Capacity Theorems for the “Z” Channel

We consider the two-user Z channel (ZC), where there are two senders and two receivers. One of the senders transmits information to its intended receiver (without interfering with the unintended receiver), while the other sender transmits information to both receivers. The complete characterization of the discrete memoryless ZC remains unknown to date. For the Gaussian ZC, the capacity has only been established for a crossover link gain of $1$. In this work, we study both the discrete memoryless ZC and the Gaussian ZC. We first establish achievable rates for the general discrete memoryless ZC. The coding strategy uses rate-splitting and superposition coding at the sender with information for both receivers. At the receivers, we use joint decoding. We then specialize the rates obtained to two different types of degraded discrete memoryless ZCs and also derive respective outer bounds to their capacity regions. We show that as long as a certain condition is satisfied, the achievable rate region is the capacity region for one type of degraded discrete memoryless ZC. The results are then extended to the two-user Gaussian ZC with different crossover link gains. We determine an outer bound to the capacity region of the Gaussian ZC with strong crossover link gain and establish the capacity region for moderately strong crossover link gain.