Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-28T23:33:47.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Skew-Hadamard Matrices of the Goethals-Seidel Type

Published online by Cambridge University Press:  20 November 2018

Edward Spence
Edward Spence*
Affiliation:
The University of Glasgow, Glasgow G12 8QW, Scotland
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.

1. Introduction. We prove, using a theorem of M. Hall on cyclic projective planes, that if g is a prime power such that either 1 + q + q2 is a prime congruent to 3, 5 or 7 (mod 8) or 3 + 2q + 2q2 is a prime power, then there exists a skew-Hadamard matrix of the Goethals-Seidel type of order 4(1 + q + q2). (A Hadamard matrix H is said to be of skew type if one of H + I, H — lis skew symmetric. ) If 1 + q + q2 is a prime congruent to 1 (mod 8), then a Hadamard matrix, not necessarily of skew type, of order 4(1 + q + q2) is constructed. The smallest new Hadamard matrix obtained has order 292.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975 , pp. 555 - 560
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Baumert, L. D., Cyclid difference sets, Springer-Verlag Lecture Notes in Mathematics, No. 182, 1971.Google Scholar
2. Elliott, J. E.H.and Butson, A. T., Relative difference sets, Illinois J. Math. 10 (1966), 517–531.Google Scholar
3. Goethals, J. M. and Seidel, J. J., A skew-Hadamard matrix of order 36, J. Austral. Math. Soc. 11 (1970), 343–344.Google Scholar
4. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar
5. Szekeres, G., Cyclotomy and complementary difference sets, Acta Arith. 18 (1971), 349–353.Google Scholar
6. Szekeres, G., Tournaments and Hadamard matrices, Enseignment Math. IS (1969), 269–278.Google Scholar