Linear Algebra and its Applications | 2021
Construction of spread codes based on Abelian non-cyclic orbit codes
Abstract
Abstract Let F q 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, GL n ( F q ) , on G q ( k , n ) (Grassmann variety). The orbits of a subgroup of GL n ( F q ) acting on the Grassmann variety are called (subspace) orbit codes. When the subgroup acting on G q ( 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.