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Steiner Triple Systems of Order 21 with a Transversal Subdesign TD(3, 6)

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Abstract

A Steiner triple system (STS) contains a transversal subdesign TD(3, w) if its point set has three pairwise disjoint subsets A, B, C of size w and w2 blocks of the STS intersect with each of A, B, C (those w2 blocks form a TD(3,w)). We prove several structural properties of Steiner triple systems of order 3w + 3 that contain one or more transversal subdesigns TD(3, w). Using exhaustive search, we find that there are 2 004 720 isomorphism classes of STS(21) containing a subdesign TD(3, 6) (or, equivalently, a 6 × 6 Latin square).

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Acknowledgments

The authors are grateful to Svetlana Topalova for useful discussions.

Funding

The research of M.J. Shi (corresponding author) was supported by the National Natural Science Foundation of China (no. 61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (no. 1808085J20), and Academic Fund for Outstanding Talents in Universities (gxbjZD03); the research of D.S. Krotov was supported within the framework of the state contract of the Sobolev Institute of Mathematics, project no. 0314-2019-0016.

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui Province, P. R. China

    Y. Guan & M. J. Shi

  2. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    D. S. Krotov

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  1. Y. Guan
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  2. M. J. Shi
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  3. D. S. Krotov
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Correspondence to M. J. Shi.

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Russian Text © The Author(s), 2020, published in Problemy Peredachi Informatsii, 2020, Vol. 56, No. 1, pp. 26–37.

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Guan, Y., Shi, M.J. & Krotov, D.S. Steiner Triple Systems of Order 21 with a Transversal Subdesign TD(3, 6). Probl Inf Transm 56, 23–32 (2020). https://doi.org/10.1134/S0032946020010032

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