## Abstract

The problem of mismatched decoding with an additive bounded metric q for a discrete memoryless channel W is addressed. We study two kinds of decoders. The δ-margin mismatched decoder outputs a message whose metric with the channel output exceeds that of all the other codewords by at least δ. The τ-threshold decoder outputs a single message whose metric with the channel output exceeds a threshold τ. It is proved that the mismatch capacity with a constant margin decoder is equal to the "product-space" improvement of the random coding lower bound on the mismatch capacity, C_{q} ^{(∞)}(W), which was introduced by Csiszár and Narayan. We next consider sequences of P -constant composition codebooks. Using the Central Limit Theorem, it is shown that for such sequences of codebooks the supremum of achievable rates with constant threshold decoding is upper bounded by the supremum of the achievable rates with a constant margin decoder, and therefore also by C_{q} ^{(∞)}(W). Further, a soft converse is proved stating that if the average probability of error of a sequence of codebooks with ordinary mismatched decoding converges to zero sufficiently fast, the rate of the code sequence is upper bounded by C_{q} ^{(∞)}(W). In particular, if q is a bounded rational metric, and the average probability of error converges to zero faster than O(n^{-1}), then R ≤ C_{q} ^{(∞)}(W). Finally, a max-min multi-letter upper bound on the mismatch capacity that bears some resemblance to C_{q} ^{(∞)}(W) is presented.

Original language | English |
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Article number | 8365148 |

Pages (from-to) | 6196-6207 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 64 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2018 |

### Bibliographical note

Publisher Copyright:© 2018 IEEE.

### Funding

Manuscript received September 30, 2015; revised November 1, 2016, June 8, 2017, and December 22, 2017; accepted February 16, 2018. Date of publication May 24, 2018; date of current version August 16, 2018. This work was supported by the Israel Science Foundation under Grant 2013/919. This paper was presented in part at the 2015 IEEE International Symposium on Information Theory and at the 2016 International Zurich Seminar on Communications.

Funders | Funder number |
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Israel Science Foundation | 2013/919 |

## Keywords

- Channel coding
- mismatch capacity
- mismatched decoding
- threshold capacity