Capacity Results for Classes of Partially Ordered $K$ -User Broadcast Channels With Two Nested Multicast Messages

Abstract

The $K$ -user discrete memoryless (DM) broadcast channel (BC) with two nested multicast messages is studied in which one common message is to be multicast to all receivers and the second private message to a subset of receivers. The receivers that must decode both messages are referred to as private receivers and the others that must decode only the common message as common receivers. For two nested multicast messages, we establish the capacity region for several classes of partially ordered DM BCs characterized by the respective associated sets of pair-wise relationships between and among the common and private receivers, each described by the well-known pair-wise more capable or less noisy condition. For three classes of partially ordered DM BCs, the capacity region is shown to be simply achieved by two-level superposition coding and the proofs of the converses rely on a recently found information inequality. The rate region achievable by two-level superposition coding is then enhanced through a multi-level superposition coding scheme after splitting the private message into as many parts as there are common receivers and indirect decoding. A closed-form two-dimensional polyhedral (polygonal) description is obtained for it for a given coding distribution in spite of the indeterminate number of split rates via a structured form of Fourier-Motzkin elimination. Through a converse result that relies on the Csiszar sum lemma and that information inequality, a specialization of this region that corresponds to splitting the private message into just two sub-messages is proved to be the capacity region for several classes of partially ordered DM BCs beyond those for which two-level superposition coding is capacity optimal, thereby underscoring the benefit of rate-splitting. All previously known capacity results for partially ordered DM BCs with two nested multicast messages for the two and three-receiver DM BCs as well as DM BCs with one private or one common receiver are subsumed in the general results obtained in this work.