Saturating systems and the rank covering radius

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 $s_{q^m\mathrm{/}q}(k,\rho )$, 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 $s_{q^m\mathrm{/}q}(k,\rho )$ and evaluate it for certain values of $k$ and $\rho$. We give constructions of rank $\rho$-saturating systems suggested from geometry.