Typical Error Exponents: A Dual Domain Derivation

TL;DR

This paper introduces a dual domain approach to analyze typical error exponents, demonstrating that the probability of error exponents falling below a known lower bound diminishes with increasing code length across various ensembles and channels.

Contribution

It provides a unified derivation of typical error exponents valid for diverse ensembles and channels, extending previous results to arbitrary alphabets and channels with memory.

Findings

Error probability tends to zero for large code lengths.

Results recover known bounds for specific ensembles and channels.

Applicable to channels with memory and arbitrary pairwise-independent ensembles.

Abstract

This paper shows that the probability that the error exponent of a given code randomly generated from a pairwise independent ensemble being smaller than a lower bound on the typical random-coding exponent tends to zero as the codeword length tends to infinity. This lower bound is known to be tight for i.i.d. ensembles over the binary symmetric channel and for constant-composition codes over memoryless channels. Our results recover both as special cases and remain valid for arbitrary alphabets, arbitrary channels -- for example finite-state channels with memory -- and arbitrary pairwise-independent ensembles. We specialize our results to the i.i.d., constant-composition and cost-constrained ensembles over discrete memoryless channels and to ensembles over finite-state channels.