Good Matrices of Orders $33,35$ and $127$ Exist

Abstract

Four $\{\pm 1\}$-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, $A{A}^T+B{B}^T+C{C}^T+D{D}^T=4n{I}_n$, and, pairwise, they satisfy $X{Y}^T=Y{X}^T$. It is known that they exist for odd $n\le 31$. 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.