On maximum distance separable group codes with complementary duals

TL;DR

This paper investigates the structural relationship between maximum distance separable (MDS) LCD group codes and their underlying groups, providing parameter estimations and generalizations using group theory and Sylow theorems.

Contribution

It offers new insights into the subgroup structures of LCD MDS group codes and generalizes existing results using Sylow theorems.

Findings

Estimation of parameters for LCD group codes under certain conditions

Enhanced understanding of the relationship between group and code structures

Generalization of Cruz and Willems' result using Sylow theorems

Abstract

Given an LCD group code $C$ in a group algebra $KG$, we inspect kinship between $C$ and $G$, more precisely between the subgroup structures of $G$ and $C$. Under some special circumstances our inspection provides an estimation for various parameters of a group code $C$. When $C$ is MDS, the inter relation between $K$ and $G$ becomes more impressive. Application of Sylow theorem facilitated us to explore the inter relation between $G$ and $K$ (when $C$ is LCD and MDS) in more general way and finally we get the result of Cruz and Willems (Lemma $4.4$) as a simple sequel.