Skew-Hadamard matrices of order 2(q + 1)

Szekeres has established the existence of a skew-Hadamard matrix of order 2(q + 1) in the case q ≡ 5 (mod 8), a prime power. His method utilized complementary difference sets in the elementary abelian group of order q. The main result of this paper is to show that, for the same q, there exist skew-Hadamard matrices of order 2(q + 1) that are of the Goethals-Seidel type. This is achieved by using a cyclic relative difference set with parameters $\text{(q + 1, 4, q,}\text{1}\text{4}\text{(q ? 1))}$.