Towards a Large Set of Steiner Quadruple Systems

Let $D( v )$ be the number of pairwise disjoint Steiner quadruple systems. A simple counting argument shows that $D( v )\leqq v - 3$. In this paper it is proved that $D( 2^k n )\geqq ( 2^k - 1 ) n,k\geqq 2$, if there exists a set of $3n$ pairwise disjoint Steiner quadruple systems of order $4n$ with a certain structure. This implies that $D( v )\geqq v - o( v )$ for infinitely many values of v. New lower bounds on $D( v )$ for many values of v that are not divisible by 4 are also given, and it is proved that $D( v )\geqq 2$ for all $v \equiv 2$ or $4(\bmod 6 ),v\geqq 8$.