Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

Abstract EN

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebras F2k[A×Z2×Z2s] 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 2s over some Galois extensions of the ring F2k+uF2k, where u2=0.

Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length ps over Fpk+uFpk are given.

Combining these results, the complete enumeration of self-dual abelian codes in F2k[A×Z2×Z2s] is therefore obtained.