Third-order Analysis of Channel Coding in the Small-to-Moderate Deviations Regime

TL;DR

This paper introduces the concept of channel skewness to improve third-order approximations of channel coding limits in small-to-moderate deviations regimes, providing more accurate bounds for practical blocklengths.

Contribution

It generalizes existing asymptotic bounds to include the moderate deviations regime and introduces the channel skewness as a key factor in third-order analysis.

Findings

The channel skewness significantly improves approximation accuracy in the MD regime.

Exact computation of channel skewness for Gaussian channels using Shannon's bounds.

Enhanced bounds for binary symmetric channels in practical blocklength and error probability ranges.

Abstract

This paper studies the third-order characteristic of nonsingular discrete memoryless channels and the Gaussian channel with a maximal-power constraint. The third-order term in our expansions employs a new quantity here called the channel skewness, which affects the approximation accuracy more significantly as the error probability decreases. For the Gaussian channel, evaluating Shannon's 1959 random coding bound and Vazquez-Vilar's 2021 meta-converse bound in the central limit theorem (CLT) regime enables exact computation of the channel skewness. For discrete memoryless channels, this work generalizes Moulin's 2017 bounds on the asymptotic expansion of the maximum achievable message set size for nonsingular channels from the CLT regime to include the moderate deviations (MD) regime, thereby refining Altu\u{g} and Wagner's 2014 MD result. For an example binary symmetric channel and most practically important $(n, \epsilon)$ pairs, including $n \in [100, 500]$ and $\epsilon \in [10^{-10}, 10^{-1}]$, an approximation up to the channel skewness is the most accurate among several expansions in the literature. A derivation of the third-order term in the type-II error exponent of binary hypothesis testing in the MD regime is also included; the resulting third-order term is similar to the channel skewness.