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,
Existence.
If "p" is prime and formula_2, then formula_3 can exist
only for formula_4 with integer "m" and
it is conjectured they exist for all such cases 
with formula_5. For formula_6, the corresponding conjecture is existence for all multiples of 4.
In general, the problem of finding all sets
formula_7 such that the Butson - type matrices
formula_8 exist, remains open.
Examples.
belong to the Butson-type,