Multi-letter converse bounds for the mismatched discrete memoryless channel with an additive metric

The problem of mismatched decoding with an additive metric q for a discrete memoryelss channel W is addressed. Two max-min multi-letter upper bounds on the mismatch capacity C_q(W) are derived. We further prove that if the average probability of error of a sequence of codebooks converges to zero sufficiently fast, then the rate of the code-sequence is upper bounded by the “product-space” improvement of the random coding lower bound on the mismatched capacity, C^(∞)_q (W), introduced by Csiszár and Narayan. In particular, if q is a bounded rational metric, and the average probability of error converges to zero faster than O(1/n), then R ≤ C^(∞)_q (W). Consequently, in this case if a sequence of codes of rate R is known to achieve average probability of error which is o(1/n), then there exists a sequence of codes operating at a rate arbitrarily close to R with average probability of error which vanishes exponentially fast. We conclude by presenting a general expression for the mismatch capacity of a general channel with a general type-dependent decoding metric.