On the capacity region of the K-user discrete memoryless broadcast channel with two degraded messages

An Achievable Rate Region for the K-Receiver Two Nested Groupcast DM Broadcast Channel and a Capacity Result for the Combination Network

“…Theorem 2 can be seen as a generalization of previously proposed achievable schemes in the context of particular examples. These include (a) the two-receiver DM-BC with degraded messages in which superposition coding alone (without rate-splitting) is sufficient to achieve the capacity region as was shown in the important work of Korner and Marton in [9] (b) the three-receiver DM-BC with two nested messages with one and two common receivers which was investigated in-depth in [11] and (c) the DM-BC with two nested messages for an arbitrary number of receivers and with an arbitrary number of common receivers studied by the authors in [12]. In particular, the achievable rate regions based on rate-splitting and superposition coding obtained in the aforementioned papers can be obtained by specific choices of F. Those rate regions were shown in those works to be the capacity regions for certain classes of channels.…”

“…By choosing F = {1, 12, 123, 1234} and Q = const, we get from Theorem 2 that the set of rates satisfying In fact, if we choose F = {1, 12, 123, 1234, · · · , 1234 · · · K}, we recover the result in [12, Thoerem 2] for any number of receivers K with any number of common and private receivers. Hence, the result in [12, Thoerem 2] is a special case of the result in Theorem 2 obtained by a specific choice of F but unlike in Theorem 2 it has a polyhedral description in terms of the two message rates [12].…”

“…Hence, the set of indices of private receivers that demand both messages is S p = {1, 2, · · · , P } and the set of indices of common receivers that demand only the common or multicast message M S 2 is S l ={P + 1, · · · , P + L}. When specialized to such a two nested message set, the inner bound of Theorem 1 gives a rate region that includes the one proposed by the authors previously for the two nested messages problem in [12] because the former is based on a more general coding scheme. We will show later that this extra generality is needed in order to obtain a succinct characterization of the capacity of the general K-receiver combination network with two nested messages and one or two common receivers.…”

“…Also, this inner bound is shown to be capacity-optimal for classes of channels characterized by certain pair-wise relationships between and among the common and private receivers. For example, the scheme is optimal for the class of four-receiver DM BCs with L=3 and P =1 in which the private receiver is less noisy than two of the three common receivers.Inspired by the general order-theoretic framework of Romero and Varanasi on rate-splitting and superposition coding for the K-receiver DM BC with general message sets in [13], we propose a new inner bound for the DM BC with two groupcast messages that, when specialized to the case of nested messages, relates to our previous results in [12] in the following ways (a) it incorporates more general message splitting, wherein the private message is now split into 2 L parts, instead of L parts, hence (b) subsuming the inner bound of [12], and thereby being a fortiori capacity-optimal for all classes of DM BCs for which capacity was characterized in our work in [12] and (c) testing its efficacy beyond those classes, by the criterion of whether or not, when specialized to the so-called combination network (cf. [14], [15]), it yields its capacity region.…”

“…In particular, each receiver decodes only the desired message uniquely and any other sub-messages assigned to the receiver due to message splitting, non-uniquely. The general rate-splitting strategy that we use here comes at the expense of the ability to eliminate the split rates for arbitrary L and K, in contrast to our work in [12]. The description of the achievable region uses the order-theoretic framework developed for the multiple-access channel with general message sets by Romero and Varanasi in [16] and its conference versions, which the same authors later applied to the DM BC in [13].…”