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A few more balanced Room squares

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

D. R. Stinson  and
S. A. Vanstone
D. R. Stinson
Affiliation:
Department of Computer, Science, University of Manibota, Winnipeg, Manitoba R3T 2N2, Canada
S. A. Vanstone
Affiliation:
Department of Combinatorics, and Optimization, St. Jerome's College, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Abstract

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The existence problem for balanced Room squares is, in general, unsolved. Recently, B. A. Anderson gave a construction for a class of these designs with side 2n − 1, where n is odd and n ≥ 3. For n even, the existence has not yet been settled. In this paper, we use the affine geometry of dimension 2 k and order 2, and a hill-climbing algorithm, to construct a number of new balanced Room squares directly. Recursive techniques based on finite geometries then give new squares of side 22k − 1 for infinitely many values of k.

MSC classification

Secondary: 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 3 , December 1985 , pp. 344 - 352
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Anderson, B. A., ‘Hyperplanes and balanced Howell rotations,’ Ars Combinatoria 13 (1982), 163–168.Google Scholar
[2]Anderson, B. A., private communication.Google Scholar
[3]Berlekamp, E. R. and Hwang, F. K., ‘Construction for balanced Howell rotations for bridge tournaments,’, J. Combin. Theory 12 (1972), 159–166.CrossRefGoogle Scholar
[4]Dinitz, J. H. and Stinson, D. R., ‘A fast algorithm for finding strong starters,’ SIAM J. Algebraic Discrete Methods 2 (1981), 50–56.CrossRefGoogle Scholar
[5]Fuji-Hara, R. and Vanstone, S. A., ‘Affine geometries obtained from projective planes and skew resolutions of AG(3, q)’, Ann. Discrete Math. 18 (1983), 355–376.Google Scholar
[6]Fuji-Hara, R. and Vanstone, S. A., ‘Recursive constructions for skew resolutions in affine geometries’, Aequationes Math. 23 (1981), 242–251.CrossRefGoogle Scholar
[7]Fuji-Hara, R. and Vanstone, S. A., ‘Some results on balanced Room squares and their generalizations’, preprint.Google Scholar
[8]Mullin, R. C. and Wallis, W. D., ‘The existence of Room squares’, Aequationes Math. 13 (1975), 1–7.CrossRefGoogle Scholar
[9]Parker, E. T. and Mood, A. M., ‘Some balanced Howell rotations for duplication bridge sessions’, Amer. Math. Monthly 26 (1955), 714–716.CrossRefGoogle Scholar
[10]Schellenberg, P. J., ‘On balanced Room squares and complete balanced Howell rotations’, Aequationes Math. 9 (1973), 75–90.CrossRefGoogle Scholar
[11]Stinson, D. R. and Vanstone, S. A., ‘Orthogonal packings in PG(5,2)’, Geom. Dedicata, submitted.Google Scholar