Some Sufficient Conditions for a Steiner Triple System to Be a Derived Triple System

A Steiner quadruple system (or more simply a quadruple system) is a pair (Q, q) where Q is a set and q is a collection of 4-element subsets of Q (called blocks) such that every 3-element subset of Q is contained in exactly one block. The number j Q 1 is called the order of the quadruple system (Q, q) and it is well known that the spectrum for quadruple systems consists of all y1 = 2 or 4 (mod 6) [2]. A Steiner triple system (or triple system) is a pair (S, t) where S is a set and t is a collection of 3element subsets (usually called triples) such that each pair of elements in S occurs in exactly one triple oft. As with quadruple systems, the order of the Steiner triple system (S, t) is the number I S 1 . It is well known that the spectrum for triple systems is the set of all IZ = 1 or 3 (mod 6). There is a natural connection between Steiner quadruple systems and Steiner triple systems. If (Q, q) is a quadruple system and x is any element in Q we will denote by Qz the set Q\{x} and the set of all triples (a, b, c} such that (x, a, b, c} E q by q(x). It is a routine matter to see that (Qa: , q(x)) is a triple system called a derived triple system (DTS) of the quadruple system (Q, q). The isomorphism classes of Steiner triple systems which are derived are largely unknown. For orders 1, 3, 7, and 9 there is only one triple system, so trivially such a triple system is derived. For IZ = 13, Hung and Mendelsohn have shown that both isomorphism classes are derived [5]. Lindner, using the generalized singular direct product, has constructed an infinite class of DTS’s [3]. What he has shown is the following.