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IEEE Transactions on Information Theory

On two probabilistic decoding algorithms for binary linear codes

Miodrag Živković

Published Jan 1, 1991• 14 citations
Publisher

Abstract

A generalization of the Sullivan inequality (1967) on the ratio of the probability of a linear code to that of any of its cosets is proved. Starting from this inequality, a sufficient condition for successful decoding of linear codes by a probabilistic method is derived. A probabilistic decoding algorithm for low-density parity-check codes is also analyzed. This method of analysis enables one to estimate experimentally the probability of successful decoding for any given linear code and for any chosen family of parity-check sets. Using the example of a linear (512,100) code, the author illustrates the dependence of the successful probability on the channel noise level.>

DOI
10.1109/18.104339

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IEEE Transactions on Information Theory•Jan 1, 2001•3,067 citations
A generalization of the Sullivan inequality (1967) on the ratio of the probability of a linear code to that of any of its cosets is proved. Starting from this inequality, a sufficient condition for successful decoding of linear codes by a probabilistic method is derived. A probabilistic decoding algorithm for low-density parity-check codes is also analyzed. This method of analysis enables one to estimate experimentally the probability of successful decoding for any given linear code and for any chosen family of parity-check sets. Using the example of a linear (512,100) code, the author illustrates the dependence of the successful probability on the channel noise level.>
A generalization of Sullivan inequality on the ratio of the probabi-lity of a linear code to that of any of its cosets is proved. Starting from this inequality, a sufficient condition for successful decoding of linear codes by a probabilistic method is derived. A probabilistic decoding algorithm for “low–density parity–check codes ” is also analyzed. The results obtained allow one to estimate experimentally the probability of successful decoding using these probabilistic algorithms. Index Terms — linear code, inequality, probabilistic decoding, low– density parity–check code
A novel analytical approach to performance evaluation of soft-decoding algorithms for binary linear block codes based on probabilistic iterative error correction is presented. A convergence condition establishing the critical noise rate below which the expected bit-error probability tends to zero is theoretically derived. It explains the capability of iterative probabilistic decoding of binary linear block codes with sparse parity-check matrices to correct, with probability close to one, error patterns with the number of errors (far) beyond half the code minimum distance. Systematic experiments conducted on truncated simplex codes seem to agree well with the convergence condition. The method may also be interesting for the theoretical analysis of the so-called turbo codes.
We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of “probabilistic expansion ” in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.
We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of ldquoprobabilistic expansionrdquo in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.
Low-density parity-check (LDPC) codes are linear block codes defined by sparse parity-check matrices. The codes exhibit excellent performance under iterative decoding, and the weight distribution is used to analyze lower error probability of their decoding performance. In this paper, we propose a probabilistic method for computing the weight distribution of LDPC codes. The proposed method efficiently finds low-weight codewords in a given LDPC code by using Stern's algorithm, and stochastically computes the low part of the weight distribution from the frequency of the found codewords. It is based on a relation between the number of codewords with a given weight and the rate of generating the codewords in Stern's algorithm. In the numerical results for LDPC codes of length 504, 1008 and 4896, we could compute the weight distribution by the proposed method with greater accuracy than by conventional methods.
We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined.