Self-Dual Abelian Codes in some Non-Principal Ideal Group Algebras

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $$\mathbb{F}$_{2^k}[A\times $\mathbb{Z}$2\times $\mathbb{Z}${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-know 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 ${\mathbb{F}}_{2^k}+u{\mathbb{F}}_{2^k}$, where $u^2=0$. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length $p^s$ over ${\mathbb{F}}_{p^k}+u{\mathbb{F}}_{p^k}$ are given. Combining these results, the complete enumeration of self-dual abelian codes in ${\mathbb{F}}_{2^k}[A\times {\mathbb{Z}}_2\times {\mathbb{Z}}_{2^s}]$ is therefore obtained.