  In this paper we provide a general method to construct four-parameter
families of complex Hadamard matrices of order six. Our approach is to write a
6-dimensional matrix as composed of four blocks, each one in the form of a
circulant matrix. Use of Sylvester method of inverse orthogonal matrices leads
to two constraints on the six parameters, and their fulfilment generates
complex Hadamard matrices when the free parameters take values on the unit
circle. We hope that the problem of mutually unbiased bases could be solved for
six-level systems by using our results.
