robust Hadamard matrix

A square $N\times N$ complex Hadamard matrix $H$ is called robust if all principal minors of $H$ of order two are extremal so that their modulus is equal to $2$ [167].

The name reflects the fact that the property of being a Hadamard matrix is robust with respect to projections $\Pi_2$ onto a subspace spanned by two vectors of the basis used, $\Pi_2 H \Pi_2 \in$ $\mathcal{H}_2$.

If a robust Hadamard matrix exists, then its size must be even.

Examples:

$F_2$
$H_4$
$$\left[\begin{array}{rrrrrrrr} 1 & 1& 1& 1& 1& 1& 1& 1\\ -1 & 1& 1& 1& -1& 1& -1& -1\\ -1 & -1& 1& 1& 1& -1& 1& -1\\ -1 & -1& -1& 1& 1& 1& -1& 1\\ -1 & 1& -1& -1& 1& 1& 1& -1\\ -1 & -1& 1& -1& -1& 1& 1& 1\\ -1 & 1& -1& 1& -1& -1& 1& 1\\ -1 & 1& 1& -1& 1& -1& -1& 1 \end{array}\right]$$