#### Abstract

R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h > 1, then there is a real Hadamard matrix of order hc.

Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1 + a^{2} + b^{2} where a, b are integers.

We give many constructions for new infinite classes of complex Hadamard matrices and show that they exist for orders 306, 650, 870, 1406, 2450 and 3782: for the orders 650, 870, 2450 and 3782, a symmetric conference matrix cannot exist.

## Publication Details

Jennifer Seberry Wallis, Complex Hadamard matrices, Linear and Multilinear Algebra, 1, (1973), 257-272.