Distance and fractional isomorphism in Steiner triple systems

In 8], Quattrochi and Rinaldi introduced the idea ofn ⁻¹-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 integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ⁻¹-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2⁻¹-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2⁻¹-isomorphic and 3⁻¹-isomorphic pairs of STS(15)s.