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Limits on Pairwise Amicable Orthogonal Designs

Published online by Cambridge University Press:  20 November 2018

Warren Wolfe
Warren Wolfe*
Affiliation:
Royal Roads, Mutiary College, Victoria, British Columbia
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An orthogonal design in order n of type (u1, …, ut) on the commuting variables x1, …, xt is an n × n matrix X with entries 0, ±x1, …, ±xt such that

In [5] Geramita and Wallis show that if n = 24a+b·n0, where n0 is odd and 0 ≦ b > 4, then t ≧ ρ(n) = 8a + 2b. The result is essentially Radon's limit on the number of anti-commuting, real, anti-symmetric, orthogonal matrices in order n. Garamita and Pullman show that this limit is sharp for orthogonal designs: i.e., given n, there exists an orthogonal design in order n with ρ(n) variables [6].

Two orthogonal designs, X and F, are called amicable if XYt = YXt.

Such pairs of orthogonal designs are especially useful in generating new orthogonal designs [5] or [6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1043 - 1054
Copyright
Copyright © Canadian Mathematical Society 1981

References

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