A construction of Abelian non-cyclic orbit codes

A constant dimension code consists of a set of k-dimensional subspaces of \(\mathbb {F}_{q}^{n}\), where \(\mathbb {F}_{q}\) is a finite field of q elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all k-dimensional subspaces of \(\mathbb {F}_{q}^{n}\). If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code. In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Our code is a partial spread and consequently its minimum subspace distance is maximal.