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LDPC Codes--Structural Analysis and Decoding Techniques.

Xiaojie Zhang

Published Jan 1, 2012• 6 citations
OpenAlex

Abstract

Low-density parity-check (LDPC) codes have been the focus of much research over the past decade thanks to their near Shannon limit performance and to their efficient message- passing (MP) decoding algorithms. However, the error floor phenomenon observed in MP decoding, which manifests itself as an abrupt change in the slope of the error-rate curve, has hindered the adoption of LDPC codes and MP decoders in some applications requiring very low error rates. As an alternative to MP decoding, linear programming (LP) decoding is an approximation to maximum-likelihood decoding by relaxing the optimal decoding problem into a linear optimization problem. It has been noticed that, when the symbols error probability in channel output is low, LP decoding has superior error correction performance. However, due to the inefficiency of general- purpose LP solvers which are not optimized for solving LP problems, LP decoding is computationally more complex than MP decoding, especially for codes of large block size. In this dissertation, we first design an efficient exhaustive search algorithm to find all small error-prone substructures, some of which are commonly blamed for certain decoding failures of MP decoding. Then, we investigate the cause of error floors in LDPC codes from the perspective of the MP decoder implementation, with special attention to limitations that decrease the numerical accuracy of messages passed during decoding, and propose a quantization method for fixed-point implementation of MP decoding which significantly improves the error-floor performance by overcoming the limitations of standard quantization rules. For LP decoding, we improve the error-correcting capability of LP decoding by using an effective algorithm to generate additional redundant parity-check constraints which eliminate certain undesired solutions to LP decoding problem. We further propose an efficient message-passing algorithm to solve the LP decoding problem. This algorithm is based on the alternating direction method of multipliers (ADMM), a classic technique in convex optimization theory, and our key contribution is a novel, efficient projection algorithm that can improve the decoding speed of the ADMM- based LP decoder. The last part of this dissertation is a separate piece of work on optimizing video transmission over distributed cognitive radio network

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ADMM Decoding of LDPC Codes: Simplification and Improvement

Haoyuan Wei

In this dissertation, we study linear programming (LP) decoding of low-density parity-check (LDPC) codes based on the alternating direction method of multipliers (ADMM) technique, or ADMM decoding for short.The decoding of LDPC codes is formulated as an LP model and solved efficiently by the ADMM technique.However, compared with traditional belief-propagation decoding, ADMM decoding suffers from higher complexity and worse error correction performance, which prevents the employment of ADMM decoding in reality.To reduce the complexity of ADMM decoding, we first focus on the simplification of the check polytope projection, the most complex operation in ADMM decoding.We propose an iterative check polytope projection algorithm without the sorting operation.The proposed algorithm converges with the increase of iterations.Moreover, for a fixed number of iterations, its average complexity and the worst-case complexity are linear in the input dimension.Another direction we propose to simplify ADMM decoding is to devise a better scheduling scheme than the standard flooding scheme.We start from the node-wise scheduling scheme which only updates the check node messages with the maximum message residual.Then we simplify the calculation of the message residual and propose a reduced-complexity node-wise scheduling scheme.To improve the error correction performance of ADMM decoding, we propose a novel ADMM penalized decoding algorithm whose penalty term is based on check nodes (CN).We investigate the properties of the CN penalty functions and show several examples.We make conclusions on its convergence properties and prove its failure probability is independent of the transmitted codewords for symmetric channels.Monte-Carlo simulation and the instanton analysis show its better error correction performance in the low and high SNR regions.Finally, we propose a post-processing technique to lower the error floor of LDPC codes.The output of the first stage decoder can be revised if the syndrome is found This is a journey with joys and tears.I encountered many amazing people.They instructed me, helped me, and accompanied me.First of all, I would like to thank Prof. Amir H. Banihashemi for his guidance.His attitudes towards the research and devotion to one specific area are impressive.All these will have a life-long influence on me.It is also a great time working with people in his coding research group.I have a great memory of learning from their presentations.Next, I should thank Prof.

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Decomposition Methods for Large Scale LP Decoding

Siddharth Barman, Xishuo Liu, Stark Christiaan Draper +1 more

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.

arXiv:1204.0556•Apr 2, 2012•5 citations

Decomposition Methods for Large Scale LP Decoding

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When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper, we draw on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a “two-slice” characterization of the parity polytope, the polytope formed by taking the convex hull of all codewords of the single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by the ADMM decoder and its solution allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher SNR than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using the ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.

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Decoding linear codes via optimization and graph-based techniques

Mohammad Hossein Taghavi

Low-density parity-check (LDPC) codes have made it possible to communicate at information rates very close to the Shannon capacity by combining sparsity with quasi- randomness, which enables the use of low-complexity iterative message-passing (IMP) decoders. So far, most systematic studies of IMP decoders have focused on evaluating the average performance of random ensembles of LDPC codes with infinite length. However, the statistical nature of IMP algorithms does not seem very suitable for rigorous analysis the decoding of individual finite-length codes. The need for finite-length studies are most critical in applications such as data storage, where the required decoding error rate is too low to be verifiable by simulation. As an alternative to IMP algorithms, linear programming (LP) decoding is based on relaxing the optimal decoding into a linear optimization. The geometric nature of this approach makes it more amenable to deterministic finite-length analysis than IMP decoding. On the other hand, LP decoding is computationally more complex than IMP decoding, due to both the large number of constraints in the relaxed problem, and the inefficiency of using general -purpose LP solvers. In this dissertation, we study several aspects of LP decoding, starting by some steps toward reducing its complexity. We introduce an adaptive implementation of LP decoding, where the relaxed problem is replaced by a sequence of subproblems of much smaller size, resulting in a complexity reduction by orders of magnitude. This is followed by a sparse implementation of an interior-point LP solver which exploits the structure of the decoding problem. We further propose a cutting- plane approach to improve the error-correcting capability of LP decoding. Along the way, several properties are proved for LP decoding and its proposed variations. We continue by investigating the application of an optimization-based approach to decoding linear codes in the presence of intersymbol interference (ISI). By relaxing the optimal detection problem into a linear program, we derive a new graphical representation for the ISI channel, which can be used for combined equalization and decoding by LP or IMP decoders. Finally, in a separate piece of work, we study the effect of nonlinearities on the multiuser capacity of optical fibers

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Efficient implementation of linear programming decoding

Mohammad Hossein Taghavi, Amin Shokrollahi, Paul H. Siegel

While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.

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Efficient Implementation of Linear Programming Decoding

Mohammad Hossein Taghavi, Amin Shokrollahi, Paul H. Siegel

While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.

IEEE Transactions on Information Theory•Sep 1, 2011•48 citations
Low-density parity-check (LDPC) codes have been the focus of much research over the past decade thanks to their near Shannon limit performance and to their efficient message- passing (MP) decoding algorithms. However, the error floor phenomenon observed in MP decoding, which manifests itself as an abrupt change in the slope of the error-rate curve, has hindered the adoption of LDPC codes and MP decoders in some applications requiring very low error rates. As an alternative to MP decoding, linear programming (LP) decoding is an approximation to maximum-likelihood decoding by relaxing the optimal decoding problem into a linear optimization problem. It has been noticed that, when the symbols error probability in channel output is low, LP decoding has superior error correction performance. However, due to the inefficiency of general- purpose LP solvers which are not optimized for solving LP problems, LP decoding is computationally more complex than MP decoding, especially for codes of large block size. In this dissertation, we first design an efficient exhaustive search algorithm to find all small error-prone substructures, some of which are commonly blamed for certain decoding failures of MP decoding. Then, we investigate the cause of error floors in LDPC codes from the perspective of the MP decoder implementation, with special attention to limitations that decrease the numerical accuracy of messages passed during decoding, and propose a quantization method for fixed-point implementation of MP decoding which significantly improves the error-floor performance by overcoming the limitations of standard quantization rules. For LP decoding, we improve the error-correcting capability of LP decoding by using an effective algorithm to generate additional redundant parity-check constraints which eliminate certain undesired solutions to LP decoding problem. We further propose an efficient message-passing algorithm to solve the LP decoding problem. This algorithm is based on the alternating direction method of multipliers (ADMM), a classic technique in convex optimization theory, and our key contribution is a novel, efficient projection algorithm that can improve the decoding speed of the ADMM- based LP decoder. The last part of this dissertation is a separate piece of work on optimizing video transmission over distributed cognitive radio network
In this dissertation, we study linear programming (LP) decoding of low-density parity-check (LDPC) codes based on the alternating direction method of multipliers (ADMM) technique, or ADMM decoding for short.The decoding of LDPC codes is formulated as an LP model and solved efficiently by the ADMM technique.However, compared with traditional belief-propagation decoding, ADMM decoding suffers from higher complexity and worse error correction performance, which prevents the employment of ADMM decoding in reality.To reduce the complexity of ADMM decoding, we first focus on the simplification of the check polytope projection, the most complex operation in ADMM decoding.We propose an iterative check polytope projection algorithm without the sorting operation.The proposed algorithm converges with the increase of iterations.Moreover, for a fixed number of iterations, its average complexity and the worst-case complexity are linear in the input dimension.Another direction we propose to simplify ADMM decoding is to devise a better scheduling scheme than the standard flooding scheme.We start from the node-wise scheduling scheme which only updates the check node messages with the maximum message residual.Then we simplify the calculation of the message residual and propose a reduced-complexity node-wise scheduling scheme.To improve the error correction performance of ADMM decoding, we propose a novel ADMM penalized decoding algorithm whose penalty term is based on check nodes (CN).We investigate the properties of the CN penalty functions and show several examples.We make conclusions on its convergence properties and prove its failure probability is independent of the transmitted codewords for symmetric channels.Monte-Carlo simulation and the instanton analysis show its better error correction performance in the low and high SNR regions.Finally, we propose a post-processing technique to lower the error floor of LDPC codes.The output of the first stage decoder can be revised if the syndrome is found This is a journey with joys and tears.I encountered many amazing people.They instructed me, helped me, and accompanied me.First of all, I would like to thank Prof. Amir H. Banihashemi for his guidance.His attitudes towards the research and devotion to one specific area are impressive.All these will have a life-long influence on me.It is also a great time working with people in his coding research group.I have a great memory of learning from their presentations.Next, I should thank Prof.
When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.
When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper, we draw on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a “two-slice” characterization of the parity polytope, the polytope formed by taking the convex hull of all codewords of the single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by the ADMM decoder and its solution allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher SNR than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using the ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.
Low-density parity-check (LDPC) codes have made it possible to communicate at information rates very close to the Shannon capacity by combining sparsity with quasi- randomness, which enables the use of low-complexity iterative message-passing (IMP) decoders. So far, most systematic studies of IMP decoders have focused on evaluating the average performance of random ensembles of LDPC codes with infinite length. However, the statistical nature of IMP algorithms does not seem very suitable for rigorous analysis the decoding of individual finite-length codes. The need for finite-length studies are most critical in applications such as data storage, where the required decoding error rate is too low to be verifiable by simulation. As an alternative to IMP algorithms, linear programming (LP) decoding is based on relaxing the optimal decoding into a linear optimization. The geometric nature of this approach makes it more amenable to deterministic finite-length analysis than IMP decoding. On the other hand, LP decoding is computationally more complex than IMP decoding, due to both the large number of constraints in the relaxed problem, and the inefficiency of using general -purpose LP solvers. In this dissertation, we study several aspects of LP decoding, starting by some steps toward reducing its complexity. We introduce an adaptive implementation of LP decoding, where the relaxed problem is replaced by a sequence of subproblems of much smaller size, resulting in a complexity reduction by orders of magnitude. This is followed by a sparse implementation of an interior-point LP solver which exploits the structure of the decoding problem. We further propose a cutting- plane approach to improve the error-correcting capability of LP decoding. Along the way, several properties are proved for LP decoding and its proposed variations. We continue by investigating the application of an optimization-based approach to decoding linear codes in the presence of intersymbol interference (ISI). By relaxing the optimal detection problem into a linear program, we derive a new graphical representation for the ISI channel, which can be used for combined equalization and decoding by LP or IMP decoders. Finally, in a separate piece of work, we study the effect of nonlinearities on the multiuser capacity of optical fibers
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.