2017/11/29 by Claro, Elias Javier Garcia, Recillas, Horacio Tapia
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1711.10671
If \mathbbFq is a finite field, C is a vector subspace of \mathbbFqn (linear code), and G is a subgroup of the group of linear automorphisms of \mathbbFqn, C is said to be G-invariant if g(C)=C for all g∈ G. A solution to the problem of computing all the G-invariant linear codes C of \mathbbFqn is offered. This will be referred as the invariance problem. When n=|G|t, we determine conditions for the existence of an isomorphism of \mathbbFq[G]-modules between \mathbbFqn and \mathbbFq[G]× ⋯ × \mathbbFq[G] (t-times), that preserves the Hamming weight. This reduces the invariance problem to the determination of the \mathbbFq[G]-submodules of \mathbbFq[G]× ⋯ × \mathbbFq[G] (t-times). The concept of Gaussian binomial coefficient for semisimple \mathbbFq[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⊆ \mathbbFqn is provided, when (|G|,q)=1.