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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.

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Such matrices do not exist as from the parity consideration already first two rows cannot be orthogonal.

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