F_q[G]-modules and G-invariant codes

11/29/2017

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by   Elias Javier Garcia Claro, et al.

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If F_q is a finite field and G is a subgroup of the linear automorphisms of F_q^n, a solution to the problem of finding all the G-invariant linear codes C of F_q^n (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 F_q[G]-modules between F_q^n and F_q[G]×...×F_q[G] (t-times), that preserves the Hamming weight, this reduce the invariance problem, to determine the F_q[G]-submodules of F_q[G]×...×F_q[G] (t-times). The concept of Gaussian binomial coefficient for semisimple F_q[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^n, when (|G|,q)=1 is provided.