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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1. Alte Landstr. 6, 35232, Dautphetal, Germany

   Michael Damm

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1. Michael Damm
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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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• Received: 16 December 2002

• Revised: 15 April 2003

• Published: 23 June 2003

• Issue Date: August 2003

• 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