$q$-analogs of group divisible designs

Abstract

A well known class of objects in combinatorial design theory are {group divisible designs}. Here, we introduce the $q$-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, $q$-Steiner systems, design packings and $q^r$-divisible projective sets. We give necessary conditions for the existence of $q$-analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a $(6,3,2,2{)}_2$ group divisible design over $\mathrm{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $\mathrm{GF}(2{)}^6$ is covered at most twice.

18 pages, 3 tables, typos corrected