  In this paper we disprove the Haagerup statement that all complex Hadamard
matrices of order five are equivalent with the Fourier matrix $F_5$ by
constructing circulant matrices that lead to new Hadamard matrices. An
important item is the construction of new mutually unbiased bases that are a
basic concept of quantum theory and play an essential role in quantum
tomography, quantum cryptografy, teleportation, construction of dense coding
schemes, classical signal proccesing, etc.
