Semi Williamson Type Matrices and the \(W(2n,n)\) Conjecture

Four \((1, -1, 0)\)-matrices of order \(m\), \(X = (x_{ij})\), \(Y = (y_{ij})\), \(Z = (z_{ij})\), \(U = (u_{ij})\) satisfying

1. \(XX^T + YY^T + ZZ^T + UU^T= 2mI_m\),
2. \(x_{ij}^2 + y_{ij}^2 + z_{ij}^2 + u_{ij}^2 = 2, i,j = 1,\ldots,m\),
3. \(X, Y, Z, U\) 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 \(n_1, \ldots, n_k\), then there exist semi Williamson type matrices of order \(N = \prod_{j=1}^k n_j^{r_j}\), where \(r_j\) 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 \(k = 0, 1, \ldots, 4n\). In addition, we find infinitely many new \(W(2n, n)\), \(n\) odd and the sum of two squares.