Abstract: An infinite family of H₁ is obtained, that is,H₁(ν)≠∅ for υ ∈ N=N₁ ∪ N₂ ∪ N₃ where
N₁={5,13,17,25,37,41,61},N₂{υ=3^2r(p₁^r…p_x^r)⁴:p₁≡3(mod 4)prime,p₁>3,r,r₁≥0，i=1,2,…,n;n=1,2,…}，N₃={υ₁υ₂:υ₁∈N₁ and υ₂∈N₂}. Using H₁(υ),we give 2-(2υ;k₁,k₂;k₁ + k₂ -υ} supplementary difference sets with υ=(k₁-υ)² + (k₂-υ)². Finally. we prove that if there exists an orthogonal design OD(4t;t,t,t,t) and H₁(υ) ≠∅ then a Hadamard matrix of order 4tυ can be constructed.

Key words: class H₁, supplementary difference sets, orthogonal design, Hadamard matrix