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.
Similar content being viewed by others
References
Bailey R.A., Cameron P.J., Connelly R.: Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. Am. Math. Monthly 115, 383–404 (2008).
Bose R.C.: On the application of the properties of Galois fields to the problem of construction of hyper-greco-latin squares. Sankhyā 3(4), 323–338 (1938).
Bruck R.H.: Finite nets. II. Uniqueness and embedding. Pacific J. Math. 13, 421–457 (1963).
Dembowski P.: Finite Geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer, Berlin, xi+375 pp (1968).
Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1985).
Lorch J.: Mutually orthogonal families of linear Sudoku solutions. J. Aust. Math. Soc. 87(3), 409–420 (2009).
Metsch K.: Improvement of Bruck’s completion theorem. Des. Codes Cryptogr. 1(2), 99–116 (1991).
Pedersen R., Vis T.: Sets of mutually orthogonal Sudoku Latin squares. College Math. J. 40(3), 174–180 (2009).
Shrikhande S.S.: A note on mutually orthogonal Latin squares. Sankhyā Ser. A 23, 115–116 (1961).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-016-0234-3