On the dimension of ideals in group algebras, and group codes

TL;DR

This paper investigates the dimensions of ideals in group algebras, deriving bounds and relations that help analyze abelian and MDS group codes, with practical examples included.

Contribution

It introduces new bounds and relations for ideal dimensions in group algebras, connecting group codes with MDS codes and providing methods to compute code parameters.

Findings

Computed elements with Hamming weight equal to their dimension for abelian codes

Established bounds on the minimum distance of certain MDS group codes

Presented examples illustrating the theoretical results

Abstract

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.