New constructions of abelian non-cyclic orbit codes based on parabolic subgroups and tensor products

Abstract

Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of $F_q^n$ under the action of some subgroup of the finite general linear group ${\mathrm{GL}}_n\left(q\right)$. The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of ${\mathrm{GL}}_n\left(q\right)$, we propose a new construction of an abelian non-cyclic orbit codes of size $q^k$ with $k\le n/2$. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7].