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Using cyclotomy to construct orthogonal designs

Published online by Cambridge University Press:  09 April 2009

Joan Cooper  and
Jennifer Seberry Wallis
Joan Cooper
Affiliation:
Department of MathematicsUniversity of Newcastle, Australia.
Jennifer Seberry Wallis
Affiliation:
Department of MathematicsInstitute of Advanced Studies Australian National University.
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An orthogonal design of order n and type (s1, s2) on the commuting variables x1, x2 is a matrix of order n with entries from {0, ± x1, ± x2} whose row vectors are formally orthogonal.

This note uses cyclotomy to construct orthogonal designs and finds several infinite families of new designs.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 1 , August 1976 , pp. 119 - 127
Copyright
Copyright © Australian Mathematical Society 1976

References

Baumert, L. D. and Marshall, Hall Jr, (1965), ‘A new construction for Hadamard matrices’, Bull. Amer. Math. Soc. 71, 169–170.CrossRefGoogle Scholar
Joan, Cooper (1972), ‘A binary composition for collections and sets’, Proceedings of the First Australian Conference on Combinatorial Mathematics, 145–161, TUNRA, Newcastle.Google Scholar
Joan, Cooper (1974), ‘A note on Hadamard arrays’, Bull. Austral. Soc. 10, 15–21.Google Scholar
Joan, Cooper and Jennifer, Wallis (1972), ‘A construction for Hadamard arrays’, Bull. Austral. Math. Soc. 7, 269–278.Google Scholar
Geramita, Anthony V., Geramita, Joan Murphy and Wallis, Jennifer Seberry (1975), ‘Orthogonal designs’, Linear and Multilinear Algebra 3.Google Scholar
Geramita, Anthony V. and Wallis, Jennifer Seberry, ‘Orthogonal designs II’, Aequationes Math. (to appear).Google Scholar
Geramita, Anthony V. and Wallis, Jennifer Seberry (1974), ‘Orthogonal designs III: weighing matrices’, Utilitas Math. 6, 209–236.Google Scholar
Geramita, Anthony V. and Wallis, Jennifer Seberry (1975), ‘Orthogonal designs IV: existence questions’, J. Combinatorial Theory. (Series A), 19, 66–83.CrossRefGoogle Scholar
Geothals, J. M. and Seidel, J. J. (1967), ‘Orthogonal matrices with zero diagonal’, Canad. J. Math. 19, 1001–1010.CrossRefGoogle Scholar
Marshall, Hall Jr, (1967), Combinatorial Theory (Blaisdell, [Ginn] Waltham, Massachusetts).Google Scholar
Hunt, David C. and Jennifer, Wallis (1973), ‘Cyclotomy, Hadamard arrays and supplementary difference sets’, Proceedings of the Second Manitoba Conference on Numerical Mathematics, 351–381, Congressus Numerantium 7, University of Manitoba, Winnipeg.Google Scholar
Thomas, Storer (1967), Cyclotomy and Difference Sets (Lectures in Advanced Mathematics, 2. Markham, Chicago, Illinois).Google Scholar
Turyn, Richard J. (1974), ‘Hadamard matrices, Baumert-Hall units, four-symbol sequence, pulse compression and surface wave encodings’, J. Combinatorial Th. (ser. A), 16, 313–333.CrossRefGoogle Scholar
Turyn, Richard J. (1972), ‘Hadamard matrices, algebras, and composition theorems’, Notices Amer. Math. Soc. 19, A–388.Google Scholar
Jennifer, Wallis (1973), ‘Hadamard matrices of order 28m, 36m, and 44m’, J. Combinatorial Theory (Series A) 15, 323–328.Google Scholar
Wallis, W. D., Street, Anne Penfold, Wallis, Jennifer Seberry (1972), Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices (Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-Heidelberg-New York).CrossRefGoogle Scholar