Covering of Subspaces by Subspaces

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.Comment: arXiv admin note: text overlap with arXiv:0805.352

On qq-covering designs

A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ . Let $\mathcal{C}_q(n, k, r)$ denote the minimum number of $(k-1)$-spaces in a $q$-covering design $\mathbb{C}_q(n, k, r)$. In this paper improved upper bounds on $\mathcal{C}_q(2n, 3, 2)$, $n \ge 4$, $\mathcal{C}_q(3n + 8, 4, 2)$, $n \ge 0$, and $\mathcal{C}_q(2n,4,3)$, $n \ge 4$, are presented. The results are achieved by constructing the related $q$-covering designs

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

Lengths of divisible codes with restricted column multiplicities

We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $q^r$-divisible codes over $\mathbb{F}_q$. We also give a few computational results for field sizes $q>2$. Non-existence results of divisible codes with restricted column multiplicities for a given length have applications e.g. in Galois geometry and can be used for upper bounds on the maximum cardinality of subspace codes.Comment: 26 pages, 7 table