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,
If p is prime then H ( p , N ) can exist only for N = m p with integer m and it is conjectured they exist for all such cases with p ≥ 3 . In general, the problem of finding all sets { q , N } such that the Butson - type matrices H ( q , N ) exist, remains open.
H ( 2 , N ) contains real Hadamard matrices of size N, H ( 4 , N ) contains Hadamard matrices composed of ± 1 , ± i - such matrices were called by Turyn, complex Hadamard matrices.in the limit q → ∞ one can approximate all complex Hadamard matrices.Fourier matrices [ F N ] j k := exp [ ( 2 π i ( j − 1 ) ( k − 1 ) / N ] f o r j , k = 1 , 2 , … , N belong to the Butson-type,
while
