#### Abstract

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebras with respect to both the Euclidean and Hermitian inner products, where and are positive integers and is an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length over some Galois extensions of the ring , where . Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length over are given. Combining these results, the complete enumeration of self-dual abelian codes in is therefore obtained.

#### 1. Introduction

Information media, such as communication systems and storage devices of data, are not percent reliable in practice because of noise or other forms of introduced interference. The art of error correcting codes is a branch of Mathematics that has been introduced to deal with this problem since the 1960s. Linear codes with additional algebraic structures and self-dual codes are important families of codes that have been extensively studied for both theoretical and practical reasons (see [1â€“10] and references therein). Some major results on Euclidean self-dual cyclic codes have been established in [11]. In [12], the complete characterization and enumeration of such self-dual codes have been given. These results on Euclidean self-dual cyclic codes have been generalized to abelian codes in group algebras [6] and the complete characterization and enumeration of Euclidean self-dual abelian codes in principal ideal group algebras (PIGAs) have been established. Extensively, the characterization and enumeration of Hermitian self-dual abelian codes in PIGAs have been studied in [9]. To the best of our knowledge, the characterization and enumeration of self-dual abelian codes in nonprincipal ideal group algebras (non-PIGAs) have not been well studied. It is therefore of natural interest to focus on this open problem.

In [6, 9], it has been shown that there exists a Euclidean (resp., Hermitian) self-dual abelian code in if and only if and is even. In order to study self-dual abelian codes, it is therefore restricted to the group algebra , where is an abelian group of odd order and is a nontrivial abelian group of two power order. In this case, is a PIGA if and only if is a cyclic group (see [13]). Equivalently, is a non-PIGA if and only if is noncyclic. To avoid tedious computations, we focus on the simplest caseâ€‰â€‰where , where is a positive integer. Precisely, the goal of this paper is to determine the algebraic structures and the numbers of Euclidean and Hermitian self-dual abelian codes in .

It turns out that every Euclidean (resp., Hermitian) self-dual abelian code in is isomorphic to a suitable Cartesian product of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length over some Galois extension of the ring , where (see Section 2). Hence, the number of self-dual abelian codes in can be determined in terms of the cyclic codes mentioned earlier. Subsequently, useful properties of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length over are given for all primes . Combining these results, the characterizations and enumerations of Euclidean and Hermitian self-dual abelian codes in are rewarded.

The paper is organized as follows. In Section 2, some basic results on abelian codes are recalled together with a link between abelian codes in and cyclic codes of length over Galois extensions of . General results on the characterization and enumeration of cyclic codes of length over are provided in Section 3. In Section 4, the characterizations and enumerations of Euclidean and Hermitian self-dual cyclic codes of length over are established. Summary and remarks are given in Section 5.

#### 2. Preliminaries

In this section, we recall some definitions and basic properties of rings and abelian codes. Subsequently, a link between an abelian code in nonprincipal ideal algebras and a product of cyclic codes over rings is given. This link plays an important role in determining the algebraic structures and the numbers of Euclidean and Hermitian self-dual abelian codes in nonprincipal ideal algebras.

##### 2.1. Rings and Abelian Codes in Group Rings

For a prime and a positive integer , denote by the finite field of order . Let be a ring, where the addition and multiplication are defined as in the usual polynomial ring over with indeterminate together with the condition . We note that is isomorphic to as rings. The Galois extension of of degree is defined to be the quotient ring , where is an irreducible polynomial of degree over . It is not difficult to see that the Galois extension of of degree is isomorphic to as rings. In the caseâ€‰â€‰where is even, the mapping is a ring automorphism of order on . The readers may refer to [14, 15] for properties of the ring .

For a commutative ring with identity and a finite abelian group , written additively, let denote the* group ring* of over . The elements in will be written as , where . The addition and the multiplication in are given as in the usual polynomial ring over with indeterminate , where the indices are computed additively in . Note that is the multiplicative identity of (resp., ), where is the identity of . We define a* conjugation * on to be the map that fixes and sends to for all ; that is, for , we set . In the caseâ€‰â€‰where there exists a ring automorphism on of order , we define for all . In the caseâ€‰â€‰where is a finite field , then can be viewed as an -algebra and it is called a* group algebra*. The group algebra is called a* principal ideal group algebra (PIGA)* if its ideals are generated by a single element.

An* abelian code* in is defined to be an ideal in . If is cyclic, this code is called a* cyclic code*, a code which is invariant under the right cyclic shift. It is well known that cyclic codes of length over can be regarded as ideals in the quotient polynomial ring .

The* Euclidean inner product* in is defined as follows. For in , we set In addition, if there exists a ring automorphism of order on , the -*inner product* of and is defined by If (resp., ) and (resp., ) for all (resp., ), the -inner product is called the* Hermitian inner product* and denoted by .

The* Euclidean dual* and* Hermitian dual* of in are defined to be the setsrespectively.

An abelian code is said to be Euclidean self-dual (resp., Hermitian self-dual) if (resp., ).

For convenience, denote by , , and the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length over , respectively.

##### 2.2. Decomposition of Abelian Codes in

In [6, 9], it has been shown that there exists a Euclidean (resp., Hermitian) self-dual abelian code in if and only if and is even. To study self-dual abelian codes, it is sufficient to focus on , where is an abelian group of odd order and is a nontrivial abelian group of two power order. In this case, is a PIGA if and only if is a cyclic group for some positive integer (see [13]). The complete characterization and enumeration of self-dual abelian codes in PIGAs have been given in [6, 9]. Here, we focus on self-dual abelian codes in non-PIGAs, or equivalently, is non-cyclic. To avoid tedious computations, we establish results for the simplest caseâ€‰â€‰where . Useful decompositions of are given in this section.

First, we consider the decomposition of . In this case, is semisimple [2] which can be decomposed using the Discrete Fourier Transform in [7] (see [6, 9] for more details). For completeness, the decompositions used in this paper are summarized as follows.

For an odd positive integer and a positive integer , let denote the multiplicative order of modulo . For each , denote by the additive order of in . A *-cyclotomic class* of containing , denoted by , is defined to be the set where in .

An* idempotent* in is a nonzero element such that . It is called* primitive* if, for every other idempotent , either or . The primitive idempotents in are induced by the -cyclotomic classes of (see [5, Propositionâ€‰â€‰II.4]). Let be a complete set of representatives of -cyclotomic classes of and let be the primitive idempotent induced by for all . From [7], can be decomposed asand hence,It is well known (see [6, 9]) that , where is a multiple of . Precisely, provided that is induced by . It follows that . Under the ring isomorphism that fixes the elements in and , is isomorphic to the ring , where . We note that this ring plays an important role in coding theory and codes over rings in this family have extensively been studied in [14â€“17] and references therein.

From (8) and the ring isomorphism discussed above, we havewhere for all .

In order to study the algebraic structures of Euclidean and Hermitian self-dual abelian codes in , the two rearrangements of â€™s in the decomposition (9) are needed. The details are given in the following two subsections.

###### 2.2.1. Euclidean Case

A -cyclotomic class is said to be of* type * if (in this case, ),* type * if and , or* type * if . The primitive idempotent induced by is said to be of type if is a -cyclotomic class of type .

Without loss of generality, the representatives of -cyclotomic classes of can be chosen such that , , and are sets of representatives of -cyclotomic classes of types , , and , respectively, where .

Rearranging the terms in the decomposition of in (7) based on the types of primitive idempotents, we havewhere for all , for all , and for all .

From (10), we haveIt follows that an abelian code in can be viewed aswhere , , , and are cyclic codes in , , , and , respectively, for all , , and .

Using the analysis similar to those in [6, Sectionâ€‰â€‰II.D], the Euclidean dual of in (12) is of the formSimilar to [6, Corollaryâ€‰â€‰2.9], necessary and sufficient conditions for an abelian code in to be Euclidean self-dual can be given using the notions of cyclic codes of length over , , and in the following corollary.

Corollary 1. *An abelian code in is Euclidean self-dual if and only if in the decomposition (12)*(i)* is a Euclidean self-dual cyclic code of length over for all ,*(ii)* is a Hermitian self-dual cyclic code of length over for all ,*(iii)* is a cyclic code of length over for all .*

Given a positive integer and an odd positive integer , the pair is said to be* good* if divides for some integer and* bad* otherwise. These notions have been introduced in [6, 12] for the enumeration of self-dual cyclic codes and self-dual abelian codes over finite fields.

Let be a function defined on the pair , where is an odd positive integer, as follows:The number of Euclidean self-dual abelian codes in can be determined as follows.

Theorem 2. *Let and be positive integers and let be a finite abelian group of odd order and exponent . Then the number of Euclidean self-dual abelian codes in is where denotes the number of elements in of order determined in [18].*

*Proof. *From (12) and Corollary 1, it suffices to determine the numbers of cyclic codes â€™s, â€™s, and â€™s such that and are Euclidean and Hermitian self-dual, respectively.

From [9, Remarkâ€‰â€‰2.5], the elements in of the same order are partitioned into -cyclotomic classes of the same type. For each divisor of , a -cyclotomic class containing an element of order has cardinality and the number of such -cyclotomic classes is . We consider the following cases.*Caseâ€‰â€‰1 (** and **).* By [6, Remarkâ€‰â€‰2.6], every -cyclotomic class of containing an element of order is of type I. Since there are such -cyclotomic classes, the number of Euclidean self-dual cyclic codes â€™s of length corresponding to is*Caseâ€‰â€‰2 (** and **).* By [6, Remarkâ€‰â€‰2.6], every -cyclotomic class of containing an element of order is of type II. Hence, the number of Hermitian self-dual cyclic codes â€™s of length corresponding to is *Caseâ€‰â€‰3 (**).* By [6, Lemmaâ€‰â€‰4.5], every -cyclotomic class of containing an element of order is of type III. Then the number of cyclic codes â€™s of length corresponding to isThe desired result follows since runs over all divisors of .

This enumeration will be completed by counting the above numbers , , and in Corollaries 22, 25, and 17, respectively.

###### 2.2.2. Hermitian Case

We focus on the caseâ€‰â€‰where is even. A -cyclotomic class is said to be of* type * if or* type * if . The primitive idempotent induced by is said to be of type if is a -cyclotomic class of type .

Without loss of generality, the representatives of -cyclotomic classes can be chosen such that and are sets of representatives of -cyclotomic classes of types and , respectively, where .

Rearranging the terms in the decomposition of in (7) based on the above types of primitive idempotents, we havewhere for all and for all .

From (19), we haveHence, an abelian code in can be viewed aswhere , , and are cyclic codes in , , and , respectively, for all and .

Using the analysis similar to those in [9, Sectionâ€‰â€‰II.D], the Hermitian dual of in (21) is of the form

Similar to [9, Corollaryâ€‰â€‰2.8], necessary and sufficient conditions for an abelian code in to be Hermitian self-dual are now given using the notions of cyclic codes of length over and in the following corollary.

Corollary 3. *An abelian code in is Hermitian self-dual if and only if in the decomposition (21)*(i)* is a Hermitian self-dual cyclic code of length over for all ,*(ii)* is a cyclic code of length over for all .*

Given a positive integer and an odd positive integer , the pair is said to be* oddly good* if divides for some odd integer and* evenly good* if divides for some even integer . These notions have been introduced in [9] for characterizing the Hermitian self-dual abelian codes in PIGAs.

Let be a function defined on the pair , where is an odd positive integer, asThe number of Hermitian self-dual abelian codes in can be determined as follows.

Theorem 4. *Let be an even positive integer and let be a positive integer. Let be a finite abelian group of odd order and exponent . Then the number of Hermitian self-dual abelian codes in is where denotes the number of elements of order in determined in [18].*

*Proof. *By Corollary 3 and (21), it is enough to determine the numbers of cyclic codes â€™s and â€™s of length in (21) such that is Hermitian self-dual.

The desired result can be obtained using arguments similar to those in the proof of Theorem 2, where [9, Lemmaâ€‰â€‰3.5] is applied instead of [6, Lemmaâ€‰â€‰4.5].

This enumeration will be completed by counting the above numbers and in Corollaries 25 and 17, respectively.

#### 3. Cyclic Codes of Length over

The enumeration of self-dual abelian codes in non-PIGAs in the previous section requires properties of cyclic codes of length over . In this section, a more general situation is discussed. Precisely, properties cyclic of length over are studied for all primes . We note that algebraic structures of cyclic codes of length over were studied in [14, 15]. Here, based on [19], we give an alternative characterization of such codes which is useful in studying self-dual cyclic codes of length over .

First, we note that there exists a one-to-one correspondence between the cyclic codes of length over and the ideals in the quotient ring . Precisely, a cyclic code of length can be represented by the ideal in .

From now on, a cyclic code will be referred to as the above polynomial presentation. Note that the map defined byis a surjective ring homomorphism. For each cyclic code in and , letFor each , is called the th* torsion code* of . The codes and are sometimes called the* residue* and* torsion codes* of , respectively.

It is not difficult to see that, for each , if and only if for some . Consequently, we have that are ideals in (cyclic codes of length over ). We note that every ideal in is of the form for some and the cardinality of is .

From the structures of cyclic codes of length over discussed above and [14, Propositionâ€‰â€‰2.5], we have the following properties of the torsion and residue codes.

Proposition 5. *Let be an ideal in and let . Then the following statements hold:*(i)* is an ideal of and for some .*(ii)*If , then .*(iii)*.*

With the notations given in Proposition 5, for each , is called the th-*torsional degree* of .

*Remark 6. *From Proposition 5 and the definition above, we have the following facts:(i)Since , we have .(ii)If , then .

Next, we determine a generator set of an ideal in .

Theorem 7. *Let be an ideal of . Thenwhere, for each ,*(i)*either or for some and ,*(ii)* if and only if and ,*(iii)*if , then .*

*Proof. *The statement can be obtained using a slight modification of the proof of [20, Theoremâ€‰â€‰6.5]. For completeness, the details are given as follows.

For each ideal in , it can be represented as or , where . If , then we are done by choosing . For , let . By abuse of notation, . Hence, satisfies conditions (i), (ii), and (iii).

Since is an ideal of the ring and is a surjective ring homomorphism, is an ideal of which implies that , where satisfies conditions (i), (ii), and (iii). If , then take . Assume that ; then , where . Then there exists such that and ; that is, . Since is an ideal of , it follows that for some . Let . Claim that . Since is an ideal of , we have . Thus, . To show that , let . Then for some . Thus, which implies that for some . Since , it follows that . Therefore, as desired.

Note that implies . Assume that . Then , where . If , then we are done. Assume that . Then , where . Hence, . Since , we have . If , then we are done. Assume that . Then for some . It follows that , a contradiction. Therefore, .

However, the generator set given in Theorem 7 does not need to be unique. The unique presentation is given in the following theorem.

Theorem 8. *Let be an ideal in , , and . Thenwherewith being either zero or a unit with .**Moreover, is unique in the sense that if there exists a pair of polynomials satisfying the conditions in the theorem, then and .*

*Proof. *If , then and which imply that and . The polynomials and have the desired properties.

Next, assume that . Then there exists the smallest nonnegative integer such that . From Theorem 7, it can be concluded thatwherefor some .*Caseâ€‰â€‰1 *. Then and . It follows that , , and . Let . Since , we havewhere for all . Since , we have for all . It follows that . Let . Then .

We show that . From the discussion above, we have . Since , it follows that for all . Hence, which implies thatTherefore, . As desired, .*Caseâ€‰â€‰2 *. Then and which implies that and . By choosing and , the result follows.

To prove the uniqueness, let be such that and satisfy the conditions in the theorem. Then .

Write , where . Then It can be seen that , where or is a unit with . If is a unit, then which implies that , a contradiction. Hence, which means that as desired.

*Definition 9. *For each ideal in , denote by the unique representation of the ideal obtained in Theorem 8.

Illustrative examples of the representations in Theorems 7 and 8 are given as follows.

*Example 10. *Consider the ideal in . Using Theorem 8, we obtain that has the unique representation . Based on Theorem 7, can be represented as , , and .

The annihilator of an ideal in is key to determine properties as well as the number of ideals in .

*Definition 11. *Let be an ideal in . The* annihilator* of , denoted by , is defined to be the set .

The following properties of the annihilator can be obtained using arguments similar to those in the caseâ€‰â€‰of Galois rings in [19].

Theorem 12. *Let be an ideal of . Then the following statements hold:*(i)* is an ideal of .*(ii)*If , then .*(iii)*.*

Theorem 13. *Let denote the set of ideals of and let and . Then the map defined by is a bijection.*

The rest of this section is devoted to the determination of all ideals in . In view of Theorem 13, it suffices to focus on the ideals in .

For each in , if , then and . Hence, the only ideal in with is of the form . In the following two theorems, we assume that .

Theorem 14. *Let be the representation of an ideal in . Then it is a representation of an ideal in if and only if , , and are integers and such that , , , , and is either zero or a unit in .*

*Proof. *Form Theorem 8, we have that , , and are integers and such that , , , , and is either zero or a unit in .

Assume that is a representation of an ideal in . Then which implies that . Hence, we have

Conversely, assume that , where , , and are integers and such that , , , , and is either zero or a unit in . Clearly, . To show that in , it remains to prove that and .

Let . It is not difficult to see that Since