Good Matrices of Orders \(33, 35\) and \(127\) Exist

Four \(\{\pm1\}\)-matrices \(A, B, C, D\) of order \(n\) are called good matrices if \(A – I_n\) is skew-symmetric, \(B, C\), and \(D\) are symmetric, \(AA^T + BB^T + CC^T + DD^T = 4nI_n\), and, pairwise, they satisfy \(XY^T = YX^T\). It is known that they exist for odd \(n \leq 31\). We construct four sets of good matrices of order \(33\) and one set for each of the orders \(35\) and \(127\).

Consequently, there exist \(4\)-Williamson type matrices of order \(35\), and a complex Hadamard matrix of order \(70\). Such matrices are constructed here for the first time. We also deduce that there exists a Hadamard matrix of order \(1524\) with maximal excess.