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On Two Probabilistic Decoding Algorithms for Binary Linear Codes

Miodrag Živković

Published Jan 1, 2005• 2 citations
OpenAlex

Abstract

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

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On two probabilistic decoding algorithms for binary linear codes

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Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes

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Abstract—Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities ” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut, ” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding. Index Terms—Iterative decoding, linear codes, linear programming (LP) decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, pseudocodewords. I.

•Jan 1, 2012•58 citations
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 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.>
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.
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.
Low-density parity-check (LDPC) codes achieve outstanding performance by the sum-product decoder. Its performance approaches the maximum-likelihood (ML) decoding performance at high SNR region. Upper bounds on the error probability of the ML decoded LDPC codes can be computed using the weight distribution. Recently, X.-Y. Hu et al. presented an approximate algorithm for estimating the weight distribution of LDPC codes. However, the weight distribution computed by their method depends on the iterative decoding algorithm, and is not always estimated exactly. In this paper, we propose a probabilistic method for computing the low-part weight distribution of LDPC codes. Using our method, the approximate weight distribution of LDPC codes can be computed with high accuracy
Abstract—Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities ” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut, ” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding. Index Terms—Iterative decoding, linear codes, linear programming (LP) decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, pseudocodewords. I.