Relations between random coding exponents and the statistical physics of random codes

The partition function pertaining to finite--temperature decoding of a (typical) randomly chosen code is known to have three types of behavior, corresponding to three phases in the plane of rate vs. temperature: the {\it ferromagnetic phase}, corresponding to correct decoding, the {\it paramagnetic phase}, of complete disorder, which is dominated by exponentially many incorrect codewords, and the {\it glassy phase} (or the condensed phase), where the system is frozen at minimum energy and dominated by subexponentially many incorrect codewords. We show that the statistical physics associated with the two latter phases are intimately related to random coding exponents. In particular, the exponent associated with the probability of correct decoding at rates above capacity is directly related to the free energy in the glassy phase, and the exponent associated with probability of error (the error exponent) at rates below capacity, is strongly related to the free energy in the paramagnetic phase. In fact, we derive alternative expressions of these exponents in terms of the corresponding free energies, and make an attempt to obtain some insights from these expressions. Finally, as a side result, we also compare the phase diagram associated with a simple finite-temperature universal decoder for discrete memoryless channels, to that of the finite--temperature decoder that is aware of the channel statistics.Comment: 26 pages, 2 figures, submitted to IEEE Transactions on Information Theor

Bit-Interleaved Coded Modulation Revisited: A Mismatched Decoding Perspective

We revisit the information-theoretic analysis of bit-interleaved coded modulation (BICM) by modeling the BICM decoder as a mismatched decoder. The mismatched decoding model is well-defined for finite, yet arbitrary, block lengths, and naturally captures the channel memory among the bits belonging to the same symbol. We give two independent proofs of the achievability of the BICM capacity calculated by Caire et al. where BICM was modeled as a set of independent parallel binary-input channels whose output is the bitwise log-likelihood ratio. Our first achievability proof uses typical sequences, and shows that due to the random coding construction, the interleaver is not required. The second proof is based on the random coding error exponents with mismatched decoding, where the largest achievable rate is the generalized mutual information. We show that the generalized mutual information of the mismatched decoder coincides with the infinite-interleaver BICM capacity. We also show that the error exponent -and hence the cutoff rate- of the BICM mismatched decoder is upper bounded by that of coded modulation and may thus be lower than in the infinite-interleaved model. We also consider the mutual information appearing in the analysis of iterative decoding of BICM with EXIT charts. We show that the corresponding symbol metric has knowledge of the transmitted symbol and the EXIT mutual information admits a representation as a pseudo-generalized mutual information, which is in general not achievable. A different symbol decoding metric, for which the extrinsic side information refers to the hypothesized symbol, induces a generalized mutual information lower than the coded modulation capacity.Comment: submitted to the IEEE Transactions on Information Theory. Conference version in 2008 IEEE International Symposium on Information Theory, Toronto, Canada, July 200