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
