Asymmetric Evaluations of Erasure and Undetected Error Probabilities

TL;DR

This paper analyzes the tradeoff between erasure and undetected error probabilities in channel coding for discrete memoryless channels, providing asymptotic decay rates and tight ensemble performance analysis using information spectrum methods.

Contribution

It introduces a detailed asymptotic analysis of error tradeoffs in erasure coding, including regimes with different decay behaviors, using advanced probabilistic techniques.

Findings

Error probabilities decay subexponentially in the moderate deviations regime.

Total error probability approaches a positive constant at the capacity rate.

Undetected error probability decays exponentially as the blocklength increases.

Abstract

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths $n$ is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analysing the ensemble performance in terms of a tradeoff between the code rate, the undetected and the total errors. This tradeoff is parametrized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than $n^{-1/2}$ corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of $n^{-1/2}$. In this case, the total error probability is asymptotically a positive constant while the undetected error probability decays as $\exp(- b n^{ 1/2})$ for some $b>0$. The proof techniques involve applications of a modified (or "shifted") version of the G\"artner-Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.