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On symmetric nets and generalized Hadamard matrices from affine designs

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Abstract

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q3, q2, q2−1/q−1) designs without symmetric (q, q)-subnets.

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Authors and Affiliations

  1. Department of Mathematics, The University of Wales, SY23 3BZ, Aberystwyth, UK

    Vassili C. Mavron

  2. Department of Mathematical Sciences, Michigan Technological University, 49931, Houghton, Michigan, USA

    Vladimir D. Tonchev

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  1. Vassili C. Mavron
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  2. Vladimir D. Tonchev
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Mavron, V.C., Tonchev, V.D. On symmetric nets and generalized Hadamard matrices from affine designs. J Geom 67, 180–187 (2000). https://doi.org/10.1007/BF01220309

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  • DOI: https://doi.org/10.1007/BF01220309

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