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On the Existence of Totally Anti-Symmetric Quasigroups of Order 4k+2

Abstract

Totally anti-symmetric quasigroups are employed in check digit systems. We give counterexamples to a conjecture of Ecker and Poch and show that there are totally anti-symmetric quasigroups for all orders n≡0,1,34, n=4(5k+2)+2, and n=4(7k+3)+2. We prove that finite totally anti-symmetric quasigroups possess a transversal, and we give a useful condition for quasigroups not to have a transversal.

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Correspondence to Michael Damm.

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Damm, M. On the Existence of Totally Anti-Symmetric Quasigroups of Order 4k+2. Computing 70, 349–357 (2003). https://doi.org/10.1007/s00607-003-0017-3

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  • DOI: https://doi.org/10.1007/s00607-003-0017-3

Classification

  • 05B15 orthogonal arrays
  • latin squares
  • room squares
  • 20N05 loops
  • quasigroups
  • 94B60 other types of codes