Support sizes of threefold quadruple systems

The support of a quadruple system QSλ(v)QSλ(v) is the set of distinct blocks in the design, and the support size is the number of distinct blocks. Let qv=v(v−1)(v−2)24 and tv=qv+5v3. In this paper we show that the set of support sizes of a threefold quadruple system QS3(v)QS3(v) with v≡2,4(mod6) elements can have any number from [qv+14,3qv]∪{qv,qv+8,qv+12}[qv+14,3qv]∪{qv,qv+8,qv+12} provided v≥38v≥38, a threefold quadruple system QS3(v)QS3(v) with v≡0(mod6) elements can have any number from [tv+14,3qv]∪{tv,tv+8,tv+12}[tv+14,3qv]∪{tv,tv+8,tv+12} provided v≥42v≥42.