Existence of Steiner quadruple systems with a spanning block design

Abstract: A Steiner system is a pair , where is a -element set and is a set of -subsets of , called blocks, with the property that every -element subset of is contained in a unique block. The sub-design in a Steiner quadruple system is said to be a spanning block design. The order of a Steiner quadruple system with a spanning block design should satisfy the necessary condition . It is proved that the above necessary condition is also sufficient. As a consequence, it is also proved that a 3-BD exists for any . [Copyright &y& Elsevier]