Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Symposium on Turbo Codes and Iterative Information Processing

Convergence of weighted min-sum decoding via dynamic programming on coupled trees

Yung-Yih Jian, Henry D. Pfister

Published Sep 1, 2010• 8 citations
Publisher

Abstract

Applying the max-product (and belief-propagation) algorithms to loopy graphs is now quite popular for constraint satisfaction problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This paper presents an analysis of weighted min-sum (a.k.a. attenuated max-product) decoding for LDPC codes that guarantees convergence to a fixed point when the weight β is sufficiently small. It also shows that, if the fixed point satisfies all the constraints, then it must be both the linear-programming (LP) and maximumlikelihood (ML) solution. For (dv,dc)-regular LDPC codes, the weight must satisfy 1/β > dν- 1 whereas the result of Koetter and Frey requires instead that 1/β > (dv- 1)(dc- 1). A counterexample is also given that shows a fixed point might not be the ML solution if 1/βν- 1. Finally, connections are explored with recent work by Arora et al. on the threshold of LP decoding.

DOI
10.1109/istc.2010.5613901

Related papers

Convergence of Weighted Min-Sum Decoding Via Dynamic Programming on Trees

Yung-Yih Jian, Henry D. Pfister

Applying the max-product (and sum-product) algorithms to loopy graphs is now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence or optimality of converged solutions or both. This paper presents an analysis of both attenuated max-product decoding and weighted min-sum decoding for low-density parity-check (LDPC) codes, which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution. For (dv, dc)-regular LDPC codes, the weight factor must satisfy β(dv-1)v-1)(dc-1) ≤ 1. In addition, the range of the weight factor for a provable ML decoding solution is extended to 0v-1) 1. In addition, counterexamples that show a fixed point might not be the ML decoding solution if β(dv-1) > 1 are given. Finally, connections are explored with recent work on the threshold of LP decoding.

arXiv:1107.3177•IEEE Transactions on Information Theory•Nov 19, 2013•3 citations

ON THE ANALYSIS OF SPATIALLY-COUPLED GLDPC CODES AND THE WEIGHTED MIN-SUM ALGORITHM

Yung-Yih Jian

This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed.\n\nA new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD.\n\nThe optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution.

•Aug 7, 2013•5 citations

Iterative Approximate Linear Programming Decoding of LDPC Codes With Linear Complexity

David Burshtein

The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm, similar to min-sum and belief propagation, for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper, the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose. In particular, we are interested in obtaining a feasible vector in the LP decoding problem that is close to optimal in the following sense. The distance, normalized by the block length, between the minimum and the objective function value of this approximate solution can be made arbitrarily small. It is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length. Combined with previous results that have shown that the LP decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity. This is achieved by first applying the iterative LP decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits, and then correcting the remaining errors using some standard method. These conclusions are also extended to generalized LDPC codes.

IEEE Transactions on Information Theory•Oct 27, 2009•67 citations

Message-Passing Algorithms and Improved LP Decoding

Sanjeev Arora, Constantinos Daskalakis, David Steurer

Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-½ codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a “locally optimal” solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a “reweighted” version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.

IEEE Transactions on Information Theory•Jul 13, 2012•24 citations

Performance and Convergence Analysis of Tree-LDPC Codes on the Min-Sum Iterative Decoding Algorithm

K. NOH

In this paper, the performance of Tree-LDPC code 1 is presented based on the min-sum algorithm with scaling and the asymptotic performance in the water fall region is shown by density evolution. We presents that the Tree-LDPC code show a significant performance gain by scaling with the optimal scaling factor 3 which is obtained by density evolution methods. We also show that the performance of min-sum with scaling is as good as the performance of sum-product while the decoding complexity of min-sum algorithm is much lower than that of sum-product algorithm.

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Jun 1, 2006•0 citations

Performance and Convergence Analysis of Tree-LDPC codes on the Min-Sum Iterative Decoding Algorithm

Kwangseok Noh, Jun Heo, Kyuhyuk Chung

In this paper, the performance of Tree-LDPC code 1 is presented based on the min-sum algorithm with scaling and the asymptotic performance in the water fall region is shown by density evolution. We presents that the Tree-LDPC code show a significant performance gain by scaling with the optimal scaling factor 3 which is obtained by density evolution methods. We also show that the performance of min-sum with scaling is as good as the performance of sum-product while the decoding complexity of min-sum algorithm is much lower than that of sum-product algorithm.

한국통신학회논문지•Jul 1, 2005•0 citations
Applying the max-product (and belief-propagation) algorithms to loopy graphs is now quite popular for constraint satisfaction problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This paper presents an analysis of weighted min-sum (a.k.a. attenuated max-product) decoding for LDPC codes that guarantees convergence to a fixed point when the weight β is sufficiently small. It also shows that, if the fixed point satisfies all the constraints, then it must be both the linear-programming (LP) and maximumlikelihood (ML) solution. For (dv,dc)-regular LDPC codes, the weight must satisfy 1/β > dν- 1 whereas the result of Koetter and Frey requires instead that 1/β > (dv- 1)(dc- 1). A counterexample is also given that shows a fixed point might not be the ML solution if 1/βν- 1. Finally, connections are explored with recent work by Arora et al. on the threshold of LP decoding.
Applying the max-product (and sum-product) algorithms to loopy graphs is now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence or optimality of converged solutions or both. This paper presents an analysis of both attenuated max-product decoding and weighted min-sum decoding for low-density parity-check (LDPC) codes, which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution. For (dv, dc)-regular LDPC codes, the weight factor must satisfy β(dv-1)v-1)(dc-1) ≤ 1. In addition, the range of the weight factor for a provable ML decoding solution is extended to 0v-1) 1. In addition, counterexamples that show a fixed point might not be the ML decoding solution if β(dv-1) > 1 are given. Finally, connections are explored with recent work on the threshold of LP decoding.
This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed.\n\nA new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD.\n\nThe optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution.
The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm, similar to min-sum and belief propagation, for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper, the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose. In particular, we are interested in obtaining a feasible vector in the LP decoding problem that is close to optimal in the following sense. The distance, normalized by the block length, between the minimum and the objective function value of this approximate solution can be made arbitrarily small. It is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length. Combined with previous results that have shown that the LP decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity. This is achieved by first applying the iterative LP decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits, and then correcting the remaining errors using some standard method. These conclusions are also extended to generalized LDPC codes.
Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-½ codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a “locally optimal” solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a “reweighted” version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.
In this paper, the performance of Tree-LDPC code 1 is presented based on the min-sum algorithm with scaling and the asymptotic performance in the water fall region is shown by density evolution. We presents that the Tree-LDPC code show a significant performance gain by scaling with the optimal scaling factor 3 which is obtained by density evolution methods. We also show that the performance of min-sum with scaling is as good as the performance of sum-product while the decoding complexity of min-sum algorithm is much lower than that of sum-product algorithm.
In this paper, the performance of Tree-LDPC code 1 is presented based on the min-sum algorithm with scaling and the asymptotic performance in the water fall region is shown by density evolution. We presents that the Tree-LDPC code show a significant performance gain by scaling with the optimal scaling factor 3 which is obtained by density evolution methods. We also show that the performance of min-sum with scaling is as good as the performance of sum-product while the decoding complexity of min-sum algorithm is much lower than that of sum-product algorithm.