x). then, there exists an integer 0 ≤ v < k such that a1qv ≡ a2 (mod qk−1). but a2 = a1± q k−1 3 , thus the last congruence implies that a1(qv−1) ≡±q k−1 3 (mod qk−1) and v ≥ 1. then 3a1(qv−1) ≡ 0 (mod qk−1) with 1 ≤ v < k. that is q k−1 q−1 |3a1 qvi−1 q−1 . but gcd(∆, 3a1) = 2, thus (qk − 1)|2(qvi − 1). however, this last condition is impossible if 1 ≤ vi < k and q ≥ 7. hence ha1 (x) 6= ha2 (x). on the other hand, note that c(a1) and c(a2) have the same length n = qk−1 d , and owing to lemma 3.1, we have that 3|n. since gcd(∆,a1) = gcd(∆,a2) = 2, clearly, in accordance with definition 2.1, c(a1) and c(a2) are two different semiprimitive two-weight irreducible cyclic codes. thus, by means of the characterization for this kind of codes in [9, theorem 7], we can see that their weight distributions are as is shown in table 1. part (b): by lemma 3.1 and since c(a2,a1) = c(a1,a2), we can assume without loss of generality that d = gcd(qk − 1,a1). clearly, the cyclic code c(a1,a2) has length n and its dimension is 2k due to part (a). now, for each α,β ∈ fqk, we define c(n,a1,a2,α,β) as the vector of length n over fq, which is given by: (trf qk /fq (α(γ a1 )i + β(γa2 )i))n−1i=0 . thanks to delsarte’s theorem ([1]), it is well known that c(a1,a2) = {c(n,a1,a2,α,β) |α,β ∈ fqk} . thus the hamming weight of any codeword c(n,a1,a2,α,β) is equal to n−z(α,β), where z(α,β) =]{ i | trf qk /fq (αγ a1i + βγa2i) = 0, 0 ≤ i < n} . now, if χ′ is the canonical additive character of fq, then, by the orthogonal property of χ′ (see, for example, [4, p. 192]), we know that for each c ∈ fq we have ∑ y∈fq χ′(yc) = { q if c = 0, 0 if c 6= 0, thus z(α,β) = 1 q n−1∑ i=0 ∑ y∈fq χ′(trf qk /fq (y(αγ a1i + βγa2i))) . if χ denotes the canonical additive character of fqk, then χ′(trf qk /fq (ε)) = χ(ε) for all ε ∈ fqk. therefore, we have z(α,β) = n q + 1 q n−1∑ i=0 ∑ y∈f∗q χ(y(αγa1i + βγa2i)) = n q + 1 q n−1∑ i=0 ∑ y∈f∗q χ(γa1iy(α + βτ�i)) , 286 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 where the last equality arises because a2 −a1 = �q k−1 3 for some integer � equal to 1 or −1. on the other hand, since d|(a2 − a1) and d = gcd(qk − 1,a1), there must exist integers l, r and s, in such way that τ� = γdl, d = r(qk − 1) + sa1 and τ�i = (γa1i)ls. therefore z(α,β) = n q + 1 q n−1∑ i=0 ∑ y∈f∗q χ(γdiy(α + βτsi)) . but 3|n. then {γdi | 0 ≤ i < n} = d(d)0 = d (3d) 0 ∪d (3d) d ∪d (3d) 2d . therefore, z(α,β) = n q + 1 q 2∑ i=0 ∑ x∈d(3d) di ∑ y∈f∗q χ(xy(α + βτsi)) . now, since d = gcd(q k−1 3 ,a1) (see proof of lemma 3.1) and gcd(∆, 3( qk−1 3 ), 3a1) = gcd(∆, 3a1), we have gcd(∆, 3d) = 2. thus, by lemma 3.2, we obtain z(α,β) = n q + 2(q − 1) 3dq 2∑ i=0 ∑ z∈d(2) di χ(z(α + βτsi)) = n q + 2(q − 1) 3dq ∑ z∈d(2)0 2∑ i=0 χ(z(α + βτsi)) , where the last equality follows from the fact that 2|d (recall that gcd(∆, 3d) = 2). now, by lemma 3.1, we have that 3d|qk − 1. therefore, note that 3 s, and this is so because if 3|s, 3d|sa1, but recall that d = r(qk − 1) + sa1, then 3d|d, and clearly this is a contradiction. then z(α,β) = n q + 2(q − 1) 3dq ∑ z∈d(2)0 2∑ i=0 χ(z(α + βτi)) . therefore the result comes from corollary 3.6 because the hamming weight of any codeword of the form c(n,a1,a2,α,β) in c(a1,a2) is equal to n−z(α,β). 5. some applications of theorem 1.1 as we saw, through this work we deals with the kind of reducible cyclic codes whose parity check polynomials are given by the products of the form ha(x)h a±q k−1 3 (x), where a is any integer and ha(x) 6= h a±q k−1 3 (x). the following result, which is the main result in [10, theorem 3.6], also deals with this kind of reducible cyclic codes: theorem 5.1. let h be a positive factor of q−1, and assume that 3 divides h. if gcd(k, 3(q−1)/h) = 2, then c ( q−1 h , q−1 h + qk−1 3 ) is an [h∆, 2k] code with the weight distribution in table 4. the following result shows that the family of codes in theorem 5.1, is included in theorem 1.1. theorem 5.2. conditions in theorem 5.1 imply conditions in theorem 1.1. 287 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 table 4. weight distribution of c ( q−1 h , q−1 h + qk−1 3 ) . weight frequency 0 1 2h 3 (qk−1 −q(k−2)/2) 3(q k−1) 2 2h 3 (qk−1 + q(k−2)/2) 3(qk−1) 2 h(qk−1 −q(k−2)/2) (q k−1)(qk−5) 8 h(qk−1 + q(k−2)/2) (qk−1)(qk−5) 8 h 3 (3qk−1 −q(k−2)/2) 3(q k−1)2 8 h 3 (3qk−1 + q(k−2)/2) 3(qk−1)2 8 table 5. weight distribution of c(a1,a2,a3). weight frequency ( ∑2 i=0 ui = 3) q−1 3δq (u1(q k + qk/2) + u2(q k −qk/2)) ( 3! u0!u1!u2! )(q k−1 2 )u1+u2 proof. clearly 3|(qk − 1), and because gcd(∆,ρ) = gcd(k,ρ) for all ρ|(q − 1) (see, for example [7, remark 3]), we have gcd(∆, 3(q − 1)/h) = 2. thus, by taking a1 = (q − 1)/h in theorem 1.1, the result follows directly. as was mentioned before, theorem 1.1 deals with the weight distribution of cyclic codes, c (a,a±q k−1 3 ) , under the condition gcd(∆, 3a) = 2, but we also believe that is interesting the weight distribution of the cyclic codes, c (a,a−q k−1 3 ,a+ qk−1 3 ) , under the same condition, because their parity-check polynomials differ by one irreducible factor. however, that kind of codes was treated in [13, corollary 10]. thus, we recall part of that result with our notation in the following: corollary 5.3. with our notation and main assumption, suppose that a1, a2, a3 and δ be integers such that δ = gcd(qk − 1,a1,a2,a3), a2 = a1 + q k−1 3 , a3 = a1 − q k−1 3 and gcd(∆, 3a1) = 2. then c(a1,a2,a2) is a [(qk − 1)/δ, 3k] cyclic code over fq with weight distribution in table 5. the following are direct applications of theorem 1.1. example 5.4. with our notation, let q = 13, k = 2, a1 = 8, a2 = a1 + qk−1 3 = 64 and a3 = a1 − q k−1 3 = −48. then ∆ = 14, d = δ = 8 and n = 21. clearly 3|(qk − 1) and gcd(∆, 3a1) = 2. by theorem 1.1, c(8) and c(64) are two different semiprimitive two-weight irreducible cyclic codes of length 21, dimension 2 and weight enumerator polynomial a(z) = 1 + 84z18 + 84z21. in addition, c(8,64) is a cyclic code of length 21, dimension 4 and weight enumerator polynomial a(z) = 1 + 252z12 + 252z14 + 3444z18 +10584z19 + 10584z20 + 3444z21 . on the other hand, by corollary 5.3, c(8,64,−48) is a cyclic code of length 21, dimension 6 and weight enumerator polynomial a(z) = 1 + 252z6 + 252z7 + 21168z12 + 42336z13 + 21168z14 +592704z18 + 1778112z19 + 1778112z20 + 592704z21 . remark 5.5. since 2|∆, clearly the length of all codes in theorem 5.1 must be an even number. therefore, the code in the previous example does not belong to the class of codes in theorem 5.1. 288 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 example 5.6. now, let q = 7, k = 2, a1 = 2, a2 = a1 − q k−1 3 = −14, and a3 = a1 + q k−1 3 = 18. then ∆ = 8, d = δ = 2 and n = 24. clearly 3|(qk − 1) and gcd(∆, 3a1) = 2. by theorem 1.1, c(2) and c(−14) are two different semiprimitive two-weight irreducible cyclic codes of length 24, dimension 2 and weight enumerator polynomial a(z) = 1 + 24z18 + 24z24. in addition, c(2,−14) is a cyclic code of length 24, dimension 4 and weight enumerator polynomial a(z) = 1 + 72z12 + 72z16 + 264z18 +864z20 + 864z22 + 264z24 . (1) on the other hand, by corollary 5.3, c(2,−14,18) is a cyclic code of length 24, dimension 6 and weight enumerator polynomial a(z) = 1 + 72z6 + 72z8 + 1728z12 + 3456z14 + 1728z16 +13824z18 + 41472z20 + 41472z22 + 13824z24 . remark 5.7. suppose again that q = 7 and k = 2. then h−14(x) = h34(x) 6= h18(x), and consequently note that despite that the weight enumerator polynomial in the previous example is exactly the same as weight enumerator polynomial in the example of [10, page 7257], the cyclic codes c(2,−14) and c(2,18) are different. in addition, also note that the cyclic code c(2,−14) = c(2,34) does not belong to the class of codes in theorem 5.1. remark 5.8. through a direct inspection, it is not difficult to see that all different reducible cyclic codes over f7 of length 24, dimension 4 and weight enumerator polynomial, as in (1), are c(2,18), c(2,34), c(18,34), c(6,10), c(6,26) and c(10,26). all these reducible cyclic codes belong to the class of codes in theorem 1.1. remark 5.9. under our main assumption, note that if gcd(∆, 3a) = 2, then gcd(∆, 3(a± q k−1 3 )) = 2. therefore, a reducible cyclic code, c (a,a±q k−1 3 ) , will belong to the family of codes in theorem 1.1, if and only if gcd(∆,a) = 2. this condition, which is easy to verify, allows us to identify all the reducible cyclic codes that satisfy conditions in theorem 1.1. in fact, if n(q,k) is the number of different reducible cyclic codes, c(a1,a2), that satisfy such conditions, then it is not difficult to see that n(q,k) = φ( ∆ 2 )(q − 1) k , where φ is the usual euler φ-function. what is interesting here is that it seems that n(q,k) is also the total number of reducible cyclic codes whose weight distribution is given in table 2. 6. conclusion in this work we found the sufficient numerical conditions in order to obtain the full weight distribution of a family of codes that belongs to the kind of reducible cyclic codes whose parity-check polynomials are given by the products of the form ha(x)h a±q k−1 3 (x). by means of the characterization of all semiprimitive two-weight irreducible cyclic codes that was presented in [9, theorem 7], we were able to identify that the codes in this family are constructed as a direct sum (as vector spaces) of two different semiprimitive two-weight irreducible cyclic codes. in addition, we also showed that the class of codes recently studied in [10] is included in this family. moreover, despite of what was stated in [10], we showed that, at least for the codes in this family, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in an easy way. finally, we believe that perhaps, following the same idea as in [8], it could be possible to develop a more general theory that allows us to present a unified explanation for an enlarged family of reducible cyclic codes of this kind. 289 j. e. cuén-ramos, g. vega / j. algebra comb. discrete appl. 4(3) (2017) 281–290 references [1] p. delsarte, on subfield subcodes of reed–solomon codes, ieee trans. inform. theory 21(5) (1975) 575–576. 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[3]. let g be a graph and x ∈ v (g), then the betweenness centrality of x in g, denoted by bg(x) or simply b(x) is defined as bg(x) = ∑ s,t∈v (g)\{x} σst(x) σst where σst(x) denotes the number of shortest s-t paths in g passing through x and σst, the total number of shortest s-t paths in g. the ratio σst(x) σst is called pair dependency or partial betweenness of (s, t) on x, denoted by δg(s, t, x). betweenness centrality of some well known graphs has been studied in [7] and we use the following definitions [8]. 2.1. betweenness centrality of a vertex in a subgraph definition 2.2. let g be a graph and h a subgraph of g. let x ∈ v (h), then the betweenness centrality of x in h denoted by bh(x) is defined as bh(x) = ∑ s,t∈v (h)\{x} σhst(x) σhst where σhst(x) and σ h st denotes the number of shortest s-t paths passing through x and the total number of shortest s-t paths respectively, lying in h. 2.2. betweenness centrality of a vertex induced by a subgraph definition 2.3. let g be a graph and h a subgraph of g. let x ∈ v (g), then the betweenness centrality of x induced by h denoted by b(x, h) is defined as b(x, h) = ∑ s,t( ̸=x)∈v (h) σst(x) σst where σst(x) and σst denotes the number of shortest s-t paths passing through x and the total number of shortest s-t paths respectively in g. the betweenness centrality of a vertex induced by a subset s ⊂ v (g) is defined likewise. 2.3. betweenness centrality of a vertex induced by a subset definition 2.4. let g be a graph and s a subset of v (g). let x ∈ v (g), then the betweenness centrality of x induced by s denoted by b(x, s) is defined as b(x, s) = ∑ s,t( ̸=x)∈s σst(x) σst where σst(x) and σst denotes the number of shortest s-t paths passing through x and the total number of shortest s-t paths respectively in g. 22 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 2.4. betweenness centrality of a vertex induced by another vertex definition 2.5. let g be a graph and s, x, t ∈ v (g), then the betweenness centrality of x induced by s in g, denoted by bg(x, s) or simply b(x, s) is defined by bg(x, s) = ∑ t∈v (g)\x σst(x) σst . it can be easily seen that in any graph g, the betweenness centrality induced by a vertex on its extreme vertex or an end vertex is zero. consider the following examples. b(xi, xj) = 0 for xi, xj ∈ kn. let pn be a path on n vertices {x1, . . . , xn}, then b(xi, xj) = { i − 1, if i < j, n − i, if j < i. if cn is a cycle on n vertices {x0, . . . , xn−1}, then if n is even, b(xi, x0) = { n−1−2i 2 , if 1 ≤ d(xi, x0) < n/2, 0, if d(xi, x0) = n/2, if n is odd, b(xi, x0) = n − 1 − 2i 2 , if 1 ≤ d(xi, x0) ≤ n − 1 2 . for a star sn with central vertex x0, b(xi, x0) = 0, b(x0, xi) = n − 2 and (1) b(xi, xj) = 0 for i, j ̸= 0. (2) for a wheel wn, n > 5 with central vertex x0, b(xi, x0) = 0, b(x0, xi) = n − 5. for i, j ̸= 0, b(xi, xj) = { 1/2, if d(xi, xj) = 1, 0, if d(xi, xj) = 2. it can be easily seen that for xi ∈ v (g), bg(xi) = 12 ∑ j ̸=i bg(xi, xj). 2.5. betweenness centrality donated by a vertex definition 2.6. let g be a graph and x0 ∈ v (g), then the betweenness centrality donated by x0 in g, denoted by dbg(x0) or simply db(x0), is defined as the sum of betweenness values induced by x0 on all other vertices in g, i.e., dbg(x0) = ∑ x∈v (g)\x0 bg(x, x0). the betweenness centrality received by a vertex, rbg(x0) is bg(x0) by definition. 23 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 2.6. betweenness centrality of a vertex induced by two disjoint subsets definition 2.7. let g be a graph and x ∈ v (g). let s, t be two disjoint subsets of v (g), then the betweenness centrality of x induced by s and t denoted by b(x, s, t) is defined as b(x, s, t) = b(x, s) + b(x, t). 2.7. betweenness centrality of a vertex induced by two disjoint subsets, one against the other definition 2.8. let g be a graph and x ∈ v (g). let s, t be two disjoint subsets of v (g) where s(̸= x) ∈ s and t(̸= x) ∈ t , then the betweenness centrality of x induced by s against t , denoted by b(x, s|t) is defined as b(x, s|t) = ∑ s∈s, t∈t σst(x) σst where σst(x) and σst denotes the number of shortest s-t paths passing through x and the total number of shortest s-t paths respectively in g. in metric graph theory, a convex subgraph of an undirected graph g is a subgraph that includes every shortest path in g between two of its vertices. a subgraph h of a graph g is an isometric subgraph, if dh(u, v) = dg(u, v) for all u, v ∈ v (h). clearly, a convex subgraph is an isometric subgraph, but the converse need not be true. 3. subgraph-amalgamation one method of constructing composite graphs is merging or pasting two or more graphs together along a common subgraph. for any finite collection of graphs gi, each with a fixed isomorphic subgraph h as common, the subgraph-amalgamation is the graph obtained by taking the union of all the gi and identifying their fixed subgraphs h’s. the simplest one is vertex-amalgamation or vertex-merging. theorem 3.1. let g be the graph obtained by merging the graphs {gi}ni=1 along n copies of isomorphic induced common convex subgraph h where h ⊂ gi ∀i. let si = v (gi − h) ∀i. then, for x ∈ h and u ∈ gk − h, bg(x) = n∑ i=1 bgi(x) − (n − 1)bh(x) + ∑ i<j b(x, si|sj), bg(u) = bgk(u) + ∑ i ̸=k b(u, si|sk). proof. consider the graphs {gi}ni=1. let g be the graph obtained by merging gi along n copies of isomorphic induced common convex subgraph h where h ⊂ gi ∀i. see figure 1. now the subgraphs gi − h and h form a partition of g. any path joining a vertex of gi − h and a vertex of gj − h for i ̸= j passes through at least one vertex of h. let x ∈ h, then its betweenness centrality bg(x) in g is due to the contribution of possible pairs of vertices from each gi and from different pairs of gi − h. since h is convex, bh(x) is repeated n times in ∑n i=1 bgi(x) and hence we get bg(x) =∑n i=1 bgi(x) − (n − 1)bh(x) + ∑ i<j b(x, si|sj). let u ∈ gk − h. to find bg(u), consider a pair of vertices s, t ∈ g which contributes to bg(u), then s ∈ sk, and t may be either in gk or in its complement g′k. those t ∈ gk gives bgk(u) and those t ∈ g ′ k gives b(u, si|sk) for i ̸= k. hence bg(u) = bgk(u) + ∑ i ̸=k b(u, si|sk). 24 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 h gk − h g1 g2 gn gk figure 1. subgraph-amalgamation algorithm 3.2. algorithm for betweenness computation of a vertex x ∈ v (h) in subgraph amalgamation require: graphs g1, g2, . . . , gn, convex common subgraph h, vertex x ∈ h ensure: betweenness of vertex x, bg(x) 1: si = v (gi − h) for i = 1, . . . , n. 2: compute betweenness of x in gi, bgi(x) for i = 1, . . . , n. add all these betweenness values to get bcomponentsum(x) 3: compute betweenness of x in h, bh(x). 4: compute betweenness of x in si, bsi(x) for i = 1, . . . , n. 5: find b(x, si|sj) = ∑ s∈si, t∈sj σst(x) σst for all i < j, i = 1, . . . , n. add all these betweenness values to get bsubsetsum(x) 6: determine bg(x) = bcomponentsum(x) − (n − 1) ∗ bh(x) + bsubsetsum(x) theorem 3.3. let g be the graph obtained by merging the graphs {gi}ni=1 along n copies of isomorphic induced common convex subgraph h where h ⊂ gi ∀i. then algorithm 3.2 correctly computes the betweenness value of vertex x ∈ h in o(n′m′) time where n′ is the order of the largest component graph gi and m′ the number of edges in gi. proof. proof of algorithm 3.2 follows from theorem 3.1. the efficient algorithm by brandes [2] compute betweenness centrality of all vertices in an unweighted graph in o(nm) time and o(m + n) space where m is the number of edges in the graph and n is the number of vertices. here we compute the betweenness centrality of a vertex using partial betweenness of different components of the amalgamation. therefore, the complexity gets reduced to o(n′m′) where n′ is the order of the largest component graph gi and m′ the number of edges in gi. algorithm 3.4. algorithm for betweenness computation of vertex u in gk − h in a subgraph amalgamation require: graphs g1, g2, . . . , gn, convex common subgraph h, vertex u ∈ v (gk) − v (h) ensure: betweenness of vertex x, bg(x) 1: si = v (gi − h) for i = 1, . . . , n 2: compute betweenness of u in gk, bgk(u). 3: compute betweenness of u in si, bsi(u) for i = 1, . . . , n. 25 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 4: find b(u, si|sk) = ∑ s∈si, t∈sk σst(u) σst for all i ̸= k. add all these betweenness values to get bsubsetsum(u) 5: determine bg(u) = bgk(u) + bsubsetsum(x) theorem 3.5. let g be the graph obtained by merging the graphs {gi}ni=1 along n copies of isomorphic induced common convex subgraph h where h ⊂ gi∀i. then algorithm 3.4 correctly computes the betweenness value of vertex u in gk − h in o(nkmk) time where nk is the order of gk and mk the number of edges in gk. proof. proof of algorithm 3.4 follows from theorem 3.1. 3.1. path-amalgamation of graphs two cycles connected by merging along a common subgraph proposition 3.6. let g be the graph obtained by merging two cycles cm and cn along a common path pp = {x1, . . . , xp} as common subgraph where p < min{⌈m2 ⌉, ⌈ n 2 ⌉} and u = {u1, . . . , um−p} and v = {v1, . . . , vn−p}, be the vertex sets of cm−pp and cn−pp respectively, then the betweenness centrality of cm in g is given by bg(xr) =bcm(xr) + bcn(xr) − (r − 1)(p − r) + b(xr, u|v ), for xr ∈ pp, bg(ur) =bcm(ur) + b(ur, u|v ), for ur ∈ u, where b(xr, u|v ) and b(ur, u|v ) are given by case 1: when cm and cn are even b(xr, u|v ) = { mn/2 − (m + n)(p − 1/2) + 2(p2 − p + 1/3), for 1 < r < p, 3mn/4 − (m + n)(p − 1/4) + (9p2 − 6p + 2)/6, for r = 1, p. b(ur, u|v ) =   (n − p)(m − p − 2r)/2, for 1 ≤ r ≤ m 2 − p, n(2p − 3)/4 − (3p2 − 3p − 2)/6, for r = m 2 − p + 1, k(k − p + 1) + (n − p)(p − 2)/2 + 1/6, for r = m 2 − p + 1 + k, 1 ≤ k ≤ p − 2. case 2: when cm and cn are odd b(xr, u|v ) = { 2[(m + 1)/2 − p][(n + 1)/2 − p], for 1 < r < p, 3mn/4 − (m + n)(p − 1/4) + (6p2 − 4p + 1)/4, for r = 1, p. b(ur, u|v ) =   (n − p)(m − p − 2r)/2, for 1 ≤ r ≤ m+1 2 − p, (p − 2)(n − p − 1)/2, for r = m+1 2 − p + 1, k(k − p + 2) + (p − 2)(n − p − 1)/2, for r = m+1 2 − p + 1 + k, 1 ≤ k ≤ p − 2. case 3: when cm is even and cn is odd b(xr, u|v ) = { (m − 2p + 1)(n − 2p + 1)/2, for 1 < r < p, 3mn/4 − (m + n)(p − 1/4) + (6p2 − 4p + 1)/4, for r = 1, p. b(ur, u|v ) =   (n − p)(m − p − 2r)/2, for 1 ≤ r ≤ m 2 − p, 1/2[(n − p)(p − 2) + (n + 1)/2 − p], for r = m 2 − p + 1, k(k − p + 1) + (n − p)(p − 2)/2, for r = m 2 − p + 1 + k, 1 ≤ k ≤ p − 2. b(vr, u|v ) =   (m − p)(n − p − 2r)/2, for 1 ≤ r ≤ n+1 2 − p, 1/2[m(p − 2) − p2 + p + 2], for r = n+1 2 − p + 1, k(k − p + 2) + (p/2)(m − p + 1) − m + 1, for r = n+1 2 − p + 1 + k, 1 ≤ k ≤ p − 2. 26 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 proof. let cm and cn be two cycles merged along a common path pp induced by the common vertices x = {x1, . . . , xp} where p < min{⌈m2 ⌉, ⌈ n 2 ⌉}. let u = {u1, . . . , um−p} and v = {v1, . . . , vn−p} be the remaining vertices in cm and cn respectively. consider the three cycles cm, cn and c′, where c′ is the cycle induced by u, v, {x1} and {xp}. by symmetry, the vertices in each of the sets x, u, v from either ends have the same betweenness centrality in g. b b b b bbb b bb b b b b b b b u1uα v1x1 xp vγ uβ vδum−p vn−p cm cnppp − 2 p − 2 m 2 − p n2 − p m 2 − p n 2 − p figure 2. two even cycles merged along a common subgraph case 1: both cm and cn are even. let uα, uβ be the eccentric vertices of xp and x1 respectively in cm and vδ, vγ be their eccentric vertices in c′ where α = m 2 − p + 1, β = m 2 , γ = n 2 − p + 1 and δ = n 2 . see figure 2. let xr ∈ x. by theorem 3.1, bg(xr) = bcm(xr) + bcn(xr) − bp (xr) + b(xr, u|v ). let us find b(xr, u|v ) as the others are known. consider u ∈ u and v ∈ v , then for each (u, v) pair there are at most three shortest u − v paths and in the case of (uα, vδ) and (uβ, vγ) there are three shortest u − v paths p1, p2 and p3 passing through the end vertices x1, xp and the whole path p . (a) consider the vertex xr for 1 < r < p, then those pairs (ui, vj) for 1 ≤ i ≤ α, δ ≤ j ≤ n − p, or (ui, vj) for β ≤ i ≤ m − p, 1 ≤ j ≤ γ contribute betweenness centrality ( m 2 − p )( n 2 − p ) + 1 2 ( m 2 − p + n 2 − p ) + 1 3 to xr. thus by theorem 3.1 we get, bg(xr) = 1 8 [ (m − 2)2 + (n − 2)2 ] − (r − 1)(p − r) + 2 [( m 2 − p )( n 2 − p ) + 1 2 ( m 2 − p + n 2 − p ) + 1 3 ] . (b) consider the vertex x1, then the pairs (ui, vj) for 1 ≤ i < α, 1 ≤ j ≤ n − p contributes (m 2 − p)(n − p); now for i = α, the pairs for 1 ≤ j < β contributes n 2 − 1, j = β contributes 2/3, β < j ≤ n−p contributes 1 2 (n 2 −p) giving the sum 1 12 (9n−6p−4) to x1. the vertices lying between uα and uβ contribute 12(n − p)(p − 2). consider uβ, it makes pairs with vj, 1 ≤ j ≤ γ and gives n 2 − p + 2 3 . again the vertices on the right of uβ with the same set give (m2 − p) [ (n 2 − p) + 1 2 ] . summing all these contributions, we get b(x1, u|v ). (c) consider the vertex ur for 1 ≤ r < α, now the vertices lying between ur and uα make pairs with all vertices of v giving the contribution (m 2 − p − r)(n − p). again uα and the vertices lying between uα and uβ contribute the sum as earlier. but uβ makes pairs with vj for 1 ≤ j ≤ γ and gives 1 2 (n 2 − p) + 1 3 and the right of uβ has no contribution. summing all the above values gives b(ur, u|v ) for 1 ≤ r < α. (d) consider the vertex uα, now the vertices ui for α < i ≤ β give the sum 12(n − p)(p − 2) + 1 2 (n 2 − p) + 1 3 . (e) consider ur for α < r ≤ ⌈m−p2 ⌉. let r = α + k where 1 ≤ k ≤ ⌈ p 2 ⌉ − 1. now uα and uβ gives uα+k the same contribution as 12( n 2 − p) + 1 3 . the vertices lying between uα and uα+k gives the sum (k − 1)(n 2 − p + 1 2 ) + k(k−1) 2 and between uα+k and uβ gives 12(n − k − p)(p − k − 2). the total of these contribution gives the expression for b(ur, u|v ) for r > α. case 2: both cm and cn are odd. 27 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 consider uα and uβ, a pair of extreme vertices of the end vertices xp and x1 of p where α = m+1 2 − p + 1 and β = m−1 2 and vδ and vγ be their eccentric vertices in c′ where δ = n−12 , γ = n+1 2 − p + 1. if ui ∈ u and vj ∈ v , then for xr ∈ p where 1 < r < p, {(ui, vj) : 1 ≤ i < α, δ < j ≤ n−p } or {(ui, vj) : β < i ≤ m−p, 1 ≤ j < γ} contributes the sum ( m+1 2 −p )( n+1 2 −p ) . since there exists no more pair, b(xr, u|v ) = 2 ( m+1 2 −p )( n+1 2 −p ) . consider the vertex x1. now the pairs {(ui, vj) : 1 ≤ i < α, 1 ≤ j ≤ n − p} contributes the sum ( m+1 2 − p ) (n − p). the vertices {ui : α ≤ i ≤ β} contributes the sum 1 2 (n−p)(p−1) and the pairs {(ui, vj) : β < i ≤ m−p, 1 ≤ j ≤ n+12 −p} contributes the sum ( m+1 2 −p )( n+1 2 −p ) . hence b(x1, u|v ) = ( m+1 2 −p )( n+1 2 −p ) +(n−p) ( m+p 2 −p ) . consider the vertex ur, 1 ≤ r < α, the vertices lying between ur and uα contributes the sum (α − r − 1)(n − p), vertices from uα to uβ as given above and no vertex from the right. hence b(ur, u|v ) = ( m+1 2 − p − r ) (n − p) + 1 2 (n − p)(p − 1) for 1 ≤ r < α. for uα, the vertices from uα+1 to uβ contributes 12(p − 2)(n − p − 1). hence b(uα, u|v ) = 1 2 (p − 2)(n − p − 1). consider a vertex on the left of uα, say uα+k, then no vertex on the right of uα is considered. the vertices from uα to uα+k−1 contributes 12k(n + k − 2p + 1) and the vertices from uα+k+1 to uβ contribute 1 2 (p − k − 2)(n − p − k − 1) and no more vertex from the right. hence, b(uα+k, u|v ) = 1 2 k(n + k − 2p + 1) + 1 2 (p − k − 2)(n − p − k − 1). b b b b bbb b bb b b b b b b b u1uα v1x1 xp vγ uβ vδum−p vn−p cm cnppp − 2 p − 3 m 2 − p n+1 2 − p m 2 − p n+1 2 − p figure 3. even and odd cycles merged along a common subgraph case 3: cm is even and cn is odd. let uα and uβ where α = m2 − p + 1, β = m 2 be the eccentric vertices in cm for the end vertices xp, x1 of p and vδ and vγ be their eccentric vertices in c′ where δ = n−12 , γ = n+1 2 − p + 1. see figure 3. consider xr ∈ p such that 1 < r < p. if ui ∈ u and vj ∈ v , then for each pair {(ui, vj) : 1 ≤ i < α, δ < j ≤ n − p or β < i ≤ m − p, 1 ≤ j < γ} there exists a geodesic passing through xr and contributes the sum (m 2 − p) ( n+1 2 − p ) and 1 2 ( n+1 2 − p ) for i = α. since there exists no such other pair, b(xr, u|v ) = 12(m − 2p + 1)(n − 2p + 1). consider the vertex x1. now {(ui, vj) : 1 ≤ i < α, 1 ≤ j ≤ n−p} contributes the sum (m2 −p)(n−p). {(uα, vj) : 1 ≤ j ≤ δ} and {(uα, vj) : δ < j ≤ n − p} contributes the sum n−12 + 1 2 ( n+1 2 − p ) . {(uβ, vj) : 1 ≤ j ≤ γ − 1} contributes the sum n+12 − p. the vertices lying between uα and uβ contributes the sum 1 2 (n − p)(p − 2) and each vertex on the right of uβ gives the sum n+12 − p. the total of these sums gives b(x1, u|v ) which is 14 [ 3mn − (4p − 1)(m + n) + 6p2 − 4p + 1 ] . consider the vertex ur, for 1 ≤ r < α. now the vertices lying between ur and uα contributes the sum (α − r − 1)(n − p). again uα and the vertices lying between uα and uβ have the same contribution as mentioned above, uβ gives 12 ( n+1 2 − p ) . for the vertex uα, vertices from uα+1 to uβ−1 and then uβ contributes to it as earlier. consider a vertex on the right of uα, say uα+k. again uα and uβ contributes the same as 12 ( n+1 2 − p ) . the vertices lying between uα and uα+k contribute (k − 1) ( n+1 2 − p ) + 1 2 k(k − 1). the vertices lying between uα+k and uβ contribute 1 2 (n − p − k)(p − k − 2). the sum of these gives b(uα+k, u|v ). consider the vertex vr in cn for 28 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 1 ≤ r < γ. the vertices lying between vr and vγ offers (n+12 −p−r)(m−p) and the vertices from vγ to vδ offers (p−1)(m−p)/2 giving b(vr, u|v ) = (m−p)(n−p−2r)/2. for vγ, the vertices from vγ+1 to vδ gives b(vγ, u|v ) = 1/2[m(p−2)−p2+p+2]. for vγ+k, 1 ≤ k ≤ p−2, the vertices from vγ+k+1 to vδ offers 12(p−k−2)(m−p−k−1) and vertices from vγ to vγ+k−1 offers k 2 (m+k−2+1) so that vertices symmetric from vγ and vδ offers the same giving b(vγ+k, u|v ) = k(k−p+2)+(p/2)(m−p+1)−m+1. corollary 3.7. let g be the graph obtained by merging m copies of cycle cn along a common path pp = {x1, . . . , xp} as common subgraph where p < ⌈n2 ⌉ and u = {u1, . . . , un−p} be the vertex sets of cn − pp. then the betweenness centrality of cn in g is given by bg(xr) = mbcn(xr) − (m − 1)(r − 1)(p − r) + ( m 2 ) b(xr, u|v ), for xr ∈ pp, bg(ur) = bcn(ur) + (m − 1)b(ur, u|v ), for ur ∈ u. where b(xr, u|v ) and b(ur, u|v ) are given by case 1: if cn is even, then b(xr, u|v ) = { n2/2 − 2n(p − 1/2) + 2(p2 − p + 1/3), for 1 < r < p, 3n2/4 − 2n(p − 1/4) + (9p2 − 6p + 2)/6, for r = 1, p. b(ur, u|v ) =   (n − p)(n − p − 2r)/2, for 1 ≤ r ≤ n 2 − p, n(2p − 3)/4 − (3p2 − 3p − 2)/6, for r = n 2 − p + 1, k(k − p + 1) + (n − p)(p − 2)/2 + 1/6, for r = n 2 − p + 1 + k, 1 ≤ k ≤ p − 2. case 2: if cn is odd, then b(xr, u|v ) = { 2 [ (n + 1)/2 − p ]2 , for 1 < r < p, 3n2/4 − 2n(p − 1/4) + (6p2 − 4p + 1)/4, for r = 1, p. b(ur, u|v ) =   (n − p)(n − p − 2r)/2, for 1 ≤ r ≤ n+1 2 − p, (p − 2)(n − p − 1)/2, for r = n+1 2 − p + 1, k(k − p + 2) + (p − 2)(n − p − 1)/2, for r = n+1 2 − p + 1 + k, 1 ≤ k ≤ p − 2. proposition 3.8. let g be the graph obtained by merging both ends of m copies of paths pn together where v (pn) = {1, 2, . . . , n}. then the betweenness centrality of g is given by b(r) = { 1 4 m(m − 1)(n − 2)2, for r = 1, n, 1 2 (n − 1)(n − 3) + 1 m + (m−2) 2 [ (n − 1 − r)2 + (r − 2)2 ] , for 1 < r < n. proof. since the ends of m copies of path pn are separately merged, any two copies of pn form an even cycle c2n−2 in g. since there are ( m 2 ) such cycles, b(r) = ( m 2 ) bc2n−2(r) = 1 4 m(m − 1)(n − 2)2 for r = 1, n. consider an internal vertex of any path pn say p (i) n . now p (i) n and p (j) n for i ̸= j form a cycle of 2n − 2 vertices in g, where the pair of end vertices gives the centrality 1 m instead of 1 2 . consider the path p (k)n where k ̸= j ̸= i. let ui and uk denotes the vertex sets of p (i) n−2 and p (k) n−2 respectively where the merged end vertices 1 and n are deleted. therefore, for any internal vertex r of pn, bg(r) = bc2n−2(r) − 1 2 + 1 m + ∑ k ̸=i ̸=j b(r, ui|uk) = 1 2 (n − 1)(n − 3) + 1 m + (m − 2) 2 [ (n − 1 − r)2 + (r − 2)2 ] 29 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 since, b(r, ui, |uk) = [ n − (r + 1) ]2 2 + 1 2 [1 + 3 + . . . (r − 2) terms] = 1 2 [ (n − 1 − r)2 + (r − 2)2 ] . 3.2. edge-amalgamation of graphs the edge amalgamation of {gi}ni=1 is the graph obtained by taking the union of all the gi and identifying their fixed edges. an amalgamation of two edges e1 = u1v1 of a graph g1 and e2 = u2v2 of a graph g2 is a graph created by identifying u1 with u2 and v1 with v2 and then deleting one of the two edges corresponding to e1 or e2. the other edge will be called the amalgamated edge. the edge amalgamation of cycles are called generalized books [5]. proposition 3.9. let two cycles cm and cn are connected by merging a pair of edges. let ui, vi be any vertices on cm and cn respectively at a distance i from the merged edge. then betweenness centrality of the resulting graph is given by case 1: if both cm and cn are even, then b(ui) =   (m−2)2 8 + (n−2)2 8 + (m−3)(n−3) 4 + (m−2)(n−2) 2 + 1 12 , for i = 0, (m−2)2 8 + (n−2)(m−2−2i) 2 , for 1 ≤ i < m 2 − 1, (m−2)2 8 + n 4 − 2 3 , for i = m 2 − 1. b(vi) are obtained on interchanging m and n. case 2: if both cm and cn are odd (see figure 4), then b(ui) =   (m−1)(m−3) 8 + (n−1)(n−3) 8 + (m−3)(n−3) 4 + (m−2)(n−2) 2 , for i = 0, (m−1)(m−3) 8 + (n−2)(m−2−2i) 2 , for 1 ≤ i < m−1 2 , (m−1)(m−3) 8 , for i = m−1 2 . b(vi) are obtained on interchanging m and n. case 3: if cm is even and cn is odd, then b(ui) =   (m−2)2 8 + (n−1)(n−3) 8 + (m−3)(n−3) 4 + (m−2)(n−2) 2 , for i = 0, (m−2)2 8 + (n−2)(m−2−2i) 2 , for 1 ≤ i < m 2 − 1, (m−2)2 8 + n 4 − 2 3 , for i = m 2 − 1. b(vi) = { (n−1)(n−3) 8 + (m − 2)(n − 2 − 2i)/2, for 1 ≤ i < n−1 2 , (n−1)(n−3) 8 , for i = n−1 2 . 30 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 b b b b b b b b b b bb b bbb b b b b bb ui ui u m−1 2 v n−1 2 vi vi cm cn figure 4. two odd cycles cm and cn identifying a pair of edges proof. consider a merged vertex. since the merged vertex lies on both cycles cm and cn, each cycle induces a value for its betweenness centrality. cm against cn also induces a value. the betweenness centrality of merged vertex is their sum. for any other vertex u, it lies on one of the cycle and betweenness centrality is the sum of the betweenness centrality induced by that cycle and the other one on it. corollary 3.10. let the odd cycles cn1, . . . , cnm share a common edge. then, for u0, an end vertex of the common edge and ui, a vertex in cnk at a distance i from the common edge, we have b(u0) = 1 8 ∑ i (ni − 1)(ni − 3) + 1 4 ∑ i<j (ni − 3)(nj − 3) + 1 2 ∑ i<j (ni − 2)(nj − 2), b(ui) = { (nk−1)(nk−3) 8 + (nk 2 − 1 − i) ∑ i ̸=k(ni − 2), for 1 ≤ i < nk−1 2 , (nk−1)(nk−3) 8 , for i = nk−1 2 . corollary 3.11. if m copies of n-cycles, each are merged at an edge, then for u0, an end vertex of the common edge and ui, a vertex at a distance i from the common edge in any n-cycle (see figure 5), then we have case 1: if n is even b(ui) =   m(n−2)2 8 + ( m 2 )[ (n−2)2 2 + (n−3)2 4 + 1 12 ] , when i = 0, (n−2)2 8 + (m − 1)(n − 2)(n 2 − 1 − i), when 1 ≤ i < n 2 − 1, (n−2)2 8 + (m − 1)(n 4 − 2 3 ), when i = n 2 − 1. case 2: if n is odd b(ui) =   m(n−1)(n−3) 8 + ( m 2 )[ (n−2)2 2 + (n−3)2 4 ] , when i = 0, (n−1)(n−3) 8 + (m − 1)(n − 2)(n 2 − 1 − i), when 1 ≤ i < n−1 2 , (n−1)(n−3) 8 , when i = n−1 2 . 31 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 b b b b bb b b b b b b b b b b b b b b b b b b u0 u0 figure 5. four 7-cycles sharing a common edge 3.3. vertex-amalgamation of graphs definition 3.12. [6] a graph g in which a vertex is distinguished from other vertices is called a rooted graph and the vertex is called a root (or terminal) of g. identifying the terminal vertices of two graphs is known as vertex-amalgamation and the new graph obtained by vertex-amalgamation of g1 and g2 is denoted by g1 · g2 [4]. proposition 3.13. let g1, . . . , gk be vertex disjoint graphs of order n1, . . . , nk respectively and g, the graph obtained by identifying the vertices vi ∈ gi, ∀i; if v0 denotes the merged vertex and v( ̸= vi) ∈ gi, then bg(v0) = k∑ i=1 bgi(vi) + ∑ i<j (ni − 1)(nj − 1) and bg(v) = bgi(v) + bgi(v, v0) ∑ j ̸=i (nj − 1). proof. consider the graph g obtained by identifying the vertices vi ∈ gi. if v0 denotes the merged vertex, i.e, vi = v0 for all i, v0 is a cut vertex (see figure 6) and the removal of v0 disconnects the graph g into k components. the betweenness centrality of v0 in g is calculated over all pairs of vertices and each pair belongs to the same component or different ones. the pairs of vertices lying in the same component give the sum ∑k i=1 bgi(vi) and the pairs of vertices lying in different components give ∑ i<j(ni−1)(nj−1). now their sum gives the result. for v ∈ gi, v ̸= vi, a pair of vertices in g provides a contribution to the centrality of v, if one of which belongs to gi. if bgi(v, v0) denotes the betweenness centrality of v induced by v0 in gi, then clearly bg(v) = bgi(vi) + bgi(v, v0) ∑ j ̸=i(nj − 1). algorithm 3.14. algorithm for betweenness computation of merged vertex v0 in a vertex amalgamation require: vertex disjoint graphs g1, g2, . . . , gk of order n1, n2, . . . , nk, and v0 the merged vertex ensure: betweenness of vertex v0, bg(v0) 1: compute betweenness of vi in gi, bgi(vi) for i = 1, . . . , k. add all these betweenness values to get bcomponentsum 2: find bcontribution = ∑ i=1,...,k−1(ni − 1) ∗ (nj − 1) for i < j. 3: determine bg(v0) = bcomponentsum + bcontribution 32 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 g1 g2 gi gk b v0 n1 n2 ni nk figure 6. vertex amalgamation of gi’s theorem 3.15. let g be the graph obtained by merging graphs {gi}ki=1, on a common root vertex v0. then algorithm 3.14 correctly computes the betweenness value of merged vertex v0 in o(n′m′) times where n′ is the order of the largest component graph gi and m′ the number of edges in gi. proof. proof of algorithm 3.14 follows from proposition 3.13. algorithm 3.16. algorithm for betweenness computation of non-terminal vertex v in gi in a vertex amalgamation require: vertex disjoint graphs g1, g2, . . . , gk of order n1, n2, . . . , nk, v0 the merged vertex, vertex v a non-terminal vertex in gi. ensure: betweenness of vertex v, bg(v) 1: compute betweenness of v in gi, bgi(v). 2: compute betweenness of v induced by v0 in gi bgi(v, v0). 3: find bcontribution = ∑ j=1,...,k;j ̸=i(nj − 1). 4: determine bg(v) = bgi(v) + bgi(v, v0) ∗ bcontribution theorem 3.17. let g be the graph obtained by merging graphs {gi}ki=1 on a common root vertex v0. then algorithm 3.16 correctly computes the betweenness value of non-terminal vertex v in o(nimi) times where ni is the order of the component graph gi and mi the number of edges in gi. proof. proof of algorithm 3.16 follows from proposition 3.13. proposition 3.18. let g1, . . . , gk be vertex disjoint graphs of order n1, . . . , nk. consider the graph g obtained on joining the vertices vi ∈ gi to a single vertex v0 by means of edges and v(̸= vi) ∈ gi, then bg(v0) = ∑ i<j ninj, bg(vi) = bgi(vi) + (ni − 1) ( ∑ j ̸=i nj + 1 ) and bg(v) = bgi(v) + bgi(v, v0) ∑ j ̸=i (nj − 1). 33 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 the above result is significant in view of the fact that networks are often connected by joining to a common node. if g is a rooted graph, the graph g(n) obtained on identifying the root of n-copies of g is called a one-point union of n copies of g [9]. we consider the following examples. 1. windmill graph the windmill graph k(m)n is the one point union of m copies of the complete graph kn; k (m) n has (n − 1)m + 1 vertices and m ( n 2 ) edges. for example, k(4)5 is given in figure 7. theorem 3.19. the betweenness centrality of windmill graph k(m)n is given by b(v) =  (n − 1) 2 ( m 2 ) , for central vertex, 0, for any other vertex. proof. the central vertex of k(m)n is a cut vertex and the removal of which disconnects into m components of kn−1. therefore, by proposition 3.13, the betweenness centrality of the central vertex is given by c(n − 1, n − 1, . . . m times) = (n − 1)2 ( m 2 ) where c(n1, n2, . . . , nk) = ∑ i<j ninj. since other vertices are vertices of the induced subgraph kn−1, their betweenness centrality is zero. b b b b b b b b b b b b b b b b b figure 7. windmill graph k(4)5 the graph k(t)3 is called a friendship graph or dutch t-windmill graph. for example, k (5) 3 is given in figure 8. every pair of vertices in k(t)3 has exactly one common neighbour. k (2) 3 is the butterfly graph and k(n)2 is the star s1,n. corollary 3.20. the betweenness centrality of friendship graph k(n)3 is given by b(v) =  4 ( n 2 ) , for central vertex, 0, for any other vertex. 2. vertex-amalgamation of cycles now let us consider the case when different cycles merged at a vertex. see figure 9. 34 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 b b b b b b b b b b b figure 8. friendship graph k(5)3 b b b b b b b b b b b b b b b b bbb bbb b c5 c6 c7 c8 figure 9. one point union of different cycles proposition 3.21. let the graph g be the union of m1 copies of odd cycle cn1 and m2 copies of even cycle cn2 with a common vertex v0. in any cycle c, let vi denotes the vertex such that d(vi, v0) = i, then, b(v0) = m1(n1 − 1)(n1 − 3) 8 + m2(n2 − 2)2 8 + (n1 − 1)2 ( m1 2 ) + (n2 − 1)2 ( m2 2 ) +(n1 − 1)(n2 − 1)m1m2, and b(vi) =   (n1−1)(n1−3) 8 + ( n1−1−2i 2 ) [ (m1 − 1)(n1 − 1) + m2(n2 − 1) ] , for vi ∈ cn1, 1 ≤ i ≤ n1−1 2 , (n2−2)2 8 + ( n2−1−2i 2 ) [ m1(n1 − 1) + (m2 − 1)(n2 − 1) ] , for vi ∈ cn2, 1 ≤ i < n2 2 , (n2−2)2 8 , for vi ∈ cn2, i = n2 2 . proof. since m1 copies of odd cycles cn1 and m2 copies of even cycles cn2 have a common vertex v0, v0 is a cut vertex of g and each cycle contributes its own betweenness centrality to v0. again v0 lies on the path joining vertices of different cycles. each pair of cycles contributes a betweenness centrality (ni −1)2 and there are ( mi 2 ) pairs of such cycles giving (ni−1)2 ( mi 2 ) to the centrality. considering the three possible combinations odd-odd, even-even and odd-even pairs of cycles, we get the centrality of v0 as their sum. let vi denotes the vertices at a distance i from v0. if vi ∈ cn1 , an odd cycle, for i ≤ n1−1 2 then from each vertex lying between vi and v n1−1 2 , there is a path passing through vi to each vertex of the remaining cycles and consequently there is an increase of ( n1−1−2i 2 ) [(m1 − 1)(n1 − 1) + m2(n2 − 1)] for b(vi). if vi ∈ cn2 , an even cycle, then from each vertex lying between vi and v n2 , there is a path passing through 35 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 b b b b b b b b b b b b bbb bb b b b b c6 c6c6 c6 b b b b b b b b b b b b bbb bb c5 c5c5 c5 figure 10. c(4)6 and c (4) 5 b b b b b b b b b bbbb b b b b b b b u0 v0 u m 2 u1 um−1 v1 vn−1 v n−1 2 v n+1 2 u m 2 −1 u m 2 +1 b bb b bb b bb bb b cm cn figure 11. an even cycle cm and an odd cycle cn merged at a vertex vi to each vertex of the remaining cycles and from the extreme vertex v n 2 there are two paths and one of which passes through vi. hence there is an increase of centrality ( n2−1−2i 2 ) [m1(n1 −1)+(m2 −1)(n2 −1)] for b(vi). since there is no path passing through the extreme vertex v n2 2 , its centrality remains the same as that of cn2 , that is, (n2−2)2 8 . corollary 3.22. let c(m)n be the one point union of m copies of cn. see figure 10. if v0 is the common vertex and vi ∈ c (m) n such that d(vi, v0) = i, then the betweenness centrality of c (m) n is given by case 1: when cn is even b(vi) =   m(n − 2)2 8 + ( m 2 ) (n − 1)2, when i = 0, (n − 2)2 8 + (m − 1)(n − 1)(n − 1 − 2i) 2 , when 1 ≤ i < n 2 , (n − 2)2 8 , when i = n 2 . case 2: when cn is odd b(vi) =   m(n − 1)(n − 3) 8 + ( m 2 ) (n − 1)2, when i = 0, (n − 1)(n − 3) 8 + (m − 1)(n − 1)(n − 1 − 2i) 2 , when 1 ≤ i ≤ n − 1 2 . corollary 3.23. let g be the graph obtained by merging the vertices u0 ∈ cm and v0 ∈ cn and let ui ∈ cm and vj ∈ cn such that d(ui, u0) = i and d(vj, v0) = j. then the betweenness centrality of g is given by 36 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 case 1: if both cm and cn are even, then b(ui) =   (m−2)2 8 + (n−2)2 8 + (m − 1)(n − 1), when i = 0, (m−2)2 8 + (n−1)(m−1−2i) 2 , when 1 ≤ i < m 2 , (m−2)2 8 , when i = m 2 . b(vi) are obtained by interchanging m and n. case 2: if both cm and cn are odd, then b(ui) = { (m−1)(m−3) 8 + (n−1)(n−3) 8 + (m − 1)(n − 1), when i = 0, (m−1)(m−3) 8 + (n−1)(m−1−2i) 2 , when 1 ≤ i ≤ m−1 2 . b(vi) are obtained by interchanging m and n. case 3: if cm is even and cn is odd (see figure 11), then b(ui) =   (m−2)2 8 + (n−1)(n−3) 8 + (m − 1)(n − 1), when i = 0, (m−2)2 8 + (n−1)(m−1−2i) 2 , when 1 ≤ i < m 2 , (m−2)2 8 , when i = m 2 . b(vj) = (n − 1)(n − 3) 8 + (m − 1)(n − 1 − 2j) 2 , when 1 ≤ j ≤ n − 1 2 . note: if the two cycles are identical, then case 1: m = n an even number. b(ui) = b(vi), where b(ui) =   (m−2)2 4 + (m − 1)2, when i = 0, (m−2)2 8 + (m−1)(m−1−2i) 2 , when 1 ≤ i < m 2 , (m−2)2 8 , when i = m 2 . case 2: m = n = an odd number. b(ui) = b(vi), where b(ui) = { (m−1)(m−3) 4 + (m − 1)2, when i = 0, (m−1)(m−3) 8 + (m−1)(m−1−2i) 2 , when 1 ≤ i ≤ m−1 2 . 4. conclusion betweenness centrality is a useful metric for analysing graph-structures and networks. when compared to other centrality measures, computation of betweenness centrality is rather difficult as it involves finding the shortest paths between pairs of vertices in a graph. therefore studying the structure based on graph operations becomes important. here new graph classes originated by subgraph amalgamation have been studied. this study can be extended to other structures and is therefore helpful for analysing larger classes of graphs. 37 s. kumar, k. balakrishnan / j. algebra comb. discrete appl. 6(1) (2019) 21–38 references [1] a. bavelas, a mathematical model for group structures, human organization 7, appl. anthropol. 7(3) (1948) 16–30. [2] u. brandes, a faster algorithm for betweenness centrality, j. math. sociol. 25(2) (2001) 163–177. [3] l. c. freeman, a set of measures of centrality based on betweenness, sociometry 40(1) (1977) 35–41. [4] r. frucht, f. haray, on the corona of two graphs, aequationes math. 4(3) (1970) 322–325. [5] j. a. gallian, a dynamic survey of graph labeling, electron. j. combin. (2009) 1–219. [6] f. harary, the number of linear, directed, rooted, and connected graphs, trans. amer. math. soc. 78(2) (1955) 445–463. [7] s. kumar, k. balakrishnan, m. jathavedan, betweenness centrality in some classes of graphs, int. j. comb. 2014 (2014) 1–12. [8] s. kumar, k. balakrishnan, on the number of geodesics of petersen graph gp(n, 2), electronic notes in discrete mathematics 63 (2017) 295–302. [9] s.-c. shee, y.-s. ho, the cordiality of one-point union of n copies of a graph, discrete math. 117(1–3) (1993) 225–243. 38 http://www.jstor.org/stable/44135428 http://www.jstor.org/stable/44135428 https://doi.org/10.1080/0022250x.2001.9990249 https://doi.org/10.2307/3033543 https://doi.org/10.1007/bf01844162 https://mathscinet.ams.org/mathscinet-getitem?mr=1668059 https://doi.org/10.2307/1993073 https://doi.org/10.2307/1993073 https://doi.org/10.1155/2014/241723 https://doi.org/10.1155/2014/241723 https://doi.org/10.1016/j.endm.2017.11.025 https://doi.org/10.1016/j.endm.2017.11.025 https://doi.org/10.1016/0012-365x(93)90337-s https://doi.org/10.1016/0012-365x(93)90337-s introduction some betweenness centrality concepts subgraph-amalgamation conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284972 j. algebra comb. discrete appl. 4(2) • 219–225 received: 14 october 2016 accepted: 28 november 2016 journal of algebra combinatorics discrete structures and applications reversible elements in rings∗ research article dilshad alghazzawi abstract: we study some properties related to zero divisors and reversibility in noncommutative rings. 2010 msc: 16u10, 16u99 keywords: reversible rings, zero divisors, right and left annihilators, strongly π-regular rings 1. introduction all our rings will have a unity. a ring r is reversible if, for any a,b ∈ r, ab = 0 if and only if ba = 0. these rings are natural generalizations of commutative rings. reversible rings were studied, in particular, by p.m. cohn [1], gutan and kisielewicz [3], kim and lee [4] and many others. our aim, in this short note, is to introduce elementwise definitions that are directly connected to reversibility but can be applied in a more flexible way. for a ∈ r, we will write r(a) = {x ∈ r | ax = 0} and l(a) = {x ∈ r | xa = 0}. an element a ∈ r is right reversible if r(a) ⊆ l(a). the set of right reversible elements will be denoted by rrev(r). a ring r is semi-commutative if for any a,b ∈ r, ab = 0 implies that arb = 0. in other words a ring r is semi-commutative when the annihilator of an element is a two sided-ideal. the subsets n(r) and u(r) will respectively stand for the set of nilpotent elements and invertible elements of the ring r. a ring r is 2-primal if the set n(r) coincides with the prime radical. other notions will be defined when and where needed. the second section is devoted to the definition, characterizations and properties of the reversible set of a ring and its behavior relative to some ring constructions. reversible nilpotent and idempotent elements are characterized and connections with 2-primal rings are established. in the third section we study some connections with other, more classical, notions such as strongly regular rings and mccoy rings. the paper ends with considerations related to elements having the property that r(a) 6= 0 implies l(a) 6= 0. ∗ supported by a ph.d. grant from king abdulaziz university (rabigh), saudi arabia. dilshad alghazzawi; department of mathematics, kau university (saudi arabia) and université d’artois, france (email: dalghazzawi@kau.edu.sa). 219 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 2. reversible set of a ring definition 2.1. let r be a ring with 1 ∈ r. for an element a in r we write rr(a) or just r(a) (resp. lr(a) or l(a)) the right (resp. left) annihilator of a. the element a is right (resp. left) reversible if r(a) ⊆ l(a) (resp. l(a) ⊆ r(a)). an element which is both left and right reversible is a reversible element. the set of right (resp. left) reversible elements of r will be denoted by rrev(r) (resp. lrev(r)). the set of reversible elements is denoted by rev(r). observe that for any ring r, rev(r) = rrev(r)∩lrev(r). a ring is reversible (cf. [1]) if rrev(r) = r (equivalently lrev(r) = r). example 2.2. 1. observe that 0 ∈ rrev(r). also if r(a) = 0 than a ∈ rrev(r). in particular, left invertible elements, right regular elements in a ring are right reversible. 2. if r is commutative or if r is domain then rrev(r) = lrev(r) = r. 3. of course, if r is reduced then rrev(r) = r = lrev(r). this is easily checked as follows: for any a ∈ r, if ab = 0 then (ba)2 = 0 and hence ba = 0, showing that rrev(r) = r. 4. a ring r is semi-commutative if, for any a ∈ r, r(a) is a 2-sided ideal. in other words r is semicommutative if for any elements a,b ∈ r we have ab = 0 implies that arb = 0. in general, if an element a ∈ rrev(r) is such that r(a) ⊆ rrev(r), then r(a) is a 2-sided ideal of r. in particular, a reversible ring is always semi-commutative. 5. let k be a field. the set of 2 × 2 lower triangular matrices over k will be denoted l2(k). we have rrev(l2(k)) = { ( α 0 β γ ) | αγ 6= 0}∪{ ( 0 0 β γ ) | β ∈ k,γ ∈ k \ {0}}. indeed, if αγ 6= 0 then the corresponding matrix is invertible and hence it belongs to rrev(r). it is easy to check that, for any α,β,γ ∈ k \{0} and any δ ∈ k, we have( 0 0 β 0 ) /∈ rrev(r), ( 0 0 δ γ ) ∈ rrev(r), ( α 0 δ 0 ) /∈ rrev(r). 6. let r be any ring. observe that for ( 1 0 0 0 ) and ( 0 0 1 0 ) in m2(r) we have ( 1 0 0 0 )( 0 0 1 0 ) = ( 0 0 0 0 ) and ( 0 0 1 0 )( 1 0 0 0 ) = ( 0 0 1 0 ) this shows that for any ring r, rrev(m2(r)) 6= m2(r). 7. it is easy to check that if two rings r,s are such that r ⊆ s then rrev(s) ∩r ⊆ rrev(r). the reverse inclusion generally does not hold. indeed, let r be a domain and σ an endomorphism of r with nonzero kernel. consider the ore extension s = r[t;σ] with polynomials of the form ∑ xiai and commutation rule ax = xσ(a). if a ∈ ker(σ) then ax = xσ(a) = 0 but xa 6= 0 this shows that a ∈ rrev(r) but a /∈ rrev(s)∩r. the following proposition provides a characterization of right reversible elements. proposition 2.3. let r be a ring. for an element a ∈ r, the following are equivalent: (i) a ∈ rrev(r), (ii) r(a) ⊆ c(a), where c(a) = {x ∈ r | ax = xa} is the centralizer of a in r, (iii) the correspondence ϕ : ar → ra defined by ϕ(ar) = ra is a well-defined additive map, 220 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 (iv) a ∈ r(r(a)), (v) for every b ∈ r we have that (ab)2 = ab implies that (ba)2 = ba. proof. (i) ⇔ (ii): this is clear. (i) ⇒ (iii): notice that if ar = ar′ then a(r−r′) = 0 and hence (r−r′)a = 0 so that ra = r′a. now we have ϕ(ar) = ra = r′a = ϕ(ar′). this shows that ϕ is well defined. the fact that the map is additive is clear. the converse implication is obvious. (iii) ⇒ (iv): for any x ∈ r(a) we have ax = 0 and 0 = ϕ(ax) = xa. this shows that a ∈ r(r(a)). (iv) ⇒ (v) suppose b ∈ r is such that (ab)2 = ab, this gives that bab − b ∈ r(a) hence by (iv) we (bab− b)a = 0, i.e., (ba)2 = ba. (v) ⇒ (i): suppose ab = 0. then (ab)2 = ab = 0 and the hypothesis shows that 0 = (ba)2 = ba and so ba = 0. corollary 2.4. if a ∈ rrev(r) is such that (ab)2 = ab then (ab)r ∼= (ba)r and r(ba) ∼= r(ab) and the idempotents ab and ba are isomorphic. proof. this is a direct consequence of proposition 2.3 above and proposition 21.20 in [5]. let us now show how the reversible notion behaves. theorem 2.5. let r be any ring. the following hold: (a) the set rrev(r) (respectively lrev(r)) is closed under product (,i.e., if a,b ∈ rrev(r), then ab ∈ rrev(r)). (b) if r and s are two rings and ϕ : r −→ s is an isomorphism of rings then ϕ(rrev(r)) = rrev(s). in particular, if u ∈ u(r) is a unit in r, then a ∈ rrev(r) if and only if uau−1 ∈ rrev(r). (c) if a ∈ rrev(r) then, for invertible elements u,v ∈ u(r), we have uav ∈ rrev(r). (d) if r is a prime ring we have {a ∈ r | r(a) = 0} = rrev(r). (e) if a is right invertible then a is right reversible if and only if a is left invertible (and hence invertible). (f) if r and s are two rings then rrev(r×s) = rrev(r)×rrev(s). (g) if r is a semisimple ring then rrev(r) = u(r), the set of invertible elements of r. (h) an idempotent e ∈ r is right reversible if and only if (e−1)re = 0. (i) let {x1, . . . ,xn} be a subset of rrev(r). if b ∈ r(x1x2 · · ·xn), then rbrx1rx2r...rxnr = 0. (j) if a ∈ r is right reversible and nilpotent, then the ideal rar is nilpotent. (k) if a ∈ r is such that the descending chain of left ideals rai stabilizes then l(a) = 0 if and only if a ∈ u(r). proof. (a) let a,b ∈ rrev(r) and let c ∈ r(ab). then we have abc = 0 and since a ∈ rrev(r), this gives bca = 0. since b ∈ rrev(r), we get cab = 0, and hence c ∈ l(ab). (b) and (c) these are left to the reader. (d) if r is prime and 0 6= a ∈ rrev(r) we have, for any b ∈ r(a) and any r ∈ r, we have abr = 0 and hence bra = 0. this gives bra = 0 and the primeness of r leads to b = 0, as required. (e) if a is right invertible then there exists b ∈ r such that ab = 1 and hence a(ba−1) = 0. since a is also right reversible we get that (ba− 1)a = 0. this gives ba2 = a and, right multiplying by b, we get ba = 1. the converse is clear. 221 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 (f) this is obvious. (g) in the light of wedderburn-artin theorem and part (f) of this theorem, we may assume that r is a matrix ring over a division ring. in particular, r is a prime ring and so the point (d) shows that the right reversible elements are nonzero divisors. now, a nonzero divisor matrix with coefficients in a division ring must be invertible, this easily yields the statement. (h) this is clear. (i) we, then, have x1x2 · · ·xnbr = 0, and since x1 ∈ rrev(r), we get x2x3 · · ·xnbrx1 = 0. so x2x3 · · ·xnbrx1r = 0. but x2 ∈ rrev(r), hence x3x4 · · ·xnbrx1rx2 = 0. continuing this process we get the desired result. (j) let us suppose that a ∈ rrev(r) is such that an = 0, for some n ∈ n . we then have a ∈ r(an−1) and the above statement (i) yields the result. (k) there exists n ∈ n and x ∈ r such that an = xan+1. since l(a) = 0, this leads to 1 = xa and hence to a = axa and also 1 = ax, showing that a ∈ u(r). let us recall that n(r) = {x ∈ r | ∃n ∈ n : xn = 0}. corollary 2.6. (1) for any ring r, rrev(r)∩n(r) is contained in the prime radical of r. (2) if all nilpotent elements of a ring are right reversible then the ring is 2-primal. (3) in a semiprime ring a nilpotent element cannot be right or left reversible. (4) if a ∈ n(r)∩rrev(r) and b ∈ n(r) then a + b ∈ n(r). (5) if a ∈ rrev(r), rr(a) is a proper ( ,i.e., different from r ) two-sided ideal. proof. 1) this is clear from theorem 2.5 (i). 2) this is an obvious consequence of corollary 2.6 (1). 3) it is enough to use the fact that a semiprime ring does not have nonzero nilpotent two sided ideal 4) let l ∈ n, be such that bl = 0. when we develop (a + b)l all monomials will be in the prime radical. 5) this is due to the fact that rr(a) is contained in l(a). we observe that the ring r of upper triangular 2 × 2 matrices over a field is 2-primal but not reversible. example 2.7. (a) observe that rrev(r) is in general not closed under addition. to give a concrete example let us consider the ring of 2 × 2 lower matrices over a field k. it is easy to check that the matrix( 1 0 1 0 ) = ( 1 0 0 1 ) + ( 0 0 1 −1 ) is not right reversible although the two matrices on the right hand side are, indeed, right reversible. (b) in connection with theorem 2.5 (j) we remark that we might have rar nilpotent even if a /∈ rrev(r). this is the case of the element a = e12 of the strictly upper 2×2 matrices over a field. 3. connections with other notions proposition 3.1. let r be a semiprime ring and a ∈ rrev(r). then: 222 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 (a) the right annihilator r(a) is a two-sided ideal of r. (b) for any n ∈ n, we have r(a) = r(an). proof. (a) if b ∈ r(a) we have bra = 0 and hence (rarb)2 = 0. since r is semiprime this leads to rarb = 0, which, in turn, implies that arb = 0. (b)for n ∈ n, n ≥ 1, and a ∈ rrev(r),b ∈ r(an) we have an−1(an−1b) = 0. since an−1 is also in rrev(r) we have that an−1bran−1b = 0. the fact that r is semiprime leads to an−1b = 0,i.e., b ∈ r(an−1). the same method leads to b ∈ r(an−2) and the desired result follows by iterations. we remark that the above statement admits a partial converse: if a ring r is such that for any a ∈ r, there exists l > 1 such that r(a) = r(al) then r is reduced and hence semiprime. we also recall that a strongly regular ring is a ring r such that for every a ∈ r there exists x ∈ r such that a = a2x. the following proposition is based on exercise 12. 6a in lam’s book [6]. proposition 3.2. the following are equivalent: (i) the ring r is strongly regular, (ii) the ring r is regular and reduced, (iii) the ring r is regular and reversible. let us now give some applications to mccoy condition on polynomials. let us first define rev(r) = rrev(r)∩ lrev(r) and say that a polynomal f(x) ∈ r[x] is right mccoy if rr[x](f(x)) 6= 0 implies that there exists a nonzero c ∈ r such that f(x)c = 0. we denote the set of right mccoy polynomials by rmc(r[x]). proposition 3.3. for any ring r we have rev(r)[x] ⊆ rmc(r[x]). proof. let f(x) = ∑n i=0 aix i ∈ rev(r)[x]. if rr[x](f(x)) = 0 then clearly f(x) ∈ rmc(r(x)). so let us suppose that 0 6= g(x) = ∑m j=0 bjx j ∈ r[x] is of minimal degree such that f(x)g(x) = 0. we, then have anbm = 0 and since an ∈ rev(r), we get bman = 0, this leads to deg(g(x)an) < deg(g(x)) and since f(x)g(x)an = 0, the minimality of deg(g(x)) shows that we have g(x)an = 0 and hence also ang(x) = 0. we now have an−1bm = 0 which leads to bman−1 = 0 and hence deg(g(x)an−1) < deg(g(x)). since we have f(x)g(x)an−1 = 0 the minimality of deg(g(x)) implies that g(x)an−1 = 0. since an−1 ∈ rev(r) we thus conclude that an−1g(x) = 0. continuing this process we will finally obtain that for all i ∈{0, . . . ,n}, aig(x) = 0. in particular, we obtain f(x)bm = 0, as desired. we also have the following properties also connected with the mccoy condition. proposition 3.4. let f(x) = ∑n i=0 aix i,g(x) = ∑m j=0 bjx j ∈ r[x] be such that f(x)g(x) = 0. then: (a) if a0 ∈ rrev(r) then g(x)am+10 = 0. in particular if a m+1 0 6= 0, then r(g(x))∩r 6= 0. (b) if b0 ∈ lrev(r), then bn+10 f(x) = 0. in particular if b n+1 0 6= 0, then l(f(x))∩r 6= 0. proof. (a) with the notation as in the statement of the theorem, it is enough to prove that, for any 0 ≤ i ≤ m, biai+10 = 0. if n = deg(f(x)) < deg(g(x)) = m, we put al = 0 for any n < l ≤ m. with this notation the equality f(x)g(x) = 0 gives, for any 0 ≤ k ≤ m, a0bk +a1bk−1 +· · ·+akb0 = 0. in particular, a0b0 = 0. since a0 ∈ rrev(r) we also have b0a0 = 0. this shows that the required equality mentioned above is valid for i = 0. let l < m and assume we have proved that bia i+1 0 = 0 for any 0 ≤ i ≤ l < m. multiplying the equation a0bl+1 +a1bl + · · ·+al+1b0 = 0 on the right by al+10 we then get a0bl+1a l+1 0 = 0 and hence, since a0 ∈ rrev(r), bl+1al+20 = 0. this yields the required equalities. (b) the second part of the theorem is proved similarly. 223 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 we now consider relations between reversible elements and other kind of classical elements. let us recall that an element a ∈ r is strongly π-regular if there exists n ∈ n such that an ∈ ran+1 ∩ an+1r. this is equivalent to asking that both chains ar ⊃ a2r ⊃ ... ⊃ anr ⊃ . . . and ra ⊃ ra2 ⊃ ... ⊃ ran ⊃ . . . stabilize. the set of strongly π-regular elements is denoted by sregπ(r). r is a π regular ring if sregπ(r) = r. let us recall that dischinger [2] showed that a ring r is strongly π-regular if and only if any descending chain condition ra ⊃ ra2 ⊃ ... stabilizes, i.e., only one of the above chain conditions is required for a ring to be strongly π-regular. let us mention that using the above, we can show that a right (resp. left) artinian ring is such that every element a ∈ r with r(a) = 0 (resp. l(a) = 0) must be invertible. in particular any left or right artinian rings is strongly π-regular. this short discussion leads quickly to the following classical result. proposition 3.5. if r is π-strongly regular then every left or right nonzero divisor is invertible. prompted by this proposition, we introduce another elementwize condition. this concept is more general than the right reversible one. for this we define the following two sets: sr(r) = {a ∈ r | r(a) 6= 0 , if l(a) 6= 0} sl(r) = {a ∈ r | l(a) 6= 0 , if r(a) 6= 0}. we say that the ring r satisfies the r (resp. l) property if sr(r) = r (resp.sl(r) = r). corollary 3.6. let r be any ring. both rrev(r) and sregπ(r) are contained in sr(r). example 3.7. consider the upper triangular matrix ring of the form( z z/2z 0 z ) it is easy to check that the element a = ( 2 1 0 1 ) is such that r(a) 6= 0 but l(a) = 0. we have seen that it was not possible to pass the right reversible property from a ring to the matrix ring. in the next proposition we show that in some cases the property sr(r) = r goes up to the matrix ring mn(r). proposition 3.8. (a) let a,u,v ∈ r such that u,v are invertible. then a ∈ sr(r) if and only if uav ∈ sr(r). a similar result is true for sl(r). (b) let r be such that sr(r) = r and suppose that every square matrix a ∈ mn(r) is diagonalizable. then sr(mn(r)) = mn(r). (c) let r be a ring with a total left ring of quotient s. if sr(s) = s then sr(r) = r. (d) let r ⊆ s be rings such that rr is essential in rs. if sr(s) = s then sr(r) = r. (e) let a ∈ r be a unit regular element (i.e., there exists an invertible element u ∈ u(r) such that a = aua). then a ∈ sr(r)∩sl(r). proof. (a) it easy to check that r(uav) = v−1r(a) and l(uav) = u−1l(a). so if we assume that r(a) 6= 0 implies l(a) 6= 0, then r(uav) 6= 0 implies l(uav) 6= 0. (b) by part (a) above, it is enough to show that a diagonal matrix a is such that r(a) 6= 0 also satisfies l(a) 6= 0. this is easy and left to the reader. (c) this is easy as follows: let a ∈ r be such that rr(a) 6= 0. hence rs(a) 6= 0 and since sr(s) = s, we have that ls(a) 6= 0, so there exists elements x,y ∈ r with ls(x) = 0 and s = x−1y ∈ ls(a). we then get that 0 6= y ∈ lr(a). (d) suppose a ∈ r is such that rr(a) 6= 0 then rs(a) 6= 0. the fact that a ∈ sr(s) implies that ls(a) 6= 0 and since r is essential in s, we obtain that lr(a) 6= 0. 224 d. alghazzawi / j. algebra comb. discrete appl. 4(2) (2017) 219–225 (e) suppose that a ∈ r is unit regular, i.e., there exists an invertible element u such that a = aua. suppose that l(a) = 0, then since (1 − au)a = 0, we have that 1 = au, and if b ∈ r(a), we get that u−1b = auu−1b = ab = 0, this gives that b = 0. so that a ∈ sr(r). the fact that a ∈ sl(r) is obtained similarly. acknowledgment: this work will be part of my ph.d. thesis under the supervision of prof. a. leroy whom i thank for his continuous guidance and help. i would like also to thank king abdulaziz university in saudi arabia for the financial supports received during the preparation of this work. references [1] p. m. cohn, reversible rings, bull. london math. soc. 31(6) (1999) 641–648. [2] f. dischinger, sur les anneaux fortement π-réguliers, c. r. acad. sci. paris sér. a–b 283(8) (1976) aii a571–a573. [3] m. gutan, a. kisielewicz, reversible group rings, j. algebra 279 (2004) 280–271. [4] n. k. kim, y. lee, extensions of reversible rings, j. pure appl. algebra, 185(1–3) (2003) 207–223. [5] t. y. lam, a first course in noncommutative rings, graduate texts in mathematics, springer verlag, new york, berlin, heidelberg, 1990. [6] t. y. lam, exercises in classical ring theory, problem books in mathematics, springer verlag, new york, berlin, heidelberg, 1994. 225 http://dx.doi.org/10.1112/s0024609399006116 http://www.ams.org/mathscinet-getitem?mr=422330 http://www.ams.org/mathscinet-getitem?mr=422330 http://dx.doi.org/10.1016/j.jalgebra.2004.02.011 http://dx.doi.org/10.1016/s0022-4049(03)00109-9 introduction reversible set of a ring connections with other notions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327375 j. algebra comb. discrete appl. 4(3) • 261–269 received: 21 february 2016 accepted: 21 march 2017 journal of algebra combinatorics discrete structures and applications on the equivalence of cyclic and quasi-cyclic codes over finite fields research article kenza guenda, t. aaron gulliver abstract: this paper studies the equivalence problem for cyclic codes of length pr and quasi-cyclic codes of length prl. in particular, we generalize the results of huffman, job, and pless (j. combin. theory. a, 62, 183–215, 1993), who considered the special case p2. this is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. this allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length pr in polynomial time. further, we characterize the set by which two quasi-cyclic codes of length prl can be equivalent, and prove that the affine group is one of its subsets. 2010 msc: 94b05, 94b15, 94b60 keywords: cyclic code, quasi-cyclic code, equivalence, automorphism, permutation 1. introduction the equivalence problem for codes has many practical applications such as code-based cryptography [8, 9, 12, 13]. as a consequence, this problem has received considerable attention in the literature [2, 3, 6, 11, 12]. however, progress in obtaining results has been slow. brand ([3]) characterized the set of permutations by which two combinatorial cyclic objects on pr elements are equivalent. using these results, huffman et al. ([6]) explicitly gave this set for the case p2, and constructed algorithms to find the equivalence between cyclic objects and extended cyclic objects. in [6], a negative answer was given to the generalization of their results to the case pr, r > 2. this is due to the fact that the permutations of brand that are crucial to the proofs do not generate a sylow subgroup of spr . babai et al. ([2]) gave an exponential time algorithm for determining the equivalence of codes. sendrier ([11]) proposed the support splitting algorithm to solve the problem of code equivalence in the binary case. however, in [12] kenza guenda (corresponding author); faculty of mathematics usthb, university of science and technology of algiers, algeria (email: ken.guenda@gmail.com). t. aaron gulliver; department of electrical and computer engineering, university of victoria, po box 1700, stn csc, victoria, bc, canada v8w 2y2 (email: agullive@ece.uvic.ca). 261 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 the authors showed that extending the support splitting algorithm to q ≥ 5 results in an exponential growth in complexity, which makes this approach impractical. in this paper, we study the equivalence problem for cyclic codes of length pr and quasi-cyclic codes of length prl over finite fields. we generalize the results of [6] (which are only for the special case p2), by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. further, the set of brand is extended to the class of quasi-cyclic codes of length prl. the remainder of this paper is organized as follows. in section 2, some preliminary results are presented. section 3 considers the equivalence of cyclic codes, in particular cyclic codes of length pr. then in section 4, the equivalence of quasi-cyclic codes of length prl is investigated. 2. preliminaries let c be a linear code of length n over the finite field of q elements, fq, and σ a permutation of the symmetric group sn, acting on {0,1, . . . ,n − 1}. for a code c, we associate another linear code σ(c) defined by σ(c) = {(xσ−1(0), . . . ,xσ−1(n−1)) ; (x0, . . .xn−1) ∈ c}. we say that the codes c and c′ are permutation equivalent if there exists a permutation σ ∈ sn such that c′ = σ(c). the automorphism group of c is the subgroup of sn given by aut(c) = {σ ∈ sn ;σ(c) = c}. a linear code c of length n over fq is called quasi-cyclic of index l or an l-quasi-cyclic code if its automorphism group contains the permutation t l given by t l : zn −→ zn i 7−→ i + l mod n, (1) where t : i 7→ i + 1 is the cyclic shift. this definition is equivalent to saying that for all c ∈ c we have t l(c) ∈ c. the index l of c is the smallest integer satisfying this property. it can easily be proven that l is a divisor of n. in the case l = 1, the code is called a cyclic code. this is a code with an automorphism group that contains the cyclic shift t. 3. equivalence of cyclic codes in this section, we consider the permutation equivalence of cyclic codes. later we will show that there is a very close link between the equivalence of some quasi-cyclic codes and cyclic codes. this provides further motivation to study the equivalence of cyclic codes. we begin with some well known results. let n be a positive integer. the set of permutations ag(n) = {τa,b : a 6= 0,(a,n) = 1,b ∈ zn} is the subgroup of sn formed by the permutations defined as follows τa,b : zn −→ zn x 7−→ (ax + b) mod n. (2) the group ag(n) is called the group of affine transformations. the affine transformations ma = τa,0, (3) 262 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 are called multipliers. the affine group agl(1,p) is the group of affine transformations over zp. for d ∈ z∗p, the generalized multiplier µd ∈ sp2 was defined in [6] as follows. let k ∈ zp2 and k = i + jp for some 0 ≤ i,j ≤ p−1, so that kµd = (id) mod p + pj. then from palfy ([10]) and alspach and parson ([1]), we have the following results. theorem 3.1. [6, theorem 1] let c and c′ be cyclic codes of length n over a finite field. suppose one of the following holds for n: (i) gcd(n,φ(n)) = 1 or n = 4, or (ii) n = pr,p > r are primes and the p-sylow subgroup of the automorphism group of c has order p. then c and c′ are equivalent by a multiplier. in the case n = p2, huffman et al. ([6]) gave the following result. theorem 3.2. [6, theorem 3.1] let c and c′ be cyclic codes of length p2 with p an odd prime, where t ∈ aut(c) and t ∈ aut(c′). then if c and c′ are equivalent, they are equivalent by a multiplier or a generalized multiplier times a multiplier. for n = pr, r > 2 the equivalence of cyclic codes of length n is very complex, but in the next section this problem is partially solved. 3.1. equivalence of cyclic codes of length pr let c be a cyclic code of length pr, p an odd prime and r > 1. further let t be the cyclic shift modulo pr and p a p-sylow subgroup of aut(c). the following subset of spr was introduced by brand ([3]) h(p) = {σ ∈ spr|σ−1tσ ∈ p}. the set h(p) is well defined since 〈t〉 is a subgroup of aut(c) of order pr, so it is a p-group of aut(c). from sylow’s theorem, there exists a p-sylow subgroup p of aut(c) such that 〈t〉 ≤ p . furthermore, in some cases the set h(p) is a group. lemma 3.3. [3, lemma 3.1] let c and c′ be cyclic codes of length pr, and p be a p-sylow subgroup of aut(c) which contains t. then c and c′ are equivalent if and only if c and c′ are equivalent by an element of h(p). lemma 3.3 shows the importance of having information on the p-sylow subgroup of aut(c). the following results provide some of this information. proposition 3.4. [4, proposition 9] let c be a cyclic code of length pr with r > 1, and mq be the multiplier defined by mq(i) = iq mod pr. then the group aut(c) contains the subgroup k = 〈t,mq〉 of order prordpr (q). let pl, l ≥ r, be the p-part of the order of k. then a p-sylow subgroup p of aut(c) has order ps such that l ≤ s ≤ pr−1 + pr−2 + · · ·+ 1. now we define the sets of brand. let p be an odd prime. for n < p, we define the following subsets of spr qn = {f : zpr → zpr|f(x) = n∑ i=0 aix i,ai ∈ zpr for each i,(p,a1) = 1, and pr−1 divides ai for i = 2,3, . . . ,n}. 263 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 qn1 = {f ∈ q n|f(x) = n∑ i=0 aix i, with a1 ≡ 1 mod pr−1}. the sets qn and qn1 are subgroups of spr [3, lemma 2.1]. note that q 1 = ag(pr). lemma 3.5. let c be a cyclic code of length pr where p is odd and m > 1. let p be a p-sylow subgroup of aut(c) which contains t. if 1 ≤ n < p, then: (i) |qn| = (p−1)p2r+n−2 and |qn1 | = pr+n, (ii) ag(pr) = nspr (〈t〉) ⊂ h(p), (iii) qn+1 = h(qn1 ), (iv) nspr (q n 1 ) = q n+1. proof. for part (i), from [3, lemma 3.2] we have the map (a0, . . . ,an) −→ f where f(x) = ∑n i=0 aix i is injective if n < p − 1. thus in qn, the coefficient a0 can take pr different values, and a1 can take pr−1(p−1) values. for 2 ≤ i ≤ n, ai can take p values. from these results we have |qn| = p2r+n−2(p−1). for qn1 , the coefficient of a0 can take p r different values, and ai for 1 ≤ i ≤ n can take p values, so that |qn1 | = pr+n. now we prove that ag(pr) = nspr (〈t〉). let σ be an element of nspr (〈t〉). then there is a j ∈ zn\{0} such that σtσ−1 = tj, or equivalently σt = tjσ. hence σt(0) = σ(1) = tjσ(0) = σ(0)+j and σt(1) = σ(1) +j = σ(0) + 2j, so that σ(k) = σ(0) +kj for any k ∈ zn. then (j,n) = 1 follows from the fact that the order of t equals the order of tj. the last inclusion is obvious. part (iii) follows from [3, lemma 3.7]. for part (iv), we begin with the ≤ condition. let h ∈ nspr (q n 1 ) and g = h −1th. as t ∈ qn1 , it must be that g ∈ qn1 . since the order of g is equal to the order of t (which is pr), from [3, lemma 3.6] there exists f ∈ qn+1 such that f−1gf = t , so then f−1h−1thf = t. the only elements of spr which commute with t (a complete cycle of length pr), are the powers of t . thus hf = tj for some j. since qn+1 is a subgroup of spr and 〈t〉≤ qn+1, h ∈ qn+1, and hence nspr (q n 1 ) ≤ qn+1. now consider the ≥ condition. let g ∈ qn+1 where g(x) = ∑n+1 i=0 gix i with p g1 and pr−1|gi for 2 ≤ i ≤ n. further, let h ∈ qn1 , where h(x) = ∑n i=0 hix i with h1 ≡ 1 mod pr−1 and pr−1|hi for 2 ≤ i ≤ n. we have hg(x) = n∑ i=0 hi  n+1∑ j=0 gjx j  i = h0 + h1 n+1∑ i=0 gjx j + n∑ i=2 hi  n+1∑ j=0 gjx j  i . since pr−1|hi, for i ≥ 2 and pr−1|gj for j ≥ 2, any terms in ∑n i=2 hi (∑n+1 j=0 gjx j )i involving gj for j ≥ 2 vanish modulo pr, so that hg(x) = h0 + h1 n+1∑ j=0 gjx j + n∑ i=2 hi (g0 + g1x) i . by [3, lemma 2.1] g−1(x) = n+1∑ i=1 bix i, with b1 = g −1 1 and bi = −gig −(i+1) 1 for 2 ≤ j ≤ n + 1. (4) we now determine g−1hg in order to prove that it is in qn1 . this is given by g−1hg(x) = n+1∑ k=1 bk  h0 + h1 n+1∑ j=0 gjx j + n∑ i=2 hi(g0 + g1x) i −g0  k 264 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 = b1  h0 + h1 n+1∑ j=0 gjx j + n∑ i=2 hi (g0 + g1x) i −g0   + n+1∑ k=2 bk  h0 + h1 n+1∑ j=0 gjx j + n∑ i=2 hi(g0 + g1x) i −g0  k. as pr−1|gj for j ≥ 2, hence pr−1|bk for k ≥ 2. furthermore, we have pr−1|hi for i ≥ 2, and thus g−1hg(x) = b1  h0 + h1 n+1∑ j=0 gjx j + n+1∑ j=0 hi(g0 + g1x) i −g0   + n+1∑ k=2 bk (h0 + h1 (g0 + g1x)−g0) k . let g−1hg(x) = ∑n+1 r=0 crx r, and note that cn+1 = b1h1gn+1 + bn+1(h1g1)n + 1. then replacing the bi with their values from (4), we obtain cn+1 = g −1 1 h1gn+1 −gn+1g −(n+2) 1 h n+1 1 g n+1 1 = g −1 1 h1(gn+1 −gn+1h n 1 ). as h1 ≡ 1 mod pr−1, we have that hn1 ≡ 1 mod pr−1. in addition, as pr−1|gn+1, it must be that gn+1hn1 ≡ gn+1 mod p r. therefore, cn+1 = 0, and pr−1|ci for 2 ≤ i ≤ n. then we only need to show that c1 ≡ 1 mod pr−1. as gj ≡ 0 mod pr−1 for j ≥ 2, hi ≡ 0 mod pr−1 for i ≥ 2, and bk ≡ 0 mod pr−1 for k ≥ 2, so then c1 ≡ b1h1g1 mod pr−1. finally, since b1 = g−11 , we have that c1 ≡ h1 ≡ 1 mod p r−1. next, we require the following theorems which characterize some p subgroups of spr . theorem 3.6. [4, theorem 10] let n be a positive integer less than p− 1. if p is a p subgroup of spr with qn1 � p ≤ qn+1, then p = q n+1 1 . further, the group q 1 1 is a normal subgroup of q 1 and is the unique p subgroup of spr of order pr+1 which contains t. theorem 3.7. [4, theorem 11] let g be a subgroup of spr and p a p-sylow subgroup of g of order ps such that t ∈ p. then the following hold: (i) if s = r, then p = 〈t〉, (ii) if r < s ≤ p + r −1, then we have p = qs−r1 . corollary 3.8. let c and c′ be two cyclic codes of length pr, and let p be a p-sylow subgroup of aut(c) such that t ∈ p. if |p| = ps and s ≤ p + r − 1, then c and c′ can be equivalent only under the action of a permutation of the following subgroups of spr : (i) ag(pr) if s = r, (ii) qs−r+1 if s > r. proof. the result follows from lemmas 3.3 and 3.5, and theorem 3.7. remark 3.9. since each affine transformation can be written as the product of a power of t and a multiplier, and t ∈ aut(c), we must have τa,b ∈ c whenever ma ∈ c. hence from corollary 3.8, if s = r then two cyclic codes of length pr are equivalent if and only if they are equivalent by a multiplier. in order to solve the equivalence problem for cyclic codes, we need the p-sylow subgroup of aut(c). to determine this, for i ≤ i ≤ p−1 consider the polynomial fi ∈ qi1 defined by fi(x) = x + p r−1(x + x2 + . . . + xi). 265 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 theorem 3.10. let g be a subgroup of spr with a p-sylow subgroup p which contains t. then the following hold: (i) if there is no fi ∈ g, then p = 〈t〉, (ii) if i is the largest value of i such that fi ∈ g and i ≤ p−2, then p = qi1. proof. if there is no fi ∈ g, then there is no fi in p . if |p | = pr, we can take p = 〈t〉, but then from theorem 3.6 any p-sylow subgroup of ps, s > r, must contain q11, which is impossible. assume that i is the largest i such that fi ∈ g and i ≤ p− 2. let p be a p-sylow subgroup of g of order s, and s be such that i + r ≤ s < p + r − 1. from theorem 3.7, we have that a p-sylow subgroup of any subgroup of g ≤ spr which contains t has order ps with m < s ≤ p + r − 1. then we have p = qs−r1 , so that s− r = i. now if s ≤ i + r ≤ p + r − 1, we have from theorem 3.7 that p = qs−r1 , so q i 1 ∩g ≤ q s−r 1 . the assumption on i gives i = s−r. assume now that s > p + r−1. since i ≤ p−2, we have that s > p + r−1 > r + i. we will prove that this case cannot occur. further, as t = f1 ∈ q11, from theorem 3.6 q11 is the unique subgroup of spr of order pr+1 which contains t , so that q11 � p . since q 1 1 � q 2 1, it must be that q 1 1 � q 2 1 ∩p ≤ q2. hence we have that q21∩p = q21, which gives q21 ≤ p . using the same approach for 2 ≤ i ≤ i, we obtain qi ≤ p . the assumption on s gives that qi � p , so qi1 � q i+1 1 ∩p ≤ q i+1 (qi+1 can be considered as it was assumed that i ≤ p− 2). hence from theorem 3.6, we obtain that qi+11 ∩p = q i+1 1 , which contradicts the assumption on i. this theorem suggests the following algorithm for i ≤ p−2. algorithm a: let p be an odd prime and c and c′ be two cyclic codes of length pm. then the equivalence of c and c′ can be determined as follows. step 1: find the order of the p-sylow subgroup of aut(c) as follows. find the largest i such that fi ∈ aut(c), and set s = i + r. step 2: find f ∈ qi+1 such that c′ = fc. remark 3.11. to find the required i in algorithm a we can use (for example), a binary search which requires checking at most dlog2(p−1)e+1 of the fi. furthermore, the cardinality of qi+1 is (p−1)p2r+i−2. this proves that the algorithm has polynomial time complexity. 4. equivalence of quasi-cyclic codes in this section, we characterize the equivalence problem for quasi-cyclic codes. consider the cycles σi = (i, i + l, i + 2l, . . . , i + (m−1)l) for 0 ≤ i ≤ l−1. the cycles σi have order m and satisfy t l = σ0 . . .σl−1. (5) this gives that |(t l)| = lcm(|σ0|, . . . , |σl−1|) = m. proposition 4.1. let n = lm with (m,l) = 1, and 〈t l〉 be the subgroup of sn generated by the permutation t l. therefore the normalizer of 〈t l〉 in sn contains the following groups: (i) q = 〈σ0, . . . ,σl−1,t〉, (ii) ag(n). proof. it is obvious that t ∈ nsn(〈t l〉). as the cycles in (5) are pairwise disjoint, it must be that σ0 . . .σl−1 = t l. furthermore, as the cycles σi are disjoint, we have that σ −1 i t lσi = t l. 266 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 we consider the affine transformation τa,b ∈ ag(n), which shows that τa,b ∈ nsn(〈t l〉) is equivalent to proving the existence of an α ∈ n∗ such that τa,bt lτ−1a,b = t lα. the permutation τa,b can be decomposed as τa,b = τ1,bτa,0. then combining this decomposition with (5), we obtain the following equality τa,bt lτ−1a,b = τ1,bτa,0σ0 . . .σl−1τ −1 a,0τ −1 1,b . (6) a well known result [5, lemma 5.1] gives that if σ = σ0 . . .σl−1 is a product of disjoint cycles and s is a permutation from sn, then sσs−1 = s(σ0)s(σ1) . . .s(σl−1), where s(σi) = (s(σ1i), . . . ,s(σmi)) for the cycle σi = (σ1i, . . . ,σmi). for ra = a mod l, we obtain that τa,0(σi) = σaira. this gives τa,0(σ0)τa,0(σ1) . . .τa,0(σl−1) = σ a 0σ a ra . . .σ a ra(l−1) = t la. for rb = b mod l, we obtain τ1,bσiτ −1 1,b = τi+rb, and hence τ1,bt lτ−11,b = l−1∏ i=0 σi+rb = t l. finally, we obtain τa,bt lτ−1a,b = τ1,bτa,0t lτ−1a,0τ −1 1,b = τ1,bt laτ−11,b = t la. this gives α = a, so that τa,b ∈ nsn(< t l >). 4.1. quasi-cyclic codes of length prl we now consider quasi-cyclic codes of length n = prl with p a prime number such that (p,l) = (p,q) = 1. in this case, 〈t l〉 ≤ aut(c) is a subgroup of order pr. hence it is contained in a p-sylow subgroup p . lemma 4.2. let c and c′ be two quasi-cyclic codes of length n = prl, and p be a p-sylow subgroup of aut(c) such that t l ∈ p. then c and c′ are equivalent only if they are equivalent by the elements of the set h′(p) = {σ ∈ sn|σ−1t lσ ∈ p}. proof. since c and c′ are equivalent, there exists a permutation σ ∈ sn such that c′ = σ(c). this gives the following relationship between the automorphism groups aut(c) and aut(c′) aut(c′) = σaut(c)σ−1. (7) let p be a sylow subgroup of aut(c). then from (7) we have that σpσ−1 = p ′′ is a sylow p-subgroup of aut(c′). from sylow’s theorem, there exists τ ∈ aut(c′) such that τp ′τ−1 = p ′′ . we can assume that 〈t l〉 ≤ p ′ since 〈t l〉 is a p-group. let γ = τ−1σ, then γ is an isomorphism between c and c′ because γ(c) = τ−1σ(c) = τ−1c′ = c′. then γ−1t lγ = σ−1τt lτ−1σ ∈ σ−1p ′′ σ = p as τt lτ−1 ∈ τp ′τ−1, and hence γ ∈ h′(p). it is obvious that if p = 〈t l〉, then we have nsn(〈t l〉) = h′(〈t l〉). the following proposition gives other properties of h′(p). 267 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 proposition 4.3. let p be a sylow p-subgroup of aut(c). then the group h′(p) has the following properties: (i) if p = 〈t l〉, then nsn(〈t l〉) = h′(〈t l〉), (ii) nsn(〈t l〉) ⊂ h′(p), (iii) nsn(p) ⊂ h′(p). proof. the first property is obtained from the definition of h′(p). to prove the second property we consider a permutation σ in nsn(〈t l〉), the normalizer of 〈t l〉 in sn. then permutation σ satisfies σ−1〈t l〉σ = 〈t l〉⊂ p . hence, we have that nsn(〈t l〉) ⊂ h′(p). (8) now consider nsn(p), the normalizer of p in sn. the permutation σ ∈ nsn(p) shows that σ−1pσ = p . thus for t l ∈ p we have σ−1t lσ ∈ p , so that nsn(p) ⊂ h ′(p). (9) corollary 4.4. the set h′(p) satisfies ag(n) ⊂ h′(p). proof. from proposition 4.1 we have ag(n) ≤ nsn(〈t l〉). furthermore, proposition 4.3 gives that nsn(〈t l〉) ⊂ h′(p), which completes the proof. from corollary 4.4, we have ag(n) ⊂ h′(p). as the multipliers are elements of ag(n), this proves that two quasi-cyclic codes of length n can be equivalent by a multiplier. this extends the results on 1-generator quasi-cyclic codes in [7]. references [1] b. alspach, t. d. parson, isomorphism of circulant graphs and digraphs, discrete math. 25(2) (1979) 97–108. [2] l. babai, p. codenotti, j. a. groshow, y. qiao, code equivalence and group isomorphism, in proc. acm-siam symp. on discr. algorithms, san francisco, ca, (2011) 1395–1408. [3] n. brand, polynomial isomorphisms of combinatorial objects, graphs combin. 7(1) (1991) 7–14. [4] k. guenda, t. a. gulliver, on the permutation groups of cyclic codes, j. algebraic combin. 38(1) (2013) 197–208. [5] m. hall, jr., the theory of groups, macmillan, new york, 1970. [6] w. c. huffman, v. job, v. pless, multipliers and generalized multipliers of cyclic objects and cyclic codes, j. combin. theory ser. a 62(2) (1993) 183–215. [7] s. ling, p. solé, on the algebraic structure of quasi-cyclic codes iii: generator theory, ieee trans. inform. theory 51(7) (2005) 2692–2700. [8] r. j. mceliece, a public-key cryptosystem based on algebraic coding theory, dsn progress report 42-44, (1978) 114–116. [9] a. otmani, j.–p. tillich, l. dallot, cryptanalysis of a mceliece cryptosystem based on quasi-cyclic ldpc codes, in proc. conf. on symbolic computation and crypt., beijing, china, (2008) 69–81. [10] p. p. palfy, isomorphism problem for relational structures with a cyclic automorphism, european j. combin. 8(1) (1987) 35–43. 268 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 http://dx.doi.org/10.1016/0012-365x(79)90011-6 http://dx.doi.org/10.1016/0012-365x(79)90011-6 http://www.ams.org/mathscinet-getitem?mr=2858409 http://www.ams.org/mathscinet-getitem?mr=2858409 http://dx.doi.org/10.1007/bf01789458 http://dx.doi.org/10.1007/s10801-012-0399-4 http://dx.doi.org/10.1007/s10801-012-0399-4 http://dx.doi.org/10.1016/0097-3165(93)90043-8 http://dx.doi.org/10.1016/0097-3165(93)90043-8 https://doi.org/10.1109/tit.2005.850142 https://doi.org/10.1109/tit.2005.850142 https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19780016269.pdf#page=123 https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19780016269.pdf#page=123 http://dx.doi.org/10.1016/s0195-6698(87)80018-5 http://dx.doi.org/10.1016/s0195-6698(87)80018-5 k. guenda, t. a. gulliver / j. algebra comb. discrete appl. 4(3) (2017) 261–269 [11] n. sendrier, finding the permutation between equivalent linear codes: the support splitting algorithm, ieee trans. inform. theory 46(4) (2000) 1193–1203. [12] n. sendrier, d.e. simos, how easy is code equivalence over fq?, in proc. int. workshop on coding theory and crypt., bergen, norway, 2013. [13] n. sendrier, d. e. simos, the hardness of code equivalence over fq and its application to codebased cryptography, in p. gaborit (ed.), post-quantum cryptography, springer lecture notes in computer science 7932, limoges, france (2013) 203–216. 269 http://orcid.org/0000-0002-1482-7565 http://orcid.org/0000-0001-9919-0323 https://doi.org/10.1109/18.850662 https://doi.org/10.1109/18.850662 https://hal.inria.fr/hal-00790861/ https://hal.inria.fr/hal-00790861/ http://dx.doi.org/10.1007/978-3-642-38616-9_14 http://dx.doi.org/10.1007/978-3-642-38616-9_14 http://dx.doi.org/10.1007/978-3-642-38616-9_14 introduction preliminaries equivalence of cyclic codes equivalence of quasi-cyclic codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.284959 j. algebra comb. discrete appl. 4(2) • 165–180 received: 14 june 2015 accepted: 21 february 2016 journal of algebra combinatorics discrete structures and applications on the graded identities of the grassmann algebra∗ review article lucio centrone abstract: we survey the results concerning the graded identities of the infinite dimensional grassmann algebra. 2010 msc: 16r10, 16p90, 16s10, 16w50 keywords: graded polynomial identities, grassmann algebra 1. introduction all algebras we refer to are assumed to be associative with unit and all fields are assumed to be infinite unless explicitely written. moreover, every group is an abelian group unless explicitely written. the grassmann algebra is the algebra of the wedge product, also called an alternating algebra or exterior algebra. it fits in several places in mathematics and, in general, in sciences as well. we recall that if f is a field and x = {x1,x2, . . .} is a countable infinite set of variables, let f〈x〉 be the free associative algebra freely generated by x over f. we shall refer to the elements of f〈x〉 as polynomials in the set of variables x. if a is an f-algebra, we say that f(x1, . . . ,xn) ∈ f〈x〉 is a polynomial identity of a if f(a1, . . . ,an) = 0 for any a1, . . . ,an ∈ a. if a has a non-trivial polynomial identity we say that a is a polynomial identity algebra or pi-algebra and we denote by t(a) the set of all polynomial identities satisfied by a. it is well known that t(a) is an ideal of f〈x〉 invariant under all endomorphisms of f〈x〉, i.e., it is a t-ideal called the t-ideal of a. concerning the mathematical aspects of the grassmann algebra and, in particular, the algebraic ones, the grassmann algebra e generated by an infinite dimensional vector space and its identities, play an important role in the structure theory of kemer on varieties of associative algebras with polynomial identities [21, 22]. more precisely, kemer proved that any associative pi-algebra over a field f of characteristic zero satisfies the same identities (is pi-equivalent) of the grassmann envelope of a finite dimensional associative superalgebra, i.e., they have the same t-ideal. moreover for any associative algebra a, t(a) is finitely generated as a t-ideal. ∗ partially supported by fapesp grant 2013/06752-4, partially supported by fapesp grant 2015/08961-5. lucio centrone; imecc, universidade estadual de campinas, rua sérgio buarque de holanda 651, campinas (sp), brazil (email: centrone@ime.unicamp.br). 165 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 one of the main goals of the theory of pi-algebras is finding a complete set of generators of the t-ideal of a given algebra. for example, it is well known that if a is a commutative unitary algebra, then t(a) is generated by the lie commutator [x,y] := xy − yx. the problem turns out to be very hard even for finite dimensional algebras. in fact, if we consider mn(f) the matrix algebra over a field f of characteristic 0, we know that its t-ideal is well known only for the case n = 2 (see [31] and [15]), provided that the case n = 1 is trivial. if f has positive characteristic p 6= 2 we have a description of a finite basis of identities for m2(f) (see [24]). we note that some further partial results for m2(f) in the case of characteristic 2 were obtained in [16] and [23] but it is still unknown if the t-ideal of m2(f) is finitely generated or not in this case. let us consider a generalization of the definition of polynomial identity for a g-graded algebra a = ⊕ g∈g a g, where g is any group. if we specialize the variable from x with a g-degree, to say ‖ · ‖, we obtain a set f〈x|g〉 of “graded polynomials”. of course we may generalize the notion of polynomial identity with the notion of g-graded polynomial identity in a natural way. we say that f(x1, . . . ,xn) ∈ f〈x|g〉 is a g-graded polynomial identity of a if f(a1, . . . ,an) = 0 for all a1, . . . ,an ∈ ⋃ g∈g a g such that ai ∈ a‖xi‖. as well as in the ungraded (or ordinary) case we denote by tg(a) the set of all g-graded polynomial identities satisfied by a. it is well known that tg(a) is an ideal of f〈x|g〉 invariant under all the g-graded endomorphisms of f〈x|g〉, i.e., it is a tg-ideal called the tg-ideal of a. in [38] vasilovsky gives a complete description of the tzn-ideal of mn(f) for a particular zn-grading and for all n ≥ 2 in characteristic 0 whereas azevedo in [3] obtained the same results without any restriction on the ground field. we recall that the works by vasilovsky and by azevedo are a generalization of the work by di vincenzo for 2 by 2 matrices (see [11]). as well as in the ordinary case, if a is an associative algebra graded by a finite group g, then tg(a) is finitely generated as a tg-ideal (see [1]). coming back to the grassmann algebra e, we know that in the ordinary case t(e) is generated by the triple commutator [x,y,z] := [[x,y],z] as shown in the papers by latyshev [27] and [26] by krakovski and regev. for this purpose, we want to point out that the latter two papers deal with characteristic 0 only even if the argument used in [27] is still valid in positive characteristic as the argument used in [17] theorem 5.1.2. for this purpose, see also the paper by giambruno and koshlukov [18]. in light of what we said until now and with the intent of a future use, in this paper we want to collect the results concerning the graded identities (and other related topics) of the grassmann algebra trying to be as exaustive as possible. we recall that in [36] the author collected the results related to z2-gradings of e. 2. graded pi-algebras we introduce the terminology for the study of graded polynomial identities. we start off with the following definition. in the sequel every algebra is associative with unit and every field is infinite unless explicitely written. definition 2.1. let g be a group and a be an algebra over a field f. we say that the algebra a is ggraded if there exist subspaces ag, g ∈ g such that a = ⊕ g∈g a g as a vector space and for all g, h ∈ g, one has agah ⊆ agh. it is easy to note that if a is any element of a it can be uniquely written as a finite sum a = ∑ g∈g ag, where ag ∈ ag. we shall call the subspaces ag the g-homogeneous components of a. accordingly, an element a ∈ a is called g-homogeneous if exists g ∈ g such that a ∈ ag. if b ⊆ a is a subspace of a, b is g-graded if and only if b = ⊕ g∈g(b ∩a g). analogously one can define g-graded algebras, subalgebras, ideals, etc. we say that a g-grading on a is homogeneous if there exists a linear basis b of a such that every element of b is a homogeneous element of a. let {xg | g ∈ g} be a family of disjoint countable sets of indeterminates. set x = ⋃ g∈g x g and denote by f〈x|g〉 the free associative algebra freely generated by x over f . an indeterminate x ∈ x is said to be of homogeneous g-degree g, written ‖x‖ = g, if x ∈ xg. we always write xg if x ∈ xg. the homogeneous g-degree of a monomial m = xi1xi2 · · ·xik is defined to be ‖m‖ = ‖xi1‖·‖xi2‖·· · ··‖xik‖. for 166 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 every g ∈ g, denote by f〈x|g〉g the subspace of f〈x|g〉 spanned by all monomials having homogeneous g-degree g. notice that f〈x|g〉gf〈x|g〉g ′ ⊆ f〈x|g〉gg ′ for all g,g′ ∈ g. thus f〈x|g〉 = ⊕ g∈g f〈x|g〉g is a g-graded algebra. the elements of the g-graded algebra f〈x|g〉 are referred to as g-graded polynomials or, simply, graded polynomials. in order to simplify the notation we shall sometimes use yi’s to denote variables of homogeneous degree 1g, zi’s to denote variables of homogenous degree different than 1g and xi’s for any variables without distinguish their homogenous degree. definition 2.2. if a is a g-graded algebra, a g-graded polynomial f(x1, . . . ,xn) is said to be a graded polynomial identity of a if f(a1,a2, . . . ,an) = 0 for all a1,a2, . . . ,an ∈ ⋃ g∈g a g such that ak ∈ a‖xk‖, k = 1, . . . ,n. we shall write f ≡ 0 in order to say that f is a graded polynomial identity for a. if a satisfies a non-trivial graded identity we say that a is a pi g-graded algebra. when the algebra a is graded by the trivial group (in fact it has no grading) we refer to polynomial identities of a and t-ideal of a. given an algebra a graded by a group g, we define tg(a) := {f ∈ f〈x|g〉|f ≡ 0 on a}, the set of g-graded polynomial identities of a. definition 2.3. an ideal i of f〈x|g〉 is said to be a tg-ideal if it is invariant under all fendomorphisms ϕ : f〈x|g〉→ f〈x|g〉 such that ϕ(f〈x|g〉g) ⊆ f〈x|g〉g for all g ∈ g. hence tg(a) is a tg-ideal of f〈x|g〉. on the other hand, it is easy to check that all tg-ideals of f〈x|g〉 are of this type. if s ⊆ f〈x|g〉, we shall denote by 〈s〉tg the tg-ideal generated by the set s, i.e., the smallest tg-ideal containing s. moreover, given a set of polynomials s ⊆ f〈x|g〉, we say that i is the tg-ideal generated by s, if i is the smallest tg-ideal containing s. in this case we say that s is a basis for i or that the elements of i follow from or are consequences of the elements of s. if s is a finite set generating the t-ideal i we say i is finitely based. notice that being a basis for a t-ideal does not imply being a minimal basis. the theory of pi g-graded algebras in characteristic zero passes through the representation theory of the symmetric group. we consider the following sn-modules. definition 2.4. let pgn = span〈x g1 σ(1) x g2 σ(2) · · ·xgn σ(n) |gi ∈ g,σ ∈ sn〉, then the elements in pgn are called multilinear polynomials of degree n of f〈x|g〉. it turns out that pgn is a left sn-module under the natural left action of the symmetric group sn; we denote the sn-character of the factor module pgn /(p g n ∩tg(a)) by χgn (a), and by cgn (a) its dimension over f . we say that ( χgn (a) ) n∈n is the g-graded cocharacter sequence of a ( cgn (a) ) n∈n is the g-graded codimension sequence of a 167 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 now, for lg1, . . . , lgr ∈ n let us consider the blended components of the multilinear polynomials in the indeterminates labeled as follows: xg11 , . . . ,x g1 lg1 , then xg2lg1 +1, . . . ,x g2 lg1 +lg2 and so on. we denote this linear space by pglg1,...,lgr . of course, this is a left slg1 ×···×slgr -module. we shall denote by χ g lg1,...,lgr (a) the character of the module pglg1,...,lgr (a)/(p g lg1,...,lgr (a) ∩tg(a)) and by cglg1,...,lgr (a) its dimension. when the algebra a is graded by the trivial group (in fact it has no grading) we refer to cocharacter sequence of a and codimension sequence of a. for a more detailed account on pi-algebras, see chapters 1 and 3 of [20] or [17]. since the ground field f is infinite, a standard vandermonde-argument yields that a polynomial f is a g-graded polynomial identity for a if and only if its multihomogeneous components are identities as well. moreover, since char(f) = 0, the well known multilinearization process shows that the tg-ideal of a g-graded algebra a is determined by its multilinear polynomials, i.e. by the various pglg1,...,lgr (a). we remark that, given the cocharacter χglg1,...,lgr (a), the graded cocharacter χ g n (a) is known as well. more precisely, the following is due to di vincenzo (see [12], theorem 2). proposition 2.5. let a be a g-graded algebra with graded cocharacter sequences χglg1,...,lgr (a). then χgn (a) = ∑ (lg1, . . . , lgr ) lg1 + . . . + lgr = n χglg1,...,lgr (a)↑sn. moreover cgn (a) = ∑ (lg1, . . . , lgr ) lg1 + . . . + lgr = n ( n lg1, . . . , lgr ) cglg1,...,lgr (a). actually, if a is a g-graded pi-algebra, it is more convenient studying pgn (a) than the whole tg(a) ∩ pgn (a). in fact, the latter grows factorially while a graded generalization of a well celebrated work by regev (see [32]) says that pgn (a) grows at most exponentially. 3. the grassmann algebra in this section we define the grassmann algebra and we list some results about its ordinary polynomial identities and the polynomial identities of some related algebras. definition 3.1. let x = {x1,x2, . . .} and let us consider f〈x〉. if i is the two-sided ideal of f〈x〉 generated by the set of polynomials {xixj + xjxi|i,j ≥ 1}, we set e := f〈x〉/i. then we say that e is the infinite dimensional grassmann algebra. indeed if the set x is finite we define analogously the finite dimensional grassmann algebra. we denote by l the vector space spanned by the ei := xi + i’s and we call it underlying vector space of e and we write e = e(l). moreover, if w = ei1 · · ·eit is a monomial in the ei’s we say t is the length of w and we write l(w) = t. the set of different ei’s appearing in w is called support of w. we observe that e has the following presentation: e = 〈1,e1,e2, . . . |eiej = −ejei, for all i,j ≥ 1〉. remark 3.2. of course over a field of characteristic 2, the grassmann algebra turns out to be commutative. hence in the sequel every field is supposed to have characteristic p 6= 2. 168 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 the set b = {1,ei1 · · ·eik|1 ≤ i1 < · · · < ik} is a basis of e over f. it is convenient to write e in the form e = e0 ⊕e1 where e0 := span{1,ei1 · · ·ei2k|1 ≤ i1 < · · · < i2k,k ≥ 0}, e1 := span{1,ei1 · · ·ei2k+1|1 ≤ i1 < · · · < i2k+1,k ≥ 0}. it is easily checked that e0e0 + e1e1 ⊆ e0 and e0e1 + e1e0 ⊆ e1. this says that the decomposition e = e0 ⊕e1 is a z2-grading of e, called natural or canonical z2-grading of e. notice that e0 coincides with the center of e whereas it is not true if l has finite dimension. for example, if l has dimension d, d odd, then e1 · · ·ed annihilates any element of e, then it is central but it does not fit in e0. the next is a well known fact. proposition 3.3. e satisfies the identity [[x,y],z] ≡ 0. let us suppose e over a field of characteristic zero, then the triple commutator is the only generator of t(e). in fact we have the following. theorem 3.4. (latyshev [27], krakowski and regev [26]) the t-ideal of e is generated by the polynomial [x1,x2,x3]. we list some results concerning the polynomial identities of some algebras related to e. unless otherwise stated the base field is supposed to be of characteristic 0. in what follows we shall denote by utn(r) the f-algebra of n×n upper triangular matrices with entries of the f-algebra r. theorem 3.5. (berele and regev [6]) the t-ideal of utn(e) is generated by the polynomial [x1,x2,x3] · · · [x3n−2,x3n−1,x3n]. more precisely, in [6] theorem 2.8 the authors give a more general version of the previous result. we also recall that in [28] latyshev proved that t(utn(e)) is finitely based. theorem 3.6. (popov [30]) the t-ideal of e ⊗e is generated by the polynomials [x1,x2, [x3,x4],x5], [[x1,x2],x 2 2]. we also want to point out that in [30] the author described the structure of the relatively free algebra of e ⊗e. if we consider the finite dimensional grassmann algebra, we have the next result by di vincenzo. theorem 3.7. (di vincenzo [10]) let e be the grassmann algebra generated by the k dimensional underlying vector space lk. then the t-ideal of e is generated by the polynomials [x1,x2,x3], [x1,x2] · · · [x2t−1,x2t], where t = [k/2] + 1 and [a] is the integer part of a. even if the characteristic zero is the most investigated case, we also have several works in positive characteristic. here we cite the analog of theorem 3.4 for any infinite field. 169 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 theorem 3.8. the t-ideal of e is generated by the polynomial [x1,x2,x3]. notice that the t-ideal of e does not depend on the characteristic of the field. this is not the case of the grassmann algebra over a finite field or, as we shall see later, of the (z2)-graded case. we may cite the work by regev [34] which gave a lot of information about the t-ideal of e, introducing the so called class identities whose definition goes far from the intent of this survey. by the way it will be interesting to note the following. proposition 3.9. let e be the infinite dimensional grassmann algebra over a field of characteristic p and let us consider e∗ = e −{1}, then e∗ satisfies the identity xp. we close this section by citing a famous result by olsson and regev about the cocharacters and the codimensions of the infinite dimensional grassmann algebra in characteristic 0. we recall that a partition of the non-negative integer n is a sequence of integers λ = (λ1, . . . ,λr) such that λ1 ≥ ···≥ λr > 0 and λ1 + · · ·+ λr = n. in this case we shall write λ ` n. we assume two partitions λ = (λ1, . . . ,λr) and µ = (µ1, . . . ,µs) to be equal if r = s and λ1 = µ1, . . . ,λr = µr. when λ = (λ1, . . . ,λk1+···+kp) and λ1 = · · · = λk1 = µ1, . . . ,λk1+···+kp−1+1 = · · · = λk1+···+kp = µp, we accept the notation λ = (µk11 , . . . ,µ kp p ). definition 3.10. given a partition λ = (λ1, . . . ,λr), we associate to λ its young diagram [λ] having r rows such that its i-th row contains λi squares. moreover we denote by λ′ the partition associated to the transpose diagram of [λ]. theorem 3.11. (olsson and regev [29] for cocharacters and krakowsky and regev [26] for codimensions) the cocharacter sequence of the grassmann algebra is the following: χn(e) = n∑ k=1 (k,1n−k), where n ≥ 1. moreover its codimension sequence is such that for each n ≥ 1 we have cn(e) = 2n−1. 4. z2-graded identities in this section we collect the results concerning the z2-graded identities of e based on the works by the author [7] and di vincenzo and da silva [14]. for the sake of completeness we want to cite the papers [37] and [2] by anisimov in which the author computes the sequence of involutive codimensions of grassmann algebra for some special involutions (see [37]), then generalized in [2]. in the latter paper the author gives also an explicit form of the sequence of involutive codimensions of the grassmann algebra 170 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 for arbitrary involution (exept one case) and for some other groups. the work by anisimov has been completed by da silva in [35] for the remaining case. it is also interesting to say that the non-homogeneous g-gradings on e are unknown as well as their corresponding ideals of graded identities. in the sequel for g-grading we mean homogeneous g-grading. in order to simplify the notation we shall use the symbols y’s for variables of z2-degree 0 and the symbols z’s for variables of z2-degree 1 as declared in section 2. let e = e(l) be the infinite dimensional grassmann algebra with underlying vector space l and let g be an abelian group. if g is finite and bl = {e1,e2, . . .} is a basis of l, let ϕ : bl → g be any map. then ϕ induces a homogeneous g-grading on e and viceversa. in this section we consider homogeneous z2-gradings only. let us consider the map ϕ : l → z2 such that ei 7→ 1. the map ϕ gives out the natural grading over e. in this case, let e0 be the homogeneous component of z2-degree 0 and let e1 be the component of degree 1. it is easy to see that e0 is the center of e and ab + ba = 0 for all a,b ∈ e1. this means that e satisfies the following graded polynomial identities: [y1,y2], [y1,z1], z1z2 + z2z1. moreover, the latter generates the whole tz2-graded ideal of e endowed with its natural z2-grading in the case of characteristic 0. in fact we have the following. theorem 4.1. (giambruno, mischenko, zaicev [19]) the tz2-graded ideal of e endowed with its natural z2-grading is generated by the polynomials [y1,y2], [y1,z1], z1z2 + z2z1, if the characteritic of the ground field is 0. now, let us consider the z2-gradings over e induced by the maps degk∗, deg∞, and degk, defined respectively by degk∗(ei) = { 1 for i = 1, . . . ,k 0 otherwise, deg∞(ei) = { 1 for i odd 0 otherwise, degk(ei) = { 0 for i = 1, . . . ,k 1 otherwise. we shall denote by ek∗, e∞, ek the grassmann algebra endowed with the z2-grading induced by the maps degk∗, deg∞, and degk. we denote by ed any of the superalgebras ek∗, e∞, ek without distinguish them. let f = z ri1 i1 · · ·zrisis [zj1,zj2] · · · [zjt−1,zjt] and consider the set s := {different homogeneous variables appearing in f}. if h = |s|, then s = {zi1, . . . ,zih}. we consider now t = {j1, . . . ,jt}⊆ s 171 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 and let us denote the previous polynomial by ft (zi1, · · · ,zih). for m ≥ 2 let gm(zi1, . . . ,zih) = ∑ t |t| even (−2)− |t| 2 ft , moreover we set g1(z1) = z1. let f be an infinite field of characteristic p > 2, then we have the next results (see [7]). theorem 4.2. let k ∈ n. if p > k, then all z2-graded polynomial identities of ek∗ are consequences of the graded identities: [x1,x2,x3], z1 · · ·zk+1. on the other side, if p ≤ k, all z2-graded polynomial identities of ek∗ are consequences of the graded identities: [x1,x2,x3], z1 · · ·zk+1, zp. theorem 4.3. all the z2-graded polynomial identities of e∞ are consequences of the graded identities: [x1,x2,x3], z p. theorem 4.4. let k ∈ n and set x = y ∪ z. then if p > k all the z2-graded identities of ek are consequences of the graded identities: • [x1,x2,x3], • [y1,y2] · · · [yk−1,yk][yk+1,x] (if k is even) • [y1,y2] · · · [yk,yk+1] (if k is odd) • gk−l+2(z1, . . . ,zk−l+2)[y1,y2] · · · [yl−1,yl] (if l ≤ k) • [gk−l+2(z1, . . . ,zk−l+2),y1][y2,y3] · · · [yl−1,yl] (if l ≤ k, l is odd) • gk−l+2(z1, . . . ,zk−l+2)[z,y1][y2,y3] · · · [yl−1,yl] (if l ≤ k, l is odd) if p ≤ k we have to add to the list above the identity • zp from theorem 4.4 it turns out that a minimal basis of the z2-graded identities of e either in positive characteristic or in characteristic zero is generated by the polynomials [y1,y2], [y1,z1], z1z2 + z2z1. the case of characteristic zero was the first case which has been considerated and it was completely solved by di vincenzo and da silva in [14]. their generators in the three cases are the ones above without the polynomial zp. we observe that the identity zp comes from the fact that the e1 component of the grassmann algebra lies in e∗, then we use proposition 3.9. in [14] the authors described the sequence of z2-graded cocharacters and codimensions in the case of characteristic 0. we collect below their results. we shall adopt the following notation. let λs = (l−s,1s) ` l, µt = (1+t,1m−t−1) ` m be the hook partition of l and m with leg s and arm t respectively. 172 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 theorem 4.5. let k ∈ n, then for each n ∈ n the n-th z2-graded codimension of ek∗ is cz2n (ek∗) = 2 n−1 k∑ t=0 ( n t ) . let k ∈ n, then the z2-graded cocharacter sequence of ek∗ is given by χl,0(ek∗) = l−1∑ s=0 λs ⊗∅ if l ≥ 1; χ0,m(ek∗) = m−1∑ t=0 ∅⊗µt if m ≥ 1; χl,m(ek∗) = l−1∑ s=0 m−1∑ t=0 2(λs ⊗µt) if l ≥ 1, 1 ≤ m ≤ k; χl,m(ek∗) = 0 if l ≥ 0, m ≥ k + 1. theorem 4.6. for each n ∈ n the n-th z2-graded codimension of e∞ is cz2n (e∞) = 4 n−1 2 . the z2-graded sequence of e∞ is given by χl,0(e∞) = l−1∑ s=0 λs ⊗∅ if l ≥ 1; χ0,m(ek∗) = m−1∑ t=0 ∅⊗µt if m ≥ 1; χl,m(ek∗) = l−1∑ s=0 m−1∑ t=0 2(λs ⊗µt) if m,l ≥ 1; theorem 4.7. let k ∈ n, then for each n ∈ n we have cn−m,m(ek) = k∑ t=0 ( n−1 t ) if m ≥ 1; cn,0(ek) = e(k)∑ t=0 ( n−1 t ) , where e(k) = { k if k is even k −1 if k is odd. 173 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 let k ∈ n, then the z2-graded cocharacter sequence of ek is given by χl,0(ek) = l−1∑ s=0 λs ⊗∅ if l ≤ k; χl,0(ek) = e(k)∑ s=0 λs ⊗∅ if l ≥ k + 1; χ0,m(ek) = m−1∑ t=0 ∅⊗µt if t ≤ k; χ0,m(ek) = e(k)∑ t=0 ∅⊗µt if t ≥ k + 1; χl,m(ek) = l−1∑ s=0 m−1∑ t=0 ms,t(λs ⊗µt) if l,m ≥ 1, where ms,t =   2 if s + t ≤ k −1 1 if s + t = k 0 otherwise. 5. g-graded identities of e now we consider the more general case of a homogeneous g-grading of e, where g is a finite abelian group. we show that in order to study the g-graded identities of e we may reduce to g′-gradings, where g′ is a group having a smaller number of elements than g. we give the proofs of the main results. the complete contents related to this section may be found in [8]. we also recall that from now on every field is supposed to be of characteristic 0. finally, if h / g we shall denote by gh or simply by g the coset of g modulo h, where g ∈ g. let us consider the following homomorphism between free graded algebras π : f〈x|g〉→ f〈y |g/h〉, such that for every g ∈ g and for every i ∈ n, π(xgi ) = y gh i , where h is a subgroup of g. definition 5.1. let g be a finite abelian group and suppose e is g-graded. we say that the subgroup h of g has the property p when for any h ∈ h, eh has infinitely many elements of even length with pairwise disjoint support. the importance of this property is given by the following proposition. proposition 5.2. let h < g having the property p and let f ∈ f〈x|g〉 be a multilinear polynomial. then f ∈ tg(e) if and only if π(f) ∈ tg/h(e). 174 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 proof. we have to prove just the only if part. let f = f(x g1 1 ,x g1 2 , . . . ,x g1 lg1 , . . . ,x gr∑r−1 i=1 lgi +1 , . . . ,x gr∑ r i=1 lgi ) ∈ tg(e) and let f = π(f). let ϕ be any g/h-graded substitution, hence ϕ(yghj ) = ∑ h∈h a gh j , and by the multilinearity of f, we can consider only substitutions ϕ such that yghj 7→ a gh j , for some h ∈ h and for any j. now, we observe that every homogeneous component eh has infinitely many elements of even length with pairwise disjoint supports because h satisfies the property p, then for every j and for every h ∈ h exists bh −1 j of even length such that ‖b h−1 j ‖ = h −1. for every h ∈ h, wgj = a gh j b h−1 j is a homogeneous element of degree g in the g-grading of e. let us consider a new substitution ψ such that xgj 7→ w g j . this is a g-graded substitution. now, since f ∈ tg(e), 0 = f(w g1 1 , . . . ,w gr lgr ) = ∏ h∈h,j b h−1 j ·f(a gh j ) because the bh −1 j ’s are in z(e) and this implies f(a gh j ) = 0. we observe that if lg is infinite dimensional and |h| = n is odd, then h = 〈g〉 has the property p. moreover even if g is a finite abelian group and h = 〈g|dimf lg = ∞ and o(g) is odd〉, then h has the property p. we shall adopt the following notation: if h is a subgroup of g, we shall denote by g the translate gh ∈ g/h. we have the following. theorem 5.3. let g be a finite abelian group of odd order and let h = 〈g|dimf lg = ∞〉. then the following properties hold: 1. for any multilinear polynomial f(x1, . . . ,xn) ∈ f〈x〉 one has that f ∈ tg(e) if and only if π(f) ∈ tg/h(e). 2. in the quotient grading of e, lg is infinite dimensional if and only if g = 1g/h. now let us consider the following subsets of g : i = {g ∈ g|dimf lg = ∞}, i1 = {g ∈i|o(g) is odd}, i2 = i−i1 and i3 = {g2|g ∈i2}−i1. we have the following. theorem 5.4. let g be a finite abelian group and let h = 〈g|g ∈i1∪i3〉. then the following properties hold: 1. for any multilinear polynomial f = f(x1, . . . ,xn) ∈ f〈x〉 one has f ∈ tg(e) if and only if π(f) ∈ tg/h(e). 2. in the quotient grading of e, if lg is infinite dimensional, then g2 = 1g/h. 175 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 proof. (1). let h ∈ h, then there exist a1, . . . ,ar ∈i1, b1, . . . ,bs ∈i3 and positive integers such that h = am11 · · ·a mr r b mr+1 1 · · ·b mr+s s . let ar+1, . . . ,ar+s ∈i2 such that bi = a2r+i, then dimf l ai = ∞ for any i = 1, . . . ,r + s. let us denote by ei the grassmann algebra generated by the subspace la mi i . for any i = 1, . . . ,r + s, ei contains infinite elements wi1,w i 2, . . . ,w i m, . . . of even length with pairwise disjoint supports. moreover, for all m ≥ 1 we have that ∥∥wim∥∥ = amii if i = 1, . . . ,r and ∥∥wim∥∥ = bmii−r for i = r + 1, . . . ,r + s. we consider in eh the elements um = w1m · · ·wr+sm , m ≥ 1; clearly the elements {um|m ≥ 1} have pairwise disjoint supports and they have even length. now h has the property p and the assertion comes by proposition 5.2. (2). let g = gh ∈ g/h be such that lg = ⊕ h∈h l gh is infinite dimensional. since g is finite there exists g′ ∈ gh such that lg ′ is infinite dimensional. if o(g′) is odd, then g′ ∈ h and so gh = g′h = 1g/h. if o(g′) is even, then g′2 ∈ h and so (gh)2 = (g′h)2 = 1g/h. in light of theorems 5.3 and 5.4, we list the results about the g-graded identities of e in the case dimf l 1g = ∞. theorem 5.5. let g = {g1, . . . ,gr} be a finite abelian group with g1 = 1g. suppose that lg1 has infinite dimension. let lg1, lg2, . . . , lgr ∈ n such that lg1 + lg2 + . . . + lgr = m. then plg1,...,lgr ⊆ tg(e) or for any f ∈ plg1,lg2,...,lgr one has f(x g1 1 , . . . ,x g1 lg1 , . . . ,x gr∑r−1 i=1 lgi +1 , . . . ,x gr∑ r i=1 lgi ) ∈ tg(e) if and only if f(x1, . . . ,xm) ∈ t(e). theorem 5.6. let g = {g1, . . . ,gr} be a finite abelian group with g1 = 1g. let l be a g-homogeneous vector space over l such that dimf lg1 = ∞ and dimf lgi = ki < ∞, if i 6= 1. if e = e(l) is the grassmann algebra generated by l, then tg(e) is generated as a tg-ideal by the following polynomials: 1. [u1,u2,u3] for any choice of the g-degree of the variables u1,u2,u3. 2. monomials of p0,t2,...,tr such that ∑r i=2 ti = 1 + ∑r i=2 ki 3. monomials of p0,t2,...,tr such that ∑r i=2 ti < 1 + ∑r i=2 ki and p0,t2,...,tr ⊆ tg(e). as a consequence of the previous results, we have, up to combinatorics, the following description of the g-graded cocharacters in the case l1g is infinite dimensional. corollary 5.7. let g = {g1, . . . ,gr} be a finite abelian group with g1 = 1g. if lg1 has infinite dimension and lg1, lg2, . . . , lgr ∈ n such that lg1 + lg2 + . . . + lgr = m, then clg1,...,lgr (e) = 0 or clg1,...,lgr (e) = 2 m−1 and in this last case, plg1,...,lgr (e) and pm(e) are isomorphic slg1 ×···×slgr -modules. 176 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 let us consider now the set s(ϕ) = { (lg1, lg2, . . . , lgr ) ∈ n r|plg1,lg2,...,lgr ⊆ tg(e) } . we note that if l1g is the only homogeneous subspace of l such that dimf l1g = ∞, then s(ϕ) 6= ∅. s(ϕ) allows us to give the complete description of the sequence of the graded cocharacters and codimensions of e. in fact, we have the following proposition. theorem 5.8. let g = {g1, . . . ,gr} be a finite abelian group and l be a g-homogeneous vector space with linear basis {e1,e2, . . .}. let ϕ : bl → g be a map such that |ϕ−1(1g)| = ∞ and consider e, the g-graded grassmann algebra obtained by ϕ. then χglg1,...,lgr (e) = 2|g|−1 ∑lg1−1 a1=0 ∑lg2−1 a2=0 · · · ∑lgr−1 ar=0 λa1 ⊗λa2 ⊗···⊗λar if (lg1, . . . , lgr ) /∈ s(ϕ), where λai is the hook partition of leg ai and arm lgi −ai + 1. moreover cgn (e) = 2 n−1 ∑ (lg1, . . . , lgr ) /∈ s(ϕ) lg1 + . . . + lgr = n ( n lg1, . . . , lgr ) . 6. other results in what follows we recall some results about the (graded) polynomial identities of structures related to the grassmann algebra. we shall consider a g-graded algebra a and the canonical z2-grading of the grassmann algebra e = e0 ⊕e1 over a field of characteristic 0, and compare the g-graded identities of a with the g×z2graded identities of the g×z2-graded algebra a⊗e with homogeneous components given by (a⊗e)(g,i) := ag ⊗ei. notice that the free algebra f〈x|g×z2〉 is both a g-graded algebra and a z2-graded algebra. referring to the z2-grading of f〈x|g×z2〉one defines the map ζ as follows. let m be a multilinear monomial in f〈x|g×z2〉 and let i1 < · · · < ik be the indexes of the variables with odd z2-degree occurring in m. then, for some σ in the symmetric group sk({i1, . . . , ik}), we may write m = m0zσ(i1)m1zσ(i2) · · ·mk−1zσ(ik)mk, where m0, . . . ,mk are multilinear monomials in even variables only and zij are odd variables. then, as in kemer [22], di vincenzo and nardozza [13] define ζ(m) := (−1)σm. note that ζ(ζ(m)) = m. we define a similar map from the free g-graded algebra to the free g×z2-graded algebra. definition 6.1. let j ⊆ n. let ϕj : f〈x〉→ f〈x|g×z2〉 be the unique g-homomorphism defined by the map ϕj(x g) = { x(g,0) if i /∈ j x(g,1) if i ∈ j. also, for a multilinear monomial m ∈ pgn , define ζj(m) := ζ(ϕj(m)). the map ϕj depends on j, of course. we may extend the map ζj by linearity to the space of all g-graded multilinear polynomials pgn . if f ∈ pgn , then ζj(f) is a multilinear element of f〈x|g×z2〉. we have the following result (see theorem 11 of [13]). theorem 6.2. let s be a system of multilinear generators for tg(a). then the system {ζj(f) ∈ f〈x〉 |f ∈ s,j ⊆ n} is a set of multilinear generators for tg×z2(a⊗e). 177 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 if we consider the case of positive characteristic, the previous result is verified for the algebra ut2(e) of upper triangular 2×2 matrices. theorem 6.3. (c. and da silva [9]) let f be a field of characteristic p > 2 and e be graded with its natural z2-grading. let us set s := {[y1,y2], [y1,z1], z1z2 ◦z2z1}, then tz2(ut2(e)) = {ζj(f)|f ∈ s}. we observe that the previous result does not depend on the characteristic of the field. moreover tz2(ut2(e)) = tz2(e)tz2(e) as in the ordinary case. we have the next related result about the g×z2-cocharacters of a⊗e. theorem 6.4. (di vincenzo and nardozza [13]) let n ∈ n and k1, l1, . . . ,kr, lr ∈ n such that ∑r i=1 ki + li = n and consider h = sk1 ×sl1 ×···skr ×slr . if (χgn (a))↓h = ∑ mλ1,µ1,...,λr,µrλ1 ⊗µ1 ⊗···⊗λr ⊗µr, then χg×z2n (a⊗e) = ∑ ∑ i (ki+li)=n ∑ λi ` ki µi ` li mλ1,µ1,...,λr,µrλ1 ⊗µ ′ 1 ⊗···⊗λr ⊗µ ′ r. we give a short account of the structure theory of t-ideals developed by kemer [22]. definition 6.5. the t-ideal s of f〈x〉 is called t-semiprime or verbally semiprime if any t-ideal u such that uk ⊆ s for some k, lies in s, i.e. u ⊆ s. the t-ideal p is t-prime or verbally prime if the inclusion u1u2 ⊆ p for some t-ideals u1 and u2 implies u1 ⊆ p or u2 ⊆ p. let e = e0 ⊕e1 be endowed with its canonical z2-grading, then the vector subspace of ma+b(e), ma,b(e) := {( r s t u ) |r ∈ ma(e(0)),s ∈ ma×b(e(1)),u ∈ mb(e(0)) } is an algebra. the building blocks in the theory of kemer are the polynomial identities of the matrix algebras over the field and over the grassmann algebra and the algebras ma,b(e). in fact, we have the following theorem. theorem 6.6. 1. for every t-ideal u of f〈x〉 there exist a t-semiprime t-ideal s and a positive integer k such that sk ⊆ u ⊆ s. 2. every t-semiprime t-ideal s is an intersection of a finite number of t-prime t-ideals q1, . . . ,qm, s = q1 ∩·· ·∩qm. 3. a t-ideal p is t-prime if and only if p coincides with one of the following t-ideals: t(mn(f)), t(mn(e)), t(ma,b(e)), (0), f〈x〉. we recall that if two algebras a and b satisfy the same polynomial identities we say that a is pi-equivalent to b and denote by a ∼ b. an important corollary to the structure theory of kemer is the tensor product theorem (tpt) which follows from the result by kemer [22]. 178 l. centrone / j. algebra comb. discrete appl. 4(2) (2017) 165–180 theorem 6.7. the tensor product of two verbally prime algebras is pi equivalent to a verbally prime algebra. more precisely, let a,b,c,d ∈ n such that a ≥ b and c ≥ d and f be a field of characteristic 0, then: 1. ma,b(e)⊗e ∼ ma+b(e); 2. ma,b(e)⊗mc,d(e) ∼ mac+bd,ad+bc(e); 3. m1,1(e) ∼ e ⊗e. the remaining pi equivalences follow from the isomorphism of the corresponding algebras. an alternative proof of the tpt can be be found in the paper by regev [33]. in [33] we also have the proof that the tpt is still valid for multilinear identities in the case e is the infinite dimensional grassmann algebra over an infinite field of characteristic p 6= 2. in the papers [4], [5] and [25] the authors deal with graded identities for certain gradings on some of the verbally prime algebras. in the paper [25] the authors constructed an appropriate model for the relatively free algebra in the variety of algebras determined by e⊗e when the field f has characteristic p > 2. this model is the generic algebra of a = f ⊕m1,1(e∗). it turned out that e ⊗e and a satisfy the same graded and hence ordinary polynomial identities. in [4] the authors used the properties of a in order to show that that t(m1,1(e)) ( t(e⊗e). hence the tpt theorem fails in positive characteristic. acknowledgments: the author wants to thank the referee for his/her fruitful comments which improved significantly the quality of the paper. the author wants to thank the organizers of the congress for the hospitality and for the inspiring atmosphere of the event. references [1] e. aldjadeff, a. kanel–belov, representability and specht problem for g−graded algebras, adv. math. 225(5) (2010) 2391–2428. 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mzuzu university, p/bag 201, luwinga, mzuzu 2, malawi (email: augulela@yahoo.com, jar@mzuzu.org). kondwani magamba; malawi university of science and technology, p.o. box 5196, limbe, malawi (email: kmagamba@must.ac.mw). 235 http://orcid.org/0000-0001-8792-6954 http://orcid.org/0000-0003-4025-9802 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 2. preliminaries as this paper is focused on irreducible goppa codes we begin with the definition of irreducible goppa codes. definition 2.1. let q be a power of a prime number and g(z) ∈ fqn [z] be irreducible of degree r. let l = fqn = {ζi : 0 ≤ i ≤ qn − 1}. then an irreducible goppa code γ(l,g) is defined as the set of all vectors c = (c0,c1, ...,cqn−1) with components in fq which satisfy the condition qn−1∑ i=0 ci z − ζi ≡ 0 (mod g(z)). the set l is called the defining set and its cardinality defines the length of γ(l,g). the polynomial g(z) is called the goppa polynomial. if the degree of g(z) is r then the code is called an irreducible goppa code of degree r. the roots of g(z) are contained in fqnr \ fqn. if α is any root of g(z) then it completely describes γ(l,g). chen in [2] described a parity check matrix h(α) for γ(l,g) which is given by h(α) = ( 1 α− ζ0 1 α− ζ1 · · · 1 α− ζqn−1 ) . we will sometimes denote this code by c(α). we next give the definition of extended irreducible goppa codes. definition 2.2. let γ(l,g) be an irreducible goppa code of length qn. then the extended code γ(l,g) is defined by γ(l,g) = {(c0,c1, ...,cqn ) : (c0,c1, ...,cqn−1) ∈ γ(l,g) and ∑qn i=0 ci = 0}. next we define the set which contains all the roots of all possible g(z) of degree r. definition 2.3. we define the set s = s(n,r) as the set of all elements in fqnr of degree r over fqn. any irreducible goppa code can be defined by an element in s. the converse is also true, that is, any element in s defines an irreducible goppa code. since an irreducible goppa code γ(l,g) is determined uniquely by the goppa polynomial g(z), or by a root α of g(z), we define the mapping below. (for further details, see [2].) definition 2.4. the relation πζ,ξ,i defined on s by πζ,ξ,i : α 7→ ζαq i + ξ for fixed i,ζ,ξ where 1 ≤ i ≤ nr, ζ 6= 0,ξ ∈ fqn is a mapping on s. this map sends irreducible goppa codes into equivalent codes and we generalise this as follows: theorem 2.5. (ryan, [8]): if α and β are related by an equation of the form α = ζβq i + ξ for some ζ 6= 0,ξ ∈ fqn, then the codes c(α) and c(β) are equivalent. the map in definition 2.4 can be broken up into the composition of two maps as follows: 1. πζ,ξ defined on s by πζ,ξ : α 7→ ζα + ξ and 2. the map σi : α 7→ αq i , where σ denotes the frobenius automorphism of fqnr leaving fq fixed. 236 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 from these two maps we define the following sets of mappings. definition 2.6. let h denote the set of all maps {πζ,ξ : ζ 6= 0,ξ ∈ fqn}. definition 2.7. let g denote the set of all maps {σi : 1 ≤ i ≤ nr}. the sets of maps h and g together with the operation composition of maps both form groups which act on s. definition 2.8. the action of h on s induces orbits denoted by a(α) where a(α) = {ζα + ξ : ζ 6= 0,ξ ∈ fqn}. we refer to a(α) as an affine set containing α where α is an element of degree r over fqn and ζ,ξ ∈ fqn. since ζ 6= 0,ξ ∈ fqn then to form the set a(α) the number of choices for ζ is qn−1 and ξ has qn choices and so |a(α)| = qn(qn − 1). definition 2.9. let a denote set of all affine sets, i.e., a = {a(α) : α ∈ s}. next, we define a mapping on s which sends extended irreducible goppa codes into equivalent extended irreducible goppa codes. definition 2.10. the relation πζ1,ζ2,ξ1,ξ2,i defined on s by πζ1,ζ2,ξ1,ξ2,i : α 7→ ζ1α qi + ξ1 ζ2αq i + ξ2 fixed i,ζj,ξj where 0 ≤ i ≤ nr, ζj,ξj ∈ fqn, j = 1, 2 and ζ1ξ2 6= ζ2ξ1 is a mapping on s. since the scalars ζj and ξj are defined up to scalar multiplication, we may assume that ζ2 = 1 or ξ2 = 1 if ζ2 = 0. we have the following generalisation: theorem 2.11. (berger, [1]): if πζ1,ζ2,ξ1,ξ2,i(α) = β then c(α) is equivalent to c(β). the map in definition 2.10 can be broken up into the composition of two maps as follows: 1. the map πζ1,ζ2,ξ1,ξ2 defined on s by πζ1,ζ2,ξ1,ξ2 : α 7→ ζ1α+ξ1 ζ2α+ξ2 , and 2. the map σi : α 7→ αq i , where σ denotes the frobenius automorphism of fqnr leaving fq fixed. from these two maps we give the following two definitions. definition 2.12. let f denote the set of all maps {πζ1,ζ2,ξ1,ξ2 : ζj,ξj ∈ fqn, j = 1, 2 and ζ1ξ2 6= ζ2ξ1}. f forms a group under the operation of composition of maps which acts on s. definition 2.13. let α ∈ s. then the orbit in s containing α under the action of f is o(α) = {ζ1α+ξ1 ζ2α+ξ2 : ζj,ξj ∈ fqn,j = 1, 2 and ζ1ξ2 − ζ2ξ1 6= 0}. the cardinality of o(α) is found in [10] and we state it in the theorem: theorem 2.14. for any α ∈ s, |o(α)| = q3n −qn = (qn − 1)(qn)(qn + 1). definition 2.15. let of denote the set of all orbits in s under the action of f, i.e., of = {o(α) : α ∈ s}. observe that of is a partition of the set s. note that g acts on the set of . 237 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 remark 2.16. from now on we take q = 2. it is shown in [9] that each of the sets o(α) in of can be partitioned into 2n + 1 sets. the theorem below provides more details. theorem 2.17. o(α) = a(α) ∪a( 1 α ) ∪a( 1 α+1 ) ∪a( 1 α+ξ1 ) ∪a( 1 α+ξ2 ) ∪ ·· ·∪a( 1 α+ξ2n−2 ) where f2n = {0, 1,ξ1,ξ2, ...,ξ2n−2}. observe that the sets of and a are different. of is a partition on s and also a is a partition on s. the number of elements in a is 2n + 1 times the number of elements in of , i.e., |a| = (2n + 1) ×|of |. g also acts on a = {a(α) : α ∈ s}. 3. counting extended irreducible binary goppa codes of degree 2m 3.1. technique of counting we wish to produce an upper bound on the number of inequivalent extended irreducible binary goppa codes of degree r = 2m. we intend to achieve this by employing the tools developed for counting the non-extended versions. in counting the non-extended irreducible goppa codes we consider the action of h on s. this gives orbits in s denoted by a(α) called affine sets. we then consider the action of g on the set a where a = {a(α) : α ∈ s}. the number of orbits in a under g gives us an upper bound on the number of inequivalent irreducible goppa codes. now to count extended irreducible goppa codes we consider the action of f on s. this action induces orbits in s denoted by o(α). next we consider the action of g on of = {o(α) : α ∈ s}. the number of orbits in of under g gives us an upper bound on the number of inequivalent extended irreducible goppa codes. to find the number of orbits in a and of we use the cauchy frobenius theorem whose proof can be found in [4]. since the cauchy frobenius theorem is central in this paper we state it as follows. theorem 3.1 (cauchy frobenius theorem). let e be a finite group acting on a set x. for any e ∈ e, let xe denote the set of elements of x fixed by e. then the number of orbits in x under the action of e is 1|e| ∑ e∈h |xe|. 3.2. cardinality of s in order to simplify our notation we denote all the factors of the degree 2m by 2i for 0 ≤ i ≤ m. now to find the number of elements in s we use the lattice of subfields of f2nm , where m = 2m as done in [7]. figure 1 shows the lattice of subfields of f2nm . remark 3.2. in figure 1 observe that the elements of degree 2m over f2n lie in f2n(2m) and f22m . so the number of elements of degree 2m in f2n(2m) is |s| = 2n(2 m) − 2n(2 m−1). 3.3. the number of fixed affine sets in a note that the group g defined in definition 2.7 is a cyclic group of order n2m, where n > 2 is prime, and it’s subgroups are all of the form 〈σk〉, where k is a factor of n2m. further, note that g acts on a. in this section, we determine the g-orbits of this action. 238 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 f2n(2m) f 2n(2 m−1) f22m f 2n(2 m−2) f22(m−1) ... ... ... f22n f222 f2n f22 f2 figure 1. we first need to know the number of affine sets a(α) which are in a. by remark 3.2, |s| = 2n(2 m) − 2n(2 m−1). since |a(α)| = 2n(2n − 1) then |a| = |s|/(2n(2n − 1)). the expected length of orbits in a under the action of g are all factors of n2m. the trivial subgroup 〈σn(2 m)〉, containing the identity, fixes every affine set in a. in the following subsections, we separately consider the remaining subgroups of g, i.e., 〈σn(2 m−1)〉, 〈σ2 m 〉, 〈σ2 m−1 〉, 〈σ2 s 〉 and 〈σn(2 s)〉 where 0 ≤ s < m− 1. 3.3.1. 〈σn(2 m−1)〉 a subgroup of g of order 2 suppose the orbit in a under the action of g containing a(α) contains n(2m−1) affine sets, i.e, {a(α),σ(a(α)),σ2(a(α)), ...,σn(2 m−1)−1(a(α))}. then a(α) is fixed by 〈σn(2 m−1)〉. that is σn(2 m−1)(a(α)) = a(α). so we have σn(2 m−1)(α) = α2 n(2m−1) = ζα + ξ for some ζ 6= 0,ξ ∈ f2n. so applying σn(2 m−1) for the second time we get α = σn(2 m)(α) = σn(2 m−1)(ζα + ξ) = ζ2 n(2m−1) α2 n(2m−1) + ξn(2 m−1) = ζα2 n(2m−1) + ξ = ζ(ζα + ξ) + ξ = ζ2α + (ζ + 1)ξ. we conclude that ζ = 1 as otherwise ζ 6= 1 would mean (1 − ζ2)α ∈ f2n contradicting the fact that α ∈ s. now consider α2 n(2m−1) = α + ξ for some ξ 6= 0 ∈ f2n. multiplying both sides by ξ−1 we get (ξ−1α)2 n(2m−1) = (ξ−1α) + 1. we may assume that α satisfies the equation x2 n(2m−1) + x + 1 = 0. (1) if α satisfies (1) then certainly all the 2n elements in the set {α + ξ : ξ ∈ f2n} also satisfy (1) while the remaining elements in a(α) do not satisfy (1). this follows because the equation (ζα + ξ)2 n(2m−1) = 239 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 ζα2 n(2m−1) + ξ = ζ(α + 1) + ξ = ζα + ζ + ξ = (ζα + ξ) + 1 holds if and only if ζ = 1. hence if α satisfies (1) then a(α) contains precisely 2n roots of (1). we now find the number of elements of s which satisfy (1). we know that x2 n(2m−1) + x + 1 = 2n(2 m−1)−1∏ i=1 (x2 + x + βi) (2) where βi denotes all the elements of f2n(2m−1) which have trace 1 over f2 [5]. we know that there are precisely 2n(2 m−1)−1 such βi. note that the trace function we are dealing with is from the field f2n(2m−1) to f2. so even if an element is in a proper subfield of f2n(2m−1), in calculating its trace we regard it as an element of f 2n(2 m−1). we further observe that if β ∈ f2n(2m−2), then tracef 2n(2 m−1)|f2 (β) = 2 · tracef 2n(2 m−2)|f2 (β). since the characteristic is 2, then we conclude that none of the βi in the decomposition of x2 n(2m−1) +x+ 1 in (2) lie in f 2n(2 m−1). however 22 m−1−1 of the βi lie in f22m−1 and the remaining 2n(2 m−1)−1 − 22 m−1 lie in f 22 m−1n (not in any of its subfields). furthermore, all the quadratic factors on the right hand side of (2) are irreducible over f 2n(2 m−1). this is due to linearity of the trace function and the fact that trace(βi) = 1 for each βi. the 22 m−1−1 quadratic equations corresponding to the βi in f22m−1 have f22m as their splitting field while the remaining 2 n(2m−1)−1 − 22 m−1−1 quadratic equations have f2n(2m) as their splitting field. so all the 2n(2 m−1) roots lie in s. conversely if α ∈ s satisfies (∗) then a(α) is fixed under 〈σn(2 m−1)〉. we may conclude that there are precisely 2 n(2m−1) 2n = 2n(2 m−1−1) affine sets a(α) fixed under 〈σn(2 m−1)〉. 3.3.2. 〈σn(2 s)〉 a subgroup of g of order 2m−s suppose the orbit in a under the action of g containing a(α) contains n(2s) affine sets where 0 ≤ s < m− 1. as in subsection 3.3.1, we have a(α) fixed by 〈σn(2 s)〉 and σn(2 s)(α) = α2 n(2s) = ζα + ξ for some ζ 6= 0,ξ ∈ f2n. applying σn(2 s) for 2m−s times to α we obtain α = σn(2 m)(α) = ζ2 m−s α + (ζ2 m−s−1 + ζ2 m−s−2 + · · · + ζ2 + ζ + 1)ξ. we conclude that ζ2 m−s = 1 otherwise ζ2 m−s 6= 1 would mean (1 −ζ2 m−s )α ∈ f2n contradicting the fact that α ∈ s. the possibilities are that ζ2 m−s = 1, ζ2 m−s−1 = 1, ζ2 m−s−2 = 1, · · · , ζ2 2 = 1, ζ2 = 1 or ζ = 1. since 2n − 1 is odd then it is not divisible by 2d where 1 ≤ d ≤ m−s. hence ζ2 m−s = 1 implies ζ = 1. so α2 n(2s) = α + ξ for some ξ 6= 0 ∈ f2n. if we multiply both sides by ξ−1 we obtain (ξ−1α)2 n(2s) = (ξ−1α) + 1. we assume that α satisfies the equation x2 n(2s) + x + 1 = 0. using similar argument to the one in subsection 3.3.1, all roots of x2 n(2s) + x + 1 = 0 lie in f 22 s+1 and f 2n(2 s+1) (and not in s). we conclude that there is no affine set a(α) fixed under 〈σn(2 s)〉. 3.3.3. 〈σ2 m 〉 a subgroup of g of order n suppose the orbit in a under the action of g containing a(α) contains 2m affine sets. then a(α) is fixed under 〈σ2 m 〉. in [8], it is proved that the number of affine sets fixed by 〈σr〉 is |s(1,r)|/(q(q − 1)). hence the number of affine sets fixed by 〈σ2 m 〉 is |s(1, 2m)|/(2(2 − 1)) = (22 m − 22 m−1 )/2 = 22 m−1 − 22 m−1−1. 3.3.4. 〈σ2 m−1 〉 a subgroup of g of order 2n suppose the orbit in a under the action of g containing a(α) contains 2m−1 affine sets. then a(α) is fixed by 〈σ2 m−1 〉. so we have σ2 m−1 (α) = α2 2m−1 = ζα + ξ for some ζ 6= 0,ξ ∈ f2n. but if a(α) is fixed under 〈σ2 m−1 〉 then it is also fixed under 〈σ2 m 〉 since 〈σ2 m 〉 ⊂ 〈σ2 m−1 〉. so a(α) contains a fixed point. that is a(α) contains some elements which satisfy x2 2m = x and these elements are in 240 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 f22m \ f22m−1 . assume α ∈ f22m \ f22m−1 then applying σ 2m−1 twice to α we obtain α = α2 2m = ζ2 2m−1 (ζα + ξ) + ξ2 2m−1 = ζ2 2m−1 +1α + ζ2 2m−1 ξ + ξ2 2m−1 = ζ2 2m−1 +1α + ζ2 2m−1 ξ + ξ2 2m−1 . we conclude that ζ2 2m−1 +1 = 1 otherwise ζ2 2m−1 +1 6= 1 would mean (1−ζ2 2m−1 +1)α ∈ f2n contradicting the fact that α is of degree 2m. now we show that 22 m−1 + 1 is relatively prime to 2n − 1. we simply show that any number of the form 2d + 1 is relatively prime to 2n − 1. that is it suffices to show that (2d + 1, 2n − 1) = 1. we show this by contradiction. assume that (2d + 1, 2n − 1) 6= 1. that is there must be some odd prime p which divides both 2d + 1 and 2n − 1. this implies that 2n ≡ 1 (mod p) and 2d ≡ −1 (mod p). so 2d ≡ −1 (mod p) implies 22d ≡ (−1)2 = 1 ≡ 2n (mod p). thus n ≡ 2d (mod (p− 1)). since p− 1 is even then n is also even. this establishes a contradiction since n is an odd prime. hence (2d + 1, 2n − 1) = 1 for odd n. it follows that (22 m−1 + 1, 2n − 1) = 1 from which we conclude that ζ2 2m−1 +1 = 1 implies ζ = 1. so α = ζ2 2m−1 +1α + ζ2 2m−1 ξ + ξ2 2m−1 implies α = α + ξ + ξ2 2m−1 . clearly, ξ is in the intersection of the fields of order 22 m−1 and 2n. since (2m−1,n) = 1 then ξ is 0 or 1. but ξ = 0 is impossible since this would mean that α ∈ f 22 m−1 . so ξ must be 1. so we have α2 2m−1 = α + 1. clearly α satisfies the equation x2 2m−1 + x + 1 = 0. (3) observe that α + 1 also satisfies (3) and one can easily check that these are the only elements in a(α) which satisfy (3). using an argument similar to the one in subsection 3.3.1, all the 22 m−1 roots of x2 2m−1 + x + 1 lie in f22m \ f22m−1 . hence we conclude that there are 2 2m−1−1 affine sets fixed under 〈σ2 m−1 〉. 3.3.5. 〈σ2 s 〉 a subgroup of g of order n(2m−s) suppose the orbit in a under the action of g containing a(α) contains 2s affine sets where 0 ≤ s < m − 1. then a(α) is fixed by 〈σ2 s 〉 and σ2 s (α) = α2 2s = ζα + ξ for some ζ 6= 0,ξ ∈ f2n. as in subsection 3.3.4, if a(α) is fixed under 〈σ2 s 〉 then it is also fixed under 〈σ2 m 〉 since 〈σ2 m 〉 ⊂ 〈σ2 s 〉. assume α ∈ f22m \f22m−1 then applying σ 2s to α for 2m−s times we obtain α = α2 2m = ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s +22 s +1α + ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s +22 s ξ + ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s ξ2 2s + ζ2 n·2s +2(n−1)·2 s +···+23·2 s ξ2 2·2s + · · · + ζ2 n·2s +2(n−1)·2 s ξ2 (n−2)·2s + ζ2 n·2s ξ2 (n−1)·2s + ξ2 n·2s (4) where n = 2m−s − 1. observe that ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s +22 s +1 must be equal to 1 otherwise it would mean that (1 − ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s +22 s +1)α ∈ f2n contradicting the fact that α is of degree 2m. we now show that 2n·2 s +2(n−1)·2 s +· · ·+23·2 s +22·2 s +22 s +1 and 2n−1 are coprime. first observe that 241 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 2n·2 s +2(n−1)·2 s +· · ·+23·2 s +22·2 s +22 s +1 = (22 s(n+1)−1)/(22 s −1) = (22 m −1)/(22 s −1). but (2m,n) = 1 so (22 m −1, 2n−1) = 1 from which we conclude that (2n·2 s +2(n−1)·2 s +· · ·+23·2 s +22·2 s +22 s +1, 2n−1) = 1. we can now conclude that ζ2 n·2s +2(n−1)·2 s +···+23·2 s +22·2 s +22 s +1 = 1 implies ζ = 1. since ζ = 1 then (4) becomes α = α + ξ + ξ2 2s + ξ2 2·2s + · · · + ξ2 (n−2)2s + ξ2 (n−1)2s + ξ2 n·2s . it is clear that ξ is in the intersection of the fields of order 22 s and 2n. since (2s,n) = 1 then ξ is 0 or 1. but ξ = 0 is impossible since this would mean that α ∈ f22s . so ξ must be 1. so we have α2 2s = α + 1. clearly α satisfies the equation x2 2s + x + 1 = 0. observe that α + 1 also satisfies the equation x2 2s + x + 1 = 0 and one can easily check that these are the only elements in a(α) which satisfy x2 2s + x + 1 = 0. using similar argument to the one in subsection 3.3.1, all the 22 s roots of x2 2s + x + 1 lie in f22s−1 not in s. hence we conclude that there is no affine set fixed under 〈σ2 s 〉. 3.4. the number of orbits in a under the action of g we use table 1 to present the information in section 3.3. this table shows the number of affine sets which are fixed under the action of various subgroups of g. the subgroups which do not fix any affine set are left out. the subgroups are listed in ascending order of the number of elements in the subgroup. so the first row is the subgroup 〈σn(2 m)〉 which is merely the trivial subgroup containing the identity. column 3 lists the number of elements in subgroup which are not already counted in subgroups in the rows above it in the table. this is to avoid repetition when we multiply column 3 by column 4 in order to get the total number of fixed affine sets by the elements in g. table 1. subgroup order of no. of elements no. of fixed product of g subgroup not in previous affine sets of columns subgroup 3 and 4 〈σ2 mn〉 1 1 2 n(2m)−2n(2 m−1) 2n(2n−1) 2n(2 m)−2n(2 m−1) 2n(2n−1) 〈σn(2 m−1)〉 2 1 2n(2 m−1−1) 2n(2 m−1−1) 〈σ2 m 〉 n n−1 22 m−1 −22 m−1−1 (n−1)(22 m−1 −22 m−1−1) 〈σ2 m−1 〉 2n n−1 22 m−1−1 (n−1)22 m−1−1 by the cauchy frobenius theorem, the number of orbits in a under the action of g is (2n(2 m)−2n(2 m−1))/(2n(2n−1))+2n(2 m−1−1)+(n−1)×22 m−1 n(2m) . remark 3.3. the number of orbits in a under the action of g gives us an upper bound on the number of irreducible goppa codes. 3.5. the number of fixed o(α) in of we are going to consider the action of g on of so that we find the number of o(α)’s which are fixed in of . this is done by acting all subgroups of g on of . we begin by finding the number of elements in of . by remark 3.2, |s| = 2n(2 m) − 2n(2 m−1). since |o(α)| = 2n(2n − 1)(2n + 1) then |of | = |s| 2n(2n−1)(2n+1). since g acts on of and its cardinality is n(2m) then the expected lengths for the orbits in of under the action of g are all the factors of n(2m). every o(α) in of is fixed by a trivial subgroup 〈σn(2 m)〉 containing the identity. as in section 3.3, we consider the remaining subgroups of g, i.e., 〈σn(2 m−1)〉, 〈σ2 m−1 〉, 〈σ2 s 〉 and 〈σn(2 s)〉 where 0 ≤ s < m− 1. 242 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 3.5.1. 〈σn(2 m−1)〉 a subgroup of g of order 2 suppose o(α) ∈ of is fixed under 〈σn(2 m−1)〉. then 〈σn(2 m−1)〉 acts on o(α) = a(α) ∪ a( 1 α ) ∪ a( 1 α+1 )∪a( 1 α+ξ1 )∪a( 1 α+ξ2 )∪a( 1 α+ξ2 )∪·· ·∪a( 1 α+ξ2n−2 ). we can consider o(α) as a set of 2n + 1 affine sets. 〈σn(2 m−1)〉 partitions this set of 2n + 1 affine sets. the only possibility are orbits of length 1 or 2. since o(α) contains an odd number of affine sets then the possibility that all orbits are of length 2 is excluded. so there has to be at least one orbit of length 1, i.e., o(α) must contain an affine set which is fixed under 〈σn(2 m)〉. by subsection 3.3.1, there are 2n(2 m−1−1) such affine sets. we claim that any fixed o(α) in of contains precisely one affine set which is fixed under 〈σn(2 m−1)〉. it suffices to show that o(α) cannot contain two affine sets which are fixed under 〈σn(2 m−1)〉. without loss of generality, suppose a(α) is fixed under 〈σn(2 m−1)〉. recall that o(α) = a(α)∪a( 1 α )∪a( 1 α+1 )∪a( 1 α+ξ1 )∪a( 1 α+ξ2 )∪a( 1 α+ξ3 )∪·· ·∪ a( 1 α+ξ2n−2 ). we show that none of the affine sets after a(α) in the above decomposition of o(α) is fixed under 〈σn(2 m−1)〉. this is done by showing that no element in any of these affine sets satisfies the equation x2 n(2m−1) + x + 1 = 0 (see equation (1) in subsection 3.3.1). by subsection 3.3.1, the 2n elements in the set {α+ξ : ξ ∈ f2n} satisfy the equation x2 n(2m−1) +x+1 = 0 from which we see that α2 n(2m−1) = α+1. it is sufficient to show that no element in a( 1 α ) satisfies x2 n(2m−1) +x+ 1 = 0. a typical element in a( 1 α ) has the form ζ α +ξ and substituting this in x2 n(2m−1) +x+ 1 we get ( ζ α +ξ)2 n(2m−1) + ( ζ α +ξ) + 1 = α 2+α+ζ α2+α 6= 0, since α is an element of degree 2m over f2n. we conclude that a( 1α) is not fixed under 〈σ n(2m−1)〉 and in fact a(α) is the only affine set in o(α) fixed under 〈σn(2 m−1)〉. it follows that the number of o(α)’s in of which are fixed under 〈σn(2 m−1)〉 is 2n(2 m−1−1). 3.5.2. 〈σn(2 s)〉 a subgroup of g of order 2m−s suppose o(α) ∈ of is fixed under 〈σn(2 s)〉 where 0 ≤ s < m. then 〈σn(2 s)〉 acts on o(α). we can consider o(α) as a set of 2n + 1 affine sets. 〈σn(2 s)〉 partitions this set of 2n + 1 affine sets. the only possible lengths of orbits are all factors of 2m−s. since o(α) contains an odd number of affine sets then the possibility that all orbits are of even length is precluded. by subsection 3.3.2, there is no affine set fixed under 〈σn(2 s)〉. so we also preclude the possibility of length 1. hence we conclude that no o(α) in of is fixed under 〈σn(2 s)〉. 3.5.3. 〈σ2 m 〉 a subgroup of g of order n suppose o(α) ∈ of is fixed under 〈σ2 m 〉. then 〈σ2 m 〉 acts on o(α) which is seen as a set of 2n + 1 affine sets. 〈σ2 m 〉 partitions this set of 2n + 1 affine sets. the only possible lengths of orbits are 1 and n. since 2n + 1 ≡ 2 + 1 = 3 (mod n) (by fermat little theorem) then n does not divide 2n + 1. so there must be at least three affine sets in o(α) fixed under 〈σ2 m 〉. we claim that there are only three affine sets in o(α) which are fixed under 〈σ2 m 〉. recall that o(α) = a(α) ∪a( 1 α ) ∪a( 1 α+1 ) ∪a( 1 α+ξ1 ) ∪a( 1 α+ξ2 ) ∪ a( 1 α+ξ3 )∪·· ·∪a( 1 α+ξ2n−2 ). without loss of generality, suppose a(α) in o(α) is fixed under 〈σ2 m 〉. so, by subsection 3.3.3, a(α) contains a fixed point, i.e., some elements of a(α) satisfy the equation x2 2m = x. it is clear that α and α + 1 in a(α) satisfy x2 2m = x. since ( 1 α )2 2m = 1 α and ( 1 α+1 )2 2m = 1 α+1 it is clear that a( 1 α ) and a( 1 α+1 ) also contain fixed points, i.e., a( 1 α ) and a( 1 α+1 ) are also fixed. we now show that no affine set after a( 1 α+1 ) in the decomposition of o(α) is fixed under 〈σ2 m 〉. first observe that, for ν ∈ f2n \ {0, 1}, we have ν2 2m 6= ν since (2m,n) = 1. so ( 1 α+ν )2 2m = 1 α+ν2 2m = 1 α+η implies that σ2 m (a( 1 α+ν )) = a( 1 α+η ) as required. therefore a(α), a( 1 α ) and a( 1 α+1 ) are the only affine sets fixed under 〈σ2 m 〉. by subsection 3.3.3, there are 22 m−1 − 22 m−1−1 affine sets which are fixed under 〈σ2 m 〉. hence the number of o(α)’s in of which are fixed under 〈σ2 m 〉 is (22 m−1 − 22 m−1−1)/3. 243 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 3.5.4. 〈σ2 m−1 〉 a subgroup of g of order 2n suppose o(α) ∈ of is fixed under 〈σ2 m−1 〉. then 〈σ2 m−1 〉 acts on o(α) which is seen as a set of 2n + 1 affine sets. 〈σ2 m−1 〉 partitions this set of 2n + 1 affine sets. o(α) = a(α) ∪ a( 1 α ) ∪ a( 1 α+1 ) ∪ a( 1 α+ξ1 )∪a( 1 α+ξ2 )∪a( 1 α+ξ3 )∪·· ·∪a( 1 α+ξ2n−2 ). the possible lengths of orbits are all factors of 2n. but the possibility that all orbits are of even length is precluded since the o(α) contains odd number of affine sets. it is also not possible to have length n for all orbits since n (2n + 1) (see subsection 3.5.3). so there must be at least one affine set fixed under 〈σ2 m−1 〉. we claim that any o(α) fixed under 〈σ2 m−1 〉 contains precisely one affine set fixed under 〈σ2 m−1 〉. by subsection 3.3.4, an affine set fixed under 〈σ2 m−1 〉 contains some elements which satisfy the equation x2 2m−1 + x + 1 = 0. suppose a(α) is fixed under 〈σ2 m−1 〉. it is clear that α and α + 1 satisfy x2 2m−1 + x + 1 = 0. we also observe that ( 1 α )2 2m−1 = 1 α+1 and ( 1 α+1 )2 2m−1 = 1 α which imply that σ2 m−1 (a( 1 α )) = a( 1 α+1 ) and σ2 m−1 (a( 1 α+1 )) = a( 1 α ). we can conclude that a( 1 α ) and a( 1 α+1 ) form an orbit of length 2. since 〈σ2 m 〉 ⊂ 〈σ2 m−1 〉 then any o(α) or affine set fixed under 〈σ2 m−1 〉 is also fixed under 〈σ2 m 〉. by subsection 3.3.3, no affine set after a( 1 α+1 ) in the decomposition of o(α) is fixed under 〈σ2 m 〉. so we conclude that 〈σ2 m−1 〉 does not fix any affine set after a( 1 α+1 ) in the decomposition of o(α). by subsection 3.3.4, there are 22 m−1−1 affine sets fixed under 〈σ2 m−1 〉. so we conclude that the number of o(α)’s in of which are fixed under 〈σ2 m−1 〉 is 22 m−1−1. 3.5.5. 〈σ2 s 〉 a subgroup of g of order n(2m−s) suppose o(α) ∈ of is fixed under 〈σ2 s 〉 where 0 ≤ s < m − 1. then 〈σ2 s 〉 acts on o(α) which is seen as a set of 2n + 1 affine sets. 〈σ2 s 〉 partitions this set of 2n + 1 affine sets. the possible lengths of orbits are all factors of n(2m−s). since o(α) contains an odd number of affine sets then the possibility that all orbits are of even length is precluded. since 2n + 1 ≡ 3 (mod n) (see subsection 3.5.3) we also preclude the possibility that all orbits are of length n. we now consider the possibility of x affine sets partitioned in orbits of length 2 and 2n + 1 − x affine sets partitioned in orbits of length n or 2n, i.e., 2n + 1 − x ≡ 0 (mod n). since 2n + 1 ≡ 3 (mod n), x has the form kn + 3 where k is an odd integer. since 2|x then k is non zero. so there are kn + 3 > 3 affine sets permuted in orbits of length 2 under 〈σs〉. since 〈σs+1〉 ⊂ 〈σs〉 it is easy to observe that two affine sets that form an orbit under 〈σs〉 are fixed under 〈σs+1〉. if s = m− 2 then it would mean that there exist a fixed o(α) under 〈σ2 m−1 〉 which contains more than 3 affine sets fixed under 〈σ2 m−1 〉, contradicting subsection 3.5.4. no affine set is fixed under 〈σ2 s+1 〉 for s < m− 2 (see subsection 3.3.5) so it would be a contradiction to say that more than 3 affine sets in each fixed o(α) are fixed under 〈σ2 s+1 〉. a similar argument can be used to show that length of multiple of 2 and n are also not possible. so there must be at least an affine set in o(α) fixed under 〈σ2 s 〉. by subsection 3.3.5, no affine set is fixed by 〈σ2 s 〉. hence there is no o(α) in of which is fixed under 〈σ2 s 〉. 3.6. the number of orbits in of under the action of g table 2 shows the number of o(α)’s in of which are fixed under the various subgroups of g. the structure of this table is similar to that of table 1. the subgroups which do not fix any o(α) are left out. the subgroups are listed in ascending order of the number of elements in the subgroup. so the first row is the subgroup 〈σn(2 m)〉 since it contains only the identity. column 3 lists the number of elements in subgroup which are not already counted in subgroups in the rows above it in the table. this is to avoid repetition when we multiply column 3 by column 4 in order to get the total number of fixed o(α)’s by the elements in g. by the cauchy frobenius theorem, the number of orbits in of under the action of g is (2n(2 m)−2n(2 m−1))/(2n(2n−1)(2n+1))+2(2 m−1−1)n+(n−1)[(22 m−1+22 m−1 )/3] n(2m) . 244 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 table 2. subgroup order of no. of elements no. of fixed product of g subgroup not in previous o(α)’s of columns subgroup 3 and 4 〈σn(2 m)〉 1 1 2 n(2m)−2n(2 m−1) 2n(2n−1)(2n+1) 2n(2 m)−2n(2 m−1) 2n(2n−1)(2n+1) 〈σn(2 m−1)〉 2 1 2(2 m−1−1)n 2(2 m−1−1)n 〈σ2 m 〉 n n−1 22 m−1 −22 m−1−1 (n−1)(22 m−1 −22 m−1−1)/3 〈σ2 m−1 〉 2n n−1 22 m−1−1 (n−1)22 m−1−1 we state this result in the following theorem: theorem 3.4 (the main theorem). let n be an odd prime number and m > 1. the number of inequivalent extended irreducible binary goppa codes of degree 2m and length 2n + 1 is at most (2n(2 m) − 2n(2 m−1))/(2n(2n − 1)(2n + 1)) + 2(2 m−1−1)n + (n− 1)[(22 m−1 + 22 m−1 )/3] n(2m) . example 3.5. we find an upper bound on the number of irreducible goppa codes and extended irreducible goppa codes of degree 4 for n = 5, 7, 11, 13 and 17 as follows: n upper bound on number upper bound on number of of irreducible goppa codes extended irreducible goppa codes 5 56 4 7 596 10 11 95420 94 13 1290872 316 17 252648992 3856 example 3.6. we find an upper bound on the number of irreducible goppa codes of degree 16 and extended irreducible goppa codes for n = 7 and 11. n upper bound on number upper bound on number of of irreducible goppa codes extended irreducible goppa codes 7 2,851,857,368,342,478,330,960,957,440 22,107,421,460,024,199,242,917,600 11 1.29813175585637777519535861321×1044 6.33544048734200963988747188270 ×1040 4. conclusion in this paper we have produced an upper bound on the number of extended irreducible binary goppa codes of degree 2m and length 2n + 1 where n is a prime number. the result is presented in the theorem 3.4. 245 http://orcid.org/0000-0001-8792-6954 a. i. musukwa et al. / j. algebra comb. discrete appl. 4(3) (2017) 235–246 references [1] t. p. berger, goppa and related codes invariant under a prescribed permutation, ieee trans. inform. theory 46(7) (2000) 2628–2633. [2] c. l. chen, equivalent irreducible goppa codes, ieee trans. inform. theory 24(6) (1978) 766–769. [3] h. dinh, c. moore, a. russell, mceliece and niederreiter cryptosystems that resist quantum fourier sampling attacks, in: rogaway p. (eds) advances in cryptology – crypto 2011. crypto 2011. lecture notes in computer science 6841 (2011) 761–779. [4] i. m. isaacs, algebra: a graduate text, brooks/cole, pacific grove, 1994. [5] r. lidl, h. niederreiter, introduction to finite fields and their applications, cambridge university press, london, 1994. [6] s. ling, c. xing, coding theory; a first course, cambridge university press, united kingdom, 2004. [7] k. magamba, j. a. ryan, counting irreducible polynomials of degree r over fqn and generating goppa codes using the lattice of subfields of fqnr , j. discrete math. 2014 (2014) 1–4. [8] j. a. ryan, irreducible goppa codes, ph.d. dissertation, university college cork, 2004. 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[10] j. a. ryan, counting extended irreducible binary quartic goppa codes of length 2n + 1, ieee trans. inform. theory 61(3) (2015) 1174–1178. 246 http://orcid.org/0000-0001-8792-6954 https://doi.org/10.1109/18.887871 https://doi.org/10.1109/18.887871 https://doi.org/10.1109/tit.1978.1055951 https://doi.org/10.1007/978-3-642-22792-9_43 https://doi.org/10.1007/978-3-642-22792-9_43 https://doi.org/10.1007/978-3-642-22792-9_43 http://dx.doi.org/10.1155/2014/263179 http://dx.doi.org/10.1155/2014/263179 http://samsa-math.org/wp-content/uploads/2013/04/samsa2012proceedingscomplete.pdf http://samsa-math.org/wp-content/uploads/2013/04/samsa2012proceedingscomplete.pdf https://doi.org/10.1109/tit.2015.2389246 https://doi.org/10.1109/tit.2015.2389246 introduction preliminaries counting extended irreducible binary goppa codes of degree 2m conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.327399 j. algebra comb. discrete appl. 5(1) • 5–18 received: 12 december 2016 accepted: 6 april 2017 journal of algebra combinatorics discrete structures and applications hermitian self-dual quasi-abelian codes research article herbert s. palines, somphong jitman, romar b. dela cruz abstract: quasi-abelian codes constitute an important class of linear codes containing theoretically and practically interesting codes such as quasi-cyclic codes, abelian codes, and cyclic codes. in particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes. based on the well-known decomposition of quasi-abelian codes, the characterization and enumeration of hermitian self-dual quasi-abelian codes are given. in the case of 1-generator quasi-abelian codes, we offer necessary and sufficient conditions for such codes to be hermitian self-dual and give a formula for the number of these codes. in the case where the underlying groups are some p-groups, the actual number of resulting hermitian self-dual quasi-abelian codes are determined. 2010 msc: 94b15, 94b60, 16a26 keywords: hermitian self-dual codes, quasi-abelian codes, 1-generator, p-groups 1. introduction quasi-cyclic codes form an important class of linear codes due to their rich algebraic structures, large number of codes with good parameters, and various applications (see [9], [10], [11], [12], [14], [17], and references therein). let fq denote a finite field of order q. it is known that quasi-cyclic codes of length ml and index l over fq can be regarded as fq[zm]-submodules of the fq[zm]-module (fq[zm])l, where zm denotes the cyclic group of order m and fq[zm] is the group algebra of zm over fq (see [10]). in a more general setting, quasi-abelian codes are defined by replacing zm with a finite abelian group. particularly, if g is a finite abelian group and h ≤ g, then an h-quasi-abelian code is defined to be an fq[h]-submodule of the fq[h]-module fq[g]. this class of codes was first introduced in [18] and further studies of their properties have been made in [4, section 7] and [1]. more recently in [6], via the herbert s. palines; institute of mathematical sciences and physics, university of the philippines los baños, college, laguna 4031, philippines, and institute of mathematics, college of science, university of the philippines diliman, quezon city 1101, philippines (email: herbertpalines@gmail.com). somphong jitman (corresponding author); department of mathematics, faculty of science, silpakorn university, nakhon pathom 73000, thailand (email: sjitman@gmail.com). romar b. dela cruz; institute of mathematics, college of science, university of the philippines diliman, quezon city 1101, philippines (email: rbdelacruz@math.upd.edu.ph). 5 http://orcid.org/0000-0003-1076-0866 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 discrete fourier transform, the structural characterization of quasi-abelian codes have been established together with the existence of asymptotically good quasi-abelian codes. quasi-abelian codes serve as the general case for quasi-cyclic codes (if h 6= g is cyclic), abelian codes (if h = g), and cyclic codes (if h = g is cyclic). since the theory of quasi-abelian codes generalizes that of quasi-cyclic codes, a link can be established between 1-generator quasi-abelian codes and irreducible or minimal cyclic codes which plays a central role in the theory of cyclic codes [2]. self-dual codes form another fascinating family of codes and are known to be closely related with other objects such as lattices and possess variety of practical applications (see [13]). moreover, both euclidean and hermitian self-dual codes have close connection with quantum stabilizer codes [8]. in [6], the authors presented necessary and sufficient conditions for quasi-abelian codes to be euclidean self-dual and gave enumeration of those codes based on the classification of q-cyclotomic classes of the underlying group. moreover, they have shown that some class of binary euclidean self-dual strictly-quasi-abelian codes are asymptotically good. to the best of our knowledge, no study has been done yet on hermitian self-dual quasi-abelian codes. it is therefore of natural interest to investigate such family of codes and compare the result of this study with that of [6]. in this work, considering finite abelian groups h ≤ g, we offer sufficient and necessary conditions for an h-quasi-abelian code in fq[g] to be hermitian self-dual using similar decomposition given in [6, section 3] (see proposition 2.3). consequently, enumeration of hermitian self-dual h-quasi-abelian codes is presented (see corollary 3.1). in similar fashion, the sufficient and necessary conditions for a 1-generator quasi-abelian code to be hermitian self-dual are obtained (see corollary 4.3). enumeration of hermitian self-dual 1-generator quasi-abelian codes is also given. in the case h ∼= (zpk)s is a p-group, p is a prime, k > 0 and s > 0, we classify completely the q-cyclotomic classes of h (see propositions 3.6 and 3.10) which lead to the actual number of the resulting hermitian self-dual h-quasi-abelian codes. the asymptotic goodness of hermitian self-dual strictly-quasi-abelian codes over f22s is guaranteed by [6, section 7] since every code over f22s with generator matrix containing only elements from f2 is hermitian self-dual if and only if such a matrix generates a euclidean self-dual code over f2. the paper is organized as follows. in section 2, we recall notations and definitions which are essential to this work as well as the well-known decomposition of semi-simple group algebras. enumeration of hermitian self-dual quasi-abelian codes, where the underlying groups are some p-groups, is established in section 3. finally in section 4, we focus on the characterization and enumeration of hermitian self-dual 1-generator quasi-abelian codes. 2. preliminaries for a prime power q and positive integer n, let fq denote a finite field of order q and let g be a finite abelian group of order n, written additively. denote by fq[g] the group algebra of g over fq. the elements in fq[g] will be written as ∑ g∈g αgy g, where αg ∈ fq. the addition and the multiplication in fq[g] are given as in the usual polynomial rings over fq with the indeterminate y , where the indices are computed additively in g. as convention, y 0 is treated as the multiplicative identity of fq[g], where 0 is the identity of g. let r be a finite commutative ring with unity. a linear code of length n over r is defined to be an r-submodule of rn. a (linear) code c in fq[g] refers to an fq-subspace of fq[g]. this can be viewed as a linear code of length n over fq by indexing the n-tuples by the elements of g. for more details, the reader is referred to [6]. consider a subgroup h of g, a code c in fq[g] is called an h-quasi-abelian code (specifically, an h-quasi-abelian code of index l, where l := [g : h]) if c is an fq[h]-module, i.e., c is closed under addition and multiplication by the elements in fq[h]. if h is a non-cyclic subgroup of g, then we say that c is a strictly-quasi-abelian code. if it is clear in the context or if h is not specified, such a code will be called simply a quasi-abelian code. an h-quasi-abelian code c is said to be of 1-generator if c is 6 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 a cyclic fq[h]-module. let {g1,g2, . . . ,gl} be a fixed set of representatives of the cosets of h in g. let r := fq[h]. define φ : fq[g] →rl by φ (∑ h∈h l∑ i=1 αh+giy h+gi ) = (α1(y ),α2(y ), . . . ,αl(y )) , where αi(y ) = ∑ h∈h αh+giy h ∈ r, for all i = 1, 2, . . . , l. it is well known that φ is an r-module isomorphism interpreted as follows. lemma 2.1. the map φ induces a one-to-one correspondence between h-quasi-abelian codes in fq[g] and linear codes of length l over r. in fnq , the euclidean inner product of u = (u1,u2, . . . ,un) and v = (v1,v2, . . . ,vn) is defined to be 〈u,v〉e := ∑n i=1 uivi. from this point, we assume q = q 2 0, where q0 is a prime power. consequently, the hermitian inner product of u and v is defined as 〈u,v〉h := ∑n i=1 uivi, where ¯ is the automorphism on fq defined by α 7→ αq0 for all α ∈ fq. for a code c of length n over fq, let c⊥e and c⊥h denote its euclidean dual and hermitian dual, respectively. the code c is said to be euclidean (resp., hermitian) self-dual if c⊥e = c (resp., c⊥h = c). the hermitian inner product in fq[g] is defined by 〈u,v〉h := ∑ g∈g αgβg for all u = ∑ g∈g αgy g and v = ∑ g∈g βgy g in fq[g]. the hermitian dual of a code c ⊆ fq[g] is given by c⊥h := {u ∈ fq[g] | 〈u,v〉h = 0 for all v ∈ c}. similarly, the code c in fq[g] is said to be hermitian self-dual if c⊥h = c. note that without confusion, we use the symbol ⊥h to indicate both the hermitian dual of a code over fq and the hermitian dual of a code in fq[g]. all throughout, the self-duality of quasi-abelian codes is studied with respect to the given hermitian inner product in fq[g]. 2.1. decomposition and hermitian dual codes the main tool of this work appears in this subsection. the idea is to have a convenient decomposition of quasi-abelian codes using the well-known decomposition of semi-simple group algebras introduced in [16]. then, combining this technique with the results of [7, proposition 2.7] and [6, proposition 4.1], we obtain characterization of hermitian self-dual quasi-abelian codes (see proposition 2.3). this will lead to enumeration of such class of codes. for completeness, we discuss the concepts of q-cyclotomic classes and primitive idempotents as appeared in [7, section ii.c]. given coprime positive integers i and j, the multiplicative order of j modulo i, denoted by ordi(j), is defined to be the smallest positive integer s such that i divides js − 1. for each a ∈ h, denote by ord(a) the additive order of a in h. from this point, we assume that gcd(|h|,q) = 1. a q-cyclotomic class of h containing a ∈ h, denoted by sq(a), is defined to be the set sq(a) :={qi ·a | i = 0, 1, . . .} = {qi ·a | 0 ≤ i < ordord(a)(q)}, where qi ·a := qi∑ j=1 a in h. 7 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 for a positive integer r and a ∈ h, denote by −r ·a the element r · (−a) ∈ h. a q-cyclotomic class sq(a) is said to be of type i if sq(a) = sq(−q0 · a) and it is of type ii if sq(−q0 · a) 6= sq(a). clearly, sq(0) is a q-cyclotomic class of type i. an idempotent in a ring is a non-zero element e such that e2 = e, and it is called primitive idempotent if, for every other idempotent f, either ef = e or ef = 0. the primitive idempotents in r := fq[h] are induced by the q-cyclotomic classes of h (see [5, proposition ii.4]). assume that h contains t q-cyclotomic classes. without loss of generality, let {a1 = 0,a2, . . . ,at} be a set of representatives of the q-cyclotomic classes of h such that {ai | i = 1, 2, . . . ,ri} and {ari+j,ari+rii+j = −q0 · ari+j | j = 1, 2, . . . ,rii} are sets of representatives of q-cyclotomic classes of types i and ii, respectively, where t = ri + 2rii. let {e1,e2, . . . ,et} be the set of primitive idempotents of r induced by {sq(ai) | i = 1, 2, . . . , t}, respectively. it is well known that rei is isomorphic to an extension field of fq of degree |sq(ai)| for each i = 1, 2, . . . , t. in [16], r := fq[h] is decomposed in terms of ei’s. later, the components in the decomposition of r are rearranged in [7] and obtain the following. r = t⊕ i=1 rei ∼= ( ri∏ i=1 ei ) ×  rii∏ j=1 (kj ×k′j)   , (1) where ei ∼= rei, kj ∼= reri+j, and k′j ∼= reri+rii+j are finite extension fields of fq for all i = 1, 2, . . . ,ri and j = 1, 2, . . . ,rii. remark 2.2. it is known that ei ∼= fqsi , kj ∼= fqtj and k ′ j ∼= f q t′ j , where si := |sq(ai)|, tj := |sq(ari+j)|, and t′j := |sq(ari+rii+j)| for i = 1, 2, . . . ,ri and j = 1, 2, . . . ,rii. note that |sq(ari+j)| = |sq(ari+rii+j)| for each j = 1, 2, . . . ,rii. thus, kj ∼= k′j for each j = 1, 2, . . . ,rii. from (1), we have fq[g] ∼= rl ∼= ( ri∏ i=1 eli ) ×  rii∏ j=1 (klj ×k ′ j l )   , (2) where the isomorphisms are r-module isomorphisms. they can be viewed as fq-linear isomorphisms as well. consequently, every quasi-abelian code c in fq[g] can be viewed as c ∼= ( ri∏ i=1 ci ) ×  rii∏ j=1 ( dj ×d′j ) , (3) where ci, dj and d′j are linear codes of length l over ei, kj, and k ′ j, respectively, for all i = 1, 2, . . . ,ri and j = 1, 2, . . . ,rii. using arguments similar to the proofs of [7, proposition 2.7] and [6, proposition 4.1], it can be concluded that the hermitian dual of c is of the form c⊥h ∼= ( ri∏ i=1 c⊥hi ) ×  rii∏ j=1 ( (d′j) ⊥e ×d⊥ej ) . (4) from (3) and (4), we have the following necessary and sufficient conditions for quasi-abelian codes to be hermitian self-dual. proposition 2.3. an h-quasi-abelian code c in fq[g] is hermitian self-dual if and only if, in the decomposition (3), i) ci is hermitian self-dual for all i = 1, 2, . . . ,ri, and ii) d′j = d ⊥e j for all j = 1, 2, . . . ,rii. 8 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 3. enumeration of hermitian self-dual quasi-abelian codes in this section, we enumerate hermitian self-dual quasi-abelian codes by using the decomposition in (3), proposition 2.3 and the following formulas. let n(q, l) (resp., nh(q, l)) denote the number of linear codes (resp., hermitian self-dual codes) of length l over fq. it is well known (see [15] and [13]) that n(q, l) = l∑ i=0 i−1∏ j=0 ql −qj qi −qj , (5) nh(q, l) =   l 2 −1∏ i=0 (qi+ 1 2 + 1) if l is even, 0 otherwise, (6) where the empty product is set to be 1. in general, to count the number of hermitian self-dual quasi-abelian codes in fq[g], in (3), we count the number of hermitian self-dual codes ci of length l over fqsi for all i = 1, 2, . . . ,ri and multiply it with the number of all possible linear codes dj of length l over fqtj for all j = 1, 2, . . . ,rii. this technique is clear in the following corollary. hereafter, the numbers si, tj, and t′j will appear frequently in the succeeding results. if needed, the reader is referred back to remark 2.2 for the definitions of si, tj, and t′j. corollary 3.1. let h ≤ g be finite abelian groups such that gcd(|h|,q) = 1 and l = [g : h]. assume that fq[h] contains ri (resp., 2rii) primitive idempotents of type i (resp., ii). assume further that the primitive idempotents of type i are induced by q-cyclotomic classes of size si for each i = 1, 2, . . . ,ri and the primitive idempotents of type ii are induced by q-cyclotomic classes of sizes tj and t′j, pair-wise, for each j = 1, 2, . . . ,rii. then the number of hermitian self-dual h-quasi-abelian codes in fq[g] is ri∏ i=1 nh(q si, l) rii∏ j=1 n(qtj, l). (7) we note that sq(0) is a q-cyclotomic class of h of type i. then ri ≥ 1, and hence, the product∏ri i=1 nh(q si, l) = 0 for all odd positive integers l. hence, there are no hermitian self-dual h-quasi-abelian codes if l = [g : h] is odd. therefore, we have the following result derived from (6) and (7). lemma 3.2. there exists a hermitian self-dual h-quasi-abelian code in fq[g] if and only if the index l = [g : h] is even. remark 3.3. from lemma 3.2, it is apparent that given a finite abelian group g and q = q20, the existence of hermitian self-dual h-quasi-abelian codes in fq[g] depends only on the choice of h, particularly on index l being even. in the theory of quasi-cyclic codes, it is practical to use a relatively small fixed value of the index l mainly for the purpose of efficient decoding [3]. moreover, this case contains the known case of double circulant codes (see [10, section vi.a] and [12, section ii.a]). since the theory of quasi-abelian codes generalizes that of quasi-cyclic codes, we can adopt those concepts. note that a quasi-cyclic code is cyclic when l = 1. thus l = 2 is the smallest index such that a code is quasi-cyclic. specifically for l = 2, one can talk about self-dual 1-generator quasi-abelian codes (see section 4). consider the example below for the number of quasi-abelian codes of index 2. example 3.4. let h ≤ g be finite abelian groups such that gcd(|h|,q) = 1 and l = [g : h] = 2. assume that fq[h] contains ri (resp., 2rii) primitive idempotents of type i (resp., ii). assume further that the primitive idempotents of type i are induced by q-cyclotomic classes of size si for each i = 1, 2, . . . ,ri and the primitive idempotents of type ii are induced by q-cyclotomic classes of sizes tj and t′j, pair-wise, for 9 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 each j = 1, 2, . . . ,rii. then the number of hermitian self-dual h-quasi-abelian codes of index 2 in fq[g] is ri∏ i=1 (qsi0 + 1) rii∏ j=1 (qtj + 3). in the next two subsections, we consider the case where the subgroups h of g are some p-groups. it is interesting to see that for this particular case, the cardinality and the number of q-cyclotomic classes of h can be explicitly determined. hence, one can obtain the actual number of resulting hermitian self-dual h-quasi-abelian codes. in this regard, we offer sufficient and necessary conditions for a q-cyclotomic class of h to be of type i or type ii. 3.1. h ∼= (z2k)s the succeeding discussion is instrumental in determining the explicit forms of ri and rii. let h ∼= (zpk)s, where k and s are positive integers, and p is prime such that gcd(p,q) = 1. define hpi := {h ∈ h| ord(h) = pi}, for each 0 ≤ i ≤ k. observe that h1,hp, . . . ,hpk are pair-wise disjoint and h = h1 ∪hp ∪ ·· ·∪hpk, where h1 = {0}. for each 1 ≤ i ≤ k, it is not difficult to see that hpi = ( pk−izpk )s \ (pk−(i−1)zpk)s. consequently, we have |h1| = 1 and, via inclusion-exclusion principle, |hpi| = pis −p(i−1)s, for each i = 1, 2, . . . ,k. recall that q = q20 where q0 is a prime power. hereafter, let νpi := ordpi(q) and µpi := ordpi(q0), for i = 0, 1, . . . ,k. note that if h ∈ hpi, |sq(h)| = ordord(h)(q) = νpi. now, consider the case where q is odd and p = 2, i.e., h ∼= (z2k)s. suppose h ∈ h2. since ord(h) = 2 for all h ∈ h2, q ≡±1 (mod ord(h)) and q0 ≡±1 (mod ord(h)), then we have h = q·h = q0·h = q0·(−h) = −q0 · h. then sq(h) = sq(−q0 · h) is of type i and having cardinality equal to 1. for the case where h ∈ h2i, 2 ≤ i ≤ k, we have the same result. suppose h ∈ h2i, for a given 2 ≤ i ≤ k, and assume sq(h) is of type i. then |sq(h)| = ν2i is odd (see [7, remark 2.6 (2)]). moreover, the elements of h2i are partitioned into q-cyclotomic classes of the same type and size (see [7, remark 2.5 (ii)]). thus, ν2i divides |h2i|. in particular, ν2i divides |2k−iz2k \ 2k−i+1z2k| = 2i − 2i−1 = 2i−1. since ν2i is odd, it must be 1. furthermore, it can be shown that µ2i = 2 for all i = 2, 3, . . . ,k. note that 2i | (q− 1) since ν2i = 1 and thus, 2i | (q20 − 1). we show that indeed, µ2i = 2. suppose contrary, i.e., µ2i = 1 = ν2i. it implies that q0 · h = h and −h = −q0 · h = q · h = h, since sq(h) is assumed to be of type i. it implies that h = 0 or ord(h) = 2 which contradicts that h ∈ h2i, i = 2, 3, . . . ,k. we state these observations in the following lemma. lemma 3.5. let h ∈ h2i, for a given 0 ≤ i ≤ k. if sq(h) is of type i, then ν2i = 1. moreover, µ2i = 2 for all i = 2, 3, . . . ,k. in the next proposition, we give the necessary and sufficient conditions for a q-cyclotomic class of h to be of type i or type ii. since all q-cyclotomic classes in h2i are of the same type and size, we characterize the q-cyclotomic classes of h through its subsets h2i, for 0 ≤ i ≤ k, keeping in mind that sq(h) is always of type i, for all h ∈ h1 ∪h2. proposition 3.6. let h ∈ h2i, for a given 0 ≤ i ≤ k. then sq(h) is of type i if and only if q0 ≡ −1 (mod 2i). equivalently, sq(h) is of type ii if and only if q0 6≡−1 (mod 2i). proof. clearly, the proposition holds for the case where h ∈ h1 ∪ h2. now, consider h ∈ h2i, for a given 2 ≤ i ≤ k, and assume sq(h) is of type i. from lemma 3.5, ν2i = 1 and µ2i = 2. thus, q ≡ 1 (mod 2i) and q0 6≡ 1 (mod 2i). hence, q0 ≡−1 (mod 2i). 10 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 on the other hand, assume q0 ≡−1 (mod 2i). thus, for each h ∈ h2i, −q0 ·h = h ∈ sq(h). hence, sq(h) is of type i. remark 3.7. using proposition 3.6, we can completely classify the sets h2i, 0 ≤ i ≤ k, that contain q-cyclotomic classes of type i or type ii. choose the largest integer 0 ≤ r′ ≤ k such that 2r ′ |(q0 + 1). hence, by proposition 3.6 h2i contains q-cyclotomic classes of type i for all i = 0, 1, . . . ,r′ and the rest of the sets h2j contain elements of type ii, for j = r′ + 1, . . . ,k. this will lead to a decomposition of fq[h]. let r′ be a positive integer as described in remark 3.7. since ν2i = 1 for all 0 ≤ i ≤ r′, then ri = r′∑ i=0 |h2i| ν2i = 2r ′s and rii = k∑ r=r′+1 |h2r| 2ν2r = k∑ r=r′+1 2rs − 2(r−1)s 2ν2r . thus, from (1), this will give the following decomposition, fq[h] ∼=  2r′s∏ i=1 fq  ×   k∏ r=r′+1   2rs−2(r−1)s 2ν2r∏ j′=1 (fqν2r ×fqν2r )     . similar with (3), every h-quasi-abelian code c in fq[g] can be written as c ∼=  2r′s∏ i=1 ci  ×   k∏ r=r′+1   2rs−2(r−1)s 2ν2r∏ j′=1 ( dr,j′ ×d′r,j′ )   , (8) where ci, dr,j′ and d′r,j′ are linear codes of length l over fq, fqν2r and fqν2r , respectively, for i = 1, 2, . . . , 2r ′s, r = r′ + 1, . . . ,k, and j′ = 1, 2, . . . , (2rs − 2(r−1)s)/2ν2r. given the decomposition of c in (8), we deduce the next proposition. proposition 3.8. let h ≤ g be finite abelian groups such that h ∼= (z2k)s, gcd(|h|,q) = 1 and l = [g : h]. let 0 ≤ r′ ≤ k be the largest integer such that 2r ′ |(q0 + 1). the number of hermitian self-dual h-quasi-abelian codes in fq[g] is    l2−1∏ i=0 (qi+ 1 2 + 1)2 r′s     k∏ r=r′+1   l∑ i=0 i−1∏ j=0 (qν2r )l − (qν2r )j (qν2r )i − (qν2r )j   2rs−2(r−1)s 2ν2r   if l is even, 0 if l is odd. proof. the result follows from (8) and proposition 2.3 by counting the number of all possible hermitian self-dual linear codes ci over fq of length l and linear codes dr,j′ over fqν2r of length l, for i = 1, 2, . . . ,r′s, r = r′ + 1, . . . ,k, and j′ = 1, 2, . . . , (2rs − 2(r−1)s)/2ν2r, then apply formulas (5) and (6). a specific case of proposition 3.8 is given in the example below, where h ∼= (z2)s (i.e., r′ = k = 1) is an elementary 2-group. 11 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 example 3.9. let h ≤ g be finite abelian groups such that h ∼= (z2)s, gcd(|h|,q) = 1 and l = [g : h]. the number of hermitian self-dual h-quasi-abelian codes in fq[g] is  l 2 −1∏ i=0 (qi+ 1 2 + 1)2 s if l is even, 0 if l is odd. table 3.1 illustrates proposition 3.8 when q = 9, l = 2, for k = 1, 2 and s = 1, 2. note that in the last column, a·b gives the number of the resulting codes. moreover, since the value of k ≤ 2 and q0 = 3, then r′ = k, for k = 1, 2. hence, the second factor in the formula given by b is empty and set to be 1. in other words, all cyclotomic classes of h is of type i, for k = 1, 2. in this case, the numbers in the last column of the table also gives the number of hermitian self-dual 1-generator h-quasi-abelian codes as presented in corollary 4.5 (i). table 1. number of hermitian self-dual h-quasi-abelian codes in fq[g], h ∼= (z2k) s, l = [g : h] = 2 and q = 9. s k |h| |g| r′ a = (q0 + 1)2 r′s b = ∏k r=r′+1(q ν2r + 3)|h2r |/2ν2r a ·b 1 1 2 4 1 16 1 16 2 4 8 2 256 1 256 2 1 4 8 1 256 1 256 2 16 32 2 416 1 416 3.2. h ∼= (zpk)s, where p is an odd prime to complete our characterization, consider h ∼= (zpk)s, k,s > 0, where p is an odd prime and gcd(p,q) = 1. recall that in the case p = 2, there is a chance that the q-cyclotomic classes of h are divided exactly into classes of type i and type ii. it is interesting to note that it is a totally different situation when p is odd. specifically, we show that all non-zero elements in h belong to just one type of q-cyclotomic classes. moreover, the necessary and sufficient conditions for them to be of type i or type ii are determined. recall that hpi is the set containing all elements of h of order pi, i = 0, 1, . . . ,k and h = h1∪hp∪·· ·∪hpk. note that sq(0) = {0} = h1 is of type i. we start with hp the characterization of q-cyclotomic classes of h. proposition 3.10. let h ∈ hp. then sq(h) is of type i if and only if ordp(q) is odd and ordp(q0) is even. equivalently, sq(h) is of type ii if and only if ordp(q) is even or ordp(q0) is odd. proof. following the notation introduced above, let νp = ordp(q). if h ∈ hp, then qνp ·h = h. assume sq(h) is of type i. then −q0 · h = qi · h = q2i0 · h for some 0 ≤ i < νp. it follows that h = −q2i−10 ·h = −q 2i−2 0 (q0 ·h) = −q 2i−2 0 (−q 2i 0 ·h) = q 2(2i−1) 0 ·h = q (2i−1) ·h which implies νp|(2i− 1). hence, νp is odd. we note that ordp(q0) ∈{νp, 2νp}. if ordp(q0) = νp, then h = q νp 0 ·h = q 2i−1 0 ·h = −h, which implies that h = 0, a contradiction. hence, ordp(q0) = 2νp, which is even. conversely, assume that ordp(q) is odd and ordp(q0) is even. it follows that ordp(q) = νp and ordp(q0) = 2νp. then h = qνp ·h = q 2νp 0 ·h, i.e., (q νp 0 − 1)(q νp 0 + 1) ·h = 0. since ordp(q0) = 2νp, we have p (qνp0 − 1), and hence, (q νp 0 + 1) ·h = 0. it follows that q0(q νp 0 + 1) ·h = (q νp+1 2 + q0) ·h = 0. since νp is odd, νp + 1 is even. which implies that −q0 ·h = q νp+1 2 ·h ∈ sq(h). therefore, sq(h) is of type i as desired. 12 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 next, we show that all q-cyclotomic classes of h \ {0} are of the same type. because of this, the q-cyclotomic classes of h are completely characterized. proposition 3.11. let a ∈ hp and b ∈ hpi, for any given 1 ≤ i ≤ k. then, sq(a) is of type i if and only if sq(b) is of type i. equivalently, sq(a) is of type ii if and only if sq(b) is of type ii. proof. let a ∈ hp and assume that sq(a) is of type i. then, by proposition 3.10, νp = ordp(q) is odd and µp = ordp(q0) = 2νp is even. we show that pi | ( qνp·p i−1 − 1 ) by induction on i. it is clear when i = 1. now, assume pi−1 | ( qνp·p i−2 − 1 ) , for 1 < i ≤ k. then, qνp·p i−2 ≡ 1 (mod pi−1) and hence, qνp·p i−2·j ≡ 1 (mod pi−1) for all j ≥ 0. thus, ∑p−1 j=0 q νp·pi−2·j ≡ ∑p−1 j=0 1 (mod p i−1). this implies that p | (∑p−1 j=0 q νp·pi−2·j ) . since qνp·p i−1 − 1 = ( qνp·p i−2 − 1 )(∑p−1 j=0 q νp·pi−2·j ) , pi−1 | ( qνp·p i−2 − 1 ) and p | (∑p−1 j=0 q νp·pi−2·j ) , it follows that pi | ( qνp·p i−1 − 1 ) . therefore, νpi | νp ·pi−1 and means νpi is odd. note that µpi ∈ {νpi, 2νpi}. since µp is even, νpi is odd and µp | µpi hence, µpi = 2νpi. hence, pi | ( q 2ν pi 0 − 1 ) and pi ( q ν pi 0 − 1 ) . it follows that pi | ( q ν pi 0 + 1 ) . in other words, q0(q ν pi 0 + 1) ·b = 0 or −q0 · b = q ν pi +1 0 · b = q ν pi +1 2 · b ∈ sq(b) for each b ∈ hpi. conversely, assume that sq(b) is of type i, for all b ∈ hpi. then, −q0 ·b = qj ·b for some 0 ≤ j < νpi. it follows that −q0(pi−1 · b) = qj(pi−1 · b), which implies sq(pi−1 · b) is of type i. since pi−1 · b ∈ hp, sq(a) and sq(pi−1 · b) are of the same type. combining propositions 3.10 and 3.11, the corollary below follows immediately. corollary 3.12. let h be a non-zero element in h ∼= (zpk)s, p is odd and gcd(p,q) = 1. then sq(h) is of type i if and only if ordp(q) is odd and ordp(q0) is even. equivalently, sq(h) is of type ii if and only if ordp(q) is even or ordp(q0) is odd. we are now ready to obtain a decomposition for fq[h]. this entails computing for ri and rii. if there exists h ∈ h \{0} such that sq(h) is of type i, then by corollary 3.12, rii = 0 and ri = k∑ i=0 | hpi | νpi = k∑ i=0 pis −p(i−1)s νpi , where νp0 = ν1 = 1 and pis − p(i−1)s is equal to 1 when i = 0. on the other hand, if there exists h ∈ h \{0} such that sq(h) is of type ii, then corollary 3.12 implies that ri = |h1| = 1 and rii = k∑ i=1 | hpi | 2νpi = k∑ i=1 pis −p(i−1)s 2νpi . recall that νp := ordp(q) and µp := ordp(q0). from the above calculations, together with corollary 3.12 and (1), we have fq[h] ∼=   fq ×   k∏ i=1   2is−2(i−1)s ν pi∏ j′=1 f q ν pi     if νp is odd and µp is even, fq ×   k∏ i=1   2is−2(i−1)s 2ν pi∏ j=1 ( f q ν pi ×fqνpi )   if νp is even or µp is odd. 13 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 it also implies that an h-quasi-abelian code c in fq[g] can be decomposed as c ∼=   c1 ×   k∏ i=1   2is−2(i−1)s ν pi∏ j′=1 ci,j′     if νp is odd and µp is even, c1 ×   k∏ i=1   2is−2(i−1)s 2ν pi∏ j=1 ( di,j ×d′i,j )     if νp is even or µp is odd, (9) where c1 and ci,j′ are linear codes of length l over fq and fqνpi , respectively, for i = 1, 2, . . . ,k and j′ = 1, 2, . . . , (2is − 2(i−1)s)/νpi. similarly, both di,j and d′i,j are linear codes of length l over fqνpi , for i = 1, 2, . . . ,k and j = 1, 2, . . . , (2is − 2(i−1)s)/2νpi. the above decomposition of the code c will lead us to the following proposition. proposition 3.13. let h ≤ g be finite abelian groups such that h ∼= (zpk)s, p is odd, gcd(|h|,q) = 1 and l = [g : h] is even. the number of hermitian self-dual h-quasi-abelian codes in fq[g] is    l2−1∏ i=0 (qi+ 1 2 + 1)     k∏ i=1   l2−1∏ r=0 ( (qνpi )r+ 1 2 + 1 ) pis−p(i−1)s ν pi   if νp is odd and µp is even,   l2−1∏ i=0 (qi+ 1 2 + 1)     k∏ i=1   l∑ r=0 r−1∏ j=0 (qνpi )l − (qνpi )j (qνpi )r − (qνpi )j   pis−p(i−1)s 2ν pi   if νp is even or µp is odd. proof. apply the same arguments as in the proof of proposition 3.8 to (9). an example is given when h ∼= (zp)s is an elementary p-group. example 3.14. let h ≤ g be finite abelian groups such that h ∼= (zp)s, p is odd, gcd(|h|,q) = 1 and the index l = [g : h] is even. then the number of hermitian self-dual h-quasi-abelian codes in fq[g] is  l 2 −1∏ i=0 (qi+ 1 2 + 1) ( (qνp)i+ 1 2 + 1 )ps−1 νp if νp is odd and µp is even,  l2−1∏ i=0 (qi+ 1 2 + 1)     l∑ r=0 r−1∏ j=0 (qνp)l − (qνp)j (qνp)r − (qνp)j   ps−1 2νp if νp is even or µp is odd. see table 3.2 for the number of hermitian self-dual h-quasi-abelian codes when p = 3, q = 4, l = 2, for k = 1, 2 and s = 1, 2. in this case, νp = 1 and µp = 2. then the q-cyclotomic classes of h are all of type i, and hence, this table also illustrates the 1-generator case given in corollary 4.5 (ii), type i case. 4. hermitian self-dual 1-generator quasi-abelian codes in this section, we study 1-generator h-quasi-abelian codes in fq[g], a cyclic fq[h]-module of fq[g], where h ≤ g are finite abelian groups such that gcd(|h|,q) = 1. the main idea here is to use [6, theorem 6.1] and combine it with the characterization of hermitian self-dual h-quasi-abelian codes obtained in 14 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 table 2. number of hermitian self-dual h-quasi-abelian codes in fq[g], h ∼= (z3k) s, l = [g : h] = 2 and q = 4. s k |h| |g| a = (q0 + 1) b = ∏k i=1 (q ν pi + 1) |h pi |/ν pi a ·b 1 1 3 6 3 9 27 2 9 18 3 729 2187 2 1 9 18 3 6561 19683 2 81 162 3 38 ·924 3 ·38 ·924 proposition 2.3. we also consider the case where h ∼= (zpk)s, for p = 2 or p is odd, and obtain explicit enumeration. from [6], we have the following characterization of 1-generator quasi-abelian codes. theorem 4.1 ([6, theorem 6.1]). let q be a prime power and let h ≤ g be finite abelian groups with l = [g : h] and gcd(|h|,q) = 1. let e1,e2, . . . ,et be the primitive idempotents of fq[h]. in the light of (3), let c ∼= t∏ i=1 ci be an h-quasi-abelian code in fq[g], where ci is a linear code of length l over li ∼= fq[h]ei. then c is 1-generator if and only if the li-dimension of ci is at most 1, for each i = 1, 2, . . . , t. since the fq-dimension of a 1-generator h-quasi-abelian code c in fq[g] cannot exceed |h|, c⊥h could never be a 1-generator if [g : h] > 2. in the case where [g : h] = 2, we have the following characterization. corollary 4.2. assume the notation in theorem 4.1. in addition, we assume that [g : h] = 2. if c is a 1-generator h-quasi-abelian code in fq[g], then the following statements are equivalent. i) c⊥h is a 1-generator h-quasi-abelian code. ii) ci has li-dimension 1 for all i = 1, 2, . . . , t. iii) the fq-dimension of c is |h|. proof. the corollary follows immediately from theorem 4.1 and observations similar to those in [12, corollary 3.2]. combining proposition 2.3 and corollary 4.2, we conclude the following characterization for hermitian self-dual 1-generator quasi-abelian codes (cf. [12, theorem 3.3]). corollary 4.3. a 1-generator h-quasi-abelian code c in fq[g] is hermitian self-dual if and only if [g : h] = 2 (i.e., g = z2 ×h) and, in (3), c is decomposed as c ∼= ( ri∏ i=1 ci ) × ( rii∏ k=1 ( dj ×d⊥ej )) , where i) ci is hermitian self-dual of length 2 over ei for all i = 1, 2, . . . ,ri, and ii) dj is a linear code of dimension 1 and length 2 over kj for all j = 1, 2, . . . ,rii. 15 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 the enumeration of hermitian self-dual 1-generator quasi abelian codes immediately follows. corollary 4.4. let h ≤ g be finite abelian groups such that gcd(|h|,q) = 1, and [g : h] = 2. assume that fq[h] is decomposed as in (1) and contains ri (resp., 2rii) primitive idempotents of type i (resp., ii). assume further that the primitive idempotents of type i are induced by q-cyclotomic classes of size si for each i = 1, 2, . . . ,ri and the primitive idempotents of type ii are induced by q-cyclotomic classes of sizes tj and t′j, pair-wise, for each j = 1, 2, . . . ,rii. then the number of hermitian self-dual 1-generator h-quasi-abelian codes in fq[g] is ri∏ i=1 (qsi0 + 1) rii∏ j=1 (qtj + 1). proof. the corollary follows from corollary 4.3, (6), and the fact that the number of 1-dimensional subspaces of f2 q tj is qtj + 1. we end this paper by considering the case of hermitian self-dual 1-generator h-quasi-abelian codes where h are some p-groups. corollary 4.5. let h ≤ g be finite abelian groups such that h ∼= (zpk)s, gcd(|h|,q) = 1 and l = [g : h] = 2 (i.e., g = z2 ×h). then one of the following statements holds. i) if p = 2, q is odd and 0 ≤ r′ ≤ k is the largest integer such that 2r ′ |(q0 + 1), then the number of hermitian self-dual 1-generator h-quasi-abelian codes in fq[g] is (q0 + 1) 2r ′s ( k∏ r=r′+1 (qν2r + 1) 2rs−2(r−1)s 2ν2r ) . ii) if p is odd and gcd(p,q) = 1, then the number of hermitian self-dual 1-generator h-quasi-abelian codes in fq[g] is  k∏ i=0 ( q ν pi 0 + 1 )pis−p(i−1)s ν pi if νp is odd and µp is even, (q0 + 1) ( k∏ i=1 (qνpi + 1) pis−p(i−1)s 2ν pi ) if νp is even or µp is odd. proof. the first statement is derived using (8) and corollary 4.3 by getting the number of hermitian self-dual codes ci over fq of length l = 2, for i = 1, 2, . . . , 2r ′s, and the number of 1-dimensional linear codes dr,j′ of length l = 2 over fqν2r which is equal qν2r + 1, for r = r′ + 1, . . . ,k and j′ = 1, 2, . . . , (2rs − 2(r−1)s)/2ν2r. suppose p is odd, gcd(p,q) = 1, νp is odd and µp is even. this case follows directly from proposition 3.13 by letting l = 2 and noting that q = q20. on the other hand, suppose νp is even or µp is odd. we apply corollary 4.3 and (9). the first factor is obtained by counting the number of hermitian self-dual codes c1 of length 2 over fq. for the second factor, we count the number of 1-dimensional linear codes di,j over fqνpi , given by q ν pi + 1, for each i = 1, 2, . . . ,k, and j = 1, 2, . . . , (pis −p(i−1)s)/2νpi. for the case where h is an elementary p-group, we have the following example. example 4.6. let h ≤ g be abelian groups such that h ∼= (zp)s, an elementary p-group, gcd(|h|,q) = 1 and l = [g : h] = 2 (i.e., g = z2 ×h). then one of the following statements holds. 16 h. s. palines et al. / j. algebra comb. discrete appl. 5(1) (2018) 5–18 i) if p = 2 and q is odd, then the number of hermitian self-dual 1-generator h-quasi-abelian codes in fq[g] is (q0 + 1) 2s. ii) if p is odd and gcd(p,q) = 1, then the number of hermitian self-dual 1-generator h-quasi-abelian codes in fq[g] is  (q0 + 1)(q νp 0 + 1) ps−1 νp if νp is odd and µp is even, (q0 + 1)(q νp + 1) ps−1 2νp if νp is even or µp is odd. 5. summary characterization and enumeration of hermitian self-dual quasi-abelian codes were established based on the well-known decomposition of quasi-abelian codes. necessary and sufficient conditions for the existence of hermitian self-dual 1-generator quasi-abelian codes were also given. for special cases where the underlying groups are some p-groups, complete classification of cyclotomic classes has been done. as a result, the actual number of resulting hermitian self-dual quasi-abelian codes has been determined. it is interesting to note that the results in this work is restricted to fq[h] being a semi-simple group algebra, i.e., the characteristic of fq and |h| are coprime, where h is a finite abelian group. acknowledgment: the authors would like to thank the anonymous referees for their helpful comments. h. s. palines wishes to thank the university of the philippines systems for providing the alternative study “sandwich” grant while doing this research at the mathematics department, faculty of science, silpakorn university, nakhon pathom, thailand. also, to the department of science and technology-science education institute (dost-sei) of the philippines for his ph.d. scholarship grant. s. jitman is supported by the thailand research fund under research grant trg5780065. references 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[18] s. k. wasan, quasi abelian codes, publ. inst. math. 21(35) (1977) 201–206. 18 https://doi.org/10.1109/tit.2003.809501 https://doi.org/10.1109/tit.2003.809501 https://doi.org/10.1109/tit.2005.850142 https://doi.org/10.1109/tit.2005.850142 https://link.springer.com/book/10.1007/3-540-30731-1 https://link.springer.com/book/10.1007/3-540-30731-1 http://dx.doi.org/10.1007/s11424-007-9053-y https://doi.org/10.1016/s0021-9800(68)80067-5 https://doi.org/10.1109/18.165458 https://doi.org/10.1109/18.165458 https://doi.org/10.1109/tit.2004.831861 https://doi.org/10.1109/tit.2004.831861 http://www.ams.org/mathscinet-getitem?mr=469498 introduction preliminaries enumeration of hermitian self-dual quasi-abelian codes hermitian self-dual 1-generator quasi-abelian codes summary references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.423733 j. algebra comb. discrete appl. 5(2) • 71–83 received: 18 october 2017 accepted: 9 march 2018 journal of algebra combinatorics discrete structures and applications the root diagram for one-point ag codes arising from certain curves with separated variables research article federico fornasiero, guilherme tizziotti abstract: heegard, little and saints introduced in [8] an encoding algorithm for a class of ag codes via gröbner basis more compact compared with the usual encoding via generator matrix. so, knowing that the main drawback of gröbner basis is the high computational cost required for its calculation, in [12], the same authors introduced the concept of root diagram that allows the construction of an algorithm for computing a gröbner basis with a lower complexity for one-point hermitian codes. in [4], farrán, munuera, tizziotti and torres extended the results obtained in [12] for codes on norm-trace curves. in this work we generalize these results by constructing the root diagram for codes arising from certain curves with separated variables that has certain special automorphism and a weierstrass semigroup generated by two elements. such family of curves includes the norm-trace curve, among other curves with recent applications in coding theory. 2010 msc: 11t71, 13p10 keywords: ag codes, gröbner basis, root diagram 1. introduction in the early 1980s, v.d. goppa constructed error-correcting codes using algebraic curves, the called algebraic geometric codes (ag codes), see [6] and [7]. the introduction of methods from algebraic geometry to construct good linear codes was one of the major developments in the theory of errorcorrecting codes. from that moment many studies and applications on this theory have emerged. in [8], little, saints and heegard noted that any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring and so that the theory of gröbner bases for modules gives a compact description and implementation of a systematic encoder, which is similar to the usual one for cyclic codes. this encoding method is efficient and also interesting from a theoretical point of view. it is known that federico fornasiero; department of mathematics, universidade federal de pernambuco, brazil (email: federico@dmat.ufpe.br). guilherme tizziotti (corresponding author); department of mathematics, universidade federal de uberlândia, brazil (email: guilhermect@ufu.br). 71 https://orcid.org/0000-0003-1026-0546 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 the main drawback of gröbner basis is the high computational cost required for its calculation. indeed, it is well known that the complexity of computing a gröbner basis is doubly exponential in general. but, in [12], using an appropriate automorphism of the hermitian curve, little et al. introduced the concept of root diagram that allows construction of an algorithm for computing a gröbner basis with a lower complexity for one-point hermitian codes. in other words, the root diagram is the key to the construction of the algorithm given in [12, proposition 4.4]. in [4], the results of [12] were extended to codes arising from the norm-trace curve, which is a generalization of the hermitian curve. in this work, using the same techniques used in [12] and [4], we will construct the root diagram for codes arising from certain curves, which we will denote by x?, with separated variables that has certain special automorphism and a weierstrass semigroup generated by two elements (see (1), (2) and (3) in section 3). in addition to hermitian and norm-trace curves, we have important examples of such curves x? with recent applications in coding theory, namely: the maximal curve with plane model yq + y = xq r+1, see [11]; a quotient of the hermitian curve with plane model yq + y = xm, see [13]; and curves on kummer extensions, see [2]. this paper is organized as follows. in section 2 we recall some background on gröbner basis for modules, ag codes and root diagram. in section 3 we present a way to construct the root diagram for one-point ag codes c arising from x?. finally, in section 4 we present examples of those curves and the necessary information to construct the root diagram studied in the previous section. 2. background we will denote a finite field with q elements by fq. let x be a projective, non-singular, geometrically irreducible algebraic curve of genus g > 0 over fq; throughout the paper we will refer to this simply as curve. if ]x(fq) is the number of fq-rational points on x , then ]x(fq) ≤ q + 1 + 2g √ q. this inequality is so-called hasse-weil bound and if ]x(fq) = q + 1 + 2g √ q the curve x is called a maximal curve. let fq(x) be the field of rational functions on x . for a fq-rational point p on x let h(p) := {n ∈ n0 ; ∃f ∈ fq(x) with div∞(f) = np}, where n0 is the set of nonnegative integers and div∞(f) denotes the divisor of poles of f. the set h(p) is a numerical semigroup, called weierstrass semigroup of x at p and its complement g(p) = n0\h(p) is called weierstrass gap set of p . as an important result, the cardinality of the set g(p) is equal to genus g of x , see theorem 1.6.8 in [15]. 2.1. gröbner basis for fq[t]-modules we will introduce some notations about gröbner basis for fq[t]-modules that are needed later. for a complete treatment on this topic see [1] and [3]. a monomial m in the free fq[t]-module fq[t]r is an element of the form m = tiej, where i ≥ 0 and e1, . . . ,er is the standard basis of fq[t]r. fixed a monomial ordering, for all element f ∈ fq[t]r, with f 6= 0, we may write f = a1m1 + · · · + a`m`, where, for 1 ≤ i ≤ `, 0 6= ai ∈ fq and mi is a monomial in fq[t]r satisfying m1 > m2 > ... > m`. the term a1m1 is called leading term of f and denoted by lt(f), the coefficient a1 and the monomial m1 are called leading coefficient, lc(f), and leading monomial, lm(f), respectively. a gröbner basis for a submodule m ⊆ fq[t]r is a set g = {g1, . . . ,gs} such that {lt(g1), . . . ,lt(gs)} generates the submodule lt(m) formed by the leading terms of all elements in m. the monomials in lt(m) are called nonstandard while those in the complement of lt(m) are the standard monomials for m. we recall that every submodule m ⊆ fq[t]n has a gröbner basis g, which induces a division algorithm: given f ∈ fq[t]r there exist a1, . . . ,as,rg ∈ fq[t]r such that f = a1g1 + . . .+ asgs + rg (see [1, algorithm 1.5.1] or [3, theorem 3]). in this work we will use the pot (position over term) ordering over fq[t]r which is defined as follows. let {e1, . . . ,er} be the standard basis in fq[t]r, with e1 > ... > er. the pot ordering on fq[t]r is defined 72 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 by tiej > t ke` if j < `, or j = ` and i > k. we say that f ∈ fq[t]r is reduced with respect to a set p = {p1, . . . ,pl} of non-zero elements in fq[t]r if f = 0 or no monomial in f is divisible by a lm(pi), i = 1, . . . , l. a gröbner basis g = {g1, . . . ,gs} is reduced if gi is reduced with respect to g −{gi} and lc(gi) = 1 for all i, and non-reduced otherwise. every submodule of fq[t]r has a unique reduced gröbner basis (see [1], theorem 3.5.22). 2.2. linking ag codes and fq[t]-modules let p1, . . . ,pn,q1, . . . ,q` be n + ` distinct fq-rational points on x and let m1, . . . ,m` be integers. consider the divisors d = p1 + · · ·+ pn and g = m1q1 + · · ·+ m`q`. the algebraic geometry code (ag code) cx(d,g) arising from the curve x is defined as cx(d,g) := {(f(p1), . . . ,f(pn)) ∈ fnq : f ∈l(g)} , (1) where l(g) is the space of rational functions f on x such that f = 0 or div(f) + g ≥ 0, where div(f) denote the (principal) divisor of the function f ∈ l(g). the number n is the length of cx(d,g), and the dimension of cx(d,g) is its dimension as an fq-vector space, which is generally denoted by dim(cx(d,λp)) := k. the elements in cx(d,g) are called codewords. if g = m1q1 the ag code cx(d,m1q1) is called one-point ag code. for more details about ag codes, see e.g. [10]. let supp = {p1, . . . ,pn} the support of the divisor d. since |supp(d)| = n, we have that the permutation group p(supp(d)) on supp(d) is isomorphic to the symmetric group sn, and each σ ∈ p(supp(d)) induces a fq-linear mapping σ̂ of the code cx(d,g) to fnq by setting σ̂(f(p1), . . . ,f(pn)) := (f(σ(p1)), . . . ,f(σ(pn))). the mapping σ̂ is an automorphism of the code cx(d,g)) if σ̂(cx(d,g)) = cx(d,g). in [7], goppa already observed that the underlying algebraic curve induces automorphism of the associated ag codes as follows. proposition 2.1. let aut(x) = {σ : x → x ; σ is birational } be the automorphism group of x over fq and, for divisors d and g, consider the subgroup autd,g(x) = {σ ∈ aut(x) : σ(d) = d and σ(g) = g} . then, each σ ∈ autd,g(x) induces an automorphism of cx(d,g) by σ̂(f(p1), . . . ,f(pn)) = (f(σ(p1)), . . . ,f(σ(pn))) . assume that x has a nontrivial automorphism σ ∈ autd,g(x) and let h be the cyclic subgroup of aut(x) generated by σ. let supp(d) = o1 ∪ . . . ∪ or be the decomposition of the support of d into disjoint orbits under the action of σ. then, by proposition 2.1, the entries of the codewords in cx(d,g) will be cyclically permuted in several blocks by σ. we will denote σ0 = id, where id is the identity automorphism, and, for a positive integer j, σj = σ ◦σ ◦ . . .◦σ︸ ︷︷ ︸ j . in this way, for each i = 1, . . . ,r, pick any one point pi,0 ∈ oi and enumerate the other points on oi as pi,j = σj(pi,0), where j runs from 1 to |oi|−1. using this fact, we get the following result. lemma 2.2. let cx(d,g) be an ag code arising from x over fq and let σ ∈ autd,g(x) be a nontrivial automorphism. if supp(d) = o1 ∪ . . .∪or is the decomposition of the support of d into disjoint orbits under the action of σ, then there is a one-to-one correspondence between cx(d,g) and a submodule c of the free module fq[t]r. 73 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 proof. suppose that supp(d) = o1 ∪ . . .∪or is the decomposition of the support of d into disjoint orbits under the action of σ. for each i = 1, . . . ,r, let oi = {pi,0, . . . ,pi,|oi|−1}, where pi,j = σ j(pi,0) for each j = 1, . . . , |oi|−1, and let hi(t) = ∑|oi|−1 j=0 f(pi,j)t j. the r-tuples (h1(t), . . . ,hr(t)) can be seen also as an element of the fq[t]-module a =⊕r i=1 fq[t]/〈t |oi| − 1〉. so, the collection c̃ of r-tuples obtained from all f ∈ l(g) is closed under sum and multiplication by t. define c := π−1(c̃), where π is the natural projection from fq[t]r onto⊕r i=1 fq[t]/〈t |oi|−1〉. thus, we get a one-to-one correspondence between cx(d,g) and c ≤ fq[t]r. by the previous lemma, an ag code cx(d,g) can be identified with a submodule c ≤ fq[t]r and thus the standard theory of gröbner basis for modules may be applied. suppose that cx(d,g) has length n and dimension k. a gröbner basis g = {g(1), . . . ,g(r)} for c ≤ fq[t]r with exactly r elements allows us to obtain a systematic encoding of c. since {lt(g(1)), . . . ,lt(g(r))} generates lt(c), it follows that the nonstandard monomials appearing in the r-tuples (h1(t), . . . ,hr(t)) can be obtained from the g(i)’s. by ordering these monomials in decreasing order we obtain the so-called information positions of (h1(t), . . . ,hr(t)), which are the first k monomials ml = tilejl, l = 1, . . . ,k. let v c(h1(t), . . . ,hr(t)) be the vector of coefficients of the terms of (h1(t), . . . ,hr(t)) listed in the pot order. we have the following systematic encoding algorithm: algorithm 2.3. input: a gröbner basis g, monomials {m1, . . . ,mk} and w = (w1, . . . ,wk) ∈ fkq. output: c(w) ∈ c = c(x ,d,g). 1. set f := w1m1 + · · ·+ wkmk. 2. compute f = a1g(1) + . . . + arg(r) + rg. 3. return c(w) := v c(f −rg). this method is more compact compared with the usual encoding via generator matrix. the total amount of computation is roughly the same and the amount of necessary stored data is lower in this method, of order r(n−k) against k(n−k) when encoding via generator matrix. more details about this encoding algorithm can be found in [8]. 2.3. the root diagram consider the one-point ag code c = cx(d,λp) and suppose that x has an automorphism σ fixing the divisors d and g = λp . suppose also that the order of σ is a factor of q−1. let c be the submodule of fq[t]r associated to c by the automorphism σ. using the pot ordering we can get a gröbner basis g = {g1, . . . ,gr} for c such that gi = (0, . . . ,0,g (i) i (t),g (i+1) i (t), . . . ,g (r) i (t)), for all i = 1, . . . ,r, see [[8], proposition ii.b.4]. note that, if deg(g(i)i (t)) = di, then g (i) i (t) has di distinct roots in f ∗ q = fq \ {0}. in fact, let qi = (t |oi| − 1)ei. note that qi ∈ π−1(0, . . . ,0) and we have that qi ∈ c. since |oi| divides the order of σ, it follows that t|oi| − 1 divides tq−1 − 1 = ∏ a∈f∗q (t − a). now, lt(g(i)) = g(i)i (t) divides lt(qi) = t |oi| −1, and the claim follows from the fact that tq−1 −1 has q −1 distinct roots in fq. for i = 1, . . . ,r, let ri ⊆ f∗q be the set of roots of t|oi| − 1. by a root diagram dc for the code c, we mean a table with r rows. for each i, the boxes on the i-th row correspond to the elements of ri. we mark the roots of g(i)i (t) on the i-th row with a x in the corresponding box. by proposition ii.c.1 in [8], there is a fq-basis for c in one-to-one correspondence with the nonstandard monomials in c. that is, terms of the form t`ej appearing as leading terms of some element of c, with ` ≤ |oj|−1. now, if there are mj empty boxes on row j of the root diagram, then g (i) j (t) has |oj|−mj roots and lt(g(j)) = t|oj|−mj. so, we obtain mj nonstandard monomials t`ej. this fact gives us the following important result. 74 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 proposition 2.4. ([12], proposition 2.3) the dimension of the code c is equal to the number of empty boxes in the root diagram dc. 3. the root diagram for certain one-point ag codes let x? be the curve defined over fq by affine equation f(y) = g(x) and that has the following conditions: 1. f(t),g(t) ∈ fq[t ], deg(f) = a and deg(g) = b, with gcd(a,b) = 1; 2. there exists a point p on x? such that div∞(x) = ap , div∞(y) = bp, and h(p) = 〈a,b〉; 3. there exists σ ∈ autd,g(x?), where g = λp for some positive integer λ, given by σ(x) = αx and σ(y) = αty, for some positive integer t and some α ∈ f∗q. we observe that if the order of α is equal to ord(α) := ν, then, by the definition of σ, it follows that the order of σ is equal to ν which divides q −1. we can ask if curves with such conditions do exist or if there are a large number of them. by [15, prop. 6.4.1] and [9, lemma 12.2 and th. 12.9], we have that the curves defined over a finite field fqs by the affine equation yq n + y = xm, with gcd(q,m) = 1, are examples of curves that satisfy the above conditions if m(qn−1) divides qs−1. we note that it is hard to study automorphisms of curves in general, especially without giving the equation that defines it. in particular, with the results on automorphisms known to date, we can not present general examples of curves satisfying the above conditions. for a study on automorphism of algebraic curves we refer the reader to [9, ch. 11 and 12], particularly in the section 12.1 results on automorphisms of curves given by separated polynomials can be found. let d = p1 + . . . + pn, with pi 6= p for all i = 1, . . . ,n, a divisor on x? and let supp(d) = o1 ∪ . . .∪or ∪or+1 ∪ . . .∪or+s be the decomposition of the support of d into disjoint orbits under the action of σ. in this section we will describe the root diagram for one-point ag codes cx?(d,λp). note that, by definition of σ, if q = (0,η) ∈ oi, for some η ∈ fq, then oi = {(0,η),(0,αtη), . . . ,(0,αt.tiη)}, where ti is the smallest nonnegative integer such that αt.(ti+1) = 1. analogously, if q = (ω,0) ∈ oi, for some ω ∈ fq, then oi = {(ω,0),(αω,0), . . . ,(αν−1ω,0)}. let or+1, . . . ,or+s be the orbits that contains fq-rational points on x? of the form (0,η) or (ω,0). we will work with the first r rows of the root diagram dc for the code cx?(d,λp), the results for the last s rows are similar can be obtained in particular cases. for each i = 1, . . . ,r, suppose that oi = {pi,0,pi,1, . . . ,pi,|oi|−1}, where pi,0 = (xi,yi), with xi 6= 0 and yi 6= 0, and pi,j = σ j(pi,0) = (αjxi,α jtyi). so, from the definition of σ it follows that |o1| = . . . = |or| = ord(α) = ν. associated with the decomposition of the support of d into disjoint orbits under the action of σ as above, let us assume the following conditions: (i) for each i = 1, . . . ,r, there exists a polynomial mi(y) such that the orbit oi is the intersection of supp(d) with the curve mi(y) = 0 and, for all i, mi(y) is a non-zero constant when restricted to each of the orbits ok, k 6= i; (ii) for each 1 ≤ i ≤ r and 0 ≤ j ≤ |oi| − 1 = ν − 1, there exists a polynomial bi,j(x,y) such that bi,j(x,y) vanishes at each point of oi except pi,j. in the proposition 3.3 and the theorem 3.4 below, the reader will see that the existence of these polynomials is fundamental to obtain the root diagram for one-point ag codes cx?(d,λp). let κ be the smallest positive integer such that ακt = 1. the next result gives us a way to get mi(y). proposition 3.1. let κ, oi and pi,j = σj(pi,0) = (αjxi,αjtyi) be as above. if (∗) α`tyi 6= α`tyk, for all ` = 0,1, . . . ,κ−1 and k 6= i, 75 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 then mi(y) = ∏κ−1 `=0 (y −α `tyi) satisfies the condition (i) above. proof. by the definition of κ, it follows that the orbit oi is the intersection of supp(d) with the curve mi(y) = 0. the condition (∗) implies that mi(y) is a non-zero constant when restricted to each of the orbits ok, k 6= i. note that the condition (∗), which is the key to getting a polynomial mi(y) as in (i), depends on the decomposition of the support of d and the coordinates of the points on such support. we obtain bi,j(x,y) in a similar way by using a solution of an interpolation problem. lemma 3.2. for i = 1, . . . ,r and j = 0, . . . , |oi|− 1, let mi(y) and bi,j(x,y) be as (i) and (ii) above. then, div∞(mi) = (ρ1b)p and div∞(bi,j) = (ρ2a + ρ3b)p, where ρ1,ρ2 and ρ3 are nonnegative integers. proof. we have that div∞(x) = ap and div∞(y) = bp . then, the result follows from the fact that mi(y) and bi,j(x,y) are polynomials. let ρ1,ρ2 and ρ3 be as the previous lemma. so, for λ ≤ (ρ2a+ρ3b)+r(ρ1b), we can get the following information about the root diagram dc. proposition 3.3. let cx?(d,λp) and σ be as above. let dc be the root diagram for cx?(d,λp). fix i, 1 ≤ i ≤ r, and let ρ1, ρ2 and ρ3 be as above. 1. if λ ≥ (i− 1)(ρ1b), then the i-th row of dc is not full, in the sense that not every box in the i-th row are marked with x; 2. if λ ≥ (ρ2a + ρ3b) + (i− 1)(ρ1b), then the row is empty, in the sense that none of the boxes in the i-th row is marked with x. proof. let c ≤ fq[t]r be the submodule associated to cx?(d,λp), where d = p1 + . . . + pn and pi 6= p for all i = 1, . . . ,n. 1. suppose that λ ≥ (i−1)(ρ1b). by lemma 3.2, the function fi(x,y) = m1(x,y) · · ·mi−1(x,y) belongs to l(λp) and hence (fi(p1), . . . ,fi(pn)) ∈ cx?(d,λp). by computing (fi(p1), . . . ,fi(pn)), we observe that c contains an element of the form (0, . . . ,0,hi(t), . . . ,hr(t)) with i−1 zeroes and hi(t) = ∑|oi|−1 j=0 fi(pi,j)t j. since fi(pi,j) = m1(pi,j) · · ·mi−1(pi,j) = constant c 6= 0 , we have hi(t) = c. ∑|oi|−1 j=0 t j and thus h(1) 6= 0 as |oi| divides q−1. therefore the i-th row of dc is not full, since g(i)i (t) divides hi(t). 2. now, suppose λ ≥ (ρ2 +ρ3b)+(i−1)(ρ1b). so, by lemma 3.2, gi(x,y) = bi,0(x,y)fi(x,y) ∈ l(λp) and gi(q) = 0 for q ∈ o1 ∪o2 ∪ . . .∪oi−1. moreover, gi(q) = 0 for all q ∈ oi \{pi,0}. then the element of c corresponding to (gi(p1), . . . ,gi(pn)) verifies h1(t) = h2(t) = . . . = hi−1(t) = 0 and hi(t) = gi(pi,0) = c 6= 0. thus, c contains the element (0, . . . ,0,c,hi+1(t), . . . ,hr(t)). so, the i-th row of dc is empty. let n be the number of fq-rational points on x?. by riemann-roch theorem, it follows that if λ < n, then the dimension of the one-point ag code cx?(d,λp) is equal to the dimension of the riemann-roch space l(λp). in this case, we complete the information about the root diagram dc. 76 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 theorem 3.4. let dc be the root diagram for cx?(d,λp). if there is i ∈{1, . . . ,r} such that (i−1)(ρ1b) ≤ λ < (ρ2a + ρ3b) + (i−1)(ρ1b), then the i-th row of dc is neither full, nor empty, and the complement of the set of roots marked on row i of the diagram is the set ei = {α−(β+γb) ∈ f∗q | 0 ≤ β ≤ b−1, 0 ≤ γ ≤ ρ1 −1, (i−1)(ρ1b) + βa + γb ≤ λ}. proof. let c ≤ fq[t]r be the submodule associated to cx?(d,λp). let di ⊂ f∗q be the set of nonmarked boxes in row i, where 1 ≤ i ≤ r. we will show that di = ei. let fi(y) be as in the previous proposition and consider fi(x,y) = fi(y)xβyγ. by lemma 3.2 and the conditions over β and γ given in the definition of ei, we have that fi(x,y) ∈ l(λp). so, associated to fi(x,y) we get an element h = (h1(t), . . . ,hr(t)) ∈ c. since fi(q) = 0 for all q ∈ o1 ∪ . . . ∪ oi−1, it follows that hk(t) = 0, for k = 1, . . . , i − 1. let pi,j = σ(pi,0) = (αjxi,α`jyi) ∈ oi. thus, fi(pi,j) = fi(pi,j)αjβx β i α `jγy γ i = fi(pi,j)x β i y γ i α j(β+`γ), for all j = 0,1, . . . , |oi|− 1. now, fi(pi,j), x β i and y γ i are all non-zero constants and independent of j. taking bi = fi(pi,j)x β i y γ i 6= 0, we have hi(t) = |oi|−1∑ j=0 fi(pi,j)t j = |oi| · bi |oi|−1∑ j=0 (α(β+`γ)t)j whose roots are all distinct from α−(β+`γ). consequently, α−(β+`γ) is not a root of g(i)i (t) and hence ei ⊆ di. by proposition 2.4, dim(cx?(d,λp)) = ∑ ]di. since h(p) = 〈a,b〉 and λ < n, we have that dim(cx?(d,λp)) = ]{(β,γ) ∈ n20 ; 0 ≤ β ≤ b−1 and βa + γb ≤ λ}. let êi = {(β,γ) ∈ n20 | 0 ≤ β ≤ b − 1,0 ≤ γ ≤ ρ1 − 1, (i − 1)(ρ1b) + βa + γb ≤ λ}. thus, ]{(β,γ) ∈ n20 ; 0 ≤ β ≤ b − 1 and βa + γb ≤ λ} = ∑ ]êi and, since ∑ ]êi = ] ∑ ei, it follows that∑ ]di = ∑ ]ei. therefore, ei = di. let fi(y) be as above. then, we have that fi(q) = ci ∈ f∗q, for all q ∈ oi. with the conditions of the above theorem, fix an index i, 1 ≤ i ≤ r, where the row i of dc is neither full, nor empty. let αk1,αk2, . . . ,αk` be the roots marked on the row i and let p(t) = ∏` j=1(t − α kj) be the unique monic polynomial of degree ` with these roots. note that, including zeroes for powers of t higher than the number of roots, we can write p(t) = ∑|oi|−1 j=0 ajt j, where aj = 0 for j > `. consider the function fi(x,y) = fi(y) ci  |oi|−1∑ j=0 aj bi,j(x,y) bi,j(pi,j)   then, by the definition of fi(y) and bi,j(x,y), it is clear that fi(x,y) ∈ l(λp) and its associated module element h ∈ c has i−1 leading zero components and i-th component hi(t) equal to p(t). using this fact and the same procedures used in [12, section 4, pp. 306] and [4, section 4, pp. 60], namely • rootdiagram[i]: returns a list of the roots corresponding to the marked boxes in line i of dc; • boxes[i]: the number of boxes in row i of dc, that is boxes[i] = |oi|; • evaluate[i,point]: a procedure which takes as input the coefficients {ak} of the unique monic polynomial over fq having the marked elements on a row number i as roots and a point pi,j on oi, and evaluates the function fi(x,y) as above at a point pi,j; we have the following algorithm, which is completely analogous to proposition 4.4 in [12] and algorithm 4.2 in [4]. 77 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 algorithm 3.5. input: the root diagram dc, the n fq-rational points pi,j of supp(d) = o1∪. . .∪or∪or+1∪or+s. output: a non-reduced gröbner basis g = {g(1),g(2), . . . ,g(r+s)} of c. 1. g := {} 2. for i from 1 to r + s do 3. if |rootdiagram[i]|< boxes[i] then 4. for k from 1 to r + s do 5. g(i)k := 0 6. if k ≥ i then 7. for j from 0 to boxes[k]−1 do 8. g(i)k := g (i) k + evaluate[i,pk,j]t j ek 9. end for 10. end if 11. end for 12. else 13. g(i) := (tboxes[i] −1) ei 14. end if 15. g := g∪{g(i)} 16. end for 17. return g remark 3.6. note that this algorithm makes use only of interpolation and function evaluation problems. as studied in [12] and [4], it has a computational complexity much lower than the complexity of general gröbner basis algorithms. in particular, it does not use divisions or reductions that would increase the complexity, as in the case of buchberger’s algorithm. 4. examples 4.1. the maximal curve yq + y = xq r+1 let xq,qr+1 be the curve defined over fq2r by the affine equation yq + y = xq r+1, where q is a prime power and r an odd integer. note that when r = 1 the curve is just the hermitian curve. the curve xq,qr+1 has genus g = qr(q−1)/2, one single singular point p∞ = (0 : 1 : 0) at infinity and other q2r+1 fq2r-rational points. thus, this curve is a maximal curve over fq,qr+1 because its number of fq2r-rational points equals the upper hasse-weil bound, namely equals q2r + 1 + 2gqr. furthermore, h(p∞) = 〈q,qr + 1〉, see [14], and σ : x 7→ αx y 7→ αq r+1y (2) with α ∈ f∗ q2r such that α(q r+1)(q−1) = 1, is an automorphism of xq,qr+1, see [11]. note that σ has order (qr + 1)(q −1). so, the order of σ divides q2r −1. note that under the action of the automorphism σ above the q2r+1 fq2r-rational points on xq,qr+1 are disposed in q(qr−1 +· · ·+q+1)+2 orbits, where q(qr−1 +· · ·+q+1) of them has length (qr +1)(q−1) and the remaining two orbits, one has length q−1 and the other has length 1. in fact, for the definition of the automorphism σ, it is clear that: • σ(0,0) = (0,0), and so we have a one orbit with a single point; 78 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 • all the q −1 fq2r-rational points (0,b), with b 6= 0, form an orbit with length q −1, since σ(0,b) = (0,αq r+1b) and α ∈ f∗ q2r is such that α(q r+1)(q−1) = 1; • the other q2r+1 −q = q(qr + 1)(qr −1) fq2r-rational points (x,y) ∈xq,qr+1, with x 6= 0 and y 6= 0, are arranged in q(qr−1 + · · ·+ q + 1) orbits of length (qr + 1)(q −1). let α be as in (2). let f∗ q2r = 〈a〉 and t ∈ {0,1, . . . ,q2r − 2} be such that α = at. so, given pi,0 = (a ti,ali) ∈ oi, the other points pi,j on oi are pi,j = σj(pi,0) = (ati+jk,ali+jk(q r+1)), with j ∈{1, . . . ,(qr + 1)(q −1)−1}. then, for i = 1, . . . ,r and j = 0, . . . ,(qr + 1)(q −1)−1, we get mi(y) := q−2∏ j=0 (y −ali+jk(q r+1)) = yq−1 −ali(q−1), (3) and bi,j(x,y); = q−2∏ s=1 (y −ali+k(q r+1)(j+s)) (qr+1)−1∏ s=1 (x−ati+k(j+s)). (4) since div∞(x) = qp∞ and div∞(y) = (qr + 1)p∞, we have that • div∞(mi(y)) = (q −1)(qr + 1)p∞; that is, mi(y) ∈ l((q −1)(qr + 1)p∞), for all i = 1, . . . ,r; • div∞(bi,j(x,y)) = ((q−2)(qr +1)+q((qr +1)−1))p∞; that is, bi,j ∈ l(q.qr +(q−2)(qr +1))p∞), for all 1 ≤ i ≤ r) e 0 ≤ j ≤ (qr + 1)(q −1)−1. with the notations on the previous section we have that: • a = q and b = qr + 1; • p = p∞; • div∞(x) = qp∞ and div∞(y) = (qr + 1)p∞; • h(p∞) = 〈q,qr + 1〉; • ρ1 = q −1, ρ2 = qr and ρ3 = q −2. thus, using the proposition 3.3 and the theorem 3.4, we can get the root diagram for one-point codes cxq,qr+1(d,λp∞) and then the gröbner basis for the module c associated to cxq,qr+1(d,λp∞) by algorithm 3.5. example 4.1. consider the curve x2,9 : y2 +y = x9 over f64 and the code c = c(x2,9,d,20p∞), where d is the sum of the 128 f64-rational points distinct of p∞ = (0 : 1 : 0). let α be a generator of f∗64. the automorphism σ(x,y) = (α7x,y) decomposes the points in supp(d) into sixteen orbits, being fourteen of 79 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 length 9 and two of length 1: o1 = {p1,0 = (α,α18),p1,1 = (α8,α18), . . . ,p1,8 = (α57,α18)}, o2 = {p2,0 = (α,α54),p2,1 = (α8,α54), . . . ,p2,8 = (α57,α54)}, o3 = {p3,0 = (α2,α36),p3,1 = (α9,α36), . . . ,p3,8 = (α58,α36)}, o4 = {p4,0 = (α2,α45),p4,1 = (α9,α45), . . . ,p4,8 = (α58,α45)}, o5 = {p5,0 = (α3,α31),p5,1 = (α10,α31), . . . ,p5,8 = (α59,α31)}, o6 = {p6,0 = (α3,α59),p6,1 = (α10,α59), . . . ,p6,8 = (α59,α59)}, o7 = {p7,0 = (α4,α9),p7,1 = (α11,α9), . . . ,p7,8 = (α60,α9)}, o8 = {p8,0 = (α4,α27),p8,1 = (α11,α27), . . . ,p8,8 = (α60,α27)}, o9 = {p9,0 = (α5,α47),p9,1 = (α12,α47), . . . ,p9,8 = (α61,α47)}, o10 = {p10,0 = (α5,α61),p10,1 = (α12,α61), . . . ,p10,8 = (α61,α61)}, o11 = {p11,0 = (α6,α55),p11,1 = (α13,α55), . . . ,p11,8 = (α62,α55)}, o12 = {p12,0 = (α6,α62),p12,1 = (α13,α62), . . . ,p12,8 = (α62,α62)}, o13 = {p13,0 = (α7,α21),p13,1 = (α14,α21), . . . ,p13,8 = (1,α21)}, o14 = {p14,0 = (α7,α42),p14,1 = (α14,α42), . . . ,p14,8 = (1,α42)}, o15 = {p15,0 = (0,1)}, o16 = {p16,0 = (0,0)}. since the set of roots of t9 − 1 in f64 is {1,α7,α14,α21,α28,α35,α42,α49,α56}, proposition 3.3 and theorem 3.4 (where a = 2, b = 9, ρ1 = 1, ρ2 = 8, ρ3 = 0 and λ = 20) give the following root diagram. α7 α14 α21 α28 α35 α42 α49 α56 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 4.2. a quotient of the hermitian curve let xq,m be the curve defined over fq2 by the affine equation yq + y = xm, where q is a prime power and m > 2 is a divisor of q + 1. this curve is a quotient of the hermitian curve and has genus g = (q−1)(m−1)/2, a single point at infinity, denoted by p∞, and other q(1 + m(q−1)) fq2-rational points. one may notice that p∞ = (0 : 1 : 0) if m = q + 1, and p∞ = (1 : 0 : 0) if m 6= q + 1. in [5], it is shown that xq,m is a maximal curve and in [13], g. matthews studied weierstrass semigroup and algebraic codes arising from xq,m. in addition, we have that h(p∞) = 〈m,q〉, see [5, theorem 3]. let f∗ q2 = 〈α〉 and k such that mk = q + 1. then, 80 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 τ : x → αkx y → αq+1y (5) is an automorphism of xq,m of order m(q −1), which divides q2 −1. it is not hard to see that under the action of the automorphism τ above the q(1 + m(q − 1)) fq2rational points on xq,m are disposed in q+2 orbits, where q of them has length m(q−1) and the remaining two orbits, one has length q −1 and the other has length 1. taking r = q and the first r orbits given by points on xq,m of the form p = (a,b) with a,b 6= 0. so, for each i = 1, . . . ,r, given pi,0 = (α`i,αti) ∈ oi, the other points pi,j on oi are pi,j = σj(pi,0) = (α`i+jk,αti+j(q+1)), with j ∈{1, . . . ,m(q −1)−1}. that is, oi = {pi,j = (α`i+jk,αti+j(q+1)) ; j = 0, . . . ,m(q −1)−1}. then, for i = 1, . . . ,r and j = 0,1, . . . ,m(q −1)−1, we get mi(y) = q−2∏ j=0 (y −αti+j(q+1)) and bi,j(x,y) = q−2∏ s=0,s6=j (y −αti+s(q+1)) m(q−1)−1∏ s=0,s6=j (x−α`i+sk). so, since div∞(x) = qp∞ and div∞(y) = mp∞, it follows that • div∞(mi(y)) = (q −1)mp∞; that is, mi(y) ∈ l((q −1)mp∞), for all i = 1, . . . ,r; • div∞(bi,j(x,y)) = ((q − 2)m + (m − 1)q)p∞; that is, bi,j ∈ l((m − 1)q + (q − 2)m)p∞), for all 1 ≤ i ≤ r) and 0 ≤ j ≤ m(q −1)−1. with the notations on the previous section we have that: • a = q and b = m; • p = p∞; • (x)∞ = qp∞ and (y)∞ = mp∞; • h(p∞) = 〈q,m〉; • ρ1 = q −1, ρ2 = q −2 and ρ3 = m−1. therefore, we can get the root diagram for one-point codes cxq,m(d,λp∞) and then the gröbner basis for the module c associated to cxq,m(d,λp∞). example 4.2. consider the curve x? : y5 +y = x3 over f25 and the code c = cx?(d,30p∞), where d is the sum of the 65 f25-rational points distinct of p∞. let α be a generator of f∗25. the automorphism τ(x,y) = (α2x,α6y) decomposes the points in supp(d) into seven orbits, being five of length 12, one of 81 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 length 4 and one of length 1: o1 = {p1,0 = (1,α),p1,1 = (α2,α7), . . . ,p1,11 = (α22,α19)}, o2 = {p2,0 = (1,α20),p2,1 = (α2,α2), . . . ,p2,11 = (α22,α14)}, o3 = {p3,0 = (1,α4),p3,1 = (α2,α10), . . . ,p3,11 = (α22,α22)}, o4 = {p4,0 = (1,α5),p4,1 = (α2,α11), . . . ,p4,11 = (α22,α23)}, o5 = {p5,0 = (1,α18),p5,1 = (α2,1), . . . ,p5,11 = (α22,α12)}, o6 = {p6,0 = (0,α3),p6,1 = (0,α9),p6,2 = (0,α15),p6,3 = (0,α21)}, o7 = {p7,0 = (0,0)}. since the set of roots of t12 −1 and t4 −1 in f25 are {1,α2,α4,α6, . . . ,α22} and {1,α6,α12,α18}, respectively. so, proposition 3.3 and theorem 3.4 (where a = 5, b = 3, ρ1 = 4, ρ2 = 3, ρ3 = 2 and λ = 30) give the following root diagram. α2 α4 α6 α8 α10 α12 α14 α16 α18 α20 α22 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x acknowledgment: the authors would like to thank the anonymous referees for very useful comments and suggestions that improved the presentation of this work. references [1] w. adams, p. loustaunau, an introduction to gröbner bases, providence, ri: american mathematical society, 1994. 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[13] g. l. matthews, weierstrass semigroups and codes from a quotient of the hermitian curve, des. codes cryptogr. 37(3) (2005) 473–492. 82 https://doi.org/10.1109/tit.2016.2583437 https://doi.org/10.1109/tit.2016.2583437 https://doi.org/10.1016/j.jsc.2012.05.001 https://doi.org/10.1016/j.jsc.2012.05.001 https://doi.org/10.1007/bf01200462 https://doi.org/10.1007/bf01200462 https://mathscinet.ams.org/mathscinet-getitem?mr=628795 https://mathscinet.ams.org/mathscinet-getitem?mr=670165 https://doi.org/10.1109/18.476247 https://doi.org/10.1109/18.476247 https://mathscinet.ams.org/mathscinet-getitem?mr=2386879 https://mathscinet.ams.org/mathscinet-getitem?mr=2386879 https://doi.org/10.1109/18.945272 https://doi.org/10.1109/18.945272 https://doi.org/10.1016/s0022-4049(96)00067-9 https://doi.org/10.1016/s0022-4049(96)00067-9 https://doi.org/10.1007/s10623-004-4038-5 https://doi.org/10.1007/s10623-004-4038-5 f. fornasiero, g. tizziotti / j. algebra comb. discrete appl. 5(2) (2018) 71–83 [14] a. sepúlveda, g. tizziotti, weierstrass semigroup and codes over the curve yq + y = xq r+1, adv. math. commun. 8(1) (2014) 67–72. [15] h. stichtenoth, algebraic function fields and codes, springer–verlag, berlin, 1993. 83 https://dx.doi.org/10.3934/amc.2014.8.67 https://dx.doi.org/10.3934/amc.2014.8.67 https://mathscinet.ams.org/mathscinet-getitem?mr=2464941 introduction background the root diagram for certain one-point ag codes examples references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.423751 j. algebra comb. discrete appl. 5(2) • 85–99 received: 19 august 2017 accepted: 26 ferbruary 2018 journal of algebra combinatorics discrete structures and applications some results on the comaximal ideal graph of a commutative ring research article subramanian visweswaran, jaydeep parejiya abstract: the rings considered in this article are commutative with identity which admit at least two maximal ideals. let r be a ring such that r admits at least two maximal ideals. recall from ye and wu (j. algebra appl. 11(6): 1250114, 2012) that the comaximal ideal graph of r, denoted by c (r) is an undirected simple graph whose vertex set is the set of all proper ideals i of r such that i 6⊆ j(r), where j(r) is the jacobson radical of r and distinct vertices i1, i2 are joined by an edge in c (r) if and only if i1 + i2 = r. in section 2 of this article, we classify rings r such that c (r) is planar. in section 3 of this article, we classify rings r such that c (r) is a split graph. in section 4 of this article, we classify rings r such that c (r) is complemented and moreover, we determine the s-vertices of c (r). 2010 msc: 13a15, 05c25 keywords: comaximal ideal graph, special principal ideal ring, planar graph, split graph, complement of a vertex in a graph 1. introduction the rings considered in this article are commutative with identity which admit at least two maximal ideals. let r be a ring. we denote the set of all maximal ideals of r by max(r). we denote the jacobson radical of r by j(r). we denote the cardinality of a set a using the notation |a|. motivated by the research work done on comaximal graphs of rings in [9, 12, 13, 15, 16] and on the annihilating-ideal graphs of rings in [5, 6], m. ye and t. wu in [18] introduced a graph structure on a ring r, whose vertex set is the set of all proper ideals i of r such that i 6⊆ j(r) and distinct vertices i1 and i2 are joined by an edge if and only if i1 + i2 = r. m. ye and t. wu called the graph introduced by them in [18] as the comaximal ideal graph of r and denoted it using the notation c (r) and investigated the influence of certain graph parameters of c (r) on the ring structure of r. let r be a ring such that |max(r)| ≥ 2. subramanian visweswaran (corresponding author), jaydeep parejiya; department of mathematics, saurashtra univesity, rajkot, gujarat, india 360 005 (email: s_visweswaran2006@yahoo.co.in, parejiyajay@gmail.com). 85 https://orcid.org/0000-0002-4905-809x https://orcid.org/0000-0002-2072-2719 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 the aim of this article is to classify rings r such that c (r) is planar; c (r) is a split graph; and c (r) is complemented. it is useful to recall the following definitions from commutative ring theory. a ring r is said to be quasilocal (respectively, semiquasilocal) if r has a unique maximal ideal (respectively, r has only a finite number of maximal ideals). a noetherian quasilocal (respectively, semiquasilocal) ring is referred to as a local ring (respectively, a semilocal ring). recall that a principal ideal ring is said to be a special principal ideal ring (spir) if r has a unique prime ideal m. it is clear that m is nilpotent. if r is a spir with m as its only prime ideal, then we denote it by saying that (r, m) is a spir. suppose that a ring t is quasilocal with m as its unique maximal ideal such that m 6= (0) and nilpotent. let n ≥ 2 be least with the property that mn = (0). if m is principal, then it follows from (iii) ⇒ (i) of [3, proposition 8.8] that {mi|i ∈{1, . . . , n− 1}} is the set of all nonzero proper ideals of t . hence, (t, m) is a spir. we next recall the following definitions and results from graph theory. the graphs considered in this article are undirected and simple. let g = (v, e) be a graph. recall from [4, definition 1.2.2] that a clique of g is a complete subgraph of g. the clique number of g, denoted by ω(g) is defined as the largest integer n ≥ 1 such that g contains a clique on n vertices [4, definition, page 185]. we set ω(g) = ∞ if g contains a clique on n vertices for all n ≥ 1. let n ∈ n. a complete graph on n vertices is denoted by kn. a graph g = (v, e) is said to be bipartite if v can be partitioned into two nonempty subsets v1 and v2 such that each edge of g has one end in v1 and the other in v2. a bipartite graph with vertex partition v1 and v2 is said to be complete if each element of v1 is adjacent to every element of v2. let m, n ∈ n. let g = (v, e) be a complete bipartite graph with vertex partition v1 and v2. if |v1| = m and |v2| = n, then g is denoted by km,n [4, definition 1.1.12]. let g = (v, e) be a graph. recall from [4, definition 8.1.1] that g is said to be planar if g can be drawn in a plane in such a way that no two edges of g intersect in a point other than a vertex of g. recall that two adjacent edges are said to be in series if their common end vertex is of degree two [7, page 9]. two graphs are said to be homeomorphic if one graph can be obtained from the other by insertion of vertices of degree two or by the merger of edges in series [7, page 100]. it is useful to note from [7, page 93] that the graph k5 is referred to as kuratowski’s first graph and the graph k3,3 is referred to as kuratowski’s second graph. the celebrated theorem of kuratowski states that a graph g is planar if and only if g does not contain either of kuratowski’s two graphs or any graph homeomorphic to either of them [7, theorem 5.9]. let g = (v, e) be a graph. it is convenient to name the following conditions satisfied by g so that it can be used throughout section 2 of this article. (i) we say that g satisfies (ku1) if g does not contain k5 as a subgraph (that is, equivalently, if ω(g) ≤ 4). (ii) we say that g satisfies (ku∗1) if g satisfies (ku1) and moreover, g does not contain any subgraph homeomorphic to k5. (iii) we say that g satisfies (ku2) if g does not contain k3,3 as a subgraph. (iv) we say that g satisfies (ku∗2) if g satisfies (ku2) and moreover, g does not contain any subgraph homeomorphic to k3,3. suppose that a graph g = (v, e) is planar. it follows from kuratowski’s theorem [7, theorem 5.9] that g satisfies both (ku∗1) and (ku ∗ 2). hence, g satisfies both (ku1) and (ku2). it is interesting to note that a graph g can be nonplanar even if it satisfies both (ku1) and (ku2). for an example of this type, refer [7, figure 5.9(a), page 101] and the graph g given in this example does not satisfy (ku∗2). it is not hard to construct an example of a graph g such that g satisfies (ku1) but g does not satisfy (ku∗1). let r be a ring such that |max(r)| ≥ 2. in section 2 of this article, we try to classify rings r such that c (r) is planar. it is proved in [18, theorem 3.1] that ω(c (r)) = |max(r)|. hence, c (r) satisfies (ku1) if and only if |max(r)| ≤ 4. in section 2 of this article, we first focus on classifying rings r such that c (r) satisfies (ku2). it is shown in lemma 2.1 that if c (r) satisfies (ku2) , then |max(r)| ≤ 3. 86 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 let r be a ring such that |max(r)| = 2. it is proved in proposition 2.7 that c (r) is planar if and only if c (r) satisfies (ku2) if and only if r ∼= r1 × r2 as rings, where (ri, mi) is a quasilocal ring for each i ∈ {1, 2} and for at least one i ∈ {1, 2}, (ri, mi) is a spir with m2i = (0). let r be a ring with |max(r)| = 3. it is shown in proposition 2.13 that c (r) satisfies (ku2) if and only if r ∼= r1×r2×r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. it is proved in theorem 2.18 that c (r) is planar if and only if r ∼= r1 × r2 × r3 as rings, where ri is a field for at least two values of i ∈{1, 2, 3} and if i ∈{1, 2, 3} is such that ri is not a field, then (ri, mi) is a spir with m2i = (0). let g = (v, e) be a graph. recall that g is a split graph if v is the disjoint union of two nonempty subsets k and s such that the subgraph of g induced on k is complete and s is an independent set of g. let r be a commutative ring with identity. in [16] p. k. sharma and s.m. bhatwadekar introduced and investigated a graph associated with r, whose vertex set is the set of all elements of r and distinct vertices x, y are joined by an edge if and only if rx + ry = r. the graph studied in [16] is named as the comaximal graph of r in [12]. in [9], m.i. jinnah and s.c. mathew classified rings r such that the comaximal graph of r is a split graph. let r be a ring such that |max(r)| ≥ 2. in section 3 of this article, we try to classify rings r such that c (r) is a split graph. it is proved in lemma 3.2 that if c (r) is a split graph, then |max(r)| ≤ 3. let r be a ring such that |max(r)| = 3. it is shown in theorem 3.3 that c (r) is a split graph if and only if r ∼= f1 × f2 × f3 as rings, where fi is a field for each i ∈ {1, 2, 3}. let r be a ring such that |max(r)| = 2. it is proved in theorem 3.5 that c (r) is a split graph if and only if r ∼= f ×s as rings, where f is a field and s is a quasilocal ring. let g = (v, e) be a graph. recall from [2, 11] that two distinct vertices u, v of g are said to be orthogonal, written u ⊥ v if u and v are adjacent in g and there is no vertex of g which is adjacent to both u and v in g; that is, the edge u − v is not an edge of any triangle in g. let u ∈ v . a vertex v of g is said to be a complement of u if u ⊥ v [2]. moreover, we recall from [2] that g is complemented if each vertex of g admits a complement in g. furthermore, g is said to be uniquely complemented if g is complemented and whenever the vertices u, v, w of g such that u ⊥ v and u ⊥ w, then a vertex x of g is adjacent to v in g if and only if x is adjacent to w in g. let r be a ring such that r is not an integral domain. recall from [1] that the zero-divisor graph of r denoted by γ(r) is an undirected graph whose vertex set is z(r)\{0} (here z(r) denotes the set of all zero-divisors of r) and distinct vertices x, y are joined by an edge if and only if xy = 0. the authors of [2] determined in section 3 of [2] rings r such that γ(r) is complemented or uniquely complemented. for a ring r, we denote the set of all units of r by u(r) and we denote the set of all nonunits of r by nu(r). the krull dimension of a ring r is simply denoted by dimr. in [15, proposition 3.11] it is proved that the subgraph of the comaximal graph of r induced on nu(r)\j(r) is complemented if and only if dim( r j(r) ) = 0. section 4 of this article is devoted to find a classification of rings r such that c (r) is complemented. let r be a ring such that |max(r)| ≥ 2. it is verified in remark 4.1(ii) that if c (r) is complemented, then it is uniquely complemented. it is shown in theorem 4.7 that c (r) is complemented if and only if r is semiquasilocal. moreover, in section 4, a discussion on the s-vertices of c (r) is included. let g = (v, e) be a graph. recall from [14, definition 2.9] a vertex a of g is said to be a smarandache vertex or simply a s-vertex if there exist distinct vertices x, y, and b of g such that a−x, a−b, and b−y are edges of g but there is no edge joining x and y in g. in [14], a.m. rahimi investigated the s-vertices of the zero-divisor graph of a commutative ring and the zero-divisor graph of a ring with respect to an ideal. for a ring r with |max(r)| ≥ 2, it is noted in remark 4.8 that if |maxr)| = 2, then no vertex of c (r) is a s-vertex. let r be a ring such that |max(r)| ≥ 3. it is shown in proposition 4.9 that a vertex i of c (r) is a s-vertex if and only if i is not contained in at least two distinct maximal ideals of r. let a, b be sets. if a is a subset of b and a 6= b, then we denote it symbolically using the notation a ⊂ b. let g be a graph. we denote the vertex set of g by v (g). let r be a ring. for a proper ideal i of r, as in [15], we denote {m ∈ max(r)|m ⊇ i} by m(i). 87 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 2. some preliminary results and on the planarity of c (r) as is already mentioned in the introduction, the rings considered in this article are commutative with identity which admit at least two maximal ideals. lemma 2.1. let r be a ring. if c (r) satisfies (ku2), then |max(r)| ≤ 3. proof. suppose that |max(r)| ≥ 4. let {mi|i ∈ {1, 2, 3, 4}} ⊆ max(r). let v1 = {m1, m2, m1m2} and let v2 = {m3, m4, m3m4}. observe that v1 ∪ v2 ⊆ v (c (r)), v1 ∩ v2 = ∅, and the subgraph of c (r) induced on v1 ∪ v2 contains k3,3 as a subgraph. hence, c (r) does not satisfy (ku2). this is in contradiction to the hypothesis that c (r) satisfies (ku2). therefore, |max(r)| ≤ 3. lemma 2.2. let r be a ring such that |max(r)| ≥ 2. if c (r) satisfies (ku2), then there exist nonzero rings r1 and r2 such that r ∼= r1 ×r2 as rings. proof. assume that c (r) satisfies (ku2). we assert that r admits a nontrivial idempotent. suppose that r does not have any nontrivial idempotent. by hypothesis, |max(r)| ≥ 2. let m1, m2 be distinct maximal ideals of r. observe that m1 + m2 = r. hence, there exist a ∈ m1 and b ∈ m2 such that a + b = 1. therefore, ra + rb = r. it is clear that for all i, j ∈ n, rai + rbj = r. since we are assuming that r has no nontrivial idempotent, we obtain that rai 6= raj and rbi 6= rbj for all distinct i, j ∈ n. let v1 = {ra, ra2, ra3} and let v2 = {rb, rb2, rb3}. note that v1∪v2 ⊆ v (c (r)) and v1∩v2 = ∅. for all i, j ∈ n, rai + raj ⊆ m1 and rbi + rbj ⊆ m2. hence, no two members of vi are adjacent in c (r) for each i ∈{1, 2}. it is clear from the above discussion that the subgraph of c (r) induced on v1∪v2 is k3,3. this is a contradiction. therefore, r admits at least one nontrivial idempotent. let e be a nontrivial idempotent of r. observe that the mapping f : r → re×r(1−e) defined by f(r) = (re, r(1−e)) is an isomorphism of rings. let us denote the ring re by r1 and r(1 − e) by r2. it is clear that r1 and r2 are nonzero rings and r ∼= r1 ×r2 as rings. remark 2.3. let r be a ring such that |max(r)| = 2. if c (r) satisfies (ku2), then we know from lemma 2.2 that there exist nonzero rings r1, r2 such that r ∼= r1 × r2 as rings. as |max(r)| = 2, it follows that ri is quasilocal for each i ∈ {1, 2}. we assume that r = r1 × r2 where (ri, mi) is a quasilocal ring for each i ∈{1, 2} and try to classify such rings r in order that c (r) satisfies (ku2). lemma 2.4. let r1, r2 be rings and let r = r1 ×r2. suppose that ri has at least two nonzero proper ideals for each i ∈{1, 2}. then c (r) does not satisfy (ku2). proof. we are assuming that ri has at least two nonzero proper ideals for each i ∈ {1, 2}. let i1, i2 be distinct nonzero proper ideals of r1 and let j1, j2 be distinct nonzero proper ideals of r2. let v1 = {i1×r2, i2×r2, (0)×r2} and let v2 = {r1×j1, r1×j2, r1×(0)}. observe that v1∪v2 ⊆ v (c (r)) and v1 ∩ v2 = ∅. as (ii × r2) + (r1 × jk) = r1 × r2 for all i, k ∈ {1, 2, 3} (where we set i3 is the zero ideal of r1 and j3 = zero ideal of r2), it follows that the subgraph of c (r) induced on v1 ∪v2 contains k3,3 as a subgraph. therefore, c (r) does not satisfy (ku2). let i be an ideal of a ring r. then the annihilator of i in r, denoted by annri is defined as annri = {r ∈ r|ir = (0)}. lemma 2.5. let (r, m) be a local ring which is not a field. the following statements are equivalent: (i) r has only one nonzero proper ideal. (ii) (r, m) is a spir with m2 = (0). proof. (i) ⇒ (ii) we are assuming that r has only one nonzero proper ideal. hence, m is the only nonzero proper ideal of r. let x ∈ m, x 6= 0. then m = rx. note that annrm is a nonzero proper ideal of r and so, annrm = m. hence, m2 = (0). therefore, (r, m) is a spir with m2 = (0). 88 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 (ii) ⇒ (i) as r is not a field, m 6= (0). thus if (r, m) is a spir with m2 = (0), then it is clear that m is the only nonzero proper ideal of r. lemma 2.6. let r = r1 × r2, where (ri, mi) is a quasilocal ring for each i ∈ {1, 2}. the following statements are equivalent: (i) c (r) satisfies (ku2). (ii) for at least one i ∈{1, 2}, (ri, mi) is a spir with m2i = (0). proof. (i) ⇒ (ii) assume that c (r) satisfies (ku2). we know from lemma 2.4 that for at least one i ∈ {1, 2}, ri has at most one nonzero proper ideal. hence, for that i, either mi = (0) or in the case mi 6= (0), we obtain from lemma 2.5 that (ri, mi) is a spir with m2i = (0). therefore, there exists at least one i ∈{1, 2} such that (ri, mi) is a spir with m2i = (0). (ii) ⇒ (i) without loss of generality, we can assume that (r1, m1) is a spir with m21 = (0). observe that {m1 = m1×r2, m2 = r1×m2} is the set of all maximal ideals of r. since |max(r)| = 2, it follows from (3) ⇒ (1) of [18, theorem 4.5] that c (r) is a complete bipartite graph with vertex partition v1 and v2, where vi is the set of all proper ideals a of r such that m(a) = {mi} for each i ∈{1, 2}. note that if a ∈ v1, then a = i × r2 for some ideal i of r1 such that i ⊆ m1. since there are at most two proper ideals of r1, we obtain that |v1| ≤ 2. it is now clear that c (r) satisfies (ku2). proposition 2.7. let r be a ring such that |max(r)| = 2. the following statements are equivalent: (i) c (r) is planar. (ii) c (r) satisfies both (ku∗1) and (ku ∗ 2). (iii) c (r) satisfies (ku2). (iv) r ∼= r1 ×r2 as rings, where (ri, mi) is a quasilocal ring for each i ∈{1, 2} and for at least one i ∈{1, 2}, (ri, mi) is a spir with m2i = (0) proof. (i) ⇒ (ii) this follows from kuratowski’s theorem [7, theorem 5.9]. (ii) ⇒ (iii) this is clear. (iii) ⇒ (iv) this follows from remark 2.3 and lemma 2.6. (iv) ⇒ (i) let us denote the ring r1 ×r2 by t . without loss of generality, we can assume that (r1, m1) is a spir with m21 = (0). let v1, v2 be as in the proof of (ii) ⇒ (i) of lemma 2.6 and it is already noted there that |v1| ≤ 2 and c (t ) is a complete bipartite graph with vertex partition v1 and v2. it is now clear that c (t ) is planar. since r ∼= t as rings, we get that c (r) is planar. let r be a ring such that |max(r)| = 3. we next try to classify such rings r in order that c (r) satisfies (ku2). lemma 2.8. let r1, r2 be rings and let r = r1 ×r2. if r1 admits at least two maximal ideals and if c (r1) does not satisfy (ku2), then c (r) does not satisfy (ku2). proof. we are assuming that c (r1) does not satisfy (ku2). then there exist subsets a = {i1, i2, i3} and b = {j1, j2, j3} of v (c (r1)) such that a ∩ b = ∅ and ii + jk = r1 for all i, k ∈ {1, 2, 3}. let v1 = {i1×r2, i2×r2, i3×r2} and let v2 = {j1×r2, j2×r2, j3×r2}. observe that v1∪v2 ⊆ v (c (r)), v1 ∩ v2 = ∅, and as (ii × r2) + (jk × r2) = r1 × r2 = r for all i, k ∈ {1, 2, 3}, it follows that the subgraph of c (r) induced on v1 ∪ v2 contains k3,3 as a subgraph. this proves that c (r) does not satisfy (ku2). lemma 2.9. let r be a ring such that |max(r)| = 3. if c (r) satisfies (ku2), then r ∼= r1 ×r2 ×r3 as rings, where ri is a quasilocal ring for each i ∈{1, 2, 3}. 89 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 proof. assume that c (r) satisfies (ku2). as |max(r)| = 3, it follows from lemma 2.2 that there exist nonzero rings t1 and t2 such that r ∼= t1 ×t2 as rings. since r has exactly three maximal ideals, it follows that either t1 or t2 is not quasilocal. without loss of generality, we can assume that t1 is not quasilocal. hence, the number of maximal ideals of t1 is exactly two. let us denote the ring t1 × t2 by t. since r ∼= t as rings, we obtain that c (t ) satisfies (ku2). now, it follows from lemma 2.8 that c (t1) satisfies (ku2). hence, we obtain from lemma 2.2 that there exist nonzero rings t11 and t12 such that t1 ∼= t11 × t12 as rings. therefore, r ∼= t11 × t12 × t2 as rings. hence, on renaming the rings t11, t12, and t2, we obtain that there exist rings r1, r2, and r3 such that r ∼= r1 × r2 × r3 as rings. since |max(r)| = 3, it is clear that ri is quasilocal for each i ∈{1, 2, 3}. lemma 2.10. let r be a ring such that |max(r)| = 3. if c (r) satisfies (ku2), then r ∼= r1×r2×r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. proof. assume that c (r) satisfies (ku2). we know from lemma 2.9 that r ∼= r1 ×r2 ×r3 as rings, where ri is a quasilocal ring for each i ∈{1, 2, 3}. let mi denote the unique maximal ideal of ri for each i ∈ {1, 2, 3}. let us denote the ring r1 × r2 × r3 by t. since r ∼= t as rings, we obtain that c (t ) satisfies (ku2). let i ∈ {1, 2, 3}. it follows from lemma 2.4 that ri has at most one nonzero proper ideal. hence, either mi = (0) in which case, ri is a field or mi 6= (0) is the only nonzero proper ideal of ri in which case, we obtain from lemma 2.5 that (ri, mi) is a spir with m2i = (0). this proves that r ∼= r1 ×r2 ×r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. lemma 2.11. let r be a ring such that |max(r)| = 3. let {m1, m2, m3} denote the set of all maximal ideals of r. let i ∈{1, 2, 3}. let us denote the set of all proper ideals i of r such that m(i) = {mi} by wi. if |wi| ≤ 2 for each i ∈{1, 2, 3}, then c (r) satisfies (ku2). proof. suppose that c (r) does not satisfy (ku2). then there exist subsets v1 = {i1, i2, i3} and v2 = {j1, j2, j3} of v (c (r)) such that v1∩v2 = ∅ and ii + jk = r for all i, k ∈{1, 2, 3}. after renaming the maximal ideals of r (if necessary), we can assume without loss of generality that i1 ⊆ m1. since i1 + jk = r for each k ∈{1, 2, 3}, it follows that jk 6⊆ m1 for each k ∈{1, 2, 3}. by hypothesis, |w2| ≤ 2 and |w3| ≤ 2. therefore, we obtain that w2∩v2 6= ∅ and w3∩v2 6= ∅. this implies that j1j2j3 ⊆ m2m3. it follows from ii + jk = r for all i, k ∈ {1, 2, 3} that ii + j1j2j3 = r for each i ∈ {1, 2, 3} and so, ii + m2m3 = r. hence, we get that ii ∈ w1 for each i ∈{1, 2, 3}. this implies that |w1| ≥ 3. this is in contradiction to the assumption that |w1| ≤ 2. therefore, c (r) satisfies (ku2). let r be a ring such that |max(r)| = 3. let i ∈ {1, 2, 3} and let mi, wi be as in the statement of lemma 2.11. in proposition 2.12, we classify such rings r in order that |wi| ≤ 2 for each i ∈{1, 2, 3}. proposition 2.12. let r be a ring such that |max(r)| = 3. let {mi|i ∈ {1, 2, 3}} denote the set of all maximal ideals of r. let i ∈ {1, 2, 3}. let us denote the set of all proper ideals i of r such that m(i) = {mi} by wi. then the following statements are equivalent: (i) |wi| ≤ 2 for each i ∈{1, 2, 3}. (ii) r ∼= r1 ×r2 ×r3 as rings, where (ri, ni) is a spir with n2i = (0) for each i ∈{1, 2, 3}. proof. (i) ⇒ (ii) let i ∈{1, 2, 3}. we claim that mi is principal. first, we verify that m1 is principal. suppose that m1 is not principal. observe that m1 6⊆ m2 ∪ m3. let a1 ∈ m1\(m2 ∪ m3). as m1 6= ra1 by assumption, it follows from [10, theorem 81] that there exists a2 ∈ m1\(ra1 ∪ m2 ∪ m3). note that m1 6= ra2 and it is clear from the choice of the elements a1, a2 that ra1 6= ra2 and {ra1, ra2, m1}⊆ w1. this is in contradiction to the assumption that |w1| ≤ 2. therefore, m1 is principal. similarly, it can be shown that m2 and m3 are principal. let i ∈ {1, 2, 3}. observe that m2i = m 3 i . suppose that m 2 i 6= m 3 i . then {mi, m2i , m 3 i} ⊆ wi. this is impossible, since |wi| ≤ 2. therefore, m 2 i = m 3 i . since mi + mj = r for all distinct i, j ∈ {1, 2, 3}, it follows from [3, proposition 1.10(i)] that j(r) = ∩3i=1mi = ∏3 i=1 mi. hence, (j(r))2 = ∏3 i=1 m 2 i = ∏3 i=1 m 3 i = (j(r)) 3. now, as j(r) = ∏3 i=1 mi is principal, there exists a ∈ j(r) such that j(r) = ra. from (j(r))2 = (j(r))3, we obtain that ra2 = ra3. hence, a2 = ra3 for some r ∈ r. since 1 − ra is a unit in r, we obtain that a2 = 0 and so, (j(r))2 = (0). since 90 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 m2i + m 2 j = r for all distinct i, j ∈ {1, 2, 3} and ∩ 3 i=1m 2 i = ∏3 i=1 m 2 i = (0), we obtain from the chinese remainder theorem [3, proposition 1.10(ii) and (iii)] that the mapping f : r → r m21 × r m22 × r m23 given by f(r) = (r + m21, r + m 2 2, r + m 2 3) is an isomorphism of rings. let i ∈ {1, 2, 3}. let us denote the ring r m2 i by ri. let us denote mim2 i by ni. since mi is a principal ideal of r, we obtain that ni is a principal ideal of ri and it is clear that n2i = (0 + m 2 i ). this shows that (ri, ni) is a spir with n 2 i is the zero ideal of ri for each i ∈{1, 2, 3} and r ∼= r1 ×r2 ×r3 as rings. (ii) ⇒ (i) assume that r ∼= r1 × r2 × r3 as rings, where (ri, ni) is a spir with n2i = (0) for each i ∈ {1, 2, 3}. let us denote the ring r1 × r2 × r3 by t . observe that t is semilocal with {n1 = n1 ×r2 ×r3, n2 = r1 ×n2 ×r3, n3 = r1 ×r2 ×n3} as its set of all maximal ideals. let us denote the set of all proper ideals a of t such that m(a) = {ni} by ui for each i ∈{1, 2, 3}. since ri has at most one nonzero proper ideal for each i ∈ {1, 2, 3}, it follows that |ui| ≤ 2. from r ∼= t as rings, we obtain that |wi| ≤ 2 for each i ∈{1, 2, 3}. proposition 2.13. let r be a ring such that |max(r)| = 3. the following statements are equivalent: (i) c (r) satisfies (ku2). (ii) r ∼= r1 ×r2 ×r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. proof. (i) ⇒ (ii) assume that c (r) satisfies (ku2). we know from lemma 2.10 that r ∼= r1×r2×r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. (ii) ⇒ (i) assume that r ∼= r1 × r2 × r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. let {mi|i ∈{1, 2, 3}} denote the set of all maximal ideals of r. let i ∈{1, 2, 3} and let us denote the set of all proper ideals i of r such that m(i) = {mi} by wi. we know from (ii) ⇒ (i) of proposition 2.12 that |wi| ≤ 2. hence, we obtain from lemma 2.11 that c (r) satisfies (ku2). let r be a ring such that |max(r)| = 3. we try to classify such rings r in order that c (r) is planar. if c (r) is planar, then we know from kuratowski’s theorem [7, theorem 5.9] that c (r) satisfies (ku2). hence, we obtain from (i) ⇒ (ii) of proposition 2.13 that r ∼= r1 × r2 × r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈{1, 2, 3}. lemma 2.14. let r = r1 × r2 × r3, where (ri, mi) is a spir with mi 6= (0) but m2i = (0) for each i ∈{1, 2, 3}. then c (r) does not satisfy (ku∗2). proof. the proof of this lemma closely follows the proof given in [17, lemma 3.13]. note that |max(r)| = 3 and {m1 = m1 × r2 × r3, m2 = r1 × m2 × r3, m3 = r1 × r2 × m3} is the set of all maximal ideals of r. let us denote the subgraph of c (r) induced on w = {v1 = m1, v2 = m21, v3 = m3, v4 = m2, v5 = m 2 2, v6 = m 2 3, v7 = m1∩m2} by h. observe that in h, the edges m3−m1∩m2 and m1∩m2−m23 are in series and moreover, in h, vi is adjacent to v4 and v5 for each i ∈{1, 2, 3}. furthermore in h, v1 and v2 are adjacent to v6. therefore, on merging the edges v3 −v7 and v7 −v6, we obtain a graph h1 which contains k3,3 as a subgraph. hence, h contains a subgraph which is homeomorphic to k3,3. this shows that c (r) does not satisfy (ku∗2). lemma 2.15. let r = f × r2 × r3, where f is a field and (ri, mi) is a spir with mi 6= (0) but m2i = (0) for each i ∈{2, 3}. then c (r) does not satisfy (ku ∗ 1). proof. the proof of this lemma closely follows the proof given in [17, lemma 3.14]. observe that |max(r)| = 3 and {m1 = (0)×r2×r3, m2 = f ×m2×r3, m3 = f ×r2×m3} is the set of all maximal ideals of r. let us denote the subgraph of c (r) induced on w = {v1 = m2, v2 = m1m3, v3 = m22, v4 = m3, v5 = m1m2, v6 = m 2 3, v7 = m1} by h. note that in h, the edges e1 : v1 −v2, e2 : v2 −v3 are edges in series and the edges e3 : v4 − v5, e4 : v5 − v6 are edges in series. observe that in h, v1 is adjacent to all the elements of w except v3 and v5; v3 is adjacent to all the elements of w except v1 and v5; v4 is adjacent to all the elements of w except v2 and v6; v6 is adjacent to all the elements of w except v2 91 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 and v4; v7 is adjacent to all the elements of w except v2 and v5. let h1 be the graph obtained from h on merging the edges e1 and e2 and on merging the edges e3 and e4. it is clear that h1 is a complete graph on five vertices. this proves that c (r) contains a subgraph h such that h is homeomorphic to k5. therefore, c (r) does not satisfy (ku∗1). lemma 2.16. let r = f1 ×f2 ×r3, where f1 and f2 are fields and (r3, m3) is a spir with m3 6= (0) but m23 = (0). then c (r) is planar. proof. observe that |max(r)| = 3 and {m1 = (0)×r2×r3, m2 = f1×(0)×r3, m3 = f1×f2×m3} is the set of all maximal ideals of r. observe that v (c (r)) equals {v1 = m1, v2 = m2, v3 = m3, v4 = m1m2, v5 = m2m3, v6 = m1m3, v7 = m 2 3, v8 = m1m 2 3, v9 = m2m 2 3}. it is not hard to verify that c (r) is the union of the cycle γ : v1 −v3 −v2 −v7 −v1, the edges e1 : v3 −v4, e2 : v4 −v7, e3 : v1 −v2, and the pendant edges e4 : v1 − v5, e5 : v1 − v9, e6 : v2 − v6, and e7 : v2 − v8. note that γ can be represented by means of a rectangle. the edges e1, e2 are edges in series and their common end vertex v4 can be plotted inside the rectangle representing γ and the edges e1, e2 can be drawn inside this rectangle. the edges ei(i ∈{3, 4, 5, 6, 7}) are such that one of their end vertices ∈{v1, v2} and they can be drawn outside the rectangle representing γ in such a way that there are no crossing over of the edges. this proves that c (r) is planar. lemma 2.17. let r = f1 ×f2 ×f3, where fi is a field for each i ∈{1, 2, 3}. then c (r) is planar. proof. note that |max(r)| = 3 and {m1 = (0) × f2 × f3, m2 = f1 × (0) × f3, m3 = f1 × f2 × (0)} is the set of all maximal ideals of r. observe that v (c (r)) equals {v1 = m1, v2 = m2, v3 = m3, v4 = m1m2, v5 = m2m3, v6 = m1m3}. it is clear that c (r) is the union of the cycle γ : v1 − v2 − v3 − v1 and the pendant edges e1 : v1−v5, e2 : v2−v6, and e3 : v3−v4. the cycle γ can be represented by means of a triangle and the one of the end vertex of ei is vi for each i ∈ {1, 2, 3} and the edges e1, e2, e3 can be drawn outside the triangle representing γ in such a way that there are no crossing over of the edges. this proves that c (r) is planar. theorem 2.18. let r be a ring such that |max(r)| = 3. the following statements are equivalent: (i) c (r) is planar. (ii) c (r) satisfies both (ku∗1) and (ku ∗ 2). (iii) r ∼= r1 × r2 × r3 as rings, where ri is a field for at least two values of i ∈ {1, 2, 3} and if i ∈{1, 2, 3} is such that ri is not a field, then (ri, mi) is a spir with m2i = (0). proof. (i) ⇒ (ii) this follows from kuratowski’s theorem [7, theorem 5.9]. (ii) ⇒ (iii) since c (r) satisfies (ku∗2), we get that c (r) satisfies (ku2). therefore, we obtain from proposition 2.13 that r ∼= r1 × r2 × r3 as rings, where (ri, mi) is a spir with m2i = (0) for each i ∈ {1, 2, 3}. let us denote the ring r1 × r2 × r3 by t. since r ∼= t as rings, we obtain that c (t ) satisfies (ku∗1) and (ku ∗ 2). as c (t ) satisfies (ku ∗ 2), it follows from lemma 2.14 that ri is a field for at least one value of i ∈ {1, 2, 3}. suppose that ri is a field for exactly one value of i ∈ {1, 2, 3}. without loss of generality, we can assume that r1 is a field and r2, r3 are not fields. in such a case, we obtain from lemma 2.15 that c (t ) does not satisfy (ku∗1). this is a contradiction. therefore, ri is a field for at least two values of i ∈{1, 2, 3}. this proves that r ∼= r1 ×r2 ×r3 as rings, where ri is a field for at least two values of i ∈ {1, 2, 3} and if i ∈ {1, 2, 3} is such that ri is not a field, then (ri, mi) is a spir with m2i = (0). (iii) ⇒ (i) suppose that r ∼= r1 × r2 × r3 as rings, where ri is a field for at least two values of i ∈ {1, 2, 3} and if i ∈ {1, 2, 3} is such that ri is not a field, then (ri, mi) is a spir with m2i = (0). let us denote the ring r1 ×r2 ×r3 by t. note that either ri is a field for each i ∈{1, 2, 3}, in which case, we obtain from lemma 2.17 that c (t ) is planar or there are exactly two values of i ∈{1, 2, 3} such that ri is a field and in such a case, we obtain from lemma 2.16 that c (t ) is planar. since r ∼= t as rings, we get that c (r) is planar. 92 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 3. when is c (r) a split graph? let r be a ring such that |max(r)| ≥ 2. the aim of this section is to classify rings r such that c (r) is a split graph. throughout this section, we assume that k is a nonempty subset of v (c (r)) such that the subgraph of c (r) induced on k is complete and s is a nonempty subset of v (c (r)) such that s is an independent set of c (r). lemma 3.1. let r be a ring such that c (r) is a split graph with v (c (r)) = k ∪s and k ∩s = ∅. if |max(r)| ≥ 3, then max(r) = k. proof. first, we claim that max(r)) ⊆ k. suppose that max(r) 6⊆ k. then there exists m ∈ max(r) such that m /∈ k. hence, m ∈ s. we are assuming that |max(r)| ≥ 3. therefore, there exist distinct maximal ideals m′, m′′ of r such that m′ 6= m and m′′ 6= m. since m+m′ = m+m′′ = m+m′m′′ = r, we get that m′, m′′, and m′m′′ are adjacent to m in c (r). as m ∈ s, we obtain that m′, m′′, m′m′′ ∈ k. hence, m′ and m′m′′ must be adjacent in c (r). this is impossible, since m′+m′m′′ = m′ 6= r. therefore, max(r) ⊆ k. we next verify that k ⊆ max(r). let i ∈ k. then i is a proper ideal of r and so, there exists a maximal ideal m of r such that i ⊆ m. we assert that i = m. suppose that i 6= m. then i and m are adjacent in c (r). this is impossible, since i + m = m 6= r. hence, i = m and this proves that k ⊆ max(r) and so, k = max(r). lemma 3.2. let r be a ring. if c (r) is a split graph, then |max(r)| ≤ 3. proof. suppose that |max(r)| ≥ 4. now, v (c (r)) = k ∪ s with k ∩ s = ∅. since c (r) is a split graph by assumption, we obtain from lemma 3.1 that max(r) = k. let {mi|i ∈{1, 2, 3, 4}}⊆ max(r). note that for all distinct i, j ∈ {1, 2, 3, 4}, mimj /∈ max(r) = k and so, mimj ∈ s. hence, both m1m2 and m3m4 must be in s. therefore, m1m2 cannot be adjacent to m3m4 in c (r). this is impossible, since m1m2 + m3m4 = r. therefore, |max(r)| ≤ 3. let r be a ring such that |max(r)| = 3. in theorem 3.3, we classify such rings r in order that c (r) is a split graph. theorem 3.3. let r be a ring such that |max(r)| = 3. the following statements are equivalent: (i) c (r) is a split graph. (ii) r ∼= f1 ×f2 ×f3 as rings, where fi is a field for each i ∈{1, 2, 3}. proof. (i) ⇒ (ii) we are assuming that c (r) is a split graph. then v (c (r)) = k∪s with k∩s = ∅. as |max(r)| = 3, we obtain from lemma 3.1 that max(r) = k. let {m1, m2, m3} denote the set of all maximal ideals of r. let i ∈ {1, 2, 3} and let us denote the set of all proper ideals i of r such that m(i) = {mi} by wi. we assert that wi = {mi} for each i ∈ {1, 2, 3}. first, we show that w1 = {m1}. it is clear that m1 ∈ w1. let i ∈ w1 be such that i 6= m1. as k = max(r), it follows that i must be in s. note that m2m3 ∈ s. it is clear that i 6= m2m3. now, i, m2m3 ∈ s and s is an independent set of c (r), we get that i and m2m3 cannot be adjacent in c (r). however, i + m2m3 = r. this is a contradiction and so, i = m1. this shows that w1 = {m1}. similarly, it can be shown that w2 = {m2} and w3 = {m3}. we next show that mi is principal for each i ∈{1, 2, 3}. note that m1 6⊆ m2∪m3. hence, there exists x1 ∈ m1\(m2 ∪ m3). observe that rx1 ∈ w1 = {m1} and so, m1 = rx1. similarly, using the facts that w2 = {m2} and w3 = {m3}, it can be proved that m2 = rx2 for any x2 ∈ m2\(m1 ∪ m3) and m3 = rx3 for any x3 ∈ m3\(m1 ∪ m2). it is clear that j(r) = ∩3i=1mi = ∏3 i=1 mi = rx1x2x3. let i ∈{1, 2, 3}. note that m2i ∈ wi = {mi} and so, mi = m 2 i . this implies that ∏3 i=1 mi = ∏3 i=1 m 2 i . hence, rx1x2x3 = rx 2 1x 2 2x 2 3. therefore, x1x2x3 = rx 2 1x 2 2x 2 3 for some r ∈ r. as x1x2x3 ∈ j(r), we obtain that 1 − rx1x2x3 is a unit in r and so, x1x2x3 = 0. since mi + mj = r for all distinct i, j ∈ {1, 2, 3} and j(r) = (0), we obtain from the chinese remainder theorem [3, proposition 1.10(ii) and (iii)] that the mapping f : r → r m1 × r m2 × r m3 given by f(r) = (r + m1, r + m2, r + m3) is an isomorphism of rings. let 93 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 us denote the field r mi by fi for each i ∈ {1, 2, 3}. therefore, r ∼= f1 × f2 × f3 as rings, where fi is a field for each i ∈{1, 2, 3}. (ii) ⇒ (i) we are assuming that r ∼= f1×f2×f3 as rings, where fi is a field for each i ∈{1, 2, 3}. let us denote the ring f1 ×f2 ×f3 by t. note that v (c (t )) = {m1 = (0)×f2 ×f3, m2 = f1 ×(0)×f3, m3 = f1×f2×(0), m1m2 = (0)×(0)×f3, m2m3 = f1×(0)×(0), m1m3 = (0)×f2×(0)}. let k = {m1, m2, m3} and let s = {m1m2, m2m3, m1m3}. it is clear that v (c (t )) = k ∪ s, k ∩ s = ∅, the subgraph of c (t ) induced on k is complete and s is an independent set of c (t ). therefore, c (t ) is a split graph. as r ∼= t as rings, we obtain that c (r) is a split graph. let r be a ring such that |max(r)| = 2. we next try to classify such rings in order that c (r) is a split graph. lemma 3.4 is well-known. we include a proof of lemma 3.4 for the sake of completeness. lemma 3.4. let g = (v, e) be a complete bipartite graph. the following statements are equivalent: (i) g is a split graph. (ii) g is a star graph. proof. (i) ⇒ (ii) assume that g is a split graph. hence, there exist nonempty subsets k, s of v such that v = k ∪ s, k ∩ s = ∅, the subgraph of g induced on k is complete, and s is an independent set of g. by hypothesis, g is a complete bipartite graph. let g be complete bipartite with vertex partition v1 and v2. we claim that s ∩ vi = ∅ for some i ∈ {1, 2}. suppose that s ∩ vi 6= ∅ for each i ∈ {1, 2}. let si ∈ s ∩ vi for each i ∈ {1, 2}. then s1 and s2 are adjacent in g. this is impossible, since s is an independent set of g. therefore, either s ∩ v1 = ∅ or s ∩ v2 = ∅. without loss of generality, we can assume that s∩v2 = ∅. hence, s = s∩v = s∩(v1 ∪v2) = (s∩v1)∪(s∩v2) = s∩v1 and so, s ⊆ v1. it follows from v = v1 ∪ v2 = s ∪ k and s ⊆ v1 that v2 ⊆ k. since no two distinct elements of v2 are adjacent in g, whereas any two distinct vertices of k are adjacent in g, it follows that |v2| = 1. this shows that g is a star graph. (ii) ⇒ (i) suppose that g is a star graph. hence, g is a complete bipartite graph with vertex partition v1 and v2 such that |vi| = 1 for at least one i ∈ {1, 2}. without loss of generality, we can assume that |v1| = 1. with k = v1 and s = v2, it is clear that v = k ∪ s, k ∩ s = ∅, the subgraph of g induced on k is complete, and s is an independent of g. therefore, g is a split graph. theorem 3.5. let r be a ring such that |max(r)| = 2. the following statements are equivalent: (i) c (r) is a split graph. (ii) r ∼= f ×s as rings, where f is a field and s is a quasilocal ring. proof. let {m1, m2} denote the set of all maximal ideals of r. (i) ⇒ (ii) assume that c (r) is a split graph. as |max(r)| = 2, we know from (3) ⇒ (1) of [18, theorem 4.5] that c (r) is a complete bipartite graph with vertex partition v1 and v2, where for each i ∈ {1, 2}, vi is the set of all proper ideals i of r such that m(i) = {mi}. as we are assuming that c (r) is a split graph, we obtain from lemma 3.4 that c (r) is a star graph. hence, there exists a vertex i of c (r) such that i is adjacent to each vertex j of c (r) with j 6= i. we can assume without loss of generality that i ∈ v1. in such a case, we obtain that v1 = {i}. it is clear that m1 ∈ v1 and so, i = m1. let a ∈ m1\m2. as ra ∈ v1, we get that m1 = ra. observe that a2 ∈ m1\m2. hence, ra2 ∈ v1 and so, ra = ra2 = m1. now, there exists r ∈ r such that a = ra2. this implies that e = ra is a nontrivial idempotent element of r and moreover, m1 = ra = re. note that the mapping f : r → r(1 − e) × re defined by f(x) = (x(1 −e), xe) is an isomorphism of rings. hence, f(m1) = (0) ×re is a maximal ideal of r(1 − e) × re. therefore, r(1 − e) is a field. since |max(r)| = 2, it follows that the ring re is quasilocal. thus with f = r(1 − e) and s = re, we obtain that f is a field and s is a quasilocal ring and r ∼= f ×s as rings. (ii) ⇒ (i) assume that r ∼= f × s as rings, where f is a field and s is a quasilocal ring. let us denote the ring f ×s by t. let v1 = {(0)×s} and v2 = {f ×i|i is a proper ideal of s}. note that c (t ) is a 94 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 complete bipartite graph with vertex partition v1 and v2 and as |v1| = 1, it follows that c (t ) is a star graph. hence, we obtain from (ii) ⇒ (i) of lemma 3.4 that c (t ) is a split graph. since r ∼= t as rings, we get that c (r) is a split graph. 4. some more results on c (r) let r be a ring such that |max(r)| ≥ 2. the aim of this section is to classify rings r such that c (r) is complemented and to determine the s-vertices of c (r). let r be a ring. let x be the set of all prime ideals of r. recall from [3, exercise 15, page 12] that for a subset e of r, the set of all prime ideals p of r such that p ⊇ e is denoted by v (e). we know from [3, exercise 15, page 12] that the collection {v (e)|e ⊆ r} satisfies the axioms for closed sets in a topological space. the resulting topology is called the zariski topology. the topological space x is called the prime spectrum of r and is denoted by spec(r). let e ⊆ r. we know from [3, exercise 15(i), page 12] that v (e) = v (i), where i is the ideal of r generated by e. the subspace of spec(r) consisting of all the maximal ideals of r with the induced topology is called the maximal spectrum of r and is denoted by max(r). the collection {v (i)∩max(r)|i varies over all ideals of r} is the collection of all closed sets of max(r). as in [15], we denote v (i)∩max(r) by m(i). thus m(i), as mentioned in the introduction, is the set of all maximal ideals m of r such that m ⊇ i. as in [15], for an element a ∈ r, we denote m(ra) simply by m(a) and max(r)\m(a) by d(a). let g = (v, e) be a simple graph. let v ∈ v . then the set of all u ∈ v such that v is adjacent to u in g is called the set of neighbours of v in g and we use the notation ng(v) to denote the set of all neighbours of v in g. remark 4.1. let r be a ring such that |max(r)| ≥ 2. the following statements hold. (i) let i, j ∈ v (c (r)) be such that i ⊥ j in c (r). then ij ⊆ j(r). (ii) if c (r) is complemented, then it is uniquely complemented. proof. (i) suppose that ij 6⊆ j(r). then there exists a maximal ideal m of r such that ij 6⊆ m. hence, i 6⊆ m and j 6⊆ m and so, i + m = j + m = r. now, m ∈ v (c (r)) is such that m is adjacent to both i and j in c (r). this is impossible, since i ⊥ j in c (r). therefore, ij ⊆ j(r). (ii) let i ∈ v (c (r)). we are assuming that c (r) is complemented. hence, there exists at least one j ∈ v (c (r)) such that i ⊥ j in c (r). let j1, j2 ∈ v (c (r)) be such that i ⊥ j1 and i ⊥ j2 in c (r). we know from (i) that iji ⊆ j(r) for each i ∈ {1, 2}. let a ∈ v (c (r)) be such that j1 is adjacent to a in c (r). hence, j1 + a = r. we claim that j2 + a = r. suppose that j2 + a 6= r. then there exists m ∈ max(r) such that j2 + a ⊆ m. it follows from i + j2 = r that i 6⊆ m. as ij1 ⊆ j(r) ⊆ m, we get that j1 ⊆ m. therefore, j1 + a ⊆ m. this is impossible, since j1 + a = r. hence, j2 + a = r. this shows that nc (r)(j1) ⊆ nc (r)(j2). similarly, it can be shown that nc (r)(j2) ⊆ nc (r)(j1). therefore, nc (r)(j1) = nc (r)(j2). this proves that c (r) is uniquely complemented. lemma 4.2. let r be a ring such that |max(r)| ≥ 2. the following statements are equivalent: (i) c (r) is complemented. (ii) m(i) is a closed and open subset of max(r) for each i ∈ v (c (r)). proof. we adapt an argument found in the proof of [15, proposition 3.10]. (i) ⇒ (ii) assume that c (r) is complemented. let i ∈ v (c (r)). it is clear that m(i) is a closed subset of max(r). since c (r) is complemented, there exists j ∈ v (c (r)) such that i ⊥ j in c (r). hence, i and j are adjacent in c (r) and there is no a ∈ v (c (r)) such that a is adjacent to both i and j in c (r). that is, i + j = r and there is no proper ideal a of r with a + i = a + j = r. note that there exist a ∈ i and b ∈ j such that a + b = 1. we claim that m(i) = d(b). let m ∈ m(i). as m ⊇ i, a ∈ i , and a + b = 1, it follows that b /∈ m. hence, m ∈ d(b). this shows that m(i) ⊆ d(b). let m ∈ d(b). 95 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 hence, m + j = r. since, i ⊥ j in c (r), it follows that i + m 6= r and so, i ⊆ m. that is, m ∈ m(i). this proves that d(b) ⊆ m(i) and so, m(i) = d(b) is a closed and open subset of max(r). (ii) ⇒ (i) let i ∈ v (c (r)). by assumption, m(i) is a closed and open subset of max(r). hence, there exists an ideal j of r such that m(i) = max(r)\m(j). this implies that i + j = r and ij ⊆ j(r). if a ∈ v (c (r)) is such that a + i = a + j = r, then a + ij = r. this is impossible, since ij ⊆ j(r). therefore, there is no a ∈ v (c (r)) such that a is adjacent to both i and j in c (r). this proves that i ⊥ j in c (r). therefore, c (r) is complemented. proposition 4.3. let r be a ring such that |max(r)| ≥ 2. if r is semiquasilocal, then c (r) is complemented. proof. let {mi|i ∈ {1, 2, . . . , n}} denote the set of all maximal ideals of r. note that v (c (r)) = {i|i is a proper ideal of r with i 6⊆ j(r)}. let i ∈ v (c (r)). let i1, . . . , it ∈ {1, 2, . . . , n} be such that m(i) = {mi1, . . . , mit}. it is clear that 1 ≤ t < n. let us denote the set {1, 2, . . . , n}\{i1, . . . , it} by {it+1, . . . , in}. consider the ideal j = ∩nj=t+1mij . it is clear that m(j) = {mij|j ∈ {t + 1, . . . , n}}. it follows from i + j = r and ij ⊆ j(r) that m(i) = max(r)\m(j) is a closed and open subset of max(r). therefore, we obtain from (ii) ⇒ (i) of lemma 4.2 that c (r) is complemented. corollary 4.4. let r be a ring such that |max(r)| ≥ 2. if r is semiquasilocal, then c (r) is uniquely complemented. proof. this follows from proposition 4.3 and remark 4.1(ii). let r be a ring. as in [12], we call the graph studied by p.k. sharma and s.m. bhatwadekar in [16] as the comaximal graph of r and as in [12], we denote it using the notation γ(r). it is useful to recall here that the vertex set of γ(r) is the set of all elements of r and distinct vertices a and b are joined by an edge in γ(r) if and only if ra + rb = r. moreover, as in [12], we use the notation γ1(r) to denote the subgraph of γ(r) induced on u(r); we use γ2(r) to denote the subgraph of γ(r) induced on nu(r); for a ring r with |max(r)| ≥ 2, we use γ2(r)\j(r) to denote the subgraph of γ(r) induced on nu(r)\j(r). it is shown in [15, proposition 3.11] that γ2(r)\j(r) is complemented if and only if dim( r j(r) ) = 0. we prove in theorem 4.7 that for a ring r with |max(r)| ≥ 2, c (r) is complemented if and only if r j(r) ∼= f1 ×f2 ×···×fn as rings, where fi is a field for each i ∈{1, 2, . . . , n}. lemma 4.5. let r be a ring such that |max(r)| ≥ 2. the following statements are equivalent: (i) c (r) is complemented. (ii) c ( r j(r) ) is complemented. proof. (i) ⇒ (ii) we are assuming that c (r) is complemented. observe that j( r j(r) ) is the zero ideal of r j(r) . let i j(r) ∈ v (c ( r j(r) )). then it is clear that i ∈ v (c (r)). as c (r) is complemented, there exists j ∈ v (c (r)) such that i ⊥ j in c (r). we know from the proof of (i) ⇒ (ii) of lemma 4.2 that there exists b ∈ j such that m(i) = d(b). it is not hard to verify that m( i j(r) ) = d(b + j(r)). hence, for any i j(r) ∈ v (c ( r j(r) )), m( i j(r) ) is a closed and open subset of max( r j(r) ). therefore, we obtain from (ii) ⇒ (i) of lemma 4.2 that c ( r j(r) ) is complemented. (ii) ⇒ (i) we are assuming that c ( r j(r) ) is complemented. let i ∈ v (c (r)). let us denote the ideal i + j(r) by a. then a j(r) ∈ v (c ( r j(r) )). since c ( r j(r) ) is complemented, it follows from the proof of (i) ⇒ (ii) of lemma 4.2 that there exists b ∈ r\j(r) such that m( a j(r) ) = d(b + j(r)). now, it is easy to verify that m(i) = d(b). thus for any i ∈ v (c (r)), m(i) is a closed and open subset of max(r). therefore, we obtain from (ii) ⇒ (i) of lemma 4.2 that c (r) is complemented. 96 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 let r be a ring. recall from [8, exercise 16, page 111] that r is said to be von neumann regular if for each element a ∈ r, there exists b ∈ r such that a = a2b. we know from (a) ⇔ (d) of [8, exercise 16, page 111] that r is von neumann regular if and only if dimr = 0 and r is reduced. hence, if r is von neumann regular, then j(r) = nil(r) = (0). let r be a von neumann regular ring. let a ∈ r. we know from (1) ⇒ (3) of [8, exercise 24, page 113] that a = ue, where u is a unit of r and e is an idempotent element of r. hence, any ideal of r is a radical ideal of r. let i be any ideal r. since the set of all prime ideals of r equals the set of all maximal ideals of r, it follows from [3, proposition 1.14] that i = r(i) is the intersection of all the maximal ideals m of r such that m ∈ m(i). lemma 4.6. let r be a von neumann regular ring such that |max(r)| ≥ 2. the following statements are equivalent: (i) c (r) is complemented. (ii) r ∼= f1 ×f2 ×···×fn as rings for some n ≥ 2, where fi is a field for each i ∈{1, 2, . . . , n}. proof. (i) ⇒ (ii) we are assuming that c (r) is complemented. since j(r) = (0), it is clear that v (c (r)) equals the set of all nonzero proper ideals of r. let i be a nonzero proper ideal of r. as c (r) is complemented, we know from the proof of (i) ⇒ (ii) of lemma 4.2 that m(i) = d(b) for some nonzero nonunit b of r. note that b = ue, where u is a unit of r and e is an idempotent element of r. therefore, m(i) = d(b) = d(e) = m(1 − e). hence, i = ∩m∈m(i)m = ∩m∈m(1−e)m = r(1 − e). this proves that each ideal of r is finitely generated and so, r is noetherian. therefore, we obtain from [8, exercise 21, page 112] that there exist n ∈ n and fields f1 . . . , fn such that r ∼= f1 ×···× fn as rings. since |max(r)| ≥ 2, it follows that n ≥ 2. (ii) ⇒ (i) assume that r ∼= f1 × f2 × ···× fn as rings for some n ≥ 2, where fi is a field for each i ∈ {1, 2, . . . , n}. let us denote the ring f1 × f2 × ··· × fn by t . note that t is semilocal with {m1 = (0) × f2 ×···× fn, m2 = f1 × (0) ×···× fn, . . . , mn = f1 ×···× fn−1 × (0)} as its set of all maximal ideals. we know from proposition 4.3 that c (t ) is complemented. as r ∼= t as rings, we get that c (r) is complemented. theorem 4.7. let r be a ring such that |max(r)| ≥ 2. the following statements are equivalent: (i) c (r) is complemented. (ii) r j(r) ∼= f1 ×f2 ×···×fn as rings for some n ≥ 2, where fi is field for each i ∈{1, 2, . . . , n}. (iii) r is semiquasilocal. proof. (i) ⇒ (ii) we are assuming that c (r) is complemented. we know from (i) ⇒ (ii) of lemma 4.5 that c ( r j(r) ) is complemented. note that j( r j(r) ) equals the zero ideal of r j(r) . let a ∈ r be such that a + j(r) is a nonzero nonunit of r j(r) . as c ( r j(r) ) is complemented, we obtain from (i) ⇒ (ii) of lemma 4.2 that m(a + j(r)) is a closed and open subset of max( r j(r) ). therefore, it follows from [15, lemma 1.2] that dim( r j(r) ) = 0. thus r j(r) is reduced and zero-dimensional and so, r j(r) is von neumann regular. it now follows from (i) ⇒ (ii) of lemma 4.6 that r j(r) ∼= f1 ×f2 ×···×fn for some n ≥ 2, where fi is a field for each i ∈{1, 2, . . . , n}. (ii) ⇒ (iii) we are assuming that r j(r) ∼= f1 × f2 × ···× fn as rings for some n ≥ 2, where fi is a field for each i ∈ {1, 2, . . . , n}. hence, r j(r) is semilocal. as |max(r)| = |max( r j(r) )|, we get that r is semiquasilocal. (iii) ⇒ (i) since r is semiquasilocal, we obtain from proposition 4.3 that c (r) is complemented. let r be a ring with |max(r)| ≥ 2. we next discuss some results regarding the s-vertices of c (r). 97 s. visweswaran, j. parejiya / j. algebra comb. discrete appl. 5(2) (2018) 85–99 remark 4.8. let r be a ring such that |max(r)| = 2. let {m1, m2} denote the set of all maximal ideals of r. we know from (3) ⇒ (1) of [18, theorem 4.5] that c (r) is a complete bipartite graph with vertex partition v1 and v2, where vi = {i|m(i) = {mi}} for each i ∈{1, 2}. hence, it is clear that no vertex of c (r) is a s-vertex of c (r). therefore, for a ring r with |max(r)| ≥ 2, in determining the s-vertices of c (r), we assume that |max(r)| ≥ 3. proposition 4.9. let r be a ring such that |max(r)| ≥ 3. let i ∈ v (c (r)). then the following statements are equivalent: (i) i is a s-vertex of c (r). (ii) |max(r)\m(i)| ≥ 2. proof. (i) ⇒ (ii) now, i ∈ v (c (r)) and we are assuming that i is a s-vertex of c (r). hence, there exist distinct i1, i2, i3 ∈ v (c (r)) such that i −i1, i −i2, i2 −i3 are edges of c (r), but there is no edge joining i1 and i3 in c (r). since i1 and i3 are not adjacent in c (r), there exists m ∈ max(r) such that i1 + i3 ⊆ m. it follows from i + i1 = i2 + i3 = r that i 6⊆ m and i2 6⊆ m. as i2 is proper ideal of r, there exists m′ ∈ max(r) such that i2 ⊆ m′. it is clear that m 6= m′ and it follows from i + i2 = r that i 6⊆ m′. hence, {m, m′}⊆ max(r)\m(i) and so, |max(r)\m(i)| ≥ 2. 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[18] m. ye, t. wu, co–maximal ideal graphs of commutative rings, j. algebra appl. 11(6) (2012) article id. 1250114 (14 pages). 99 https://doi.org/10.1080/00927872.2012.744025 https://doi.org/10.1080/00927872.2012.744025 http://dx.doi.org/10.4153/cmb-2013-033-7 https://doi.org/10.1006/jabr.1995.1236 https://doi.org/10.1006/jabr.1995.1236 https://doi.org/10.1007/s40863-017-0065-9 https://doi.org/10.1007/s40863-017-0065-9 https://doi.org/10.1142/s0219498812501149 https://doi.org/10.1142/s0219498812501149 introduction some preliminary results and on the planarity of c(r) when is c(r) a split graph? some more results on c(r) references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.369865 j. algebra comb. discrete appl. 5(2) • 51–63 received: 6 may 2017 accepted: 27 october 2017 journal of algebra combinatorics discrete structures and applications fourier matrices of small rank research article gurmail singh abstract: modular data is an important topic of study in rational conformal field theory. cuntz, using a computer, classified the fourier matrices associated to modular data with rational entries up to rank 12, see [3]. here we use the properties of c-algebras arising from fourier matrices to classify complex fourier matrices under certain conditions up to rank 5. also, we establish some results that are helpful in recognizing c-algebras that not arising from fourier matrices by just looking at the first row of their character tables. 2010 msc: 05e30, 05e99, 81r05 keywords: fourier matrices, modular data, fusion rings, c-algebras 1. introduction fourier matrices are a fundamental ingredient of modular data. modular data is a basic component of rational conformal field theory, see [5]. further, rational conformal field theory has important applications in physics, see [4] and [8]. in particular, it has nice applications to string theory, statistical mechanics, and condensed matter physics, see [10] and [13]. modular data give rise to fusion rings, c-algebras and c∗-algebras, see [3] and [11]. these rings and algebras are interesting topics of study in their own right. a unitary and symmetric matrix whose first column has positive real entries is called a fourier matrix if its columns under entrywise multiplication produce integral structure constants. the set of columns of a fourier matrix under entrywise multiplication and usual addition generate a fusion algebra, see [3]. but a two-step rescaling on fourier matrices gives rise to self-dual c-algebras. cuntz, using a computer, classified the fourier matrices with rational entries up to rank 12, see [3]. but rational fourier matrices do not include some other important matrices, see sections 4, 5 and 6. here we use c-algebra perspective to classify the complex fourier matrices up to rank 5 under certain conditions. also, we establish some results that are helpful in recognizing the c-algebras that are not arising from fourier matrices by mere looking at the first row of their character tables. gurmail singh; department of mathematics and statistics, university of regina, canada, s4s 0a2 (email: gurmail.singh@uregina.ca). 51 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 in section 2, we collect the definitions and introduce a two-step rescaling of fourier matrices. in section 3, we summarize the results that are useful to recognize the c-algebras that are not arising from fourier matrices. in section 4, we classify fourier matrices of rank 2 and 3. in sections 5 and 6, we classify non-homogeneous fourier matrices of rank 4 and rank 5, respectively. for the classification of homogenous fourier matrices see [11, theorem 13]. 2. c-algebras arising from fourier matrices a scaling of the rows of a fourier matrix gives the basis of a fusion algebra that contains the identity element. but a two-step rescaling of a fourier matrix gives the standard basis of c-algebra. definition 2.1. let a be a finite dimensional and commutative algebra over c with distinguished basis b = {b0 := 1a,b1, . . . ,br−1}, and an r-linear and c-conjugate linear involution ∗ : a → a. let δ : a → c be an algebra homomorphism. then the triple (a,b,δ) is called a c-algebra if it satisfies the following properties: 1. for all bi ∈ b, (bi)∗ = bi∗ ∈ b, 2. for all bi,bj ∈ b, we have bibj = ∑ bk∈b λijkbk, for some λijk ∈ r, 3. for all bi,bj ∈ b,λij0 6= 0 ⇐⇒ j = i∗, 4. for all bi ∈ b,λii∗0 = λi∗i0 > 0. 5. for all bi ∈ b, δ(bi) = δ(bi∗ ) > 0. the algebra homomorphism δ is called a degree map, and the values δ(bi), for all bi ∈ b, are called the degrees of a. for i 6= 0, δ(bi) is called a nontrivial degree. if δ(bi) = λii∗0, for all bi ∈ b, we say that b is a standard basis. the order of a c-algebra is denfined as δ(b+) := ∑r−1 i=0 δ(bi). a c-algebra is called symmetric if bi∗ = bi, for all i. a c-algebra with rational structure constants is called a rational c-algebra. the readers interested in c-algebras are directed to [1], [2] and [7]. to keep the generality, in the following definition of a fourier matrix we assume the structure constants to be integers instead of nonnegative integers, see [3, definition 2.2]. definition 2.2. let r ∈ z+ and i an r ×r identity matrix. then s is called a fourier matrix if 1. s is a unitary and symmetric matrix, that is, ss̄t = i,s = st , 2. si0 > 0, for 0 ≤ i ≤ r − 1, where s is indexed by {0, 1, 2, . . . ,r − 1}, 3. nijk = ∑ l slisljs̄lks −1 l0 ∈ z, for all 0 ≤ i,j,k ≤ r − 1. let s be a fourier matrix. let s = [sij] be a matrix with entries sij = sij/si0, for all i,j, we call it an s-matrix associated to s (briefly, s-matrix). since s is a unitary matrix, ss̄t =diag(d0,d1, . . . ,dr−1) is a diagonal matrix, where di = ∑ j sijs̄ij. the numbers di are called norms of s-matrix. the relation sij = sij/si0 implies the structure constants nijk = ∑ l slisljslkd −1 l , for all i,j,k. since the structure constants nijk are integers, the numbers sij are algebraic integers, see [3, section 3]. therefore, if s has only rational entries then entries of s-matrix are rational integers, and such s-matrices are known as integral fourier matrices, see [3, definition 3.1]. cuntz classified the integral fourier matrices up to rank 12 by using a computer, see [3]. in this paper, we consider the broader class of s-matrices that have algebraic integer entries. there is an interesting row-and-column operation (two-step rescaling) procedure that can be applied to a fourier matrix s that results in the first eigenmatrix, the character table, of a self-dual c-algebra. the steps of the procedure are reversed to obtain the fourier matrix s from the first eigenmatrix. the 52 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 explanation of the procedure is as follows. let s = [sij] be a fourier matrix indexed with {0, 1, . . . ,r−1}. we divide each row of s with its first entry and obtain the s-matrix. the multiplication of each column of the s-matrix with its first entry gives the p-matrix associated to s (briefly, p-matrix), the first eigenmatrix of a self-dual c-algebra. that is, sij = sijs −1 i0 and pij = sijs0j, for all i,j, where pij denotes the (i,j)-entry of the p-matrix. conversely, to obtain the s-matrix from a p-matrix, divide each column of the p-matrix with the squareroot of its first entry. further, the fourier matrix s is obtained from the s-matrix by dividing the ith row of s-matrix by √ di, where di = ∑ j |sij| 2. that is, sij = pij/ √ p0j, and sij = sij/ √ di, for all i,j. since the entries of an s-matrix are algebraic integers, the entries of a p-matrix are also algebraic integers. remark 2.3. throughout this paper, unless mentioned explicitly, the sets of columns of a p-matrix and an s-matrix are denoted by b = {b0,b1, . . . ,br−1} and b̃ = {b̃0, b̃1, . . . , b̃r−1}, respectively. the structure constants generated by the columns, with entrywise multiplication, of a p-matrix and an s-matrix are denoted by λijk and nijk, respectively. mt denotes the transpose of a matrix m. let s be a fourier matrix and a := cb, a c-span of b. define a map ∗ : a −→ a by ( ∑ j ajbj) ∗ =∑ j ājbj∗ = ∑ j āj[p̄0j, p̄1j, . . . , p̄r−1,j] t . this map ∗ is an involution on a, and the map δ : a −→ c defined as δ( ∑ j ajb̃j) = ∑ j ājs0j, that is, δ(bi) = δ(s0ib̃i) = s 2 0i for all i, is a positive degree map of a. since bi = s0ib̃i, the structure constants generated by the basis b are given by λijk = nijks0is0js −1 0k , for all i,j,k. s is a unitary matrix, therefore, nij0 = ∑ l slisljs̄l0s −1 l0 6= 0 ⇐⇒ j = i ∗ and nii∗0 = 1 > 0, for all i,j. thus λij0 6= 0 ⇐⇒ j = i∗ and λii∗0 > 0, for all i,j. therefore, the vector space a := cb is a c-algebra of order d0, b is the standard basis of a, and p-matrix is the first eigenmatrix of a, see [11, theorem 4], and we say (a,b,δ) is a c-algebra arising from a fourier matrix s. note that, entries of the first eigenmatrix p are the entries of the character table a. thus at some places we consider the p-matrix of a as the character table of a and the ith row of p-matrix as the ith irreducible character of a. let (a,b,δ) be a c-algebra arising from a fourier matrix s. since s is a symmetric matrix, a is a self-dual c-algebra and d0 = djδ(bj), for all j. the entries of an s-matrix and the associated p-matrix are algebraic integers. therefore, if a has rational degrees then both the degrees and norms are rational integers and both divide the order of a, see [11, proposition 5]. note that, a c-algebra arising from a fourier matrix s is a symmetric c-algebra if and only if s is a real matrix. a c-algebra that has at least two different nontrivial degrees is called a non-homogeneous c-algebra and we call the associated fourier matrix (s-matrix) a non-homogeneous fourier matrix (non-homogeneous s-matrix, respectively). every self-dual c-algebra not necessarily have rational degrees. for example, a self-dual c-algebra of rank 2 with basis {1,x}, and the structure constants given by the equation x2 = 1 + x does not have rational degrees. we remark that this c-algebra does not arise from a fourier matrix. but a rational c-algebra arising from a fourier matrix has integral degrees, see [11, proposition 5]. lemma 2.4. let (a,b,δ) be a c-algebra arising from a fourier matrix s of rank r. 1. if a has rational order then the order of a is an integer. 2. if a has nonnegative structure constants then degrees of a are greater or equal to 1. 3. if a has rational order and all the degrees of a different from 1 are all equal then the degrees of a are integers. (note: the algebra a need not be homogeneous.) proof. (i). since degrees of a are algebraic integers and order of a is the sum of degrees of a, the order of a is a rational integer. (ii). for any i, δ(bi) is the first entry of column vector (b̃i)2 and nii0 = 1. note that, nijk ≥ 0 for all i,j,k, because a has nonnegative structure constants. therefore, δ(bi) = 1 + m, where m is a nonnegative algebraic integer. (iii). let all the degrees of a different from 1 be a positive real number t. therefore, d0 = m + nt, where m is the number of degrees equal to 1 and n is the number of degrees equal to t. since the order d0 is an integer, t is an integer. 53 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 3. recognition of c-algebras arising from fourier matrices the following results are useful for recognizing c-algebras that are not arising from fourier matrices by mere looking at the degrees of c-algebras, that is, the first row of the character tables. all the character tables of the association schemes used here are produced by hanaki and miyamoto, see [6]. lemma 3.1. let (a,b,δ) be a c-algebra arising from a fourier matrix s with nonnegative structure constants. let l(b) = {b ∈ b : δ(b) = 1}. then l(b) is an abelian group. proof. since δ(b0) = 1, b0 ∈ l. let bi,bj ∈ l(b). therefore, bi = b̃i and bj = b̃j. thus bibj = b̃ib̃j =∑ k nijkb̃k implies 1 = δ(bibj) = ∑ k nijkδ(b̃k). by lemma 2.4, δ(bl) ≥ 1, thus δ(b̃l) = √ δ(bl) ≥ 1, for all l. therefore, bibj = b̃k, for some bk ∈ b. also, for all bi ∈ l(b), bibi∗ = b0 + ∑ j λii∗jbj and λii∗j ≥ 0 imply bibi∗ = b0, that is, b −1 i = bi∗ ∈ l(b). hence l(b) is an abelian group. proposition 3.2. let (a,b,δ) be a c-algebra arising from a fourier matrix s with nonnegative structure constants. let s be a real fourier matrix. 1. let the order of a be a rational number. if all the degrees of a different from 1 are equal to t then t might be a power of 2. 2. if rank of a is an even integer then a cannot have only one degree different from 1. 3. let the order of a be a rational number. if the rank of a is greater than 3 then a cannot have only one degree greater or equal to r and all other degrees equal to 1. proof. (i). since s is a real fourier matrix, by lemma 3.1, the elements of b with degree 1 form an elementary abelian group. thus the number of elements of b with degree 1 is a power of 2. by lemma 2.4, t is an integer. the result follows from the fact that t divides the order of the algebra, see [11, proposition 5 (ii)]. (ii). if rank of a is 2 then both degrees are equal to 1, see section 4. by lemma 3.1, the set of elements of b with degree 1 form an elementary abelian group. therefore, the order of the group is a power of 2, say 2m, where m is a nonnegative integer. since a has only one degree different from 1, 2m = r − 1, a contradiction to the fact that r is an even integer. (iii). suppose a has only one degree k that is greater or equal to r and the remaining degrees are equal to 1. without loss of generality, let the first row of the character table be [1, 1, . . . , 1,k], where k ≥ r. since δ(b0) = . . . = δ(br−2), d0 = . . . = dr−2. the structure constants are nonnegative, therefore, |pij| ≤ p0j for all i,j, see [12, proposition 4.1]. therefore, the only possible entries of row 2, . . ., row r − 1 of p-matrix are [1, 1, . . . , 1,−k], which is not possible as p nonsingular. the adjacency algebras of the association schemes as12(9), as14(4), as16(10), as16(20), as16(21) and as16(62) have the degree patterns that violate the above result, see [6]. therefore, they are not arising from fourier matrices. the character table of the association scheme as4(2) [6] is an example where part (iii) of the above proposition fails for the rank 3. the above proposition also helps to sieve out a lot of c-algebras even if their self-duality is not known. the next proposition helps to recognize the c-algebras not arising from fourier matrices. proposition 3.3. let (a,b,δ) be a c-algebra arising from a fourier matrix s of rational order. let the number of i’s such that δ(bi) = 1 be t and the remaining r − t degrees are equal to k. then the possible values of k are the divisors of t. proof. by lemma 2.4, the order and degrees of a are integers. since entries of s-matrix are algebraic integers, all the norms are also integer, that is, d0k−1 ∈ z. therefore, d0 = t+ (d0−t)k implies tk−1 ∈ z. hence k is in the subset of the divisors of t. 54 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 the above proposition illustrates that the adjacency algebras of the association schemes as9(3), as9(8), as10(6), as16(20), as16(21) and as16(62) [6] are not arising from fourier matrices. in the next proposition we examine the possible number of occurrences of a degree if it is one of the degrees and satisfy a certain criteria. proposition 3.4. let (a,b,δ) be a c-algebra arising from a fourier matrix s with integral degrees. 1. if for a given j, δ(bj) (6= 1) is a smallest nontrivial degree that divides all the nontrivial degrees δ(bl) (6= 1) then the number of degrees equal to 1 is a multiple of δ(bj). 2. let δ(bt) be a degree divisible by all the smaller degrees and divides all the bigger degrees. let δ(bs) be the largest degree among all the degrees strictly less than δ(bt). let the sum of the degrees less than δ(bs) be β1δ(bs) and the number of degrees equal to δ(bs) be β2. then β1δ(bs) + β2δ(bs) is divisible by δ(bs). (note: β1δ(bs) is not equal to zero only if δ(bs) > 1.) 3. suppose a has nonnegative structure constants. if for all i < j, δ(bi) divides δ(bj) then a has integral structure constants. proof. (i). since degrees of a are integers, the norms are integers. thus, d0δ(bj)−1 ∈ z. therefore, ( 1 + r−1∑ i=1 δ(bi) ) δ(bj) −1 = ( ∑ δ(bi)=1 δ(bi) + ∑ δ(bi)≥δ(bj) δ(bi) ) δ(bj) −1 = ( ∑ δ(bi)=1 δ(bi) ) δ(bj) −1 + α ∈ z, where α ∈ z. hence the number of degrees equal to 1 are multiple of δ(bj). (ii). the degrees and norms of a are integers. therefore, d0δ(bt) −1 = (r−1∑ i=0 δ(bi) ) δ(bt) −1 = ( ∑ δ(bi)<δ(bs) δ(bi) + ∑ δ(bi)=δ(bs) δ(bi) + ∑ δ(bi)>δ(bs) δ(bi) ) δ(bt) −1 = ( β1δ(bs) + ∑ δ(bi)=δ(bs) δ(bi) ) δ(bt) −1 + γ ∈ z, where γ ∈ z. thus β1δ(bs) + β2δ(bs) is divisible by δ(bt). (iii). since λijk are nonnegative, nijk are nonnegative, because λijk = nijks0is0js −1 0k , for all i,j,k. let b̃ib̃j = ∑ k nijkb̃k. on comparing the first entry of both sides, we conclude that b̃k cannot occur with nonzero coefficient whenever s0k > s0is0j, that is, √ δ(bk) > √ δ(bi)δ(bj) implies nijk = 0. hence the assertion follows from the relation between λijk and nijk. for example, the adjacency algebras of association schemes as7(2), as8(5), as8(6), as9(8), as9(9) and adjacency algebras of homogenous schemes have the degree patterns that violate the above proposition, see [6]. therefore, they are not arising from fourier matrices. lemma 3.5. let (a,b,δ) be a c-algebra arising from a rational fourier matrix s of odd rank and odd order. let the odd degree among all the degrees of a be maximum. then the rank of a must be at least 11. proof. let d0 = δ(bi)ai. by [3, lemma 3.7], d0 is an odd square, thus ai is a square. let δ(b1) be an odd integer and δ(b1) ≥ δ(bi) for each i. therefore, d0 ≥ 9δ(b1), and d0 ≤ 1 + (r − 1)δ(b1). thus 9δ(b1) ≤ 1 + (r − 1)δ(b1) implies δ(b1)(9 − (r − 1)) ≤ 1 implies δ(b1) ∈ z+ only if r − 1 ≥ 9. 55 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 4. fourier matrices of rank 2 and 3 in this section we classify fourier matrices of rank 2 and 3. in fact, we find the p-matrices of c-algebras arising from fourier matrices of rank 2 and 3. but the associated fourier matrix s can be recovered easily from the p-matrix as described in section 2. since the row sum of a character table is zero, the character table of a c-algebra of rank 2 with standard basis b = {b0,bi} is given by p = [ 1 n 1 −1 ] , and the structure constants are given by b21 = nb0 + (n − 1)b1. therefore, the structure constant n111 is integer only for n = 1, and the associated fourier matrix s = 1 √ 2 [ 1 n 1 −1 ] . let p be the character table for a symmetric c-algebra arising from a fourier matrix of rank 3 with standard basis b = {b0,b1,b2}. let b1b2 = ub1 + vb2. then p =   1 k1 k21 φ1 φ2 1 ψ1 ψ2   , where φ1 = (v −u− 1 + √ (u−v − 1)2 + 4u)/2, φ2 = (u−v − 1 − √ (u−v − 1)2 + 4u)/2, and ψ1 = (v −u− 1 − √ (u−v − 1)2 + 4u)/2, ψ2 = (u−v − 1 + √ (u−v − 1)2 + 4u)/2. therefore, d0 = 1 + k1 + k2, d1 = 1 + |φ1|2 k1 + |φ2|2 k2 , d2 = 1 + |ψ1|2 k1 + |ψ2|2 k2 . lemma 4.1. there is no symmetric homogenous c-algebra of rank 3 arising from a fourier matrix s. proof. suppose (a,b,δ) is a symmetric homogenous c-algebra arising from a fourier matrix. since st = s, φ2k = ψ1l implies u = v. the structure constant n210 = 0 implies k = 2u. thus φ1 = (−1 + √ 1 + 2k)/2 and φ2 = (−1 − √ 1 + 2k)/2. a homogenous c-algebra arising from a fourier matrix has all degrees equal to 1, see [11, proposition 12]. but the structure constants n112 = 1 (2k + 1) √ k [k2 + k 2 ] and n222 = 1 (2k + 1) √ k [k2 − 1 − 3 2 k] are not integers for k = 1, a contradiction. theorem 4.2. let (a,b,δ) be a symmetric non-homogeneous c-algebra of rank 3 arising from a fourier matrix s with integral degrees. then the corresponding matrices p, s and s are as follows. p =   1 1 21 1 −2 1 −1 0   , s =   1 1 √ 2 1 1 − √ 2 1 −1 0   and s =   1/2 1/2 1/ √ 2 1/2 1/2 −1/ √ 2 1/ √ 2 −1/ √ 2 0   . proof. let δ(bi) = ki, for all i. since the integral degrees of a divide the order, k1 divides 1 + k2, and k2 divides 1 + k1. therefore, the only possible degree pattern of a are [1, 1, 2] and [1, 2, 3], up to the permutations. since n012 = 0, 1− v k1 − u k2 = 0. therefore, the degree patterns [1, 1, 2] and [1, 2, 3] imply v = 1 − u 2 and v = 2 − 2u 3 , respectively. case 1. let the degree pattern be [1, 1, 2], that is, k1 = 1 and k2 = 2. therefore, v = 1 − u 2 . since n011 = 1, u3(u− 1) = 0. hence (u,v) = (0, 1), or (u,v) = (1, 1/2). subcase 1. let (u,v) = (0, 1). 56 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 therefore, p =   1 1 21 1 −2 1 −1 0   , s =   1 1 √ 2 1 1 − √ 2 1 −1 0   and s =   1/2 1/2 1/ √ 2 1/2 1/2 −1/ √ 2 1/ √ 2 −1/ √ 2 0   . subcase 2. let (u,v) = (1, 1 2 ). therefore, p =   1 1 2 1 −3 + √ 17 4 −1 − √ 17 4 1 −3 − √ 17 4 −1 + √ 17 4   . note that δ(b1) = δ(b0), but d1 6= d0, a contradiction. hence u = 1 and v = 1 2 is not a possible case. case 2. let the degree pattern be [1, 2, 3], that is, k1 = 2 and k2 = 3. therefore, v = 2 − 2u 3 . since n011 = 1, 625u4 − 1850u3 + 2520u2 − 1296u + 243 = 0. but it has no real roots, see [9]. thus we rule out [1, 2, 3] degree pattern, because an s-matrix associated with a symmetric c-algebra might be a real matrix. remark 4.3. the above p-matrix is given by the character table of the adjacency algebra of an association scheme as4(2), see [6]. in the next theorem we prove that there is only one asymmetric c-algebra of rank 3 arising from a fourier matrix. moreover, the following theorem shows that for rank 3 it is not necessary to assume |sij| ≤ s0j to prove that the homogeneous c-algebra arising from a fourier matrix is a group algebra, see [11, theorem 13]. theorem 4.4. let (a,b,δ) be an asymmetric c-algebra arising from a fourier matrix s of rank 3. then the p-matrix is the first eigenmatrix of the group algebra of a group of order 3. proof. the p-matrix of an asymmetric c-algebra of rank 3 is as follows, p =   1 k k1 α ᾱ 1 ᾱ α   , where α = (−1 + i √ 1 + 2k)/2. since a is homogenous, k = 1, see [11, proposition 12]. therefore, p(= s) =   1 1 11 ζ3 ζ23 1 ζ23 ζ3   and s = 1√ 3   1 1 11 ζ3 ζ23 1 ζ23 ζ3   . 5. fourier matrices of rank 4 in this section we classify the fourier matrices under certain conditions, and we show that there is no non-homogeneous integral fourier matrix of rank 4. for homogenous fourier matrices see [11]. 57 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 lemma 5.1. let (a,b,δ) be a c-algebra arising from a fourier matrix s of rank r. let |sij| ≤ s0j, for all j. let δ(bj) = kj for all j, and ki = 1 for some i > 0. 1. then |pij| = kj for all j. 2. if s is a real matrix then pij = ±kj for all j. proof. (i). since δ(bi) = 1, di = d0. therefore, the row 1 and row i of the s-matrix are [1, 1, √ k2, . . . ,√ kr−1] and [1,pi1,pi2/ √ k2, . . . ,pi,r−1/ √ kr−1], respectively. since d0 = di, |pij|/ √ kj = √ kj, for all j. hence |pij| = kj, for all j. (ii). by part (i), |pij| = kj for all j. since s is a real matrix, pij = ±kj for all j. the next proposition classify the non-homogeneous fourier matrices of rank 4 with one nontrivial degree equal to 1. proposition 5.2. let (a,b,δ) be a non-homogeneous rational c-algebra arising from a fourier matrix s of rank 4. let |sij| ≤ s0j, for all j. let δ(bj) = kj, for all j, and ki = 1, for some i > 0. then the associated p-matrix is   1 1 4 6 1 1 4 −6 1 1 −2 0 1 −1 0 0   . proof. since a is a rational c-algebra, the degrees of a are integers, see [11, proposition 5]. let δ(bi) = ki, for all i. without loss of generality, suppose k1 = 1. therefore, we have p =   1 1 k2 k3 1 p11 p12 p13 1 p21 p22 p23 1 p31 p32 p33   . case 1. if p11,p12 and p13 are not rational integers. since d0 = d1, |p11| = 1, |p12| = k2 and |p13| = k3. thus p11,p12 and p13 cannot be irrational real numbers. therefore, they can be non-real algebraic integers. since the structure constants are rational numbers, a complex conjugate of an irreducible character of a is an irreducible character of a. without loss of generality, assume that the third irreducible character is a complex conjugate of the second character. thus k2 = 1. but a is non-homogenous and d0 = diδ(bi), therefore, k3 = 3 and d3 = 2. since s is a symmetric matrix, |p31| = |p32| = 1, thus d3 > 3, a contradiction. case 2. if p11,p12 and p13 are rational integers. by lemma 5.1, p11 = ±1,p12 = ±k2 and p13 = ±k3. since the row sum of p-matrix is zero, the second row of p-matrix is either [1,−1,k2,−k2] or [1, 1,k2,−(k2 + 2)]. subcase 1. let the second row of p be [1,−1,k2,−k2]. then, the first row of p-matrix is [1, 1,k2,k2]. since k2 divides d0 and a is non-homogenous, k2 = 2. thus, by the symmetry of fourier matrix s and orthogonality of characters, we have p =   1 1 2 2 1 −1 2 −2 1 1 −1 −1 1 −1 −1 1   but the s-matrix associated to the above p-matrix does not have integral structure constants. subcase 2. let the second row of p be [1, 1,k2,−(k2 + 2)]. 58 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 then, the first row of p-matrix is [1, 1,k2,k2 + 2]. since k2 divides d0 and a is non-homogenous, k2 = 2 or 4. thus, by the symmetry of fourier matrix s and orthogonality of characters, we have p =   1 1 2 4 1 1 2 −4 1 1 −2 0 1 −1 0 0   or p =   1 1 4 6 1 1 4 −6 1 1 −2 0 1 −1 0 0   . but the s-matrix associated to the first p-matrix does not have integral structure constants. remark 5.3. the above p-matrix is the first eigenmatrix of the adjacency algebra of an association scheme as12(8), see [6]. cuntz, with a computer, shows that there is no non-homogenous rational fourier matrix of rank 4, see [3]. in the next theorem, we use c-algebra perspective to show that there is no non-homogeneous rational fourier matrix of rank 4, that is, there is no non-homogeneous s-matrix with integral entries. unlike the above proposition, we do not assume any nontrivial degree equal to 1. theorem 5.4. there is no non-homogenous rational fourier matrix s of rank 4. proof. let (a,b,δ) be a c-algebra arising from a rational fourier matrix s of rank 4. let δ(bi) = ki, for all i > 0. since s-matrix is integral, k1,k2 and k3 are square integers, see [11, proposition 5 (iii)]. as d0 = 1 + k1 + k2 + k3 and k1,k2 and k3 divide d0, therefore, k2 + k3 ≡−1 mod k1, k1 + k3 ≡−1 mod k2 and k1 + k2 ≡−1 mod k3. claim: k1 = k2 = k3 = 1. without loss of generality, suppose k1 is an even integer. since k1,k2 and k3 are squares, k1 ≡ 0 mod 4 and d0 6≡ 0 mod 4, a contradiction to the fact that k1 divides d0. therefore k1,k2 and k3 are odd integer. suppose k1 ≥ k2,k3 and k1 > 1. now, if all k1,k2 and k3 are odd integers then k1,k2,k3 ≡ 1 mod 4. but d0 ≡ 0 mod 4 implies d0 = k1a, a ≥ 4. therefore, k1(a− 1) = 1 + k2 + k3 implies 3k1 ≤ 1 + k2 + k3. thus k2 or k3 > k1, again a contradiction. 6. fourier matrices of rank 5 in this section we prove that there is no non-homogenous s-matrix with integral entries (integral fourier matrix) of rank 5. but the following proposition shows, under certain conditions, that there are three s-matrices of rank 5 with algebraic integer entries. recall that, a fourier matrix s with rational entries has associated integral s-matrix, and a complex fourier matrix s has associated s-matrix with algebraic integer entries. proposition 6.1. let (a,b,δ) be a non-homogeneous rational c-algebra arising from a fourier matrix of rank 5. if |sij| ≤ s0j for all j. if δ(bi) = 1 for one i > 0 and δ(bj) = kj for all j 6= i. then up to simultaneous row and column permutations the p-matrices are as follows,  1 1 2 2 2 1 1 2 −2 −2 1 1 −2 0 0 1 −1 0 √ 2 − √ 2 1 −1 0 − √ 2 √ 2   ,   1 1 2 4 8 1 1 2 4 −8 1 1 2 −4 0 1 1 −2 0 0 1 −1 0 0 0   and   1 1 4 3 3 1 1 4 −3 −3 1 1 −2 0 0 1 −1 0 √ 3 − √ 3 1 −1 0 − √ 3 √ 3   . proof. since a is a rational c-algebra, the degrees of a are integers. let δ(bi) = ki, for all i. without loss of generality, let k1 = 1. therefore, first eigenmatrix of a is given by p =   1 1 k2 k3 k4 1 p11 p12 p13 p14 1 p21 p22 p23 −(1 + p21 + p22 + p23) 1 p31 p32 p33 −(1 + p31 + p32 + p33) 1 p41 p42 p43 −(1 + p41 + p42 + p43)   . 59 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 case 1. if p11,p12,p13 and p14 are not all rational integers. since d0 = d1, |p11| = 1, |p12| = k2, |p13| = k3 and |p14| = k4. since the row sum is zero, at least two of these p11,p12,p13 and p14 can be non-real algebraic integers. since the structure constants are rational numbers, a complex conjugate of an irreducible character is an irreducible character. without loss of generality, we assume that the third irreducible character is a complex conjugate of the second character, thus k2 = 1. without loss of generality, let k3 ≤ k4. therefore, d0 = dimi implies (k3,k4) ∈ {(1, 2), (1, 4), (2, 5), (3, 6), (6, 9)}. if k3 = 1 then d3 = d0 = d1 = d2 and |sij| ≤ s0j imply all the entries of the rows 1, 2, 3 and 4 are nonzero. since k3 6= k4, the entries of the fifth row are rational integers because galois conjugate of an irreducible character is an irreducible character, and rows of s-matrix corresponding to the conjugate characters have equal norm. but k3 = 1 implies d4 ≤ 3. thus there might be at least two zero entries in the fifth row of p-matrix. but s is a symmetric matrix, we get a contradiction. for (k3,k4) ∈{(2, 5), (3, 6), (6, 9)}, k3 6= k4, thus the entries of the row 5 are rational integers because k2 = k1 = k0 = 1 and the galois conjugate of an irreducible character is an irreducible character. each entry of the row 1, 2 and 3 of p-matrix is non-zero. but for each of the above pair there are exactly 3 zeros in the fifth row. since s is a symmetric matrix, we get a contradiction. case 2. if p11,p12,p13 and p14 are all rational integers. by lemma 5.1, the only possible degree patterns are: [1,−1,k2,k3,−(k2 + k3)], [1, 1,k2,k3,−(k2 + k3 + 2)], [1, 1, k2, −k3, − (k2 −k3 + 2)], [1, − 1, k2, −k3, − (k2 −k3)], [1, 1, −k2, −k3, k2 + k3 − 2], [1, − 1, −k2, −k3, k2 + k3]. subcase 1. let the second row of p be [1,−1,k2,k3,−(k2 + k3)]. then the first row of the character table is [1, 1,k2,k3,k2 + k3] and (k2 + k3) ∣∣(2 + k2 + k3). thus (k2,k3) = (1, 1). therefore, by the orthogonality of characters, we have p =   1 1 1 1 2 1 −1 1 1 −2 1 1 p22 p23 −1 1 1 p32 p33 −1 1 −1 p42 p43 1   . since k2 = k3 = 1, d2 = d3 = 6. but each of |p22|, |p23|, |p32| and |p33| can be at most 1. thus both d2 and d3 are strictly less than 6, a contradiction. hence this case is not possible. subcase 2. let the second row of p be [1, 1,k2,k3,−(k2 + k3 + 2)]. then the first row of the character table is [1, 1,k2,k3,k2 + k3 + 2]. therefore, by the orthogonality of the irreducible characters and symmetry of the matrix s, we have p =   1 1 k2 k3 k2 + k3 + 2 1 1 k2 k3 −(k2 + k3 + 2) 1 1 p22 −2 −p22 0 1 1 p32 −2 −p32 0 1 −1 0 0 0   . without loss of generality, let k2 ≤ k3. since k2 ∣∣(2k3 + 4) and k3∣∣(2k2 + 4), we have (k2,k3) ∈{(1, 2), (1, 3), (1, 6), (2, 4), (2, 8), (3, 10), (4, 6), (4, 12), (6, 16), (8, 10), (12, 28)}. note that k2 6= k3, thus d3 6= d4. the structure constants are rational. therefore, if p22 or p33 is not rational then row 3 and 4 of p-matrix should be galois conjugates. but the rows of s-matrix corresponding to conjugate irreducible characters should have equal norm. thus p22 and p32 are rational integers. therefore, det(p) ∈ z and (detp)2 = n5. thus n = 2(k2 + k3 + 2) need to be a square. but the only two pairs (2, 4), (4, 12) do not fail this test. for (k2,k3) = (2, 4), d2 = 8 = 1 + 1 + (p22√ 2 )2 + (−2 −p22√ 4 )2 . 60 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 since the entries of p-matrix are algebraic integers, we have p22 = 2. similarly, d3 = 4 implies p32 = −2. therefore, p =   1 1 2 4 8 1 1 2 4 −8 1 1 2 −4 0 1 1 −2 0 0 1 −1 0 0 0   . for (k2,k3) = (4, 12), d2 = 9 and d3 = 3. therefore, we have p22 = 4,−5 and p32 = 1,−2. since s is a symmetric matrix, p22 = 4 and p32 = −2 is the only possibility. but for p22 = 4 and p32 = −2, the associated s-matrix does not have integral structure constants. subcase 3. let the second row of p be [1, 1, k2, −k3, − (k2 −k3 + 2)]. then, the first row of the character table is [1, 1,k2,k3,k2 −k3 + 2]. therefore, by the orthogonality of characters, symmetry of the fourier matrix s and pp̄ = ni, we have p =   1 1 k2 k3 k2 −k3 + 2 1 1 k2 −k3 −(k2 −k3 + 2) 1 1 −2 0 0 1 −1 0 p33 −p33 1 −1 0 p43 −p43   . therefore, k2 ∣∣4, k3∣∣2k2 + 4, (k2 − k3 + 2)∣∣(k2 + k3 + 2) and k2 − k3 + 2 > 0. hence (k2,k3) ∈ {(1, 1), (1, 2), (2, 2), (4, 2), (4, 3), (4, 4)}. since |sij| ≤ s0j, (k2,k3) 6∈ {(1, 1), (1, 2)}. for (k2,k3) = (2, 2), d3 = d4 = 4. thus p33p̄33 = 2, p43p̄43 = 2. but the integrality of the structure constants of s-matrix and orthogonality of characters forces p33 = ± √ 2 and p43 = ∓ √ 2. therefore, up to simultaneous permutation of row 4 and row 5, and column 4 and column 5, we have p =   1 1 2 2 2 1 1 2 −2 −2 1 1 −2 0 0 1 −1 0 √ 2 − √ 2 1 −1 0 − √ 2 √ 2   . for (k2,k3) = (4, 2), d3 = 6. thus |p33| = 4 √ 3 > 2, a contradiction. for (k2,k3) = (4, 3), k4 = 3, d3 = 4 and d4 = 4. thus |p33| = √ 3 and |p43| = √ 3. but the integrality of structure constants and orthogonality of characters forces p33 = ± √ 3 and p43 = ∓ √ 3. therefore, up to simultaneous permutation of row 4 and row 5, and column 4 and column 5, we have p =   1 1 4 3 3 1 1 4 −3 −3 1 1 −2 0 0 1 −1 0 √ 3 − √ 3 1 −1 0 − √ 3 √ 3   . although the structure constants are not all integers, for example λ342 = 3/2, but the associated s-matrix has integral structure constants. subcase 4. let the second row of p be [1, − 1, k2, −k3, − (k2 −k3)]. then, the first row of the character table is [1, 1, k2, k3, k2 −k3]. therefore, by the orthogonality of the characters and symmetry of the matrix s, we have p =   1 1 k2 k3 k2 −k3 1 −1 k2 −k3 −(k2 −k3) 1 1 0 p23 −(2 + p23) 1 −1 p32 p33 −(p32 + p33) 1 −1 p42 p43 −(p42 + p43)   . 61 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 since pp̄ = ni, from row 1, 2 and column 3, we get k2 = 0, a contradiction. subcase 5. let the second row of p be [1, 1, −k2, −k3, k2 + k3 − 2]. then the first row of the character table is [1, 1, k2, k3,k2 + k3 − 2]. therefore, by the symmetry of the matrix s and orthogonality of characters, we have p =   1 1 k2 k3 k2 + k3 − 2 1 1 −k2 −k3 k2 + k3 − 2 1 −1 p22 −p22 0 1 −1 p32 −p32 0 1 1 0 0 −2   . therefore k2 ∣∣2k3, k3∣∣2k2, (k2 + k3 − 2)∣∣(k2 + k3 + 2) and k2 + k3 − 2 > 0. without loss of generality, let k2 ≤ k3. hence (k2,k3) ∈ {(1, 2), (2, 2)}. since |sij| ≤ s0j, (k2,k3) 6= (1, 2). for (k2,k3) = (2, 2), d2 = d3 = 4. thus p22p̄22 = 2, p32p̄32 = 2. but the integrality of structure constants and orthogonality of characters forces p22 = ± √ 2 and p32 = ∓ √ 2. therefore, up to simultaneous permutation of rows and columns, we have p =   1 1 2 2 2 1 1 −2 −2 2 1 −1 √ 2 − √ 2 0 1 −1 − √ 2 √ 2 0 1 1 0 0 −2   . subcase 6. let the second row of p be [1, − 1, −k2, −k3, k2 + k3]. then the first row of the character table is [1, 1, k2, k3, k2 + k3]. therefore, by the symmetry of the matrix s and orthogonality of the characters, we have p =   1 1 k2 k3 k2 + k3 1 −1 −k2 −k3 k2 + k3 1 −1 p22 p23 −1 1 −1 p32 p33 −1 1 1 p42 p43 0   . similar to the subcase 1, k2 = k3 = 1 implies d2 = d3 = 6, and we get a contradiction. remark 6.2. we note that the associated s-matrices to the p-matrices in the above proposition are not integral fourier matrices. the first two matrices are the character tables of as08(10), as16(24), respectively, see [6]. the third matrix is not a first eigenmatrix of an adjacency algebra of any association scheme because the structure constants generated by the columns of p-matrix are not all nonnegative integers. in the next theorem, by using the properties of c-algebras, we show that there is no non-homogeneous rational fourier matrix of rank 5. theorem 6.3. there is no non-homogeneous rational fourier matrix s of rank 5. proof. let (a,b,δ) be a non-homogenous c-algebra arising from a rational fourier matrix s of rank 5. let δ(bi) = ki, for all i. since s-matrix is integral, ki are perfect square integers, see [11, proposition 5]. by [3, lemma 3.7], d0 is a square. therefore, d0 ≡ 0, 1 mod 4. let d0 = k4a,d0 = k3b,d0 = k2c,d0 = k1d. then each of a,b,c and d is greater than 1 and a square integer because a is non-homogeneous. case 1. if each of k1,k2,k3,k4 is an odd integer, then d0 is odd. also, the fact that k1,k2,k3,k4 are odd implies a,b,c,d are odd and greater than or equal to 9. without loss of generality, let k4 ≥ k1,k2,k3. therefore, d0 ≥ 9k4,d0 = 1 + k1 + k2 + k3 + k4 ≤ 1 + 4k4, a contradiction. 62 http://orcid.org/0000-0002-0819-8221 g. singh / j. algebra comb. discrete appl. 5(2) (2018) 51–63 case 2. if three of k1,k2,k3,k4 are odd and one is even, then d0 is even. without loss of generality, suppose k4 is even. thus d0 = k4a implies a ≥ 4. subcase 1. if k4 > k1,k2,k3, then d0 ≥ 4k4, d0 ≤ 1 + (k4 −1) + (k4 −1) + (k4 −1) + k4 = 4k4 −2, a contradiction. subcase 2. if k4 < one of k1,k2,k3, say k3, so k1,k2 ≤ k3, then d0 ≥ 4k3 because b is an even square. thus d0 ≤ 1 + k3 + k3 + k3 + (k3 − 1) = 4k3 implies k1 = k2 = k3 and k4 = k3 − 1, d0 = 4k3. now d0 = 4x 2 because d0 is a square and an even integer. hence k1 = k2 = k3 = x2 and k4 = x2 − 1. since x2 − 1 divides 4x2 and x is an odd integer, x2 − 1 divides 4, we get a contradiction. case 3. if two of k1,k2,k3,k4 are odd and two are even, then d0 ≡ 3 mod 4, a contradiction. case 4. if one of k1,k2,k3,k4 is odd and three are even, then d0 ≡ 2 mod 4, a contradiction. acknowledgment: the author would like to thank professor allen herman whose valuable suggestions helped him to improve this paper. references [1] z. arad, e. fisman, m. muzychuk, generalized table algebras, israel j. math. 114(1) (1999) 29–60. [2] h. i. blau, table algebras, european j. combin. 30(6) (2009) 1426–1455. [3] m. cuntz, integral modular data and congruences, j. algebraic combin. 29(3) (2009) 357–387. [4] p. francesco, p. mathieu, d. sénéchal, conformal field theory, springer–verlag, new york, 1997. [5] t. gannon, modular data: the algebraic combinatorics of conformal field theory, j. algebraic combin. 22(2) (2005) 211–250. [6] a. hanaki, i. miyamoto, classification of association schemes with small vertices, 2017, available at: math.shinshu-u.ac.jp/ hanaki/as/. [7] d. g. higman, coherent algebras, linear algebra appl. 93 (1987) 209–239. [8] j. d. qualls, lectures on conformal field theory, arxiv:1511.04074 [hep-th]. [9] e. l. rees, graphical discussion of the roots of a quartic equation, amer. math. monthly 29(2) (1922) 51–55. [10] m. schottenloher, a mathematical introduction to conformal field theory, springer–verlag, berlin, heidelberg, 2nd edition, 2008. [11] g. singh, classification of homogeneous fourier matrices, arxiv:1610.05353 [math.ra]. [12] b. xu, characters of table algebras and applications to association schemes, j. combin. theory ser. a 115(8) (2008) 1358–1373. [13] a. zahabi, applications of conformal field theory and string theory in statistical systems, ph.d. dissertation, university of helsinki, helsinki, finland, 2013. 63 http://orcid.org/0000-0002-0819-8221 https://doi.org/10.1007/bf02785571 https://doi.org/10.1016/j.ejc.2008.11.008 https://doi.org/10.1007/s10801-008-0139-y http://www.springer.com/us/book/9780387947853#otherversion=9781461274759 https://doi.org/10.1007/s10801-005-2514-2 https://doi.org/10.1007/s10801-005-2514-2 http://math.shinshu-u.ac.jp/~hanaki/as/ http://math.shinshu-u.ac.jp/~hanaki/as/ https://doi.org/10.1016/s0024-3795(87)90326-0 https://arxiv.org/abs/1511.04074 https://doi.org/10.2307/2972804 https://doi.org/10.2307/2972804 http://www.springer.com/gp/book/9783540686255 http://www.springer.com/gp/book/9783540686255 https://arxiv.org/abs/1610.05353 https://doi.org/10.1016/j.jcta.2008.02.005 https://doi.org/10.1016/j.jcta.2008.02.005 https://helda.helsinki.fi/bitstream/handle/10138/40728/zahabi_dissertation.pdf?...1 https://helda.helsinki.fi/bitstream/handle/10138/40728/zahabi_dissertation.pdf?...1 introduction c-algebras arising from fourier matrices recognition of c-algebras arising from fourier matrices fourier matrices of rank 2 and 3 fourier matrices of rank 4 fourier matrices of rank 5 references jacodesmath / issn 2148-838x j. algebra comb. discrete appl. 2(1) • 17-24 received: 21 august 2014; accepted: 21 november 2014 doi 10.13069/jacodesmath.15518 journal of algebra combinatorics discrete structures and applications maximal induced paths and minimal percolating sets in hypercubes research article anil m. shende∗ roanoke college abstract: for a graph g, the r-bootstrap percolation process can be described as follows: start with an initial set a of "infected” vertices. infect any vertex with at least r infected neighbours, and continue this process until no new vertices can be infected. a is said to percolate in g if eventually all the vertices of g are infected. a is a minimal percolating set in g if a percolates in g and no proper subset of a percolates in g. an induced path, p , in a hypercube qn is maximal if no induced path in qn properly contains p. induced paths in hypercubes are also called snakes. we study the relationship between maximal snakes and minimal percolating sets (under 2-bootstrap percolation) in hypercubes. in particular, we show that every maximal snake contains a minimal percolating set, and that every minimal percolating set is contained in a maximal snake. 2010 msc: 05c38, 68r10 keywords: hypercubes, induced paths, percolating sets 1. introduction the problem of finding longest induced paths (often called snakes in the literature) in hypercubes has been studied since 1958 [5]. definite values for the lengths of longest snakes in n-dimensional hypercubes are known only for dimensions n ≤ 7 [1]. several properties of maximal snakes have been found useful in establishing better bounds on the lengths of longest snakes in hypercubes [1, 2]. the notion of r-bootstrap percolation in graphs has been studied since 1979 [3]. riedl considers the 2-bootstrap percolation process in hypercubes; in particular, he studies minimal percolating sets (under 2-bootstrap percolation) in hypercubes and provides an expression for the size of largest minimal percolating sets in hypercubes [4]. in this paper we show the relationship between maximal snakes and minimal percolating sets (under 2-bootstrap percolation) in hypercubes. in particular we show that every maximal snake contains a minimal percolating set, and that every minimal percolating set is contained in a maximal snake. ∗ e-mail: shende@roanoke.edu 17 minimal percolating sets in hypercubes 2. notation most of the definitions and notation in this section are directly from [2] and [4]. we will label the vertices of an n-dimensional hypercube, qn = (vn, e) by the 2n distinct n-bit labels such that the labels on two vertices differ in exactly one bit position if and only if the two vertices are neighbours in qn. the rightmost bit in a label will be called bit number 0 and the leftmost bit will be called bit number (n−1). [n] denotes the set {0, . . . , n−1}. for any vertex v ∈ vn, for each i ∈ [n], (v)i will denote the bit in bit number i of the label for v. for each i ∈ [n], lni denotes the label that has all zeroes except a 1 as bit number i. 0n denotes the label with all zeroes. un will denote the set {lni | i ∈ [n]}, i.e., the set of all vertices at unit distance from 0 n. for a bit b in a label, b will denote the bit complement of b. we will use an asterisk to denote a wildcard in labels. a sequence of k asterisks in a label will be denoted by (∗)k. thus an n-bit label with one asterisk denotes two vertices; moreover, these two vertices are adjacent, i.e., they are a 1-subcube of qn. in general, for each k, 0 ≤ k ≤ n, a label with k asterisks denotes a k-dimensional subcube. for example, the label (∗)201 is a 2-dimensional subcube of q4 such that all the vertices in the subcube have 1 for bit number 0 and 0 for bit number 1. for a subcube s of qn, dim(s) will denote the dimension of s. when vertices u and v differ in bit position d, we will denote this as u d←→ v. for any two vertices x, y ∈ qn, ham(u, v) denotes the hamming distance between x and y. for a path p = v0, v1, . . . , vl, p̂ will denote the path v1, . . . , vl−1, i.e., the path p without its endpoints. for a path p , we will use p to denote the sequence of vertices as well as the set of vertices. an n-snake is an induced path in the n-dimensional, hypercubical, undirected graph. a maximal n-snake, p , is an n-snake such that no other n-snake properly contains p . the r-bootstrap percolation process in g = (v, e) is described as follows: let a ⊆ v be a set of “infected” vertices. let a0 = a. then, let at be the set of vertices in at−1 union the set of vertices which have at least r neighbours in at−1. the set 〈a〉 = ∪iai is the set of vertices infected by a. a set a is said to percolate in g if 〈a〉 = v . a percolating set a is said to be minimal if for all v ∈ a, a\v does not percolate in g. in the case r = 2 and g = qn = (vn, e), the progress of the percolation process can be described as follows: given a set a ⊆ vn, let a0 = ⋃ u∈a{ {u} }, i.e., the set of all the 0-subcubes represented by the vertices in a. then, choose a sequence of sets of subcubes a1, a2, . . . , ak so that at is identical to at−1 except that two subcubes b, c ∈ at−1 that are within distance 2, in qn, of each other are replaced by the subcube 〈b ∪ c〉, and so that ak is a set of subcubes all of which are distance at least 3, in qn, from each other. clearly, then, a percolates in qn if ak = {vn}. a0, . . . , ak is called an execution path of the percolation process. in the rest of the paper, unless specified otherwise, “distance” will refer to “distance in qn”, and by “percolating” we will mean 2-percolating. 3. maximal snakes percolate theorem 3.1. every maximal n-snake is a percolating set. proof. the set of maximal n-snakes that have 0n as one of the end points, and ln0 as the next vertex is equal to the set of all maximal n-snakes up to isomorphism [2]. thus, without loss of generality, let p be a maximal n-snake v0, v1, . . . , vk where v0 = 0n and v1 = ln0 . since the snake is maximal, none of the vertices lni , 1 ≤ i < n are on the snake (since 0 n is an end point of the snake and ln0 is on the snake), and each of these vertices has at least two neighbours on the snake. (if not, then the snake could have been extended from 0n and thus would not be maximal.) thus, if all the vertices of the snake are infected, then in one percolation step, each of the vertices, lni , 1 ≤ i < n, gets infected. thus, each vertex in the set un = {lni | 0 ≤ i < n} is infected. by proposition 7 in [4], u n is a minimal percolating set in qn for n ≥ 2. thus, every maximal n-snake is a percolating set. 18 a. m. shende 4. minimal percolating sets and snakes lemma 4.1. suppose p = v0, v1, . . . , vl is an n-snake. then, g′ = (qn\p̂) is connected. proof. let d1, d2, . . . , dl ∈ [n] such that for each i, 1 ≤ i ≤ l, vi−1 di←→ vi. let np (vi) be defined as: np (vi) = {u | u 6∈ p and ham(u, vi) = 1 and (∀j ∈ [i])ham(u, vj) > 1}. it suffices to show that for each vertex u at distance 1 from p , there is a path in g′ from v0 to u, i.e., for each i ∈ [l], for each vertex u in np (vi), there is a path in g′ from v0 to u. (to show that there is a path in g′ from vl to u, we can reverse the labeling of vertices on p and apply the argument below.) we will use mathematical induction on i. base case : i = 0. every vertex in np (v0) is a neighbour of v0, and thus the claim is trivially true. induction hypothesis : for some k ∈ [n], for all k′, 0 ≤ k′ ≤ k, for each vertex u in np (vk′), there is a path in g′ from v0 to u. to show that : for each vertex u in np (vk+1), there is a path in g′ from v0 to u. let u′ be the vertex such that u dk+1←→ u′. clearly, then, u′ is a neighbour of vk, and u′ 6∈ p by the definition of np . (if u′ ∈ p , then, since p is an induced path, u′ = vk−1, a neighbour of u, contradicting u ∈ np (vk+1).) thus, for some k′ ≤ k, u′ ∈ np (v′k). then, by the induction hypothesis there is a path in g′ from v0 to u′ and thus, there is a path in g′ from v0 to u. as a corollary of the above result we have corollary 1. suppose p = v0, v1, . . . , vl is an n-snake. then for each vertex w ∈ vn\p , there exist n-snakes, p1 and p2, in qn\p̂ such that p1 has end points v0 and w, and p2 has end points vl and w. lemma 4.2. every set a ⊆ vn that is isomorphic to un is contained in an n-snake, and both the end points of the snake are in a. proof. for each i 6= j ∈ [n], let lnij denote the n-bit label that has ones in bit positions i and j and zeroes everywhere else. then, consider the path defined by the sequence of vertices ln0 , l n 01, l n 1 , l n 12, . . . , l n (n−2), l n (n−2)(n−1), l n (n−1) it can easily be verified that this path is an n-snake. moreover, un is contained in this path and both the end points of the path are in un. lemma 4.3. let a ⊆ vn be a minimal percolating set in qn such that a = b ∪ c where 〈b〉 = qn−1 and c = {u} such that u 6∈ 〈b〉. suppose p is a snake in 〈b〉 such that p contains each vertex in b, and both the end points of p are in b. then, there is a snake p ′ in qn that contains each vertex in a, and both the end points of p ′ are in a. proof. suppose p = v0, v1, . . . , vk. we consider the two cases: u is at distance 1 in qn from p , and u is at distance greater than 1 in qn from p . case 1 suppose u is at distance 1 in qn from p . since u 6∈ 〈b〉, u is a neighbour of at most one vertex in p . 19 minimal percolating sets in hypercubes if u is a neighbour of one of the two end points, say v0, of p, then p ′ is simply p extended by the edge (v0, u). clearly, p ′ is a snake in qn that contains each vertex of a, and both the end points of p ′ are in a. suppose u is a neighbour of an internal vertex vi in p, and vi d←→ u. there are two cases to consider: vi ∈ b and vi 6∈ b. vi ∈ b then, by the minimality of a, vi−1 6∈ a and vi+1 6∈ a. since vi−1 6∈ a, it is not an end point of p . then, consider the vertices u1 and u2 such that vi−1 d←→ u1 and vi−2 d←→ u2. (note that vi−2 exists since vi−1 is not an end point of p .) u1 and u2 are not in 〈b〉, u1 is a neighbour of u and u2 is a neighbour of u1. let p ′ be the path p with the edges (vi−2, vi−1) and (vi−1, vi) replaced by the path vi−2, u2, u1, u, vi. then, p ′ is a snake in qn that contains each vertex of a, and both the end points of p ′ are in a. vi 6∈ b consider the vertices u1 and u2 such that vi−1 d←→ u1 and vi+1 d←→ u2. u1 and u2 are not in 〈b〉, u1 is a neighbour of u and u2 is a neighbour of u, and u1 and u2 are not neighbours (since they are neighbours, along the same direction d, of two vertices at distance 2 in p , and p is a snake). let p ′ be the path p with the path vi−1, vi, vi+1 replaced by the path vi−1, u1, u, u2, vi+1. then, p ′ is a snake in qn that contains each vertex of a, and both the end points of p ′ are in a. case 2 suppose u is at distance greater than 1 in qn from p . since 〈b〉 = qn−1, there is one bit position that has the same value for the labels on all the vertices in 〈b〉. without loss of generality, let bit position 0 for the labels on each vertex in 〈b〉 be 0. let u′ be the vertex such that u 0←→ u′. since u 6∈ 〈b〉, u′ ∈ 〈b〉, and u′ 6∈ p . then, by corollary 1 there is a snake (vl =)w′0, w′1, . . . w′m(= u′) in 〈b〉\p̂ . for each i, 0 ≤ i ≤ m, let wi be the vertex such that wi 0←→ w′i. then, p ′ = v0, . . . , vl, w0, w1, . . . , wm(= u) is a snake in qn that contains each vertex of a, and both the end points of p ′ are in a. lemma 4.4. let a ⊆ vn be a minimal percolating set in qn such that a = b ∪ c where 〈b〉 = qn−2 and c = {u} such that u 6∈ 〈b〉, and u is at distance 2 in qn from 〈b〉. suppose p is a snake in 〈b〉 such that p contains each vertex in b, and both the end points of p are in b. then, there is a snake p ′ in qn that contains each vertex in a, and both the end points of p ′ are in a. proof. without loss of generality, let 〈b〉 = ∗∗ ·· · ∗ ∗00. then, u is in the subcube ∗∗ ·· · ∗ ∗11. let v0 and vl be the end points of p . consider the vertices u1 and u2 such that vl 0←→ u1 and u1 1←→ u2. clearly, u1 is in the subcube ∗∗ · · · ∗ ∗01, and u2 is in the subcube ∗∗ · · · ∗ ∗11. moreover, since p is a snake, u1 is at distance at least 2 from all the vertices in p , except vl, and u2 is at distance at least 2 in qn from each vertex in p . let pu be a snake in the subcube ∗∗ · · · ∗ ∗11 from u2 to u. clearly, each vertex of pu is at distance at least 2 from p . now let p ′ be the path p extended by the edge (vl, u1), followed by the edge (u1, u2) and then followed by pu. it can be easily verified that p ′ is a snake in qn, and both the end points of p ′ are in a. in what follows, we will use some additional notation: for a set c ⊆ [n], qn|c denotes the subcube bn−1 . . . b0 of qn where for each i 6∈ c, bi is an asterisk, and for each i ∈ c, bi ∈{0, 1}. lemma 4.5. for n ≥ 6, suppose c, d, e ⊆ [n] such that |c| = 2, |d| = s ≥ 2, |e| = t ≥ 2, and c, d and e are mutually exclusive. let f = [n]\(c ∪ d ∪ e). let p = v0, . . . , vl be a snake in the (n − (s + 2))-dimensional subcube qn|c∪d, characterised by the bits ci and di for i ∈ c and i ∈ d, respectively. then, for any vertex u in the subcube qn|c∪d∪f , characterised by the bits ci, di, and (vl)i, for i ∈ c, i ∈ d and i ∈ f , respectively, p can be extended to an (n − 2)-dimensional snake in the subcube qn|c, characterised by the bits ci for i ∈ c, with end points v0 and u. 20 a. m. shende (an example may help clarify the statement of the lemma. let n = 8, c = {0, 1}, d = {2, 3} and e = {4, 5, 6}. thus, s = 2, t = 3 and f = {7}. let qn|c∪d be the subcube ∗∗∗∗ 1001. suppose p is a snake in this subcube with end points v0 = 00001001 and vl = 10111001. then, for any vertex u in the subcube 1∗∗∗0101, say the vertex 11010101, p can be extended to an (n−2)-snake in the subcube ∗∗∗∗∗∗01 such that the end points of this snake are v0 = 00001001 and u = 11010101.) proof. without loss of generality, let c = {0, 1}, d = {2, . . . , s + 1} and e = {s + 2, . . . , s + t + 1}. then, f = {s + t + 2, . . . , n−1}. let p = v0, . . . , vl be a snake in the subcube given by (∗)n−(s+t+2) (∗)t bs+1 · · ·b2 b1b0, and let u be a vertex in the subcube given by (vl)n−1 · · ·(vl)(s+t+2) (∗)t bs+1 · · ·b2 b1b0. now consider the following vl = (vl)n−1 · · ·(vl)(s+t+2) (vl)(s+t+1) · · ·(vl)(s+2) bs+1 · · ·b2 b1b0y h1 w = (vl)n−1 · · ·(vl)(s+t+2) (vl)(s+t+1) · · ·(vl)(s+2) bs+1 · · ·b2 b1b0y h2 u = (vl)n−1 · · ·(vl)(s+t+2) (u)(s+t+1) · · ·(u)(s+2) bs+1 · · ·b2 b1b0 where h1 is a shortest path between vl and w, and h2 is a shortest path between w and u. since p is a snake in the subcube s, each vertex vi, i < l differs from vl in at least one bit position ai where ai 6∈ {0, . . . , (s + 1)}. each vertex of h1, except the vertex vl, differs from vl in at least one bit position j ∈ {0, . . . , (s + 1)}. since s ≥ 2, the length of h1 is at least 2, and so each vertex of h2, differs from each vertex of p in at least two bit positions. moreover, each vertex h of h1, except w, differs from w in at least one bit position jh 6∈ {(s + 2), . . . , (s + t + 1)}, and each vertex g of h2 differs from w in at least one bit position jg ∈ {(s + 2), . . . , (s + t + 1)}. thus, p concatenated with h1 concatenated with h2 is a snake that has end points v0 and u, and is contained in the (n−2)-dimensional subcube given by (∗)(n−(s+t+2)) (∗)t (∗)s b1b0. the following lemma follows from the proof of proposition 9 in [4]. our proof is essentially the proof from [4]; for completeness we provide the relevant parts of the proof here. lemma 4.6. let a ⊆ vn be a minimal percolating set in qn. then, there exists an execution path⋃ u∈a{{u}} = a0, . . . , ak = {vn}, where ak−1 consists of exactly two subcubes s1 and s2 such that dim(s1) ≥ dim(s2) and exactly one of the following is true: c1 : dim(s1) = n−1, s2 = {v}, and v 6∈ s1. c2 : dim(s1) = n−2 and s2 = {v} at distance 2 from s1. c3 : dim(s1) ≤ n−4 and dim(s2) ≤ n−4, and s1 and s2 are at distance 2 from each other. proof. since a percolates in qn, for any execution path of the percolation process ak = vn, and so ak−1 must consist of exactly two subcubes, say s1 and s2, which together infect qn. amongst all execution paths with dim(s1) ≥ dim(s2), choose one where dim(s1) is the largest. by minimality of a, dim(s1) ≤ n−1. we consider the cases depending on dim(s1). 21 minimal percolating sets in hypercubes case 1 dim(s1) = n−1. then, by the minimality of a, there must be a single vertex v in a∩(qn\s1) such that a percolates in qn. thus, s2 = {v} in this case. case 2 dim(s1) = n − 2. by the choice of the execution path that has dim(s1) the largest possible, there cannot be a vertex v in a∩s2 such that the distance of v from s1 is at most 1, since otherwise s1 could be extended to a subcube of dimension n − 1. thus, by the minimality of a, there must be a single vertex v in a∩(qn\s1) such that the distance of v from s1 is 2 and a percolates in qn. case 3 dim(s1) = n−3. there cannot be a vertex of a∩s2 within distance 2 of s1, as this contradicts the maximality of dim(s1) in our choice of execution path. hence, a∩s2 is contained in a subcube of qn at distance 3 from s1, as the set of vertices which are at distance 3 from s1 is a subcube of dimension (n − 3). thus s1 and s2 are at distance (in qn) 3 from each other, contradicting the fact that a percolates. hence, this case cannot occur. case 4 dim(s1) ≤ n−4. then, by the maximality of dim(s1) in the choice of execution path, dim(s2) ≤ n−4. moreover, since a percolates in qn, s1 and s2 must be at distance 2 from each other. theorem 4.7. every minimal percolating set a ⊆ vn is contained in an n-snake. proof. we will use mathematical induction on the dimension n to show that every minimal percolating set a ⊆ vn is contained in an n-snake, and both the end points of the snake are in a. base cases n ≤ 3 : we will consider each of the three cases, n = 1, n = 2 and n = 3, separately. for n = 1, it is easy to see that the only minimal percolating set is v1, and that is contained in the unique 1-snake, and both the end points of the snake are in v1. for n = 2, the only (up to isomorphism) minimal percolating set in q2 is u2, and by lemma 4.2, this minimal percolating set is contained in a 2-snake, and both the end points of the snake are in u2. for n = 3, there are two (up to isomorphism) minimal percolating sets: u3 and the set a = {000, 001, 111}. lemma 4.2 asserts the existence of a 3-snake containing u3 and the 3-snake 000, 001, 011, 111 contains a. moreover, in each case, both the end points of the snake are in the minimal percolating set. induction hypothesis : for some m ≥ 3, for all m′ ≤ m, every minimal percolating set a ⊆ vm′ in qm′ is contained in an m′-snake, and both the end points of the snake are in a. to show that : every minimal percolating set a ⊆ vm+1 in qm+1 is contained in an (m + 1)-snake, and both the end points of the snake are in a. by lemma 4.6, there exists an execution path ⋃ u∈a{u} = a0, . . . , ak = vm+1, where ak−1 consists of exactly two subcubes s1 and s2 such that dim(s1) ≥ dim(s2), satisfying one of the three mutually exclusive cases listed in the lemma. we show that in each case, a is contained in an (m + 1)-snake. (c1, c2 and c3 refer to the three cases listed in the statement of lemma 4.6.) c1 in this case, by the induction hypothesis there is an m-snake in s1 that contains each vertex in s1 ∩a, and both the end points of the snake are in s1 ∩a. lemma 4.3 applies and asserts that a is contained in an (m + 1)-snake, and both the end points of this snake are in a. c2 in this case, by the induction hypothesis there is an (m − 1)-snake in s1 that contains each vertex in s1 ∩a, and both the end points of the snake are in s1 ∩a. lemma 4.4 applies and asserts that a is contained in an (m + 1)-snake, and both the end points of this snake are in a. 22 a. m. shende c3 in this case, there exist integers s and t such that dim(s1) = ((m + 1)−s) ≤ ((m + 1)−4) and dim(s2) = ((m + 1)− t) ≤ ((m + 1)−4). then, s ≥ 4 and t ≥ 4. let d′, e′ ⊆ [(m+1)] such that s1 is the subcube qn|d′, characterised by pi for each i ∈ d′, and s2 is the subcube qn|e′, characterised by ri for each i ∈ e′. since s1 and s2 are at distance 2 from each other, |d′ ∩ e′| = 2 and pi 6= ri for i ∈ (d′ ∩ e′). let c = d′ ∩ e′, d = d′\c, e = e′\c and f = [(m+1)]\(c ∪ d ∪ e). clearly, then |c| = 2, |d| = s ≥ 2, |e| = t ≥ 2, and |f| = (m + 1) − (s + t + 2). without loss of generality, let c = {0, 1}, d = {2, . . . , s + 1}, and e = {s + 2, . . . , s + t + 1}, and let p0 = p1 = 0 and r0 = r1 = 1, i.e., s1 = (∗)(m+1)−(s+t+2) (∗)t ps+1 · · ·p2 00, and s2 = (∗)(m+1)−(s+t+2) rs+t+1 · · ·rs+2 (∗)s 11. by the induction hypothesis, there is an (m + 1)− (s + 2)-snake, p1 in s1 that contains each vertex in s1 ∩a, and both the end points, say v0 and va, of p1 are in s1 ∩a, and there is an (m + 1)−(t + 2)-snake, p2 in s2 that contains each vertex in s2 ∩a, and both the end points, say w0 and wb, of p2 are in s2 ∩a. by lemma 4.5 p1 can be extended to a snake p∗1 with end points v0 and the vertex u1 given by u1 = (va)m . . . (va)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 00. similarly, by lemma 4.5 p2 can be extended to a snake p∗2 with end points w0 and the vertex u2 given by u2 = (wb)m . . . (wb)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 11. now consider u1 = (va)m . . . (va)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 00 ↓ ux = (va)m . . . (va)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 01y p3 uy = (wb)m . . . (wb)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 01 ↓ u2 = (wb)m . . . (wb)(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 11 where p3 is a shortest path in the subcube (∗)(m+1)−(s+t+2) rs+t+1 · · ·rs+2 ps+1 · · ·p2 01. each vertex of this subcube differs from p1 in at least s ≥ 2 bit positions, and differs from p2 in at least t ≥ 2 bit positions. by the reasoning of lemma 4.5, p3 differs from p∗1 and p∗2 in at least two bit positions. thus, the concatenation of p∗1 with the edge (u1, ux) followed by p3 is a snake. similarly, the concatenation of p∗2 with the edge (u2, uy) is a snake. moreover, these two snakes concatenated together is a snake with end points v0 and w0, both of which are in a. 23 minimal percolating sets in hypercubes 5. conclusions and open questions as noted earlier, our interest in maximal snakes derives from properties of maximal snakes in lower dimensions that are useful heuristics in generating long maximal snakes in higher dimensions. the heuristics essentially seed the exhaustive search with an initial segment of the snake. our hope is that minimal percolating sets can prove to be better seeds and will speed up the exhaustive search somewhat. our intuition comes from the observation that 1) minimal pecolating set vertices are sprinkled throughout the hypercube, and 2) at each step of the proofs above where we use a shortest path or any snake between two vertices, we could use a suitable longest path instead. we would also like to explore the possibility of stronger results that may characterize longest maximal snakes in terms of minimal percolating sets. for example, we would like to consider questions such as 1. is there a difference in the set of minimal percolating sets contained in a maximal, but not longest, snake, and the set of minimal percolating sets contained in a longest snake? 2. in some dimensions, notably 4 and 6 of the known ones, there is a unique (up to isomorphism) longest snake. is there a stronger relationship between the longest snake and minimal percolating sets in these dimensions? references [1] d. kinny. a new approach to the snake-in-the-box problem., in proceedings of the 20th european conference on artificial intelligence, ecai-2012, 462-467, 2012. [2] dayanand s. rajan and anil m. shende. maximal and reversible snakes in hypercubes, in 24th annual australasian conference on combinatorial mathematics and combinatorial computing, 1999. [3] eric riedl. largest and smallest minimal percolating sets in trees, the electronic journal of combinatorics, 19, 2010. [4] eric riedl. largest minimal percolating sets in hypercubes under 2-bootstrap percolation, the electronic journal of combinatorics, 17, 2010. [5] w. h. kautz, unit distance error checking codes, in ire transaction on electronic computers, 7, 179–180, 1958. 24 introduction notation maximal snakes percolate minimal percolating sets and snakes conclusions and open questions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.369863 j. algebra comb. discrete appl. 5(1) • 29–43 received: 17 december 2016 accepted: 6 october 2017 journal of algebra combinatorics discrete structures and applications a module minimization approach to gabidulin decoding via interpolation research article anna-lena horlemann-trautmann,∗ margreta kuijper abstract: we focus on iterative interpolation-based decoding of gabidulin codes and present an algorithm that computes a minimal basis for an interpolation module. we extend existing results for reed-solomon codes in showing that this minimal basis gives rise to a parametrization of elements in the module that lead to all gabidulin decoding solutions that are at a fixed distance from the received word. our module-theoretic approach strengthens the link between gabidulin decoding and reed-solomon decoding, thus providing a basis for further work into gabidulin list decoding. 2010 msc: 11t71, 94b35 keywords: gabidulin codes, linearized polynomials, interpolation, minimal basis, parametrization, polynomial modules, rank metric, iterative algorithm. 1. introduction over the last decade there has been increased interest in gabidulin codes, mainly because of their relevance to network coding [12, 23] and distributed storage [20]. gabidulin codes are optimal rank-metric codes over a field fqm (where q is a prime power). they are named after the work of gabidulin in [9] and have independently been presented earlier by delsarte in [6]. these codes can be seen as the q-analog of reed-solomon codes, using q-linearized polynomials instead of arbitrary polynomials. they are optimal in the sense that they are not only mds codes with respect to the hamming metric, but also achieve the singleton bound with respect to the rank metric and are thus mrd (maximum rank distance) codes. the decoding of gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding within the unique decoding radius in [9, 10] and more recently [16, 19, 21, 25]. if ∗ this author was partially supported by swiss national science foundation fellowship no. 147304. anna-lena horlemann-trautmann; faculty of mathematics and statistics, university of st. gallen, st. gallen, switzerland (email: anna-lena.horlemann@unisg.ch). margreta kuijper (corresponding author); department of electrical and electronic engineering, university of melbourne, vic 3010, australia (email: mkuijper@unimelb.edu.au). 29 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 n is the length of the gabidulin code and k denotes the dimension of the code as a linear space over the field fqm, the unique decoding radius is given by b(n − k)/2c. a main open question is whether there exist parameter sets for which gabidulin codes can be (list) decoded beyond the unique decoding radius efficiently. this paper seeks to contribute to current research efforts on this open question. using the close resemblance between reed-solomon codes and gabidulin codes, the paper [16] translates gabidulin decoding into a set of polynomial interpolation conditions. essentially, this setup is also used in the papers [12, 27] that present iterative algorithms that perform gabidulin list decoding with a list size of 1. in this paper we present an iterative algorithm that bears similarity to the ones in [12, 16, 27]. as new results we show that the algorithm computes a minimal basis for an interpolation module that we associate with the received word. this result enables a parametrization of elements in the module that lead to all gabidulin decoding solutions that are at a fixed distance from the received word. thus we present a module minimization interpretation of the pioneering work by loidreau [16]. the paper is structured as follows. in the next section we present several preliminaries on q-linearized polynomials and gabidulin codes, including the polynomial interpolation conditions from [16]. subsection 2.3 deals with modules over the ring of linearized polynomials and draws attention to minimal bases of these modules and their predictable leading monomial property. in section 3 we reformulate the gabidulin decoding requirements in terms of a module represented by four q-linearized polynomials and present our polynomial-time algorithm. we conclude this paper in section 4. 2. preliminaries 2.1. q-linearized polynomials let q be a prime power and let m be a positive integer. denote the finite field with q elements by fq and denote a primitive element of the extension field fqm by α. since fqm is isomorphic (as a vector space) to the vector space fmq , matrices over the base field fq can be interpreted as vectors over the extension field, i.e., we have the isomorphism fm×nq ∼= fnqm. in the sequel we denote the rank of a matrix x over fq by rankq(x). for a vector (v1, . . . ,vn) ∈ fnqm we denote the k ×n moore matrix by mk(v1, . . . ,vn) :=   v1 v2 . . . vn v [1] 1 v [1] 2 . . . v [1] n ... v [k−1] 1 v [k−1] 2 . . . v [k−1] n   , (1) where [i] := qi. a q-linearized polynomial over fqm is defined to be of the form f(x) = n∑ i=0 aix [i] , ai ∈ fqm, where, assuming that an 6= 0, n is called the q-degree of f(x), denoted by qdeg(f). this class of polynomials was studied in detail by ore in [17]. one can easily check that f(x1 +x2) = f(x1)+f(x2) and f(λx1) = λf(x1) for any x1,x2 ∈ fqm and λ ∈ fq, hence the name linearized. the set of all q-linearized polynomials over fqm is denoted by lq(x,qm). this set is a non-commutative ring with the normal addition + and with composition ◦ of polynomials. because of the non-commutativity, products and quotients of elements of lq(x,qm) have to be specified as being “left" or “right". to not be mistaken with the standard division, we call the inverse of the composition symbolic division. thus f(x) is symbolically divisible by g(x) with right quotient m(x) if g(x) ◦m(x) = g(m(x)) = f(x). 30 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 efficient algorithms for all these operations (left and right symbolic multiplication and division) can be found e.g. in [12]. lemma 2.1 (cf. [15] theorem 3.50). let f(x) ∈lq(x,qm) and let fqs be the smallest extension field of fqm that contains all roots of f(x). then the set of all roots of f(x) is a fq-linear vector space in fqs. definition 2.2. let u be a fq-linear subspace of fqm. we call πu (x) := ∏ g∈u (x−g) the q-annihilator polynomial of u. lemma 2.3 ([15] theorem 3.52). let u be a fq-linear subspace of fqm. then πu (x) is an element of lq(x,qm). note that, if g1, . . . ,gn is a basis of u, one can rewrite πu (x) = λ det(mn+1(g1, . . . ,gn,x)) for some constant λ ∈ fqm; clearly its q-degree equals n. the notion of q-lagrange polynomial is as follows: definition 2.4. let g = (g1, . . . ,gn) ∈ fnqm, where g1,g2, . . . ,gn are fq-linearly independent. let r = (r1, . . . ,rn) ∈ fnqm. define the matrix di(g,x) as mn(g1, . . . ,gn,x) without the i-th column. we define the q-lagrange polynomial corresponding to g and r as λg,r(x) := n∑ i=1 (−1)n−iri det(di(g,x)) det(mn(g)) ∈ fqm [x]. it can be easily verified that the above polynomial is q-linearized and that λg,r(gi) = ri for i = 1, . . . ,n. throughout the paper we use matrix composition, which is defined analogously to matrix multiplication: [ a(x) b(x) c(x) d(x) ] ◦ [ e(x) f(x) g(x) h(x) ] := [ a(e(x)) + b(g(x)) a(f(x)) + b(h(x)) c(e(x)) + d(g(x)) c(f(x)) + d(h(x)) ] . let g1, . . . ,gn ∈ fqm be linearly independent over fq; as before denote g := (g1, . . . ,gn). throughout the remainder of the paper we use the standard notation 〈g1, . . . ,gn〉 for the fq-linear span of g1,g2, . . .gn. furthermore we abbreviate the notation π〈g1,g2,...,gn〉(x) by πg(x). we need the following fact for our investigations in section 3. lemma 2.5. let g1, . . . ,gn ∈ fqm be linearly independent over fq and let l(x) ∈lq(x,qm) be such that l(gi) = 0 for all i. then ∃h(x) ∈lq(x,qm) : l(x) = h(x) ◦ πg(x). proof. we know from lemma 2.3 that πg(x) ∈ lq(x,qm). moreover unique left and right division in lq(x,qm) holds, i.e. in this case there exist unique polynomials h(x),r(x) ∈ lq(x,qm) such that l(x) = h(x) ◦ πg(x) + r(x) and qdeg(r(x)) < qdeg(πg(x)) = n. since any α ∈ 〈g1, . . . ,gn〉 is a root of l(x) as well as πg(x), they must also be a root of r(x). hence we have qn distinct roots for r(x) and deg(r) < qn, thus r(x) ≡ 0 and the statement follows. 2.2. gabidulin codes let g1, . . . ,gn ∈ fqm be linearly independent over fq. a gabidulin code c ⊆ fnqm of dimension k is defined as the linear block code with generator matrix mk(g1, . . . ,gn), as defined in (1). using the 31 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 isomorphic matrix representation, we can interpret c as a matrix code in fm×nq . the rank distance dr on fm×nq is defined by dr(x,y ) := rankq(x −y ) , x,y ∈ fm×nq and analogously for the isomorphic extension field representation. the code c has dimension k over fqm and minimum rank distance (over fq) n−k + 1. in fact, an equivalent definition of the code is c = {(m(g1), . . . ,m(gn)) ∈ fnqm | m(x) ∈lq(x,q m)<k}, where lq(x,qm)<k := {m(x) ∈ lq(x,qm) | qdeg(m(x)) < k}. for more information on bounds and constructions of rank-metric codes the interested reader is referred to [9]. consider a received word r = (r1, . . . ,rn) ∈ fnqm as the sum r = c + e, where c = (c1, . . . ,cn) ∈ c is a codeword and e = (e1, . . . ,en) ∈ fnqm is the error vector. we now recall the polynomial interpolation setup from [16] via a more general formulation in the next theorem. theorem 2.6. let m(x) ∈lq(x,qm), qdeg(f(x)) < k and ci = m(gi) for i = 1, . . . ,n. then dr(c,r) = t if and only if there exists a d(x) ∈lq(x,qm), such that qdeg(d(x)) = t and d(ri) = d(m(gi)) ∀i ∈{1, . . . ,n}. proof. let d(x) ∈ lq(x,qm) be such that d(ri) = d(f(gi)) and qdeg(d(x)) = t. this implies that d(ri −f(gi)) = 0 for all i. define ei := ri −f(gi), then ei ∈ fqm and every element of 〈e1, . . . ,en〉 is a root of d(x) (see lemma 2.1). since d(x) is non-zero and has degree qt, it follows that the linear space of roots has q-dimension t, which implies that (e1, . . . ,en) has rank t. this means that the rank distance between (c1, . . . ,cn) and (r1, . . . ,rn) is equal to t. thus, one direction is proven. for the other direction let (c1, . . . ,cn), (r1, . . . ,rn) have rank distance t, i.e. (e1, . . . ,en) := (c1 − r1, . . . ,cn − rn) has rank t. then by lemma 2.3 there exists a non-zero d(x) ∈ lq(x,qm) of degree qt such that d(ei) = 0 for all i. by linearity we get that d(ci) = d(ri) for i = 1, . . . ,n. since ci = f(gi) the statement follows. remark 2.7. theorem 2.6 states that the set of roots of d(x) is a vector space of degree t which is equal to the span of e1, . . . ,en (for this note that ei = m(gi) − ri). this is why d(x) is unique up to scalar multiplication (for given codeword and received word) and is also called the error span polynomial (cf. e.g. [22]). the analogy in the classical hamming metric set-up is the error locator polynomial, whose roots indicate the error locations. 2.3. modules over lq(x, qm) as mentioned before, the set of q-linearized polynomials lq(x,qm) is a ring with addition and composition. hence, for any positive integer `, the set lq(x,qm)` is a (right or left) module. in this work we will consider lq(x,qm)` as a left module and investigate its (left) submodules. in this section, we give some general definitions and results on lq(x,qm)` and present the terminology of the predictable leading monomial property. all of these are analogous to the definitions and results for modules over fq[x] (equipped with normal polynomial multiplication) from [3], see also the early work by fitzpatrick [7] and the textbooks [2, 5]. linearized polynomials belong to the class of skew polynomials, for which the general theory of linear algebra and gröbner bases is well established, see e.g. [1, 4, 11]. for reasons of clear exposition and self-containedness, we formulate the results that we need explicitly in terms of rings with composition, in the language of linearized polynomials. thus, compared to the fq[x]-case, multiplication is replaced by composition. to avoid confusion, we denote polynomials by f(x), while vectors of polynomials are denoted by f. if we need to index polynomials, we use the notation f1(x), . . . ,fs(x), while for vectors of polynomials we use the notation f(1), . . . ,f(s). 32 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 elements of lq(x,qm)` are of the form f := [f1(x) . . . f`(x)] = ∑̀ i=1 fi(x)ei where fi(x) = ∑ j fijx [j] ∈ lq(x,qm) and e1, . . . ,e` are the unit vectors of length `. analogous to polynomial multiplication on fqm [x]` we define for h(x) ∈lq(x,qm) the left operation h(x) ◦f := [h(f1(x)) . . . h(f`(x))] = ∑̀ i=1 h(fi(x))ei. the monomials of f are of the form x[k]ei for all k such that fik 6= 0. definition 2.8. a subset m ⊆ lq(x,qm)` is a (left) submodule of lq(x,qm)` if it is closed under addition and composition with lq(x,qm) on the left. definition 2.9. consider the non-zero elements f(1), . . . ,f(s) ∈ lq(x,qm)`. we say that f(1), . . . ,f(s) are linearly independent if for any a1(x), . . . ,as(x) ∈lq(x,qm) s∑ i=1 ai(x) ◦f(i) = [ 0 . . . 0 ] =⇒ a1(x) = · · · = as(x) = 0. a generating set of a submodule m ⊆ lq(x,qm)` is called a basis of m if all its elements are linearly independent. one can easily see that b = {xe1,xe2 . . . ,xe`} is a basis of lq(x,qm)`, thus lq(x,qm)` is a free and finitely generated module. we need the notion of monomial order for the subsequent results, which we will define in analogy to [2, definition 3.5.1]. definition 2.10. a monomial order < on lq(x,qm)` is a total order on lq(x,qm)` that fulfills the following two conditions: • x[k]ei < x[j] ◦ (x[k]ei) for any monomial x[k]ei ∈lq(x,qm)` and j ∈ n>0. • if x[k]ei < x[k ′]ei′, then x[j]◦(x[k]ei) < x[j]◦(x[k ′]ei′ ) for any monomials x[k]ei,x[k ′]ei′ ∈lq(x,qm)` and j ∈ n0. we have different choices for monomial orders, of which the following is of interest for our investigations. definition 2.11. the (k1, . . . ,k`)-weighted term-over-position monomial order is defined as x[i1]ej1 <(k1,...,k`) x [i2]ej2 : ⇐⇒ i1 + kj1 < i2 + kj2 or [i1 + kj1 = i2 + kj2 and j1 < j2]. note that this monomial order for lq(x,qm)` coincides with the weighted term-over-position monomial order for fqm [x], since one could replace the q-degrees with normal degrees and get the classical cases. we furthermore need the following definition in analogy to the weighted term-over-position monomial order: 33 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 definition 2.12. the (k1, . . . ,k`)-weighted q-degree of [f1(x) . . . f`(x)] is defined as max{ki + qdeg(fi(x)) | i = 1, . . . ,`}. in the following we will not fix a monomial order. the results, if not noted differently, hold for any chosen monomial order. definition 2.13. we can order all monomials of an element f ∈ lq(x,qm)` in decreasing order with respect to some monomial order. rename them such that x[i1]ej1 > x [i2]ej2 > ... . then 1. the leading monomial lm(f) = x[i1]ej1 is the greatest monomial of f. 2. the leading position lpos(f) = j1 is the vector coordinate of the leading monomial. 3. the leading term lt(f) = fj1,i1x [i1]ej1 is the complete term of the leading monomial. in order to define minimality for submodule bases we need the following notion of reduction, in analogy to [2, definition 4.1.1]. definition 2.14. let f,h ∈ lq(x,qm)` and let f = {f(1), . . . ,f(s)} be a set of non-zero elements of lq(x,qm)`. we say that f reduces to h modulo f (in one step) if and only if h = f − ((b1x[a1]) ◦f(1) + · · · + (bkx[ak]) ◦f(k)) for some a1, . . . ,ak ∈ n0 and b1, . . . ,bk ∈ fqm, where lm(f) = x[ai] ◦ lm(f(i)), i = 1, . . . ,k, and lt(f) = (b1x [a1]) ◦ lt(f(1)) + · · · + (bkxa[k] ) ◦ lt(f(k)). we say that f is minimal with respect to f if it cannot be reduced modulo f. definition 2.15. a module basis b is called minimal if all its elements b are minimal with respect to b\{b}. proposition 2.16. let b be a basis of a module m ⊆ lq(x,qm)`. then b is a minimal basis if and only if all leading positions of the elements of b are distinct. proof. let b be minimal. if two elements of b have the same leading position, the one with the greater leading monomial can be reduced modulo the other element, which contradicts the minimality. hence, no two elements of a minimal basis can have the same leading position. the other direction follows straight from the definition of reducibility and minimality of a basis, since if the leading positions of all elements are different, none of them can be reduced modulo the other elements. the property outlined in the following theorem is well-established for minimal gröbner bases for modules in fq[x]` with respect to multiplication. it extends to non-commutative gröbner bases of solvable type, see e.g. [11, lemma 1.5]. as a result, it also holds over the ring of linearized polynomials. it was labeled predictable leading monomial (plm) property in [13] to emphasize its closeness to forney’s predictable degree property [8]. it captures the exact property that is needed in subsequent proofs. note that in [13] minimal bases were addressed as minimal gröbner bases. it can be shown that in our current setting these are the same. theorem 2.17 (plm property). let m be a module in lq(x,qm)` with minimal basis b = {b(1), . . . ,b(l)}. then for any 0 6= f ∈ m, written as f = a1(x) ◦ b(1) + · · · + al(x) ◦ b(l), 34 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 where a1(x), . . . ,al(x) ∈lq(x,qm), we have lm(f) = max 1≤i≤l;ai(x) 6=0 {lm(ai(x)) ◦ lm(b(i))} where lm(ai(x)) is the term of ai(x) of highest q-degree. proof. since b is minimal, all leading positions and thus also all leading monomials of its elements are distinct (by proposition 2.16). without loss of generality assume that lm(b(1)) > lm(b(2)) > · · · > lm(b(l)) and that all ai(x) are non-zero. since lq(x,qm) contains no zero divisors, we have that lpos(ai(x)◦b(i)) = lpos(b(i)) for 1 ≤ i ≤ l. as a result, all leading positions and therefore all leading monomials of the ai(x) ◦ b(i)’s are distinct. thus there exist j1, . . . ,jl such that lm(aj1 (x) ◦ b (j1)) > lm(aj2 (x) ◦ b (j2)) > · · · > lm(ajl (x) ◦ b (jl)). it follows that lm(f) = lm(aj1 (x) ◦ b (j1)) = lm(aj1 (x)) ◦ lm(b (j1)) = max 1≤i≤l {lm(ai(x)) ◦ lm(b(i))}. proposition 2.18. the leading positions and weighted q-degrees of all elements of two distinct minimal bases for the same module in lq(x,qm)` have to be the same. this implies that the cardinality of both bases are equal as well. proof. let b1 = {b(i) | i = 1, . . . ,l} and b2 = {c(i) | i = 1, . . . ,l′} be two different minimal bases of the same module in lq(x,qm)`. then b(j) must be a linear combination of the c(i) for j = 1, . . . ,l. similarly, c(i) must be a linear combination of the b(j) for i = 1, . . . ,l′. hence, by the plm property and since all leading positions are different in the bases, there exist j′ ∈{1, . . . ,l′} and a(x),a′(x) ∈lq(x,qm) such that lm(a(x) ◦ c(j ′)) = lm(b(j)) and lm(a′(x) ◦ b(j)) = lm(c(j ′)). this implies on the one hand that lpos(b(j)) = lpos(c(j ′)) and on the other that qdeg(a(x)) = qdeg(a′(x)) = 0, which implies that qdeg(b(j)) = qdeg(c(j ′)). 3. iterative decoding of gabidulin codes for the remainder of the paper let g1, . . . ,gn ∈ fqm be linearly independent over fq and let mk(g1, . . . ,gn) be the generator matrix of the gabidulin code c ⊆ fnqm. denote g = (g1, . . . ,gn) and let r = (r1, . . . ,rn) ∈ fnqm be the received word. throughout the remainder of this paper our monomial order will be the (0,k − 1)-weighted term-over-position monomial order. 3.1. parametrization in the following we abbreviate the row span of a (polynomial) matrix a by rs(a). definition 3.1. the interpolation module m(r) for r is defined as the left submodule of lq(x,qm)2, given by m(r) := rs [ πg(x) 0 −λg,r(x) x ] . we identify any [f(x) g(x)] ∈ m(r) with the bivariate linearized q-polynomial q(x,y) = f(x)+g(y). the following theorem shows that the name interpolation module is justified for m(r): 35 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 theorem 3.2. m(r) consists exactly of all q(x,y) = f(x) + g(y) with f(x),g(x) ∈lq(x,qm), such that q(gi,ri) = 0 for i = 1, . . . ,n. proof. for the first direction let q(x,y) = f(x) + g(y) be an element of m(r). then there exist β(x),γ(x) ∈lq(x,qm) such that f(x) = β(x) ◦ πg(x) −γ(x) ◦ λg,r(x) and γ(x) = g(x), thus q(gi,ri) = β(πg(gi)) −γ(λg,r(gi)) + γ(ri) = 0 −γ(ri) + γ(ri) = 0. for the other direction let f(x),g(x) ∈ lq(x,qm) be such that q(gi,ri) = f(gi) + g(ri) = 0 for i = 1, . . . ,n. to show that q(x,y) ∈ m(r) we need to find β(x),γ(x) ∈lq(x,qm) such that β(x) ◦ πg(x) −γ(x) ◦ λg,r(x) = f(x) and γ(x) = g(x). we substitute the second into the first equation to get β(x) ◦ πg(x) = f(x) + g(x) ◦ λg,r(x). (2) by assumption, the equation f(gi) + g(λg,r(gi)) = f(gi) + g(ri) = 0 holds for all i. then, by lemma 2.5, it follows that f(x) + g(x)◦λg,r(x) is symbolically divisible on the right by πg(x) and hence there exists β(x) ∈lq(x,qm) such that (2) holds. the above leads to the following characterization of codewords with distance t to the received word: theorem 3.3. the elements f = [n(x) −d(x)] of m(r) that fulfill 1. qdeg(n(x)) ≤ t + k − 1, 2. qdeg(d(x)) = t, 3. n(x) is symbolically divisible on the left by d(x), i.e. there exists m(x) ∈ lq(x,qm) such that d(m(x)) = n(x), are in one-to-one correspondence with the codewords of rank distance t to the received word r. proof. to prove the first direction let c ∈ fnqm be a codeword such that dr(c,r) = t with the corresponding message polynomial m(x) ∈lq(x,qm)<k. then by theorem 2.6 there exists d(x) ∈lq(x,qm) of q-degree t such that d(m(gi)) = d(ri) for i = 1, . . . ,n. by theorem 3.2 we know that [d(m(x)) −d(x)] is in m(r). it holds that qdeg(d(m(x))) ≤ t + k− 1 and that (d(m(x))) is symbolically divisible on the left by d(x). for the other direction let [n(x) − d(x)] ∈ m(r) fulfill conditions 1) − 3). then the divisor m(x) ∈ lq(x,qm) has q-degree less than k and n(x) = d(m(x)). since it is in m(r) it follows from theorem 3.2 that d(m(gi)) − d(ri) = 0 for all i. define c := (m(g1), . . . ,m(gn)), then it follows from theorem 2.6 that dr(c,r) = t. note that conditions 1) and 2) of theorem 3.3 can alternatively be formulated as the condition that lpos(f) = 2 with (0,k − 1)-weighted q-degree of f being equal to t + k − 1. it follows from theorem 3.3 that decoding within rank radius t is equivalent to finding all elements f = [n(x) −d(x)] in m(r) with (0,k−1)-weighted q-degree less than t+k and leading position 2, such that n(x) is symbolically divisible on the left by d(x). the following theorem presents a parametrization that is helpful in order to find such elements. theorem 3.4. (parametrization) let b = {b(1),b(2)} be a minimal basis of m(r) with respect to the (0,k − 1)-weighted degree, with lpos(b(1)) = 1 and lpos(b(2)) = 2. define `1 and `2 as the (0,k − 1)weighted q-degrees of b(1),b(2), respectively. let t be a nonnegative integer. then all elements f ∈ m(r) with lpos(f) = 2 and (0,k − 1)-weighted q-degree equal to t + k − 1 are given by f = β(x) ◦ b(1) + γ(x) ◦ b(2), where β(x) is a q-linearized polynomial with qdeg(β(x)) ≤ t + k − 1 − `1 and γ(x) is a q-linearized polynomial with qdeg(γ(x)) = t + k − 1 − `2. 36 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 proof. the parametrization follows straightforwardly from theorem 2.17. 3.2. construction of a minimal basis we now present an iterative algorithm for the construction of a minimal basis for the interpolation module. the algorithm is similar to the ones in[12, 16, 27]. our main contribution is the recognition, via theorem 3.4, that such an algorithm essentially computes a minimal basis for the interpolation module rather than just one solution corresponding to the received word. a preliminary version of this result is the short conference paper [14]. we first need the following result: lemma 3.5. for i = 1, . . . ,n denote by mi the interpolation module for (g1, . . . ,gi) and (r1, . . . ,ri). let [ p(x) −k(x) n(x) −d(x) ] be a basis for mi−1 and γi := p(gi) −k(ri) , ∆i := n(gi) −d(ri). if γi 6= 0, then the set of row vectors of[ b(1) b(2) ] := [ xq − γq−1i x 0 ∆ix −γix ] ◦ [ p(x) −k(x) n(x) −d(x) ] is a basis of mi. if ∆i 6= 0, then the set of row vectors of[ ∆ix −γix 0 xq − ∆q−1i x ] ◦ [ p(x) −k(x) n(x) −d(x) ] is a basis of mi. proof. we first consider the first case and show that both b(1) and b(2) are in mi. from the assumptions it follows that p(gj) = k(rj) and that n(gj) = d(rj) for 1 ≤ j < i. moreover, the two entries of b(1) are given by (xq − γq−1i x) ◦p(x) = p(x) q − γq−1i p(x), (xq − γq−1i x) ◦k(x) = k(x) q − γq−1i k(x), thus p(gj)q − γ q−1 i p(gj) −k(rj) q + γ q−1 i k(rj) = 0 for 1 ≤ j ≤ i. for b (2) we get ∆ip(gj) − γin(gj) − ∆ik(rj) + γid(rj) = ∆i(p(gj) −k(rj)) − γi(n(gj) −d(rj)) = ∆iγi − γi∆i = 0 for 1 ≤ j ≤ i. thus, b(1) and b(2) are elements of mi. it remains to show that b(1) and b(2) span the entire interpolation module (and not just a submodule of it). for this, it is sufficient to show that [ πi−1(x) 0 ] and [ λi−1(x) −x ] are linear combinations of b(1) and b(2). since [ p(x) −k(x) ] and [ n(x) −d(x) ] are a basis of mi, there exist β̄(x), γ̄(x) ∈lq(x,qm) such that β̄(x) ◦ [ p(x) −k(x) ] + γ̄(x) ◦ [ n(x) −d(x) ] = [ πi−1(x) 0 ]. 37 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 let β(x),γ(x) ∈lq(x,qm) be such that β(x) ◦ (xq − γq−1i x) = (x q − πi−1(gi)q−1x) ◦ ( γ̄( ∆i γi x) + β̄(x) ) , γ(x) = −(xq − πi−1(gi)q−1x) ◦ γ̄( 1 γi x). note that it can easily be checked that γi is a root of the right side of the previous equation, thus β(x) is well-defined by lemma 2.5 . denote the first and second row of the new basis by b(1) and b(2), respectively. then β(x) ◦ b(1)1 + γ(x) ◦ b (2) 1 = β(x) ◦ (x q − γq−1i x) ◦p(x) + γ(x) ◦ (∆ip(x) − γin(x)) = (xq − πi−1(gi)q−1x) ◦ ( γ̄( ∆i γi x) + β̄(x) ) ◦p(x) + γ(∆ip(x)) −γ(γin(x)) = ( (xq − πi−1(gi)q−1x) ◦ β̄(x) −γ(∆ix) ) ◦p(x) + γ(∆ip(x)) −γ(γin(x)) = ( (xq − πi−1(gi)q−1x) ◦ β̄(x) ) ◦p(x) + (xq − πi−1(gi)q−1x) ◦ γ̄(n(x)) = (xq − πi−1(gi)q−1x) ◦ ( β̄(p(x)) + γ̄(n(x)) ) = (xq − πi−1(gi)q−1x) ◦ πi−1(x) = πi(x), and β(x) ◦ b(1)2 + γ(x) ◦ b (2) 2 = −β(x) ◦ (x q − γq−1i x) ◦k(x) −γ(x) ◦ (∆ik(x) − γid(x)) = ( (xq − πi−1(gi)q−1x) ◦ β̄(x) −γ(∆ix) ) ◦k(x) + γ(∆ik(x)) −γ(γid(x)) = (xq − πi−1(gi)q−1x) ◦ ( β̄(k(x)) + γ̄(d(x)) ) = 0. thus β(x)◦b(1) + γ(x)◦b(2) = [ πi(x) 0 ], i.e. [ πi(x) 0 ] is in the module spanned by the new basis. analogously, if we have that c̄(x) ◦ [ p(x) −k(x) ] + d̄(x) ◦ [ n(x) −d(x) ] = [ λi−1(x) −x ] and define c(x),d(x) ∈lq(x,qm) such that d(x) = − ( d̄(x) + λi−1(gi) −ri πi−1(gi) γ̄(x) ) ◦ ( 1 γi x) c(x) ◦ (xq − γq−1i x) = c̄(x) + λi−1(gi) −ri πi−1(gi) β̄(x) −d(∆ix) we get c(x) ◦ b(1) + d(x) ◦ b(2) = [ λi(x) −x ]. hence, we have shown that the new basis {b(1),b(2)} spans the entire interpolation module. for the second case note that rs ([ ∆ix −γix 0 xq − ∆q−1i x ] ◦ [ p(x) −k(x) n(x) −d(x) ]) = rs ([ xq − ∆q−1i x 0 γix −∆ix ] ◦ [ n(x) −d(x) p(x) −k(x) ]) , which corresponds to the first case after exchanging p(x) with n(x) and k(x) with d(x) (and vice versa). 38 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 using lemma 3.5 as our main ingredient, we now set out to design an iterative algorithm that computes a minimal basis for mi at each step i. algorithm 1 iterative computation of a minimal basis of m(r). require: positive integers k, n; g = (g1, . . . , gn) ∈ fnqm, received word r = (r1, . . . , rn) ∈ fnqm. initialize j := 0, b0 := [ x 0 0 x ] . we denote bi := [ pi(x) −ki(x) ni(x) −di(x) ] . for i from 1 to n do γi := pi−1(gi) −ki−1(ri) , ∆i := ni−1(gi) −di−1(ri). if [qdeg(pi−1(x)) ≤ qdeg(di−1(x)) + k − 1 and γi 6= 0] or ∆i = 0 then bi := [ xq − γq−1i x 0 ∆ix −γix ] ◦bi−1 else bi := [ ∆ix −γix 0 xq − ∆q−1i x ] ◦bi−1 end if end for return bn theorem 3.6. algorithm 1 yields a minimal basis of the interpolation module m(r), where the leading position of the first row equals 1 and the leading position of the second row equals 2. proof. denote by m1 the matrix we multiply by on the left in the first if statement and by m2 the one in the else statement of the algorithm. we know from lemma 3.5 that at each step, bi is a basis for the interpolation module mi. we now show that it is a minimal basis with respect to the (0,k − 1)weighted term-over-position monomial order via induction on i. assume that at step i the first row has leading position 1 and the second row has leading position 2, i.e. qdeg(pi(x)) > qdeg(ki(x)) + k−1 and qdeg(ni(x)) ≤ qdeg(di(x)) + k − 1. if qdeg(pi(x)) ≤ qdeg(di(x)) + k − 1 we composite on the left by m1. hence, qdeg(pi+1(x)) = qdeg(pi(x)) + 1 and qdeg(ki+1(x)) = qdeg(ki(x)) + 1 < qdeg(pi(x)) −k + 2 = qdeg(pi+1(x)) −k + 1. thus, the leading position of the first row of bi+1 is still 1. moreover, qdeg(ni+1(x)) ≤ max{qdeg(pi(x)), qdeg(ni(x))}≤ qdeg(di(x)) + k − 1 and, since the assumptions imply that qdeg(ki(x)) < qdeg(di(x)), qdeg(di+1(x)) = max{qdeg(ki(x)), qdeg(di(x))} = qdeg(di(x)). thus the leading position of the second row is 2. since the assumptions are true for b0 the statement follows via induction. analogously one can prove that composition with m2 yields a basis of mi with different leading positions in the two rows. thus at each step we get a basis of mi with different leading positions, which is by proposition 2.16 a minimal basis. thus, after n steps, bn is a minimal basis for the interpolation module m(r). remark 3.7. it can be verified that, due to the linear independence of g1, . . . ,gk, up to a constant, at step k the algorithm has computed the q-annihilator polynomial and the q-lagrange polynomial corresponding to the data so far. 39 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 proposition 3.8. algorithm 1 has computational complexity order oqm (n2). proof. for the iterative computation of the minimal basis from algorithm 1 we need n steps. in each step we need • four evaluations and differences of linearized polynomials of q-degree at most n (to compute ∆i and γi and in the update of bi), • two multiplications of a linearized polynomial of q-degree at most n by a scalar (in the update of bi), • a composition of a linearized polynomial of q-degree at most n with (xq −gq−1i x) (for this note that we compute a (q − 1)-th power of gi with a q-th power and one division). all of these operations can be done with oqm (n) operations. overall we get an upper bound on the complexity of oqm (n2). 3.3. decoding algorithm we can now set up the decoding algorithm which will find the closest codeword(s) to a given received word. for this we need the following lemma. lemma 3.9. consider a gabidulin code c ⊆ fnqm of dimension k. let m(r) be the interpolation module of the received word r ∈ fnqm with minimal basis b = {b(1),b(2)} where lpos(b(i)) = i for i = 1, 2. denote the (0,k − 1)-weighted q-degree of b(i) by `i for i = 1, 2. then `1 + `2 = n + k − 1. (3) proof. by proposition 2.18 the q-degrees of any minimal basis of the interpolation module m(r) have to add up to the same number, hence it is enough to show that they add up to n+k−1 for one particular basis. consider the iterative construction of a minimal basis from algorithm 1. it is easy to see that the initial basis has weighted q-degrees 0 and k − 1. moreover, at each step the q-degree of one row is increased by one, whereas the q-degree of the other row remains the same. thus, the sum of the two q-degrees is increased by 1 at each step. since we get the desired basis of m(r) at the n-th step, equation (3) follows. in the following theorem we pay specific attention to the unique decoding case. theorem 3.10. consider the setting of theorem 3.4. if t = min{dr(c,r) | c ∈ c} and t ≤ (n−k)/2 then f = γ(x) ◦ b(2) with qdeg(γ(x)) = 0 and symbolic division on the two components of the vector b(2) produces the message polynomial corresponding to the unique closest codeword to r. proof. let m(x) ∈lq(x,qm) be the message polynomial corresponding to the unique closest codeword c. then by theorem 3.3, there exist d(x) ∈lq(x,qm) of q-degree t such that f = [ d(m(x)) −d(x) ] is an element of the interpolation module with leading position 2. note that then the (0,k−1)-weighted degree of f equals t + k − 1 ≤ (n + k − 2)/2. theorem 2.17 now implies that lm(f) ≥ lm(b(2)), which implies (since the leading positions of both elements are 2) that `2 ≤ (n + k − 2)/2. it now follows from lemma 3.9 that `1 ≥ (n + k)/2. thus, in the parametrization we get β(x) ≡ 0, which means that there exists γ(x) such that f = γ(x) ◦ b(2). but since γ(x) cancels out when we divide the right entry of f by the left, we can simply choose γ(x) = x to recover the message polynomial. this also implies that qdeg(b(2)) = t + k− 1, and the last statement of the theorem follows. 40 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 note that, if the received word is within the unique decoding radius, the main result of loidreau’s algorithm [16] is similar, namely the symbolic division on two components to produce the message polynomial, as in the above theorem. we present the general structure of the decoding algorithm in algorithm 2 below. the outer if loop is based on theorem 3.10, whereas the corresponding else loop uses theorems 3.3 and 3.4 to enumerate all message polynomials corresponding to the codewords of distance t to the received word (for increasing t). algorithm 2 iterative decoding of a gabidulin code c ⊆ fnqm. require: positive integers k, n; received word r ∈ fnqm. use algorithm 1 to compute a minimal basis bn = {b(1), b(2)} of m(r). set `1 := qdeg(b(1)), `2 := qdeg(b(2)), list:= [ ], j := 0 . if `2 ≤ (n + k − 2)/2 and b (2) 1 is symbolically divisible by b (2) 2 on the left then add the symbolic quotient of b(2)1 and b (2) 2 to list. else while list= [ ] do for all a(x) ∈lq(x, qm), qdeg(a(x)) ≤ `2 − `1 + j do for all monic c(x) ∈lq(x, qm), qdeg(b(x)) = j do f := a(x) ◦ b(1) + c(x) ◦ b(2) if f1(x) is symb. (right) divisible by f2(x) then add the respective symb. quotient to list end if end for end for j := j + 1 end while end if return list remark 3.11. algorithm 2 is an extension of loidreau’s algorithm, in the sense that it has the same performance in the unique decoding case, but it can also find all closest codewords if the received word is beyond the unique decoding radius of the gabidulin code. this is due to the parametrization derived in subsection 3.1 and the fact that loidreau’s algorithm actually computes a minimal basis of the interpolation module, as shown in subsection 3.2. we conclude this section with a final example. example 3.12. consider the gabidulin code in f23 ∼= f2[α] (with α3 = α + 1) of length n = 3 and dimension k = 2 with generator matrix g = ( 1 α α2 1 α2 α4 ) . thus g = (g1,g2,g3) = (1,α,α2). suppose that the received word equals r = ( α + 1 0 α ). using algorithm 1, we iteratively compute b1 = [ x2 + x 0 (α + 1)x x ] , b2 = [ x4 + (α2 + α + 1)x2 + (α2 + α)x 0 (α2 + α)x2 + (α2 + α + 1)x (α2 + α)x ] , 41 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 b3 = [ α2x4 + α5x x αx4 + α4x2 + x αx2 + α6x ] . b3 is a minimal (0, 1)-weighted basis of the interpolation module m(r). in the notation of theorem 3.4, we get `1 = `2 = 2. as a result, by theorem 3.4, any monic f ∈ m(r) that has (0, 1)-weighted q-degree 2 and fulfills lpos(f) = 2 can be written as f = β(x) ◦ b(1) + γ(x) ◦ b(2), where β(x) is a q-linearized polynomial with qdeg(β(x)) ≤ 0 and γ(x) is a monic q-linearized polynomial with qdeg(γ(x)) = 0. thus f = b0x◦ b(1) + x◦ b(2) = b0b (1) + b(2) = [ (b0α 2 + α)x4 + α4x2 + (b0α 5 + 1)x αx2 + (b0 + α 6)x ] . in fact, in this example we can use this basis to obtain a complete list of codewords of rank distance 1 away from r = ( α + 1 0 α ), as follows. it is easily checked that for b0 ∈ f23\{α6} we get divisibility. thus it follows from theorem 3.3 that the complete list of all message polynomials & codewords of rank distance 1 away from r = ( α + 1 0 α ) is given by m1(x) = x 2 + αx c1 = ( α + 1 0 α 2 + 1), m2(x) = α 5x2 + α2x c2 = ( α + 1 α α), m3(x) = α 3x2 + α4x c3 = ( α 2 + 1 0 α2), m4(x) = α 4x2 c4 = ( α 2 + α α2 + 1 α), m5(x) = α 6x2 + α6x c5 = ( 0 α + 1 1), m6(x) = α 2x2 + α3x c6 = ( α 2 + α + 1 0 α), m7(x) = αx 2 + x c7 = ( α + 1 1 α + 1). note that the corresponding hamming distances to r vary from 1 to 3. 4. conclusions we extended the welch-berlekamp type algorithm given in the pioneering work by loidreau [16], to be able to decode also beyond the unique decoding radius. for this we derived a parametrization of all codewords within a given radius of the received words, based on a minimal basis of the interpolation module. to compute such a minimal basis we presented a polynomial-time iterative algorithm with simple update steps, similar to loidreau’s algorithm. the main contribution of our paper is the recognition that such algorithms actually compute a minimal basis of the interpolation module which can then be used to provide a parametrization of all solutions corresponding to the received word. in the reed-solomon case, massey’s parametrization resulting from the berlekamp-massey algorithm was used by wu [26] as the foundation to his polynomial-time reed-solomon list decoding algorithm. this was used in [3] as the foundation for a polynomial time reed-solomon list decoding method via welchberlekamp type interpolation. in this paper we strengthened the link between reed-solomon decoding and gabidulin decoding in providing a similar parametrization from a welch-berlekamp type algorithm for gabidulin decoding. currently no polynomial-time list decoding algorithms exist for general gabidulin codes; on the contrary, it is shown that polynomial list sizes are not possible for certain parameter sets (see e.g. [18, 24]). however, there are still many parameters for which it is an open question whether polynomial-time list decoding of gabidulin codes is possible. it is a topic of future research to build on the results of this paper in extending the parametrization-based methods of [3, 26] to a possibly polynomial-time gabidulin list decoding algorithm. 42 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 a.-l. horlemann-trautmann, m. kuijper / j. algebra comb. discrete appl. 5(1) (2018) 29–43 references [1] s. abramov, m. bronstein, linear algebra for skew–polynomial matrices, technical report inria, rr–4420, 2002. 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[27] h. xie, z. yan, b. w. suter, general linearized polynomial interpolation and its applications, international symposium on network coding (netcod) proceedings (2011) 1–4. 43 http://orcid.org/0000-0003-2685-2343 http://orcid.org/0000-0001-9223-9550 https://hal.inria.fr/file/index/docid/72168/filename/rr-4420.pdf https://hal.inria.fr/file/index/docid/72168/filename/rr-4420.pdf http://dx.doi.org/10.1090/gsm/003 http://dx.doi.org/10.1090/gsm/003 https://doi.org/10.1109/tit.2011.2165803 https://doi.org/10.1109/tit.2011.2165803 https://doi.org/10.1016/j.jsc.2005.10.002 https://doi.org/10.1016/j.jsc.2005.10.002 https://doi.org/10.1007/b138611 https://doi.org/10.1007/b138611 https://doi.org/10.1016/0097-3165(78)90015-8 https://doi.org/10.1016/0097-3165(78)90015-8 https://doi.org/10.1109/18.412677 https://doi.org/10.1137/0313029 https://doi.org/10.1137/0313029 http://www.ams.org/mathscinet-getitem?mr=791529 http://www.ams.org/mathscinet-getitem?mr=791529 https://doi.org/10.1007/bfb0034349 https://doi.org/10.1007/bfb0034349 https://doi.org/10.1016/s0747-7171(08)80003-x https://doi.org/10.1016/s0747-7171(08)80003-x https://doi.org/10.1109/tit.2008.926449 https://doi.org/10.1109/tit.2008.926449 https://doi.org/10.1016/j.laa.2010.08.030 https://doi.org/10.1016/j.laa.2010.08.030 https://doi.org/10.1109/itw.2014.6970898 https://doi.org/10.1109/itw.2014.6970898 https://doi.org/10.1007/11779360_4 https://doi.org/10.1007/11779360_4 https://doi.org/10.1090/s0002-9947-1933-1501703-0 https://doi.org/10.1109/tit.2016.2532343 https://doi.org/10.1109/tit.2016.2532343 https://doi.org/10.1109/isit.2004.1365435 https://doi.org/10.1109/isit.2004.1365435 https://doi.org/10.1109/allerton.2012.6483348 https://doi.org/10.1109/allerton.2012.6483348 https://doi.org/10.1109/isit.2009.5205272 https://doi.org/10.1109/isit.2009.5205272 https://doi.org/10.1109/tit.2009.2032817 https://doi.org/10.1109/tit.2009.2032817 https://doi.org/10.1109/tit.2008.928291 https://doi.org/10.1109/tit.2008.928291 https://doi.org/10.1109/tit.2013.2274653 https://doi.org/10.1109/tit.2013.2274653 https://doi.org/10.1007/s10623-012-9659-5 https://doi.org/10.1007/s10623-012-9659-5 https://doi.org/10.1109/tit.2008.926355 https://doi.org/10.1109/tit.2008.926355 https://doi.org/10.1109/isnetcod.2011.5978942 https://doi.org/10.1109/isnetcod.2011.5978942 introduction preliminaries iterative decoding of gabidulin codes conclusions references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.466634 j. algebra comb. discrete appl. 5(3) • 153–165 received: 24 october 2017 accepted: 5 september 2018 journal of algebra combinatorics discrete structures and applications batch codes from hamming and reed-muller codes∗ research article travis baumbaugh, yariana diaz, sophia friesenhahn, felice manganiello, alexander vetter abstract: batch codes, introduced by ishai et al., encode a string x ∈ σk into an m-tuple of strings, called buckets. in this paper we consider multiset batch codes wherein a set of t-users wish to access one bit of information each from the original string. we introduce a concept of optimal batch codes. we first show that binary hamming codes are optimal batch codes. the main body of this work provides batch properties of reed-muller codes. we look at locality and availability properties of first order reed-muller codes over any finite field. we then show that binary first order reed-muller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary reed-muller codes which have order less than half their length. 2010 msc: 14g50 keywords: batch codes, hamming codes, reed-muller codes 1. introduction consider the situation where a certain amount of data, such as information to be downloaded, is distributed over a number of devices. we could have multiple users who wish to download this data. in order to reduce wait time, we look at locally repairable codes with availability as noted in [4]. a locally repairable code, with locality r and availability δ, provides us the opportunity to reconstruct a particular bit of data using δ disjoint sets of size at most r [12]. when we want to reconstruct a certain bit of information, this is the same as a private information retrieval (pir) code. however, we wish to examine a scenario where we reconstruct not necessarily distinct bits of information. a possible answer to the more complex scenario seems to be a family of codes called batch codes, introduced by ishai et al. in [6]. batch codes were originally studied as a schematic for distributing data ∗ this work was supported by nsf grant dms #1547399. travis baumbaugh (corresponding author), felice manganiello; clemson university, united states (email: tbaumba@g.clemson.edu, manganm@clemson.edu). yariana diaz; the university of iowa, united states (email: yariana-diaz@uiowa.edu). sophia friesenhahn; willamette university, united states (email: sophmf@gmail.com). alexander vetter; villanova university, united states (email: avetter@villanova.edu). 153 https://orcid.org/0000-0002-3539-9758 https://orcid.org/0000-0003-0603-9750 https://orcid.org/0000-0001-8847-8962 https://orcid.org/0000-0002-5764-569x https://orcid.org/0000-0003-2539-9815 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 across multiple devices and minimizing the load on each device and total amount of storage. we study (n,k,t,m,τ) batch codes, where n is the code length, k is the dimension of the code, t is the number of bits we wish to retrieve, m is the number of buckets, and τ is the maximum number of bits used from each bucket for any reconstruction of t bits. in this paper we seek to minimize the number of devices in the system and the load on each device while maximizing the amount of reconstructed data. in other words, we want to minimize mτ while maximizing t. in section 2, we formally introduce batch codes and summarize results from previous work on batch codes. we then introduce the concepts of locality and availability of a code. we conclude the section by introducing a concept of optimal batch codes. after the background, we study the batch properties of binary hamming codes and reed-muller codes. section 3 focuses on batch properties of binary hamming codes. we show that hamming codes are optimal (2s−1, 2s − 1 −s, 2,m,τ) batch codes for m,τ ∈ n such that mτ = 2s−1. section 4 is the main body of this work and provides batch properties of reed-muller codes. we first study the induced batch properties of a code c given that c⊥ is of a (u | u + v)-code construction with determined batch properties. in section 4.1 we study the locality and availability properties of first order reed-muller codes over any finite field. we find that the locality of rmq(1,µ) is 2 when q 6= 2 and 3 when the q = 2. furthermore, we also show that its availability is ⌊ qµ−1 2 ⌋ when q 6= 2, whereas when q = 2, the availability is 2 µ−1 3 if µ is even and at least 2 µ−4 4 otherwise. in section 4.2 we show that binary first order reed-muller codes are optimal batch codes for t = 4. we first look at the specific rm(1, 4) case and achieve parameters (16, 5, 4,m,τ) such that mτ = 10. we then prove a general result that any reed-muller code with ρ = 1 and µ ≥ 4 has batch properties (2µ,µ + 1, 4,m,τ) for any m,τ ∈ n such that mτ = 10. finally, in section 4.3 we generalize our study of reed-muller codes and look at properties of rm(ρ,µ) for all values of ρ and conclude our study by presenting batch properties (2µ,k, 4,m,τ) such that mτ = 10 · 22ρ−2 for rm(ρ′,µ) where µ ∈{2ρ + 2, 2ρ + 3} and ρ′ ≤ ρ. 2. background in 2004, ishai et al. [6] introduced the following definition of batch codes: definition 2.1. an (n,k,t,m,τ) batch code over an alphabet σ encodes a string x ∈ σk into an m-tuple of strings, called buckets, of total length n such that for each t-tuple of distinct indices i1, . . . , it ∈ [k] = {1, ...,k}, the entries xi1, . . . ,xit can be decoded by reading at most τ symbols from each bucket. we can view the buckets as servers and the symbols used from each bucket as the maximal load on each server. in the above scenario, a single user is trying to reconstruct t bits of information. this definition naturally leads to the concept of multiset batch codes which have nearly the same definition as above, but the indices i1, . . . , ik ∈ [k] are not necessarily distinct. this means we have t users who each wish to reconstruct a single element. this definition in turn relates to private information retrieval (pir) codes which are similar to batch codes but instead look to reconstruct the same bit of information t times. another notable type of batch code defined in [6] is a primitive multiset batch code where the number of buckets is m = n. in this research, the queries are considered to happen at the same time, while the asynchronous case is considered in [10]. the following are useful lemmas proven in [6]: lemma 2.2. an (n,k,t,m,τ) batch code for any τ implies an (nτ,k,t,mτ, 1) batch code. lemma 2.3. an (n,k,t,m, 1) batch code implies an (n,k,t,dm τ e,τ) batch code. much of the related research involves primitive multiset batch codes with a systematic generator matrix. in [6], the authors give results for some multiset batch codes using subcube codes and reedmuller codes. they use a systematic generator matrix, which often allows for better parameters. their 154 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 goal was to maximize the efficiency of the code for a fixed number of queries t. the focus of research on batch codes then shifted to combinatorial batch codes. these were first introduced by [9]. they are replication based codes using various combinatorial objects that allow for efficient decoding procedures. we do not consider combinatorial batch codes but some relevant results can be found in [9], [2], [3], and [11]. next, the focus of research turned to linear batch codes, which use classical error-correcting codes. the following useful results are proven in [7]: theorem 2.4. let c be an [n,k,t,n, 1] linear batch code over f2 with generator g. then, g is a generator matrix of the classical error-correcting [n,k,d]2 linear code where d ≥ t. theorem 2.5. let c1 be an [n1,k,t1,n1, 1]q linear batch code and c2 be an [n2,k,t2,n2, 1]q linear batch code. then, there exists an [n1 + n2,k,t1 + t2,n1 + n2, 1]q linear batch code. theorem 2.6. let c1 be an [n1,k1, t1,n1, 1]q linear batch code and c2 be an [n2,k2, t2,n2, 1]q linear batch code. then, there exists an [n1 + n2,k1 + k2,min(t1, t2),n1 + n2, 1]q linear batch code. because of the vast amount of information on classical error-correcting codes, we use these as our central focus in this paper. as is often the case, studying the properties of the dual codes helps us find efficient batch codes. next, [13] considers restricting the size of reconstruction sets. these are similar to codes with locality and availability: definition 2.7. a code c has locality r ≥ 1 if for any c ∈c, any entry in c can be recovered by using at most r other entries of c. definition 2.8. a code c with locality r ≥ 1 has availability δ ≥ 1 if for any c ∈ c, any entry of c can be recovered by using one of δ disjoint sets of at most r other entries of y the restriction on the size of reconstruction sets can be viewed as trying to minimize total data distribution. we restrict the size of our reconstruction sets to the locality of the code. by using this restriction, we find multiset batch codes with optimal data distribution. for cyclic codes, the locality can be derived from the following in [5]: lemma 2.9. let c be an [n,k,d] cyclic code, and let d′ be the minimum distance of its dual code c⊥. then, the code c has all symbol locality d′ − 1. this relies on each entry being in the support of a minimal weight dual code word. we generalize this lemma to the following one. lemma 2.10. let c ⊆ fnq be a linear code and let d′ be the minimum distance of c⊥. if c⊥ is generated by its minimum weight codewords and ⋃ λ∈c⊥ supp(λ) = [n], (1) then c has all symbol locality d′ − 1. proof. condition (1) implies that no coordinate of c is independent on the others. if the minimum weight codewords generate c⊥, then each coordinate of c is in the support of at least one minimum weight codeword of c⊥. this implies the all symbol locality d′ − 1 of c. condition (1) is a reasonable condition for a code. without it the code c would have non recoverable coordinates. we give here a bound that relates the locality property of a linear code with its batch properties. 155 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 lemma 2.11. let c be an [n,k,t,m,τ] linear batch code with minimal locality r. then it holds that mτ ≥ (t− 1)r + 1. (2) proof. we consider such a code c. if each entry has at least one reconstruction set with fewer than r elements, then by the definition of locality, c has locality less than r, a contradiction to r being the minimal locality. therefore, there exists at least one entry for which all recovery sets are of size at least r. if we wish to recover this entry t times, then we may read the entry itself and then make use of t− 1 disjoint recovery sets, each of size r. this implies reading at least (t− 1)r + 1 entries, and since we may read only τ entries from each of the m buckets, we must have that mτ ≥ (t− 1)r + 1. from the perspective of individual devices storing bits of data, mτ represents the total amount of data read to provide t pieces of the original data. to minimize bandwidth usage in the case where the entries of a codeword represent nodes on a network, we must minimize mτ. definition 2.12. a [n,k,t,m,τ] linear batch code c with minimal locality r is optimal if it satisfies condition (2) with equality. we now show that binary hamming codes are optimal linear batch codes. 3. hamming codes hamming codes were first introduced in 1950 by richard hamming. in what follows, we consider binary hamming codes over f2. the parameters of binary hamming codes are shown in [8]. definition 3.1. for some s ≥ 2, let h ∈ f2 s−1×s 2 be a matrix whose columns are all of the nonzero vectors of fs2. let n = 2 s − 1. we use h as our parity check matrix and define the binary hamming code: hs := {c ∈ fn2 | ch t = 0} it is well-known that hs is a [2s − 1, 2s − 1 −s, 3] cyclic code. its dual code, the simplex code, is a [2s − 1,s, 2s−1] cyclic code. thus, by lemmma 2.9, the locality of hs is 2s−1 − 1. we now present the batch properties of binary hamming codes. theorem 3.2. a binary [n = 2s − 1,k = 2s − 1 − s] hamming code is an optimal batch code with properties [2s − 1, 2s − 1 −s, 2,m,τ], for any m,τ ∈ n such that mτ = 2s−1. proof. first, note that mτ ≥ (2 − 1)(2s−1 − 1) + 1 = 2s−1. the buckets for m = 2s−1, τ = 1 are constructed as follows. let h be the parity check matrix of a binary hamming code, h, with columns hj ∈ fs2 for 1 ≤ j ≤ n. if ha + hb = 1 (the all ones column), then we place a and b into the same bucket. note that because h` = 1 in h, ` is placed into its own bucket. let rd ∈ fn2 be the rows in h such that 1 ≤ d ≤ (n − k) = s. for any c ∈ h, rd · ct = 0, and thus ∑ i∈supp(rd) ci = 0. as a result of this construction, if a,b ∈ supp(rd), then entry d of ha + hb is 0, so a and b cannot be in the same bucket. therefore, ∑ i∈supp(rd) ci = 0 only involves bits in separate buckets. hence, any bit in a codeword can be written as a linear combination of bits in separate buckets. now, we show how to reconstruct any two bits of information. • case 1: if a and b are in separate buckets, then use ca and cb. • case 2: suppose a and b are in the same bucket. we can take ca itself. to construct cb, we choose an rd such that b ∈ supp(rd). then, we can write cb = ∑ i∈supp(rd)\{b} ci, which we know only contains bits in disjoint buckets. 156 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 every bucket has cardinality 2 aside from the bucket corresponding to the all ones column in h, so this construction gives us exactly 2s−1 buckets. thus, we have shown that the batch properties hold for m = 2s−1 and τ = 1. further, lemma 2.3 implies that this is true for any m,τ such that mτ = 2s−1. note that the locality of h is 2s−1 − 1, and therefore, t = 2 is also maximal. suppose instead that we could have t ≥ 3. then in particular, each entry must be reconstructible at least 3 times. we may take the entry itself, but then there must be at least 2 other reconstruction sets used which are disjoint and of size 2s−1 − 1. these would correspond to two codewords in the dual code of weight 2s−1 with the intersection of their support being only the given entry. the sum of these codewords will thus have weight 2s−1 + 2s−1 −2 = 2s−2. however, the all ones vector is also in the dual code. adding this vector to the sum will produce a codeword of weight one, a contradiction. thus, t = 2 is maximal. example 3.3. we now give an example for s = 3. this hamming code is a [7, 4]-linear code, and the dual code is a [7, 3]-linear code. the parity check matrix h is as follows: h =  1 0 1 0 1 0 10 1 1 0 0 1 1 0 0 0 1 1 1 1   thus, the buckets are: {1, 6},{2, 5},{3, 4},{7} we note that for general batch codes we are only interested in reconstructing information bits. however, we are able to obtain any pair of bits in the codeword. additionally, we note that although mτ is optimized, we wish to find batch codes where t > 2. we desire a larger value as we are concerned with practical applications, and the goal is to quickly distribute data. thus we move on to reed-muller codes, where we are able to obtain larger t values. 4. reed-muller codes reed-muller codes are well known linear codes. we give some basic properties of these codes, but an interested reader can find more information in [1]. definition 4.1. let fq[x1, . . . ,xµ] be the ring of polynomials in µ variables with coefficients in fq and let fµq = {p1, . . . ,pn} (so n = qµ). the q-ary reed-muller code, rmq(ρ,µ) is defined as: rmq(ρ,µ) := {(f(p1), . . . ,f(pn)) | f ∈ fq[x1, . . . ,xµ]ρ}, where fq[x1, . . . ,xµ]ρ is the set of all multivariate polynomials over fq of total degree at most ρ. it is known that if ρ < µ(q − 1) then the dual of a reed-muller code rmq(ρ,µ) is the reed-muller code rmq(µ(q − 1) − 1 −ρ,µ) [1]. in the binary case, reed-muller codes can be equivalently defined using the (u | u+v)-code construction. for completeness, we first give a description of the (u | u + v)-code construction and the related generator matrix, which can be found in [8]. definition 4.2. given two linear codes c1,c2 with identical alphabets and block lengths, we construct a new code c as follows: c := {(u | u + v) | u ∈c1, v ∈c2}. 157 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 let g, g1, and g2 be the generator matrices for the codes c, c1, and c2, respectively, where c is obtained from c1 and c2 via the (u | u + v)-construction. then we have g := ( g1 g1 0 g2 ) from this, we have the following proposition. proposition 4.3. let c1 be an [n,k1,d1]-linear code, c2 an [n,k2,d2]-linear code, and c the code obtained from c1,c2 via the (u | u + v)-construction. then, c is an [2n,k,d]-code where k = k1 + k2 and d = min{2d1,d2}. later, our focus will be on q = 2, so when referring to rm2(ρ,µ) we omit the 2 for convenience. we obtain the following equivalent definition of a binary reed-muller code. definition 4.4. let ρ < µ. a binary reed-muller code rm(ρ,µ) is defined as follows: rm(ρ,µ) := {(u | u + v) | u ∈rm(ρ,µ− 1), v ∈rm(ρ− 1,µ− 1)} where rm(0,µ) := 1 of length 2µ, and rm(µ,µ) := i2µ. as a consequence if gρ,µ is the generator matrix of the code rm(ρ,µ), then gρ,µ := ( gρ,µ−1 gρ,µ−1 0 gρ−1,µ−1 ) we now examine the batch properties of the (u | v + u)-code construction. the first notable result comes from codes that are contained in other codes: theorem 4.5. let c1,c2 be codes of length n and dimension k1 and k2, respectively such that c2 ⊆c1. if c1 is a (n,k1, t,m,τ) batch code, then c2 is a (n,k2, t,m,τ) batch code. proof. note that c⊥1 ⊆ c⊥2 because c2 ⊆ c1. therefore the same parity check equations for c1 apply to c2. thus, to recover information in c2, we may use the same parity check equations we would in c1, which implies c2 is at least a (n,k2, t,m,τ) batch code. we now introduce results for a (u | u + v)-code construction. theorem 4.6. let n,k1,k2 ∈ n such that n ≥ k1 ≥ k2, and let c be a [n,k1 + k2] linear code. then let c⊥ be a (u | u + v)-code construction of c⊥1 and c⊥2 , where • c1⊥ is an [n,n−k1] linear code and c2⊥ is an [n,n−k2] linear code, and • c⊥2 ⊆c⊥1 . if c2 is a (n,k,t,mτ) batch code, then c is a (2n,k1 + k2, t,m,τ) batch code. proof. the first two parameters of c follow from the definition of a (u | u + v) construction. let c be constructed as described, and let c2 be an (n,k,t,m,τ) batch code. then consider any t-tuple of indices i1, . . . , it ∈ [2n] and let i′j = ij if ij ∈ [n] and i ′ j = ij −n otherwise. then we know that i ′ 1, . . . , i ′ t ∈ [n], and so by the batch properties of c2 there exist t disjoint recovery sets for the entries in these indices, the union of which consists of at most τ entries in each of the m buckets. if for all i ∈ {n + 1, . . . , 2n}, we place i in the same bucket as i−n, then this results in m buckets for c. if we consider the recovery sets from above, if ri′ j is the recovery set for i′j, and ij ∈ [n], then we claim that r′ i′ j = ri′ j is a recovery set for ij in c. this is because the recovery set comes from a vector 158 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 v ∈ c⊥2 , and by construction, since c⊥2 ⊆ c⊥1 , we have that (v|0), (0|v) ∈ c⊥. likewise, if ij /∈ [n], then by using (0|v), we find that r′ i′ j = {i + n|i ∈ ri′ j } is a recovery set for ij. since the original ri′ j are all disjoint, so are the r′ i′ j , and so we have t disjoint recovery sets, the union of which consists of at most τ elements from each of m buckets, and so c is a (2n,k1 + k2, t,m,τ) batch code. because binary reed-muller codes have parity check matrices that satisfy the above properties, we turn to that family of codes. 4.1. locality and availability properties of rmq(1,µ) reed-muller codes for which ρ = 1 are known as first order reed-muller codes. we look at the properties using the polynomial evaluation definition of reed-muller codes. we begin with a result in the q-ary case. theorem 4.7. let fµq = {pi | 1 ≤ i ≤ 2µ = n} be the set of evaluation points for rmq(1,µ). then (λ1, . . . ,λn) ∈rmq(1,µ)⊥ if and only if n∑ i=1 λipi = 0 and n∑ i=1 λi = 0. (3) proof. first, if (λ1, . . . ,λn) is in the dual code, then by definition, n∑ i=1 λif(pi) = 0 (4) for every polynomial f ∈ fq[x1, . . . ,xµ]1. in particular, note that for any 1 ≤ k ≤ µ, from fk(x1, . . . ,xµ) = xk, we have n∑ i=1 λifk(pi) = n∑ i=1 λipi,k = 0, where pi,k is the kth entry of point pi. we may gather these equations together for 1 ≤ k ≤ µ to write the linear combination n∑ i=1 λipi = 0. we then consider f0 = 1, and equation (4) becomes ∑n i=1 λi = 0, and so we have this direction. for the other direction, assume that n∑ i=1 λipi = 0 and n∑ i=1 λi = 0. then in particular, for any 1 ≤ k ≤ µ, we have ∑n i=1 λipi,k = 0, and so for any f ∈ fq[x1, . . . ,xµ] 1, by linearity we have that n∑ i=1 λif(pi) = n∑ i=1 λi [ f0 + µ∑ k=1 fkpi,k ] = f0 n∑ i=1 λi + µ∑ k=1 fk n∑ i=1 λipi,k = 0, and thus (λ1, . . . ,λµ) ∈rm(1,µ)⊥. 159 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 from theorem 4.7 we obtain the following corollaries: corollary 4.8. the minimum distance of rmq(1,µ)⊥ is 4 if q = 2 and 3 otherwise. proof. let q = 2 and suppose by way of contradiction that the minimum weight is 2. then there exist two distinct points that sum to zero. this is not possible, and thus the minimum weight must be greater than 2. note that the only choice of λi is 1, and thus the sum is 0 if and only if there are an even number of points. therefore, the weight of the codewords is a multiple of 2, and thus the minimum weight is not 3. the following points are in p (for µ ≥ 2): p0 = (0, 0, 0, . . . , 0) t ,p1 = (1, 0, 0, . . . , 0) t ,p2 = (0, 1, 0, . . . , 0) t , and p3 = p1 + p2. these points satisfy the conditions, and thus the minimum distance for characteristic 2 is 4. if q 6= 2, let p1 = (0, 0, 0, 0, . . . , 0)t ,p2 = (1, 0, 0, 0, . . . , 0)t ,p3 = (−a, 0, 0, 0, . . . , 0)t ∈ fµq and the entries of λ be −(a + 1),a, and 1 corresponding to the positions of p1,p2, and p3, respectively, and 0 otherwise. then, if a 6= −1, 0, λ satisfies equations (3). suppose there exists a λ ∈rmq(1,µ)⊥ with weight 2. then we have two distinct points pi,pj ∈ fµq and λi,λj ∈ fq such that λj = −λi and λk = 0 for all k 6= i,j. our two conditions imply: λipi −λipj = 0 =⇒ pi = pj, a contradiction to the two points being distinct. corollary 4.9. when q = 2, every codeword in rm(ρ,µ) satisfies equation 3 for ρ ≤ µ− 2. proof. the dual code of rm(1,µ) is rm(µ− 2,µ) and rm(ρ1,µ) ⊂ rm(ρ2,µ) if ρ1 < ρ2. thus, any codeword in rm(ρ,µ) is also in rm(µ− 2,µ). therefore, theorem 4.7 implies our claim. theorem 4.10. let q 6= 2. then rmq(1,µ) has locality 2 and availability δ = ⌊ qµ−1 2 ⌋ . proof. let pa ∈ fµq be an evaluation point. then consider any α ∈ fq such that α 6= 0,−1. we have that 1 + α + (−α−1) = 0, and will find corresponding points to use in the reconstruction of pa. for any choice of pb ∈ fµq , pb 6= pa, take pc = (α + 1) −1(pa + αpb). upon rearrangement, we have that pa + αpb + (−α− 1)pc = 0. furthermore, we find that pc 6= pa,pb. if pc = pa, then our equation becomes pa + αpb + (−α− 1)pa = 0, which simplifies to αpb −αpa = 0, which would contradict pb 6= pa. likewise, pc = pb would imply pa + αpb + (−α − 1)pb = 0, which becomes pa −pb = 0, another contradiction. a simple counting argument tells us that there are ⌊ qµ−1 2 ⌋ choices of pairs pb,pc for pa, and each of these corresponds to a unique λ ∈ rmq(1,µ)⊥ of weight 3 that can be used to recover ca, the supports of which intersect only in {a}. thus, the locality is 2 and the availability is ⌊ qµ−1 2 ⌋ . as proven in [1] the minimum-weight codewords are generators of a reed-muller code rmps(µ,ρ) where p is a prime number and 0 ≤ ρ ≤ µ(ps − 1) if and only if either m = 1 or µ = 1 or ρ < p or ρ > (µ− 1)(ps − 1) + ps−1 − 2. together with lemma 2.10 it follows that these reed-muller codes have all symbol availability. thus, in the following theorems, we only consider the availability of p1 as this implies all symbol availability. theorem 4.11. rm(1,µ) has availability δ = 2 µ−1 3 when µ is even. 160 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 proof. an inductive argument on µ satisfies this claim. we look at the sum of evaluation points to prove our claim. we are looking for 2 µ−1 3 disjoint sets of three points of fµq that sum to (0, . . . , 0)t ∈ fµq . it is easy to verify the claim for µ = 2 since there is only one equation for which this is true (0, 1)t + (1, 0)t + (1, 1)t = (0, 0)t . now assume the claim is true for µ = 2k, we show that it is true also for µ = 2k + 2. we have 2 2k−1 3 disjoint sets of three points that all sum to p̄1 = (0, . . . , 0)t ∈ f2kq . let p1 = (0, . . . , 0)t ∈ f2k+2q . for any choice of set of points {s1,s2,s3}⊂ f2kq that sum to p̄1 in f2kq it holds that (st1 |0, 0) t + (st2 |0, 0) t + (st3 |0, 0) t = p1 (st1 |1, 0) t + (st2 |0, 1) t + (st3 |1, 1) t = p1 (st1 |0, 1) t + (st2 |1, 1) t + (st3 |1, 0) t = p1 (st1 |1, 1) t + (st2 |1, 0) t + (st3 |0, 1) t = p1. (5) additionally it also holds that (p̄t1 |1, 1) t + (p̄t1 |1, 0) t + (p̄t1 |0, 1) t = pt1 . (6) the four equations in (5) all use distinct sets of points because s1,s2, and s3 are distinct. now, there are 2 2k−1 3 sets of distinct points like s1,s2, and s3. thus, we have a total of 4 · 22k − 1 3 + 1 = 22k+2 − 1 3 . note that the extra one comes from equation (6). also note that our total is an integer as 22k+2 ≡ 4k+1 ≡ 1k+1 ≡ 1 (mod 3). because of this, every single coordinate is used, and thus we have maximal availability. theorem 4.12. rm(1,µ) has availability δ at least 2 µ−4 4 when µ is odd. proof. we prove this by induction on µ. for µ = 3, let s1 = (1, 0, 0)t , s2 = (0, 1, 0)t and s3 = s1 + s2, then s1 + s2 + s3 = p̄1 = (0, 0, 0) t . no combination of the four remaining points of f32 sum to p̄1. here we have availability 1 = 23−4 4 , and so we have our base case. now assume that, for µ = 2k + 1, where k ≥ 1, we have that the availability of rm(1,µ) is at least 2µ−4 4 , and there are at least 3 points that are not used in any recovery set for p1. let s1,s2, and s3 be any three points in a recovery set of p1 ∈ f µ 2 . then for µ + 2, the disjoint sets of three points that sum to p̃1 = (0, . . . , 0) ∈ f µ+5 2 can be defined by the equations: (st1 |0, 0) t + (st2 |0, 0) t + (st3 |0, 0) t = p̃1 (st1 |1, 0) t + (st2 |0, 1) t + (st3 |1, 1) t = p̃1 (st1 |0, 1) t + (st2 |1, 1) t + (st3 |1, 0) t = p̃1 (st1 |1, 1) t + (st2 |1, 0) t + (st3 |0, 1) t = p̃1 161 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 this results in at least 4 ( 2µ−4 4 ) = 2µ−4 possible recovery sets. however, we also have points t1,t2, and t3 that are not used in f µ 2 , and so we may also define the following equations: (p̄t1 |1, 0) t + (tt1 |0, 1) t + (tt1 |1, 1) t = p̃1 (p̄t1 |0, 1) t + (tt2 |1, 1) t + (tt2 |1, 0) t = p̃1 (p̄t1 |1, 1) t + (tt3 |1, 0) t + (tt3 |0, 1) t = p̃1 adding these, we have availability of at least 2µ − 4 + 3 = 2µ − 1 = 2 µ+2−4 4 , and so our property holds for µ + 2 as well. we also note that 3(2µ −1) + 3 = 3 ·2µ < 4 ·2µ = 2µ+2, and so there are at least 3 unused points. thus, by induction, we have that for every k ≥ 1, when µ = 2k + 1, the availability of rm(1,µ) is at least 2 µ−4 4 thus we have achieved a lower bound on δ. note, however, that we have not shown that this is necessarily an optimal construction. we now study the batch properties of reed-muller codes. 4.2. batch properties of rm(1,µ) theorem 4.13. the linear code rm(1, 4) is a (16, 5, 4,m,τ) batch code for any m,τ ∈ n such that mτ = 10. proof. first, note that the dual code of rm(1, 4) is rm(2, 4), which informs us how to reconstruct elements of the codewords. the generator matrix for rm(1, 4) can be recursively constructed as follows: g1,4 = ( g1,3 g1,3 0 g0,3 ) it can be verified that any query of 4 coordinates of a codeword in rm(1, 4) is possible with the following partition into buckets: {1}, {2}, {3}, {4}, {5, 6}, {7, 8}, {9, 11}, {10, 12}, {13, 16}, {14, 15}. in the above case, m = 10 and τ = 1. by lemma 2.3, this holds for any m,τ ∈ n such that mτ = 10. we now show how to extend this construction to rm(1,µ) for any µ ≥ 4. theorem 4.14. any first order reed-muller code, rm(1,µ), with µ ≥ 4, has batch properties (n,k, 4,m,τ) for any m,τ ∈ n such that mτ = 10. proof. we will proceed by induction. first, we have just shown this for the base case where µ = 4. now, assume that for some µ − 1 ≥ 4, we have that rm(1,µ − 1) has batch properties (n,k, 4,m,τ). recall that the dual code of c = rm(1,µ) is c⊥ = rm(µ − 2,µ). then as reed-muller codes follow the (u | u + v)-construction, c⊥ is the (u | u + v)-code construction of c⊥1 = rm(µ − 2,µ − 1) and c⊥2 = rm(µ−3,µ−1). since rm(µ−3,µ−1) ⊆rm(µ−2,µ−1), we have c⊥2 ⊆c⊥1 . this means we may apply theorem 4.6. since c2 = rm(1,µ− 1), we know that c is also a (n,k, 4,m,τ) batch code. by induction, this is now true for any µ ≥ 4. since the locality of these codes is r = 3, for t = 4, we have mτ = 10 = (4−1) ·3 + 1 = (t−1)r + 1, and thus we have optimal batch properties. we now extend this even further for most rm(ρ,µ). 162 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 4.3. batch properties of rm(ρ,µ) we begin with a preliminary result that uses the recursive construction of reed-muller binary codes. lemma 4.15. let a ∈rm(ρ− 1,µ− 2). then (a|a|0|0), (a|0|a|0), (a|0|0|a), (0|a|a|0), (0|a|0|a), (0|0|a|a) ∈rm(ρ,µ), where 0 ∈ fµ2 . proof. let g = ( gρ,µ−1 gρ,µ−1 0 gρ−1,µ−1 ) be the generator of rm(ρ,µ). recursively, we obtain that g =   gρ,µ−2 gρ,µ−2 gρ,µ−2 gρ,µ−2 0 gρ−1,µ−2 0 gρ−1,µ−2 0 0 gρ−1,µ−2 gρ−1,µ−2 0 0 0 gρ−2,µ−2   (7) from the second and third block rows in matrix (7), we see that for any a ∈ rm(ρ − 1,µ − 2), the second row implies (0|a|0|a) ∈ rm(ρ,µ), and the third row implies (0|0|a|a) ∈ rm(ρ,µ). note that our code is linear, and thus (0|a|0|a) + (0|0|a|a) = (0|a|a|0) ∈ rm(ρ,µ). finally, note that since rm(ρ − 1,µ − 2) ⊆ rm(ρ,µ − 2), the first row implies (a|a|a|a) ∈ rm(ρ,µ), and so combining this with the previous vectors, we find that (a|a|0|0), (a|0|a|0), (a|0|0|a) ∈rm(ρ,µ). we now show the batch properties of rm(ρ,µ) for ρ ≥ 1 and µ ≥ 2ρ + 2. theorem 4.16. let ρ ≥ 1 and µ ∈ {2ρ + 2, 2ρ + 3}. then, for ρ′ ≤ ρ, rm(ρ′,µ) is a (n,k, 4,m,τ) linear batch code for any m,τ ∈ n such that mτ = 10 · 22ρ−2. proof. we focus on the case where µ = 2ρ + 2 as the case µ = 2ρ + 3 proceeds with similar steps. if rm(ρ, 2ρ + 2) is a (n,k, 4,m,τ) linear batch code for any m,τ ∈ n such that mτ = 10 · 22ρ−2, then it follows from theorem 4.5 that for any ρ′ ≤ ρ, the code rm(ρ′, 2ρ + 2) is a (n,k, 4,m,τ) batch code for any m,τ ∈ n such that mτ = 10 · 22ρ−2. thus, we need only prove that this property holds for rm(ρ, 2ρ + 2). we proceed by induction on ρ. note that in section 4.2, the claim is true for ρ = 1, the base cases with µ = 4, 5. assume that ρ ≥ 1. we show that the properties hold for rm(ρ + 1, 2(ρ + 1) + 2) = rm(ρ + 1, 2ρ + 4), assuming that rm(ρ, 2ρ + 2) is a (n,k, 4,m,τ) linear batch code for any m,τ ∈ n such that mτ = 10 · 22ρ−2. in particular, we may choose τ = 1 and have m = 10 · 22ρ−2 buckets. we now examine rm(ρ + 1, 2ρ + 4). the dual code of rm(ρ + 1, 2ρ + 4) is rm(ρ + 2, 2ρ + 4). by lemma 4.15, for any a ∈rm(ρ + 1, 2ρ + 2) = rm(ρ, 2ρ + 2)⊥, we have (a|a|0|0), (a|0|a|0), (a|0|0|a), (0|a|a|0), (0|a|0|a), (0|0|a|a) ∈rm(ρ + 2, 2ρ + 4). this provides a way to produce parity check equations for rm(ρ+1, 2ρ+4) from those for rm(ρ, 2ρ+1), which in turn provides a way to make recovery sets for the former from those for the latter, as each vector corresponds to a recovery set for every index in its support. each bucket b = {i1, . . . , i`} for rm(ρ, 2ρ + 2) can be made into 4 buckets for rm(ρ + 1, 2ρ + 4): b1 = b, b2 = b + n = {i1 + n,. . . , i` + n}, b3 = b + 2n, and b4 = b + 3n. this results in 4 · 10 · 22ρ−2 = 10 · 22ρ = 10 · 22(ρ+1)−2 buckets, and we must show that any set of 4 indices may be recovered by drawing at most 1 entry from each bucket. 163 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 consider any tuple of 4 indices i1, i2, i3, i4 ∈ [4n] = ⋃3 s=0([n] + sn) and let sk = ⌊ ik−1 n ⌋ for k ∈ [4]. then let i′k = ik − skn, so that i ′ 1, i ′ 2, i ′ 3, i ′ 4 ∈ [n]. by the induction hypothesis, we have recovery sets r′1,r ′ 2,r ′ 3,r ′ 4 ⊆ [n] for these indices in rm(ρ, 2ρ + 2). these recovery sets are sets such that i′k ∈ r ′ k for all k ∈ [4] and 1. (r′k \{i ′ k}) ∩ (rj \{i ′ j}) = ∅ for k,j ∈ [4] with k 6= j 2. ⋃4 k=1(r ′ k \{i ′ k}) consists of at most 1 index in each bucket each r′k is either {i ′ k} or the support of some vector a ∈ rm(ρ + 1, 2ρ + 2). if |r ′ k| = 1, then let rk = r ′ k + skn. otherwise, let rk = (r ′ k + skn)∪(r ′ k + s ′ kn). by lemma 4.15, we know that if s ′ k 6= sk, rk is the support of some vector in rm(ρ + 2, 2ρ + 4), and so this is a valid recovery set. we must now show that these recovery sets have the desired properties given the correct choice of s′k values. we now note that since indices are being recreated from d = |{s1,s2,s3,s4}| different quarters of [4n], we can take at least d of the recovery sets to be singletons. further, assume that we take as many recovery sets to be singletons as possible. in particular, this means that no recovery set will contain more than one index in each bucket. this is because the only way rk could contain two indices in a bucket would be if r′k did. since r ′ k is a proper recovery set, it could only contain two indices in one bucket if one of those indices was i′k. then that bucket is not used in any other recovery set, and so we could instead take r′k = {i ′ k}, as per our assumption. this leaves at most 4 − d recovery sets which are not singletons and require a subset in a second quarter. assume without loss of generality that these are r1, . . . ,rd−4. then we may let s′1, . . . ,s ′ d−4 be the elements of {0, 1, 2, 3}\{s1,s2,s3,s4}. since these are distinct, the only way some rk \{ik} and rj \ {ij} could have a nonempty intersection would be if condition 1 was violated. thus, condition 1 also holds for the rk. we have already covered the fact that none of the r′k + s ′ kn will not contain more than one index in each bucket, and since these are in separate quarters, the only way ⋃4 k=1(rk \{ik}) would contain more than one index in a bucket would be if some elements being recovered in the same quarter had (r′k \{i ′ k}) ∪ (r ′ j \{ij}) consisting of more than 1 index in some bucket. this would violate condition 2, and so we know that the rks also satisfy 2. thus, this code is a (4n,k′, 4, 10·22(ρ+1)−2, 1) batch code, and by lemma 2.3, we know that rm(ρ+ 1, 2(ρ + 1) + 2) is a (4n,k′, 4,m,τ) batch code for any m,τ ∈ n such that mτ = 10 · 22(ρ+1)−2. this completes the induction step, and so for any ρ ≥ 1, rm(ρ, 2ρ + 2) is a (4n,k′, 4,m,τ) batch code for any m,τ ∈ n such that mτ = 10 · 22ρ−2. 5. conclusion this work focuses on batch properties of binary hamming and reed-muller code. the high locality of binary hamming codes implies their availability to be at most 1. binary hamming codes can be viewed as linear batch codes retrieving queries of at most 2 indices, the trivial case. nonetheless, we prove that for t = 2, binary hamming codes are actually optimal (2s−1, 2s−1−s, 2,m,τ) batch codes for m,τ ∈ n such that mτ = 2s−1. we turn to binary reed-muller codes for optimal batch codes that allow larger queries, meaning t-tuples with t larger than 2. this research direction is motivated by the large availability of first order reed-muller codes as showed in the paper. we prove the optimality of first order reed-muller codes for t = 4. finally we generalize our study to reed-muller codes rm(ρ,µ) which have order less than half their length by proving that they have batch properties (2µ,k, 4,m,τ) such that mτ = 10·22ρ−2 for rm(ρ′,µ) where µ ∈{2ρ + 2, 2ρ + 3} and ρ′ ≤ ρ. acknowledgment: the authors are grateful to clemson university for hosting the reu at which this work was completed. the reu was made possible by an nsf research training group (rtg) grant 164 t. baumbaugh et al. / j. algebra comb. discrete appl. 5(3) (2018) 153–165 (dms #1547399) promoting coding theory, cryptography, and number theory at clemson. references [1] e. f. assmus, j. d. key, designs and their codes, volume 103 of cambridge tracts in mathematics, cambridge university press, cambridge, 1992. 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[13] e. k. thomas, v. skachek, constructions and bounds for batch codes with small parameters, in coding theory and applications, springer, cham, (2017) 283–295. 165 http://dx.doi.org/10.1017/cbo9781316529836 http://dx.doi.org/10.1017/cbo9781316529836 http://dx.doi.org/10.3934/amc.2012.6.165 http://dx.doi.org/10.3934/amc.2012.6.165 https://mathscinet.ams.org/mathscinet-getitem?mr=2856861 https://mathscinet.ams.org/mathscinet-getitem?mr=2856861 https://doi.org/10.1109/jproc.2010.2096170 https://doi.org/10.1109/jproc.2010.2096170 http://dx.doi.org/10.1109/tit.2016.2605119 http://dx.doi.org/10.1109/tit.2016.2605119 http://dx.doi.org/10.1145/1007352.1007396 http://dx.doi.org/10.1145/1007352.1007396 https://doi.org/10.1007/978-3-319-17296-5_26 https://doi.org/10.1007/978-3-319-17296-5_26 https://mathscinet.ams.org/mathscinet-getitem?mr=465509 https://mathscinet.ams.org/mathscinet-getitem?mr=465509 http://dx.doi.org/10.3934/amc.2009.3.13 http://dx.doi.org/10.3934/amc.2009.3.13 https://arxiv.org/abs/1806.00592 https://arxiv.org/abs/1806.00592 http://dx.doi.org/10.1007/s10623-014-0007-9 http://dx.doi.org/10.1007/s10623-014-0007-9 https://doi.org/10.1007/978-3-319-70293-3_16 https://doi.org/10.1007/978-3-319-70293-3_16 https://link.springer.com/chapter/10.1007/978-3-319-66278-7_24 https://link.springer.com/chapter/10.1007/978-3-319-66278-7_24 introduction background hamming codes reed-muller codes conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.508968 j. algebra comb. discrete appl. 6(1) • 13–20 received: 21 november 2017 accepted: 11 december 2018 journal of algebra combinatorics discrete structures and applications new linear codes over gf(3), gf(11), and gf(13)∗ research article nuh aydin, derek foret abstract: explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. cyclic codes and their various generalizations, such as quasi-twisted (qt) codes, are known to contain many codes with best known parameters. despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets f3, f11, and f13 with better parameters. a total of 38 new linear codes over the three alphabets are presented. 2010 msc: 94b15, 94b60 keywords: best known codes, constacyclic codes, quasi-cyclic codes, quasi-twisted codes 1. introduction let fq denote the finite field with q elements (which is also denoted by gf(q)). a linear code c of length n over fq is a vector subspace of fnq . over a given finite field fq, a linear code has three fundamental parameters: the length n, the dimension k, and the minimum (hamming) distance d. a code with these parameters is denoted as an [n,k,d]q-code. one of the main problems of coding theory is a discrete optimization problem: determine the optimal values of these parameters and construct codes whose parameters attain the optimal values. over a given finite field fq we can fix two of the parameters, and look for the optimal value of the remaining one. for example, dq[n,k] stands for the largest minimum distance of a linear code of length n and dimension k over fq. this optimization problem is very difficult. in general, it is only solved for the cases where either k or n − k is small. there is a database of best known linear codes [14] over small finite fields which is a major source of reference for coding theory researchers. while there exist theoretical upper bounds on the value of d for a given n and k, in general there do not exist explicit constructions for linear codes attaining these parameters. a code is called “good" if it has best known parameters; it is called “new" or “record-breaking" if its parameters are better than ∗ this research was partially supported by kenyon summer science scholars program. nuh aydin (corresponding author), derek foret; department of mathematics, kenyon college, united states (email: aydinn@kenyon.edu, foretd@kenyon.edu). 13 https://orcid.org/0000-0002-5618-2427 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 the currently best known codes; and called “optimal" if its parameters attain the equality in a theoretical bound. the good codes are also referred to as the best known linear codes (bklcs). the database [14] of bklc’s includes the parameters of best known codes together with their constructions over finite fields up to size 9. computer algebra system magma also contains a similar database [8]. more recently, tables of best known codes over gf(11) and gf(13) have been published in [10] and [9]. the majority of the bklcs do not attain the theoretical upper bounds on minimum distance. therefore, it is still possible to improve the parameters of bklcs. it should be noted however that it may not always be possible to attain the theoretical upper bound on the minimum distance for a given length and dimension. computers are often used in searching for codes with best parameters but there is an inherent difficulty: computing the minimum distance of a linear code is computationally intractable (np-hard)[17]. since it is not possible to conduct exhaustive searches for linear codes if the dimension is large, researchers often focus on promising subclasses of linear codes with rich mathematical structures. the class of cyclic codes and its various generalizations, such as constacyclic codes, quasi-cyclic (qc) codes, and quasitwisted (qt) codes, lend themselves well to the problem of finding codes with good parameters. many record-breaking codes have been obtained in the class of qc codes in the last few decades ([1–3, 5–7, 11– 13, 15, 16]). these are the codes we consider in this paper. the search algorithm introduced in [6] has been highly effective and used in several subsequent works (e.g., [1–3, 5, 7, 12, 13]). we begin by reviewing the structure of constacyclic and qt codes and then describe how we implemented an improved version of the search algorithm that was used in previous works to search for new linear codes. we conclude with a list of new record breaking codes over the fields f3, f11, and f13 with generators and other necessary information to construct them. 2. constacyclic search method a linear code c over fq that is closed under a constacyclic shift is called a constacyclic code, that is, if (c0,c1, . . . ,cn−1) ∈ c, then (a · cn−1,c0,c1, . . . ,cn−2) ∈ c as well. here, a ∈ f∗q is a non-zero field element, and it is called the shift constant. the important special case of cyclic codes are obtained when we take a = 1. it is well known that constacyclic codes are ideals in fq[x]/〈xn −a〉 if we represent their elements as polynomials, and that each constacyclic code c contains at least one polynomial that generates it as an ideal. while generating polynomials are not unique, the monic generator polynomial of least degree is. this is what we call the generator polynomial g(x) of c. it is also well known that g(x) is the generator polynomial for a constacyclic code of length n with shift constant a if and only if g(x) divides xn −a, i.e. xn −a = g(x)h(x) for some h(x) ∈ fq[x]/〈xn −a〉. hence, there is a one-to-one correspondence between the divisors of xn −a and the constacyclic codes of length n with shift constant a. the polynomial h(x) is called the check polynomial of c. a constacyclic code is therefore uniquely determined by either its generator polynomial or its check polynomial. from the generator polynomial g(x) = g0+g1x+· · ·+gmxm of a constacyclic code c we obtain its generator matrix as a circulant (twistulant) matrix  g0 g1 · · · gm 0 · · · 0 0 g0 g1 · · · gm 0 · · · 0 . . . 0 . . . 0 g0 g1 · · · gm   where each row is the cyclic (constacyclic) shift of the row above. using the computational algebra system magma, we are able to factor xn − a and compute the minimum distance d of all constacyclic codes generated by the divisors of xn −a. we follow the method in [5] with a few variations to be more comprehensive in the constacyclic search, and the qt search that follows. 14 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 we first consider cyclic codes with length-dimension pairs for which lower bounds are available (found in either [14], [10], or [9]). we are then able to reduce the number of constacyclic codes to be examined by using the following theorems. theorem 2.1. [6] let a ∈ fq where a = ωi for some primitive element ω in fq. then, a has an n-th root in fq if and only if gcd(n,q −1)|i. theorem 2.2. [6] if fq contains an n-th root of a, then a constacyclic code with length n and shift constant a is equivalent to a cyclic code of length n. the above two theorems immediately lead to the following corollary: corollary 2.3. let ω be a primitive element of fq and let a = ωi for some positive integer i. if gcd(n,q−1)|i then a constacyclic code with length n and shift constant a is equivalent to a cyclic code of length n. therefore, once we examine all cyclic codes of length n over fq, we do not need to examine constacyclic codes of length n for shift constant a = ωi if gcd(n,q − 1)|i, as the resulting constacyclic code is equivalent to a cyclic code that we have already computed and equivalent codes have the same parameters. of the remaining constacyclic codes with shift constant a 6= 1, some of them turn out to be equivalent to each other. the next theorem specifies which shift constants still need to be examined once all cyclic codes are computed. theorem 2.4. [6] let a,b have the same order in the multiplicative group f∗q and let ca be the constacyclic code [n,k,d]q with shift a. then there exists a constacyclic code cb that is equivalent to ca. given the theorem above, we classify the elements of f∗q according to their multiplicative orders and examine only one constant from each class. the following table combines the results of corollary 2.3 and theorem 2.4 to show which constacyclic codes of length n and shift constant a still need to be checked after computing all cyclic codes. any constacyclic code with parameters not found in the table would be equivalent to a cyclic or a constacyclic code. table 1. constacyclic codes searched q a 6= 0,1 n 3 2 all n = 2m 11 2 all n = 2m or n = 5m 3 all n = 5m 10 all n = 2m 13 2 all n = 2m or n = 3m 3 all n = 3m 4 all n = 3m or n = 4m 5 all n = 2m 12 all n = 4m two parts of our search method that have the largest impact on the running time are the number factors of xn −a considered and the time to compute the minimum distance of each code generated by a divisor of xn −a. when xn −a has many factors, we only consider a predetermined amount of factors of xn − a for each degree. we also allocate a maximum computation time for each minimum distance calculation and if the minimum distance is not computed within the time limit, then the computation is aborted. after these initial searches, our results are compiled into a master list which contains the best found minimum distance for each length and dimension pair, and the list is compared with the databases to identify any record-breaking codes. when there are multiple codes with the same highest minimum distance for a given length and dimension [n,k], we just keep the generator polynomial of one of them. this master list also forms the basis of the next step in searching for new qt codes. 15 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 3. quasi-twisted search method quasi-twisted (qt) codes are a generalization of constacyclic codes. definition 3.1. let πa denote the constacyclic shift operator. a linear code c is said to be `-quasitwisted (`-qt) if it is invariant under π`a, where ` is some positive integer which divides the length of the code. that is, if (c0,c1, . . . ,cn−1) ∈ c, then (acn−`, . . . ,acn−1,c0,c1, . . . ,cn−`−1) ∈ c as well. the class of qt codes contains several other classes as special cases. when a = 1, we obtain the class of quasi-cyclic (qc) codes, when ` = 1 the class of constacyclic codes, and when a = ` = 1, the class of cyclic codes. algebraically, a qt code of length n = ml is an r-submodule of r` where r = fq[x]/〈xm −a〉. a generator matrix of a qt code can be obtained in the form  g1,1 g1,2 . . . g1,` g2,1 g2,2 . . . g2,` ... ... ... ... gr,1 gr,2 . . . gr,`   where each gi,j is a twistulant matrix. such a code is called an r-generator qt code. most of the work on qt codes is focused on the 1-generator case. we also consider 1-generator qt codes in the paper. a method of constructing qt codes from constacyclic codes is based on the following theorem [6]. theorem 3.2. [3, 6] let c be a 1-generator qt code of length n = m` over fq with a generator of the form (g(x)f1(x),g(x)f2(x), . . . ,g(x)f`(x)) where xm −a = g(x)h(x) and (h(x),fi(x)) = 1 for all i = 1, . . . ,`. then dim(c) = m− deg(g(x)), and d(c) ≥ ` ·d where d is the minimum distance of the constacyclic code generated by g(x). search methods based on this theorem have been implemented in a number of articles and many record-breaking codes have been obtained [1–3, 5, 7, 12, 13]. we continue this line of work by refining the search method even further. we build up qt codes from the best constacyclic codes obtained in the first stage. to do so, we take the generator polynomial g(x) of one of the [m,k,d]q constacyclic codes and construct a qt code of index ` with generating polynomial of the form (g(x),f1(x)g(x),f2(x)g(x), . . . ,f`(x)g(x)) so that n = m`, the length of the qt code, does not exceed the maximum lengths of codes found in the databases. polynomials fi(x) that are relatively prime to the check polynomial h(x) = (xm −a)/g(x)are randomly chosen from the elements of fq[x]/〈xm −a〉 with deg(fi(x)) < deg(h(x)). the generator polynomial of the best constacyclic code for every length/dimension pair that our constacyclic search found is used for as many values of ` as possible. unlike in our constacyclic search, where one program examines numerous factors of xn − a, our qt search programs examine only one factor of xm −a at a time; that is, the given generator polynomial g(x). as the dimension of an `-qt code in this method is m − k where k is the degree of g(x), each program only considers codes of one length-dimension pair at a time. programs are run as long as possible and aborted after a certain period of time if they had not been completed. we used magma software in the implementation of the search algorithms [8]. 16 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 4. new codes 4.1. new constacyclic codes the following are the parameters of new record breaking constacyclic codes. for each code we give its shift constant a, the parity check polynomial h, and w, the improvement on the minimum distance. the parity check polynomial is given in place of the generator polynomial g(x) due to considerations of space. the coefficients of the parity check polynomial are listed in ascending order. thus, the parity check polynomial for the first code in the table is h(x) = 10 + x + 7x2 + 9x3 + 2x4 + 4x5 + 10x6 + x7. table 2. record-breaking gf(11) and gf(13) constacyclic codes [n,k,d]q a h w [111,7,93]11 1 [10,1,7,9,2,4,10,1] 4 [133,7,109]11 1 [10,9,4,6,5,7,2,1] 1 [157,6,136]13 1 [1,0,7,12,7,0,1] 3 [170,5,150]13 1 [1,2,5,5,2,1] 1 [183,6,159]13 3 [9,6,5,10,11,2,1] 2 [244,6,213]13 1 [5,7,12,5,12,2,1] 3 4.2. new quasi-twisted codes the following are the parameters of new record breaking qt codes together with their generators. the shift constants a of the original constacyclic codes along with index ` are also given. each `-qt code is generated by a matrix of the form [g1,g2, . . . ,g`] where g1 is the generator matrix of a constacyclic code with generator polynomial g(x), and each one of the remaining gi is a twistulant matrix defined by g(x)fi(x) mod (x n −a), where fi(x) ∈ fq[x]/〈xn −a〉 is relatively prime with the check polynomial. to identify each new code, we have also given g(x) (or h(x) depending on space as above) as well as each fi(x). all of these codes have been included in the database [8]. we have special comments for 3 of the codes in the list. we found the code with parameters [150,22,67]3 in the summer of 2015. later we found a [150,22,68]3 code using a more comprehensive search strategy in the summer of 2018 [4]. for the code with parameters [105,28,37] over f3, an alternative construction that involves many steps is given in [14] 1. since our construction is more direct, hence preferable, we include this code in the list. the same is the case for the [108,19,48]3-code in the list. it is possible that additional new codes might be obtained from these by applying the standard constructions such as extending, shortening, and puncturing. 1 the database contains a code with these parameters with the date of 2016-02-17. the construction given for this code is indirect and involves many steps. obtaining this code directly as a qc code is preferable. moreover, we found this code in the summer of 2015 but it was not reported until the spring of 2016. 17 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 table 3. record-breaking qt codes over gf(3) and gf(11). [n,k,d]q α ` polynomials [22,7,14]11 1 2 g [1,7,6,7,1] f1 [2,7,4,1,3,10] [44,5,35]11 1 4 g [1,5,4,2,4,5,1] f1 [9,1,1,4,8] f2 [9,3,10,3,9] f3 [7,6,0,8,3] [105,7,84]11 1 3 h [7,0,2,7,9,2,10,1] f1 [10,0,2,1,9,10,4] f2 [4,9,2,9,1,7,1] [140,7,114]11 1 4 h [7,0,2,7,9,2,10,1] f1 [5,10,3,9,7,4,10] f2 [6,10,6,3,2,6,1] f3 [8,3,8,9,1,1,10] [180,7,149]11 1 3 h [6,9,9,9,4,6,2,1] f1 [4,9,8,5,0,5] f2 [8,5,1,4,9,6,6] [222,7,187]11 1 2 h [10,1,7,9,2,4,10,1] f1 [0,5,9,7,7,0,9] [54,17,21]3 1 3 g [11] f1 [02100011120022022] f2 [1120000001110211] [72,19,28]3 1 3 g [211211] f1 [2220112222001220221] f2 [022010100001221121] [72,23,25]3 1 3 g [11] f1 [01122221012112210211222] f2 [11100101010100221020221] [96,22,38]3 1 4 g [121] f1 [2000000000202101001122] f2 [222012220001011221112] f3 [121020102210122001022] [96,23,37]3 1 4 g [11] f1 [21120000122222112010021] f2 [21011202100201022220122] f3 [0012021200000220121121] [108,23,43]3 1 4 g [12021] f1 [20011120102211120210221] f2 [10100220002102121101201] f3 [21201210011020112222011] [60,21,21]3 1 2 g [1201221021] f1 [220221221121001220021] 18 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 [60,23,19]3 1 2 g [11122111] f1 [0020111110201210001122] [99,25,37]3 1 3 g [112020201] f1 [2211100112220202001001102] f2 [0222112121210222002101221] [99,26,36]3 1 3 g [21220011] f1 [21222102220010010101000122] f2 [101020102111000111001201] [70,18,28]3 1 2 g [211210212200020101] f1 [2211102102202212] [70,22,25]3 1 2 g [22012112222121] f1 [0212020110210011112001] ∗[150,22,67]3 1 5 g [200201001] f1 [201010011201220212022] f2 [2011010222200200112] f3 [2100200100212110012201] f4 [0201221221220110212101] ∗[105,28,37]3 1 3 g [20000001] f1 [022221201110020200110110102] f2 [1102100001000202101200221212] ∗[108,19,48]3 1 3 g [220121102102212011] f1 [0020100200212000022] f2 [110122020011221022] [111,19,50]3 1 3 g [1020220100010220201] f1 [2221211012111220022] f2 [1001122120001122121] [156,23,70]3 1 4 g [12112222110200211] f1 [11002122111110022021221] f2 [0110010122211021001021] f3 [01122011220011011020211] [80,21,31]3 1 2 g [22001100011012102021] f1 [12021022220212212121] [200,21,98]3 1 5 g [22001100011012102021] f1 [0200012220000112202] f2 [022200002211101112011] f3 [22002101002112110211] f4 [201000120020000002] [132,22,58]3 1 3 g [12200202120210211201011] f1 [0000112102010210210212] f2 [0112112020201110011222] [176,22,84]3 1 4 g [12200202120210211201011] f1 [2100210110120022211222] f2 [0001102221222001011022] f3 [0220001200100220201221] [104,25,41]3 1 2 g [1110022112022011221200212111] f1 [0010001100101201212222201] 19 n. aydin, d. foret / j. algebra comb. discrete appl. 6(1) (2019) 13–20 [104,28,37]3 1 2 g [2020210012121010221210211] f1 [1001022102211210110102110112] [112,22,48]3 1 2 g [20202120101202010212102101201200011] f1 [0212012112010121220002] [201,22,98]3 1 3 g [2201102112100222011112211222201110021221022011] f1 [002012112212211210222] f2 [0110100221011221001122] [140,26,58]3 1 2 g [122102210220022121110112021222210021022102211] f1 [021001020200200211002001] references [1] r. ackerman, n. aydin, new quinary linear codes from quasi–twisted codes and their duals, appl. math. lett. 24(4) (2011) 512–515. 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[17] a. vardy, the intractability of computing the minimum distance of a code, ieee trans. inform. theory 43(6) (1997) 1757–1766. 20 https://doi.org/10.1016/j.aml.2010.11.003 https://doi.org/10.1016/j.aml.2010.11.003 https://doi.org/10.1007/s00200-017-0327-x https://doi.org/10.1007/s00200-017-0327-x http://dx.doi.org/10.3934/amc.2017016 http://dx.doi.org/10.3934/amc.2017016 https://doi.org/10.1016/j.jfranklin.2013.11.019 https://doi.org/10.1016/j.jfranklin.2013.11.019 https://doi.org/10.1023/a:1011283523000 https://doi.org/10.1023/a:1011283523000 https://doi.org/10.1016/s0893-9659(02)00050-2 http://dx.doi.org/10.1006/jsco.1996.0125 http://dx.doi.org/10.1006/jsco.1996.0125 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.13069/jacodesmath.36947 https://doi.org/10.1080/02522667.2014.961788 https://doi.org/10.1080/02522667.2014.961788 https://doi.org/10.1109/18.333888 https://doi.org/10.1109/tit.2003.819337 https://doi.org/10.1109/tit.2003.819337 https://doi.org/10.1016/s0012-365x(03)00126-2 https://doi.org/10.1016/s0012-365x(03)00126-2 http://www.codetables.de/ http://www.codetables.de/ https://doi.org/10.1023/a:1018090707115 https://doi.org/10.1023/a:1018090707115 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.850694 https://doi.org/10.1109/18.641542 https://doi.org/10.1109/18.641542 introduction constacyclic search method quasi-twisted search method new codes references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.514339 j. algebra comb. discrete appl. 6(1) • 39–51 received: 1 april 2017 accepted: 9 december 2018 journal of algebra combinatorics discrete structures and applications codes over zp[u]/〈ur〉×zp[u]/〈us〉 research article ismail aydogdu abstract: in this paper we generalize z2z2[u]-linear codes to codes over zp[u]/〈ur〉× zp[u]/〈us〉 where p is a prime number and ur = 0 = us. we will call these family of codes as zp[ur, us]-linear codes which are actually special submodules. we determine the standard forms of the generator and parity-check matrices of these codes. furthermore, for the special case p = 2, we define a gray map to explore the binary images of z2[ur, us]-linear codes. finally, we study the structure of self-dual z2[u2, u3]-linear codes and present some examples. 2010 msc: 94b05, 94b60 keywords: linear codes, self-dual codes, z2z2[u]-linear codes, zp[ur, us]-linear codes 1. introduction linear codes are the most important family of codes in coding theory. because they have some advantages compared to the non-linear codes. for example, linear codes have an easier encoding and decoding process than non-linear arbitrary codes. in the beginning, studies on linear codes were mainly over fields, especially binary fields. later, codes over extension fields and, in general, finite fields were studied by many researchers. in 1994, a remarkable paper written by hammons et al.[8] brought a new direction to researches on coding theory. in this paper, they showed that some well known non-linear codes are actually binary images of some linear codes over the ring of integers modulo 4, i.e., z4. after this work a lot of research has been directed towards codes over rings. the structure of binary linear codes and quaternary linear codes have been studied in details for the last sixty years. however, in 2010, borges et al. have introduced a new class of error correcting codes, actually mixed alphabet codes, over z2 × z4 called z2z4-additive codes in [7]. these family of codes generalizes the class of binary linear codes and the class of quaternary linear codes. for positive integers α,β, a z2z4-additive code c is defined as a subgroup of zα2 × z β 4 with α + 2β = n. another important ring with four elements which is not isomorphic to z4 is the ring r = z2 + uz2 = {0, 1,u, 1 + u} where u2 = 0. there are many studies related to codes over the ring r = z2 + uz2 in the literature, the reader ismail aydogdu; department of mathematics, yildiz technical university, istanbul, turkey (email: iaydogdu@yildiz.edu.tr). 39 https://orcid.org/0000-0001-9308-4829 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 may see some of them in [1, 2, 6]. recently, aydogdu et al. have introduced z2z2[u]-linear codes in [3] where z2[u] = {0, 1,u, 1 + u} = r. even though the structures of these codes and the structures of z2z4-additive codes are similar z2z2[u]-linear codes have some advantages compared to z2z4-additive codes. one of these advantages is that, the gray (binary) images of z2z2[u]-linear codes are always linear, but this is not always the case for codes over zα2 × z β 4 . another advantage of working with such linear codes is that the factorization of polynomials in z2[x] is also valid since z2 is a subring of r (see [6]) and hensel’s lift is not necessary. the generalizations of z2z4-additive codes are considered in the literature. for instance, aydogdu and siap introduced z2z2s and zpr zps additive codes in [4] and [5] respectively. this paper attempts to study a further generalization. an important reason motivating us to study these families of additive codes is that they can be mapped to codes over finite fields via gray maps and moreover they offer some interesting algebraic structures. in this paper, we generalize z2z2[u]-linear codes to codes over zp[u]/〈ur〉×zp[u]/〈us〉 = zp[ur,us] where p is a prime number and ur = 0 = us. we also determine the structure of these codes by giving the standard forms of generator and parity-check matrices. finally, we present binary images of these codes for the special case p = 2 and illustrate some examples. throughout the paper we assume that r ≤ s. let rr = zp + uzp + u2zp + · · ·+ ur−1zp = zp[u]/〈ur〉 and rs = zp + uzp + u2zp + · · ·+ us−1zp = zp[u]/〈us〉 be the finite rings with ur = 0 = us. let us define the zp[u]-module zp[ur,us] = {(a,b) | a ∈rr and b ∈rs} . being inspired by the definition of z2z2[u]-linear codes, we give the following definition. definition 1.1. let c be the non-empty subset of zαp [u]/〈ur〉×zβp [u]/〈us〉. if c is a rr-submodule of zαp [u]/〈ur〉×zβp [u]/〈us〉 then c is called a zp[ur,us]-linear code. we understand from the definition of zp[ur,us]-linear codes that the first α coordinates of the zp[ur,us]-linear code c are elements from rr and the remaining β coordinates are elements from rs. also, it can be easily concluded that this code is isomorphic to an abelian group z kr−1 p × z2kr−2p × ··· × zrk0p × zsl0p × z (s−1)l1 p × ··· × z ls−1 p . considering all these parameters we will say c is of type (α,β; k0,k1, . . . ,kr−1; l0, l1, . . . , ls−1). further, the number of the codewords of the c is |c| = prk0p(r−1)k1 · · ·pkr−1 · · ·psl0p(s−1)l1 · · ·pls−1. definition 1.2. let c be a zp[ur,us]-linear code. let us define cα(respectively cβ) as the punctured code of c by deleting the coordinates outside α(respectively β). if c = cα ×cβ then c is called separable. we also note that for a z2z2[u]-linear code c of type (α,β; k0; k1,k2), the standard forms of the generator and the parity-check matrices are given by g =   ik0 a1 0 0 ut0 s ik1 a b1 + ub2 0 0 0 uik2 ud   , (1) h =   −a t 1 iα−k0 −ust 0 0 −tt 0 −(b1 + ub2)t + dtat −dt iβ−k1−k2 0 0 −uat uik2 0   (2) where a, a1, b1, b2, d, s and t are matrices over z2. 40 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 2. standard form of the generator matrices of zp[ur, us]-linear codes in this section of the paper, we give standard form of the generator matrix of a zp[ur,us]-linear code c. a generator matrix g for a linear code c is the matrix such that the rows are basis vectors of c. we can put this generator matrix in a special form by elementary row operations, and we say this matrix is the standard form of the generator matrix. generator matrices give us information about the structure of a linear code. if we know the standard form, we can easily read the type of a code and then find the number of elements. further, we say any two codes are permutation equivalent or only equivalent if one can be obtained from the other by permutation of their coordinates or (if necessary) changing the coordinates by their unit multiples. the below theorem determines the generator matrix of a zp[ur,us]-linear code c. theorem 2.1. let c be a zp[ur,us]-linear code then c is permutation equivalent to a zp[ur,us]-linear code which has the following standard generator matrix of the form. g = [ r a b s ] (3) where r =   ik0 r01 r02 r03 · · · r0,r−2 r0,r−1 r0,r 0 uik1 ur12 ur13 · · · ur1,r−2 ur1,r−1 ur1,r 0 0 u2ik2 u 2r23 · · · u2r2,r−2 u2r2,r−1 u2r2,r ... ... ... ... ... ... ... ... 0 0 0 0 · · · ur−2ikr−2 ur−2rr−2,r−1 ur−2rr−2,r 0 0 0 0 · · · 0 ur−1ikr−1 ur−1rr−1,r   , a =   0 0 · · · us−ra01 us−ra02 · · · us−ra0,r−2 us−ra0,r−1 us−ra0,r 0 0 · · · 0 us−r+1a12 · · · us−r+1a1,r−2 us−r+1a1,r−1 us−r+1a1,r 0 0 · · · 0 0 · · · us−r+2a2,r−2 us−r+2a2,r−1 us−r+2a2,r ... ... ... ... ... ... ... ... ... 0 0 · · · 0 0 · · · 0 us−2ar−2,r−1 us−2ar−2,r 0 0 · · · 0 0 · · · 0 0 us−1ar−1,r   , b =   0 b01 b02 b03 · · · b0,r−2 b0,r−1 b0,r 0 b11 b12 b13 · · · b1,r−2 b1,r−1 b1,r ... ... ... ... ... ... ... ... 0 bs−r−1,1 bs−r−1,2 bs−r−1,3 · · · bs−r−1,r−2 bs−r−1,r−1 bs−r−1,r 0 0 ubs−r,2 ubs−r,3 · · · ubs−r,r−2 ubs−r,r−1 ubs−r,r 0 0 0 u2bs−r+1,3 · · · u2bs−r+1,r−2 u2bs−r+1,r−1 u2bs−r+1,r ... ... ... ... ... ... ... ... 0 0 0 0 · · · 0 0 ur−1bs−2,r 0 0 0 0 · · · 0 0 0   , 41 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 s =   il0 s01 s02 s03 · · · s0,s−2 s0,s−1 s0,s 0 uil1 us12 us13 · · · us1,s−2 us1,s−1 us1,s 0 0 u2il2 u 2s23 · · · u2s2,s−2 u2s2,s−1 u2s2,s ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 · · · us−3ss−3,s−2 us−3ss−3,s−1 us−3ss−3,s 0 0 0 0 · · · us−2ils−2 us−2ss−2,s−1 us−2ss−2,s 0 0 0 0 · · · 0 us−1ils−1 us−1ss−1,s   . in this standard form of the generator matrix, rij’s are matrices over rr and aij’s are matrices over rs for 0 ≤ i < j ≤ r. and for 0 ≤ t < s− 1, 0 ≤ p < q ≤ s,btj’s are matrices over rr and spq’s are matrices over rs. further, ik’s and il’s are identity matrices with given sizes. proof. let c ⊆rαr ×rβs be a zp[ur,us]-linear code of type (α,β; k0,k1, . . . ,kr−1; l0, l1, . . . , ls−1). since the first α coordinates of c is a submodule of rr and the last β coordinates of c is a submodule of rs then from [9] we can write the generator matrix for c as follows:  r̄ 1 2 s̄   where r̄ =   ik0 r̄01 r̄02 r̄03 · · · r̄0,r−2 r̄0,r−1 r̄0,r 0 uik1 ur̄12 ur̄13 · · · ur̄1,r−2 ur̄1,r−1 ur̄1,r 0 0 u2ik2 u 2r̄23 · · · u2r̄2,r−2 u2r̄2,r−1 u2r̄2,r ... ... ... ... ... ... ... ... 0 0 0 0 · · · ur−2ikr−2 ur−2r̄r−2,r−1 ur−2r̄r−2,r 0 0 0 0 · · · 0 ur−1ikr−1 ur−1r̄r−1,r   and s̄ =   il0 s̄01 s̄02 s̄03 · · · s̄0,s−2 s̄0,s−1 s̄0,s 0 uil1 us̄12 us̄13 · · · us̄1,s−2 us̄1,s−1 us̄1,s 0 0 u2il2 u 2s̄23 · · · u2s̄2,s−2 u2s̄2,s−1 u2s̄2,s ... ... ... ... ... ... ... ... 0 0 0 0 · · · us−2ils−2 us−2s̄s−2,s−1 us−2s̄s−2,s 0 0 0 0 · · · 0 us−1ils−1 us−1s̄s−1,s   . now, we have to determine the forms of the matrices in 1 and 2. we will put codewords to 1 such that they do not change the form of the matrix r̄. so we have 42 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51   us−rā01 u s−rā02 · · · us−rā0,s+1 us−r+1ā11 u s−r+1ā12 · · · us−r+1ā1,s+1 r̄1 ... ... ... ... us−2ār−2,1 u s−2ār−2,2 · · · us−2ār−2,s+1 us−1ār−1,1 u s−1ār−1,2 · · · us−1ār−1,s+1 2 s̄1   . similarly, putting codewords to 2 by protecting the form of the matrix s̄ we get   us−rā01 u s−rā02 · · · us−rā0,s+1 us−r+1ā11 u s−r+1ā12 · · · us−r+1ā1,s+1 r̄2 ... ... ... ... us−2ār−2,1 u s−2ār−2,2 · · · us−2ār−2,s+1 us−1ār−1,1 u s−1ār−1,2 · · · us−1ār−1,s+1 b̄01 b̄02 · · · b̄0,r+1 b̄11 b̄12 · · · b̄1,r+1 ... ... ... ... b̄s−r−1,1 b̄s−r−1,2 · · · b̄s−r−1,r+1 ub̄s−r,1 ub̄s−r,2 · · · ub̄s−r,r+1 s̄2 u2b̄s−r+1,1 u 2b̄s−r+1,2 · · · u2b̄s−r+1,r+1 ... ... ... ... ur−1b̄s−2,1 u r−1b̄s−2,2 · · · ur−1b̄s−2,r+1 0 0 · · · 0   . finally, by applying necessary elementary row operations to above matrix, we have the standard form in (3). example 2.2. let c be a z2[u2,u3]-linear code with the following generator matrix  1 u 1 + u 1 + u + u2 0 u u 0 u u u + u2 u 1 + u u 1 1 + u2 1 u + u2 1 0 u u u + u2 0 1 + u u u u2 u2 u2   . (4) then, by applying elementary row operations we have the standard form of this generator matrix as,  1 0 0 0 0 0 0 u u 0 0 0 0 0 1 + u 1 0 0 0 0 0 0 1 0 0 0 0 0 0 u   . therefore, • c is of type (3, 3; 1, 1; 2, 1, 0). 43 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 • c has 22·1 · 21 · 23·2 · 22·1 · 20 = 211 = 2048 codewords. example 2.3. now, we consider any zp[u2,u3]linear code of type (α,β; k0,k1; l0, l1, l2). so, r = 2 and s = 3. then the generator matrix of c is permutation equivalent to a matrix of the form g =   ik0 r01 r02 0 0 ua01 ua02 0 uik1 ur12 0 0 0 u 2a12 0 b01 b02 il0 s01 s02 s03 0 0 ub12 0 uil1 us12 us13 0 0 0 0 0 u2il2 u 2s23   . (5) corollary 2.4. taking p = 2, r = 1 and s = 2 we have z2[u,u2]-linear code which is equal to a z2z2[u]linear code introduced in [3]. then if c is a z2[u,u2]-linear code of type (α,β; k0; l0, l1), its generator matrix is g =  ik0 r01 0 0 ua010 b01 il0 s01 s02 0 0 0 uil1 us12   (6) which is permutation equivalent to a matrix of the form (1). 3. duality on zp[ur, us]-linear codes and parity-check matrices we have well known concept of duality over finite fields and rings. in this part, we define an inner product for codes over rαr ×rβs and we determine the structure of the dual space of a zp[ur,us]-linear code c using this inner product. the inner product for any two vectors v,w ∈rαr ×rβs is defined by v ·w = us−r ( α∑ i=1 viwi ) + α+β∑ j=α+1 vjwj. moreover, we can easily define the dual code c⊥ of zp[ur,us]-linear code c in the following standard way: c⊥ = { w ∈rαr ×r β s | v ·w = 0 for all v ∈c } . it is also clear that c⊥ is an rr-submodule of rαr ×rβs . so, c⊥ is also a zp[ur,us]-linear code. we know that the dual space of any linear code is generated by a parity-check matrix of c or equivalently generator matrix of c⊥. to establish the parity-check matrix of a zp[ur,us]-linear code c we first give the following definitions. let k(r) and l(s) be the number of rows of the matrices r and s, respectively. for i = 0, 1, ..., r− 1(s− 1), let ki(r) (li(s)) denote the number of rows of r (s) that are divisible by ui but not by ui+1. then, k(r) = r−1∑ i=0 ki(r) and l(s) = s−1∑ i=0 li(s). therefore, we give the following theorem that determines the standard form of the parity-check matrix of a zp[ur,us]-linear code c. theorem 3.1. the parity-check matrix for a zp[ur,us]-linear code c of type (α,β; k0,k1, . . . ,kr−1; l0, l1, . . . , ls−1) with generator matrix in (3) is given by the following standard form matrix. h = [ r̄ + f m n s̄ + e ] 44 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 where r̄, f, m, n, s̄ and e are matrices of the following forms. r̄ =   r̄0,r r̄0,r−1 r̄0,r−2 · · · r̄0,3 r̄0,2 r̄0,1 iα−k(r) ur̄1,r ur̄1,r−1 ur̄1,r−2 · · · ur̄1,3 ur̄1,2 uikr−1(r) 0 ... ... ... ... ... ... ... ... ur−2r̄r−2,r u r−2r̄r−2,r−1 u r−2ik2(r) · · · 0 0 0 0 ur−1r̄r−1,r u r−1ik1(r) 0 · · · 0 0 0 0   , f =   f0,r−2 f0,r−3 · · · f0,2 f0,1 0 0 0 uf1,r−2 uf1,r−3 · · · uf1,2 0 0 0 0 ... ... ... ... ... ... ... ... ur−4fr−4,r−2 u r−4fr−4,r−3 · · · 0 0 0 0 0 ur−3fr−3,r−2 0 · · · 0 0 0 0 0 0 0 · · · 0 0 0 0 0 0 0 · · · 0 0 0 0 0   , m =   m0,s−1 m0,s−2 · · · m0,3 m0,2 m0,1 0 0 um1,s−1 um1,s−2 · · · um1,3 um1,2 0 0 0 ... ... ... ... ... ... ... ... ur−2mr−2,s−1 u r−2mr−2,s−2 · · · 0 0 0 0 0 ur−1mr−1,s−1 u r−1mr−1,s−2 · · · 0 0 0 0 0   , n =   n0,r n0,r−1 n0,r−2 · · · n0,3 n0,2 n0,1 0 un1,r un1,r−1 un1,r−2 · · · un1,3 un1,2 0 0 ... ... ... ... ... ... ... ... ur−1nr−1,r 0 0 · · · 0 0 0 0 0 0 0 · · · 0 0 0 0 ... ... ... ... ... ... ... ... 0 0 0 · · · 0 0 0 0   , s̄ =   s̄0,s s̄0,s−1 s̄0,s−2 · · · s̄0,3 s̄0,2 s̄0,1 iβ−l(s) us̄1,s us̄1,s−1 us̄1,s−2 · · · us̄1,3 us̄1,2 uils−1(s) 0 ... ... ... ... ... ... ... ... us−2s̄s−2,s u s−2s̄s−2,s−1 u s−2il2(s) · · · 0 0 0 0 us−1s̄s−1,s u s−1il1(s) 0 · · · 0 0 0 0   , e =   e0,s−2 e0,s−3 e0,s−4 · · · e0,2 e0,1 0 0 0 ue1,s−2 ue1,s−3 ue1,s−4 · · · ue1,2 0 0 0 0 ... ... ... ... ... ... ... ... ... ur−2er−2,s−2 u r−2er−2,s−3 u r−2er−2,s−4 · · · 0 0 0 0 0 0 0 0 · · · 0 0 0 0 0 ... ... ... ... ... ... ... ... ... 0 0 0 · · · 0 0 0 0 0   . 45 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 also, r̄i,j = − j−1∑ k=i+1 r̄i,kr t r−j,r−k −r t r−j,r−i, for 0 ≤ i < j ≤ r, s̄i,j = − j−1∑ k=i+1 s̄i,ks t s−j,s−k −s t s−j,s−i, for 0 ≤ i < j ≤ s, mi,j = − j−1∑ k=i+1 mi,ks t s−j−1,s−k−1 −q1 ( j−1∑ l=i+1 r̄i,lb t s−j−1,r−l + j−3∑ m=i+1 fi,mb t s−j−1,r−m−2 + b t s−j−1,r−i ) , l ≤ r − 1, m ≤ r − 3 and q1 = { u, j < r uj−r+1, j ≥ r } , ni,j = − j−1∑ k=i+1 ni,kr t r−j,r−k − j−1∑ k=i+1 s̄i,ka t r−j,r−k − j−3∑ k=i+1 ei,i+j−k−2a t r−j,r+k−i−j −a t r−j,r−i , ei,j = − j−1∑ k=i+1 ei,ks t s−j−2,s−k−2 −q2 j∑ l=i+1 ni,lb t s−j−2,r−l , l ≤ r − 1 and q2 = { u, j < r uj−r+2, j ≥ r } , fi,j = − j−1∑ k=i+1 fi,kr t r−j−2,r−k−2 − j∑ k=i+1 mi,ka t r−j−2,r−k−1. proof. firstly, we can check that ght = 0. so, 〈h〉⊆c⊥. further, |c| = prk0p(r−1)k1 · · ·pkr−1 · · ·psl0p(s−1)l1 · · ·pls−1. since k(r) = r−1∑ i=0 ki(r) and ki(r) = ki, then |c⊥| = pr(α−(k0+k1+···+kr−1))p(r−1)kr−1 · · ·pk1ps(β−(l0+l1+···+ls−1))p(s−1)ls−1 · · ·pl1. let |c||c⊥| = pn, hence n = rk0 + (r − 1)k1 + · · · + (r − (r − 1))kr−1 + sl0 + (s− 1)l1 + · · · + (s− (s− 1))ls−1 +rα−rk0 −···−rkr−1 + (r − 1)kr−1 + · · · + k1 + sβ −sl0 −···−sls−1 +(s− 1)ls−1 + · · · + l1 = rα + sβ. consequently, |c||c⊥| = prα+sβ. therefore, the rows of the matrix h are not only orthogonal to c but also they generate all dual space. so, the proof is completed. 46 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 example 3.2. let c be a zp[u2,u3]-linear code of type (α,β; k0,k1; l0, l1, l2) with the standard form of the generator matrix in (5). we calculate the parity-check matrix of this code step by step as follows. h = [ r̄ + f m n s̄ + e ] . since r = 2, then f = 0. now, r̄ = [ r̄02 r̄01 iα−k0−k1 ur̄12 uik1 0 ] where r̄01 = −rt12, r̄02 = −r̄01r t 01 −r t 02, r̄12 = −r t 01. n =   n02 n01 0un12 0 0 0 0 0   where n01 = −at12, n02 = −n01rt01 − s̄01at01 −at02, n12 = −at01. m =   m02 m01 0 0um12 0 0 0 0 0 0 0   where m01 = −ubt12, m02 = −m01st01 −u(r̄01bt01 + bt02), m12 = −ubt01. e =  e01 0 0 00 0 0 0 0 0 0 0   where e01 = −unt01bt01, and s̄ =   s̄03 s̄02 s̄01 iβ−l0−l1−l2us̄13 us̄12 uil2 0 u2s̄23 u 2il1 0 0   where s̄01 = −st23, s̄02 = −s̄01s t 12 −s t 13, s̄03 = −(s̄01st02 + s̄02s t 01) −s t 03, s̄12 = −st12, s̄13 = −s̄12s t 01 −s t 02, s̄23 = −st01. so, h =   r̄02 r̄01 iα−k0−k1 m02 m01 0 0 ur̄12 uik1 0 um12 0 0 0 n02 n01 0 s̄03 + e01 s̄02 s̄01 iβ−l0−l1−l2 un12 0 0 us̄13 us̄12 uil2 0 0 0 0 u2s̄23 u 2il1 0 0   . we can easily determine the type of c⊥ from the above matrix as (α,β; α−k0−k1,k1; β−l0−l1−l2, l2, l1). example 3.3. now, let c be a z2[u2,u3]-linear code with the generator matrix in (4). we have found the standard form of this generator matrix before in example 2.2. therefore, g =   1 0 0 0 0 0 0 u u 0 0 0 0 0 1 + u 1 0 0 0 0 0 0 1 0 0 0 0 0 0 u   =   ik0 r01 r02 0 0 0 uik1 ur12 0 0 0 b01 b02 il0 s01 0 0 ub12 0 uil1   . 47 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 hence, with the help of previous example we have the parity-check matrix of c as  0 1 1 u + u 2 0 0 0 u 0 0 0 0 0 0 0 0 0 u2   . so, c⊥ is of type (3, 3; 1, 1; 0, 0, 1) and has |c⊥| = 22·12121 = 16 codewords. corollary 3.4. let c be a z2[u,u2]-linear code of type (α,β; k0; l0, l1) with generator matrix in (6) then the parity-check matrix of c is −r t 01 iα−k0 −ubt01 0 0 −at01 0 −st02 + st12st01 −st12 iβ−l0−l1 0 0 −ust01 uil1 0   . note that this matrix is permutation equivalent to a matrix in (2). 4. binary images of z2[ur, us]-linear codes binary linear codes are the most important member of the family of linear codes. so, taking p = 2 we have z2[ur,us]-linear codes. then we can look at z2[ur,us]-linear codes as a binary codes under the special gray map, defined as follows. definition 4.1. let rs = z2 + uz2 + u2z2 + · · · + us−1z2 with us = 0. define the mapping φs : rs −→ z2 s−1 2( a0 + ua1 + · · · + us−1as−1 ) −→ (as−1,a0 ⊕as−1,a1 ⊕as−1, . . . ,as−2 ⊕as−1, a0 ⊕a1 ⊕as−1, . . . ,a0 ⊕as−2 ⊕as−1, . . . , a0 ⊕a1 ⊕···⊕as−1) where ai ⊕aj = ai + aj mod 2. this is a linear gray map from rs to z2 s−1 2 . as an example, let us consider φ3 : r3 −→ z42 0 −→ (0000) 1 −→ (0101) u −→ (0011) 1 + u −→ (0110) u2 −→ (1111) 1 + u2 −→ (1010) u + u2 −→ (1100) 1 + u + u2 −→ (1001). this map can be extended to rαr ×rβs as follows. definition 4.2. let x = (x0,x1, . . . ,xα−1) ∈ rαr and y = (y0,y1, . . . ,yβ−1) ∈ rβs and let φ : rαr × rβs −→ zn2 be the map defined by φ(x,y) = (φr(x0), . . . , φr(xα−1), φs(y0), . . . , φs(yβ−1)). we called the binary image φ(c) = c as a z2[ur,us]−linear code of length n = 2r−1α + 2s−1β. 48 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 example 4.3. let c be a z2[u,u3]-linear code of type (7, 7; 0; 0, 1, 3) with the following generator matrix  0 1 0 1 1 0 0 u + u2 u u + u2 u + u2 u u u 0 1 1 1 0 1 0 u2 u2 u2 0 u2 0 0 0 0 1 1 1 0 1 0 u2 u2 u2 0 u2 0 1 0 0 1 1 1 0 0 0 u2 u2 u2 0 u2   . using the gray map that we defined above, we have the gray image φ(c) = c is a binary linear code with the parameters [35, 5, 16]. it is worth mentioning that c with this parameters is an optimal code, that is, c has the best minimum distance d = 16 compared to the existing and known bounds for n = 35 and k = 5. 5. self-dual z2[u2, u3]-linear codes we know that if c = c⊥ then c is called a self-dual code. in this section, we investigate the structure of self-dual codes over z2[u]/〈u2〉×z2[u]/〈u3〉. we give some conditions for a z2[u2,u3]-linear code c to be a self-dual. further, we present some examples of self-dual codes. lemma 5.1. let c be a self-dual z2[u2,u3]-linear code. then c is of type (2k0 + k1, 2(l0 + l1); k0,k1; l0, l1, l2) . proof. since c is self-dual then the dual code c⊥ has the same type with c. therefore, we have (α,β; k0,k1; l0, l1, l2) = (α,β; α−k0 −k1,k1; β − l0 − l1 − l2, l2, l1) α−k0 −k1 = k0, β − l0 − l1 − l2 = l0, l1 = l2 α = 2k0 + k1, β = 2l0 + 2l1. corollary 5.2. if c is a self-dual code of type (α,β; k0,k1; l0, l1, l2), then β is even. corollary 5.3. if c is a separable z2[u2,u3]-linear code of type (α,β; k0,k1; l0, l1, l2), then it has the standard form of the following generator matrix g =   ik0 r01 r02 0 0 0 0 0 uik1 ur12 0 0 0 0 0 0 0 il0 s01 s02 s03 0 0 0 0 uil1 us12 us13 0 0 0 0 0 u2il2 u 2s23   . theorem 5.4. let c be a self-dual z2[u2,u3]-linear code of type (2k0 + k1, 2(l0 + l1); k0,k1; l0, l1, l2). then the following statements are equivalent. i) cα is a self-dual code over z2[u]/〈u2〉. ii) cβ is a self-dual code over z2[u]/〈u3〉. iii) c is separable and |cα| = 22k0+k1, |cβ| = 23(l0+l1). corollary 5.5. let c1 be a self-dual code of length α over z2[u]/〈u2〉 and c2 be a self-dual code of length β over z2[u]/〈u3〉. then, c1 ×c2 is a self-dual z2[u2,u3]-linear code of length α + β. 49 i. aydogdu / j. algebra comb. discrete appl. 6(1) (2019) 39–51 example 5.6 (separable). let c be a z2[u2,u3]-linear code of type (3, 4; 1, 1; 1, 1, 1) with the following generator matrix   1 0 1 0 0 0 0 0 u 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 u 0 u 0 0 0 0 0 u2 u2   . hence, c is a separable self-dual code. example 5.7 (non-separable). a z2[u2,u3]-linear code c with the below generator matrix is a nonseparable self-dual code of type (4, 4; 1, 2; 1, 1, 1).   1 1 1 1 + u 0 0 u + u2 u 0 u 0 u 0 0 0 u2 0 0 u u 0 0 0 0 0 0 1 1 1 1 1 1 + u2 0 0 u u 0 u 0 u 0 0 0 0 0 0 u2 u2   . 6. conclusion this paper generalizes z2z2[u]-linear codes to zp[ur,us]-linear codes. the original study was introduced for the special case p = 2, r = 1 and s = 2. we give the standard forms of the generator and parity-check matrices of these codes. further, we study z2[ur,us]-linear codes, for the special case p = 2 and relate these codes to binary codes by using a special gray map. also, we investigate self-dual codes over z2[u]/〈u2〉× z2[u]/〈u3〉. since this family of linear codes has been introduced recently many more properties of these families of codes await explorations, such as cyclicity. references [1] t. abualrub, i. siap, cyclic codes over the rings z2 +uz2 and z2 +uz2 +u2z2, des. codes cryptogr. 42(3) (2007) 273–287. 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[9] g. h. norton, a. sălăgean, on the structure of linear and cyclic codes over a finite chain ring, appl. algebra engrg. comm. comput. 10(6) (2000) 489–506. 51 https://doi.org/10.1109/18.312154 https://doi.org/10.1109/18.312154 https://doi.org/10.1007/pl00012382 https://doi.org/10.1007/pl00012382 introduction standard form of the generator matrices of zp[ur,us]-linear codes duality on zp[ur,us]-linear codes and parity-check matrices binary images of z2[ur,us]-linear codes self-dual z2[u2,u3]-linear codes conclusion references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.451229 j. algebra comb. discrete appl. 5(3) • 117–127 received: 16 october 2016 accepted: 25 april 2018 journal of algebra combinatorics discrete structures and applications characterization of 2×2 nil-clean matrices over integral domains research article kota nagalakshmi rajeswari, umesh gupta abstract: let r be any ring with identity. an element a ∈ r is called nil-clean, if a = e + n where e is an idempotent element and n is a nil-potent element. in this paper we give necessary and sufficient conditions for a 2×2 matrix over an integral domain r to be nil-clean. 2010 msc: 15a24 keywords: nil-clean matrix, idempotent matrix, nil-potent matrix, diophantine equation 1. introduction let r be an associative ring with identity. an element a ∈ r is called clean if a = e + u, where e, u ∈ r, e is an idempotent and u is a unit. we shall call such a representation a clean representation of a. further, a is called strongly clean (uniquely clean) if it has a clean representation in which the idempotent and the unit commute (it has only one clean representation). the ring r is said to be clean (strongly clean, uniquely clean) if each of its element is. the notion of clean rings was introduced by w. k. nicholson [10] while studying rings with exchange property and observed that these rings have the exchange property. the notions of strongly clean and uniquely clean ring were introduced subsequently. ever since, many mathematicians have studied these rings extensively. in an attempt to unify several variants of strongly clean property a. j. diesl [7] has introduced the notion of nil-clean rings. a ring r is nil-clean if each of its elements is the sum of an idempotent and a nil-potent. the notion of strongly nil-clean (uniquely nil-clean) were defined analogous to those of strongly clean (uniquely clean). these rings have been studied in great detail in [7]. among other things, the author has shown [[7], prop 3.4] that a nil-clean ring is clean and immediately posed the question: "whether or not a nil-clean element in a ring is clean?" k. n. rajeswari (corresponding author); school of mathematics, vigyan bhawan, khandwa road, devi ahilya university, indore-452 001, india (email: knr k@yahoo.co.in). umesh gupta; sanghvi institute of management and science, behind iim, pigdamber, indore-453 331, india (email: ugupta19@gmail.com). 117 https://orcid.org/0000-0002-7722-2491 https://orcid.org/0000-0001-7732-5768 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 this question was answered in negative by andrica dorin and calugareanu grigore in [3]. they have given an example of a matrix in m2(zz) which is nil-clean but not clean. in [2] they have also given example of a unit regular matrix (matrix which is the product of an idempotent matrix and a unit) which is not nil-clean and that of a nil-clean matrix which is not unit regular. to tackle questions as above, characterization of elements having a prescribed decomposition, nilclean decomposition in the present context, are essential. as matrices are good sources of examples, it is nice to have characterization of matrices having nil-clean decomposition. another motivation to study nil-clean matrices stems from lie theory, as strongly nil-clean decomposition of matrices over fields is the classical jordan-chevalley decomposition which is of central importance in lie theory. the nil-clean (strongly nil-clean) ring theory has application in various branches of mathematics. for instance in [6] nil-cleanness of endomorphism ring of a group of finite rank and in [12] nil-cleanness of group rings have been studied. in [4], the authors have introduced the notion of nil-clean graph of a ring r for which r is the set of vertices and a, b ∈ r are adjacent if a + b is nil-clean. the characterization of nil-clean elements will certainly help understand such applications. in this paper, we give a characterization of nil-clean 2 × 2 matrices over an integral domain. the idempotents, unipotents (element of the form 1 + n, where n is nil-potent) and nil-potents in any ring r are nil-clean. by nil-clean we mean, nil-clean elements other than these. it is well known that when r is an integral domain, an idempotent 2 × 2 matrix other than 0 and i is of the form [ x y w 1−x ] with x(1 − x) = yw. for the sake of completeness, a proof is given [see: lemma 2.1]. similarly, 2 × 2 nil-potent matrices are of the form [ a b c −a ] with a2 + bc = 0 [see: lemma 2.2]. thus a nil-clean matrix a = [ a b c d ] satisfies the conditions (i) tr(a) = 1. (ii) det [ a−x b−y c−w d−1 + x ] = 0, where x(1−x) = yw. expanding the determinant and simplifying gives (2a−1)x + cy + bw = a2 + bc. (1) thus, if [ a b c d ] is nil-clean as above, then it has trace 1 and the equation (1) has a solution x, y, w with x(1−x) = yw. we shall show these two conditions characterize a nil-clean matrix [see: theorem 3.1]. in [9], the authors have characterized nil-clean matrices over ufd. but, our approach is different from their approach. the authors of [9] have used characterization of rank one 2 × 2 matrices over a ufd r to arrive at the characterization while we have used solutions of certain diophantine equation to get the characterization. when the entries are integers, the equation (1) is a linear diophantine equation in 3 variables which has a solution if and only if gcd(2a − 1, c, b) divides (a2 + bc) and this forces gcd(2a − 1, b, c) = 1. the condition that equation (1) has a solution satisfying x(1 − x) = yw yields a second degree diophantine equation in two variables. this equation has been analyzed for its solutions in section 4. discussion in the sections 3 and 4 has been summarized in section 5 along with illustrations. it has been shown [[7], cor. 5.4] that central idempotents and central nil-potents in a ring are uniquely nil-clean. in section 6, we let r be a principal ideal domain (pid) and show using the characterization of section 3 that if an idempotent in m2(r) is not central then it is not uniquely nil-clean. thus in m2(r), the only uniquely nil-clean idempotents are central idempotents i.e., the zero matrix and the identity matrix. we observe that nilpotents in m2(r) are uniquely nil-clean. 118 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 throughout the paper r is an integral domain, mn(r) is the ring of n × n matrices over r. for a ∈ mn(r), tr(a) denotes the trace of a, r(a) denotes its rank and |a| its determinant. let rn be the free r-module of n tuples of elements of r. 2. preliminaries let r be an integral domain. in this section we recall some results and facts. lemma 2.1. e = [ x y w z ] ∈ m2(r) (e 6= i, e 6= 0) is idempotent if and only if it is of the form[ x y w 1−x ] with x(1−x) = yw. proof. let e = [ x y w z ] (e 6= i, e 6= 0) be an idempotent matrix. as idempotent matrix, which is not identity is singular, we have xz = yw and as tr(e) = r(e) = 1, x + z = 1 i.e., z = 1 − x. hence the matrix is [ x y w 1−x ] with x(1−x) = yw. converse can be checked by direct calculation. lemma 2.2. n = [ a b c d ] ∈ m2(r) is nil-potent if and only if it is of the form [ a b c −a ] with a2+bc = 0. proof. let n = [ a b c d ] be a nil-potent matrix. since trace and determinant of a nil-potent matrix are always zero, a + d = 0 and bc − ad = 0 i.e., d = −a and bc + a2 = 0. hence the matrix is [ a b c −a ] with bc + a2 = 0. converse is easy to check. proposition 2.3. the equation x2 −y2 = c admits integer solutions if and only if c belongs to 4zz or it is odd. proof. (see [[13], property 1]). proposition 2.4. the equation x2−dy2 = c2 (where d is not a perfect square) admits an infinite number of solutions in in. proof. (see [[13], property 2]). theorem 2.5. let r be a pid and a ∈ mn(r) be an idempotent matrix. then a is similar to the matrix [ ir 0 0 0 ] where ir is the identity matrix of order r and r = r(a). proof. let b be a basis of rn, the free r-module of rank n. then a determines a module homomorphism t : rn → rn which is idempotent and whose matrix with respect to b is a. let i be the identity transformation, n(t) and n(i − t) be the null spaces of t and i − t respectively. as r is a pid, n(t) and n(i − t) are free. as t is idempotent, for x ∈ rn, tx ∈ n(i − t) and (i − t)x ∈ n(t). hence rn = n(i − t) + n(t). clearly, n(i − t) ∩ n(t) = 0. thus rn = n(i − t) ⊕ n(t). let {v1, ....vr} be a basis of n(i−t) and {vr+1, ....vn} be a basis of n(t). then b′ = {v1, ....vr, vr+1, ....vn} 119 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 is a basis of rn and the matrix of t with respect to b′ is [ ir 0 0 0 ] . therefore, a is similar to [ ir 0 0 0 ] and r = r(a). it is well known that a polynomial equation with integral coefficients for which integral solutions are desired is called a diophantine equation. in the later part we will use two types of diophantine equations, one linear diophantine equation in three variables and other is quadratic diophantine equation in two variables. a linear diophantine equation with three variables is of the form ax + by + cz = d (2) where a, b, c and d are integers. this equation has a solution if and only if gcd(a, b, c) divides d (see [11]). it is also that once it has a solution, it has infinite number of solutions. solution to the equation (2) is obtained as follows (see [8]): let gcd(a, b) = p, a′ = a/p and b′ = b/p. let (u0, v0) be any one solution of a′u + b′v = c, (z0, t0) be any one solution of cz + pt = d and (x0, y0) be any one solution of a′x + b′y = t0, then the general solution of ax + by + cz = d is x = x0 + b ′k −u0m y = y0 −a′k −v0m z = z0 + pm k, m ∈ zz we shall denote quadratic diophantine equation in two variables by ax2 + bxy + cy2 + dx + ey + f = 0 (3) where a, b, c, d, e and f are integers. we shall use the notations of [3] and follow the method of solution given there. let d = b2 − 4ac, g = gcd(b2 − 4ac, 2ae − bd) and 4 = 4acf + bde − ae2 −cd2 −fb2. then the equation reduces to gx2 − d g y 2 = −4a 4 g (4) where x = d g y + 2ae−bd g and y = 2ax + by + d. now, we have the following cases: if d > 0 and perfect square, the equation (4) admits finite number of integer solutions provided right hand side of (4) is non zero [see: proposition 2.3]. if d > 0 and not a perfect square, the equation (4) admits an infinite number of solutions in in[see: proposition 2.4] if d < 0 i.e., b2 − 4ac < 0. (see [1]) since the ellipse is a closed figure, the number of solutions will be finite(if a solution exists). the equation is ax2 + bxy + cy2 + dx + ey + f = 0 (5) ⇒ cy2 + (bx + e)y + (ax2 + dx + f) = 0 ⇒ y = −(bx + e)± √ (bx + e)2 −4c(ax2 + dx + f) 2c (6) 120 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 for a value of x there will be two values of y except at the left and right extremes of the ellipse. in this case there will be only one value of y. to determine the location of the left and right extremes we should equal the square root to zero. i.e., (bx + e)2 −4c(ax2 + dx + f) = 0 ⇒ (b2 −4ac)x2 + 2(be −2cd)x + (e2 −4cf) = 0 so the value of x should be between the roots of this equation. if the roots are not real, there will be no solution to the equation (5), all integer values of x should be replaced in equation (6) in order to find an integer value of y. 3. main result let r be an integral domain. it is clear that idempotent matrices, nil-potent matrices and unit matrices of the form i + n, where i is the identity matrix and n is a nil-potent matrix are nil-clean. we shall call these trivially nil-clean. so from now onwards, nil-clean means non-trivially nil-clean. in this section, we shall discuss necessary and sufficient conditions for a matrix a ∈ m2(r) to be nil-clean. theorem 3.1. let a = [ a b c d ] be any matrix in m2(r). then a is nil-clean if and only if tr(a) = 1 and (2a−1)x + cy + bw = a2 + bc has a solution satisfying x(1−x) = yw. proof. let a be nil-clean. say a = e + n, where e 6= 0, e 6= id, is an idempotent matrix and n is a nil-potent matrix. then a−e = [ a−x b−y c−w d−1 + x ] is nil-potent. hence (a−e)2 = [ a−x b−y c−w d−1 + x ]2 = 0. since, the trace of a nil-potent matrix is zero, a − x + d − 1 + x = 0 ⇒ a + d = 1 i.e. tr(a) = 1. also, determinant of a nil-potent matrix is zero, hence (a−x)(d−1 + x)− (b−y)(c−w) = 0 ⇒ (a−x)(x−a)− (b−y)(c−w) = 0 ⇒ (a−x)2 + (b−y)(c−w) = 0 ⇒ x2 −2ax + a2 + bc−yc− bw + yw = 0 ⇒ x2 −2ax + a2 + bc−yc− bw + x−x2 = 0 ⇒ x(1−2a)−yc−wb + a2 + bc = 0 ⇒ (2a−1)x + cy + bw = a2 + bc. conversely, let tr(a) = 1 and (2a−1)x + cy + bw = a2 + bc have a solution x, y, w with x(1−x) = yw. let e = [ x y w 1−x ] . then a−e = [ a−x b−y c−w d−1 + x ] . now, tr(a−e) = (a−x) + (d−1 + x) = a + d−1 = 0 (∵ tr(a) = 1) 121 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 and |a−e| = ∣∣∣∣∣ a−x b−yc−w d−1 + x ∣∣∣∣∣ = (a−x)(d−1 + x)− (b−y)(c−w) = (a−x)(x−a)− (b−y)(c−w) = −x2 −a2 + 2ax− bc + yc + bw −yw = −x2 −a2 + 2ax− bc + yc + bw − (x−x2) (∵ yw = x(1−x)) = (2a−1)x + cy + bw − (a2 + bc) = 0 [∵ (2a−1)x + cy + bw = a2 + bc] hence a−e is nil-potent, which proves a is nil-clean. 4. analysis on main result when r = zz in this section, we shall analyze the linear diophantine equation (2a − 1)x + cy + bw = a2 + bc as to when it has a solution satisfying x(1−x) = yw. let a1 = 2a−1 and d1 = a2 + bc, then the equation becomes a1x + cy + bw = d1. let gcd(a1, c) = p, a′1 = a1/p and c ′ = c/p. let (u0, v0) be any one solution of a′1u + c ′v = b, (w0, t0) be any one solution of bw + pt = d1 and (x0, y0) be any one solution of a′1x + c ′y = t0, then the general solution of a1x + cy + bw = d1 is x = x0 + c ′k −u0m y = y0 −a′1k −v0m w = w0 + pm the condition x−x2 = yw ⇒(x0 + c′k −u0m)− (x0 + c′k −u0m)2 = (y0 −a′1k −v0m)(w0 + pm) ⇒(x0 + c′k −u0m)− (x20 + c ′2k2 + u20m 2 + 2x0c ′k −2x0u0m− 2c′ku0m) = y0w0 −a′1kw0 −v0mw0 + y0pm−a ′ 1kpm−v0m 2p ⇒c′2k2 − (2c′u0 + a′1p)km + (u 2 0 −v0p)m 2 + (2x0c ′ − c′ −a′1w0)k + (u0 −2x0u0 −v0w0 + y0p)m + (x20 −x0 + y0w0) = 0 ⇒ak2 + bkm + cm2 + dk + em + f = 0 (7) where a = c′2, b = −(2c′u0 + a′1p), c = (u 2 0 −v0p), d = (2x0c ′ − c′ −a′1w0), e = (u0 −2x0u0 −v0w0 + y0p), and f = (x 2 0 −x0 + y0w0) now, d = b2 −4ac = (2c′u0 + a ′ 1p) 2 −4c′2(u20 −v0p) = 4c′2u20 + a ′2 1 p 2 + 4c′u0a ′ 1p−4c′2u20 + 4c′2v0p = a21 + 4c ′u0a ′ 1p + 4c ′2v0p = a21 + 4c ′u0a ′ 1p + 4c ′p(b−a′1u0) (∵ a′1u0 + c′v0 = b) = a21 + 4cb. 2ae −bd = 2c′2(u0 −2x0u0 −v0w0 + y0p) + (2c′u0 + a′1p)(2x0c′ − c′ −a′1w0) 122 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 = 2c′2u0 −4c′2x0u0 −2c′2v0w0 + 2c′2y0p + 4c′2u0x0 −2c′2u0 −2c′u0a′1w0 + 2a′1px0c′ −a′1pc′ −a′21 pw0 = −2c′w0(a′1u0 + c′v0) + 2c′p(a′1x0 + c′y0)−a′1pc′ −a′21 pw0 = −2c′w0b + 2c′pt0 −a′1pc′ −a′21 pw0 ( ∵ a′1x0 + c ′y0 = t0 a′1u0 + c ′v0 = b ) = −2c′w0b + 2c′(d1 − bw0)−a′1pc′ −a′21 pw0 = 2c′d1 −4c′w0b−a′1pc′ −a′21 pw0 = 1 p (2cd1 −4cw0b−a1c−a21w0) let g = gcd(d, 2ae −bd). 4 = 4acf + bde −ae2 −cd2 −fb2 = 4c′2(u20 −v0p)(x20 −x0 + y0w0)− (2c′u0 + a′1p)(2x0c′ − c′ −a′1w0) (u0 −2x0u0 −v0w0 + y0p)− c′2(u0 −2x0u0 −v0w0 + y0p)2 − (u20 −v0p)(2x0c′ − c′ −a′1w0)2 − (x20 −x0 + y0w0)(2c′u0 + a′1p)2 = 4c′2u20x 2 0 −4c′2u20x0 + 4c′2u20y0w0 −4c′2v0px20 + 4c′2v0px0 −4c′2v0py0w0 −4c′2x0u20 + 2c′2u20 + 2c′a′1w0u20 + 8c′2u20x20 −4c′2u20x0 −4c′u20a′1w0x0 +4c′2u0v0w0x0 −2c′2u0v0w0 −2c′u0a′1w20v0 −4c′2u0y0px0 + 2c′2u0y0p +2c′u0a ′ 1y0pw0 −2a′1px0c′u0 + a′1pc′u0 + a′21 pw0u0 + 4a′1x20c′u0p −2a′1pc′x0u0 −2a′21 pw0x0u0 + 2a′1pv0w0x0c′ −a′1pc′v0w0 −a′21 w20v0p −2a′1p2y0x0c′ + a′1p2c′y0 + a′21 p2y0w0 − c′2u20 −4c′2x20u20 − c′2v20w20 −c′2y20p2 + 4c′2x0u20 + 2c′2u0v0w0 −2c′2u0y0p−4c′2x0u0v0w0 +4c′2x0u0y0p + 2c ′2v0w0y0p−4u20x20c′2 −u20c′2 −u20a′21 w20 + 4u20x0c′2 +4u20x0c ′a′1w0 −2u20c′a′1w0 + 4v0px20c′2 + c′2v0p + a′21 w20pv0 −4c′2x0v0p −4c′a′1w0x0v0p + 2c′a′1w0v0p−4x20c′2u20 −x20a′21 p2 −4c′a′1x20u0p + 4c′2u20x0 +a′21 p 2x0 + 4c ′a′1u0x0p−4c′2u20y0w0 −a′21 p2y0w0 −4c′a′1u0y0w0p = (4c′2u20x 2 0 + 8c ′2u20x 2 0 −4c′2u20x20 −4c′2u20x20 −4c′2u20x20) + (−4c′2u20x0 −4c′2u20x0 −4c′2u20x0 + 4c′2u20x0 + 4c′2u20x0 + 4c′2u20x0) + (4c′2u20y0w0 −4c′2u20y0w0) + (4c′2v0px20 −4c′2v0px20) + (4c′2v0px0 −4c′2v0px0) +(2c′2v0py0w0 −4c′2v0py0w0) + (2c′2u20 − c′2u20 − c′2u20) + (2c′a′1w0u20 −2c′a′1w0u20) + (4c′u20a′1w0x0 −4c′u20a′1w0x0) + (4c′2u0v0w0x0 −4c′2u0v0w0x0) + (2c′2u0v0w0 −2c′2u0v0w0)−2c′u0a′1w20v0 + (4c′2u0y0px0 −4c′2u0y0px0) + (2c′2u0y0p−2c′2u0y0p) + (2c′u0a′1y0pw0 −4c′u0a′1y0pw0) +(4a′1px0c ′u0 −2a′1px0c′u0 −2a′1px0c′u0) + a′1pc′u0 + a′21 pw0u0 +(4a′1x 2 0c ′u0p−4a′1x20c′u0p)−2a′21 pw0x0u0 + (2a′1pv0w0x0c′− 4a′1pv0w0x0c ′)−a′1pc′v0w0 + (a′21 w20v0p−a′21 w20v0p)−2a′1p2y0x0c′ 123 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 +a′1p 2c′y0 + (a ′2 1 p 2y0w0 −a′21 p2y0w0)− c′2v20w20 − c′2y20p2 −u20a′21 w20 +c′2v0p + 2c ′a′1w0v0p−x20p2a′21 + a′21 p2x0 = −2c′2v0py0w0 −2c′u0a′1w20v0 −2c′a′1u0y0w0p + a′1pc′u0 + a′21 pw0u0 −2a′21 pw0x0u0 −2c′a′1pw0v0x0 −a′1pc′v0w0 −2a′1p2y0x0c′ + a′1p2c′y0 −c′2v20w20 − c′2y20p2 −u20a′21 w20 + c′2v0p + 2c′a′1w0v0p−x20p2a′21 + a′21 p2x0 = −(c′2y20p2 + c′2v20w20 + 2c′2v0py0w0)− (x20p2a′21 + u20a′21 w20 + 2a′21 px0u0w0) −2(a′1p2y0x0c′ + c′a′1y0u0w0p + c′v0w0x0pa′1 + c′u0w20v0a′1) + a′1pc′u0 +a′21 pw0u0 + a ′ 1p 2c′y0 + c ′2v0p + c ′a′1w0v0p + a ′2 1 p 2x0. = −(c′y0p + c′v0w0 + x0pa′1 + u0a′1w0)2 + c′p(a′1u0 + c′v0) + a′1pw0 (a′1u0 + c ′v0) + a ′ 1p 2(a′1x0 + c ′y0) = −[p(a′1x0 + c′y0) + w0(c′v0 + a′1u0)]2 + c′pb + a′1pw0b +a′1p 2t0 [ ∵ a′1u0 + c ′v0 = b a′1x0 + c ′y0 = t0 ] = −(pt0 + bw0)2 + cb + a′1p(bw0 + pt0) = −d21 + cb + a1d1 (∵ pt0 + bw0 = d1) = −(a2 + bc)2 + (2a−1)(a2 + bc) + bc = −(a2 + bc)2 + 2a(a2 + bc)−a2 − bc + bc = −(a2 + bc−a)2 = −(d1 −a)2 the equation (7) reduces to gx2 − d g y 2 = −4a4 g or x21 − dy 2 = [2c′(d1 −a)]2 (8) where x1 = gx = dm + (2ae − bd) = (a21 + 4bc)m + 1 p (2cd1 − 4cw0b − a1c − a21w0) and y = 2ak + bm + d = 2c′2k − (2c′u0 + a1)m + (2x0c′ − c′ − a′1w0). the general methods of solutions of (8) have already been discussed in section 2. 5. examples in this section, we shall briefly summarize the discussion in section 3 and 4. for a given matrix a = [ a b c d ] to be nil-clean the following conditions are necessary and sufficient : (1) tr(a) = 1 (2) gcd(2a−1, b, c) = 1 (3) the equation (8) of section 4 has a solution. the existence of solutions and the number of solutions is determined by the discriminant d of the equation (8) which is (2a−1)2 + bc. (i) if d > 0, a perfect square and right hand side of (8) is non zero, then it has finitely many solutions (see example 5.1 below). (ii) if d > 0 and not a perfect square (8) has infinitely many solutions (see example 5.2 below). (iii) if d < 0, equation (8) may or may not have a solution. if (8) has a solution then it has finitely many (see example 5.3 below). finally, example 5.4 illustrates the case where (8) does not have a solution. 124 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 example 5.1. consider the matrix a = [ 1 3 2 0 ] . the linear diophantine equation (2a−1)x+cy+bw = a2 + bc for a is x + 2y + 3w = 7. the solutions are x = 1 + 2k − m, y = 3 − k − m, w = m where k and m are integers. using condition x(1 − x) = yw, we get the quadratic diophantine equation 4k2−5km + 2k + 2m = 0. here d = 25 which is square of 5 and solutions are (4, 4), (-2, -1), (0, 0) and (1, 2). for the solution (4, 4), we get the solution of linear diophantine equation x = 5, y = −5, w = 4. let e = [ 5 −5 4 −4 ] . then a−e is nil-potent. example 5.2. consider the matrix a = [ 2 3 2 −1 ] . the linear diophantine equation (2a − 1)x + cy + bw = a2 + bc for a is 3x + 2y + 3w = 10. the solutions are x = 2k − m, y = 5 − 3k, w = m where k and m are integers. using condition x(1 − x) = yw, we get the quadratic diophantine equation 4k2 −7km + m2 −2k + 6m = 0. here d = 33 which is not a perfect square and this equation has infinite number of solutions, one of the solution is (2, 2) for which, we get the solution of linear diophantine equation x = 2, y = −1, w = 2. let e = [ 2 −1 2 −1 ] . then a−e is nil-potent. example 5.3. consider the matrix a = [ 2 3 −1 −1 ] the linear diophantine equation (2a−1)x+cy+bw = a2 + bc for a is 3x − y + 3w = 1. the solutions are x = −k, y = −1 − 3k + 3m, w = m where k and m are integers. using condition x(1 − x) = yw, we get the quadratic diophantine equation k2 − 3km + 3m2 + k − m = 0. here d = −3 < 0 and solutions are (-2, -1), (-1, 0) and (0, 0). for the solution (-2, -1), we get the solution of linear diophantine equation x = 2, y = 2, w = −1. let e = [ 2 2 −1 −1 ] . then a−e is nil-potent. example 5.4. consider the matrix a = [ 2 3 −5 −1 ] . the linear diophantine equation (2a−1)x + cy + bw = a2 + bc for a is 3x − 5y + 3w = −11. the solutions are x = 3 − 5k − m, y = 4 − 3k, w = m where k and m are integers. using condition x(1 − x) = yw, we get the quadratic diophantine equation 25k2 + 7km + m2 −25k −m + 6 = 0. here d = −51 < 0. this equation has no solution as the equation (b2 −4ac)k2 + 2(be −2cd)k + (e2 −4cf) = 0 i.e., 51k2 −86k + 23 = 0 has roots 0.33 and 1.35 and the integer k = 1 does not yield integer value for m. hence 3x−5y + 3w = −11 does not have a solution satisfying x(1−x) = yw. 6. uniquely nil-clean matrices in [7], it has been shown that central idempotents and central nilpotents are uniquely nil-clean. when r is a pid, we show that only uniquely nil-clean idempotent elements of m2(r) are central idempotents and observe that all nilpotents in m2(r) are uniquely nil-clean. remark 6.1. nil-clean property is preserved under similarity. proof. let a be nil-clean and a = e + n, where e, an idempotent matrix and n, a nilpotent matrix. say nk = 0 . let b be similar to a. then b = cac−1 for some invertible matrix c. therefore, b = cec−1 + cnc−1, where (cec−1)2 = ce2c−1 = cec−1 and (cnc−1)k = cnkc−1 = 0. hence b is nil-clean. lemma 6.2. let r be a pid and e ∈ m2(r) be an idempotent which is not central. then e is not uniquely nil-clean. 125 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 proof. since e is not central, e 6= 0 and e 6= i. then e = [ x y w 1−x ] , where yw = x(1 − x). as every idempotent matrix of rank r over a pid is similar to the matrix of the form [ ir 0 0 0 ] (by theorem 2.5), e is similar to [ 1 0 0 0 ] . let cec−1 = [ 1 0 0 0 ] . the equation (2a − 1)x + cy + bw = a2 + bc of section 3, for [ 1 0 0 0 ] reduces to x = 1. as we are interested in the solution x, y, w with yw = x(1−x), we have yw = 0. hence either y = 0 and w is arbitrary or y is arbitrary and w = 0. hence all the nil-clean decomposition of [ 1 0 0 0 ] are[ 1 0 0 0 ] = [ 1 0 w 0 ] + [ 0 0 −w 0 ] , w is arbitrary and [ 1 0 0 0 ] = [ 1 y 0 0 ] + [ 0 −y 0 0 ] , y is arbitrary. therefore, e = c−1 [ 1 0 w 0 ] c + c−1 [ 0 0 −w 0 ] c, w is arbitrary and e = c−1 [ 1 y 0 0 ] c + c−1 [ 0 −y 0 0 ] c, y is arbitrary. hence e is not uniquely nil-clean. example 6.3. let r = zz and let a = [ 9 −6 12 −8 ] and c = [ 3 −2 −4 3 ] then c−1ac = [ 1 0 0 0 ] . the nil-clean decomposition of [ 1 0 0 0 ] is [ 1 0 0 0 ] = [ 1 0 w 0 ] + [ 0 0 −w 0 ] where w is arbitrary. therefore, [ 9 −6 12 −8 ] = [ 3 2 4 3 ]([ 1 0 w 0 ] + [ 0 0 −w 0 ])[ 3 −2 −4 3 ] = ([ 3 + 2w 0 4 + 3w 0 ] + [ −2w 0 −3w 0 ])[ 3 −2 −4 3 ] = [ 9 + 6w −6−4w 12 + 9w −8−6w ] + [ −6w 4w −9w 6w ] where w ∈ zzis arbitrary hence a is not uniquely nil-clean. lemma 6.4. if a ∈ m2(r) is nilpotent, then it is uniquely nil-clean. proof. let a ∈ m2(r) be nilpotent. suppose a = e + n , where e, an idempotent matrix and n, a nilpotent matrix, is a nil-clean representation of a. comparing the traces on both sides, we have tr(e) = 0. therefore, e = 0 and a = n. in [5], the notion of nil-clean elements has been generalized to ideals of a ring r. an ideal i of r is nil-clean if i = e +n, where e and n are ideals of r, e2 = e and n is a nil ideal(i.e., all elements of n 126 k. n. rajeswari, u. gupta / j. algebra comb. discrete appl. 5(3) (2018) 117–127 are nil-potent). while the ideals generated by idempotents are idempotent, those generated by nilpotent elements need not be nil or nil-potent when the ring r is non-commutative. in view of the results of the paper, we ask the following: question 6.4. if i is the principal ideal generated by a nil-clean element in m2(r), r an integral domain, when is i nil-clean? acknowledgment: the authors thank the referees immensely for their suggestions which have improved the earlier version of the paper. references [1] d. alpern, generic two integer variable equation solver, 2018, available at www.alpertron.com.ar/quad.htm. [2] t. andreescu, d. andrica, quadratic diophantine equations, springer, new york, 2015. [3] d. andrica, g. călugăreanu, a nil–clean 2×2 matrix over the integers which is not clean, j. algebra appl. 13(6) (2014) 1450009. [4] d. k. basnet, j. bhattacharyya, nil clean graph of rings, arxiv:1701.07630 [math.ra], https://arxiv.org/abs/1701.07630. [5] a. t. block gorman, generalizations of nil clean to ideals, wellesley college, honors thesis collection, (388) 2016. [6] s. breaz, g. călugăreanu, p. danchev, t. micu, nil–clean matrix rings, linear algebra appl. 439(10) (2013) 3115–3119. [7] a. j. diesl, nil–clean rings, j. algebra 383(1) (2013) 197–211. [8] diophantine equation ax + by + cz = d solver, www.mathafou.free.fr/ex e_en/exedioph3.html. [9] s. hadjirezaei, s. karimzadeh, on the nil–clean matrix over a ufd, j. alg. struc. appl. 2(2) (2015) 49–55. [10] w. k. nicholson, lifting idempotents and exchange rings, trans. amer. math. soc. 229 (1977) 269–278. [11] i. niven, h. s. zuckerman, an introduction to the theory of numbers, john wiley–sons, 3rd edition, 1972. [12] s. sahinkaya, g. tang, y. zhou, nil–clean group rings, j. algebra appl. 16(7) (2017) 1750135. [13] f. smarandache, existence and number of solutions of diophantine quadratic equations with two unkowns in zz and in, arxiv:0 704.3716 [math.gm], http://arxiv.org/abs/0704.3716. 127 https://www.alpertron.com.ar/quad.htm https://www.alpertron.com.ar/quad.htm https://doi.org/10.1007/978-0-387-54109-9 http://dx.doi.org/10.1142/s0219498814500091 http://dx.doi.org/10.1142/s0219498814500091 https://repository.wellesley.edu/thesiscollection/388/ https://repository.wellesley.edu/thesiscollection/388/ https://doi.org/10.1016/j.laa.2013.08.027 https://doi.org/10.1016/j.laa.2013.08.027 https://doi.org/10.1016/j.jalgebra.2013.02.020 https://doi.org/10.1090/s0002-9947-1977-0439876-2 https://doi.org/10.1090/s0002-9947-1977-0439876-2 https://mathscinet.ams.org/mathscinet-getitem?mr=344181 https://mathscinet.ams.org/mathscinet-getitem?mr=344181 https://doi.org/10.1142/s0219498817501353 introduction preliminaries main result analysis on main result when r= zz examples uniquely nil-clean matrices references issn 2148-838xhttp://dx.doi.org/10.13069/jacodesmath.427968 j. algebra comb. discrete appl. 5(2) • 101–116 received: 18 may 2016 accepted: 25 april 2018 journal of algebra combinatorics discrete structures and applications non–existence of some 4–dimensional griesmer codes over finite fields∗ research article kazuki kumegawa, tatsuya maruta abstract: we prove the non–existence of [gq(4, d), 4, d]q codes for d = 2q3 − rq2 − 2q + 1 for 3 ≤ r ≤ (q + 1)/2, q ≥ 5; d = 2q3 − 3q2 − 3q + 1 for q ≥ 9; d = 2q3 − 4q2 − 3q + 1 for q ≥ 9; and d = q3 − q2 − rq − 2 with r = 4, 5 or 6 for q ≥ 9, where gq(4, d) = ∑3 i=0 ⌈ d/qi ⌉ . this yields that nq(4, d) = gq(4, d) + 1 for 2q3−3q2−3q+1 ≤ d ≤ 2q3−3q2, 2q3−5q2−2q+1 ≤ d ≤ 2q3−5q2 and q3−q2−rq−2 ≤ d ≤ q3−q2−rq with 4 ≤ r ≤ 6 for q ≥ 9 and that nq(4, d) ≥ gq(4, d) + 1 for 2q3 − rq2 − 2q + 1 ≤ d ≤ 2q3 − rq2 − q for 3 ≤ r ≤ (q + 1)/2, q ≥ 5 and 2q3 − 4q2 − 3q + 1 ≤ d ≤ 2q3 − 4q2 − 2q for q ≥ 9, where nq(4, d) denotes the minimum length n for which an [n, 4, d]q code exists. 2010 msc: 94b05, 94b27, 51e20 keywords: optimal linear codes, griesmer bound, arcs in pg(r, q) 1. introduction an [n,k,d]q code c is a linear code of length n, dimension k and minimum hamming weight d over fq, the field of q elements. a fundamental problem in coding theory is to find nq(k,d), the minimum length n for which an [n,k,d]q code exists. the griesmer bound gives a lower bound on nq(k,d) as nq(k,d) ≥ gq(k,d) = k−1∑ i=0 ⌈ d/qi ⌉ , where dxe denotes the smallest integer ≥ x. an [n,k,d]q code is called griesmer if n = gq(k,d). the values of nq(k,d) are determined for all d only for some small values of q and k, see [22]. for k = 4, ∗ this research was partially supported by jsps kakenhi grant number jp16k05256. kazuki kumegawa; department of mathematics and information sciences, osaka prefecture university, sakai, osaka 599-8531, japan (email: hmjwj674@yahoo.co.jp). tatsuya maruta (corresponding author); department of mathematical sciences, osaka prefecture university, sakai, osaka 599-8531, japan (email: maruta@mi.s.osakafu-u.ac.jp). 101 https://orcid.org/0000-0001-7858-0787 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 the exact value of nq(4,d) is known for all d only for q = 2, 3, 4. recently, one of the open cases for (q,k) = (5, 4) was solved in [16]. for general q, see [18] and [11] for known results on nq(4,d). we have recently proved the following. theorem 1.1 ([12]). there exists no [gq(4,d), 4,d]q code for (1) d = q3/2 −q2 − 2q + 1 for q = 2h, h ≥ 3, (2) d = 2q3 − 3q2 − 2q + 1 for q ≥ 5, (3) d = 2q3 −rq2 −q + 1 for 3 ≤ r ≤ q −q/p, q = ph with p prime. as a continuation on the non–existence of griesmer codes for k = 4, we prove the following four theorems. theorem 1.2. there exists no [gq(4,d), 4,d]q code for d = 2q3 − rq2 − 2q + 1 for 3 ≤ r ≤ (q + 1)/2, q ≥ 5. theorem 1.3. there exists no [gq(4,d), 4,d]q code for d = 2q3 − 3q2 − 3q + 1 for q ≥ 9. theorem 1.4. there exists no [gq(4,d), 4,d]q code for d = 2q3 − 4q2 − 3q + 1 for q ≥ 9. theorem 1.5. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 −rq − 2 with 4 ≤ r ≤ 6 for q ≥ 9. theorem 1.2 is a generalization of the non–existence of griesmer [209, 4, 166]5 codes. we note that the existence of a [gq(4,d), 4,d]q code for d = 2q3 − 3q2 − 3q + 1 is known for q = 4 but unknown and still open for q = 5, 7, 8. the existence of a [gq(4,d), 4,d]q code for d = 2q3 − 4q2 − 3q + 1 is known for q = 5 but unknown and still open for q = 7, 8. for the non–existence of [gq(4,d), 4,d]q codes for q3 − q2 − 4q + 1 ≤ d ≤ q3 − q2 − q, see [23]. the non–existence of [gq(4,d), 4,d]q codes for d = q3 − q2 − rq − 2 is known for (q,r) = (8, 4) but unknown and still open for (q,r) = (8, 5) and (q,r) = (8, 6). while the existence of a [gq(4,d) + 1, 4,d]q code for d = 2q3 − rq2 − q with 4 ≤ r ≤ q − 1 and for d = 2q3 − 4q2 − 2q is unknown in general, such a code exists for d = 2q3 − 5q2 −sq with 0 ≤ s ≤ q − 4, q ≥ 7 [21]. the existence of a [gq(4,d) + 1, 4,d]q code for d = 2q3 − 3q2 − 2q and for d = q3 − q2 − rq with 1 ≤ r ≤ q − 1 follows from the recent result from [10]. it is also known that nq(4,d) = gq(4,d) for d ≥ 2q3 − 3q2 + 1 for all q and that nq(4,d) = gq(4,d) + 1 for 2q3 − 3q2 − 2q + 1 ≤ d ≤ 2q3 − 3q2 for q ≥ 5 [12]. since the existence of an [n,k,d]q code implies the existence of an [n− 1,k,d− 1]q code by puncturing, we get the following results from theorems 1.2-1.5. corollary 1.6. nq(4,d) = gq(4,d) + 1 for (1) 2q3 − 3q2 − 3q + 1 ≤ d ≤ 2q3 − 3q2 for q ≥ 9, (2) 2q3 − 5q2 − 2q + 1 ≤ d ≤ 2q3 − 5q2 for q ≥ 9, (3) q3 −q2 −rq − 2 ≤ d ≤ q3 −q2 −rq with 4 ≤ r ≤ 6 for q ≥ 9. corollary 1.7. nq(4,d) ≥ gq(4,d) + 1 for (1) 2q3 −rq2 − 2q + 1 ≤ d ≤ 2q3 −rq2 −q for 4 ≤ r ≤ (q + 1)/2, q ≥ 7, (2) 2q3 − 4q2 − 3q + 1 ≤ d ≤ 2q3 − 4q2 − 2q for q ≥ 9. the remainder of the paper is organized as follows. in section 2, we give the geometric preliminaries and some results on linear codes of dimension 3. we prove theorems 1.2, 1.3 and 1.5 in sections 3, 4 and 5, respectively. the proof of theorem 1.4 is similar to that of theorem 1.3 and therefore skipped. we give some remarks in section 6 as conclusion. 102 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 2. preliminaries in this section, we give the geometric method and preliminary results to prove the non–existence of some griesmer codes. we denote by pg(r,q) the projective geometry of dimension r over fq. the 0-flats, 1-flats, 2-flats, (r−2)-flats and (r−1)-flats in pg(r,q) are called points, lines, planes, secundums and hyperplanes, respectively. let c be an [n,k,d]q code having no coordinate which is identically zero. the columns of a generator matrix of c can be considered as a multiset of n points in σ = pg(k−1,q), denoted by mc. an i-point is a point of σ which has multiplicity i in mc. denote by γ0 the maximum multiplicity of a point from σ in mc and let ci be the set of i-points in σ, 0 ≤ i ≤ γ0. for any subset s of σ, the multiplicity of s with respect to mc, denoted by mc(s), is defined as mc(s) = ∑γ0 i=1 i·|s∩ci|, where |t | denotes the number of elements in a set t . a line l with t = mc(l) is called a t-line. a t-plane and so on are defined similarly. then we obtain the partition σ = ⋃γ0 i=0 ci such that n = mc(σ) and n−d = max{mc(π) | π ∈fk−2}, where fj denotes the set of j-flats of σ. conversely, such a partition σ = ⋃γ0 i=0 ci as above gives an [n,k,d]q code in the natural manner. for an m-flat π in σ, we define γj(π) = max{mc(∆) | ∆ ⊂ π, ∆ ∈fj} for 0 ≤ j ≤ k − 2. we denote simply by γj instead of γj(σ). then γk−2 = n−d, γk−1 = n. for a griesmer [n,k,d]q code, it is known (see [19]) that γj = j∑ u=0 ⌈ d qk−1−u ⌉ for 0 ≤ j ≤ k − 1. (1) so, every griesmer [n,k,d]q code is projective if d ≤ qk−1. we denote by λs the number of s-points in σ. note that we have λ2 = λ0 + n−θk−1 (2) when γ0 = 2. denote by ai the number of i-hyperplanes in σ. the list of ai’s is called the spectrum of c. we usually use τj’s for the spectrum of a hyperplane of σ to distinguish from the spectrum of c. let θj be the number of points in a j-flat, i.e., θj = (qj+1 − 1)/(q− 1). simple counting arguments yield the following. lemma 2.1 ([15]). (1) n−d∑ i=0 ai = θk−1. (2) n−d∑ i=1 iai = nθk−2. (3) n−d∑ i=2 i(i− 1)ai = n(n− 1)θk−3 + qk−2 γ0∑ s=2 s(s− 1)λs. when γ0 ≤ 2, the above three equalities yield the following: n−d−2∑ i=0 ( n−d− i 2 ) ai = ( n−d 2 ) θk−1 −n(n−d− 1)θk−2 + ( n 2 ) θk−3 + q k−2λ2. (3) if ai = 0 for all i < n−d, then every point in σ is an s-point for some integer s. this fact is known as follows. lemma 2.2 ([2]). any linear code over a finite field with constant hamming weight is a replication of simplex (i.e., dual hamming) codes. 103 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 lemma 2.3 ([27]). let π be an w-hyperplane through a t-secundum δ. then (1) t ≤ γk−2 − (n−w)/q = (w + qγk−2 −n)/q. (2) aw = 0 if a [w,k − 1,d0]q code with d0 ≥ w − ⌊ w+qγk−2−n q ⌋ does not exist, where bxc denotes the largest integer less than or equal to x. (3) γk−3(π) = ⌊ w+qγk−2−n q ⌋ if a [w,k − 1,d1]q code with d1 ≥ w − ⌊ w+qγk−2−n q ⌋ + 1 does not exist. (4) let cj be the number of j-hyperplanes through δ other than π. then ∑ j cj = q and∑ j (γk−2 − j)cj = w + qγk−2 −n−qt. (4) (5) for a γk−2-hyperplane π0 with spectrum (τ0, · · · ,τγk−3 ), τt > 0 holds if w + qγk−2 −n−qt < q. the next two lemmas are needed to prove theorems 1.3 and 1.4. lemma 2.4 ([11]). the spectrum of a [2q2 − 2q − 4, 3, 2q2 − 4q − 2]q code with q ≥ 8 is one of the followings: (a) (aq−4,aq−2,a2q−3,a2q−2) = (1, 3, 2q,q2 −q − 3), (b) (aq−3,aq−2,a2q−4,a2q−3,a2q−2) = (2, 2, 1, 2q − 2,q2 −q − 2), (c) (aq−3,aq−2,a2q−4,a2q−3,a2q−2) = (1, 3, 1, 2q − 1,q2 −q − 3), (d) (aq−2,a2q−4,a2q−3,a2q−2) = (4, 1, 2q,q2 −q − 4) or (e) (aq−2,a2q−4,a2q−2) = (4,q + 1,q2 − 4). lemma 2.5 ([11]). the spectrum of a [2q2−q−3, 3, 2q2−3q−2]q code with q ≥ 7 is one of the followings: (a) (aq−3,aq−1,a2q−2,a2q−1) = (1, 2, 2q,q2 −q − 2), (b) (aq−2,aq−1,a2q−3,a2q−2,a2q−1) = (2, 1, 1, 2q − 2,q2 −q − 1), (c) (aq−2,aq−1,a2q−3,a2q−2,a2q−1) = (1, 2, 1, 2q − 1,q2 −q − 2), (d) (aq−1,a2q−3,a2q−2,a2q−1) = (3, 1, 2q,q2 −q − 3) or (e) (aq−1,a2q−3,a2q−1) = (3,q + 1,q2 − 3). an n-set k in pg(2,q) is an (n,r)-arc if every line meets k in at most r points and if some line meets k in exactly r points. let mr(2,q) denote the largest value of n for which an (n,r)-arc exists in pg(2,q). see table 1 for the known values and bounds on mr(2,q) for 3 ≤ q ≤ 13 [1]. an (n, 2)-arc is simply called an n-arc in pg(2,q), see [8]. a set l of s lines in σ = pg(2,q) is called an s-arc of lines in σ if l forms an s-arc in the dual space σ∗ of σ, that is, no three lines of l are concurrent. lemma 2.6 ([9]). (1) mr(2,q) ≤ (r − 1)q + r. (2) mr(2,q) ≤ (r − 1)q + r − 3 for 4 ≤ r < q with r 6 |q. (3) mr(2,q) ≤ (r − 1)q + r − 4 for 9 ≤ r < q with r 6 |q. (4) mq−2(2,q) = q2 − 2q − 3 √ q − 2 for odd square q > 112. (5) mq−2(2,q) ≤ q2 − 2q −pedp e+1+1 pe+1 e− 2 for q = p2e+1 > 17. 104 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 table 1. the known values and bounds on mr(2, q). q 3 4 5 7 8 9 11 13 r 2 4 6 6 8 10 10 12 14 3 9 11 15 15 17 21 23 4 16 22 28 28 32 38–40 5 29 33 37 43–45 49–53 6 36 42 48 56 64–66 7 49 55 67 79 8 65 78 92 9 89–90 105 10 100–102 118–119 11 132–133 12 145–147 (6) mq−2(2,q) ≤ q2 − 2q − 2 √ q − 2 for q = 22e > 4 or q ∈{52, 72, 92, 112}. (7) mq−1(2,q) = q2 −q − 2 √ q − 1 for square q > 4. (8) mq−1(2,q) ≤ q2 −q −pedp e+1+1 pe+1 e− 1 for q = p2e+1 > 19. lemma 2.7 ([12]). let c be a [gq(3,d), 3,d]q code for d = 2q2 − rq, 3 ≤ r ≤ q − q/p, q = ph with p prime. then, (1) the multiset mc consists of two copies of the plane with an r-arc of lines deleted, (2) c has spectrum (aq−r+2,a2q−r+2) = (r,θ2 −r). lemma 2.8 ([25]). let c be an [n,k,d]q code and let ∪ γ0 i=0ci be the partition of σ = pg(k − 1,q) obtained from c. if ∪i≥1ci contains a t-flat ∆ and if d > qt, then there exists an [n−θt,k,d′]q code c′ with d′ ≥ d−qt. the punctured code c′ in lemma 2.8 can be constructed from c by removing the t-flat ∆ from the multiset mc. the method to construct new codes from a given [n,k,d]q code by deleting the coordinates corresponding to some geometric object in pg(k − 1,q) is called geometric puncturing, see [21]. an [n,k,d]q code with generator matrix g is called extendable if there exists a vector h ∈ fkq such that the extended matrix [g,ht] generates an [n + 1,k,d + 1]q code. the following theorems will be applied to prove that a [gq(3,d), 3,d = 2q2 −rq − 1]q code is extendable in lemma 2.12. theorem 2.9 ([6, 7]). let c be an [n,k,d]q code with d ≡ −1 (mod q), k ≥ 3. then c is extendable if ai = 0 for all i 6≡ 0, −1 (mod q). theorem 2.10 ([20, 28]). let c be an [n,k,d]q code with d ≡ −2 (mod q), k ≥ 3, q ≥ 5. then c is extendable if ai = 0 for all i 6≡ 0,−1,−2 (mod q). theorem 2.11 ([26]). let c be an [n,k,d]q code with gcd(d,q) = 1. then c is extendable if∑ i 6≡n,n−d (mod q) ai < q k−2. lemma 2.12. the spectrum of a [2q2−(r−2)q−(r−1), 3, 2q2−rq−1]q code for 3 ≤ r ≤ q+12 , q = p h with p prime is (aq−r+1,aq−r+2,a2q−r+1,a2q−r+2) = (1,r − 1,q,q2 − r + 1) or (aq−r+2,a2q−r+1,a2q−r+2) = (r,q + 1,q2 −r). 105 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 proof. let c be an [n = 2q2 − (r − 2)q − (r − 1), 3,d = 2q2 − rq − 1]q code for 3 ≤ r ≤ q+12 , q = p h with p prime. note that c is extended to the code in lemma 2.7 if c is extendable. from (1), we have γ0 = 2 and γ1 = 2q − (r − 2). since (γ1 − γ0)θ1 + γ0 − 1 = n, the lines through a fixed 2-point is one (γ1 −1)-line and q γ1-lines. hence ai = 0 for θ1 + 1 ≤ i ≤ γ1 −2. let l be a t-line containing a 1-point p . considering the lines through p , we get n = 2q2−(r−2)q−(r−1) ≤ (γ1−1)q + t, giving q−(r−1) ≤ t. so, ai = 0 for 1 ≤ i ≤ q −r. suppose aθ1 > 0. then, c is not extendable by lemma 2.7. let l be a θ1-line. since n = (γ1 −1)q + θ1−r, the lines (6= l) through a fixed 1-point on l are r (γ1−1)-lines and (q−r) γ1-lines if q ≥ 2r. then, c is extendable from theorem 2.11, a contradiction. when q = 2r − 1, the lines ( 6= l) through a fixed 1-point on l are either “one θ1-line and (q − 1) γ1-lines" or “r (γ1 − 1)-lines and (q − r) γ1-lines". if a 0-point exists, we have n ≥ (γ1 − 1)θ1 = n + q, a contradiction. hence [q2 − (r− 1)q −r, 3,q2 −rq − 1]q code exists by lemma 2.8. however, there exists no (q2 − (r − 1)q −r,q − (r − 1))-arc from lemma 2.6 (2) when q = 2r − 1 ≥ 7 and from table 1 when (q,r) = (5, 3), a contradiction. thus aθ1 = 0. next, suppose a0 > 0. then, c is not extendable by lemma 2.7. let l be a 0-line. since n = γ1q + 0 − (r − 1) and γ1 − (r − 1) > θ1, the lines ( 6= l) through a fixed 0-point on l are (γ1 − 1)-lines or γ1-lines. hence aj > 0 implies j ∈ {0,γ1 − 1,γ1} and a0 = 1. then, c is extendable by theorem 2.11, a contradiction. hence a0 = 0. finally, suppose ai > 0 for some q − r + 3 ≤ i ≤ q. then, c is not extendable by lemma 2.7. let l be a (q −e)-line with 0 ≤ e ≤ r− 3 and let q be a 0-point on l. if four of the lines through q have multiplicities at most q, then we have n ≤ 4q + (q − 3)γ1 = n− 2q + 4(r − 2) < n, a contradiction. so, at most two of the lines ( 6= l) through q have no 2-point and∑ i6≡n,n−d (mod q) ai ≤ 2(e + 1) + 1 ≤ 2r − 3 < 2r − 1 ≤ q. then, applying theorem 2.11, c is extendable, a contradiction. hence ai = 0 for all i 6∈ {q − r + 1,q − r + 2, 2q − r + 1, 2q − r + 2}. applying theorem 2.9, c is extendable. hence c can be obtained from a [2q2−(r−2)q−(r−2), 3, 2q2−rq]q code c′ by removing one coordinate. let r be the point corresponding to the coordinate. there are two possible spectra (aq−r+1,aq−r+2,a2q−r+1,a2q−r+2) = (1,r−1,q,q2−r+ 1) or (aq−r+2,a2q−r+1,a2q−r+2) = (r,q + 1,q2 − r), according to the cases r is a 1-point or a 2-point, respectively. lemma 2.13. the spectrum of a [q2 − r, 3,q2 − q − r]q code with 1 ≤ r ≤ q − 2 satisfies ai = 0 for 1 ≤ i ≤ q −r − 1. proof. let c be a [q2−r, 3,q2−q−r]q code, which is griesmer. from (1), we have γ0 = 1 and γ1 = q. let l be an i-line with i > 0 containing a 1-point p . counting the 1-points on the lines through p , we get n = q2 −r ≤ (q − 1)q + i, whence q −r ≤ i. 3. proof of theorem 1.2 we assume q ≥ 7 since the theorem is already known for (r,q) = (3, 5) [14]. we first prove the non–existence of [gq(4,d), 4,d]q code for d = 2q3 −rq2 − 2q + 2. lemma 3.1. there exists no [n = 2θ3 −rθ2 − 2θ1 + 3, 4,d = 2q3 −rq2 − 2q + 2]q code for 3 ≤ r ≤ q+12 , q = ph ≥ 7 with p prime. proof. let c be a putative [n = 2q3 − (r − 2)q2 − rq − (r − 3), 4,d = 2q3 − rq2 − 2q + 2]q code with 3 ≤ r ≤ (q+ 1)/2, q ≥ 5. note that n = gq(4,d) and hence γ0 = 2, γ1 = 2θ1−r, γ2 = n−d = 2θ2−rθ1−1 from (1). let ∆ be a γ2-plane. since γ2 = (γ1 −2)(q + 1) + 2−1 and n = (γ1 −2)θ2 + 2−(2q−1), every line on ∆ through a 2-point is a γ1-line or a (γ1 − 1)-line, and any i-plane through a 2-point satisfies (γ1−2)(q + 1) + 2−(2q−1) = γ2−2(q−1) ≤ i ≤ γ2. by lemma 2.3 (1), any t-line in an i-plane satisfies t ≤ i + r − 3 q + 1. (5) 106 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 the spectrum of ∆ is either (a) (τq−r+1,τq−r+2,τ2q−r+1,τ2q−r+2) = (1,r − 1,q,q2 − r + 1) or (b) (τq−r+2,τ2q−r+1,τ2q−r+2) = (r,q + 1,q 2 −r) by lemma 2.12. let δ be an i-plane. it follows from (5) and ∆’s possible spectra that q − r + 1 ≤ i+q+r−3 q , i.e., q2−rq−(r−3) ≤ i. assume i ≤ θ2. since δ has no 2-point, δ∩∆ is a (q−r + 1)-line or a (q−r + 2)-line. so, i ≤ θ2 − r + 1. now, let i = q2 −uq − (r − 3) + s with 0 ≤ u ≤ r − 2, 0 ≤ s ≤ q − 1. from (5), we have t ≤ q −u + 1. if t = q −u + 1, then i + q + r − 3 − qt = s ≤ q − 1, and the γ2-plane ∆ contains a t-line by lemma 2.3 (5), a contradiction. hence t ≤ q −u. considering the lines in δ through a fixed 1-point of δ∩ ∆, i ≤ (q−u− 1)q + (q−r + 2) = q2 −uq− (r− 2) < i, a contradiction. thus, ai = 0 for q2 − (r − 2)q − (r − 3) ≤ i ≤ θ2, and ai > 0 implies q2 −rq − (r − 3) ≤ i ≤ q2 − (r − 2)q − (r − 2) or γ2 − 2q + 2 ≤ i ≤ γ2. from (3), we get ∑ i ( γ2 − i 2 ) ai = q 2λ2 −q5 + 3r − 2 2 q4 − r2 − 3r − 4 2 q3 − r2 + 6 2 q2 − 2q + 3. (6) for any w-plane through a t-line, (4) gives ∑ j cj = q and∑ j (2q2 − (r − 2)q − (r − 1) − j)cj = w + q + r − 3 −qt. (7) suppose ai > 0 for i = q2 − rq − (r − 3) + e with 0 ≤ e ≤ q − 1. since δ ∩ ∆ is a (q − r + 1)-line by (5), ∆ has spectrum (a). if ai > 0, the rhs of (7) is q2 − (r − 1)q + e− qt ≤ q2 − (r − 2)q − 1. since the coefficient of cq2−(r−2)q−(r−2) in (7) is q2 − 1 > q2 − (r − 2)q − 1, we get ai = 1 and aj = 0 for q2 − rq − (r − 3) ≤ j ≤ q2 − (r − 2)q − (r − 2) with j 6= i. setting w = n − d, the maximum possible contributions of cj’s in (7) to the lhs of (6) on ∆ are (cq2−rq−(r−3)+e,cn−d−e,cn−d) = (1, 1,q − 2) for t = q−r+ 1; (cγ2−2(q−1),cγ2−(q−1),cn−d) = ( q+1 2 , 1, q−3 2 ) if q is odd and (cγ2−2(q−1),cn−d) = ( q 2 + 1, q 2 −1) if q is even for t = q−r + 2; (cγ2−2(q−1),cn−d) = (1,q−1) for t = 2q−r + 1; (cγ2−(q−2),cn−d) = (1,q−1) for t = 2q −r + 2. hence, when q is odd, we get (lhs of (6)) ≤ (( q2 + 2q − 2 −e 2 ) + ( e 2 )) τq−r+1 + (q + 1 2 ( 2(q − 1) 2 ) + ( q − 1 2 )) τq−r+2 + ( 2(q − 1) 2 ) τ2q−r+1 + ( q − 2 2 ) τ2q−r+2 ≤ ( q2 + 2q − 2 2 ) τq−r+1 + (q + 1 2 ( 2(q − 1) 2 ) + ( q − 1 2 )) τq−r+2 + ( 2(q − 1) 2 ) τ2q−r+1 + ( q − 2 2 ) τ2q−r+2, giving λ2 < q 3 − 3r − 4 2 q2 + r2 −r − 3 2 q + r2 − 3r + 4 2 . when q is even, we can similarly obtain λ2 < q 3 − 3r − 4 2 q2 + r2 −r − 3 2 q + r2 − 2r + 3 2 . on the other hand, since λ0 ≥ |δ ∩c0|, we have λ2 = n−θ3 + λ0 ≥ n−θ3 + θ2 − (q2 −rq − (r − 3) + e) ≥ q3 − (r − 1)q2 −q + 1 107 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 giving a contradiction for q ≥ 2r − 1 with q ≥ 7 and r ≥ 3. thus, ai = 0 for q2 − rq − (r − 3) ≤ i ≤ q2 − (r − 1)q − (r − 2). by a similar argument using lemma 2.3, (6) and (7), we can get ai = 0 for all q2−(r−1)q−(r−3) ≤ i ≤ q2 − (r − 2)q − (r − 2). hence, ai > 0 implies γ2 − 2q + 2 ≤ i ≤ γ2. finally, we investigate (6) and (7) with i = n − d again. we only give the proof when ∆ has spectrum (a) since one can prove similarly for spectrum (b). assume q is odd. the maximum possible contributions of cj’s in (7) to the lhs of (6) on ∆ are (cγ2−2(q−1),cn−d−1,cn−d) = ( q+3 2 , 1, q−5 2 ) for t = q−r + 1; (cγ2−2(q−1),cγ2−(q−1),cn−d) = ( q+1 2 , 1, q−3 2 ) for t = q−r + 2; (cγ2−2(q−1),cn−d) = (1,q−1) for t = 2q −r + 1; (cγ2−(q−2),cn−d) = (1,q − 1) for t = 2q −r + 2. hence we get (lhs of (6)) ≤ q + 3 2 ( 2(q − 1) 2 ) τq−r+1 + ( q + 1 2 ( 2(q − 1) 2 ) + ( q − 1 2 ) )τq−r+2 + ( 2(q − 1) 2 ) τ2q−r+1 + ( q − 2 2 ) τ2q−r+2, giving λ2 < q 3 − 3r − 3 2 q2 + r2 −r − 5 2 q + r2 − 3r + 6 2 . on the other hand, we have λ2 = n−θ3 + λ0 ≥ (2θ3 −rθ2 − 2θ1 + 3) −θ3 = q3 − (r − 1)q2 − (r + 1)q − (r − 2), giving a contradiction for q ≥ 2r−1. one can get a contradiction similarly when q is even. this completes the proof. in the above proof, we often obtain a contradiction to rule out the existence of some i-plane by eliminating the value of λ2 using (4), (3) and the possible spectra for a fixed w-plane. we refer to this proof technique as "(λ2,w)-ruling out method ((λ2,w)-rom)" in what follows. proof of theorem 1.2. let c be a putative [n = 2q3−(r−2)q2−rq−(r−2), 4,d = 2q3−rq2−2q + 1]q code with 3 ≤ r ≤ (q + 1)/2, q ≥ 5. by lemma 1, γ0 = 2, γ1 = 2q−(r−2), γ2 = n−d = 2θ2−rθ1−1. by lemma 2.12, the spectrum of a γ2-plane ∆ is (a) (τq−r+1,τq−r+2,τ2q−r+1,τ2q−r+2) = (1,r−1,q,q2−r+1) or (b) (τq−r+2,τ2q−r+1,τ2q−r+2) = (r,q + 1,q2 −r). so a j-line on ∆ satisfies j ∈{q −r + 1,q −r + 2, 2q −r + 1, 2q −r + 2}. (8) by lemma 2.3, an i-plane satisfies i ≥ (q−r + 1)q−(q + r−2) = q2 −rq−(r−2). hence ai = 0 for any i < q2 −rq− (r− 2). assume that an i-plane contains a 2-point. since (γ1 − 2)θ2 + 2 = n + 2q, we have i ≥ (γ1 − 2)θ1 + 2 − 2q = (2q −r)θ1 + 2 − 2q = 2q2 −rq − (r − 2) > θ2 for q ≥ 2r−1. hence an i-plane with i ≤ θ2 = q2 +q+ 1 has no 2-point. thus ai = 0 if i < q2−rq−(r−2) or θ2 < i < 2q2 −rq − (r − 2). let δ be a i-plane, s = γ1(δ). then, δ ∩c is an (i,s)-arc, corresponding to an [i, 3, i−s]q code. by lemma 2.3(1), s ≤ i + r − 2 q + 1 (9) by lemma 2.3 (5), δ contains a t-line if i + q + (r − 2) −qt < q. (10) (case 1) assume q2 −rq − (r − 2) ≤ i < q2 − (r − 1)q − (r − 2). we have s ≤ q−(r−1) by (9). since δ∩∆ is a j-plane satisfying (8), we get s = q−(r−1). by lemma 108 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 2.6 (2), i ≤ (q −r)q + (q −r + 1) − 3 = q2 − (r − 1)q − (r + 2). (case 2) assume q2 − (r − 1)q − (r − 2) ≤ i < q2 − (r − 2)q − (r − 2). by (9), s ≤ q − (r − 2). it follows from (case 1) that s = q − (r − 2). by lemma 2.6 (2), we get i ≤ q2 − (r − 2)q − (r + 1). (case 3) assume q2 − (r − 2)q − (r − 2) ≤ i < q2 − (r − 3)q − (r − 2). by (9), s ≤ q − (r− 3). it follows from (case 2) that s = q − (r− 3). then, by lemma 2.3 (5), ∆ has a (q − (r − 3))-line, a contradiction. hence ai = 0. (case 4) assume q2 −uq − (r − 2) ≤ i < q2 − (u− 1)q − (r − 2), 0 ≤ u ≤ r − 3. by (9), s ≤ q − u + 1. if s = q − u + 1, then ∆ contains a (q − u + 1)-line by lemma 2.3 (5), a contradiction. hence s ≤ q − u, and δ ∩ ∆ is a (q − r + 1)-line or a (q − r + 2)-line. considering the lines in δ through a fixed 1-point on δ ∩ ∆, we have i ≤ (q − u − 1)q + q − r + 2 = q2 − uq − (r − 2). hence i = q2 − uq − (r − 2), and δ ∩ ∆ is a (q − r + 2)-line. let p be any 1-point in δ. then, there exists a γ2-plane through p meeting δ in a (q − r + 2)-line. otherwise, one can get an [n + 1, 4,d + 1]q code by adding p to the multiset for c, which contradicts theorem 3.1. thus, the lines through p in δ are one (q − r + 2)-line and q (q −u)-lines, and other possible lines in δ are 0-lines. let ci be the code corresponding to δ. then ci is an [i, 3, i− (q −u) = q2 − (u + 1)q − (r − 2 −u)]q code with spectrum (µ0,µq−r+2,µq−u) = ( q ( r − 2 q −u − r −u− 3 q −r + 2 ) , i q −r + 2 , iq q −u ) , (11) where µj is the number of j-lines in δ. since (q −u)(q − 1) < i from the assumption q ≥ 2r − 1, we get µ0 = 0 or 1. take a 0-point q not on a 0-line in δ. it follows from (q − u)q + q − r + 2 = i + q that r − 2 −u divides q. so, r − 2 −u = pm (12) for some integer m ≥ 0. if m = 0, then u = r−3 and i = q2−(r−3)q−(r−2). since gcd(q−r+2,q−r+3) = 1, (q −r + 2)|i implies (q −r + 3)|q. from (11), µ0 = q q −r + 3 (r− 2) 6= 0. hence µ0 = 1, r = 3, u = 0, and i = q2 − 1. assume m > 0. then h ≥ 2 and 1 ≤ m ≤ h− 1, for r − 2 ≤ (q − 3)/2. suppose µ0 = 0. from (11) and (12), we have (u + 1)q = p2m + u(r − 1). (13) if h ≤ 2m, then, from (12) and (13), q divides either u or r − 1, a contradiction. hence 2m ≤ h − 1. from (13), we get q = p2m u + 1 + u(r − 1) u + 1 < ph−1 + r − 1 ≤ q 2 + q − 1 2 < q, a contradiction. hence µ0 = 1. since (q −u)(q − 1) + q − r + 2 = i + u, the number of (q − r + 2)-lines through a fixed 0-point on the 0-line in δ is 1 + u/(r− 2 −u). so, pm divides u and r− 2 also from (12). from µ0 = 1 and (11), we have (q −u)(q −r + 2) q = q(u + 1) −u(r − 1) −p2m. (14) suppose h ≤ 2m. then, from (14), we obtain (r − 2)((1 −u)q −u) ≡ 0 (mod q2). (15) since q divides u(r − 2), (15) yields (r − 2)(q − u) ≡ 0 (mod q), a contradiction. hence 2m ≤ h − 1. if u = 0, then (14) gives r − 2 = p2m, which contradicts (12). thus, u > 0. then, from (14), we have q(u + 1) −u(r − 1) −p2m < q −r + 2, giving qu < u(r − 1) − (r − 1) + 1 + p2m, i.e., q ≤ (u− 1)(r − 1) u + p2m u < pm + r − 1 ≤ √ q p + q − 1 2 < q, 109 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 a contradiction. hence ai = 0 except for the case (r,u) = (3, 0). (case 5) assume q2 + q − (r − 2) ≤ i ≤ θ2. by (9), s ≤ q + 2. if s = q + 2, then ∆ contains a (q + 2)-line by lemma 2.3 (5), a contradiction. hence s ≤ q + 1. so, δ ∩ ∆ is a (q − r + 1)-line or a (q − r + 2)-line. considering the lines through a fixed 1-point on δ ∩ ∆, we get i ≤ q · q + (q − r + 2) = q2 + q − (r − 2). hence i = q2 + q − (r − 2). since θ2 − (q2 + q − (r− 2)) = r− 1 and θ1 − (r− 1) = q −r + 2, a t-line on δ satisfies θ1 ≥ t ≥ q −r + 2. so, δ ∩ ∆ is a (q −r + 2)-line. hence, the spectrum of δ is (τq−r+2,τq,τq+1) = (1, (r − 1)q, (q − r + 2)q). then any point of δ\∆ is not contained in a γ2-plane, and c is extendable, which contradicts lemma 2.11. hence ai = 0. from the above (case 1) (case 5), ai > 0 implies i ∈{q2 −rq − (r − 2), · · · ,q2 − (r − 1)q − (r + 2),q2 − (r − 1)q − (r − 2), · · · , q2 − (r − 2)q − (r + 1), 2q2 −rq − (r − 2), · · · , 2q2 − (r − 2)q − (r − 1)}, or i = q2 − 1 when r = 3. by (3), we get ∑ j ( γ2 − j 2 ) = q2λ2 −q5 + 3r − 2 2 q4 − r2 − 3r − 4 2 q3 − r2 + 2 2 q2 − 2q + 1. (16) note that the lhs of (16) contains the term ( q2−q−1 2 ) aq2−1 only for r = 3. for any w-plane through a t-line, (4) gives ∑ j cj = q and∑ j (2q2 − (r − 2)q − (r − 1) − j)cj = w + q + (r − 2) −qt. (17) now, we rule out the possible i-planes for q2−rq−(r−2) ≤ i ≤ q2−(r−1)q−r−2 by (λ2,γ2)-rom. suppose ai > 0 for i = q2 −rq − (r− 2) + e with 0 ≤ e ≤ q − 4 and let δ be an i-plane. we may assume that ∆ has spectrum (a) since δ∩∆ is a (q−r + 1)-line. it follows from (4) that ai = 1 and that aj = 0 for q2 − rq − (r − 2) ≤ j ≤ q2 + q − (r − 2) with j 6= i. assume q is odd. setting w = n − d, the maximum possible contributions of cj’s in (17) to the lhs of (16) are (cq2−rq−(r−2)+e,cn−d−e,cn−d) = (1, 1,q − 2) for t = q − r + 1; (c2q2−rq−(r−2),c2q2−(r−3 2 )q−(r−3 2 ),cn−d) = ( q+1 2 , 1, q−3 2 ) for t = q − r + 2; (c2q2−rq−(r−2),cn−d) = (1,q−1) for t = 2q−r + 1; (c2q2−(r−1)q−(r−2),cn−d) = (1,q−1) for t = 2q−r + 2. hence we get (lhs of (16)) ≤ ( ( q2 + 2q − 1 −e 2 ) + ( e 2 ) )τq−r+1 + ( q + 1 2 ( 2q − 1 2 ) + (q−1 2 2 ) )τq−r+2 + ( 2q − 1 2 ) τ2q−r+1 + ( q − 1 2 ) τ2q−r+2 ≤ ( q2 + 2q − 1 2 ) τq−r+1 + ( q + 1 2 ( 2q − 1 2 ) + (q−1 2 2 ) )τq−r+2 + ( 2q − 1 2 ) τ2q−r+1 + ( q − 1 2 ) τ2q−r+2, giving λ2 < q 3 + 4 − 3r 2 q2 + r2 −r − 1 2 q + 4r2 − 7r + 3 8 . on the other hand, since λ0 ≥ |δ ∩c0| = θ2 − i, we have λ2 = n−θ3 + λ0 ≥ n−θ3 + (θ2 − (q2 − (r − 1)q − (r + 2))) = q3 + (1 −r)q2 −q + 4, 110 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 giving a contradiction. one can get a contradiction similarly when q is even. hence ai = 0. one can also rule out possible i-planes for i = q2 − (r − 1)q − (r − 2) + e with 0 ≤ e ≤ q − 3 by (λ2,γ2)-rom. next, we rule out the possible (q2−1)-plane by (λ2,q2−1)-rom. suppose aq2−1 > 0 for r = 3. the spectrum of a (q2−1)-plane is (τ0,τq−1,τq) = (1,q+1,q2−1) since it corresponds to a [q2−1, 3,q2−q−1]q code. from (17) we have aq2−1 = 1 and aj = 0 for q2−2q−1 ≤ j ≤ q2−q−5. then, the maximum possible contributions of cj’s in (17) with w = q2 − 1 to the lhs of (16) are (ci,c2q2−2q−5,cn−d−1) = (1, 1,q − 2) for t = 0; (c2q2−3q−1,cn−d−1,cn−d) = (1, 1,q − 2) for t = q − 1; c2q2−q−3 = q for t = q. hence we get (lhs of (16)) ≤ ( q2 −q − 1 2 ) + ( ( q2 −q − 1 2 ) + ( q + 3 2 ) )τ0 + ( 2q − 1 2 ) τq−1 + 0 · τq giving λ2 < q3 − 5q2/2 − 2q + 4. on the other hand, since λ0 ≥ θ2 − i, we have λ2 = n−θ3 + λ0 ≥ 2q3 −q2 − 3q − 1 −θ3 + θ2 − (q2 − 1) = q3 − 2q2 − 3q, giving a contradiction. hence aq2−1 = 0. finally, we apply (λ2,γ2)-rom for i = γ2 to get a contradiction. we only give the proof when ∆ has spectrum (a) since one can prove similarly for spectrum (b). assume q is odd. the maximum possible contributions of cj’s in (17) to the lhs of (16) on ∆ are (c2q2−rq−(r−2),c2q2−(r−1 2 )q−(r−3 2 ),cn−d) = (q+1 2 , 1, q−3 2 ) for t = q − r + 1; (c2q2−rq−(r−2),c2q2−(r−3 2 )q−(r−3 2 ),cn−d) = ( q+1 2 , 1, q−3 2 ) for t = q − r + 2; (c2q2−rq−(r−2),cn−d) = (1,q−1) for t = 2q−r + 1; (c2q2−(r−1)q−(r−2),cn−d) = (1,q−1) for t = 2q−r + 2. hence (lhs of (16)) ≤ ( q + 1 2 ( 2q − 1 2 ) + (3q−1 2 2 ) )τq−r+1 + ( q + 1 2 ( 2q − 1 2 ) + (q−1 2 2 ) )τq−r+2 + ( 2q − 1 2 ) τ2q−r+1 + ( q − 1 2 ) τ2q−r+2, giving λ2 < q 3 + 3 − 3r 2 q2 + r2 −r − 3 2 q + 4r2 − 7r + 4 4 . on the other hand, it follows from λ0 ≥ θ2 − i that λ2 = n−θ3 + λ0 ≥ n−θ3 = q3 − (r − 1)q2 − (r + 1)q − (r − 1), giving a contradiction. one can get a contradiction similarly when q is even. this completes the proof. 4. proof of theorem 1.3 to prove theorems 1.3 and 1.4, the possible spectra of some 3-dimensional codes in table 2 are needed. we omit the proof of theorem 1.4 as noted in section 1. see [13] for the proof of theorem 1.3 for q = 9. let c be a putative [n = 2q3−q2−4q−2, 4,d = 2q3− 3q2−3q+1]q code for q ≥ 11. it follows from (1) that γ0 = 2, γ1 = 2q−1, γ2 = 2q2−q−3. the spectrum of a γ2-plane ∆ is one of the followings by lemma 2.5: (a) (τq−3,τq−1,τ2q−2,τ2q−1) = (1, 2, 2q,q2 −q− 2), (b) (τq−2,τq−1,τ2q−3,τ2q−2,τ2q−1) = (2, 1, 1, 2q − 2,q2 − q − 1), (c) (τq−2,τq−1,τ2q−3,τ2q−2,τ2q−1) = (1, 2, 1, 2q−1,q2−q−2), (d) (τq−1,τ2q−3,τ2q−2,τ2q−1) = (3, 1, 2q,q2−q−3), or (e) (τq−1,τ2q−3,τ2q−1) = (3,q + 1,q2 − 3). hence, a j-line on ∆ satisfies j ∈{q − 3,q − 2,q − 1, 2q − 3, 2q − 2, 2q − 1}. (18) 111 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 table 2. the spectra of some [n, 3, d]q codes for q ≥ 9 ([5, 8]). parameters possible spectra [q2 − 3, 3,q2 −q − 3]q, q ≥ 11 (a0,aq−3,aq−1,aq) = (1, 1, 3q,q2 − 2q − 1) (a0,aq−2,aq−1,aq) = (1, 3, 3q − 3,q2 − 2q) [q2 − 3, 3,q2 −q − 3]q, q = 9 (a0,a6,a8,a9) = (1, 1, 27, 62) (a0,a7,a8,a9) = (1, 3, 24, 63) (a6,a9) = (13, 78) [q2 − 2, 3,q2 −q − 2]q (a0,aq−2,aq−1,aq) = (1, 1, 2q,q2 −q − 1) [q2 − 1, 3,q2 −q − 1]q (a0,aq−1,aq) = (1,q + 1,q2 − 1) [q2, 3,q2 −q]q (a0,aq) = (1,q2 + q) [q2 + q − 3, 3,q2 − 4]q (aq−3,aq,aq+1) = (1, 4q,q2 − 3q) (aq−2,aq−1,aq,aq+1) = (1, 3, 4q − 5,q2 − 3q + 2) (aq−1,aq,aq+1) = (6, 4q − 8,q2 − 3q + 3) [q2 + q − 2, 3,q2 − 3]q (aq−2,aq,aq+1) = (1, 3q,q2 − 2q) (aq−1,aq,aq+1) = (3, 3q − 3,q2 − 2q + 1) [q2 + q − 1, 3,q2 − 2]q (aq−1,aq,aq+1) = (1, 2q,q2 −q) [q2 + q, 3,q2 − 1]q (aq,aq+1) = (q + 1,q2) [q2 + q + 1, 3,q2]q aq+1 = q 2 + q + 1 from lemma 2.1 (3), we have λ0(∆) = 5, 5, 4, 3, 4 for the cases a,b,c,d,e, respectively. by lemma 2.3, an i-plane satisfies i ≥ q(q−3)−(q+2) = q2−4q−2. hence ai = 0 for any i < q2−4q−2. assume that an i-plane contains a 2-point. since (γ1−2)θ2+2 = n+3q+1, we have i ≥ (γ1−2)θ1+2−(3q+1) = 2q2−4q−2. let δ be an i-plane, r = γ1(δ). then, δ∩c is an (i,r)-arc, corresponding to an [i, 3, i−r]q code. lemma 2.3 (1) gives r ≤ i + 2 q + 1. (19) for any w-plane through a t-line, (4) gives∑ j (γ2 − j)cj = w + q + 2 −qt (20) with ∑ j cj = q. the equality (2) yields: λ2 = q 3 − 2q2 − 5q − 3 + λ0. (21) assume q2−4q−2 ≤ i < q2−3q−2. from (19), and (18) we have r = q−3. then, i ≤ (q−4)q+(q−3)−4 = q2 − 3q − 7 for q ≥ 13 by lemma 2.6 (3) and i ≤ 78 for q = 11 by table 1. we also have that q2 − 3q − 2 ≤ i < q2 − 2q − 2 implies r = q − 2 and i ≤ (q − 3)q + (q − 2) − 4 = q2 − 2q − 6 and that q2−2q−2 ≤ i < q2−q−2 implies r = q−1 and i ≤ (q−2)q+ (q−1)−4 = q2−q−5. hence, i > q2−q−5 implies r ≥ q. assume q2−q−2 ≤ i < q2−2. by (19), r = q. (20) with (w,t) = (i,q) yields that cγ2 > 0, which contradicts that a γ2-plane has no q-line. hence ai = 0. similarly, q2 − 2 ≤ i < q2 + q − 2 implies r = q and i ≤ q2. the spectrum of a q2-plane is (τ0,τq) = (1,q2 + q) from table 2, which contradicts (18). hence qq2 = 0. we have aq2+q = aθ2 = 0 similarly. thus, we have ai = 0 for all i /∈{q2 − 4q − 2, . . . ,q2 − 3q − 7,q2 − 3q − 2, . . . ,q2 − 2q − 6,q2 − 2q − 2, . . . , q2 −q − 5,q2 − 2,q2 − 1,q2 + q − 2,q2 + q − 1, 2q2 − 4q − 2, . . . , 2q2 −q − 3}. note that a79 = a80 = a81 = 0 for q = 11. from (3), we get γ2−2∑ i=0 ( γ2 − i 2 ) ai = q 2λ2 − (q5 − 7 2 q4 − 7 2 q3 + 13 2 q2 + 7 2 q − 1). (22) 112 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 we first rule out possible (q2 +q−2)-planes by (λ2,q2 +q−2)-rom. suppose aq2+q−2 > 0. the spectrum of a [q2+q−2, 3,q2−3]q code is (x) (τq−2,τq,τq+1) = (1, 3q,q2−2q) or (y) (τq−1,τq,τq+1) = (3, 3q−3,q2− 2q+1) from table 2. setting w = q2 +q−2 in (20), the maximum possible contributions of cj’s to the lhs of (22) are (c2q2−4q−2,c2q2−2q−4,cn−d) = (1, 1,q−2) for t = q−2; (c2q2−4q−2,cn−d−1,cn−d) = (1, 1,q−2) for t = q − 1; (c2q2−2q−4,cn−d−1) = (1,q − 1) for t = q; cn−d−1 = q for t = q + 1. estimating the lhs of (22) for spectrum (x), we get (lhs of (22)) ≤ ( q2 − 2q − 1 2 ) + ( ( 3q − 1 2 ) + ( q + 1 2 ) )τq−2 + ( 3q − 1 2 ) τq−1 + ( q + 1 2 ) τq, giving λ2 ≤ (2q3 − 6q2 − 8q + 27)/2. on the other hand, (21) gives λ2 ≥ q3 − 2q2 − 5q, a contradiction. we also get a contradiction similarly for spectrum (y). hence aq2+q−2 = 0. one can prove aq2+q−1 = aq2−2 = aq2−1 = 0 for q ≥ 11 using the spectra in table 2, similarly. next, we rule out the possible i-planes for q2−4q−2 ≤ i ≤ q2−3q−7 for q ≥ 13 and for 75 ≤ i ≤ 78 for q ≥ 11 by (λ2,γ2)-rom. suppose ai > 0 for i = q2 −4q−2 + e with 0 ≤ e ≤ q−5, q ≥ 13. then, we may assume that ∆ has spectrum (a). note that rhs of (20) is at most q2 − 3q + e−qt ≤ q2 − 4q − 5. since ∆ has no 0-line and the coefficient of cq2−q−5 in (20) is q2 + 2, we get ai = 1 and aj = 0 for j ≤ q2 − q − 5 with j 6= i. setting w = n−d in (20), the maximum possible contributions of cj’s to the lhs of (22) are (ci,cn−d−e,cn−d) = (1, 1,q−2) for t = q−3; (c2q2−4q−2,cn−d−y,cn−d) = (x, 1,q−x−1) for t = q − 1; (c2q2−3q−2,cn−d) = (1,q − 1) for t = 2q − 2; (c2q2−2q−2,cn−d) = (1,q − 1) for t = 2q − 1, where (x,y) = (q/3, 4q/3 − 1), (x,y) = ((q− 1)/3, (7q− 4)/3), (x,y) = ((q + 1)/3, (q− 2)/3) if q ≡ 0, 1, 2 (mod 3), respectively. estimating the lhs of (22), we get (lhs of (22)) ≤ ( ( q2 + 3q − 1 −e 2 ) + ( e 2 ) )τq−3 + ( ( 3q − 1 2 ) x + ( y 2 ) )τq−1 + ( 2q − 1 2 ) τ2q−2 + ( q − 1 2 ) τ2q−1 ≤ ( q2 + 3q − 1 2 ) τq−3 + ( ( 3q − 1 2 ) q + 1 3 + (q−2 3 2 ) )τq−1 + ( 2q − 1 2 ) τ2q−2 + ( q − 1 2 ) τ2q−1, giving λ2 ≤ (18q3−45q2 + 81q + 92)/18. on the other hand, since ∆ has five 0-points and one (q−3)-line, say l, ∆\l has one 0-point. since cn−d ≥ q−e−1 ≥ 4 for t = q−3, there are at least four γ2-planes with spectrum (a) through l and (21) yields λ2 ≥ q3 − 2q2 − 5q − 3 + (θ2 − (q2 − 3q − 7)) + 4 = q3 − 2q2 −q + 5, giving a contradiction for q ≥ 13. for q = 11, we consider a putative i-plane with i = q2 − 4q − 2 + e with 0 ≤ e ≤ 3 in the same way. since cn−d ≥ q − e − 1 ≥ 7 for t = q − 3, we can get a contradiction as above. hence ai = 0 for q2 − 4q − 2 ≤ i ≤ q2 − 3q − 7. one can similarly prove that ai = 0 for q2 − 3q − 2 ≤ i ≤ q2 − 2q − 6 and for q2 − 2q − 2 ≤ i ≤ q2 −q − 5 by (λ2,γ2)-rom. thus, we have proved that ai = 0 for all i < 2q2 −4q−2. finally, applying (λ2,γ2)-rom for i = λ2, we get a contradiction as follows. setting w = n−d, the maximum possible contributions of cj’s in (20) to the lhs of (22) are (c2q2−4q−2,cn−d−w,cn−d) = (z, 1,q−z−1) for t = q−3; (c2q2−4q−2,cn−d−b,cn−d) = (a, 1,q−a−1) for t = q−2; (c2q2−4q−2,cn−d−y,cn−d) = (x, 1,q−x−1) for t = q−1; (c2q2−4q−2,cn−d) = (1,q − 1) for t = 2q − 3; (c2q2−3q−2,cn−d) = (1,q − 1) for t = 2q − 2; (c2q2−2q−2,cn−d) = (1,q − 1) for t = 2q − 1, where (a,b,x,y,z,w) = (q/3, 7q/3 − 1,q/3, 4q/3 − 1,q/3 + 1,q/3), ((q + 2)/3, (q − 1)/3, (q − 1)/3, (7q− 4)/3, (q + 2)/3, (4q− 1)/3), ((q + 1)/3, (4q− 2)/3, (q + 1)/3, (q− 2)/3, (q + 1)/3, (7q− 2)/3) if q ≡ 0, 1, 2 (mod 3), respectively. estimating the lhs of (22), we get (lhs of (22)) ≤ ( ( 3q − 1 2 ) z + ( w 2 ) )τq−3 + ( ( 3q − 1 2 ) a + ( b 2 ) )τq−2 +( ( 3q − 1 2 ) x + ( y 2 ) )τq−1 + ( 3q − 1 2 ) τ2q−3 + ( 2q − 1 2 ) τ2q−2 + ( q − 1 2 ) τ2q−1 113 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 giving λ2 ≤ (6q3−18q2 + 24q + 37)/6 if ∆ has spectrum (d) and if q ≡ 2 mod 3. on the other hand, (21) yields λ2 ≥ q3 −2q2 −5q−3, giving a contradiction for q ≥ 11. one can get a contradiction similarly for the other cases. this completes the proof. 5. proof of theorem 1.5 lemma 5.1. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 4q − 2 for q ≥ 9. proof. let c be a putative [n = q3 − 4q− 6, 4,d = q3 −q2 − 4q− 2]q code. note that n = gq(4,d) and hence γ0 = 1, γ1 = q, γ2 = n − d = q2 − 4 from (1). let ∆ be a γ2-plane and let δ be an i-plane. by lemma 2.13, the spectrum of ∆ satisfies τj = 0 for 1 ≤ j ≤ q−5. since a t-line in δ satisfies t ≤ (i+ 6)/q, we have ai = 0 for 1 ≤ i ≤ q − 7. assume i = sq − 6 + e with 0 ≤ e ≤ q − 1. for 2 ≤ s ≤ q − 5, we have γ1(δ) ≤ s−1 by lemma 2.3 (5). then, it follows from lemma 2.6 (1) that i ≤ (s−2)q+s−1 < sq−6+e, a contradiction. for s = q−4, from lemma 2.6 (2), we have i ≤ (q−5)q+q−4−3 < i, a contradiction again. similarly, using lemma 2.6 and table 1, we can deduce that ai = 0 for all i /∈ {0,q2 − 6,q2 − 5,q2 − 4} for q ≥ 11 and that ai = 0 for all i /∈ {0, 48, 75, 76, 77} for q = 9. for q = 9, a 48-plane has a 0-line [24], but the equation (4) with (i,t) = (48, 0) has no solution. hence a48 = 0. from (4), we have a0 = 0 or 1. the equality (3) gives aq2−6 = (q 4 + 4q3 − 9q2 + 14q + 2)/2 − ( q2 − 4 2 ) a0 ≥ 2q3 + 7q − 9 > θ3, a contradiction. lemma 5.2. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 6q for q ≥ 9. proof. let c be a putative [n = q3 − 6q− 6, 4,d = q3 −q2 − 6q]q code. then, n = gq(4,d) and γ0 = 1, γ1 = q, γ2 = n−d = q2 − 6 from (1). let ∆ be a γ2-plane and let δ be an i-plane. by lemma 2.13, the spectrum of ∆ satisfies τj = 0 for 1 ≤ j ≤ q− 7. since a t-line in δ satisfies t ≤ (i + 6)/q, we have ai = 0 for 1 ≤ i ≤ q−7. using lemmas 2.3, 2.6 and table 1 similarly to the proof of lemma 5.1, we can deduce that ai = 0 for all i /∈ {0,q2 − 6} for q ≥ 11 and that ai = 0 for all i /∈ {0, 48, 75} for q = 9. since the equation (4) with (i,t) = (0, 0) has no solution for q ≥ 9, we obtain a0 = 0. then, the three equations in lemma 2.1 have no solution for q = 9, a contradiction. we also get a contradiction for q ≥ 11 from lemma 2.2. lemma 5.3. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 6q − 1 for q ≥ 9. proof. let c be a putative [n = q3 − 6q− 7, 4,d = q3 −q2 − 6q− 1]q code. then, n = gq(4,d), γ0 = 1, γ1 = q, γ2 = n − d = q2 − 6 from (1). let ∆ be a γ2-plane and let δ be an i-plane. by lemma 2.13, the spectrum of ∆ satisfies τj = 0 for 1 ≤ j ≤ q − 7. since a t-line in δ satisfies t ≤ (i + 7)/q, we have ai = 0 for 1 ≤ i ≤ q − 8. using lemmas 2.3, 2.6 and table 1, it can be shown that ai = 0 for all i /∈{0,q2 − 3q − 7,q2 − 7,q2 − 6} for q ≥ 11 and that ai = 0 for all i /∈{0, 47, 48, 65, 74, 75} for q = 9. suppose a0 > 0. it follows from (4) that a0 = 1 and that aj = 0 for 0 < j < q2 − 7. then, c is extendable by theorem 2.11, contradicting lemma 5.2. hence a0 = 0. then, c is extendable by theorem 2.9, a contradiction again. lemma 5.4. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 6q − 2 for q ≥ 9. proof. let c be a putative [n = q3 − 6q− 8, 4,d = q3 −q2 − 6q− 2]q code. then, n = gq(4,d), γ0 = 1, γ1 = q, γ2 = n−d = q2 − 6 from (1). let ∆ be a γ2-plane and let δ be an i-plane. by lemma 2.13, the spectrum of ∆ satisfies τj = 0 for 1 ≤ j ≤ q− 7. since a t-line in δ satisfies t ≤ (i + 8)/q, we have ai = 0 for 1 ≤ i ≤ q − 9 for q ≥ 11. using lemmas 2.3, 2.6 and table 1, it can be shown that 114 k. kumegawa, t. maruta / j. algebra comb. discrete appl. 5(2) (2018) 101–116 ai = 0 for all i /∈{0,q2 − 4q − 8,q2 − 3q − 8,q2 − 3q − 7,q2 − 8,q2 − 7,q2 − 6} for q ≥ 13, ai = 0 for all i /∈{0, 102, 113, 114, 115} for q = 11, ai = 0 for all i /∈{0, 28, 37, 46, 47, 48, 55, 64, 65, 73, 74, 75} for q = 9. suppose a0 > 0. it follows from (4) that a0 = 1 and that aj = 0 for 0 < j < q2 − 8 for q ≥ 9. then, the equality (3) gives aq2−8 = 3q3 + 10q − 20 > θ3, a contradiction. hence a0 = 0. then, c is extendable by theorem 2.10, a contradiction again. the following three lemmas can be proved similarly to lemmas 5.2, 5.3, 5.4, respectively. lemma 5.5. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 5q for q ≥ 9. lemma 5.6. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 5q − 1 for q ≥ 9. lemma 5.7. there exists no [gq(4,d), 4,d]q code for d = q3 −q2 − 5q − 2 for q ≥ 9. now, theorem 1.5 follows from lemmas 5.1, 5.4, 5.7. this completes the proof. 6. conclusion to solve the problem finding the exact values of nq(k,d) for all d for fixed q and k, it is sufficient to determine nq(k,d) for finite values of d since nq(k,d) = gq(k,d) for all d ≥ (k−2)qk−1 −(k−1)qk−2 + 1, k ≥ 3 for all q [17]. for k = 4, it is known that nq(4,d) = gq(4,d) for q3 − q2 − q + 1 ≤ d ≤ q3 + q2 + q, d ≥ 2q3 − 3q2 + 1 for all q and for 2q3 − 5q2 + 1 ≤ d ≤ 2q3 − 5q2 + 3q for q ≥ 7 ([18, 21]). the key contribution here is showing the non–existence of [gq(4,d), 4,d]q codes for many values of d close to these "griesmer area", and it seems reasonable to seek a generalization for larger k. to this direction, see 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