A Hadamard matrix $H$ is skew Hadamard if $H+H^T=2I$

A collection of skew Hadamard matrices, including at least one example of every order $n \le 100$ and also including every equivalence class of order $\le 28$ is available at this web page. It has been conjectured that one exists for every positive order divisible by 4.

Reid and Brown in 1972 showed that there exists a ``doubly regular tournament of order n'' if and only if there exists a skew Hadamard matrix of order n+1.

Bibliography

1

S. Georgiou, C. Koukouvinos, J. Seberry, Hadamard matrices, orthogonal designs and construction algorithms, pp. 133-205 in DESIGNS 2002: Further computational and constructive design theory, Kluwer 2003.

2

K.B. Reid, E. Brown, Doubly regular tournaments are equivalent to skew Hadamard matrices, J. Combinatorial Theory A 12 (1972) 332-338.

3

J. Seberry, M.Yamada, Hadamard matrices, sequences, and block designs, pp. 431-560 in Contemporary Design Theory, a collection of surveys (J.H.Dinitz & D.R.Stinson eds.), Wiley 1992.