Jitman

Journal of Algebra Combinatorics Discrete Structures and Applications

Self-dual and complementary dual abelian codes over

Galois rings

∗

Research Article

Somphong Jitman

,

San Ling

Abstract: Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide appli-cations. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ringGR(pr_{, s)[G], whereGis a finite abelian group andGR(p}r_{, s)is a Galois ring. Characterizations} of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code inGR(pr, s)[G]. A general formula for the number of such self-dual codes is established. In the case where gcd(|G|, p) = 1, the number of self-dual abelian codes inGR(pr, s)[G]is completely and explicitly determined. Applying known results on cyclic codes of lengthpaoverGR(p2, s), an explicit formula for the number of self-dual abelian codes inGR(p2, s)[G] are given, where the Sylowp-subgroup ofGis cyclic. Subsequently, the characterization and enumer-ation of complementary dual abelian codes inGR(pr, s)[G]are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

2010 MSC: 94B15, 94B05, 16A26

Keywords: Abelian codes, Galois rings, Self-dual codes, Complementary dual codes, Codes over rings

1.

Introduction

Algebraically structured codes over finite fields with self-duality and complementary duality are important families of linear codes that have been extensively studied for both theoretical and practical reasons (see [1], [3], [11], [13], [15], [21], [26], [27], and references therein). Codes over finite rings have been interesting since it was proven that some binary non-linear codes such as the Kerdock, Preparata, and Goethal codes are the Gray images of linear codes overZ4 [10]. Algebraically structured codes such

∗

S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. S. Ling was supported by Nanyang Technological University Research Grant M4080456.

Somphong Jitman(Corresponding Author); Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand (email: [email protected]).

as cyclic, constacyclic, and abelian codes have extensively been studied overZpr, Galois rings, and finite

chain rings in general (see [7],[18], and references therein).

The characterization and enumeration of Euclidean self-dual cyclic codes over finite fields have been established in [11] and generalized to Euclidean and Hermitian self-dual abelian codes over finite fields in [13] and [15], respectively. Over some finite rings, a characterization of self-dual cyclic, constacyclic and abelian codes has been done (see, for example, [1], [7],[16], [17], [24], and [26]). In [1], [5], [4] and [23], characterization and enumeration of Euclidean and Hermitian self-dual cyclic codes over finite chain rings have been discussed. Euclidean complementary dual cyclic codes over finite fields have been studied in [27]. Recently, they have been generalized to Euclidean and Hermitian complementary dual abelian codes over finite fields in [3]. The complete characterization and enumeration of complementary dual abelian codes over finite fields have been established in the said paper.

In this paper, we focus on abelian codes over Galois rings GR(pr_{, s), i.e., ideals in the group ring}

GR(pr_{, s)[G] of an abelian group G over a Galios ring GR(p}r_{, s). Specifically, we study self-dual and}

complementary dual abelian codes in GR(pr_{, s)[G] with respect to both the Euclidean and Hermitian}

inner products. We characterize such self-dual abelian codes and determine necessary and sufficient conditions for the existence of a self-dual abelian code inGR(pr, s)[G]. We give a formula for the number of self-dual abelian codes inGR(pr_{, s)[G]. Under the restrictioni) gcd(|G|, p) = 1; or ii)r= 2and the}

Sylow p-subgroup of G is cyclic, the numbers of self-dual abelian codes in GR(pr_{, s)[G] are explicitly}

determined. Subsequently, the characterization and enumeration of complementary dual abelian codes in GR(pr_{, s)[G]are given. The number of complementary dual abelian codes inGR(p}r_{, s)[G]is shown to}

be independent ofr and the Sylowp-subgroup ofG.

We note that the Hermitian duality is meaningful only when s is even. Since we study Euclidean and Hermitian self-dual codes in parallel, the assumption that sis even is included whenever we refer to the Hermitian duality.

The paper is organized as follows. In Section 2, we recall and prove some basic results for group rings, abelian codes, and their duals. In Section3, we present the characterization and a general set up for the enumeration of self-dual abelian codes inGR(pr, s)[G]. The complete enumeration of Euclidean and Hermitian self-dual abelian codes in GR(pr_{, s)[G]is given in the special cases where i) gcd(p,|G|) = 1;}

andii)r= 2and the Sylowp-subgroup ofGis cyclic. In Section4, the characterization and enumeration of complementary dual abelian codes in GR(pr_{, s)[G]are given.}

2.

Preliminaries

In this section, we recall some definitions and basic properties of abelian codes and prove some results on their Euclidean and Hermitian duals.

2.1.

Abelian codes

For a finite commutative ring R with identity and a finite abelian group G, written additively, let R[G] denote the group ring of Gover R. The elements inR[G] will be written as P

g∈GαgYg, where

αg ∈R. The addition and the multiplication inR[G] are given as in the usual polynomial rings overR

with the indeterminateY, where the indices are computed additively inG. By convention,Y0= 1is the identity of R, where0 is the additive identity ofG.

Anabelian codeinR[G]is defined to be an ideal ofR[G]. IfG={g1, g2, . . . , gn}is an abelian group of

ordern, it is not dificult to see that the mapπ:R[G]→Rn_{defined by}Pn i=1αgiY

gi _7→(α

g1, αg2, . . . , αgn)

is anR-module isomorphism. Hence, an abelian codeC inR[G]can be viewed as anR-submoduleπ(C)

in Rn_{. Precisely, π(C) is a linear code of lengthnoverR. If Gis cyclic of ordern, an abelian code in}

in the quotient polynomial ringR[X]/hXn_−1i.

From now on, we focus on the case where the ring is a Galois ring GR(pr, s), a Galois extension of degree sof an integer residue ring Zpr. Letξ be an element inGR(pr, s)that generates a Teichmüller

set Ts ofGR(pr, s). In other words,Ts={0,1, ξ, ξ2, . . . , ξp

s₋₂

}. Then every element in GR(pr, s)has a uniquep-adic expansion of the form

α=α0+α1p+· · ·+αr−1pr−1,

where αi ∈ Tsfor all i= 0,1, . . . , r−1. In the case wheresis even, the map¯ : GR(pr, s)→GR(pr, s)

defined by

α=αp₀s/2+αp₁s/2p+· · ·+αp_r−s/₁2pr−1 (1) is a ring automorphism onGR(pr_{, s). For more details on Galois rings, we refer the readers to [25].}

Let P be the Sylow p-subgroup of G and let A be a complementary subgroup of P in G. Then G∼=A×P. LetR:= GR(pr_{, s)[A]. Then the mapΦ : GR(p}r_{, s)[G]→ R[P]given by}

Φ(X

a∈A X

b∈P

αa+bYa+b) = X

b∈P

αb(Y)Yb,

where αb(Y) =Pa∈Aαa+bYa ∈ R,is a well-known ring isomorphism (see [19, Section 3.2, Exercise 4]).

Then Φinduces a one-to-one correspondence between the ideals in GR(pr_{, s)[G]and the ideals in R[P].}

Since an abelian code is an ideal in a group ring, the above discussion can be interpreted in terms of abelian codes as follows.

Lemma 2.1. The mapΦinduces a one-to-one correspondence between the abelian codes inGR(pr_{, s)[G]} and the abelian codes in R[P].

An abelian codeC inGR(pr_{, s)[G]is said to beEuclidean self-dual(resp.,Euclidean complementary} dual) ifC=C⊥E _{(resp.,C ∩ C}⊥E _{={0}), whereC}⊥E _{is the dual ofC with respect to the form}

hu,viE:=

X

g∈G

αgβg,

where u=P

g∈GαgYg andv=Pg∈GβgYg.

Define aninvolution _b onR to be theGR(pr_{, s)-module homomorphism that fixesGR(p}r_{, s)and}

sends Ya _toY−a _{for alla∈A. An abelian codeD in R[P]is said to be}

b-self-dualifD=D

⊥

b, where

D⊥_b is the dual ofD with respect to the form

hx,yi

b

:=X

b∈P

xb(Y)y\b(Y),

where x=P

b∈Pxb(Y)Yb andy=Pb∈Pyb(Y)Y b_.

In addition, if s is even, an abelian codeC in GR(pr_{, s)[G] is said to beHermitian self-dual (resp.,} Hermitian complementary dual) if C = C⊥H _{(resp., C ∩ C}⊥H _{= {0}), where C}⊥H _{is the dual of C with} respect to the form

hu,viH:=

X

g∈G

αgβg,

where u=P

Define an involution_eon Rto be the GR(pr_{, s)-module homomorphism that sendsα to α for all}

α∈GR(pr_{, s)and sendsY}a _toY−a _{for alla∈A. An abelian codeDinR[P]is said to be∼-self-dualif}

D=D⊥∼, whereD⊥∼ is the dual ofD with respect to the form hx,yi∼ :=

X

b∈P

xb(Y)y^b(Y),

where x=P

b∈Pxb(Y)Y

b _andy=P

b∈Pyb(Y)Yb.

Similarly to the finite field case, the following relations among the above forms can be verified using arguments similar to those in [13, Proposition 2.4] and [15, Proposition 2.4].

Lemma 2.2. Let rand sbe positive integers and let pbe a prime number. LetG∼=A×P be as above andu,v∈GR(pr_{, s)[G]. Then the following statements hold.}

i) hYgu,viE= 0for allg∈Gif and only if hYbΦ(u),Φ(v)i

b

= 0for all b∈P.

ii) If sis even, then hYg_u,vi

H= 0for allg∈Gif and only if hYbΦ(u),Φ(v)i∼ = 0for allb∈P.

The next corollary follows immediately.

Corollary 2.3. Let r ands be positive integers and letp be a prime number. LetC be an abelian code in GR(pr_{, s)[G]. Then the following statements hold.}

i) Φ(C)⊥b= Φ(C

⊥E_{). In particular,C is Euclidean self-dual if and only ifΦ(C)is}

b-self-dual.

ii) If s is even, then Φ(C)⊥∼ = Φ(C⊥H_{). In particular, C is Hermitian self-dual if and only if Φ(C) is} ∼-self-dual.

Therefore, to study Euclidean and Hermitian self-dual abelian codes inGR(pr_{, s)[G], it is sufficient}

to consider _b-self-dual and∼-self-dual abelian codes inR[P], respectively.

2.2.

Decomposition and dualities

Recall thatprepresents a prime number,sis a positive integer, andAis a finite abelian group such that gcd(p,|A|) = 1.

For coprime positive integersi, j, letordi(j)denote the multiplicative order ofj moduloi. Fora∈

A, denote byord(a)the additive order ofainA. For eacha∈A, aps_{-cyclotomic classof Acontainingais}

defined to be the set

Sps(a) :={psi·a|i= 0,1, . . .}={psi·a|0≤i <ord_ord(a)(ps)},

where psi_·a:= Ppsi

j=1ain A. A p

s_{-cyclotomic class S}

ps(a) is said to be of type I if a =−a, type II if

Sps(a) =S_ps(−a)anda6=−a, ortypeIII ifS_ps(−a)6=S_ps(a). Ifsis even, aps-cyclotomic classS_ps(a)is

said to be oftypeII0 ifSps(a) =S_ps(−ps/2·a)ortypeIII0 ifS_ps(−ps/2·a)6=S_q(a), where−ps/2·adenotes

ps/2·(−a).

Remark 2.4. We have the following facts for the ps-cyclotomic classes ofA (see [13, Remark 2.5] and [15, Remark 2.6]).

1. Aps-cyclotomic class of typeI has cardinality one.

2. Sps(0)is aps-cyclotomic class of both typesI andII0.

3. If a ps_{-cyclotomic class of type II exists, then its cardinality is even. Moreover, if S}

ps(a) is a

4. A ps_{-cyclotomic class of A of type II}0 _{has odd cardinality. Moreover, if S}

ps(a) is a ps-cyclotomic

class of type II0 of cardinalityν, then−a=psν/2_·aand−ps/2_·a=ps(ν+1)/2_·a.

Assume thatA has cardinalitymand exponentM. By the Fundamental Theorem of finite abelian groups, A can be written as a direct product of finite cyclic groups A = QN

i=1Zmi, where Zmi =

{0,1, . . . , mi−1}denotes the additive cyclic group of order mi≥2 for all1≤i≤N. Then an element

b∈Acan be written asb= (b1, b2, . . . , bN), wherebi∈Zmi. For eachh∈A, letγh :A→Z be defined

by

γh(b) = N X

i=1

bihi(M/mi), (2)

where the sum is a rational sum.

Letµbe the order of ps _{moduloM. Denote byζ a primitiveMth root of unity inGR(p}r_{, sµ). For}

a givenc=P

a∈AcaYa∈ R:= GR(pr, s)[A], its Discrete Fourier Transform (DFT) is˘c=Ph∈Ac˘hYh,

where

˘

ch= X

a∈A

caζγh(a)∈GR(pr, sµ). (3)

Moreover, if Sps(h)has cardinalityν, then it is not difficult to verify that˘c_h is contained in a subring of

GR(pr_{, sµ)which is isomorphic toGR(p}r_{, sν).}

Using this DFT, the decomposition ofR:= GR(pr_{, s)[A], where gcd(p,|A|) = 1, has been given in}

[18] in terms of the mix-radix representation of the elements inA. In order to utilize the decomposition in [18] for characterizing self-dual codes, we need to consider a suitable rearrangement of the terms in the decomposition.

2.2.1. Euclidean case

For the Euclidean self-duality, we consider the rearrangement based on theps_{-cyclotomic classes of}

types I−III as follows. Assume thatAcontains L ps-cyclotomic classes. Without loss of generality, let {a1, a2, . . . , aL} be a set of representatives of theps-cyclotomic classes such that {ai | i = 1,2, . . . , tI}, {atI+j | j = 1,2, . . . , tII} and {atI+tII+k, atI+tII+tIII+k = −atI+tII+k | k = 1,2, . . . , tIII} are sets of repre-sentatives of ps_{-cyclotomic classes of types I,II, and III, respectively, whereL=t}

I+tII+ 2tIII. From the definition,|Sps(a_i)|= 1for alli= 1,2, . . . , t_I. From Remark2.4, the order of theps-cyclotomic classes of

type II is even order. Forj= 1,2, . . . , tII, let2ej denote the cardinality ofSps(a_t

I+j). Fork= 1,2, . . . , tIII, Sps(a_t

I+tII+k)andSps(atI+tII+tIII+k)have the same cardinality and denote it byfk.

Rearranging the terms in the decomposition in [18] based on theps_{-cyclotomic classes ofAof types}

I−III, we have

R ∼=

tI

Y

i=1

GR(pr, s)

!

×



 tII

Y

j=1

GR(pr,2sej) 

× tIII

Y

k=1

(GR(pr, sfk)×GR(pr, sfk)) !

, (E1)

where GR(pr,2sej) is induced by Sps(a_t

I+j) for all j = 1,2, . . . , tII and GR(p

r_{, sf}

k)×GR(pr, sfk) is

induced by (Sps(a_t

I+tII+k), Sps(−atI+tII+k)) for all k = 1,2, . . . , tIII. For more details and the explicit isomorphism, the readers may refer to [18, Section II].

It follows that

R[P]∼=

tI

Y

i=1

GR(pr, s)[P]

!

×



 tII

Y

j=1

GR(pr,2sej)[P] 

× tIII

Y

k=1

(GR(pr, sfk)[P]×GR(pr, sfk)[P]) !

.

Therefore, by Lemma2.1, every abelian code in GR(pr_{, s)[G]}∼_{=R[P] can be written in the form}

C ∼=

tI Y i=1 Ui ! ×   tII Y j=1 Vj  × tIII Y k=1

(Wk×Wk0) !

, (E3)

where Ui is an abelian code in GR(pr, s)[P], Vj is an abelian code in GR(pr,2sej)[P], andWk, Wk0 are

abelian codes inGR(pr_{, sf}

k)[P]for alli= 1,2, . . . , tI,j= 1,2, . . . , tII, andk= 1,2, . . . , tIII. The Euclidean dual ofC in (E3) can be viewed to be of the form

C⊥E ∼₌

tI

Y

i=1

U⊥E

i ! ×   tII Y j=1

V⊥H

j  × tIII Y k=1

(W_k0)⊥E_×W⊥E

k

!

. (E4)

The detailed justification for (E4) is provided in AppendixA.1.

2.2.2. Hermitian case

In the case where s is even, we consider the other rearrangement of the decomposition of R in terms of the ps_{-cyclotomic classes of A of types II}0 _{and III}0_{. Let {b}

1 = 0, b2, . . . , bL} denote a set of

representatives of the ps_{-cyclotomic classes such that {b}

j | j = 1,2, . . . , tII0} and {b_t

II0+k, btII0+tIII0+k = −ps/2_·b

t_II0+k | k = 1,2, . . . , tIII0} representps-cyclotomic classes of types II0 and III0, respectively, where

L = tII0 + 2t_III0. For j = 1,2, . . . , t_II0, let e´_j denote the cardinality of S_ps(b_j). For k = 1,2, . . . , t_III0,

Sps(b_t

II0+k)andSps(b_t

II0+tIII0+k)have the same cardinality and denote it byf´k.

Rearranging the terms in the decomposition in [18] based on theps_{-cyclotomic classes ofAof types}

II0 and III0, we have

R ∼=



 t_II0

Y

j=1

GR(pr, s´ej) 

× t_III0

Y

k=1

GR(pr, sf´k)×GR(pr, sf´k)

!

, (H1)

whereGR(pr_{, se´}

j)is induced bySps(b_j)for allj= 1,2, . . . , t_II0 and GR(pr, sf´_k)×GR(pr, sf´_k)is induced

by Sps(b_t

II0+k), Sps(−ps/2·b_t

II0+k)

for allk= 1,2, . . . , tIII0.

Consequently,

R[P]∼=



 t_II0

Y

j=1

GR(pr, s´ej)[P] 

× t_III0

Y

k=1

GR(pr, sf´k)[P]×GR(pr, sf´k)[P]

!

, (H2)

and, by Lemma 2.1, every abelian code inGR(pr_{, s)[G]}∼_{=R[P]can be viewed as}

C ∼=



 t_II0

Y

j=1

Ej 

× t_III0

Y

k=1

(Fk×Fk0) !

, (H3)

where Ej is an abelian code in GR(pr, s´ej)[P] and Fk, Fk0 are abelian codes in GR(p r_{, sf}´

k)[P] for all

j = 1,2, . . . , tII0 andk= 1,2, . . . , t_III0.

Then the Hermitian dual ofC in (H3) has the form

C⊥H ∼₌



 t_II0

Y

j=1

E⊥H

j 

× t_III0

Y

k=1

(F_k0)⊥E_×F⊥E

k

!

. (H4)

3.

Self-dual abelian codes in

GR(p

r

, s)[G]

In this section, we characterize and enumerate Euclidean and Hermitian self-dual abelian codes in

GR(pr_{, s)[G]. We determine necessary and sufficient conditions for the existence of self-dual abelian codes}

in GR(pr_{, s)[G] in Subsection 3.1 and followed by general results for the enumeration of such self-dual}

codes in Subsection3.2. Some special cases will be discussed in Subsections3.3 and3.4.

3.1.

The existence of self-dual abelian codes

The characterizations of Euclidean and Hermitian self-dual abelian codes in GR(pr_{, s)[G]are given}

as follows.

From (E3) and (E4), the characterization of Euclidean self-dual abelian codes in GR(pr_{, s)[G] is}

given in the next proposition.

Proposition 3.1. Let r and s be positive integers and let p be a prime number. An abelian code C in

GR(pr_{, s)[G] is Euclidean self-dual if and only if, in the decomposition (E3),}

i) Ui is Euclidean self-dual for alli= 1,2, . . . , tI,

ii) Vj is Hermitian self-dual for allj= 1,2, . . . , tII, and

iii) W_k0 =W⊥E

k for allk= 1,2, . . . , tIII.

The characterization of Hermitian self-dual abelian codes inGR(pr, s)[G] follows immediately from (H3) and (H4).

Proposition 3.2. Letrandsbe positive integers such thatsis even and letpbe a prime number. Then an abelian code C inGR(pr_{, s)[G] is Hermitian self-dual if and only if, in the decomposition (H3),}

i) Ej is Hermitian self-dual for allj = 1,2, . . . , tII0, and

ii) F_k0 =F⊥E

k for allk= 1,2, . . . , tIII0.

Necessary and sufficient conditions for the existence of Euclidean and Hermitian self-dual abelian codes in GR(pr_{, s)[G] are given as follows. The conditions for the Euclidean case have been proven in}

[26, Theorem 1.1]. Here, we provide an alternative constructive proof.

Proposition 3.3. Letrandsbe positive integers and letpbe a prime number. LetGbe a finite abelian group. Then there exists a Euclidean self-dual abelian code in GR(pr, s)[G] if and only if one of the following statements holds,

i) r is even, or

ii) p= 2 and|G| is even.

In addition, ifs is even, then the conditions are equivalent to the existence of a Hermitian self-dual abelian code in GR(pr_{, s)[G].}

Proof. Assume thatGis decomposed as G=A⊕P, wherep-|A|andP is the Sylow p-subgroup of

Gof orderpa_{, wherea≥0.}

From (E3), assume that the code

C ∼=

tI

Y

i=1

Ui !

×



 tII

Y

j=1

Vj 

× tIII

Y

k=1

is Euclidean self-dual inGR(pr_{, s)[G]. Then, by Proposition3.1,U}

1is Euclidean self-dual inGR(pr, s)[P].

It follows that|U1|= (ps)rp

a_/2

andrpa_{/2 is an integer. Hence,r is even, orp= 2anda≥1.}

For the converse, if ris even, then pr/2GR(pr, s)[G]is Euclidean self-dual. Assume that r is odd, p= 2and|G| is even. Letr0 =dr

2e. Then|P|= 2

a _{witha≥1 andr= 2r}0_{−1. Since the order ofP is}

even, P contains an elementxof order2. Define

C ∼=

tI

Y

i=1

2r0GR(2r, s)[P] + 2r0−1GR(2r, s)[P](Yx+ 1)

! ×   rII Y j=1

2r0GR(2r,2sej)[P] + 2r

0₋₁

GR(2r,2sej)[P](Yx+ 1)   × tIII Y k=1

(GR(2r, sfk)[P]× {0}) !

.

We prove that C is Euclidean self-dual. By Proposition3.1, it is sufficient to show that

U := 2r0GR(2r, s)[P] + 2r0−1GR(2r, s)[P](Yx+ 1)

is Euclidean self-dual and

Vj := 2r

0

GR(2r,2sej)[P] + 2r

0

−1_GR(2r_,2se

j)[P](Yx+ 1)

is Hermitian self-dual for allj = 1,2. . . , tII.

Let u= 2r0_{e+ 2}r0−1_e0_(Yx_{+ 1)and v = 2}r0_{f + 2}r0−1_f0_(Yx_{+ 1)be elements inU, wheree, e}0_,f,

andf0 are inGR(2r_{, s)[P]. Sincer= 2r}0_{−1 andx=−x, we have}

hu,viE=h2r

0

e,2r0fiE+h2r

0

e,2r0−1f0(Yx+ 1)iE

+h2r0−1e0(Yx+ 1),2r0fiE+h2r

0₋₁

e0(Yx+ 1),2r0−1f0(Yx+ 1)iE

= 2r−1he0(Yx+ 1),f0(Yx+ 1)iE

= 2r−1 he0Yx,f0YxiE+he0Yx,f0iE+he0,f0YxiE+he0,f0iE

= 2r−1 2he0,f0iE+ 2he0Yx,f0iE

= 0. It is not difficult to verify that

|U|= |2

r0_GR(2r_{, s)[P]||2}r0−1_GR(2r_{, s)[P](Y}x_{+ 1)|} |(2r0

GR(2r_{, s)[P])∩(2}r0₋₁

GR(2r_{, s)[P](Y}x_{+ 1))|} =

(2s₎(r0−1)2a₍₂s₎r02a/2

(2s₎(r0₋₁₎₂a_/2 = (2

s₎r2a−1 .

Therefore,U is Euclidean self-dual.

Using arguments similar to the above, we can see thatVjis Hermitian self-dual for allj= 1,2. . . , tII.

For the Hermitian case, we assume that s is even. The proof of the sufficiency is similar to the Euclidean case, except that (H3) and Proposition3.2are applied instead of (E3) and Proposition3.1.

For the converse, if r is even, thenpr/2_GR(pr_{, s)[G]is Hermitian self-dual. Assume that r is odd,}

p= 2and|G|is even. ThenP contains an elementxof order2. Letr0 =dr

2eand define

C ∼=



 rII0

Y

j=1

2r0GR(2r, se´j)[P] + 2r

0₋₁

GR(2r, se´j)[P](Yx+ 1)



× tIII0

Y

k=1

GR(2r, sf´k)[P]× {0}

!

.

By arguments similar to those in the proof of the Euclidean case, we can verity that2r0_GR(2r_{, se´} j)[P] +

2r0−1_GR(2r_{, se´}

j)[P](Yx+ 1) is Hermitian self-dual for all j = 1,2, . . . , tII0. Therefore, C is Hermitian

3.2.

Enumeration of self-dual abelian codes

We aim to characterize and enumerate Euclidean and Hermitian self-dual cyclic and abelian codes over Galois rings. For convenience, we fix the following notations.

• N C(GR(pr_{, s), n) – the number of cyclic codes of lengthnoverGR(p}r_{, s),}

• N EC(GR(pr_{, s), n) – the number of Euclidean self-dual cyclic codes of lengthnoverGR(p}r_{, s),} • N HC(GR(pr, s), n) – the number of Hermitian self-dual cyclic codes of lengthnoverGR(pr, s), • N A(GR(pr_{, s)[G]) – the number of abelian codes inGR(p}r_{, s)[G],}

• N EA(GR(pr_{, s)[G]) – the number of Euclidean self-dual abelian codes inGR(p}r_{, s)[G],} • N HA(GR(pr_{, s)[G]) – the number of Hermitian self-dual abelian codes inGR(p}r_{, s)[G],}

where sis assumed to be even in cases ofN HC(GR(pr_{, s), n)andN HA(GR(p}r_{, s)[G]).}

To determine the numbers of Euclidean and Hermitian self-dual abelian codes, we need some group-theoretic and number-group-theoretic results. For completeness, we recall the following results.

For a finite groupAand a positive integerd, letNA(d)denote the number of elements inAof order

d. The explicit expression ofNA(d)is completely determined in [2].

Letq be a prime power and letj be a positive integer. The pair(j, q)is said to beoddly goodifj divides qt+ 1for some odd integert≥1, andevenly goodifj dividesqt+ 1for some even integert≥2. It is said to be goodif it is oddly good or evenly good, andbad otherwise. The characterization of good and oddly-good pairs of integers can be found in [12], [11], [13], [15], and [20].

Letχandλbe functions defined on the pair (j, q), where j is a positive integer, as follows.

χ(j, q) =

(

0 if(j, q)is good,

1 otherwise, (4) and

λ(j, q) =

(

0 if(j, q)is oddly good,

1 otherwise. (5) The following two lemmas are extended from the case where qis a power of 2 in [13] and [15] and the proof is omitted. The readers may refer to [13, Lemma 4.5] and [15, Lemma 7] for the idea of the proofs. Lemma 3.4. Let s be a positive integer and let pbe a prime number. Let A be a finite abelian group such that gcd(|A|, p) = 1and leth∈A. Then Sps(h)is of typeIII if and only if(ord(h), ps)is bad.

Lemma 3.5. Let s be an even positive integer and let pbe a prime number. Let A be a finite abelian group such that gcd(|A|, p) = 1and leth∈A\ {0}. ThenSps(h)is of typeIII0 if and only if(ord(h), ps/2)

is evenly good or bad.

Utilizing the decomposition in Section2and the discussion above, we obtain the following formulas for the numbers of Euclidean and Hermitian self-dual abelian codes in GR(pr_{, s)[G], where G is an}

arbitrary finite abelian group. Without loss of generality, we assume thatG=A⊕P, whereP is a finite abelianp-group andAis a finite abelian group such thatp-|A|.

Theorem 3.6. Let pbe a prime and let s, r be integers such that 1 ≤s and 1 ≤r. Let A be a finite abelian group of exponent M such that p-M and letP be a finite abelianp-group. Then

N EA(GR(pr, s)[A⊕P]) = (N EA(GR(pr, s)[P]))

P

d|M,ord_d(ps)=1

(1−χ(d,ps_))NA(d)

× Y

d|M

ordd(ps)6=1

(N HA(GR(pr, s·ordd(ps))[P]))

(1−χ(d,ps₎₎ N_A(d) ord_d(ps)

×Y

d|M

(N A(GR(pr, s·ordd(ps))[P]))

In addition, ifs is even, then

N HA(GR(pr, s)[A⊕P]) =Y

d|M

(N HA(GR(pr, s·ordd(ps))[P]))

(1−λ(d,ps2)) NA(d)

ordd(ps)

×Y

d|M

(N A(GR(pr, s·ordd(ps))[P]))

λ(d,ps2) NA(d)

2ordd(ps)_.

Proof. First, we consider the Euclidean case. From (E3) and Proposition3.1, it is sufficient to count the numbers of Euclidean self-dual abelian codes Ui’s, the numbers of Hermitian self-dual abelian codes

Vi’s, and the numbers of abelian codesWi’s which correspond to the ps-cyclotomic classes of types I, II,

and III, respectively.

From [14, Remark 2.5], we note that the elements in A of the same order are partitioned into ps

-cyclotomic classes of the same type. For each divisordofM, aps_{-cyclotomic class containing an element}

of order dhas cardinality ordd(ps), and hence, the number of suchps-cyclotomic classes is NA

(d) ordd(ps).

For each divisordofM, we consider the following3 cases. Case 1. χ(d, ps_{) = 0 and ord}

d(ps) = 1. By Lemma 3.4, every ps-cyclotomic class of A containing

an element of order d is of type I. Since there are _ordNA(d)

d(ps) such p

s_{-cyclotomic classes, the number of}

Euclidean self-dual abelian codesUi’s corresponding to dis

(N EA(GR(pr, s·ordd(ps))[P]))

N_A(d)

ord_d(ps) _{= (N EA(GR(p}r_{, s)[P]))}(1−χ(d,p

s_))NA(d)

.

Case 2. χ(d, ps) = 0 and ordd(ps) 6= 1. By Lemma 3.4, every ps-cyclotomic class of A containing

an element of order d is of type II. Since there are NA(d)

ordd(ps) such p

s_{-cyclotomic classes, the number of}

Hermitian self-dual abelian codesVi’s corresponding todis

(N HA(GR(pr, s·ordd(ps))[P]))

N_A(d)

ordd(ps) _{= (N HA(GR(p}r_{, s·ord}

d(ps))[P]))

(1−χ(d,ps)) NA(d) ordd(ps)_.

Case 3. χ(d, ps_{) = 1. By Lemma 3.4, every p}s_{-cyclotomic class of A containing an element of order}

d is of type III. Since there are _ordNA(d)

d(ps) such p

s_{-cyclotomic classes, the number of abelian codes W} i’s

corresponding to dis

(N A(GR(pr, s·ordd(ps))[P]))

N_A(d)

ord_d(ps)_{(N A(GR(p}r_{, s·ord}

d(ps))[P]))

χ(d,ps₎ N_A(d) 2ord_d(ps)_.

Sincedruns over all divisors ofM, we conclude the desired result.

For the Hermitian case, by Proposition 3.2, it suffices to count the numbers of Hermitian self-dual abelian codes Ei’s and the numbers of abelian codesFi’s in (H3) which correspond to theps-cyclotomic

classes of types II0and III0, respectively. Considering the cases whereλ(d, p2s) = 1and whereλ(d, ps2) = 0, the desired result can be obtained similarly to the Euclidean case, where Lemma 3.5 is applied instead of Lemma 3.4.

Note that, ifAis a cyclic group, the exponentM is just the cardinality ofAandNA(d)is justφ(d),

where φis an Euler’s totient function.

In Theorem 3.6, if P is cyclic of orderpa_{, then the values N A, N EA and N HA may be replaced}

3.3.

Self-dual abelian codes in

GR(

p

_{, s}

_)[

_A

_]

_,

_gcd(

_p,

_|

_A

_|

_{) = 1}

In this subsection, we complete the enumeration of Euclidean and Hermitian self-dual abelian codes in GR(pr, s)[A], where gcd(p,|A|) = 1, or equivalently,GR(pr, s)[A]is a principal ideal group ring (see Proposition 3.7). IfA is cyclic, this case is identical with that of simple root cyclic codes.

Proposition 3.7. Let p be a prime number and let r, s be positive integers. Let G be a finite abelian group. Then one of the following statements holds.

i) If r= 1, thenGR(pr_{, s)[G]}∼₌

Fps[G] is a principal ideal ring if and only if the Sylowp-subgroup of

Gis cyclic.

ii) If r≥2, thenGR(pr_{, s)[G]is a principal ideal ring if and only ifgcd(p,|G|) = 1.}

For r= 1, the statement has been proven in [9]. Forr≥2, using notion of morphic rings (see the definition in [6]), it has been shown that Zpr[G] is principal ideal if and only if gcd(p,|G|) = 1 (see [8,

Theorem 1.2] and [6, Theorem 3.12 and Corollary 3.13]). The statements can be extended naturally to the case of GR(pr_{, s)[G].}

The enumerations of Euclidean and Hermitian self-dual abelian codes in a principal ideal group ring

GR(pr_{, s)[A]is given as follows.}

Theorem 3.8. Let p be a prime and let s, r be positive integers. Let A be a finite abelian group of exponent M such that gcd(p,|A|) = 1. Then

N EA(GR(pr, s)[A]) =







(1 +r)

P

d|M

χ(d,ps) NA(d) 2ord_d(ps)

ifr is even,

0 ifr is odd.

In addition, ifs is even, then

N HA(GR(pr, s)[A]) =







(1 +r)

P

d|M

λ(d,ps/2₎ N_A(d) 2ord_d(ps)

ifr is even,

0 ifr is odd.

Proof. InGR(pr, s), every ideal can be regarded as an abelian code inGR(pr, s)[G]withG={0}, and we have the following facts.

i) The number of abelian codes inGR(pr_{, s)isr+ 1.}

ii) Ifr is odd, then there are neither Euclidean self-dual abelian codes nor Hermitian self-dual abelian codes inGR(pr, s).

iii) If r is even, then rr/2GR(pr, s) is the only Euclidean self-dual abelian code and it is the only Hermitian self-dual abelian code if sis even.

The above results hold true for any Galois extension of GR(pr, s).

By consideringP ={0} in Theorem3.6, the result follows immediately.

Note that, ifAis cyclic,M andNA(d)can be replaced by the cardinality ofAandφ(d), respectively,

where φis the Euler’s totient function.

If A is cyclic of ordern withgcd(n, p) = 1, then the number of Euclidean self-dual cyclic codes of lengthnoverGR(pr_{, s)obtained in Theorem3.8is a special case of [1, Theorem 5.7] by viewingGR(p}r_{, s)}

3.4.

Self-dual abelian codes in

GR(

p

_{, s}

_)[

_A

_⊕

_C

pa

]

In this section, we restrict our study to the case where r= 2andP =Cpa, a cyclic group of order

pa. The enumerations of Euclidean and Hermitian self-dual abelian codes in GR(p2, s)[A⊕Cpa]can be

obtained as an application of Theorem 3.6 and some known results on cyclic codes of length pa _over

GR(p2_{, s).}

We recall some results on cyclic codes of lengthpaoverGR(p2, s). The next lemma follows immedi-ately from [16, Corollary 3.9] and [16, Theorem 3.6].

Lemma 3.9. The number of cyclic codes of lengthpa _overGR(p2_{, s) is}

N C(GR(p2, s), pa) = 2

pa−1

X

d=0

ps(min{bd

2c,p

a−1_}+1) −1

ps₋₁ +

ps(pa−1+1)₋₁

ps₋₁ . (6)

Proposition 3.10([17, Corollary 3.5]). The number of Euclidean self-dual cyclic codes of length2a _over

GR(22_{, s)is}

N EC(GR(22, s),2a) =



  

  

1 ifa= 1,

1 + 2s _{ifa= 2,}

1 + 2s_{+ 2}2s+1

(2s)(2a−2−1)−1 2s₋₁

ifa≥3.

If pis an odd prime, then the number of Euclidean self-dual cyclic codes of lengthpa overGR(p2, s)is

N EC(GR(p2, s), pa) = 2 (p

s₎(pa−1₊₁₎ /2−1

ps₋₁

!

.

Proposition 3.11 ([14, Theorem 3.5]). Let p be a prime and let s, abe positive integers such that s is even. Then the number of Hermitian self-dual cyclic codes of length pa _overGR(p2_{, s)is}

N HC(GR(p2, s), pa) =

pa−1

X

i1=0

psi1/2₌ p

s(pa−1_+1)/2 −1

ps/2₋₁ .

Remark 3.12. For cyclic codes of length pa _{over GR(p}2_{, s), the numbers N C,N EC, and N HC have}

already been determined in Lemma 3.9, Proposition3.10, and Proposition3.11, respectively. Combining these results and Theorem 3.6, the numbers N EA(GR(p2_{, s)[A⊕C}

pa]) and N HA(GR(p2, s)[A⊕C_pa])

are explicitly determined.

The numbers of Euclidean and Hermitian self-dual cyclic codes of arbitrary lengthnoverGR(p2_{, s)}

can be obtained as a corollary of Remark 3.12. Some parts of the formulas can be simplified as in the next corollary.

Corollary 3.13. Let p be a prime and let s, n be positive integers. Write n =mpa_{, where a≥0 and}

p_-m. Then

N EC(GR(p2, s), n) = N EC(GR(p2, s), pa)η(m)

× Y

d|m d6∈{1,2}

N HC(GR(p2, s·ordd(ps)), pa)

(1−χ(d,ps))ordφ_d(d(ps) )

×Y

d|m

N C(GR(p2, s·ordd(ps)), pa)

where

η(m) =

(

1 ifm is odd,

2 ifm is even.

In addition, ifs is even, then

N HC(GR(p2, s), n) =Y

d|m

N HC(GR(p2, s·ordd(ps)), pa)

(1−λ(d,p

s

2))_ordφ(d)

d(ps)

×Y

d|m

N C(GR(p2, s·ordd(ps)), pa) λ(d,p

s

2)_2ordφ(d)

d(ps)_.

Proof. Setting r = 2 and A a cyclic group of orderm in Theorem 3.6, the exponent ofA ism and NA(d)is justφ(d), whereφis the Euler’s function.

Note that Sps(0) is the only ps-cyclotomic class of Aof type I if mis odd, andS_ps(0)and S_ps(m

2)

are the only ps-cyclotomic classes ofAof type I ifmis even. Therefore, the values ofη(m)follows.

4.

Complementary dual abelian codes in

GR(p

r

, s)[G]

In this section, the characterization and enumeration of complementary dual abelian codes in the group ringGR(pr, s)[G]are given based on the decomposition in Section2and the theory of local group rings.

4.1.

Characterization and enumeration of complementary dual abelian codes

in

GR(

p

_{, s}

_)[

_P

_]

In this subsection, we focus on complementary dual abelian codes and direct summand ideals in each component of GR(pr_{, s)[P]in the decompositions (E1) and (H1), whereP is a finite abelianp-group.}

First, we recall some useful definitions and properties in ring theory. For a finite commutative ring R with identity, the Jacobson radical of R, denoted by J ac(R), is defined to be the intersection of all maximal ideals ofR. The ring Ris said to be local if it has a unique maximal ideal.

A local group ring has been characterized in the following lemma.

Lemma 4.1 ([22, Theorem]). Let R be a commutative ring with identity and let G be a finite abelian group. Then R[G]is local if and only ifR is local, Gis ap-group and p∈J ac(R).

Proposition 4.2. Let p be a prime number and let r, s be positive integers. Let P be a finite abelian p-group. ThenGR(pr_{, s)[P] is a local group ring.}

Proof. Since the idealhpiis the unique maximal ideal ofGR(pr_{, s), the ringGR(p}r_{, s)is local.}

More-over, p∈ hpi=J ac(GR(pr_{, s). By Lemma4.1,GR(p}r_{, s)[P]is a local group ring.}

By Proposition4.2,GR(pr, s)[P]is local. Denote byM the maximal ideal ofGR(pr, s)[P]. The char-acterizations of the Euclidean and Hermitian complementary dual abelian codes and the direct summands in a local group ring GR(pr_{, s)[P]are given in the following theorems.}

Proof. Clearly,{0}andGR(pr_{, s)[P]are Euclidean complementary dual abelian codes inGR(p}r_{, s)[P].}

LetC be an abelian code inGR(pr_{, s)[P]such that{0}}

(C(GR(pr, s)[P]. ThenC ⊆M. It follows that

M⊥E ⊆ C⊥E ⊆_{M which impliesM}⊥E ⊆ C ⊆_{M. Hence,} {₀} 6_=M⊥E ⊆ C ∩ C⊥E ⊆_{M. Consequently,} C is not Euclidean complementary dual. Therefore, the ideals{0} andGR(pr, s)[P]are the only Euclidean complementary dual abelian codes in GR(pr, s)[P].

It is not difficult to see that the proof of Theorem 4.3is independent of the inner product. Hence, we have the following corollary.

Corollary 4.4. Let p be a prime number and letr, s be positive integers such that sis even. Let P be a finite abelianp-group. Then{0} andGR(pr_{, s)[P]are the only Hermitian complementary dual abelian} codes in GR(pr_{, s)[P].}

Theorem 4.5. Letpbe a prime number and letr, sbe positive integers. LetP be a finite abelianp-group. Then ideals {0}andGR(pr_{, s)[P] are the only direct summands in GR(p}r_{, s)[P].}

Proof. Clearly,{0}andGR(pr_{, s)[P]are direct summands inGR(p}r_{, s)[P]. Let{0}}

(C(GR(pr, s)[P]

be an ideal in GR(pr_{, s)[P]. Suppose that C is a direct summand. Then there exists an ideal C}0 _in

GR(pr_{, s)[P] such that C ∩ C}0 _{= {0} and C+C}0 _{= GR(p}r_{, s)[P]. Since M is the maximal ideal in}

GR(pr, s)[P], we have C ⊆ M and C0 _{⊆ M. Hence, C +C}0 _{⊆ M}

( GR(pr, s)[P], a contradiction.

Therefore, the ideals {0}andGR(pr, s)[P]are the only direct summands inGR(pr, s)[P].

The above results can be summarized as follows.

Corollary 4.6. Letpbe a prime number and letr, sbe positive integers. LetP be a finite abelianp-group. Then the following statements hold.

1. The number of Euclidean complementary dual abelian codes inGR(pr_{, s)[P] is2.}

2. Ifs is even, the number of Hermitian complementary dual abelian codes inGR(pr_{, s)[P]is2.} 3. The number of direct summand ideals inGR(pr, s)[P] is2.

4.2.

Characterization and enumeration of complementary dual abelian codes

in

GR(

p

, s

)[

G

]

In this subsection, we focus on the characterization and enumeration of complementary dual abelian codes inGR(pr, s)[G], whereGis an arbitrary finite abelian group. Using the decompositions in Section

2 and results in the previous subsection, the characterization and enumeration of such complementary dual codes are given independent of rand the Sylowp-subgroup ofG.

Recall thatG∼=A×P, whereP is the Sylowp-subgroup ofGand Ais its complement subgroup. The group ring GR(pr, s)[G] is viewed as GR(pr, s)[G] ∼= R[P], where R = GR(pr, s)[A]. Using the decomposition ofGR(pr_{, s)[G] in (E2), the characterization of a Euclidean complementary dual abelian}

code inGR(pr_{, s)[G]can be concluded via (E3) and (E4) as follows.}

Proposition 4.7. Letpbe a prime number and letr, sbe positive integers. Let Abe finite abelian group such that p _- |A| and let P be a finite abelian p-group. Then an abelian code C in GR(pr_{, s)[A×P]} decomposed as in (E3)is Euclidean complementary dual if and only if the following statements hold.

1. Ui is Euclidean complementary dual for all i= 1,2, . . . , rI. 2. Vj is Hermitian complementary dual for allj= 1,2, . . . , rII. 3. Wk∩(Wk0)⊥E ={0} andWk0 ∩W

⊥E

The next corollary follows immediately from Theorem4.3, Proposition4.7, and Corollary4.4.

Corollary 4.8. Let p be a prime number and let r, sbe positive integers. Let A be finite abelian group such that p _- |A| and let P be a finite abelian p-group. Then an abelian code C in GR(pr_{, s)[A×P]} decomposed as in (E3)is Euclidean complementary dual if and only if the following statements hold.

1. Ui∈ {{0},GR(pr, s)[P]} for alli= 1,2, . . . , rI. 2. Vj ∈ {{0},GR(pr,2sej)[P]} for allj = 1,2, . . . , rII. 3. (Wk, Wk0)∈ {({0},{0}),(GR(p

r_{, sf}

k)[P],GR(pr, sfk)[P])} for allk= 1,2, . . . , rIII.

From Corollary4.8, it is not difficult to see that the number of Euclidean complementary dual abelian codes inGR(pr_{, s)[A×P]is independent ofrand the groupP and it can be determined in the following}

corollary.

Corollary 4.9. Let p be a prime number and let r, sbe positive integers. Let A be finite abelian group such thatp-|A|and letP be a finite abelianp-group. If the exponent of AisM andGR(pr, s)[A×P]is

decomposed as in (E2), then the number of Euclidean complementary dual abelian codes inGR(pr_{, s)[A×}

P] is

2rI+rII+rIII_{= 2}

P

d|M

(1−χ(d,ps)) NA(d)

ordd(ps)+

P

d|M

χ(d,ps) NA(d)

2ordd(ps) ,

where NA(d)denote the number of elements in Aof orderd.

Proof. The first part follows from Corollary 4.8. The equality can be derived similar to the proof of Theorem 3.6.

Using the decomposition ofGR(pr_{, s)[G]in (H2), the characterization of a Hermitian complementary}

dual abelian code inGR(pr_{, s)[G] can be concluded via (H3) and (H4) in the following proposition.} Proposition 4.10. Let pbe a prime number and letr, s be positive integers such thats is even. LetA be finite abelian group such that p-|A| and letP be a finite abelianp-group. Then an abelian codeC in

GR(pr_{, s)[A×P] decomposed as in (H3) is Hermitian complementary dual if and only if the following} statements hold.

1. Ej is Hermitian complementary dual for allj = 1,2, . . . , r0II. 2. Fk∩(Fk0)

⊥E ₌{₀} _andF0

k∩F

⊥E

k ={0} for allj= 1,2, . . . , rIII0.

The following result follows directly from Proposition4.10and Corollary 4.4.

Corollary 4.11. Let pbe a prime number and letr, s be positive integers such thats is even. Let A be finite abelian group such that p_- |A| and let P be a finite abelian p-group. Then an abelian code C in

GR(pr_{, s)[A×P] decomposed as in (H3) is Hermitian complementary dual if and only if the following} statements hold.

1. Ej∈ {{0},GR(pr, se´j)[P]} for allj= 1,2, . . . , r0II. 2. (Fk, Fk0)∈ {({0},{0}),(GR(p

r_{, sf}´

k)[P],GR(pr, sf´k)[P])} for allk= 1,2, . . . , rIII0.

From Corollary4.11, the number of Hermitian complementary dual abelian codes inGR(pr_{, s)[A×P]}

Corollary 4.12. Let pbe a prime number and letr, s be positive integers such that sis even. Let A be finite abelian group such that p_-|A|and let P be a finite abelianp-group. If the exponent ofA isM and

GR(pr_{, s)[A×P] is decomposed as in (H2), then the number of Hermitian complementary dual abelian} codes in GR(pr, s)[A×P] is

2rI0+rII0 _{= 2}

P

d|M

(1−λ(d,ps2)) NA(d)

ord_d(ps)+

P

d|M

λ(d,ps2) NA(d)

2ord_d(ps) ,

where NA(d)denote the number of elements in Aof orderd.

Proof. The first part follows from Corollary4.11. The equality can be derived similar to the proof of Theorem 3.6.

5.

Conclusion

Self-dual and complementary dual abelian codes in GR(pr_{, s)[G], a group ring of a finite abelian}

group Gover a Galois ringGR(pr_{, s), have been studied with respect to the Euclidean and Hermitian}

inner products. We have characterized such self-dual codes as well as determined necessary and sufficient conditions for GR(pr_{, s)[G] to contain a Euclidean (resp, Hermitian) self-dual abelian code. For any}

finite abelian group G and Galois ring GR(pr, s), the enumerations of such self-dual codes have been given. In the case where gcd(|G|, p) = 1, the enumeration has been completed by restricting the Sylow p-subgroup to be{0}. Applying some known results on cyclic codes of lengthpa _overGR(p2_{, s), we have}

determined explicitly the numbers of Euclidean and Hermitian self-dual abelian codes in GR(p2_{, s)[G]if}

the Sylowp-subgroup ofGis cyclic. As corollaries, analogous results on Euclidean and Hermitian self-dual cyclic codes overGR(pr, s)have been concluded. Subsequently, the characterization and enumeration of complementary dual abelian codes inGR(pr, s)[G]have been given. The number of complementary dual abelian codes inGR(pr_{, s)[G]has been shown to be independent ofrand the Sylowp-subgroup ofG.}

It would be interesting to study the unknown terms in Theorem3.6and extend the results to abelian codes over finite chain rings or the case where the Sylow p-subgroup of the group is not cyclic.

Appendix A

In this appendix, we discuss the Euclidean and Hermitian duals of abelian codes in GR(pr, s)[G]. First, we recall thatG∼=A×P, wherePis the Sylowp-subgroup ofGandAis a complementary subgroup ofP inG. The group ringR:= GR(pr_{, s)[A]is decomposed as in (E1) or (H1), andGR(p}r_{, s)[G]}∼_=R[P].

A.1.

Euclidean duality

Letψdenote the isomorphism in (E1). For each element x∈ R, we can write

ψ(x) = (x1, . . . , xrI, y1, . . . , yrII, z1, z

0

1, . . . , zrIII, z

0

rIII), (7)

where xi∈GR(pr, s),yj ∈GR(pr,2sej), andzk, zk0 ∈GR(p r_{, sf}

k)for alli= 1,2, . . . , rI,j= 1,2, . . . , rII, andk= 1,2, . . . , rIII.

We are going to view x_b defined in Section 2 in terms of (7). We note that, for c=P a∈AcaY

a _∈

GR(p2_{, s)[A], we have}

b

c=X

a∈A

caY−a= X

a∈A

Then_bc˘=P

a∈Ad˘aY

a_,where

˘

da= X

h∈A

c−hζγa(h).

From (3), we can see that, ifSps(h)is of type I, then

˘

dh= ˘ch, (8)

and ifSps(h)is of type II with cardinality2ν, then−h=psν·hby Remark2.4. It follows that

˘

dh= X

a∈A

c−aζγh(a)= X

a∈A

caζγ−h(a)= X

a∈A

caζγpsν·h(a)= X

a∈A

ca

ζγh(a)

psν

=θ(˘ch), (9)

where θ(α) =αp₀sν +αp₁sνp+· · ·+αp_rsν₋₁pr−1 _{for allα=α}

0+α1p+· · ·+αr−1pr−1.

Therefore, by the isomorphismψ(see also [18]), the following properties are obtained.

1. From (8), the involution_binduces the identity automorphism onGR(pr, s).

2. From (9), the involution_binduces the ring automorphism¯onGR(pr,2sej)as defined in (1),i.e.,

α=α₀psej +αp₁sejp+· · ·+αp_rsej₋₁pr−1

for allα=α0+α1p+· · ·+αr−1pr−1 in GR(pr,2sej), where αi∈ T2sej for alli= 0,1, . . . , r−1.

3. For each pair(z, z0)∈GR(pr, sfk)×GR(pr, sfk), we have

\

ψ−1_{(z, z}0_{) =ψ}−1_(z0_{, z).}

From the discussion, we have

ψ(_bx) = (x1, . . . , xrI, y1, . . . , yrII, z

0

1, z1, . . . , z0rIII, zrIII), where¯is induced as above in an appropriate Galois extension.

Proposition A.1. Let x=P

b∈PxbY

b _andu=P b∈PubY

b _{be elements inR[P]. Decomposingx} b,ub using (7), we have

ψ(xb) = (xb,1, . . . , xb,rI, yb,1, . . . , yb,rII, zb,1, z

0

b,1, . . . , zb,rIII, z

0

b,rIII) and

ψ(ub) = (ub,1, . . . , ub,rI, vb,1, . . . , vb,rII, wb,1, w

0

b,1, . . . , wb,rIII, w

0

b,rIII). Then

ψ(hx,ui

b

) =ψ X

b∈P

xbcub !

=X

b∈P

ψ(xb)ψ(cub)

= X

b∈P

xb,1ub,1, . . . ,

X

b∈P

xb,rIub,rI,

X

b∈P

yb,1vb,1, . . . ,

X

b∈P

yb,rIIvb,rII,

X

b∈P

zb,1wb,01,

X

b∈P

z_b,0₁wb,1, . . . ,

X

b∈P

zb,rIIIw

0

b,rIII,

X

b∈P

z_b,r0 IIIwb,rIII

!

In particular, hx,ui

b

= 0if and only if ψ(hx,ui

b

) =0, or equivalently,

X

b∈P

xb,jub,j= 0 for allj= 1,2, . . . , rI, X

b∈P

yb,jv_gb,j= 0 for allj= 1,2, . . . , rII,

and

X

b∈P

zb,jw0b,j= 0 = X

b∈P

z_b,j0 wb,j for allj= 1,2, . . . , rIII.

Using the orthogonality in Proposition A.1, the Euclidean dual ofCin (E3) can be viewed to be of the form

C⊥E ∼₌

tI

Y

i=1

U⊥E

i ! ×   tII Y j=1

V⊥H

j  × tIII Y k=1

(W_k0)⊥E_×W⊥E

k

!

. (10)

A.2.

Hermitian duality

Letψdenote the isomorphism in (H1). Then each element x∈ R, we can write

ψ(x) = (x1, . . . , xt_II0, y1, y10, . . . , yt_III0, y 0

tIII0), (11)

where xj ∈GR(pr, se´j)andyk, y0k∈GR(p r_{, sf´}

k)for allj= 1,2, . . . , tII0 andk= 1,2, . . . , t_III0.

We note that, for c=P a∈AcaY

a _∈GR(pr_{, s)[A], we have}

e

c=X

a∈A

caY−a= X

a∈A

c−aYa,

where α0+pα1+· · ·+pr−1αr−1 = α

ps/2

0 +pα

ps/2

1 +· · ·+pr−1α

ps/2

r−1. Then e˘c = P

a∈Aw˘aYa, where

˘

wa=Ph∈Ac−hζγa(h).

From (3), if Sps(h)is of type II0 with cardinality ν, then −a=psν/2·a by Remark2.4. Since ν is

odd, we have

˘

wh= X

a∈A

c−aζγh(a)= X

a∈A

caζγ−h(a)= X

a∈A

caζ

γ_psν/2·h(a)₌X

a∈A

ca

ζγh(a)

psν/2

=θ(˘ch), (12)

where θ(α) =αp₀sν/2+α₁psν/2p+· · ·+αp_rsν/₋₁2pr−1for allα=α0+α1p+· · ·+αr−1pr−1.

By the isomorphismψ(see also [18]), we have the following properties.

1. By (12), the involution_einduces the ring automorphism¯onGR(pr_{, se´}

j)as defined in (1),i.e.,

α=α₀ps´ej /2+αp₁s´ej /2p+· · ·+αp_r−s´ej /₁ 2pr−1

for allα=α0+α1p+· · ·+αr−1pr−1 in GR(pr, s´ej), whereai∈ Tse´j for alli= 0,1, . . . , r−1.

2. For each pair(z, z0)∈GR(pr, sf´k)×GR(pr, sf´k), we have

^

ψ−1_{(z, z}0_{) =ψ}−1_(z0_{, z).}

Hence,x_e defined in Section2can be viewed in terms of (11) as

ψ(x_e) = (x1, . . . , xt_II0, y 0

Proposition A.2. Let x=P

b∈PxbY

b _andu=P b∈PubY

b _{be elements inR[P]. Decomposingx} b,ub using (11), we have

ψ(xb) = (xb,1, . . . , xb,t_II0, yb,1, y0b,1, . . . , yb,t_III0, y 0

b,t_III0)

and

ψ(ub) = (ub,1, . . . , ub,t_II0, vb,1, v0b,1, . . . , vb,t_III0, v 0

b,t_III0).

Then

ψ(hx,ui∼) =ψ

X

b∈P

xbufb !

=X

b∈P

ψ(xb)ψ(fub)

= X

b∈P

xb,1ub,1, . . . ,

X

b∈P

xb,t_II0ub,tII0,

X

b∈P

yb,1v0b,1,

X

b∈P

y_b,0₁vb,1, . . . ,

X

b∈P

yb,t_III0v 0

b,t_III0,

X

b∈P

y0_b,t III0vb,tIII0

!

.

In particular, hx,ui∼= 0 if and only ifψ(hx,ui∼) =0, or equivalently,

X

b∈P

xb,jub,j= 0 for allj= 1,2, . . . , tII0 and

X

b∈P

yb,kv0b,k= 0 = X

b∈P

yb,k0 vb,k for all k= 1,2, . . . , tIII0.

Using the orthogonality in Proposition A.2, the Hermitian dual ofC in (H3) can be viewed of the form

C⊥H ∼₌



 t_II0

Y

j=1

E⊥H

j 

× t_III0

Y

k=1

(F_k0)⊥E_×F⊥E

k

!

. (13)

Acknowledgment: The authors would like to thank anonymous referees for useful comments.

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