Butson-type Hadamard matrix

In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity,


 * $$(H_{jk})^q = 1 \quad\text{for}\quad j,k = 1,2,\dots,N.$$

Existence
If p is prime and $$N>1$$, then $$H(p,N)$$ can exist only for $$N = mp$$ with integer m and it is conjectured they exist for all such cases with $$p \ge 3$$. For $$p=2$$, the corresponding conjecture is existence for all multiples of 4. In general, the problem of finding all sets $$\{q,N \}$$ such that the Butson-type matrices $$H(q,N)$$ exist, remains open.

Examples
\text{ for }j,k = 1,2,\dots,N $$
 * $$H(2,N)$$ contains real Hadamard matrices of size N,
 * $$H(4,N)$$ contains Hadamard matrices composed of $$\pm 1, \pm i$$ – such matrices were called by Turyn, complex Hadamard matrices.
 * in the limit $$q \to \infty $$ one can approximate all complex Hadamard matrices.
 * Fourier matrices $$ [F_N]_{jk}:= \exp[(2\pi i (j-1)(k-1)/N]
 * belong to the Butson-type,


 * $$F_N \in H(N,N),$$


 * while


 * $$F_N \otimes F_N \in H(N,N^2),$$


 * $$F_N \otimes F_N\otimes F_N \in H(N,N^3).$$


 * $$D_6 := \begin{bmatrix}

1 & 1  & 1  & 1 & 1  & 1 \\                1 & -1  & i  & -i& -i & i \\ 1 & i  &-1  &  i& -i &-i \\ 1 & -i & i  & -1&  i &-i \\ 1 & -i &-i  &  i& -1 & i \\ 1 & i  &-i  & -i&  i & -1 \\ \end{bmatrix} \in\, H(4,6)$$,


 * $$S_6 := \begin{bmatrix}

1 & 1  & 1  & 1 & 1  & 1  \\                1 &  1  & z  & z & z^2 & z^2 \\ 1 & z  & 1  & z^2&z^2 & z \\ 1 & z  & z^2&  1&  z & z^2 \\ 1 & z^2& z^2&  z&  1 & z \\ 1 & z^2& z  & z^2& z & 1 \\ \end{bmatrix} \in\, H(3,6)$$
 * where $$z =\exp(2\pi i/3).$$