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Families of weighing matrices

Published online by Cambridge University Press:  17 April 2009

Anthony V. Geramita ,
Norman J. Pullman  and
Jennifer S. Wallis
Anthony V. Geramita
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, Canada.
Norman J. Pullman
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, Canada.
Jennifer S. Wallis
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, Canada.
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A weighing matrix is an n × n matrix W = W(n, k) with entries from {0, 1, −1}, satisfying = WWt = KIn. We shall call k the degree of W. It has been conjectured that if n ≡ 0 (mod 4) then there exist n × n weighing matrices of every degree k ≤ n.

We prove the conjecture when n is a power of 2. If n is not a power of two we find an integer t < n for which there are weighing matrices of every degree ≤ t.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 1 , February 1974 , pp. 119 - 122
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Taussky, Olga, “(1, 2, 4, 8)-sums of squares and Hadamard matrices”, Combinatorics, 229–233 (Proc. Symposia Pure Math., 19. Amer. Math. Soc., Providence, Rhode Island, 1971).Google Scholar
[2]Wallis, Jennifer, “Orthogonal (0, 1, −1)-matrices”, Proc. First Austral. Conf. Combinatorial Math., Newcastle, 1972, 61–84 (TUNRA, Newcastle, 1972).Google Scholar
[3]Wallis, W.D., Street, Anne Penfold, Wallis, Jennifer Seberry, Combinatorics: Room squares, sum-free sets, Hadamard matrices (Lecture Notes in Mathematics, 292. Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar