A Neyman - Pearson approach to universal erasure and list decoding

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 Csiszár and Körner's maximum mutual information (MMI) decoder with an 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 Csiszár-Körner rule and of the simpler empirical-mutual-information thresholding rule are identified. The error exponents are evaluated numerically for the binary symmetric channel.