Jitman

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. Two1-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 groupsH ≤Gand a finite fieldFq, anH-quasi-abelian codeis defined to be an Fq[H]-submodule ofFq[G]. Note thatH-quasi-abelian codes are not only a generalization of quasi-cyclic codes (see [7], [8], [9], and [15]) ifH is cyclic but also of abelian codes (see [1] and [2]) ifG=H, and of cyclic codes (see [12]) ifG=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 all1-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: [email protected]).

In this paper, we give an alternative discussion on the algebraic structure of1-generator quasi-abelian codes and an algorithm to find all1-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 Section3 together with an algorithm to find all 1-generator quasi-abelian codes and the number of such codes. Examples of new codes derived from1-generator quasi-abelian codes are presented in Section4.

2.

Preliminaries

Let_Fq denote a finite field of orderqand letGbe a finite abelian group of ordern, written additively. Denote byFq[G]thegroup ringofGoverFq. The elements inFq[G]will be written asPg∈GαgYg, where αg∈Fq. The addition and the multiplication inFq[G]are given as in the usual polynomial rings overFq with the indeterminateY, where the indices are computed additively in G. We note that Y0_{= 1is the} identity of _Fq[G], where 1is the identity in Fq and0 is the identity ofG.

Given a ringR, a linear code of lengthnoverRrefers to a submodule of theR-moduleRn_{. Alinear}

codeinFq[G]refers to anFq-subspaceCofFq[G]. This can be viewed as a linear code of lengthnoverFq by indexing then-tuples by the elements in G. TheHamming weightwt(u)ofu=P

g∈GugYg ∈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 = 06 }. A linear codeC in Fq[G]is referred to as an [n, k, d]q code if ChasFq-dimensionkand minimum Hamming distanced.

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 study1-generatorH-quasi-abelian codes.

Assume that H ≤Gsuch that |H|=m and the index [G:H] = _mn =l. Let{g1,g2, . . . ,gl} be a fixed set of representatives of the cosets of H in G. LetR:=Fq[H]. DefineΦ :Fq[G]→Rl by

Φ X

h∈H l X

i=1 αh+giY

h+gi !

= (α1(Y),α2(Y), . . . ,αl(Y)), (1)

where αi(Y) = Ph∈Hαh+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 mapΦ induces a one-to-one correspondence betweenH-quasi-abelian codes in _Fq[G]

and linear codes of length l overR.

Throughout, assume thatgcd(q,|H|) = 1, or equivalently,Fq[H]is semisimple. Following [7, Section 3], the group ringR=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 bySq(h), is defined to be the set

Sq(h) :={qi·h|i= 0,1, . . .}={qi·h|0≤i≤νh},

where qi_·h:=Pqi

j=1hinH andνh is the multiplicative order ofqinZord(h).

are induced by the q-cyclotomic classes ofH (see [4, Proposition II.4]). Every idempotent e in R can be viewed as a unique sum of primitive idempotents inR. The_Fq-dimensionof an idempotente∈Ris defined to be the_Fq-dimension ofRe.

From [7, Subsection 3.2],R:=Fq[H]can be decomposed as

R=Re1+Re2+· · ·+Res,

where e1, e2, . . . , es are the primitive idempotents inR. Moreover, every ideal in R is of the form Re, where eis an idempotent inR.

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 3.1allows us to express an algorithm to find all 1-generator H-quasi-abelian codes inFq[G]in Subsection3.2.

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-submodulesRa, wherea= (a1, a2, . . . , al)∈Rl.

For each a = (a1, a2, . . . , al)∈Rl, there exists a unique idempotente ∈R such thatRe =Ra1+ Ra2+· · ·+Ral. The elementeis called theidempotent generator elementforRa. An idempotentf ∈R of largest Fq-dimension such that

fa= 0

is called theidempotent check element forRa.

LetS =Fql[H], where_Fqlis an extension field of_Fq of degreel. Let{α1, α2, . . . , αl}be a fixed basis ofFql over_Fq. Letϕ:Rl→S be anR-module isomorphism defined by

a= (a1, a2, . . . , al)7→A= l X

i=1

αiai. (2)

Using the mapϕ, the codeRacan be regarded as an R-moduleRAin S.

Lemma 3.1 ([7, Lemma 6.1]). Let a∈Rl _{and let eandf be the idempotent generator and idempotent}

check elements of Ra, respectively, Then

e+f = 1

and

dim_Fq(Ra) = dim_Fq(Re) =m−dim_Fq(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. LetA=ϕ(a)and B=ϕ(b),}

Proof. Writea= (a1, a2, . . . , al)andb= (b1, b2, . . . , bl), whereai, bi∈R for alli∈ {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 haveai=riefor someri∈R. Thenuai= (ve)(rie) =vrie2=v(rie) =vai=bi for alli∈ {1,2, . . . , l}. Hence,b=uaand

Re=Ra=Rb=R(ua) =uRa=uRe.

Sinceu∈Re andRe=uRe, we haveu∈(Re)×.

Conversely, assume that there existsu∈(Re)× _{such thatb=ua. ThenRb=Rua⊆Ra. We need}

to show thatdim_Fq(Ra) = dim_Fq(Rb). Lete0 be an idempotent generator ofRb. We have

Re0 =Rb=R(ub) =u(Rb) =u(Re) =Re

sinceu∈(Re)×. Hence, by Lemma3.1, we have

dim_Fq(Ra) = dim_Fq(Re) = dim_Fq(Re0) = dim_Fq(Rb).

Therefore,Rb=Raas desired.

3.1.

The enumeration of

1

-generator quasi-abelian codes

First, we focus on the number of1-generatorH-quasi-abelian codes of a given idempotent generator in Fq[H]. Using the fact that the idempotents inFq[H]are known, the number of1-generatorH -quasi-abelian codes in_Fq[G]can be concluded.

Proposition 3.3. Let {e1, e2, . . . , er} be a set of primitive idempotents ofR ande=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 ofSej for allj∈ {1,2, . . . , r}.

iii) e is the identity ofSe.

iv) Se=Se1⊕Se2⊕ · · · ⊕Ser.

Proof. Fori), it is clear thate1, e2, . . . , er are pairwise orthogonal (non-zero) idempotents inS. They are inSesinceej=eje∈Sefor allj∈ {1,2, . . . , r}. The statementsii)andiii)follow sincesej=se2j = (sej)ej for allsej ∈Sej andse=se2= (se)efor 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 ofRej for allj∈ {1,2, . . . , r}.

iii) e is the identity ofRe.

iv) Re=Re1⊕Re2⊕· · ·⊕Rer, whereRejis isomorphic to an extension field ofFqfor allj∈ {1,2, . . . , r}.

LetΩ =nPr j=1Aj

Aj ∈(Sej)

∗o_{⊂Se.Then we have the following results.}

Lemma 3.5. Let A = Pl

Proof. Assume that RA⊆Sb. Then A=Bb for someB ∈S. Write B =Pl

i=1αibi, wherebi ∈R. Thenai=bbi for alli∈ {1,2, . . . , l}. Hence, we have

l X

i=1 riai=

l X

i=1 ribbi=

l X

i=1 ribi

!

b∈Rb

for allPl

i=1riai∈Ra1+Ra2+· · ·+Ral.

Conversely, it suffices to show thatA∈Sb. SinceRa1+Ra2+· · ·+Ral⊆Rb, we haveai∈Rbfor alli∈ {1,2, . . . , l}. Then, for eachi∈ {1,2, . . . , l}, there existsri∈Rsuch thatai =rib. Hence,

A= l X

i=1 αiai=

l X

i=1

αirib= l X

i=1 αiri

!

b∈Sb

as desired.

Lemma 3.6. LetA=Pl

i=1αiai∈Se, where ai ∈R. Then A∈Ωif and only if Re=Ra1+Ra2+· · ·+Ral.

Proof. First, we note thatRA⊆SesinceA∈Se. ThenRa1+Ra2+· · ·+Ral⊆Reby Lemma3.5. Assume that A∈Ω. Then A =A1+A2+· · ·+Ar, where Aj ∈ (Sej)∗. We haveAej =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⊆Redj=R(e−ej)

for somej∈ {1,2, . . . , r}, whereRedj:=Re1⊕ · · · ⊕Rej−1⊕Rej+1⊕ · · · ⊕Rer.By Lemma3.5, we have

06=Aj=Aej ∈RA⊆S(e−ej),

a contradiction. Therefore,Ra1+Ra2+· · ·+Ral=Re.

Conversely, assume thatRe=Ra1+Ra2+· · ·+Ral. ThenRA⊆Seby Lemma3.5. SinceA∈Se, by Theorem3.3, we haveA=A1+A2+· · ·+Ar, whereAj∈Sej for allj∈ {1,2, . . . , r}. Suppose that Aj= 0for some j∈ {1,2, . . . , r}. ThenRA=RAdj⊆Sedj=S(e−ej). By Lemma3.5, we have

Re=Ra1+Ra2+· · ·+Ral⊆R(e−ej)

which is a contradiction. Hence,Aj ∈(Sej)∗for allj∈ {1,2, . . . , r}.

Corollary 3.7. Let A = Pl

i=1αiai ∈ Sej, where ai ∈ R. Then A ∈ (Sej)∗ if and only if Rej = Ra1+Ra2+· · ·+Ral.

Let j ∈ {1,2, . . . , r} and let kj denote theFq-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 thatA=uB.

ForA∈(Sej)∗, denote by[A]the equivalence class ofAand let[(Sej)∗] ={[A]|A∈(Sej)∗}.

Proof. LetA∈(Sej)∗ and defineρ: (Rej)×→[A],

u7→uA.

From the definition of ∼, ρ is a well-defined surjective map. For eachu1, u2 ∈(Rej)×, if u1A =u2A, then(u1−u2)A= 0. WriteA=Pl

i=1αiai, where ai∈R. Thenai(u1−u2) = 0 for alli∈ {1,2, . . . , l}. SinceA∈(Sej)∗, by Corollary3.7, we can writeej =P

i

i=1riai, whereri∈R. Hence,

ej(u1−u2) = i X

i=1 riai

!

(u1−u2) = i X

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∗_qkj|=qkj −1.

Corollary 3.9. For each i∈ {1,2, . . . , r}, we have

|[(Sej)∗]|=

|(Sej)∗|

|[A]| =

qlkj ₋₁ qkj −₁.

Let [Ω] = r Y

j=1

[(Sej)∗].Then|[Ω]|= r Y

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 all1-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 havingeas their idempotent generator is

r Y

j=1

qlkj₋₁ qkj₋₁.

Proof. Defineσ: [Ω]→C,

([A1],[A2], . . . ,[Ar])7→Ra,

where A:=A1+A2+· · ·+Ar∈Seis viewed asA=Pl_i=1αiai anda:= (a1, a2, . . . , al).

Since Aj ∈(Sej)∗ for all j ∈ {1,2, . . . , r}, we have A∈Ω. Then Re=Ra1+Ra2+· · ·+Ral by Lemma3.6, and hence,Rais a1-generator quasi-abelian code with idempotent generatore, i.e.,Ra∈C. For([A1],[A2], . . . ,[Ar]) = ([B1],[B2], . . . ,[Br])∈[Ω], there existsuj ∈(Rej)× such thatAj=ujBj for allj∈ {1,2, . . . , r}. Letu:=u1+u2+· · ·+ur. Then

u u−₁1+u₂−1+· · ·+u−_r1

=e1+e2+· · ·+er=e

is the identity of Re(see Corollary3.4), whereu−_j1 refers to the inverse ofuj in Rej. Hence, uis a unit in (Re)×_{. LetB:=}Pr

j=1Bj. Then

A= r X

j=1 Aj=

r X

j=1

ujBj =uB.

For ([A1],[A2], . . . ,[Ar]),([B1],[B2], . . . ,[Br])∈ [Ω], ifRa=Rb, then, by Lemma3.2, there exists u∈(Re)× such thatA=uB. ThenAj=uBj=uejBj sinceej is the identity ofSej by Proposition3.3. SinceAj∈(Sej)∗,uej is a non-zero inRej which is a finite field. Thus uej is a unit in (Rej)×. Hence,

([A1],[A2], . . . ,[Ar]) = ([B1],[B2], . . . ,[Br])

which implies thatσis an injective map.

To verify thatσis surjective, letRa∈C, wherea= (a1, a2, . . . , al)∈Rl. ThenRe=Ra1+Ra2+

· · ·+Ral. Hence, by Lemma3.6, we conclude that

A:= l X

i=1

αiai∈Ω.

WriteA=Pr

j=1Aj, 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 all1-generatorH-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 studyH-quasi-abelian codes of a given idempotent generator.

Lete=e1+e2+· · ·+erbe an idempotent inR, where, for eachj∈ {1,2, . . . , r},ej is the primitive idempotent in Rinduced by aq-cyclotomic classSq(hj)for somehj∈H.

For each j∈ {1,2, . . . , r}, assume thatej 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 somehji∈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 S_ql(hj). For each j ∈ {1,2, . . . , r}, let kj anddj denote the Fq-dimension of ej and the Fql-dimension ofej1, respectively. Thenkj anddj are the smallest positive integers such that

qkj·hj =hj and qldj·hj=hj.

Then kj|ldj which implies that _gcd(kj_l,k

j)|dj. Since q l_gcd(kj

l,kj) _·h j = q

kj l gcd(l,kj) _·h

j = hj, we have dj|

kj

gcd(l,kj). It follows that dj = kj

gcd(l,kj). Hence, eji’s have the same q l_{-size d}

j = kj

gcd(l,kj) and sj =

gcd(l, kj).

Using arguments similar to those in the proof of Proposition3.3, 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.

iii) ej =ej1+ej2+· · ·+ejsj is the identity ofSej for allj∈ {1,2, . . . , r}.

iv) Forj∈ {1,2, . . . , r}, we haveSej =Sej1⊕Sej2⊕ · · · ⊕Sejsj, where Sejiis an extension field of Fq

of orderqldj _{for alli∈ {1,2, . . . , s} j}.

Theorem 3.12. Letj∈ {1,2, . . . , r}be fixed. Fori∈ {1,2, . . . , sj}, letπibe a primitive element ofSeji,

a finite field of qldj _{elements. LetL} j =q

ldj₋₁

qkj−1 andTj={∞,0,1,2, . . . , q

ldj _{−2}. Then the elements}

πνt t +π

νt+1

t+1 +· · ·+π ν_sj

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, πi∞= 0.)

Proof. Note that the number of elements in (3) is

Ljqldj(sj−1)+Ljqldj(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=πνt t +π

νt+1

t+1 +· · ·+π ν_sj

sj andB =πµxx +π µx+1

x+1 +· · ·+π µ_sj sj ,

where0≤νt, µx≤Lj−1,νt+1, νt+2, . . . , νsj ∈Tj, andµx+1, µx+2, . . . , µsj ∈Tj. Assume that[A] = [B]. Then there existsu∈(Rej)× such that

πνt t +π

νt+1

t+1 +· · ·+π ν_sj

sj =A=uB=uπµxx +uπ µx+1

x+1 +· · ·+uπ µ_sj sj .

Since πνt_t ∈ (Sejt)× and uπxµx ∈ (Sejx)×, by the decomposition in Proposition 3.11, t = xand πνtt = uπµt

t ∈ Sejt. Then uejt = π νt−µt

t . Since u∈ (Rej)×, we have uq kj₋₁

= ej, and hence, ejt =ejtej = π(νt−µt)(qkj−1)

t .Since 0 ≤νt, µt ≤Lj −1 and πt has orderqldj −1, we conclude that νt =µt. Hence, uejt=ejt=ejejtwhich implies (u−ej)ejt= 0 inSejt. 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 andR(u−ej)(Rej. Hence,R(u−ej) is the zero ideal, i.e., u=ej. Therefore,A=uB=ejB=B sinceej is the identity ofSej.

The following corollary now follows from Theorem 3.10and Theorem3.12.

Corollary 3.13. Let {e1, e2, . . . , er} be a set of primitive idempotents of R ande=e1+e2+· · ·+er.

Then all 1-generator quasi-abelian codes havinge 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 all1-generatorH-quasi-abelian codes in Fq[G]as in Algorithm 1. We note that the1-generatorH-quasi-abelian codes inFq[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 Algorithm1is given as follows.

For abelian groupsH ≤Gand a finite fieldFq withgcd(q,|H|) = 1and[G:H] =l, do the following steps.

1. Compute theq-cyclotomic classes ofH inG.

2. Compute the set{e1, e2, . . . , er}of primitive idempotents ofR=Fq[H](see [4, Propo-sition II.4]).

3. For each1≤j≤r, compute a setBj of a complete set of representatives of[(Sej)∗] (see Theorem3.12).

4. Compute the idempotents ofR,i.e.,the set

T = ( t

X

j=1

eij

1≤t≤rand1≤i1< i2 <· · ·< it≤r

) .

5. For eache=Pt_j=1eij ∈T, compute the1-generator quasi-abelian codes havingeas their idempotent generator of the form

A1+A2+· · ·+At,

whereAj∈Bi_j (see Corollary3.13).

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 all1-generatorH-quasi-abelian codes inFq[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=Ya0+Ya1+Ya2+Ya3+Ya4+Ya5+Ya6+Ya7+Ya8, e2=Ya1_+Ya2_+Ya4_+Ya5_+Ya7_+Ya8_,

e3=Ya3_+Ya4_+Ya5_+Ya6_+Ya7_+Ya8_, e4=Ya1+Ya2+Ya3+Ya4+Ya6+Ya8, e5=Ya1_+Ya2_+Ya3_+Ya5_+Ya6_+Ya7

are primitive idempotents of R:=_F2[H] induced by S2(a0), S2(a1),S2(a3), S2(a4), andS2(a5),

respec-tively.

Let e:=e1+e2+e3. From Theorem3.10, it follows that the number of1-generatorH-quasi abelian codes in _F2[G] with idempotent generatoreis 3·5·5 = 75.

Let S:=F4[H], whereF4={0,1, α, α2= 1 +α}. Thene2=e21+e22 ande3=e31+e32, where

e21=Ya0_+α2_Ya1_+αYa2_+Ya3_+α2_Ya4_+αYa5_+Ya6_+α2_Ya7_+αYa8_, e22=Ya0_+αYa1_+α2_Ya2_+Ya3_+αYa4_+α2_Ya5_{+ 1Y}a6_+αYa7_+α2_Ya8_, e31=Ya0_+Ya1_+Ya2_+α2_Ya3_+α2_Ya4_+α2_Ya5_+αYa6_+αYa7_+αYa8_, e32=Ya0+Ya1+Ya2+αYa3+αYa4+αYa5+α2Ya6+α2Ya7+α2Ya8

are primitive idempotents in S induced by4-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= 2₂2₋−₁1 = 3,L2=L3= ₂222−₋1₁ = 1, andT1=T2=T3={∞,0,1,2}.

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-generatorH-quasi abelian codes in F2[G] with idempotent generatoreare of the form

A1+A2+A3,

where Ai∈Bi for alli∈ {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 ≤Gbe finite abelian groups of index [G:H] =l and let {αqi |1≤i≤l} be a fixed basis ofFql over_Fq. IfA=P

l i=1aiαq

i

∈Se, thenAandAq ₌Pl i=1a

q iαq

i+1

generate permutation equivalent H-quasi abelian codes (viewed inFq[G]) with the same idempotent generator.

Proof. Letebe the idempotent generator of a quasi-abelian codeRA. Then

Raq₁+Raq₂+· · ·+Ra_lq⊆Ra1+Ra2+· · ·+Ral=Re

Assume thate=Pl

i=1riai, whereri∈R. It follows that

e=eq= l X

i=1

rq_iaq_i ∈Raq₁+Raq₂+· · ·+Raq_l.

Hence, we haveRe=Raq₁+Raq₂+· · ·+Raq_l. Therefore,AandAq generate1-generatorH-quasi-abelian codes with the same idempotent generatore.

Letψ:R→R be a ring homomorphism defined by

γ7→γq.

Let γ = P

h∈HγhYh and β = Ph∈HβhYh be elements in R, where γh and βh are elements in Fq. If ψ(γ) =ψ(β), then

0 =γq−βq = (γ−β)q = X h∈H

(γh−βh)Yq·h.

By comparing the coefficients, we haveγh=βhfor allh∈H,i.e.,γ=β. Hence,ψis a ring automorphism and

R(aq_l, aq₁, . . . , aq_l−1) =R(ψ(al), ψ(a1), . . . , ψ(al−1)) = Ψ(R(al, a1, . . . , al−1)), (4)

where Ψis a natural extension ofψto Rl_. Since ψ(γ) = P

h∈HγhYq·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)

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 other2new codes from1-generator quasi-abelian codes together with1new code obtained by shortening of one of these codes.

Given an abelian groupH=Zn1×Zn2 of ordern=n1n2, denote byu= (u0, u1, u2, . . . , un−1)∈F n q the vector representation of

u= n2−1

X

j=0 n1−1

X

i=0

ujn1+iY(i,j)inFq[H].

Let

C(a,b):={(f a, f b)|f ∈Fq[H]}, (5)

whereaandbare elements in_Fq[H]. Using (5),2quasi-abelian codes whose minimum distance improves on Grassl’s online table [5] can be found. The codesC1andC2are presented in Table1and the generator matrices ofC1 andC2 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 puncturingC2at the first coordinate, a[35,11,17]5code can be obtained with minimum distance improved by1 from Grassl’s online table [5]. All the computations are done using MAGMA [3].

Table 1. New codes from quasi-abelian codes

name C(a,b) H aandb

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)

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