An improved Recursive Construction for Disjoint Steiner Quadruple Systems

Abstract

Let $D(n)$ be the number of pairwise disjoint Steiner quadruple systems. A simple counting argument shows that $D(n) \leq n-3$ and a set of $n-3$ such systems is called a large set. No nontrivial large set was constructed yet, although it is known that they exist if $n \equiv 2$ or $4~(\text{mod}~6)$ is large enough. When $n \geq 7$ and $n \equiv 1$ or $5~(\text{mod}~6)$, we present a recursive construction and prove a recursive formula on $D(4n)$, as follows: $$ D(4n) \geq 2n + \min \{D(2n) ,2n-7\}. $$ The related construction has a few advantages over some of the previously known constructions for pairwise disjoint Steiner quadruple systems.