Capacity Results via Message Merging and Superposition Coding in the K-Receiver Broadcast Channel with General Message Sets

A K-receiver discrete memoryless (DM) broadcast channel (BC) with general message sets is studied. A general message set is one that contains any subset of or all possible 2^K-1 groupcast messages, with each such message intended for a distinct subset of receivers. Message merging is a simple idea of bijectively mapping multiple messages into a single message, and hence with a rate that is the sum of the rates of the merged messages. Using a natural form of message merging it is shown that superposition coding of the merged messages and successive decoding of a subset of those messages at each receiver achieves the capacity region of the class of K-receiver interleavable DM BCs for any general message set. The interleavable class of DM BCs subsumes the K-receiver degraded DM BC and the less noisy DM BC in the three-receiver case as special cases. A generalization of that result is also given. For certain classes of message sets, message merging, superposition coding and successive decoding is shown to again achieve the capacity region, but for corresponding classes of channels that are larger than the interleavable DM BC. In each such class, there is a group of receivers that are not constrained to be ordered in strength by any notion of order (i.e., degraded, less noisy or more capable). Most known results on the optimality of superposition coding and successive decoding, including notably, for the interleavable DM BC with private messages, and for the recently found classes of DM BCs for two nested multicast messages with one private or one common receiver, are subsumed by the general -yet simply established -result of this paper.