Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?


Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:


\begin{align}
 \begin{bmatrix} 
x\_1 & x\_2 & \cdots & x\_{n-1} & x\_n \\ 
-x\_n & x\_1 & \cdots & x\_{n-2} & x\_{n-1} \\ 
\vdots & \vdots& \ddots & \vdots & \vdots \\ 
-x\_3 & -x\_4 & \cdots & x\_1 & x\_2 \\ 
-x\_2 & -x\_3 & \cdots & -x\_n & x\_1 \\ \end{bmatrix}. 
\end{align}


An $n \times n$ Hadamard matrix is a matrix whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.



One question about nega-cyclic Hadamard matrices
Such matrices do not exist as from the parity consideration already first two rows cannot be orthogonal.


