Mismatched Decoding: Finite-Length Bounds, Error Exponents and Approximations

This paper considers the problem of channel coding with a given (possibly suboptimal) decoding rule. Finite-length upper and lower bounds on the random-coding error probability for a general codeword distribution are given. These bounds are applied to three random-coding ensembles: i.i.d., constant-composition, and cost-constrained. Ensembletight error exponents are presented for each ensemble, and achievable second-order coding rates are given. Connections are drawn between the ensembles under both maximum likelihood decoding and mismatched decoding. In particular, it is shown that the error exponents and second-order rates of the constant-composition ensemble can be achieved using cost-constrained coding with at most two cost functions. Finally, saddlepoint approximations of the randomcoding bounds are given. These are demonstrated to be more accurate than the approximations obtained from the error exponents and second-order coding rates, while having a similar computational complexity.