Construction of spread codes based on Abelian non-cyclic orbit codes

Abstract

Let Fq be the finite field with q elements (where q is a prime power). Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear group, GLn(Fq), on Gq(k,n) (Grassmann variety). The orbits of a subgroup of GLn(Fq) acting on the Grassmann variety are called (subspace) orbit codes. When the subgroup acting on Gq(k,n) is cyclic the associated codes are called cyclic orbit codes. We make a construction abelian non-cyclic orbit codes by making full use of the companion matrix of a primitive polynomial over finite fields, it is a partial spread code. Based on this code, an optimal partial spread code is obtained. Our results answer the first of two open problems presented by Climent et al.