The Anti-Field-Descent Method

A circulant Hadamard matrix of order $v$ is a matrix of the form \[H=\begin{pmatrix} a_1 & a_2 & \cdots & a_v \\ a_v & a_1 & \cdots & a_{v-1} \\ \cdots & \cdots & \cdots &\cdots \\ a_2 & a_3 & \cdots & a_1 \\ \end{pmatrix}\] with $a_i=\pm 1$ such that any two rows of $H$ are orthogonal with respect to the standard inner product. It is conjectured that there is no circulant Hadamard matrix of order larger than $4$.