On the number of SQS

TL;DR

This paper determines the asymptotic growth rate of the number of Steiner quadruple systems of order n, showing it is proportional to n^3 log n as n becomes large, for all admissible n.

Contribution

It establishes the precise asymptotic order of the logarithm of the number of SQS(n), refining previous bounds and confirming the growth rate as Θ(n^3 log n).

Findings

Logarithm of the number of SQS(n) is Θ(n^3 log n) as n→∞.

Confirms the growth rate for all admissible n where n mod 6= 2 or 4.

Provides a tight asymptotic estimate for the count of Steiner quadruple systems.

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 $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.