Covering of Subspaces by Subspaces

TL;DR

This paper investigates bounds and constructions for covering subspaces in Grassmann graphs, with applications in coding theory, combinatorics, and geometry, introducing new methods and tables for specific parameters.

Contribution

It provides new bounds, constructions, and tables for covering subspaces in Grassmann graphs, especially for q=2, r=2,3, and discusses open questions.

Findings

Bounds on covering sizes are established.

New constructions for q=2, r=2,3 are presented.

Tables of best known coverings for specific parameters are included.

Abstract

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $\cG_q(n,r)$ by subspaces from the Grassmann graph $\cG_q(n,k)$, $k \geq r$, 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 $5 \leq n \leq 10$, are presented. We present some questions concerning possible constructions of new coverings of smaller size.