Group codes over symmetric groups

Let ${\mathbb{F}}_q$ be a finite field of characteristic $q$ and $S_n$ a symmetric group of order $n!$. In this paper, group codes in the symmetric group algebras ${\mathbb{F}}_q{S}_n$ with $q>3$ and $n=3,4$ are proposed. We compute the unique (linear and nonlinear) idempotents of ${\mathbb{F}}_q{S}_n$ corresponding to the characters of symmetric groups and use the results to characterize the minimum distances and dimensions of group codes. Furthermore, we construct MDS group codes and almost MDS group codes in ${\mathbb{F}}_q{S}_3$ and ${\mathbb{F}}_q{S}_4$.