Abstract
The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order \(n=m^2\) is \(n-m\). In this paper, we construct for \(n=q^2\), q a prime power, a set of \(q^2-q-1\) MOSLS of order \(q^2\) that cannot be extended to a set of \(q^2-q\) MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of \(n-2\) mutually orthogonal Latin Squares (MOLS) can be extended to a set of \(n-1\) MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size \(q^2-q+1\) in \(\mathrm {PG}(3,q)\) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.
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G. Van de Voorde is a postdoctoral fellow of the Research Foundation Flanders (Belgium) (FWO).
This is one of several papers published in Designs, Codes and Cryptography comprising the special issue in honor of Andries Brouwer’s 65th birthday.
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D’haeseleer, J., Metsch, K., Storme, L. et al. On the maximality of a set of mutually orthogonal Sudoku Latin Squares. Des. Codes Cryptogr. 84, 143–152 (2017). https://doi.org/10.1007/s10623-016-0234-3
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DOI: https://doi.org/10.1007/s10623-016-0234-3