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Integral quadratic forms and orthogonal designs

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Peter Eades
Peter Eades
Affiliation:
Department of Pure Mathematics, School of General Studies, Australian National University, Box 4, Canberra, A.C.T. 2600, Australia Department of Computer Science, University of Queensland, St. Lucia Queensland 4067, Australia
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Abstract

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Warren W. Wolfe obtained necessary conditions for the existence of orthogonal designs in terms of rational matrices. In this paper it is shown that these necessary conditions can be obtained in terms of integral matrices. In the integral form, Wolfe's theory is more useful in the construction of orthogonal designs.

MSC classification

Secondary: 05B20: Matrices (incidence, Hadamard, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 3 , February 1981 , pp. 297 - 306
Copyright
Copyright © Australian Mathematical Society 1981

References

[1] Eades, P. (1977), ‘Orthogonal designs constructed from circulants’, Utilitas Math. 11, 43–55.Google Scholar
[2]Estes, D. and Pall, G. (1970), ‘The definite octonary quadratic forms of determinant l’, Illinois J. Math. 14, No. I.CrossRefGoogle Scholar
[3]Geramita, A. V. and Pullman, N. J. (1974), ‘A theorem of Hurwitz and Radon and orthogonal projective modules’, Proc. Amer. Math. Soc. 42, 51–56.CrossRefGoogle Scholar
[4]Geramita, A. V. and Seberry, J. (1979), Orthogonal designs (Lecture Notes in Pure and Applied Mathematics 45, Marcel Dekker, New York and Basel).Google Scholar
[5]Goethals, J. M. and Seidel, J. J. (1970), ‘A skew-Hadamard matrix of order 36’, J. Austral. Math. Soc. 11, 343–344.CrossRefGoogle Scholar
[6]Hsia, J. S. (1977–1978), ‘Two theorems on integral matrices’, Linear and Multilinear Algebra 5, 257–264.CrossRefGoogle Scholar
[7]Jones, B. W. (1950), The arithmetic theory of quadratic forms, Carus Mathematical Monographs 10 (John Wiley and sons).Google Scholar
[8]Kneser, M. (1957), ‘Klassenzahlen definiter quadratischer Formen’, Arch. Math. 8, 241–250.Google Scholar
[9]Radon, J. (1922), ‘Linear scharen orthogonaler matrixen’, Abh. Math. Sem. Univ. Hamburg. 1, 1–14.CrossRefGoogle Scholar
[10]Shapiro, D. (1974), Similarities, quadratic forms and Clifford algebras (Ph.D. Thesis, University of California, Berkeley).Google Scholar
[11]Wolfe, W. (1977), ‘Rational quadratic forms and orthogonal designs’, Number theory and algebra edited by Zassenhaus, H. (Academic Press, New York, San Franciso, London).Google Scholar