Support sizes of threefold quadruple systems

Abstract

The support of a quadruple system ${\text{QS}}_{\lambda }\left(v\right)$ is the set of distinct blocks in the design, and the support size is the number of distinct blocks. Let $q_v=\frac{v(v-1)(v-2)}{24}$ and $t_v=q_v+\frac{5v}{3}$. In this paper we show that the set of support sizes of a threefold quadruple system ${\text{QS}}_3\left(v\right)$ with $v\equiv 2,4\left(\text{mod}6\right)$ elements can have any number from $[q_v+14,3q_v]\cup \{q_v,q_v+8,q_v+12\}$ provided $v\ge 38$, a threefold quadruple system ${\text{QS}}_3\left(v\right)$ with $v\equiv 0\left(\text{mod}6\right)$ elements can have any number from $[t_v+14,3q_v]\cup \{t_v,t_v+8,t_v+12\}$ provided $v\ge 42$.