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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 oflinear automorphisms of $\mathbb{F}_{q}^{n}$, $C$ is said to be $G$-invariantif $g(C)=C$ 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 bereferred as the invariance problem. When $n=|G|t$, we determine conditions forthe 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 reducesthe 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 coefficientfor semisimple $\mathbb{F}_{q}[G]$-modules, which is useful for counting$G$-invariant codes, is introduced. Finally, a systematic way to compute allthe $G$-invariant linear codes $C\subseteq \mathbb{F}_{q}^{n}$ is provided,when $(|G|,q)=1$.

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