Enumeration of MOLS of small order

Egan, Judith; Wanless, Ian

doi:10.1090/mcom/3010

Enumeration of MOLS of small order
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by Judith Egan and Ian M. Wanless;

Math. Comp. 85 (2016), 799-824

DOI: https://doi.org/10.1090/mcom/3010

Published electronically: July 14, 2015

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Abstract:

We report the results of a computer investigation of sets of mutually orthogonal Latin squares (MOLS) of small order. For $n\leqslant 9$ we:

determine the number of orthogonal mates for each species of Latin square of order $n$;
calculate the proportion of Latin squares of order $n$ that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random;
classify all sets of MOLS of order $n$ up to various different notions of equivalence.

We also provide a triple of Latin squares of order 10 that is the closest to being a set of MOLS so far found.

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