Construction of Steiner quadruple systems having large numbers of nonisomorphic associated Steiner triple systems

Abstract:

If $(Q,q)$ is a Steiner quadruple system and $x$ is any element in $Q$ it is well known that the set ${Q_x} = Q\backslash \{ x\}$ equipped with the collection $q(x)$ of all triples $\{ a,b,c\}$ such that $\{ a,b,c,x\} \in q$ is a Steiner triple system. A quadruple system $(Q,q)$ is said to have at least $n$ nonisomorphic associated triple systems (NATS) provided that for at least one subset $X$ of $Q$ containing $n$ elements the triple systems $({Q_x},q(x))$ and $({Q_y},q(y))$ are nonisomorphic whenever $x \ne y \in X$. Prior to the results in this paper the maximum number of known NATS for any quadruple system was 2. The main result in this paper is the construction for each positive integer $t$ of a quadruple system having at least $t$ NATS.