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 with integer m and it is conjectured exist, remains open.

Examples

• contains real Hadamard matrices of size N,

• – such matrices were called by Turyn, complex Hadamard matrices.

• one can approximate all complex Hadamard matrices.

• Fourier

• belong to the Butson-type,

  while

  1. ,

References

• 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).

External links

• Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006