Generalized bilinear forms graphs and MDR codes

Abstract

We investigate the generalized bilinear forms graph $\Gamma_d$ over a residue class ring $\mathbb{Z}_{p^s}$. We show that $\Gamma_d$ is a connected vertex transitive graph, and completely determine its independence number, clique number, chromatic number and maximum cliques. We also prove that cores of both $\Gamma_d$ and its complement are maximum cliques. The graph $\Gamma_d$ is useful for error-correcting codes. We show that every largest independent set of $\Gamma_d$ is both an MRD code over $\mathbb{Z}_{p^s}$ and a usual MDS code. Moreover, there is a largest independent set of $\Gamma_d$ to be a linear code over $\mathbb{Z}_{p^s}$.