Sparse Steiner triple systems of order 21

Janne I. Kokkala, Patric R.J. Östergård*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

A (Formula presented.) -configuration is a set of (Formula presented.) blocks on (Formula presented.) points. For Steiner triple systems, (Formula presented.) -configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a (Formula presented.) -configuration. A Steiner triple system of order (Formula presented.), an STS (Formula presented.), is (Formula presented.) -sparse if it does not contain any (Formula presented.) -configuration for (Formula presented.). The existence problem for anti-Pasch Steiner triple systems has been solved, but these have been classified only up to order 19. In the current work, a computer-aided classification shows that there are 83,003,869 isomorphism classes of anti-Pasch STS(21)s. Exploration of the classified systems reveals that there are three 5-sparse STS(21)s but no 6-sparse STS(21)s. The anti-Pasch STS(21)s lead to 14 Kirkman triple systems, none of which is doubly resolvable.

Original languageEnglish
Pages (from-to)75-83
Number of pages9
JournalJournal of Combinatorial Designs
Volume29
Issue number2
Early online date17 Nov 2020
DOIs
Publication statusPublished - Feb 2021
MoE publication typeA1 Journal article-refereed

Keywords

  • automorphism group
  • Kirkman triple system
  • Pasch configuration
  • quadrilateral
  • Steiner triple system

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