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Some results on weighing matrices

Published online by Cambridge University Press:  17 April 2009

Jennifer Seberry Wallis  and
Albert Leon Whiteman
Jennifer Seberry Wallis
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
Albert Leon Whiteman
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California, USA.
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Abstract

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It is shown that if q is a prime power then there exists a circulant weighing matrix of order q2 + q + 1 with q2 nonzero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that “given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N”.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 3 , June 1975 , pp. 433 - 447
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Elliott, J.E.H. and Butson, A.T., “Relative difference sets”, Illinois J. Math. 10 (1966), 517–531.CrossRefGoogle Scholar
[2]Geramita, Anthony V., Geramita, Joan Murphy, Wallis, Jennifer Seberry, “Orthogonal designs”, J. Lin. Multilin. Algebra (to appear).Google Scholar
[3]Geramita, Anthony V. and Wallis, Jennifer Seberry, “Orthogonal designs II”, Aequationes Math. (to appear).Google Scholar
[4]Geramita, Anthony V. and Wallis, Jennifer Seberry, “Orthogonal designs III: weighing matrices”, Utilitas Math. 6 (1974), 209–236.Google Scholar
[5]Geramita, Anthony V. and Wallis, Jennifer Seberry, “Orthogonal designs IV: existence questions”, J. Combinatorial Theory Ser. A (to appear).Google Scholar
[6]Geramita, Anthony V., Pullman, Norman S. and Wallis, Jennifer S., “Families of weighing matrices”, Bull. Austral. Math. Soc. 10 (1974), 119–122.CrossRefGoogle Scholar
[7]Goethals, J.M. and Seidel, J.J., “Orthogonal matrices with zero diagonal”, Canad. J. Math. 19 (1967), 1001–1010.CrossRefGoogle Scholar
[8]Ryser, H.J., “Variants of cyclic difference sets”, Proc. Amer. Math. Soc. 41 (1973), 45–50.CrossRefGoogle Scholar
[9]Spence, Edward, “Skew-Hadamard matrices of the Goethals-Seidel type”, Canad. J. Math. (to appear).Google Scholar
[10]Wallis, Jennifer, “Orthogonal (0, 1, −1) matrices”, Proc. First Austral. Conf. Combinatorial Math., Newcastle, 1972, 61–84 (TUNRA, Newcastle, 1972).Google Scholar
[11]Wallis, Jennifer Seberry, “Orthogonal designs V: orders divisible by eight”, Utilitas Math. (to appear).Google Scholar