A search for Hadamard matrices of Williamson type

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1,1)$-matrices $A,B,C,D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $A{A}^{\mathrm{\top }}+B{B}^{\mathrm{\top }}+C{C}^{\mathrm{\top }}+D{D}^{\mathrm{\top }}=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.