Common-Message Broadcast Channels with Feedback in the Nonasymptotic Regime: Stop Feedback

TL;DR

This paper analyzes the maximum coding rate for common-message broadcast channels with feedback, revealing that zero dispersion is not always achievable and providing bounds that improve understanding of convergence rates in the nonasymptotic regime.

Contribution

It extends nonasymptotic bounds to multi-user broadcast channels with feedback, generalizing previous two-user results and identifying conditions where zero dispersion is unattainable.

Findings

Zero dispersion cannot be achieved for certain broadcast channels.

Second-order bounds closely approximate the maximum coding rate.

Convergence to capacity is slower than in point-to-point channels.

Abstract

We investigate the maximum coding rate for a given average blocklength and error probability over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message K-user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.