On circulant best matrices and their applications

Call four type 1 (1, - 1) matrices, X-1, X-2, X-3, X-4, on the same group o f order m (odd) with the properties (i) (X-i-I)(T)= -(X-i-I), i = 1, 2, 3, (ii) X-4(T) = X-4 and the diagonal elements are positive, (iii) XiXj= XjYi and (iv) X1X1T + X2X2T + X3X3T + X4X4T = 4mI(m), best matrices. We use a co mputer to give, for the first time, all inequivalent best matrices of odd o rder m less than or equal to 31. Inequivalent best matrices of order m, m o dd, can be used to find inequivalent skew-Hadamard matrices of order 4m. We use best matrices of order (1/4) (s(2) + 3) to construct new orthogonal de signs, including new OD(2s(2) + 6; 1, 1, 2, 2, s(2), s(2)).