Comprehensive Analysis on Exact Asymptotics of Random Coding Error Probability

TL;DR

This paper derives the exact asymptotics of random coding error probabilities for general memoryless channels, extending previous results limited to special cases and revealing complex expressions for diverse channel types.

Contribution

It provides the first comprehensive derivation of exact asymptotics for all memoryless channels, including previously unsolved cases beyond symmetric and nonlattice channels.

Findings

Derived exact asymptotics for general memoryless channels.

Unified analysis covering symmetric and nonlattice channels.

Complex expressions for error probability asymptotics in general channels.

Abstract

This paper considers error probabilities of random codes for memoryless channels in the fixed-rate regime. Random coding is a fundamental scheme to achieve the channel capacity and many studies have been conducted for the asymptotics of the decoding error probability. Gallager derived the exact asymptotics (that is, a bound with asymptotically vanishing relative error) of the error probability below the critical rate. On the other hand, exact asymptotics for fixed rate above the critical rate has been unknown except for symmetric channels (in the strong sense) and strongly nonlattice channels. This paper derives the exact asymptotics for general memoryless channels covering all previously unsolved cases. The analysis reveals that strongly symmetric channels and strongly nonlattice channels correspond to two extreme cases and the expression of the asymptotics is much complicated for general channels.