Rationality of generating functions of rook polynomials and permanents of Kronecker products of Toeplitz matrices and circulants with the matrix $J_k$ and their evaluation. I

The Classification of Circulant Weighing Matrices of Weight 16 and Odd Order

In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one matrix in CW(21,16) and another one in CW(63,16) (not obtainable by Kronecker product from CW(21,16)). The classification uses a multiplier existence theorem.