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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Upper Bounds on the Error Probability for the Ensemble of Linear Block Codes with Mismatched Decoding

Toshihiro Niinomi, Hideki Yagi, Shigeichi Hirasawa

Published Oct 7, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.1587/transfun.2021tap0001

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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.
The ensemble of regular low-definition parity-check (LDPC) codes is considered. Using concentration results on the weight distribution, lower bounds on the error rate of a random code in the ensemble are derived. These bounds hold with some confidence level. Combining these results with known lower bounds on the error exponent, confidence intervals on the error exponent, under maximum-likelihood (ML) decoding, are obtained. Over a large range of channel parameter and transmission rate values, when the graph connectivity is sufficiently large, the upper bound of the interval approaches the lower bound, and the probability that the error exponent is within the interval can be arbitrarily close to one. In fact, in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents, with probability that approaches one.
In this paper, we derive an upper bound for the average block error probability of a standard irregular low-density parity-check (LDPC) code ensemble under the maximum-likelihood (ML) decoding. Moreover, we show that the upper bound asymptotically decreases polynomially with the code length. Furthermore, when we consider several regular LDPC code ensembles as special cases of standard irregular ones over an additive white Gaussian noise channel, we numerically show that the signal-to-noise ratio (SNR) thresholds at which the proposed bound converges to zero as the code length tends to infinity are smaller than those for a bound provided by Miller et al. We also give an example of a standard irregular LDPC code ensemble which has a lower SNR threshold than a given regular LDPC code ensemble.
Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>
We derive bounds on the error probability of ML decoded LDPC (low density parity check) codes, for any binary-input symmetric-output channel. For appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. The lower and upper bounds coincide asymptotically, indicating a polynomially decreasing ensemble averaged error probability. For ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing. Furthermore, the error exponent can be set arbitrarily close to the standard random coding exponent.
Recently, Hof et al. extended the type-2 Duman and Salehi (DS2) bound to generalized decoding, which was introduced by Forney, with decision criterion FR. From this bound, they derived two significant bounds. One is the Shulman-Feder bound for generalized decoding (GD) with the binary-input output-symmetric channel. The other is an upper bound for an ensemble of linear block codes, by applying the average complete weight distribution directly to the DS2 bound for GD. For the Shulman-Feder bound for GD, the authors derived a condition under which an upper bound is minimized at an intermediate step and show that this condition yields a new bound which is tighter than Hof et al.'s bound. In this paper, we first extend this result for non-binary linear block codes used over a class of symmetric channels called the regular channel. Next, we derive a new tighter bound for an ensemble of linear block codes, which is based on the average weight distribution.
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.