Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

Abstract

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras F 2 k [A × Z 2 × Z 2 s ] with respect to both the Euclidean and Hermitian inner products, where k and s are positive integers and A 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 2 s over some Galois extensions of the ring F 2 k + uF 2 k , where u 2 = 0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length p s over F p k + uF p k are given. Combining these results, the complete enumeration of self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] is therefore obtained.

An explicit representation and enumeration for self-dual cyclic codes over F2m+uFChen1, Cao2, Dinh3 et al. 2019Discrete MathematicsSelf Cite260140View full textAdd to dashboardCite

“…From these and by Theorem 2(iv), we list all distinct self-dual cyclic codes over F 2 m + uF 2 m of length 2 4 as follows: Remark. (i) In Corollary 22(ii) of [5], the authors claimed that the number of Euclidean self-dual cyclic codes of length 2 s over F 2 k + uF 2 k (u 2 = 0) is…”

“…We have given an explicit representation for self-dual cyclic codes of length 2 s over the finite chain ring F 2 m + uF 2 m (u 2 = 0) and a clear mass formula to count the number of these codes, by a new way different from that used in [5]. Especially, we provide an effective method to list precisely all distinct self-dual cyclic codes of length 2 s over F 2 m + uF 2 m by use of properties for Kronecker product of matrices and calculation of linear equations over F 2 m .…”

“…In Section 4 of [5], Choosuwan et al given an alternative characterization for cyclic codes over F p k + uF p k of length p s for any positive integer k, s and all primes p. The paper only provided a form for the Euclidean dual code of any cyclic code over F p k + uF p k of length p s (see Page 9, Theorem 19 in [5]). On that basis, the paper provided formulas to count the number of Euclidean self-dual cyclic codes and Hermitian self-dual cyclic codes over F p k + uF p k of length p s respectively.…”

“…On that basis, the paper provided formulas to count the number of Euclidean self-dual cyclic codes and Hermitian self-dual cyclic codes over F p k + uF p k of length p s respectively. For example, in Corollary 22(ii) of [5], the number of Euclidean self-dual cyclic codes of length 2 s over F 2 k + uF 2 k (u 2 = 0) was…”

An efficient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

“…Choosuwan et al [6] done the following: ♦ In pages 9 and 10, they proved that every (Euclidean) self-dual cyclic code over F p m + uF p m of length p s is given by…”