Complex Hadamard matrices

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 + a2 + b2 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.