Existence of Steiner quadruple systems with a spanning block design

A Steiner system S ( t , k , v ) is a pair ( X , B ) , where X is a v -element set and B is a set of k -subsets of X , called blocks, with the property that every t -element subset of X is contained in a unique block. The sub-design S ( 2 , 4 , v ) in a Steiner quadruple system S ( 3 , 4 , v ) is said to be a spanning block design. The order v of a Steiner quadruple system with a spanning block design should satisfy the necessary condition v ≡ 4 ( mod 12 ) . It is proved that the above necessary condition is also sufficient. As a consequence, it is also proved that a 3-BD S ( 3 , { 4 , 5 } , v ) exists for any v ≡ 5 ( mod 12 ) .