J Austral Math Soc Ser A 46 pp371--383, 1989.

Some New Series of Hadamard Matrices

Mieko Yamada

Abstract

The purpose of this paper is to prove (1) if q º 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 º 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 º 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 forst, the second and third theorem respectively, which were known to the list of Germita and Seberry. We prove these theorems by using an adaption of generalized quaternion type array and relative Gauss sums.

1980 AMS Subject Classification (1985 Revision): 05B20

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Authors

Mieko Yamada

Department of Mathematics, Tokyo Woman's Christian University, 2-6-1 Zempukuji, Suginamiku, Tokyo 167, Japan.