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      Orthogonal trades in complete sets of MOLS

      Cavenagh, Nicholas J.; Donovan, Diane M.; Demirkale, Fatih
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       www.combinatorics.org
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      Cavenagh, N. J., Donovan, D. M., & Demirkale, F. (2017). Orthogonal trades in complete sets of MOLS. The Electronic Journal of Combinatorics, 24(3).
      Permanent Research Commons link: https://hdl.handle.net/10289/11516
      Abstract
      Let Bₚ be the Latin square given by the addition table for the integers modulo an odd prime p (i.e. the Cayley table for (Zₚ, +)). Here we consider the properties of Latin trades in Bₚ which preserve orthogonality with one of the p−1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (log p)² / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in Bₚ hits the main diagonal either p or at most p − log₂ p – 1 times. Finally, if p ≡ 1 (mod 6) we show the existence of a Latin square which is orthogonal to Bₚ and which contains a 2 × 2 subsquare.
      Date
      2017
      Type
      Journal Article
      Publisher
      The Electronic Journal of Combinatorics
      Collections
      • Computing and Mathematical Sciences Papers [1454]
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