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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9020173, 12 pages
http://dx.doi.org/10.1155/2016/9020173
Research Article

Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

1Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand

Received 25 June 2016; Accepted 27 September 2016

Academic Editor: Kishin Sadarangani

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.

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.