  The transfer matrix and matrix multiplication ansatz, when applied to
nonequilibrium steady states in asymmetric exclusion processed and traffic
models, has given many exact results for phase diagrams, bulk densities and
fluxes, as well as density profiles and spatial and temporal correlation
functions for models with a dynamics that is updated in (random) sequential and
sublattice-parallel order. Here we consider fully parallel or synchronous
dynamics, for which only partial results are known, due to the appearance of
complicated strong short range correlations, that invalidate simple mean field
approximations.
  This paper is based on two new ingredients: (i) a microscopic
characterization of order parameters and local configurations in the relevant
phases, based on the microdynamics of the model, and (ii) an improved mean
field approximation, which neglects certain four point - and higher order
correlation functions. It is conjectured that the density profiles, obtained
here, are exact up to terms that are exponentially small in the system size.
