Abstract: In this paper we prove that for any prime power q≡3 (mod 8) there exist 4-{q²; k, k, k, k; λ} supplementary difference sets (SDSs) with k=q(q-1)/2, λ=4k-q²,and Hadamard matrices of order 4q², and give several constructions of these SDSs. Moreover, combining the results of reference[1],we conclude that for any prime p≡3 (mod 8) anti integer τ> 1 there exists an Hadamard matrix of order 4p^2r.

Key words: abelian group, finite field, supplementary difference sets, Hadamard matrix