Note on Williamson Matrices of Orders $25$ and $37$

Abstract

All Williamson matrices in this Note are symmetric circulants. Eight non-equivalent sets of Williamson matrices of order $25$ are known. They were discovered by Williamson ($2$ sets), Baumert and Hall ($2$ sets), and Sawade ($4$ sets). Sawade carried out a complete search and reported that there are exactly eight non-equivalent such sets of matrices. Subsequently, this was confirmed by Koukouvinos and Kounias. It is surprising that we have found two more such sets. Hence, there are ten non-equivalent sets of Williamson matrices of order $25$.

Only three non-equivalent sets of Williamson matrices of order $37$ were known so far. One of them was discovered by each of Williamson, Turyn, and Yamada. We have found one more such set.