Characterization and Enumeration of Complementary Dual Abelian Codes

TL;DR

This paper characterizes and counts complementary dual abelian codes over group algebras, revealing their structure, independence from Sylow p-subgroups, and providing enumeration formulas, extending known cyclic code results.

Contribution

It offers a comprehensive characterization and enumeration of complementary dual abelian codes in group algebras, generalizing previous cyclic code findings.

Findings

Number of such codes is independent of Sylow p-subgroup.

Complete enumeration formulas for all finite abelian groups.

Extension of cyclic code results to general abelian group codes.

Abstract

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra $\mathbb{F}_{p^\nu}[G]$ has been studied under both the Euclidean and Hermitian inner products, where $p$ is a prime, $\nu$ is a positive integer, and $G$ is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in $\mathbb{F}_{p^\nu}[G]$ has been shown to be independent of the Sylow $p$-subgroup of $G$ and it has been completely determined for every finite abelian group $G$. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.