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.
Categories: math.CO cs.IT math.IT