On an outer bound and an inner bound for the general broadcast channel

In this paper, we study the Nair-El Gamal outer bound and Marton's inner bound for general two-receiver broadcast channels. We show that the Nair-El Gamal outer bound can be made fully computable. For the inner bound, we show that, unlike in the Gaussian case, for a degraded broadcast channel even without a common message, Marton's coding scheme without a superposition variable is in general insufficient for obtaining the capacity region. Further, we prove various results that help to restrict the search space for computing the sum-rate for Marton's inner bound. We establish the capacity region along certain directions and show that it coincides with Marton's inner bound. Lastly, we discuss an idea that may lead to a larger inner bound.