On the asymptotic accuracy of the union bound

TL;DR

This paper explores the relationship between the union bound and the asymptotic error probability of maximum likelihood decoding on the binary symmetric channel, revealing new insights into code rate regions where reliability is precisely characterized.

Contribution

It establishes a connection between a new lower bound on error probability and the union bound, clarifying the conditions for asymptotic tightness of the random coding exponent.

Findings

Identifies a new region where the random coding exponent is tight

Provides a lower bound on error probability for ML decoding

Clarifies the relation between the union bound and error exponents

Abstract

A new lower bound on the error probability of maximum likelihood decoding of a binary code on a binary symmetric channel was proved in Barg and McGregor (2004, cs.IT/0407011). It was observed in that paper that this bound leads to a new region of code rates in which the random coding exponent is asymptotically tight, giving a new region in which the reliability of the BSC is known exactly. The present paper explains the relation of these results to the union bound on the error probability.