A Simple Converse of Burnashev's Reliability

TL;DR

This paper presents a simple converse proof for Burnashev's reliability function, highlighting the fundamental role of communication and confirmation phases in reliable variable-length coding with feedback.

Contribution

It offers a simpler, more intuitive converse proof that aligns with Yamamoto and Itoh's achievability scheme, clarifying the necessity of Burnashev's exponent terms.

Findings

The converse parallels Yamamoto-Itoh's scheme.

Communication and confirmation phases are essential in low-error schemes.

Provides intuitive understanding of Burnashev's exponent terms.

Abstract

In a remarkable paper published in 1976, Burnashev determined the reliability function of variable-length block codes over discrete memoryless channels with feedback. Subsequently, an alternative achievability proof was obtained by Yamamoto and Itoh via a particularly simple and instructive scheme. Their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct. We provide a converse that parallels the Yamamoto-Itoh achievability construction. Besides being simpler than the original, the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent. The proposed converse also makes it intuitively clear why the terms that appear in Burnashev's exponent are necessary.