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.