  We present the computation of invariants that arise in the strong coupling
expansion of lattice QCD. These invariants are needed for Monte Carlo
simulations of Lattice QCD with staggered fermions in a dual, color singlet
representation. This formulation is in particular useful to tame the finite
density sign problem. The gauge integrals in this limiting case
$\beta\rightarrow 0$ are well known, but the gauge integrals needed to study
the gauge corrections are more involved. We discuss a method to evaluate such
integrals.
  The phase boundary of lattice QCD for staggered fermions in the $\mu_B-T$
plane has been established in the strong coupling limit. We present numerical
simulations away from the strong coupling limit, taking into account the higher
order gauge corrections via plaquette occupation numbers. This allows to study
the nuclear and chiral transition as a function of $\beta$.
