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The Existence of Latin Squares without Orthogonal Mates

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Abstract

A latin square is a bachelor square if it does not possess an orthogonal mate; equivalently, it does not have a decomposition into disjoint transversals. We define a latin square to be a confirmed bachelor square if it contains an entry through which there is no transversal. We prove the existence of confirmed bachelor squares for all orders greater than three. This resolves the existence question for bachelor squares.

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Vic 3800, Australia

    Ian M. Wanless

  2. Department of Mathematics, The Open University, Milton Keynes, MK7 6AA, UK

    Bridget S. Webb

Authors
  1. Ian M. Wanless
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  2. Bridget S. Webb
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Corresponding author

Correspondence to Ian M. Wanless.

Additional information

Communicated by D. Jungnickel

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Wanless, I.M., Webb, B.S. The Existence of Latin Squares without Orthogonal Mates. Des Codes Crypt 40, 131–135 (2006). https://doi.org/10.1007/s10623-006-8168-9

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  • DOI: https://doi.org/10.1007/s10623-006-8168-9

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