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.

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Abstract

Quattrochi and Rinaldi introduced the idea of $n^{-1}$ - 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 integer $N$ , there exists $v_0(N)$ such that for all admissible $v \ge v_0(N)$ and for each STS $(v)$ (say $S$ ), there exists an STS $(v)$ (say $S'$ ) such that for some $n > N$ , $S$ is strictly $n^{-1}$ -isomorphic to $S'$ . We also prove that for all admissible $v \ge 13$ , there exist two STS $(v)$ s which are strictly $2^{-1}$ -isomorphic. Define the distance between two Steiner triple systems $S$ and $S'$ of the same order to be the minimum volume of a trade $T$ which transforms $S$ into a system isomorphic to $S'$ . 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.

Item Type:	Journal Article
ISSN:	0009-725X
Academic Unit/Department:	Mathematics, Computing and Technology > Mathematics and Statistics
Item ID:	22759
Depositing User:	Mike Grannell
Date Deposited:	18 Aug 2010 12:38
Last Modified:	02 Dec 2010 21:02
URI:	http://oro.open.ac.uk/id/eprint/22759

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