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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Construction of Steiner quadruple systems having large numbers of nonisomorphic associated Steiner triple systems


Author: Charles C. Lindner
Journal: Proc. Amer. Math. Soc. 49 (1975), 256-260
MSC: Primary 05B05
MathSciNet review: 0389617
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Abstract: If $ (Q,q)$ is a Steiner quadruple system and $ x$ is any element in $ Q$ it is well known that the set $ {Q_x} = Q\backslash \{ x\} $ equipped with the collection $ q(x)$ of all triples $ \{ a,b,c\} $ such that $ \{ a,b,c,x\} \in q$ is a Steiner triple system. A quadruple system $ (Q,q)$ is said to have at least $ n$ nonisomorphic associated triple systems (NATS) provided that for at least one subset $ X$ of $ Q$ containing $ n$ elements the triple systems $ ({Q_x},q(x))$ and $ ({Q_y},q(y))$ are nonisomorphic whenever $ x \ne y \in X$. Prior to the results in this paper the maximum number of known NATS for any quadruple system was 2. The main result in this paper is the construction for each positive integer $ t$ of a quadruple system having at least $ t$ NATS.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0389617-6
PII: S 0002-9939(1975)0389617-6
Keywords: Steiner quadruple system, Steiner triple system, non-isomorphic associated triple systems
Article copyright: © Copyright 1975 American Mathematical Society