Construction of Williamson type matrices

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, - 1) matrices A, B, C, D of order m which are of Williamson type, that is they pair-wise satisfy i) MNT = NMT, M, N E {A, B, C, D} and ii) AAT + BBT + CCT + DDT = 4mIm. It is shown that Williamson type matrices exist for the orders m = s(4s - 1), m = s(4s + 3) for s E {1, 3, 5, ... ,25} and also for m = 93. This gives Williamson matrices for several new orders including 33, 95, 189. These results mean there are Hadamard matrices of order i) 4s(4s - 1)t,20s(4s - 1)t, s E {1, 3, 5, ... , 25}; ii) 4s(4s + 3)t, 20s(4s + 3)t, s E {1, 3, 5, ... , 25}; iii) 4.93t,20.93t; for t E {1, 3, 5, ... , 61} u {1 + 2a10b26c, a, b, c nonnegative integers}, which are new infinite families. Also, it is shown by considering eight-Williamson-type matrices, that there exist Hadamard matrices of order 4(p + 1)(2p + 1)r and 4(p + 1)(2p + 5)r when p = 1 (mod 4) is a prime power, 8r is the order of a Plotkin array, and, in the second case 2p + 6 is the order of a symmetric Hadamard matrix. These classes are new.