In mathematics, a complex Hadamard matrixH of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-typeH(q, N) if all its elements are powers of q-th root of unity,
Existence
If p is prime and , then can exist
only for with integer m and
it is conjectured they exist for all such cases
with . For , the corresponding conjecture is existence for all multiples of 4.
In general, the problem of finding all sets
such that the Butson - type matrices
exist, remains open.
A. T. Butson, Generalized Hadamard matrices, Proc. Am. Math. Soc. 13, 894-898 (1962).
A. T. Butson, Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Can. J. Math. 15, 42-48 (1963).
R. J. Turyn, Complex Hadamard matrices, pp. 435–437 in Combinatorial Structures and their Applications, Gordon and Breach, London (1970).