# Distance and fractional isomorphism in Steiner triple systems

Forbes, Anthony; Grannell, Mike and Griggs, Terry (2007). Distance and fractional isomorphism in Steiner triple systems. Rendiconti del Circolo Matematico di Palermo Serie II, 56, pp. 17–32.

## Abstract

Quattrochi and Rinaldi introduced the idea of - isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integer , there exists such that for all admissible and for each STS (say ), there exists an STS (say ) such that for some , is strictly -isomorphic to . We also prove that for all admissible , there exist two STSs which are strictly -isomorphic. Define the distance between two Steiner triple systems and of the same order to be the minimum volume of a trade which transforms into a system isomorphic to . We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly -isomorphic and -isomorphic pairs of STSs.