A Neyman-Pearson Approach to Universal Erasure and List Decoding

TL;DR

This paper introduces a Neyman-Pearson based decoding strategy with erasure and list options for unknown channels, optimizing error and erasure tradeoffs, and providing explicit solutions and numerical evaluations.

Contribution

It generalizes existing decoding methods by incorporating a parameterized weighting function for universal erasure and list decoding over unknown channels.

Findings

Explicit expressions for optimal error exponents are derived.

The proposed decoder matches classical results for symmetric channels at low erasure exponents.

Numerical evaluations demonstrate the decoder's performance on binary symmetric channels.

Abstract

When information is to be transmitted over an unknown, possibly unreliable channel, an erasure option at the decoder is desirable. Using constant-composition random codes, we propose a generalization of Csiszar and Korner's Maximum Mutual Information decoder with erasure option for discrete memoryless channels. The new decoder is parameterized by a weighting function that is designed to optimize the fundamental tradeoff between undetected-error and erasure exponents for a compound class of channels. The class of weighting functions may be further enlarged to optimize a similar tradeoff for list decoders -- in that case, undetected-error probability is replaced with average number of incorrect messages in the list. Explicit solutions are identified. The optimal exponents admit simple expressions in terms of the sphere-packing exponent, at all rates below capacity. For small erasure exponents, these expressions coincide with those derived by Forney (1968) for symmetric channels, using Maximum a Posteriori decoding. Thus for those channels at least, ignorance of the channel law is inconsequential. Conditions for optimality of the Csiszar-Korner rule and of the simpler empirical-mutual-information thresholding rule are identified. The error exponents are evaluated numerically for the binary symmetric channel.