Abstract We introduce the concept of a sum–rank saturating system and outline its correspondence to covering properties of a sum–rank metric code. We consider the problem of determining the shortest length of a sum–rank- $$\rho $$ ρ -saturating system of a fixed dimension, which is equivalent to the covering problem in the sum–rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.

In this paper, we generalize constructions in two recent works of Ding, Heng, and Zhou to any field Fq, q odd, providing infinite families of minimal codes for which the Ashikhmin-Barg bound does not hold.