Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T14:06:12.586Z Has data issue: false hasContentIssue false
  • English
  • Français

A skew Hadamard matrix of order 36

Published online by Cambridge University Press:  09 April 2009

J. M. Goethals  and
J. J. Seidel
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 3 , August 1970 , pp. 343 - 344
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Blatt, D. and Szekeres, G., ‘A skew Hadamard matrix of order 52’, Canadian J. Math., to appear.Google Scholar
[2]Goethals, J. M. and Seidel, J. J., ‘Orthogonal matrices with zero diagonal’, Canadian J. Math. 19 (1967), 1001–1010.CrossRefGoogle Scholar
[3]Hall, M., Combinatorial theory (Blaisdell 1967).Google Scholar
[4]Johnsen, E. C., ‘Integral solutions to the incidence equation for finite projective plane cases of orders n ≡ 2 (Mod 4)’, Pacific J. Math. 17 (1966), 97–120.CrossRefGoogle Scholar
[5]Reid, K. B., and Parker, E. T., ‘Disproof of a conjecture of Erdös and Moser on tournaments’, J. Combinatorial Theory, to appear.Google Scholar
[6]Szekeres, G., ‘Tournaments and Hadamard matrices’, l'Enseign. Math., 15 (1969), 269–278.Google Scholar
[7]Williamson, J., ‘Hadamard's determinant theorem and the sum of four squares’, Duke Math. J. 11, (1944), 65–81.CrossRefGoogle Scholar