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Room n-cubes of lowe order

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Jeffrey H. Dinitz
Jeffrey H. Dinitz
Affiliation:
Department of Mathematics University of VermontBurlington, Vermont 05405, U.S.A.
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Abstract

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A Room n-cube of side t is an n dimensional array of side t which satisfies the property that each two dimensional projection is a Room square. The existence of a Room n-cube of side t is equivalent to the existence of n pairwise orthgonal symmetric Latin squares (POSLS) of side t. The existence of n pairwise orthogonal starters of order t implies the existence of n POSLS of side t. Denote by v(n) the maximum number of POSLS of side t. In this paper, we use Galois fields and computer constructions to construct sets of pairwise orthogonal starters of order t ≤ 101. The existence of these sets of starters gives improved lower bounds for v(n). In particular, we show v(17) ≥ 5, v(21) ≥ 5, v(29) ≥ 13, v(37) ≥ 15 and v(41) ≥ 9, among others.

MSC classification

Secondary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 237 - 252
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Dinitz, J. H., ‘New lower bounds for the number of pairwise orthogonal symmetric Latin squares,’ Proc. 10th S-E Conf. on Combinatorics, Graph Theory and Computing, Boca Raton, Florida, (1979), 393–398.Google Scholar
[2]Dinitz, J. H., ‘Pairwise orthogonal symmetric Latin squares’, Congressus Numerantium 32 (1981), 261–265.Google Scholar
[3]Dinitz, J. H. and Stinson, D. R., ‘A fast algorithm for finding strong starters,’ SIAM J. Algebraic and Discrete Methods 2 (1981), 50–56.CrossRefGoogle Scholar
[4]Dinitz, J. H. and Stinson, D. R., ‘The spectrum of Room cubes,’ European J. Combinatorics 2 (1981), 221–230.CrossRefGoogle Scholar
[5]Gross, K. B., ‘Some new classes of strong starters,’ Discrete Math. 12 (1975), 225–243.CrossRefGoogle Scholar
[6]Gross, K. B., Mullin, R. C., and Wallis, W. D., ‘The number of pairwise orthogonal symmetnc Latin squares,’ Utilitas Math. 4 (1973), 239–251.Google Scholar
[7]Horton, J. D., ‘Room designs and one-factorizations’, Aequationes Math., to appear.Google Scholar
[8]Mullin, R. C. and Nemeth, E., ‘An existence theorem for Room squares,’ Canad. Math. Bull. 12 (1969), 493–497.CrossRefGoogle Scholar
[9]Mullin, R. C. and Wallis, W. D., ‘The existence of Room squares,’ Aequationes Math. 13 (1975), 1–7.CrossRefGoogle Scholar
[10]Wallis, W. D., Street, A. P. and Wallis, I. S., Combinatorics: Room squares, sum-free sets, Hadamard matrices (Springer-Verlag, 1972).CrossRefGoogle Scholar