Small Group Divisible Steiner Quadruple Systems

Abstract

A group divisible Steiner quadruple system, is a triple $(X, {\cal H}, {\cal B})$ where $X$ is a $v$-element set of points, ${\cal H} = \{H_1,H_2,\ldots,H_r\}$ is a partition of $X$ into holes and ${\cal B}$ is a collection of $4$-element subsets of $X$ called blocks such that every $3$-element subset is either in a block or a hole but not both. In this article we investigate the existence and non-existence of these designs. We settle all parameter situations on at most 24 points, with 6 exceptions. A uniform group divisible Steiner quadruple system is a system in which all the holes have equal size. These were called by Mills G-designs and their existence is completely settled in this article.