Saturating systems and the rank covering radius

TL;DR

This paper introduces rank saturating systems, explores their connection to rank-metric codes with specific covering radii, and provides bounds and constructions for these systems using geometric methods.

Contribution

It defines rank saturating systems, relates them to rank-metric covering problems, and offers bounds and geometric constructions for these systems.

Findings

Established bounds on the minimum dimension of rank saturating systems.

Provided explicit constructions of rank saturating systems based on geometric ideas.

Evaluated the bounds for specific parameters of the systems.

Abstract

We introduce the concept of a rank saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of $s_{q^m/q}(k,\rho)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $\rho$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,\rho)$ and evaluate it for certain values of $k$ and $\rho$. We give constructions of rank $\rho$-saturating systems suggested from geometry.