# Saturating systems and the rank covering radius

Matteo Bonini, Martino Borello, Eimear Byrne

Publikation: Working paper/PreprintPreprint

## Abstrakt

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.
Originalsprog Udefineret/Ukendt Udgivet - 29 jun. 2022

## Emneord

• math.CO
• cs.IT
• math.IT
• 05B40, 11T71, 51E20, 52C17, 94B75