  In this paper we construct a new, previously unknown four-parameter family of
complex Hadamard matrices of order 6, the entries of which are described by
algebraic functions of roots of various sextic polynomials. We conjecture that
the new, generic family G together with Karlsson's degenerate family K and
Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of
order 6. Such a complete characterization might finally lead to the solution of
the famous MUB-6 problem.
