  We consider a matrix Riemann-Hilbert problem for the sextic nonlinear
Schr\"{o}dinger equation with a non-zero boundary conditions at infinity.
Before analyzing the spectrum problem, we introduce a Riemann surface and
uniformization coordinate variable in order to avoid multi-value problems.
Based on a new complex plane, the direct scattering problem perform a detailed
analysis of the analytical, asymptotic and symmetry properties of the Jost
functions and the scattering matrix. Then, a generalized Riemann-Hilbert
problem (RHP) is successfully established from the results of the direct
scattering transform. In the inverse scattering problem, we discuss the
discrete spectrum, residue condition, trace formula and theta condition under
simple poles and double poles respectively, and further solve the solution of a
generalized RHP. Finally, we derive the solution of the equation for the cases
of different poles without reflection potential. In addition, we analyze the
localized structures and dynamic behaviors of the resulting soliton solutions
by taking some appropriate values of the parameters appeared in the solutions.
