On the number of SQS

Abstract

A Steiner quadruple system (briefly $SQS(n)$) is a pair $(X,B)$ where $|X|=n$ and $B$ is a collection of 4-element blocks such that every 3-subset of $X$ is contained in exactly one member of $B$. Hanani \cite{Hanani} proved that the necessary condition $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$ for the existence of a Steiner quadruple systems of order $n$ is also sufficient. Lenz \cite{Lenz} proved that the logarithm of the number of different $SQS(n)$ is greater than $cn^3$ where $c>0$ is a constant and $n$ is admissible. We prove that the logarithm of the number of different $SQS(n)$ is $\Theta(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.