Ljubljana,\
Jadranska 19, SI-1000 Ljubljana, Slovenia. ]{}

Introduction.
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Systems of particles in interaction on a one-dimensional lattice have attracted lots of attention in the last decades. The reason is twofold: on one hand they seem to capture the essential physical features of out-of-equilibrium systems, and on the other hand they allow in some particular cases for exact computations of physical quantities. In particular the study of exclusion processes, for which particles experience a hard-core interaction (there is at most one particle on each site of the lattice), turned out to be very fruitful. The Asymmetric Simple Exclusion Process (ASEP) is a paradigmatic example of such model [@CMZ; @DerrReview]. The phase diagram of the continuous time model with open boundaries was exactly computed using a matrix product construction of the stationary state [@DEHP; @Sandow94]. The ASEP was also studied in the discrete time setting using various stochastic update rules [@RajewskySSS] (see also for instance [@AppertRollandCH] and references therein for more recent developments and applications to traffic flow). The matrix product ansatz was successfully used to solve the ASEP with discrete time parallel update and sequential update [@Hinrichsen; @HoneckerP; @RajewskySS; @RajewskySSS; @EvansRS; @deGierN; @WoelkiS].

The reason behind the exact solutions of 