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Broadcast Channel Synthesis from Shared Randomness

Malhar A. Managoli, Vinod M. Prabhakaran

TL;DR

This work addresses the problem of synthesizing a two-user broadcast channel using a common message when the input terminal shares independent randomness with each decoder. It provides an OSRB-based inner bound on the achievable (communication, shared randomness) region and a lower bound on the minimum communication rate $R_{opt}$; both bounds are tight in several special cases. The results recover and extend known bounds for point-to-point and no-input reductions, and also yield a new, tight characterization for the Y=Z case, with an explicit binary erasure broadcast-channel example illustrating the gains from distributed randomness. The approach combines the Output Statistics of Random Binning framework with XOR-based randomness harvesting and Slepian-Wolf decoding to establish tractable, information-theoretic guarantees for channel synthesis under distributed randomness constraints.

Abstract

We study the problem of synthesising a two-user broadcast channel using a common message, where each output terminal shares an independent source of randomness with the input terminal. This generalises two problems studied in the literature (Cuff, IEEE Trans. Inform. Theory, 2013; Kurri et.al., IEEE Trans. Inform. Theory, 2021). We give an inner bound on the tradeoff region between the rates of communication and shared randomness, and a lower bound on the minimum communication rate. Although the bounds presented here are not tight in general, they are tight for some special cases, including the aforementioned problems.

Broadcast Channel Synthesis from Shared Randomness

TL;DR

This work addresses the problem of synthesizing a two-user broadcast channel using a common message when the input terminal shares independent randomness with each decoder. It provides an OSRB-based inner bound on the achievable (communication, shared randomness) region and a lower bound on the minimum communication rate ; both bounds are tight in several special cases. The results recover and extend known bounds for point-to-point and no-input reductions, and also yield a new, tight characterization for the Y=Z case, with an explicit binary erasure broadcast-channel example illustrating the gains from distributed randomness. The approach combines the Output Statistics of Random Binning framework with XOR-based randomness harvesting and Slepian-Wolf decoding to establish tractable, information-theoretic guarantees for channel synthesis under distributed randomness constraints.

Abstract

We study the problem of synthesising a two-user broadcast channel using a common message, where each output terminal shares an independent source of randomness with the input terminal. This generalises two problems studied in the literature (Cuff, IEEE Trans. Inform. Theory, 2013; Kurri et.al., IEEE Trans. Inform. Theory, 2021). We give an inner bound on the tradeoff region between the rates of communication and shared randomness, and a lower bound on the minimum communication rate. Although the bounds presented here are not tight in general, they are tight for some special cases, including the aforementioned problems.
Paper Structure (22 sections, 10 theorems, 76 equations, 2 figures)

This paper contains 22 sections, 10 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and Related Works
  3. Notation
  4. Problem Statement
  5. Results
  6. Inner Bound
  7. Lower Bound
  8. Special Cases and An Example
  9. Special Cases
  10. Point-to-Point Channel
  11. No Input
  12. $Y=Z$
  13. An Example: Binary Erasure Broadcast Channel
  14. Proofs
  15. Proof Sketch of Inner Bound
  16. ...and 7 more sections

Key Result

Theorem 1

Define the rate region where Then, $\mathcal{S}\subseteq\mathcal{R}$.

Figures (2)

  • Figure 1: Synthesis of a broadcast channel. $P_1$ observes $X^n\sim q_X$ i.i.d.. $P_2$ and $P_3$ must output $Y^n$ and $Z^n$ respectively, such that the joint distribution is close to $q_X\,q_{Y,Z|X}$ i.i.d.. $P_1$ can send an $nR$ bits long common message to $P_2$ and $P_3$. $P_1$ and $P_2$ have access to $nR_1$ uniformly distributed random bits $\Theta_1$. Similarly, $P_1$ and $P_3$ have access to $nR_2$ uniformly distributed random bits $\Theta_2$. Here $X^n,\Theta_1,\Theta_2$ are independent.
  • Figure :

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 2
  • Lemma 1
  • Lemma 2: Cardinality Bound
  • Lemma 3