Improved Gilbert-Varshamov bound for sum-rank-metric codes via graph theory

TL;DR

This paper introduces a graph-theoretic method to improve the Gilbert-Varshamov bound for sum-rank-metric codes, showing that larger codes can be constructed asymptotically, and explores their connection to set-coloring Ramsey numbers.

Contribution

It presents a novel graph-theoretic approach that yields improved bounds for sum-rank-metric codes and links these codes to set-coloring Ramsey numbers.

Findings

Asymptotic partitioning of space into larger sum-rank-metric codes

Logarithmic factor improvement over the GV bound

Connection established between codes and set-coloring Ramsey numbers

Abstract

We use a graph-theoretic approach which yields improvements on the known Gilbert-Varshamov (GV) bound for sum-rank-metric codes for certain parameters. In particular, we show that asymptotically $\mathbb{F}_q^{\mathbf{n} \times \mathbf{m}}$ can be partitioned into sum-rank-metric codes whose average size is bigger than the GV bound by a logarithmic factor for these parameters. Finally, we discuss the connection of such codes to set-coloring Ramsey numbers.