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The classification of Steiner triple systems on 27 points with 3-rank 24

  • Dieter JungnickelEmail author
  • Spyros S. Magliveras
  • Vladimir D. Tonchev
  • Alfred Wassermann
Article
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

We show that there are exactly 2624 isomorphism classes of Steiner triple systems on 27 points having 3-rank 24, all of which are actually resolvable. More generally, all Steiner triple systems on \(3^n\) points having 3-rank at most \(3^n-n\) are resolvable. Combining this observation with the lower bound on the number of such \({\mathrm {STS}}(3^n)\) recently established by two of the present authors, we obtain a strong lower bound on the number of Kirkman triple systems on \(3^n\) points. For instance, there are more than \(10^{99}\) isomorphism classes of \({\mathrm {KTS}}(81)\).

Keywords

Steiner triple system Linear code Kirkman triple system 

Mathematics Subject Classification

05B05 51E10 94B27 

Notes

Acknowledgements

Vladimir Tonchev acknowledges support by the Alexander von Humboldt Foundation and NSA Grant H98230-16-1-0011.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of AugsburgAugsburgGermany
  2. 2.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA
  3. 3.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  4. 4.Mathematical InstituteUniversity of BayreuthBayreuthGermany

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