Support sizes of threefold quadruple systems

The support of a quadruple system QS λ ( v ) is the set of distinct blocks in the design, and the support size is the number of distinct blocks. Let q v = v ( v − 1 ) ( v − 2 ) 24 and t v = q v + 5 v 3 . In this paper we show that the set of support sizes of a threefold quadruple system QS 3 ( v ) with v ≡ 2 , 4 ( mod 6 ) elements can have any number from [ q v + 14 , 3 q v ] ∪ { q v , q v + 8 , q v + 12 } provided v ≥ 38 , a threefold quadruple system QS 3 ( v ) with v ≡ 0 ( mod 6 ) elements can have any number from [ t v + 14 , 3 q v ] ∪ { t v , t v + 8 , t v + 12 } provided v ≥ 42 .