The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation mechanism, and by making use of an explicit representation of the matrix algebra we easily find closed expressions for the correlation functions in the general probabilistic case. Asymptotic expressions, obtained by making use of earlier results, allow us to derive the exact phase diagram
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