Covering of Subspaces by Subspaces
Abstract
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph q(n,r) by subspaces from the Grassmann graph q(n,k), k ≥ 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 ≤ n ≤ 10, are presented. We present some questions concerning possible constructions of new coverings of smaller size.
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