# Generalized bilinear forms graphs and MDR codes

## TL;DR

This paper studies the properties of generalized bilinear forms graphs over residue class rings, revealing their structure, maximum cliques, and their connection to maximum rank distance (MRD) and maximum distance separable (MDS) codes.

## Contribution

It characterizes the graph-theoretic properties of $ ext{Gamma}_d$ over $ ext{Z}_{p^s}$ and links largest independent sets to optimal error-correcting codes.

## Key findings

- $ ext{Gamma}_d$ is connected and vertex transitive.
- Largest independent sets are MRD and MDS codes.
- Largest independent sets can be linear 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 $\mathbb{Z}_{p^s}$.