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

Orbit code is a class of constant dimension code which is defined as orbit of a subgroup of the general linear group \(GL_n(\mathbb{F}q)\), 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 \(GL{n+l, n}(\mathbb{F}_q)\) 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.