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 $$s_{q^m/q}(k,\rho )$$ s q m / q ( k , ρ ) , which is the minimum $$\mathbb {F}_q$$ F q -dimension of a q-system in $$\mathbb {F}_{q^m}^k$$ F q m k that is rank-$$\rho $$ ρ -saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $$s_{q^m/q}(k,\rho )$$ s q m / q ( k , ρ ) and evaluate it for certain values of k and $$\rho $$ ρ . We give constructions of rank-$$\rho $$ ρ -saturating systems suggested from geometry.