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Pairs of MOLS of order ten satisfying non-trivial relations

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Abstract

A relation on a \(k\text {{-}net}(n)\) (or, equivalently, a set of \(k-2\) mutually orthogonal Latin squares of order n) is an \({\mathbb {F}}_{2}\) linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any \(6\text {{-}net}(10)\) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are \(18\,526\,320\) pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of 3-MOLS from Dukes and Howard’s classification.

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Acknowledgements

This research was supported by the Monash eResearch Centre and eSolutions-Research Support Services through the use of the MonARCH HPC Cluster. The second author is extremely grateful for the generous hospitality of Wendy Myrvold and Peter Dukes, who taught him about relations on nets during his visits to University of Victoria, BC.

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  1. School of Mathematics, Monash University, Clayton, VIC, 3800, Australia

    Michael J. Gill & Ian M. Wanless

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  1. Michael J. Gill
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  2. Ian M. Wanless
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Correspondence to Michael J. Gill.

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Communicated by M. Buratti.

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Gill, M.J., Wanless, I.M. Pairs of MOLS of order ten satisfying non-trivial relations. Des. Codes Cryptogr. 91, 1293–1313 (2023). https://doi.org/10.1007/s10623-022-01149-6

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