Minimal group codes over alternating groups

Abstract

In this work we show that every minimal code in a semisimple group algebra ${\mathbb{F}}_{q}G$ is essential if $G$ is a simple group. Since the alternating group $A_n$ is simple if $n=3$ or $n\ge 5$, we present some examples of minimal codes in ${\mathbb{F}}_q{A}_n$. For this purpose, if $char({\mathbb{F}}_q)>n$, we present the Wedderburn-Artin decomposition of ${\mathbb{F}}_q{S}_n$ and ${\mathbb{F}}_q{A}_n$ and explicit some of the centrally primitive idempotents of ${\mathbb{F}}_q{S}_n$ and ${\mathbb{F}}_q{A}_n$.

16 pages