2. 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, packing designs and qr-divisible projective sets. We give necessary conditions for the existence of q-analogs of group divisible designs, construct an infinite series of examples and provide further existence results with the help of a computer search. One example is a (6, 2, 3, 2)2 group divisible design over GF(2) which is a packing design consisting of 180 blocks that such every 2-dimensional subspace in GF(2)6 is covered at most twice.

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