On the Achievable Rate Region of the $ K $-Receiver Broadcast Channels via Exhaustive Message Splitting

TL;DR

This paper introduces a comprehensive inner bound on the capacity region of K-receiver broadcast channels using exhaustive message splitting and a K-level Marton's coding, extending hierarchical covering lemmas to multiple users.

Contribution

It proposes a novel exhaustive message splitting scheme combined with a K-level modified Marton's coding, providing a new achievable rate region for K-receiver broadcast channels.

Findings

Derived a hierarchical covering lemma for K-level codebooks.

Established an achievable rate region that surpasses previous bounds for K ≥ 3.

Provided conditions for arbitrary input distributions and improved capacity bounds.

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 respectively. 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 $ \sum_{j=1}^K {K\choose j} $ submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages through codewords that are jointly typical with each other. 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 including $(2^{K}-1)$ random variables with more intricate dependence. As the number of auxiliary random variables and rate constraints both increase linearly with $(2^{K}-1)$, the standard Fourier-Motzkin elimination procedure becomes infeasible when $K$ is large. To tackle this problem, we obtain the final form of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of $ \{1,\dots,K\}$. The proposed achievable rate region allows arbitrary input probability mass functions (pmfs) and improves over all previously known ones for $ K$-receiver ($K\geq 3$) BCs whose input pmfs should satisfy certain Markov chain(s).