Existence of Steiner quadruple systems with a spanning block design

A Steiner system S(t,k,v) is a pair (X,B)(X,B), where XX is a vv-element set and BB is a set of kk-subsets of XX, called blocks  , with the property that every tt-element subset of XX 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 vv of a Steiner quadruple system with a spanning block design should satisfy the necessary condition v≡4(mod12). 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(mod12).