Saturating systems and the rank-metric 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/q(k,ρ ) , which is the minimum 𝔽_q -dimension of a q -system in 𝔽_q^m^k that is rank- ρ -saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on s_q^m/q(k,ρ ) and evaluate it for certain values of k and ρ . We give constructions of rank- ρ -saturating systems suggested from geometry.