## 2015 |

## Journal Articles |

Olmos, Pablo M; Urbanke, Rudiger L A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes Journal Article IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015c, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. |

Olmos, Pablo M; Urbanke, Rudiger L A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes Journal Article IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015bb, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. |

Salamanca, Luis; Murillo-Fuentes, Juan José; Olmos, Pablo M; Perez-Cruz, Fernando; Verdu, Sergio Approaching the DT Bound Using Linear Codes in the Short Blocklength Regime Journal Article IEEE Communications Letters, 19 (2), pp. 123–126, 2015, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding @article{Salamanca2014bb, title = {Approaching the DT Bound Using Linear Codes in the Short Blocklength Regime}, author = {Luis Salamanca and Juan José Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6957577}, doi = {10.1109/LCOMM.2014.2371032}, issn = {1089-7798}, year = {2015}, date = {2015-02-01}, journal = {IEEE Communications Letters}, volume = {19}, number = {2}, pages = {123--126}, abstract = {The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits.}, keywords = {binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding}, pubstate = {published}, tppubtype = {article} } The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits. |

## Inproceedings |

Olmos, Pablo M; Mitchell, David G M; Costello, Daniel J Analyzing the Finite-Length Performance of Generalized LDPC Codes Inproceedings 2015 IEEE International Symposium on Information Theory (ISIT), pp. 2683–2687, IEEE, Hong Kong, 2015, ISBN: 978-1-4673-7704-1. Abstract | Links | BibTeX | Tags: BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs @inproceedings{Olmos2015b, title = {Analyzing the Finite-Length Performance of Generalized LDPC Codes}, author = {Pablo M Olmos and David G M Mitchell and Daniel J Costello}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7282943}, doi = {10.1109/ISIT.2015.7282943}, isbn = {978-1-4673-7704-1}, year = {2015}, date = {2015-06-01}, booktitle = {2015 IEEE International Symposium on Information Theory (ISIT)}, pages = {2683--2687}, publisher = {IEEE}, address = {Hong Kong}, abstract = {In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved.}, keywords = {BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved. |

Stinner, Markus; Olmos, Pablo M Finite-Length Performance of Multi-Edge Protograph-Based Spatially Coupled LDPC Codes Inproceedings 2015 IEEE International Symposium on Information Theory (ISIT), pp. 889–893, IEEE, Hong Kong, 2015, ISBN: 978-1-4673-7704-1. Abstract | Links | BibTeX | Tags: binary erasure channel, Block codes, Couplings, Decoding, Error analysis, finite length performance, finite-length performance, graph theory, Iterative decoding, low density parity check codes, multiedge protograph, parity check codes, spatially coupled LDPC codes, spatially-coupled LDPC codes, Steady-state @inproceedings{Stinner2015, title = {Finite-Length Performance of Multi-Edge Protograph-Based Spatially Coupled LDPC Codes}, author = {Markus Stinner and Pablo M Olmos}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7282583}, doi = {10.1109/ISIT.2015.7282583}, isbn = {978-1-4673-7704-1}, year = {2015}, date = {2015-06-01}, booktitle = {2015 IEEE International Symposium on Information Theory (ISIT)}, pages = {889--893}, publisher = {IEEE}, address = {Hong Kong}, abstract = {The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies.}, keywords = {binary erasure channel, Block codes, Couplings, Decoding, Error analysis, finite length performance, finite-length performance, graph theory, Iterative decoding, low density parity check codes, multiedge protograph, parity check codes, spatially coupled LDPC codes, spatially-coupled LDPC codes, Steady-state}, pubstate = {published}, tppubtype = {inproceedings} } The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies. |

## 2014 |

## Journal Articles |

Salamanca, Luis; Murillo-Fuentes, Juan José; Olmos, Pablo M; Perez-Cruz, Fernando; Verdu, Sergio Near DT Bound Achieving Linear Codes in the Short Blocklength Regime Journal Article IEEE Communications Letters, PP (99), pp. 1–1, 2014, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding @article{Salamanca2014b, title = {Near DT Bound Achieving Linear Codes in the Short Blocklength Regime}, author = {Luis Salamanca and Juan José Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6957577}, issn = {1089-7798}, year = {2014}, date = {2014-01-01}, journal = {IEEE Communications Letters}, volume = {PP}, number = {99}, pages = {1--1}, abstract = {The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits.}, keywords = {binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding}, pubstate = {published}, tppubtype = {article} } The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits. |

## Inproceedings |

Stinner, Markus; Olmos, Pablo M Analyzing Finite-length Protograph-Based Spatially Coupled LDPC Codes Inproceedings 2014 IEEE International Symposium on Information Theory, pp. 891–895, IEEE, Honolulu, 2014, ISBN: 978-1-4799-5186-4. Abstract | Links | BibTeX | Tags: binary erasure channel, covariance analysis, covariance evolution, Decoding, degree-one check nodes, Error analysis, finite-length protograph, mean evolution, Monte Carlo methods, parity check codes, peeling decoding, protograph-based SC-LDPC codes, spatially coupled low-density parity-check codes, stable decoding phase, Steady-state, Vectors @inproceedings{Stinner2014, title = {Analyzing Finite-length Protograph-Based Spatially Coupled LDPC Codes}, author = {Markus Stinner and Pablo M Olmos}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6874961}, isbn = {978-1-4799-5186-4}, year = {2014}, date = {2014-01-01}, booktitle = {2014 IEEE International Symposium on Information Theory}, pages = {891--895}, publisher = {IEEE}, address = {Honolulu}, abstract = {The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite-length scaling laws for these constructions are given and verified by numerical simulations.}, keywords = {binary erasure channel, covariance analysis, covariance evolution, Decoding, degree-one check nodes, Error analysis, finite-length protograph, mean evolution, Monte Carlo methods, parity check codes, peeling decoding, protograph-based SC-LDPC codes, spatially coupled low-density parity-check codes, stable decoding phase, Steady-state, Vectors}, pubstate = {published}, tppubtype = {inproceedings} } The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite-length scaling laws for these constructions are given and verified by numerical simulations. |

## 2013 |

## Journal Articles |

Salamanca, Luis; Olmos, Pablo M; Perez-Cruz, Fernando; Murillo-Fuentes, Juan Jose Tree-Structured Expectation Propagation for LDPC Decoding over BMS Channels Journal Article IEEE Transactions on Communications, 61 (10), pp. 4086–4095, 2013, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, BEC, belief propagation, binary erasure channel, binary memoryless symmetric channels, BMS channels, Channel Coding, Complexity theory, convolutional codes, convolutional low-density parity-check codes, Decoding, decoding block, expectation propagation, finite-length codes, LDPC decoding, message-passing algorithm, parity check codes, Probability density function, sparse linear codes, TEP algorithm, tree-structured expectation propagation, trees (mathematics), Vegetation @article{Salamanca2013a, title = {Tree-Structured Expectation Propagation for LDPC Decoding over BMS Channels}, author = {Luis Salamanca and Pablo M Olmos and Fernando Perez-Cruz and Juan Jose Murillo-Fuentes}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6587624}, issn = {0090-6778}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Communications}, volume = {61}, number = {10}, pages = {4086--4095}, abstract = {In this paper, we put forward the tree-structured expectation propagation (TEP) algorithm for decoding block and convolutional low-density parity-check codes over any binary channel. We have already shown that TEP improves belief propagation (BP) over the binary erasure channel (BEC) by imposing marginal constraints over a set of pairs of variables that form a tree or a forest. The TEP decoder is a message-passing algorithm that sequentially builds a tree/forest of erased variables to capture additional information disregarded by the standard BP decoder, which leads to a noticeable reduction of the error rate for finite-length codes. In this paper, we show how the TEP can be extended to any channel, specifically to binary memoryless symmetric (BMS) channels. We particularly focus on how the TEP algorithm can be adapted for any channel model and, more importantly, how to choose the tree/forest to keep the gains observed for block and convolutional LDPC codes over the BEC.}, keywords = {Approximation algorithms, Approximation methods, BEC, belief propagation, binary erasure channel, binary memoryless symmetric channels, BMS channels, Channel Coding, Complexity theory, convolutional codes, convolutional low-density parity-check codes, Decoding, decoding block, expectation propagation, finite-length codes, LDPC decoding, message-passing algorithm, parity check codes, Probability density function, sparse linear codes, TEP algorithm, tree-structured expectation propagation, trees (mathematics), Vegetation}, pubstate = {published}, tppubtype = {article} } In this paper, we put forward the tree-structured expectation propagation (TEP) algorithm for decoding block and convolutional low-density parity-check codes over any binary channel. We have already shown that TEP improves belief propagation (BP) over the binary erasure channel (BEC) by imposing marginal constraints over a set of pairs of variables that form a tree or a forest. The TEP decoder is a message-passing algorithm that sequentially builds a tree/forest of erased variables to capture additional information disregarded by the standard BP decoder, which leads to a noticeable reduction of the error rate for finite-length codes. In this paper, we show how the TEP can be extended to any channel, specifically to binary memoryless symmetric (BMS) channels. We particularly focus on how the TEP algorithm can be adapted for any channel model and, more importantly, how to choose the tree/forest to keep the gains observed for block and convolutional LDPC codes over the BEC. |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC Journal Article IEEE Transactions on Information Theory, 59 (6), pp. 3354–3377, 2013, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: Algorithm design and analysis, Approximation algorithms, Approximation methods, BEC, belief propagation, Belief-propagation (BP), binary erasure channel, Complexity theory, decode low-density parity-check codes, Decoding, discrete memoryless channels, expectation propagation, finite-length analysis, LDPC codes, LDPC decoding, parity check codes, peeling-type algorithm, Probability density function, random graph evolution, Tanner graph, tree-structure expectation propagation @article{Olmos2013b, title = {Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6451276}, issn = {0018-9448}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {59}, number = {6}, pages = {3354--3377}, abstract = {We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). Expectation propagation generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regimes over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.}, keywords = {Algorithm design and analysis, Approximation algorithms, Approximation methods, BEC, belief propagation, Belief-propagation (BP), binary erasure channel, Complexity theory, decode low-density parity-check codes, Decoding, discrete memoryless channels, expectation propagation, finite-length analysis, LDPC codes, LDPC decoding, parity check codes, peeling-type algorithm, Probability density function, random graph evolution, Tanner graph, tree-structure expectation propagation}, pubstate = {published}, tppubtype = {article} } We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). Expectation propagation generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regimes over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization. |

Salamanca, Luis; Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree Expectation Propagation for ML Decoding of LDPC Codes over the BEC Journal Article IEEE Transactions on Communications, 61 (2), pp. 465–473, 2013, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: approximate inference, Approximation algorithms, Approximation methods, BEC, binary codes, binary erasure channel, code graph, Complexity theory, equivalent complexity, Gaussian elimination method, Gaussian processes, generalized tree-structured expectation propagatio, graphical message-passing procedure, graphical models, LDPC codes, Maximum likelihood decoding, maximum likelihood solution, ML decoding, parity check codes, peeling decoder, tree expectation propagation, tree graph, Tree graphs, tree-structured expectation propagation, tree-structured expectation propagation decoder, trees (mathematics) @article{Salamanca2013b, title = {Tree Expectation Propagation for ML Decoding of LDPC Codes over the BEC}, author = {Luis Salamanca and Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6384612}, issn = {0090-6778}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Communications}, volume = {61}, number = {2}, pages = {465--473}, abstract = {We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the binary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical message-passing procedure and, most interesting, the algorithm can be generalized to other channels.}, keywords = {approximate inference, Approximation algorithms, Approximation methods, BEC, binary codes, binary erasure channel, code graph, Complexity theory, equivalent complexity, Gaussian elimination method, Gaussian processes, generalized tree-structured expectation propagatio, graphical message-passing procedure, graphical models, LDPC codes, Maximum likelihood decoding, maximum likelihood solution, ML decoding, parity check codes, peeling decoder, tree expectation propagation, tree graph, Tree graphs, tree-structured expectation propagation, tree-structured expectation propagation decoder, trees (mathematics)}, pubstate = {published}, tppubtype = {article} } We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the binary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical message-passing procedure and, most interesting, the algorithm can be generalized to other channels. |

## Inproceedings |

Salamanca, Luis; Murillo-Fuentes, Juan Jose; Olmos, Pablo M; Perez-Cruz, Fernando Improving the BP Estimate over the AWGN Channel Using Tree-Structured Expectation Propagation Inproceedings 2013 IEEE International Symposium on Information Theory, pp. 2990–2994, IEEE, Istanbul, 2013, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, AWGN channels, BEC, belief propagation decoding, BI-AWGN channel, binary additive white Gaussian noise channel, binary erasure channel, BP estimation, Channel Coding, Complexity theory, error rate reduction, error statistics, Expectation, finite-length codes, Iterative decoding, LDPC codes, LDPC decoding, low-density parity-check decoding, Maximum likelihood decoding, parity check codes, posterior distribution, Propagation, TEP algorithm, tree-structured expectation propagation algorithm, trees (mathematics) @inproceedings{Salamanca2013, title = {Improving the BP Estimate over the AWGN Channel Using Tree-Structured Expectation Propagation}, author = {Luis Salamanca and Juan Jose Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6620774}, issn = {2157-8095}, year = {2013}, date = {2013-01-01}, booktitle = {2013 IEEE International Symposium on Information Theory}, pages = {2990--2994}, publisher = {IEEE}, address = {Istanbul}, abstract = {In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the binary additive white Gaussian noise (BI-AWGN) channel. By approximating the posterior distribution by a tree-structure factorization, the TEP has been proven to improve belief propagation (BP) decoding over the binary erasure channel (BEC). We show for the AWGN channel how the TEP decoder is also able to capture additional information disregarded by the BP solution, which leads to a noticeable reduction of the error rate for finite-length codes. We show that for the range of codes of interest, the TEP gain is obtained with a slight increase in complexity over that of the BP algorithm. An efficient way of constructing the tree-like structure is also described.}, keywords = {Approximation algorithms, Approximation methods, AWGN channels, BEC, belief propagation decoding, BI-AWGN channel, binary additive white Gaussian noise channel, binary erasure channel, BP estimation, Channel Coding, Complexity theory, error rate reduction, error statistics, Expectation, finite-length codes, Iterative decoding, LDPC codes, LDPC decoding, low-density parity-check decoding, Maximum likelihood decoding, parity check codes, posterior distribution, Propagation, TEP algorithm, tree-structured expectation propagation algorithm, trees (mathematics)}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the binary additive white Gaussian noise (BI-AWGN) channel. By approximating the posterior distribution by a tree-structure factorization, the TEP has been proven to improve belief propagation (BP) decoding over the binary erasure channel (BEC). We show for the AWGN channel how the TEP decoder is also able to capture additional information disregarded by the BP solution, which leads to a noticeable reduction of the error rate for finite-length codes. We show that for the range of codes of interest, the TEP gain is obtained with a slight increase in complexity over that of the BP algorithm. An efficient way of constructing the tree-like structure is also described. |

## 2012 |

## Journal Articles |

Olmos, Pablo M; Salamanca, Luis; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando On the Design of LDPC-Convolutional Ensembles Using the TEP Decoder Journal Article IEEE Communications Letters, 16 (5), pp. 726–729, 2012, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: belief propagation decoding, binary erasure channel, channel capacity, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, design, Error analysis, finite-length analysis, Iterative decoding, LDPC-convolutional ensemble design, LDPCC code decoding, low-density parity-check convolutional code, parity check codes, tree-expectation propagation decoder, tree-structured expectation propagation, window-sliding scheme @article{Olmos2012b, title = {On the Design of LDPC-Convolutional Ensembles Using the TEP Decoder}, author = {Pablo M Olmos and Luis Salamanca and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6168872}, issn = {1089-7798}, year = {2012}, date = {2012-01-01}, journal = {IEEE Communications Letters}, volume = {16}, number = {5}, pages = {726--729}, abstract = {Low-density parity-check convolutional (LDPCC) codes asymptotically achieve channel capacity under belief propagation (BP) decoding. In this paper, we decode LDPCC codes using the Tree-Expectation Propagation (TEP) decoder, recently proposed as an alternative decoding method to the BP algorithm for the binary erasure channel (BEC). We show that, for LDPCC codes, the TEP decoder improves the BP solution with a comparable complexity or, alternatively, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency.}, keywords = {belief propagation decoding, binary erasure channel, channel capacity, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, design, Error analysis, finite-length analysis, Iterative decoding, LDPC-convolutional ensemble design, LDPCC code decoding, low-density parity-check convolutional code, parity check codes, tree-expectation propagation decoder, tree-structured expectation propagation, window-sliding scheme}, pubstate = {published}, tppubtype = {article} } Low-density parity-check convolutional (LDPCC) codes asymptotically achieve channel capacity under belief propagation (BP) decoding. In this paper, we decode LDPCC codes using the Tree-Expectation Propagation (TEP) decoder, recently proposed as an alternative decoding method to the BP algorithm for the binary erasure channel (BEC). We show that, for LDPCC codes, the TEP decoder improves the BP solution with a comparable complexity or, alternatively, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency. |

## Inproceedings |

Olmos, Pablo M; Perez-Cruz, Fernando; Salamanca, Luis; Murillo-Fuentes, Juan Jose Finite-Length Analysis of the TEP Decoder for LDPC Ensembles over the BEC Inproceedings 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2346–2350, IEEE, Cambridge, MA, 2012, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: Approximation methods, BEC, binary codes, binary erasure channel, Decoding, Error analysis, error probability, finite-length analysis, LDPC ensembles, low-density parity check ensembles, parity check codes, TEP decoder, Trajectory, tree-expectation propagation algorithm, waterfall region @inproceedings{Olmos2012a, title = {Finite-Length Analysis of the TEP Decoder for LDPC Ensembles over the BEC}, author = {Pablo M Olmos and Fernando Perez-Cruz and Luis Salamanca and Juan Jose Murillo-Fuentes}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6283932}, issn = {2157-8095}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE International Symposium on Information Theory Proceedings}, pages = {2346--2350}, publisher = {IEEE}, address = {Cambridge, MA}, abstract = {In this work, we analyze the finite-length performance of low-density parity check (LDPC) ensembles decoded over the binary erasure channel (BEC) using the tree-expectation propagation (TEP) algorithm. In a previous paper, we showed that the TEP improves the BP performance for decoding regular and irregular short LDPC codes, but the perspective was mainly empirical. In this work, given the degree-distribution of an LDPC ensemble, we explain and predict the range of code lengths for which the TEP improves the BP solution. In addition, for LDPC ensembles that present a single critical point, we propose a scaling law to accurately predict the performance in the waterfall region. These results are of critical importance to design practical LDPC codes for the TEP decoder.}, keywords = {Approximation methods, BEC, binary codes, binary erasure channel, Decoding, Error analysis, error probability, finite-length analysis, LDPC ensembles, low-density parity check ensembles, parity check codes, TEP decoder, Trajectory, tree-expectation propagation algorithm, waterfall region}, pubstate = {published}, tppubtype = {inproceedings} } In this work, we analyze the finite-length performance of low-density parity check (LDPC) ensembles decoded over the binary erasure channel (BEC) using the tree-expectation propagation (TEP) algorithm. In a previous paper, we showed that the TEP improves the BP performance for decoding regular and irregular short LDPC codes, but the perspective was mainly empirical. In this work, given the degree-distribution of an LDPC ensemble, we explain and predict the range of code lengths for which the TEP improves the BP solution. In addition, for LDPC ensembles that present a single critical point, we propose a scaling law to accurately predict the performance in the waterfall region. These results are of critical importance to design practical LDPC codes for the TEP decoder. |

## 2011 |

## Inproceedings |

Olmos, Pablo M; Urbanke, Rudiger Scaling Behavior of Convolutional LDPC Ensembles over the BEC Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 1816–1820, IEEE, Saint Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: BEC, binary codes, binary erasure channel, Bit error rate, convolutional codes, convolutional LDPC ensembles, coupled sparse graph codes, Couplings, Decoding, error probability, Iterative decoding, parity check codes, scaling behavior @inproceedings{Olmos2011, title = {Scaling Behavior of Convolutional LDPC Ensembles over the BEC}, author = {Pablo M Olmos and Rudiger Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6033863}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {1816--1820}, publisher = {IEEE}, address = {Saint Petersburg}, abstract = {We study the scaling behavior of coupled sparse graph codes over the binary erasure channel. In particular, let 2L+1 be the length of the coupled chain, let M be the number of variables in each of the 2L+1 local copies, let ℓ be the number of iterations, let Pb denote the bit error probability, and let ∈ denote the channel parameter. We are interested in how these quantities scale when we let the blocklength (2L + 1)M tend to infinity. Based on empirical evidence we show that the threshold saturation phenomenon is rather stable with respect to the scaling of the various parameters and we formulate some general rules of thumb which can serve as a guide for the design of coding systems based on coupled graphs.}, keywords = {BEC, binary codes, binary erasure channel, Bit error rate, convolutional codes, convolutional LDPC ensembles, coupled sparse graph codes, Couplings, Decoding, error probability, Iterative decoding, parity check codes, scaling behavior}, pubstate = {published}, tppubtype = {inproceedings} } We study the scaling behavior of coupled sparse graph codes over the binary erasure channel. In particular, let 2L+1 be the length of the coupled chain, let M be the number of variables in each of the 2L+1 local copies, let ℓ be the number of iterations, let Pb denote the bit error probability, and let ∈ denote the channel parameter. We are interested in how these quantities scale when we let the blocklength (2L + 1)M tend to infinity. Based on empirical evidence we show that the threshold saturation phenomenon is rather stable with respect to the scaling of the various parameters and we formulate some general rules of thumb which can serve as a guide for the design of coding systems based on coupled graphs. |

Salamanca, Luis; Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando MAP Decoding for LDPC Codes over the Binary Erasure Channel Inproceedings 2011 IEEE Information Theory Workshop, pp. 145–149, IEEE, Paraty, 2011, ISBN: 978-1-4577-0437-6. Abstract | Links | BibTeX | Tags: binary erasure channel, Channel Coding, computational complexity, Decoding, generalized peeling decoder, generalized tree-structured expectation propagatio, graphical models, Iterative decoding, LDPC codes, MAP decoding, MAP decoding algorithm, Maximum likelihood decoding, parity check codes, TEP decoder, tree graph theory, Tree graphs, tree-structured expectation propagation, trees (mathematics) @inproceedings{Salamanca2011a, title = {MAP Decoding for LDPC Codes over the Binary Erasure Channel}, author = {Luis Salamanca and Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6089364}, isbn = {978-1-4577-0437-6}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE Information Theory Workshop}, pages = {145--149}, publisher = {IEEE}, address = {Paraty}, abstract = {In this paper, we propose a decoding algorithm for LDPC codes that achieves the MAP solution over the BEC. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), extends the idea of our previous work, the TEP decoder. The GTEP modifies the graph by eliminating a check node of any degree and merging this information with the remaining graph. The GTEP decoder upon completion either provides the unique MAP solution or a tree graph in which the number of parent nodes indicates the multiplicity of the MAP solution. This algorithm can be easily described for the BEC, and it can be cast as a generalized peeling decoder. The GTEP naturally optimizes the complexity of the decoder, by looking for checks nodes of minimum degree to be eliminated first.}, keywords = {binary erasure channel, Channel Coding, computational complexity, Decoding, generalized peeling decoder, generalized tree-structured expectation propagatio, graphical models, Iterative decoding, LDPC codes, MAP decoding, MAP decoding algorithm, Maximum likelihood decoding, parity check codes, TEP decoder, tree graph theory, Tree graphs, tree-structured expectation propagation, trees (mathematics)}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we propose a decoding algorithm for LDPC codes that achieves the MAP solution over the BEC. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), extends the idea of our previous work, the TEP decoder. The GTEP modifies the graph by eliminating a check node of any degree and merging this information with the remaining graph. The GTEP decoder upon completion either provides the unique MAP solution or a tree graph in which the number of parent nodes indicates the multiplicity of the MAP solution. This algorithm can be easily described for the BEC, and it can be cast as a generalized peeling decoder. The GTEP naturally optimizes the complexity of the decoder, by looking for checks nodes of minimum degree to be eliminated first. |