Small group divisible Steiner quadruple systems (Q1010751)

Summary: A group divisible Steiner quadruple system, is a triple \((X, {\mathcal H}, {\mathcal B})\) where \(X\) is a \(v\)-element set of points, \({\mathcal H} = \{H_1,H_2,\dots,H_r\}\) is a partition of \(X\) into holes and \({\mathcal B}\) 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.