An Achievable Rate Region for Two Groupcast Messages over the K-User Broadcast Channel and Capacity Results for the Combination Network
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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.
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