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In this paper we study the "Z" channel with side information non-causally available at the encoders. We use Marton encoding along with Gelf'and-Pinsker random binning scheme and Chong-Motani-Garg-El Gamal (CMGE) jointly decoding to find an achievable rate region. We will see that our achievable rate region gives the achievable rate of the multiple access channel with side information and also degraded broadcast channel with side information. We will also derive an inner bound and an outer bound on the capacity region of the state-dependent degraded discrete memoryless Z channel and also will observe that our outer bound meets the inner bound for the rates corresponding to the second transmitter. Also, by assuming the high signal to noise ratio and strong interference regime, and using the lattice strategies, we derive an achievable rate region for the Gaussian degraded Z channel with additive interference non-causally available at both of the encoders. Our method is based on lattice transmission scheme, jointly decoding at the first decoder and successive decoding at the second decoder. Using such coding scheme we remove the effect of the interference completely. I. INTRODUCTIONThe Z channel is a two-transmitter two-receiver model shown in Fig. 1 where the first sender only wishes to send information to the first receiver whereas the second transmitter sends information to both of the receivers. The Z channel was first studied by Viswanath et al [1] where they introduced the model and found the capacity region of a specialclass of Z channels and the achievable rate of a special case of the Gaussian Z channel (GZC). In [2], Liu and Ulukus obtained several capacity bounds for a class of GZC. Chong-Motani-Garg (CMGE) [3] studied three different types of degraded Z channel and characterized the capacity region in one type. They also characterized the capacity region of GZC with moderately strong crossover link.The capacity region of the general Z channel is still an open problem. The best achievable rate region for the discrete memoryless Z channel until today is due to Do et al [4].Channels with side information were first studied by Shannon [5] where he characterized the capacity of a point-to-point channel with side information causally available at the transmitters. Gelf 'and and Pinsker [6] found the capacity of a single-user channel with side information non-causally available at the encoders. State-dependent multiuser settings have been studied in [7], [8], [9], [10], and [11].In this paper we study the Z channel with channel state information non-causally available at the encoders that is depicted in Fig. 2. The reason to study this channel model is buttressed by the applications it has in some wireless communication scenarios such as the case where two communication-involved cells are interfering with each other and thus suffer from a common interference modeled by some S non-causally available to two distinct destination base stations as shown in Fig. As in Fig. 2

In this study, the authors study a two-user Gaussian doubly dirty compound multiple-access channel with partial side information (GDD-CMAC-PSI) where two independent additive interference signals are considered, each one known noncausally and partially to one of the encoders but unknown to either of the receivers. This channel, first, can model two users communicating with two base stations suffering from interference, and second, includes many previously studied channels as its special cases. For such a communication scenario, first, a general capacity outer bound is derived. Depending on the values of cross link gains, they classify the channel into three classes: weak, strong and mixed GDD-CMAC-PSI. Next, assuming that the interference signals have infinite variances, they obtain capacity outer bounds for these classes. Then, an achievable sum-rate is derived for the GDD-CMAC-PSI using Costa's strategy and thereby, they show that when both interference signals have infinite variances, this achievable sum-rate vanishes. Later, by utilising the lattice strategies and deriving achievable rate regions, independent of the interference powers, they show that in contrast with Costa's strategy, lattice-strategies can achieve positive rates. Finally, depending on signal-to-noise ratio gaps at receivers, various achievable rates are obtained.

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