Small Group Divisible Steiner Quadruple Systems

Melissa Keranen∗, Donald Kreher, Artem Zhuravlev, Michigan Technological University A group divisible Steiner quadruple system, is a triple (X,H,B) where X is a v-element set of points, H = {H1, H2, . . . , Hr} is a partition of X into holes and 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. 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 G-designs by Mills, and their existence is completely settled.