The geometry of covering codes in the sum-rank metric
Matteo Bonini, Martino Borello, Eimear Byrne
TL;DR
This work develops a geometric framework for understanding covering properties in the sum-rank metric by introducing sum-rank saturating systems and linking them to sum-rank covering radii. It provides equivalent characterizations that connect generator matrices, $q$-systems, and linear sets, and proves bounds on the minimal dimension required for saturating systems, including asymptotic inequalities in terms of $\|\mathbf{n}\|$ and $m$. It then delivers constructive methods from partitions of projective space and cutting designs to build saturating systems and, consequently, covering codes, with additional discussion of minimal sum-rank codes via strong blocking sets. Overall, the paper extends geometric approaches from the rank and Hamming metrics to the sum-rank setting, yielding practical tools for constructing saturating and covering codes with potential impact on network coding and distributed storage.
Abstract
We introduce the concept of a sum-rank saturating system and outline its correspondence to a covering properties of a sum-rank metric code. We consider the problem of determining the shortest sum-rank-$ρ$-saturating systems of a fixed dimension, which is equivalent to the covering problem in the sum-rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.