Abstract
This paper focuses on K-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey K private messages to K receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a K-level modified Marton’s coding. The key idea is to split every message into submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a K-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks’ sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the K-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with K, the standard Fourier–Motzkin elimination procedure becomes infeasible when K is large. To tackle this problem, we obtain a closed form of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of . The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for K-receiver () BCs.
Highlights
The 2-receiver discrete-time memoryless broadcast channel (DM-BCs) was first introduced by Cover [1], who proposed the prestigious superposition coding that outperforms the traditional time-division strategy
To enlarge the achievable rate region, the submitted codewords are jointly typical which is guaranteed by a sufficient condition on the sizes of subcodebooks established by the covering lemma [2]
The main difference between recursive mutual covering lemma [27] is that our hierarchical covering lemma reduces the number of variables in almost each inequalities compared with those in [27], which makes it easier for Fourier-Motzkin elimination to derive a closed-form achievable rate region
Summary
The 2-receiver discrete-time memoryless broadcast channel (DM-BCs) was first introduced by Cover [1], who proposed the prestigious superposition coding that outperforms the traditional time-division strategy. The best known inner bound on the capacity region of DM-BCs is achieved by Marton’s coding with message splitting [4,5]. For the K-receiver (K ≥ 2) DM-BCs, most previous works are based either on superposition coding (see in [19,20,21]) which requires certain constraints on Markov chains for auxiliary random variables (RVs), or on the 2-level Marton’s coding where each message is split into one common part and one private part [2,22]. We enlarge the subcodebooks sizes and use a K-level Marton’s coding to send all codewords that are jointly typical with each other This allows arbitrary dependence among the input RVs, rather than satisfying certain Markov chains as in [19,20,21,22].
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