A search for Hadamard matrices of Williamson type

TL;DR

This paper introduces near Williamson matrices, a special class of matrices, and uses computational methods to find all inequivalent examples up to order 35 and demonstrate existence up to order 63, leading to a new quaternary Hadamard matrix of order 118.

Contribution

The paper defines near Williamson matrices and computationally classifies them for small orders, also establishing their existence for larger orders and discovering a new Hadamard matrix of order 118.

Findings

All inequivalent near Williamson matrices for odd orders up to 35 were found.

Existence of near Williamson matrices was shown for all odd orders up to 63.

First known quaternary Hadamard matrix of order 118 was constructed.

Abstract

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 $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. 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 $118$.