Distance and fractional isomorphism in Steiner triple systems (Q997560)

The distance between two Steiner triple systems of the same order is the minimum volume of a trade that transforms one system into an isomorphic copy of the other. Distances among all Steiner triple systems of order 15 are reported here. One can require in addition that the trade consist of two isomorphic configurations, in which case there exist pairs of Steiner triple systems for which neither transforms to the other. Results for all pairs of Steiner triple systems of order 15 are also reported for this more stringent notion of distance. Finally, two Steiner triple systems are said to be \(\frac{1}{n}\)-isomorphic if each can be partitioned into \(n\) subsets of blocks, and for \(1 \leq i \leq n\), the \(i\)th configuration of blocks in one system is isomorphic to the \(i\)th configuartion of the other. Every two nonisomorphic Steiner triple systems of order 15 are shown here to be \(\frac{1}{n}\)-isomorphic for \(n=2\) or \(n=3\). In general, however, the existence of systems that are \(\frac{1}{n}\)-isomorphic but not \(\frac{1}{n-1}\)-isomorphic is established for all \(n \geq 2\).