matrix of Butson type
A complex Hadamard
matrix of size $N$ is called of Butson
type $BH(N, q)$ if all the
entries of $H$ are $q^{\rm th}$ roots of
unity [51], [52],
$$\left(\left[H\right]_{j,k}\right)^q=1.$$
Note that another convention $BH(q, N)$ for Butson matrices is also used in the literature.
Examples:
- $BH(N, 2)$ all real Hadamard matrices
- $BH(N, 4)$ contain entries $\pm\{ 1, i\}$ and in the literature are
also called complex Hadamard matrices [90]
- $F_N\in BH(N, N)$ Fourier matrices
- $F_N\otimes F_N\in BH(N^2, N)$
- $F_N\otimes F_N\otimes F_N\in BH(N^3, N)$
- $\quad\vdots$
- $\bigotimes_{n=1}^kF_N\in BH(N^k, N)$