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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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