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Algorithmic Approach to Non-symmetric 3-class Association Schemes

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Algorithmic Algebraic Combinatorics and Gröbner Bases

Summary

There are 24 feasible parameter sets for a primitive non-symmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove non-existence for three feasible parameter sets. Ten cases are still open.

In the imprimitive case, we survey the known results including some constructions of infinite families of schemes. In the smallest case that has been open up to now, we use computer search to find new schemes. These schemes are equivalent to “skew” Bush-type Hadamard matrices of order 36. We also consider directed graphs that satisfy only some of the conditions required for a non-symmetric association scheme with 3 classes.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7, 9220, Aalborg, Denmark

    Leif K. Jørgensen

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  1. Leif K. Jørgensen
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Correspondence to Leif K. Jørgensen .

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Editors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, POB 653, 84633, Beer Sheva, Israel

    Mikhail Klin

  2. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    Gareth A. Jones

  3. Faculty of Computer Science, Laboratory for Cryptography and Computer Security, Tržaška cesta 25, 1000, Ljubljana, Slovenia

    Aleksandar Jurišić

  4. Department of Mathematics and Computer Science, Netanya Academic College, University st. 1, 42 365, Netanya, Israel

    Mikhail Muzychuk

  5. V.A. Steklov Mathematical Institute, Laboratory of Representation Theory and Computational Mathematics, Fontanka 27, 191023, St. Petersburg, Russia

    Ilia Ponomarenko

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Jørgensen, L.K. (2009). Algorithmic Approach to Non-symmetric 3-class Association Schemes. In: Klin, M., Jones, G.A., Jurišić, A., Muzychuk, M., Ponomarenko, I. (eds) Algorithmic Algebraic Combinatorics and Gröbner Bases. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01960-9_8

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