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Some classes of Hadamard matrices with constant diagonal

Published online by Cambridge University Press:  17 April 2009

Jennifer Wallis  and
Albert Leon Whiteman
Jennifer Wallis
Affiliation:
University of Newcastle, Newcastle, New South Wales;
Albert Leon Whiteman
Affiliation:
University of Southern California, Los Angeles, California, USA.
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Abstract

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The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.

A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 2 , October 1972 , pp. 233 - 249
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Baumert, Leonard D., Cyclic difference sets (Lecture Notes in Mathematics, 182. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[2]Blatt, D. and Szekeres, G., “A skew Hadamard matrix of order 52”, Canad. J. Math. 21 (1969), 1319–1322.CrossRefGoogle Scholar
[3]Goethals, J.M. and Seidel, J.J., “Orthogonal matrices with zero diagonal”, Canad. J. Math. 19 (1967), 1001–1010.CrossRefGoogle Scholar
[4]Goethals, J.M. and Seidel, J.J., “A skew Hadamard matrix of order 36”, J. Austral. Math. Soc. 11 (1970), 343–344.CrossRefGoogle Scholar
[5]Goethals, J.M. and Seidel, J.J., “Strongly regular graphs derived from combinatorial designs”, Canad. J. Math. 22 (1970), 597–614.CrossRefGoogle Scholar
[6]van Lint, J.H. and Seidel, J.J., “Equilateral point sets in elliptic geometry”, K. Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 335–348.CrossRefGoogle Scholar
[7]Shrikhande, S.S., “On a two-parameter family of balanced incomplete block designs”, Sankhyā Ser. A 24 (1962), 33–40.Google Scholar
[8]Szekeres, G., “Tournaments and Hadamard matrices”, Enseignement Math. (2) 15 (1969), 269–278.Google Scholar
[9]Szekeres, G., “Cyclotomy and complementary difference sets”, Acta Arith. 18 (1971), 349–353.CrossRefGoogle Scholar
[10]Turyn, Richard J., “On C-matrices of arbitrary powers”, Canad. J. Math. 23 (1971), 531–535.CrossRefGoogle Scholar
[11]Wallis, Jennifer, “Some (1, -1) matrices”, J. Combinatorial Theory Ser. B 10 (1971), 1–11.CrossRefGoogle Scholar
[12]Wallis, Jennifer, “Amicable Hadamard matrices”, J. Combinatorial Theory Ser. A 11 (1971), 296–298.CrossRefGoogle Scholar
[13]Wallis, Jennifer, “Complex Hadamard matrices”, University of Newcastle, Mathematics Research Report No. 63, 1972.CrossRefGoogle Scholar
[14]Wallis, Jennifer, “On supplementary difference sets”, Aequationes Math. (to appear).Google Scholar
[15]Wallis, Jennifer, “A note on BIBDs”, J. Austral. Math. Soc. (to appear).Google Scholar
[16]Whiteman, Albert Leon, “An infinite family of skew Hadamard matrices”, Pacific J. Math. 38 (1971), 817–822.CrossRefGoogle Scholar
[17]Whiteman, Albert Leon, “Skew Hadamard matrices of Goethals-Seidel type”, (to appear).Google Scholar