The Capacity Region of a Class of Three-Receiver Broadcast Channels With Degraded Message Sets

Korner and Marton established the capacity region for the two-receiver broadcast channel with degraded message sets. Recent results and conjectures suggest that a straightforward extension of the Korner-Marton region to more than two receivers is optimal. This paper shows that this is not the case. We establish the capacity region for a class of three-receiver broadcast channels with two-degraded message sets and show that it can be strictly larger than the straightforward extension of the Korner-Marton region. The idea is to split the private message into two parts, superimpose one part onto the "cloud center" representing the common message, and superimpose the second part onto the resulting "satellite codeword." One of the receivers finds the common message directly by decoding the "cloud center," a second receiver finds it indirectly by decoding a satellite codeword, and a third receiver finds it by jointly decoding the transmitted codeword. This idea is then used to establish new inner and outer bounds on the capacity region of the general three-receiver broadcast channel with two and three degraded message sets. We show that these bounds are tight for some nontrivial cases. The results suggest that finding the capacity region of the three-receiver broadcast channel with degraded message sets is at least as hard finding as the capacity region of the general two-receiver broadcast channel with common and private message.