Using an order-theoretic framework, a novel achievable rate region is obtained for the general K-receiver discrete memoryless broadcast channel over which two groupcast messages are to be transmitted, with each message required by an arbitrary group of receivers. The associated achievability scheme is an amalgamation of random coding techniques with novel features including up-set message-splitting, message set expansion including the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding with subset inclusion order, partial interference decoding at all receivers in general, joint unique decoding at receivers that desire both messages, and non-unique or indirect decoding at receivers that desire only one of the two messages. While the generality of such a scheme implies that its rate region coincides with all previously found capacity regions for special classes of broadcast channels with two private or two nested groupcast messages, wherein the group of receivers desiring one message is contained in that desiring the other, we show that, when specialized to the so-called combination network, our inner bound coincides with the capacity region for three different scenarios, namely, (a) the two messages are intended for two distinct sets of K−1 receivers each and (b) two nested messages in which one message is intended for one or (c) two (common) receivers and both messages are intended for all other (private) receivers. Moreover, we show that there is a trade-off between the complexity of the coding scheme and that of the distribution of the auxiliary random variables and the encoding function that must be chosen to achieve the capacity region in these scenarios.In the K-receiver BC with two nested messages the receivers can be classified into L common receivers that require only one (common) message and P private receivers that require both messages (with P +L = K). The result of Korner and Marton in [9] might suggest that the nested structure of the messages might render a straightforward extension of their superposition coding scheme to be capacity-optimal even in this K-receiver setting. However, the authors of [10] and [11] showed that superposition coding alone is not optimal for the three-receiver DM BC with one and two common receivers, respectively. In the latter case, they proposed a more general scheme that involves a simple form of rate-splitting along with superposition coding [11]. However, even this scheme was only shown to achieve capacity for the restricted class of DM BCs wherein the private receiver is less noisy than one of the two common receivers. One of the challenges of obtaining capacity results for rate-splitting based schemes beyond the threereceiver case is the difficulty of obtaining a closed-form polyhedral description for the inner bound in terms of the message rates due to the large number of split rates possible. We make progress on this problem in [12] where an achievable rate region that generalizes in one directi...
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