q-analogs of group divisible designs

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 $\operatorname{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $\operatorname{GF}(2)^6$ is covered at most twice.Comment: 18 pages, 3 tables, typos correcte

Generalized vector space partitions

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.Comment: 12 pages, typos correcte