Mismatched Decoding: Error Exponents, Second-Order Rates and Saddlepoint Approximations

TL;DR

This paper introduces a new cost-constrained random-coding ensemble for channel coding with mismatched decoding, achieving optimal error exponents and second-order rates, and provides accurate saddlepoint approximations for non-asymptotic bounds.

Contribution

It proposes a novel ensemble with auxiliary costs that matches optimal error exponents and second-order rates, applicable to channels with infinite or continuous alphabets.

Findings

Achieves error exponents and second-order rates matching constant-composition coding.

Shows at most two auxiliary costs are needed for optimal performance.

Provides highly accurate saddlepoint approximations for finite block lengths.

Abstract

This paper considers the problem of channel coding with a given (possibly suboptimal) maximum-metric decoding rule. A cost-constrained random-coding ensemble with multiple auxiliary costs is introduced, and is shown to achieve error exponents and second-order coding rates matching those of constant-composition random coding, while being directly applicable to channels with infinite or continuous alphabets. The number of auxiliary costs required to match the error exponents and second-order rates of constant-composition coding is studied, and is shown to be at most two. For i.i.d. random coding, asymptotic estimates of two well-known non-asymptotic bounds are given using saddlepoint approximations. Each expression is shown to characterize the asymptotic behavior of the corresponding random-coding bound at both fixed and varying rates, thus unifying the regimes characterized by error exponents, second-order rates and moderate deviations. For fixed rates, novel exact asymptotics expressions are obtained to within a multiplicative 1+o(1) term. Using numerical examples, it is shown that the saddlepoint approximations are highly accurate even at short block lengths.