Pasch configurations in Mollard Steiner triple systems

Abstract

For a Steiner triple system S and its point i by Vi(S) we denote the number of Pasch configurations having two triples containing i. For S with point set P(S) by ν(S) we denote the multiset {vi(S) : i ∊ P (S)}. The Mollard construction for Steiner triple systems is considered in the paper. It is shown that for the Mollard STS MS, S' the values for elements of ν(MS, S') are known given the sets ν(S) and v(S'). There are 80 different multisets v(S), where S runs through 80 different isomorphism classes of STSs of order 15, whereas there are just 27 different values for the total number of Pasches for such STSs. The formulas for ν (MS, S') are used to show that there are exactly 3240 isomorphism classes of Mollard Steiner triple systems MS, S' of order 255 that are separated by v(MS, S').