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**1 - 6**of**6**### Distance and fractional isomorphism in Steiner triple systems

"... In [8], Quattrochi and Rinaldi introduced the idea of n−1- isomor-phism 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 v0(N) such that for all admissible v ≥ v0(N) and for each ST ..."

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In [8], Quattrochi and Rinaldi introduced the idea of n−1- isomor-phism 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 v0(N) such that for all admissible v ≥ v0(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 ≥ 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 trans-forms 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. This is a preprint of an article accepted for publication in Rendiconti del Circolo Matematico di Palermo. c©2006 (copyright owner as specified in the journal).

### The Cycle Switching Graph of the Steiner Triple Systems of Order 19 is Connected

- GRAPHS AND COMBINATORICS

"... Switching is a local transformation that when applied to a combinatorial object gives another object with the same parameters. It is here shown that the cycle switching graph of the 11 084 874 829 isomorphism classes of Steiner triple systems of order 19 as well as the cycle switching graph of the ..."

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Switching is a local transformation that when applied to a combinatorial object gives another object with the same parameters. It is here shown that the cycle switching graph of the 11 084 874 829 isomorphism classes of Steiner triple systems of order 19 as well as the cycle switching graph of the

### Trades and t-designs

"... Trades, as combinatorial objects, possess interesting combinatorial and algebraic properties and play a considerable role in various areas of combinatorial designs. In this paper we focus on trades within the context of t-designs. A pedagogical review of the applications of trades in constructing ha ..."

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Trades, as combinatorial objects, possess interesting combinatorial and algebraic properties and play a considerable role in various areas of combinatorial designs. In this paper we focus on trades within the context of t-designs. A pedagogical review of the applications of trades in constructing halving t-designs is presented. We also consider (N, t)-partitionable sets as a generalization of trades. This generalized notion provides a powerful approach to the construction of large sets of t-designs. We review the main recursive constructions and theorems obtained by this approach. Finally, we discuss the linear algebraic representation of trades and present two applications.

### Combinatorial Intricacies of Labeled Fano Planes

, 2015

"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.