Mutually orthogonal Latin squares based on general linear groups
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Given a finite group G, how many squares are possible in a set of mutually orthogonal Latin squares based on G? This is a question that has been answered for a few classes of groups only, and for no nonsoluble group. For a nonsoluble group G, we know that there exists a pair of orthogonal Latin squares based on G. We can improve on this lower bound when G is one of GL(2, q) or SL(2, q), q a power of 2, q ≠ 2, or is obtained from these groups using quotient group constructions. For nonsoluble groups, that is the extent of our knowledge. We will extend these results by deriving new lower bounds for the number of squares in a set of mutually orthogonal Latin squares based on the group GL(n, q), q a power of 2, q ≠ 2.
KeywordsMOLS General linear group Orthomorphism
Mathematics Subject Classification (2010)05B15 20F99
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- 2.Beth T., Jungnickel D., Lenz H.: Design theory, 2nd ed. Cambridge University Press, Cambridge (1999).Google Scholar
- 3.Colbourn C.J., Dinitz J.H. (eds.): Handbook of Combinatorial Designs, 2nd ed. Chapman and Hall/CRC, Boca Raton (2007).Google Scholar
- 5.Dénes J., Keedwell A.D.: Latin Squares and Their Applications. Akadémiai Kiadó, Budapest (1974).Google Scholar
- 6.Dénes J., Keedwell A.D.: Latin squares: new developments in the theory and applications. Annals of Discrete Mathematics, vol. 46. North Holland, New York (1991).Google Scholar
- 7.Evans A.B.: Mutually orthogonal Latin squares based on linear groups. In: Jungnickel, D. et al. (eds) Coding Theory, Design Theory, Group Theory (Burlington, VT, 1990), pp. 171–175. . Wiley-Interscience, New York (1993).Google Scholar
- 8.Evans A.B.: Orthomorphism graphs of groups. Lecture Notes in Mathematics, vol. 1535. Springer, Berlin (1992).Google Scholar