In channel decoding, a decoder with suboptimal metrics may be used because of the uncertainty of the channel statistics or the limitations of the decoder. In this case, the decoding metric is different from the actual channel metric, and thus it is called mismatched decoding. In this paper, applying the technique of the DS2 bound, we derive an upper bound on the error probability of mismatched decoding over a regular channel for the ensemble of linear block codes, which was defined by Hof, Sason and Shamai. Assuming the ensemble of random linear block codes defined by Gallager, we show that the obtained bound is not looser than the conventional bound. We also give a numerical example for the ensemble of LDPC codes also introduced by Gallager, which shows that our proposed bound is tighter than the conventional bound. Furthermore, we obtain a single letter error exponent for linear block codes.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constant
E(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.