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 half- century. 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 Univer- sity, 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 char- acterization 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-quasi- abelian 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 ql- cyclotomic 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 cor- responding 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, Propo- sition 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), respec- tively. 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 gener- ators 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 Nether- lands: 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