Covering of Subspaces by Subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph ${\text{cG}}_q(n,r)$ by subspaces from the Grassmann graph ${\text{cG}}_q(n,k)$, $k\ge r$, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, $q$-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for $q=2$ with $r=2$ or $r=3$. We discuss the density for some of these coverings. Tables for the best known coverings, for $q=2$ and $5\le n\le 10$, are presented. We present some questions concerning possible constructions of new coverings of smaller size.