Abstract
Two orthogonal latin squares of order n have the property that when they are superimposed, each of the n 2 ordered pairs of symbols occurs exactly once. In a series of papers, Colbourn, Zhu, and Zhang completely determine the integers r for which there exist a pair of latin squares of order n having exactly r different ordered pairs between them. Here, the same problem is considered for latin squares of different orders n and m. A nontrivial lower bound on r is obtained, and some embedding-based constructions are shown to realize many values of r.
Similar content being viewed by others
References
Belyavskaya G.B.: Secret-sharing schemes and orthogonal systems of k-ary operations. Quasigr. Relat. Syst. 17, 161–176 (2009)
Colbourn, C.J., Zhu, L.: The spectrum of r-orthogonal latin squares. In: Combinatorics Advances. Kluwer Academic Press, Dordrecht, pp. 49–75 (1995)
Dénes J., Keedwell A.D.: Latin Squares and Their Applications. English Universities Press, London (1974)
Howell, J.: The intersection problem and different pairs problem for latin squares. Ph.D. dissertation, University of Victoria (2010)
Ryser H.J.: A combinatorial theorem with an application to Latin rectangles. Proc. Am. Math. Soc. 2, 550–552 (1951)
Stinson D.R., Wei R., Zhu L.: New constructions for perfect hash families and related structures using combinatorial designs and codes. J. Combin. Des. 8, 189–200 (2000)
Zhu L., Zhang H.: A few more r-orthogonal Latin squares. Discret. Math. 238, 183–191 (2001)
Zhu L., Zhang H.: Completing the spectrum of r-orthogonal Latin squares. Discret. Math. 268, 343–349 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by NSERC.
Rights and permissions
About this article
Cite this article
Dukes, P., Howell, J. The Orthogonality Spectrum for Latin Squares of Different Orders. Graphs and Combinatorics 29, 71–78 (2013). https://doi.org/10.1007/s00373-011-1092-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-011-1092-4