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The K-user discrete memoryless (DM) broadcast channel (BC) with two nestedmulticast messages is studied in which one common message is to be multicast toall receivers and the second private message to a subset of receivers. Thereceivers that must decode both messages are referred to as private receiversand the others that must decode only the common message as common receivers.For two nested multicast messages, we establish the capacity region for severalclasses of DM BCs characterized by the respective associated sets of pair-wiserelationships between and among the common and private receivers, eachdescribed by the well-known more capable or less noisy conditions.  For three classes of DM BCs, the capacity region is simply achieved bysuperposition coding and the proofs of the converses rely on a recently foundinformation inequality. The achievable rate region is then enhanced through theaddition of a splitting of the private message into as many parts as there arecommon receivers and indirect decoding. A closed-form two-dimensionalpolyhedral description is obtained for it for a given coding distribution.Through a converse result that relies on the well-known Csiszar sum lemma andthe information inequality, a specialization of this region that involvessplitting the private message into just two sub-messages is proved to be thecapacity region for several classes of DM BCs, beyond those for whichsuperposition coding alone is capacity optimal, thereby underscoring thebenefit of rate-splitting.  All previously known capacity results for DM BCs with two nested multicastmessages for the two and three-receiver DM BCs as well as DM BCs with oneprivate or one common receiver are included in the general framework presentedin this work.

Capacity Results for the K-User Broadcast Channel with Two Nested Multicast Messages