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

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.