Orthogonal (0,1,-1) matrices

We study the conjecture: There exists a square (0,l,-l)-matrix W = W(w,k) of order w satisfying WWT= kIw for all k = 0, 1,..., w when w = 0 (mod 4). We prove the conjecture is true for 4, 8, 12, 16, 20, 24, 28, 32, 40 and give partial results for 36, 44, 52, 56.