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

IEEE Journal on Selected Areas in Communications

Nonlinear programming approaches to decoding low-density parity-check codes

Kai Yang, J. Feldman, Xiaodong Wang

Published Aug 1, 2006• 54 citations
Publisher

Abstract

We consider the decoding problem for low-density parity-check codes, and apply nonlinear programming methods. This extends previous work using linear programming (LP) to decode linear block codes. First, a multistage LP decoder based on the branch-and-bound method is proposed. This decoder makes use of the maximum-likelihood-certificate property of the LP decoder to refine the results when an error is reported. Second, we transform the original LP decoding formulation into a box-constrained quadratic programming form. Efficient linear-time parallel and serial decoding algorithms are proposed and their convergence properties are investigated. Extensive simulation studies are performed to assess the performance of the proposed decoders. It is seen that the proposed multistage LP decoder outperforms the conventional sum-product (SP) decoder considerably for low-density parity-check (LDPC) codes with short to medium block length. The proposed box-constrained quadratic programming decoder has less complexity than the SP decoder and yields much better performance for LDPC codes with regular structure

DOI
10.1109/jsac.2006.879405

Related papers

Multistep linear programming approaches for decoding low-density parity-check codes

Haiyang Liu, Lianrong Ma, Jie Chen

The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LOpe codes suggest that the M LP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.

Tsinghua Science & Technology•Oct 1, 2009•2 citations

Low Complexity Linear Programming Decoding of Nonbinary Linear Codes

Mayur Punekar, Mark F. Flanagan

Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has also been extended to nonbinary linear codes by Flanagan et al. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation proposed by Flanagan et al. as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. In contrast to the binary low-complexity LP decoding algorithm, our proposed algorithm is not directly related to the nonbinary SP algorithm. Nevertheless, the complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a $[80,48]$ LDPC code defined over $\mathbb{Z}_4$ using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.

arXiv:1007.1368•Jul 8, 2010•0 citations

Low complexity Linear Programming decoding of nonbinary linear codes

Mayur Punekar, Mark F. Flanagan

Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has been derived for both binary and nonbinary decoding frameworks. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation for the nonbinary codes as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. The complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80, 48] LDPC code defined over ℤ4using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.

Allerton Conference on Communication, Control, and Computing•Sep 1, 2010•8 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

Iterative Linear Programming Decoding of Nonbinary LDPC Codes With Linear Complexity

Dina Goldin, David Burshtein

The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.

IEEE Transactions on Information Theory•Aug 6, 2012•17 citations

Modified multistage linear programming decoding of LDPC codes

Xian Zhang, Piming Ma

Linear programming (LP) decoding is getting more and more attention because of the maximum-likelihood (ML) certificate property. As one of the improved LP decoding methods, multistage LP (MLP) decoding has been proposed to improve the performance of LP decoding but at the cost of higher complexity. In this paper, the problem of achieving a balance between the performance and the complexity of LP decoding is considered. First, a criterion used for searching new constraints adaptively is proposed. Then a modified MLP decoding is presented by incorporating the adaptive decoding algorithm to decrease the complexity. The presented modified MLP decoding can obtain a good tradeoff between the performance and the complexity and is proved also has the ML certificate characteristic. Simulations with moderate block length LDPC codes show that the proposed MLP decoding achieves the same BER performance as the original MLP decoding with far fewer used constraints and faster convergence speed consequently.

IEEE International Conference on Communication Technology (ICCT)•Sep 1, 2011•1 citation
We consider the decoding problem for low-density parity-check codes, and apply nonlinear programming methods. This extends previous work using linear programming (LP) to decode linear block codes. First, a multistage LP decoder based on the branch-and-bound method is proposed. This decoder makes use of the maximum-likelihood-certificate property of the LP decoder to refine the results when an error is reported. Second, we transform the original LP decoding formulation into a box-constrained quadratic programming form. Efficient linear-time parallel and serial decoding algorithms are proposed and their convergence properties are investigated. Extensive simulation studies are performed to assess the performance of the proposed decoders. It is seen that the proposed multistage LP decoder outperforms the conventional sum-product (SP) decoder considerably for low-density parity-check (LDPC) codes with short to medium block length. The proposed box-constrained quadratic programming decoder has less complexity than the SP decoder and yields much better performance for LDPC codes with regular structure
The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LOpe codes suggest that the M LP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has also been extended to nonbinary linear codes by Flanagan et al. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation proposed by Flanagan et al. as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. In contrast to the binary low-complexity LP decoding algorithm, our proposed algorithm is not directly related to the nonbinary SP algorithm. Nevertheless, the complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80,48][80,48][80,48] LDPC code defined over Z4\mathbb{Z}_4Z4​ using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has been derived for both binary and nonbinary decoding frameworks. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation for the nonbinary codes as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. The complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80, 48] LDPC code defined over ℤ4using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
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.
The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.
Linear programming (LP) decoding is getting more and more attention because of the maximum-likelihood (ML) certificate property. As one of the improved LP decoding methods, multistage LP (MLP) decoding has been proposed to improve the performance of LP decoding but at the cost of higher complexity. In this paper, the problem of achieving a balance between the performance and the complexity of LP decoding is considered. First, a criterion used for searching new constraints adaptively is proposed. Then a modified MLP decoding is presented by incorporating the adaptive decoding algorithm to decrease the complexity. The presented modified MLP decoding can obtain a good tradeoff between the performance and the complexity and is proved also has the ML certificate characteristic. Simulations with moderate block length LDPC codes show that the proposed MLP decoding achieves the same BER performance as the original MLP decoding with far fewer used constraints and faster convergence speed consequently.