On exact asymptotics of the error probability in channel coding: symmetric channels

TL;DR

This paper precisely characterizes the sub-exponential decay factors in error probabilities for symmetric channels, revealing a dichotomy and advancing the understanding of normal approximation terms.

Contribution

It determines the exact order of the sub-exponential factor in error bounds for symmetric channels, with and without feedback, and establishes the third-order term in normal approximation.

Findings

Sub-exponential factor exhibits a dichotomy in symmetric channels.

Exact order of the sub-exponential decay factor is identified.

Third-order term in normal approximation is derived.

Abstract

The exact order of the optimal sub-exponentially decaying factor in the classical bounds on the error probability of fixed-length codes over a Gallager-symmetric discrete memoryless channel with and without ideal feedback is determined. Regardless of the availability of feedback, it is shown that the order of the optimal sub-exponential factor exhibits a dichotomy. Moreover, the proof technique is used to establish the third-order term in the normal approximation for symmetric channels, where a similar dichotomy is shown to exist.