Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

Choosuwan, Parinyawat; Jitman, Somphong; Udomkavanich, Patanee

doi:https://doi.org/10.1155/2016/9020173

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2016 | Article ID 9020173 | https://doi.org/10.1155/2016/9020173

Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

Parinyawat Choosuwan,¹Somphong Jitman,²and Patanee Udomkavanich¹

Academic Editor: Kishin Sadarangani

Received25 Jun 2016

Accepted27 Sept 2016

Published27 Oct 2016

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 [1–10] 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 case 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 case 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 case where there exists a ring automorphism on of order , we define for all . In the case 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 case 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, Proposition 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 [14–17] 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 ’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, Section II.D], the Euclidean dual of in (12) is of the formSimilar to [6, Corollary 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 ’s, ’s, and ’s such that and are Euclidean and Hermitian self-dual, respectively.
From [9, Remark 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.
Case 1 ( and ). By [6, Remark 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 ’s of length corresponding to isCase 2 ( and ). By [6, Remark 2.6], every -cyclotomic class of containing an element of order is of type II. Hence, the number of Hermitian self-dual cyclic codes ’s of length corresponding to is Case 3 (). By [6, Lemma 4.5], every -cyclotomic class of containing an element of order is of type III. Then the number of cyclic codes ’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 case 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, Section II.D], the Hermitian dual of in (21) is of the form

Similar to [9, Corollary 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 ’s and ’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, Lemma 3.5] is applied instead of [6, Lemma 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, Proposition 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, Theorem 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 .
Case 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, .
Case 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 case 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 Sincewe have By Proposition 5, we obtain that and for all . Hence, and . Since , we have . Therefore, and as desired.

Since every polynomial in is either or , where is a unit in and , Theorem 14 is rewritten as follows.

Theorem 15. The expression represents an ideal in if and only if and are integers such that , , , and for all .

The number of distinct ideals of of a fixed is given in the following proposition.

Proposition 16. Let . Then the number of distinct ideals in with iswhere .

Proof. Let and be fixed.
Case 1 . Then By Theorem 15, it follows that . Then the choice for is . By Theorem 15 again, we also have . Since , we obtain that , and hence, . Now, vary from to ; we obtain that there are ideals with .
Case 2 . If , then the only ideal with is the ideal represented by . If , then we have ideals by arguments similar to those in Case 1.

For a cyclic code in , we have whenever . In the case where , by the proof of Theorem 14, the annihilator of the cyclic code is of the form . If is odd, then occurs only in the case . In the case where , is always true. By Proposition 16 and the bijection given in Theorem 13, the number of cyclic codes of length over can be summarized as follows.

Corollary 17. The number of cyclic codes of length over is

Proof. From Theorem 13, the number of cyclic codes of length over is . The desired results follow immediately form the discussion above.

4. Self-Dual Cyclic Codes of Length over

In this section, characterization and enumeration self-dual cyclic codes of length over are given under the Euclidean and Hermitian inner products.

4.1. Euclidean Self-Dual Cyclic Codes of Length over

Characterization and enumeration of Euclidean self-dual cyclic codes of length over are given in this subsection.

For each subset of , denote by the set of polynomials for all in , where is viewed as the conjugation on the group ring defined in Section 2. From the definition of the annihilator, the next theorem can be derived similar to [21, Proposition 2.12].

Theorem 18. Let be an ideal in . Then .

Using the unique generators of an ideal in determined in Theorem 15, the Euclidean dual of can be given in the following theorem.

Theorem 19. Let be an ideal in , where for all . Then is in the form of

Proof. From the proof of Theorem 14, we haveBy Theorem 18, it follows that . Hence, contains the elements and By writing and using the Binomial Theorem, it follows that contains the elementHence, By counting the number of elements, the two sets are equal as desired. Updating the indices, it can be concluded that

Assume that an ideal in is Euclidean self-dual; that is, . By Theorem 18, it is equivalent to andin for all .

We note that if , then it is not difficult to see that only the ideal generated by is Euclidean self-dual.

For the case , the situation is more complicated. First, we recall an matrix over defined in [22] as

It is not difficult to see that equations from (46) are equivalent to the matrix equationwhere and .

Moreover, it can be concluded that the ideal is Euclidean self-dual if and only if and satisfy (48). Since is a solution of (48), the corresponding idea is Euclidean self-dual in . Hence, for a fixed first torsion degree , a Euclidean self-dual ideal in always exists. By solving (48), all Euclidean self-dual ideals in can be constructed. Therefore, for a fixed first torsion degree , the number of Euclidean self-dual ideals in equals the number of solutions of (48) which is , where is the nullity of determined in [22].

Proposition 20 (see [22, Proposition 3.3]). Let be the nullity of . Then

The number of Euclidean self-dual cyclic codes in with first torsional degree is given in terms of the nullity of as follows.

Proposition 21. Let and let be the nullity of over . Then the number of Euclidean self-dual cyclic codes of length over with first torsional degree is .

From Theorem 14, we have since . Hence, the number of Euclidean self-dual cyclic codes of length over is given by the following corollary.

Corollary 22. Let be a prime and let and be positive integers. Then the following statements hold: (i)If is odd, then the number of Euclidean self-dual cyclic codes of length over is (ii)If , then the number of Euclidean self-dual cyclic codes of length over is

Proof. From Propositions 20 and 21, the number of Euclidean self-dual cyclic codes of length over is . Apply a suitable geometric sum; the results follow.

4.2. Hermitian Self-Dual Cyclic Codes of Length over

Under the assumption that is even, characterization and enumeration of Hermitian self-dual cyclic codes of length over are given in this section.

For a subset of , let where .

Based on the structural characterization of given in Theorem 15, the Hermitian dual of is determined as follows.

Theorem 23. Let be an ideal in and where . Then has the representation

Proof. From Theorem 19 and the fact that , the result follows.

Assume that is Hermitian self-dual. Then which implies that and .

If , then it is not difficult to see that the ideal generated by is only Hermitian self-dual cyclic code of length over .

Assume that . Thenfor all

From equations above and the definition of , we havewhere , , and .

From Theorem 14, we have since .

Proposition 24. Let be an even positive integer and let be a positive integer such that . Then the number of solutions of (56) in is .

Proof. Let defined by for all . Using the fact that and arguments similar to those in [23, Proposition 3.3], the result follows.

For a prime number , a positive integer , and an even positive integer , the number of Hermitian self-dual cyclic codes of length over can be determined in the following corollary.

Corollary 25. Let be a prime and let and be positive integers such that is even. Then the number of Hermitian self-dual cyclic codes of length over is

5. Conclusions and Remarks

Euclidean and Hermitian self-dual abelian codes in non-PIGAs are studied. The complete characterization and enumeration of such abelian codes are given and summarized as follows.

In Corollaries 1 and 3, self-dual abelian codes in are shown to be 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 . Subsequently, the characterizations and enumerations of cyclic and self-dual cyclic codes of length over are studied for all primes . Combining these results, the following enumerations of Euclidean and Hermitian self-dual abelian codes in are rewarded.

For each abelian group of odd order and positive integers and , the number of Euclidean self-dual abelian codes in is given in Theorem 2 in terms of the numbers , , and of cyclic codes, Euclidean self-dual cyclic codes, and Hermitian self-dual cyclic codes of length over a Galois extension of , respectively.

In addition, if is even, the number of Hermitian self-dual abelian codes in is given in Theorem 4 in terms of the numbers and of cyclic codes and Hermitian self-dual cyclic codes of length over a Galois extension of , respectively.

We note that all numbers , , and are determined in Corollaries 17, 22, and 25, respectively. Therefore, the complete enumerations of Euclidean and Hermitian self-dual abelian codes in are established.

One of the interesting problems concerning the enumeration of self-dual abelian codes in , where is an abelian group of odd order, is the case where is a -group of other types.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Thailand Research Fund under Research Grant TRG5780065 and the DPST Research Grant 005/2557.

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Copyright © 2016 Parinyawat Choosuwan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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