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

TL;DR

This paper provides a complete enumeration of self-dual abelian codes in certain non-principal ideal group algebras over finite fields, extending known characterizations to new algebraic structures.

Contribution

It introduces a method to enumerate self-dual abelian codes in non-principal ideal group algebras using cyclic code counts and extends results on cyclic codes over Galois extensions.

Findings

Enumeration formulas for self-dual abelian codes

Characterization of cyclic codes over Galois extensions

Complete enumeration in specific algebraic settings

Abstract

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.