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On Classifying Steiner Triple Systems by Their 3-Rank

  • Dieter Jungnickel
  • Spyros S. MagliverasEmail author
  • Vladimir D. Tonchev
  • Alfred Wassermann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)

Abstract

It was proved recently by Jungnickel and Tonchev (2017) that for every integer \(v=3^{m-1}w\), \(m\ge 2\), and \(w\equiv 1,3 \pmod 6\), there is a ternary linear \([v,v-m]\) code C, such that every Steiner triple system \({{\mathrm{STS}}}(v)\) on v points and having 3-rank \(v-m\), is isomorphic to an \({{\mathrm{STS}}}(v)\) supported by codewords of weight 3 in C. In this paper, we consider the ternary \([3^n, 3^n -n]\) code \(C_n\) (\(n\ge 3\)), that supports representatives of all isomorphism classes of \({{\mathrm{STS}}}(3^n)\) of 3-rank \(3^n -n\). We prove some structural properties of the triple system supported by the codewords of \(C_n\) of weight 3. Using these properties, we compute the exact number of distinct \({{\mathrm{STS}}}(27)\) of 3-rank 24 supported by the code \(C_3\). As an application, we prove a lower bound on the number of nonisomorphic \({{\mathrm{STS}}}(27)\) of 3-rank 24, and classify up to isomorphism all \({{\mathrm{STS}}}(27)\) supported by \(C_3\) that admit a certain automorphism group of order 3.

Notes

Acknowledgements

The authors wish to thank the unknown referees for reading carefully the manuscript and making several useful remarks. Vladimir Tonchev acknowledges support by the Alexander von Humboldt Foundation and NSA Grant H98230-16-1-0011.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Dieter Jungnickel
    • 1
  • Spyros S. Magliveras
    • 2
    Email author
  • Vladimir D. Tonchev
    • 3
  • Alfred Wassermann
    • 4
  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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