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Some new series of Hadamard matrices

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Mieko Yamada
Mieko Yamada
Affiliation:
Department of Mathematics, Tokyo Woman's Christian University, 2-6-1 Zempukuji, Suginamiku, Tokyo 167, Japan
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Abstract

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The purpose of this paper is to prove (1) if q ≡ 1 (mod 8) is a prime power and there exists a Hadamard matrix of order (q − 1)/2, then we can construct a Hadamard matrix of order 4q, (2) if q ≡ 5 (mod 8) is a prime power and there exists a skew-Hadamard matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2), (3) if q ≡ 1 (mod 8) is a prime power and there exists a symmetric C-matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2).

We have 36, 36 and 8 new orders 4n for n ≤ 10000, of Hadamard matrices from the first, the second and third theorem respectively, which were known to the list of Geramita and Seberry. We prove these theorems by using an adaptation of generalized quaternion type array and relative Gauss sums.

MSC classification

Secondary: 05B20: Matrices (incidence, Hadamard, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 371 - 383
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Geramita, A. V. and Seberry, J., Orthogonal designs (Lecture Notes in Pure and Applied Math. 45, Marcel Dekker, New York–Basel, 1979).Google Scholar
[2]Kiyasu, Z., Hadamard matrices and their applications (Denshi-Tsushin Gakkai, Tokyo, 1980, in Japanese).Google Scholar
[3]Kiyasu, Z., private communication.Google Scholar
[4]Spense, E., ‘Hadamard matrices from relative difference sets’, J. Combin. Theory Ser. A 19 (1975), 287–300.CrossRefGoogle Scholar
[5]Whiteman, A. L., ‘Hadamard matrices of order 4(2p+1)’, J. Number Theory 8 (1976), 1–11.CrossRefGoogle Scholar
[6]Yamada, M., ‘Hadamard matrices generated by an adaptation of generalized quaternion type array’, Graphs and Combinatorics 2 (1986), 179–187.CrossRefGoogle Scholar
[7]Yamamoto, K. and Yamada, M., ‘Williamson Hadamard matrices and Gauss sums’, J. Math. Soc. Japan 37 (1985), 703–717.CrossRefGoogle Scholar
[8]Yamamoto, K., ‘On congruences arising from relative Gauss sums’, Number Theory and Combinatorics, Japan 1984 (World Scientific Publ., Singapore, 1985).Google Scholar