Circulant Weighing Matrices

A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{i,j}$ in {$-1,0, +1$} such that $WW^T=kI_n$. For example, when $k=n$ this is a Hadamard matrix.

A circulant weighing matrix $CW(n,k)$ is a special type of weighing matrix in which every row except for the first is a right cyclic shift of the previous row. Equivalently, the first row viewed as a sequence has constant autocorrelation $0$.

Let $P$ be the set of locations with a $+1$ in the first row, and $N$ be the locations with a $-1$. Then $|P|+|N|=k$. It is known that

• $k = s^2$ for some integer $s$,
• $ |P| = (s^2+s)/2$,
• $|N| = (s^2-s)/2$.

A circulant weighing matrix $W$ is proper if the nonzeros in $P$ and $N$ are not all in a coset of a proper subgroup of ${\mathbb Z}/n{\mathbb Z}$. The existence of proper matrices is the focus of this database. See my paper with Arasu and Zhang for some of the methods used to determine existence of these matrices.

As with difference sets, there are numerous results about the existence of circulant weighing matrices. Strassler in 1997 gave a table of results for $n≤200$ and $s≤ 10$. This database attempts to bring together all known results about cases in Strassler’s table, and extend the table further.