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 prac- tically 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 Univer- sity, 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 idempo- tents 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 Hermi- tian 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 Proposi- tion 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. 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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