Semi Williamson type matrices and the W(2n, n) conjecture

Four (1, -1, 0)-matrices of order m, X = (Xij), Y = (Yij), Z = (Zij), U = (Uij) satisfying (i) XXT + yyT + ZZT + UUT = 2mIm , (ii) x2ij + y2ij + z2ij + U2ij = 2, i, j = 1, ... ,m, (iii) X, Y, Z, U mutually amicable, will be called semi Williamson type matrices of order m. In this paper we prove that if there exist Williamson type matrices of order n1,...nk. then there exist semi Williamson type matrices of order N = IIkj=1 nr j j, where rj are non-negative integers. As an application, we obtain a W(4N,2N). Although the paper presents no new W(4n,2n) for n, odd, n < 3000, it is a step towards proving the conjecture that there exists a W(4n, 2n) for any positive integer n. This conjecture is a sub-conjecture of the Seberry conjecture [4, page 92] that W(4n, k) exist for all k = 0,1, ... , 4n. In addition we find infinitely many new W(2n, n), n odd and the sum of two squares.