$\mathbb{F}_{q}[G]$-modules and $G$-invariant codes

Abstract

If $\mathbb{F}_{q}$ is a finite field, $C$ is a vector subspace of $\mathbb{F}_{q}^{n}$ (linear code), and $G$ is a subgroup of the group of linear automorphisms of $\mathbb{F}_{q}^{n}$, $C$ is said to be $G$-invariant if $g(C)=C$ for all $g\in G$. A solution to the problem of computing all the $G$-invariant linear codes $C$ of $\mathbb{F}_{q}^{n}$ is offered. This will be referred as the invariance problem. When $n=|G|t$, we determine conditions for the existence of an isomorphism of $\mathbb{F}_{q}[G]$-modules between $\mathbb{F}_{q}^{n}$ and $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times), that preserves the Hamming weight. This reduces the invariance problem to the determination of the $\mathbb{F}_{q}[G]$-submodules of $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times). The concept of Gaussian binomial coefficient for semisimple $\mathbb{F}_{q}[G]$-modules, which is useful for counting $G$-invariant codes, is introduced. Finally, a systematic way to compute all the $G$-invariant linear codes $C\subseteq \mathbb{F}_{q}^{n}$ is provided, when $(|G|,q)=1$.