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arXiv:cond-mat/0310177

The polynomial error probability for LDPC codes

Jort van Mourik, Yoshiyuki Kabashima

Published Oct 8, 2003• 4 citations
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Abstract

We obtain exact expressions for the asymptotic behaviour of the average probability of the block decoding error for ensembles of regular low density parity check error correcting codes, by employing diagrammatic techniques. Furthermore, we show how imposing simple constraints on the code ensemble (that can be practically implemented in linear time), allows one to suppress the error probability for codes with more than 2 checks per bit, to an arbitrarily low power of $N$. As such we provide a practical route to a (sub-optimal){\em expurgated} ensemble.

DOI
10.48550/arxiv.cond-mat/0310177

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We obtain exact expressions for the asymptotic behaviour of the average probability of the block decoding error for ensembles of regular low density parity check error correcting codes, by employing diagrammatic techniques. Furthermore, we show how imposing simple constraints on the code ensemble (that can be practically implemented in linear time), allows one to suppress the error probability for codes with more than 2 checks per bit, to an arbitrarily low power of NNN. As such we provide a practical route to a (sub-optimal){\em expurgated} ensemble.
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.
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.
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.
This thesis addresses the topic of Low-Density Parity-Check (LDPC) code analysis, both asymptotically and for finite block lengths. Since in general it is a difficult problem to analyze individual code instances, ensemble averages are studied by following Gallager's original idea and results of Luby et al. Often, one can relate the insights gained by studying ensemble averages to statements regarding individual codes by proving that most elements in the ensemble behave "close" to the average. One important such average is the average weight distribution, another is the average pseudo-codeword distribution, where pseudo-codewords play the same role under iterative decoding as codewords do for maximum likelihood decoding. One of the main contributions of this thesis is the calculation of such averages for fully irregular LDPC code ensembles. Much of classical coding theory is aimed at the construction of codes with large minimum distance. This is so since away from capacity accurate bounds on the performance of a code can be given in terms of its minimum distance (or more generally, in terms of its weight distribution). Therefore, it is of interest to investigate what role the minimum distance plays for iteratively decoded LDPC code ensembles. In particular, sequences of capacity-achieving LDPC code ensembles of increasing length when transmission takes place over the Binary Erasure Channel (BEC) are investigated. It is shown that, under certain technical conditions, the minimum distance of such ensembles grows sub-linearly in the block length, a result which is somewhat surprising from a classical point of view. Specific attention is also given to the design of LDPC code ensembles, where an inherent trade-off is observed between achieving large minimum distance (which is relevant for the so called error-floor regime) and achieving a large threshold (which in the limit of long block lengths determines the worst channel on which transmission can be accomplished reliably). Most results on iterative coding systems to date address their asymptotic performance, i.e., their performance when the block length tends to infinity. For small and moderate block lengths the behavior of a code can deviate significantly from its asymptotic limit. It is therefore of high practical value to be able to analyze the finite-length performance of LDPC code ensembles. In this thesis, an exact such analysis is presented for iteratively decoded LDPC code ensembles over the BEC. In particular, expressions for the exact average bit and block erasure probabilities are computed by solving a set of recursions. Such an analysis is the starting point for a finite-length optimization, a topic which is slated for future work. The methods used in this thesis include a combinatorial approach (familiar to the coding society) as well as the powerful techniques developed in the statistical physics community, for example, the replica method or the mean-field approximation. Although the statistical physics techniques are ideally suited for the analysis of iterative coding systems, they are to date only accessible to a relatively small community. Therefore, an overview of these techniques is first presented using a language familiar to the coding theory community. Next, in order to highlight the differences and common points between the combinatorial and the statistical physics approaches, both techniques are applied to the weight distribution problem. It turns out that for regular ensembles, both methods yield the same result, while for irregular ensembles, they do differ in general.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.
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.