Nonexistence of two circulant weighing matrices of weight 81$^{1}$

In this paper, we prove the nonexistence of two weighing matrices of weight 81, namely $CW(88,81)$ and $CW(99,81)$. We will apply two very different methods to do so; for the case of $CW(88,81)$, we will use almost purely counting methods, while for $CW(99,81)$, we will use algebraic methods.