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.

中文翻译：

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具有两个嵌套多播消息的部分有序 K 用户广播信道类的容量结果

研究了具有两个嵌套多播消息的 $K$ -user 离散无记忆 (DM) 广播信道 (BC)，其中一个公共消息要多播到所有接收者，第二个私有消息要多播到接收者的子集。必须解码两个消息的接收器被称为私有接收器，而其他必须只解码公共消息的接收器被称为公共接收器。对于两个嵌套的多播消息，我们为几类部分有序的 DM BC 建立容量区域，其特征在于公共和私有接收器之间的各自关联的成对关系集，每个由众所周知的成对关系描述有能力或噪音较小的条件。对于三类部分有序的 DM BC，容量区域被证明是通过两级叠加编码简单实现的，而逆的证明依赖于最近发现的信息不等式。然后，在将私人消息分成与公共接收器和间接解码一样多的部分后，通过多级叠加编码方案增强由两级叠加编码可实现的速率区域。尽管分裂率的数量不确定，但通过傅立叶-莫茨金消元的结构形式，对于给定的编码分布，获得了封闭形式的二维多面体（多边形）描述。通过依赖于 Csiszar 和引理和信息不等式的相反结果，与将私人消息分成两个子消息相对应的该区域的特殊化被证明是几类部分有序 DM BC 的容量区域，超出了那些两级叠加编码是容量最佳的区域，从而强调了好处率分裂。对于具有两个和三个接收器 DM BC 的两个嵌套多播消息的部分有序 DM BC 以及具有一个私有或一个公共接收器的 DM BC，所有先前已知的容量结果都包含在这项工作中获得的一般结果中。