Saturating systems and the rank-metric covering radius

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 that 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.