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 \({\mathrm {KTS}}(81)\).
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Acknowledgements
Vladimir Tonchev acknowledges support by the Alexander von Humboldt Foundation and NSA Grant H98230-16-1-0011.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.
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Jungnickel, D., Magliveras, S.S., Tonchev, V.D. et al. The classification of Steiner triple systems on 27 points with 3-rank 24. Des. Codes Cryptogr. 87, 831–839 (2019). https://doi.org/10.1007/s10623-018-0502-5
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DOI: https://doi.org/10.1007/s10623-018-0502-5