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

If Fq is a finite field and G is a subgroup of the linear automorphisms of F q , a solution to the problem of finding all the G-invariant linear codes C of F q (i.e. such that g(C) = C for all g ∈ G) is offered. This will be referred as the invariance problem. When n = |G|t, we determine conditions for the existance of an isomorphism of Fq[G]modules between F q and Fq[G]× · · · × Fq[G] (t-times), that preserves the Hamming weight, this reduce the invariance problem, to determine the Fq[G]-submodules of Fq[G] × · · · × Fq[G] (t-times). The concept of Gaussian binomial coefficient for semisimple Fq[G]-modules which is useful for counting G-invariant codes is introduced, and a systematic way to compute all the G-invariant linear codes C ≤ F q , when (|G|, q) = 1 is provided.