Doubly Regular Tournaments are Equivalent to Skew Hadamard Matrices

Abstract Doubly regular tournaments and homogeneous tournaments are defined and shown to be equivalent. Existence of such a tournament of order n is equivalent to the existence of a skew Hadamard matrix of order n + 1. Thus, such a tournament of order n exists whenever n + 1 = 2 t k 1 , … k s , each k i of the form p r + 1 ≡ 0 (mod 4), p a prime, and for various other orders.