$q$-analogs of group divisible designs

TL;DR

This paper introduces the concept of $q$-analogs of group divisible designs, exploring their properties, existence conditions, and providing explicit examples and computational results in the context of finite fields.

Contribution

It is the first to define and analyze $q$-analogs of group divisible designs, establishing existence conditions and constructing explicit examples.

Findings

Established necessary conditions for existence.

Constructed an infinite series of examples.

Provided computational existence results, including a specific design over GF(2).

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 $\operatorname{GF}(2)$ 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.