The Construction of Orbit Codes Based on Singular Linear Space over Finite Fields

Abstract

Orbit code is a class of constant dimension code which is defined as orbit of a subgroup of the general linear group $G{L}_n(\mathbb{F}q)$, acting on the set of all the subspaces of vector space ${\mathbb{F}}_q^n$. In this paper, the construction of orbit codes based on the singular general linear group $GLn+l,n({\mathbb{F}}_q)$ acting on the set of all the subspaces of type $(m,k)$ in singular linear spaces ${\mathbb{F}}_q^{n+l}$ is given. We give a characterization of orbit code constructed in singular linear space ${\mathbb{F}}_q^{n+l}$, and then give some basic properties of the constructed orbit codes. Finally, two examples about our orbit codes for understanding these properties explicitly are presented.