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Enumeration of Steiner triple systems with subsystems


Authors: Petteri Kaski, Patric R. J. Östergård and Alexandru Popa
Journal: Math. Comp. 84 (2015), 3051-3067
MSC (2010): Primary 05B07, 51E10
DOI: https://doi.org/10.1090/mcom/2945
Published electronically: April 7, 2015
MathSciNet review: 3378862
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Abstract: A Steiner triple system of order $ v$, an STS($ v$), is a set of $ 3$-element subsets, called blocks, of a $ v$-element set of points, such that every pair of distinct points occurs in exactly one block. A subsystem of order $ w$ in an STS($ v$), a sub-STS($ w$), is a subset of blocks that forms an STS($ w$). Constructive and nonconstructive techniques for enumerating up to isomorphism the STS($ v$) that admit at least one sub-STS($ w$) are presented here for general parameters $ v$ and $ w$. The techniques are further applied to show that the number of isomorphism classes of STS($ 21$)s with at least one sub-STS($ 9$) is $ 12661527336$ and of STS($ 27$)s with a sub-STS($ 13$) is $ 1356574942538935943268083236$.


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Additional Information

Petteri Kaski
Affiliation: Helsinki Institute for Information Technology HIIT, Department of Information and Computer Science, Aalto University, P.O. Box 15400, 00076 Aalto, Finland

Patric R. J. Östergård
Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland

Alexandru Popa
Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland

DOI: https://doi.org/10.1090/mcom/2945
Keywords: Classification, enumeration, Steiner triple system, subsystem
Received by editor(s): November 19, 2013
Received by editor(s) in revised form: March 4, 2014
Published electronically: April 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society