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Integral quadratic forms and orthogonal designs

  • Peter Eades (a1) (a2)
Abstract

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.

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Copyright
COPYRIGHT: © Australian Mathematical Society 1981
References
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[1] Eades, P. (1977), ‘Orthogonal designs constructed from circulants’, Utilitas Math. 11, 43–55.
[2]Estes, D. and Pall, G. (1970), ‘The definite octonary quadratic forms of determinant l’, Illinois J. Math. 14, No. I.
[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.
[4]Geramita, A. V. and Seberry, J. (1979), Orthogonal designs (Lecture Notes in Pure and Applied Mathematics 45, Marcel Dekker, New York and Basel).
[5]Goethals, J. M. and Seidel, J. J. (1970), ‘A skew-Hadamard matrix of order 36’, J. Austral. Math. Soc. 11, 343–344.
[6]Hsia, J. S. (1977–1978), ‘Two theorems on integral matrices’, Linear and Multilinear Algebra 5, 257–264.
[7]Jones, B. W. (1950), The arithmetic theory of quadratic forms, Carus Mathematical Monographs 10 (John Wiley and sons).
[8]Kneser, M. (1957), ‘Klassenzahlen definiter quadratischer Formen’, Arch. Math. 8, 241–250.
[9]Radon, J. (1922), ‘Linear scharen orthogonaler matrixen’, Abh. Math. Sem. Univ. Hamburg. 1, 1–14.
[10]Shapiro, D. (1974), Similarities, quadratic forms and Clifford algebras (Ph.D. Thesis, University of California, Berkeley).
[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).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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