Asymptotically good LDPC convolutional codes based on protographs

Mitchell, David G. M.; Pusane, Alí Emre; Zigangirov, Kamil Sh.; Costello, Daniel J.

doi:10.1109/isit.2008.4595143

Cited by 23 publications

(40 citation statements)

References 19 publications

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“…It was shown in [64], [67], [68], [73] that if we couple component codes whose Hamming distance grows linearly in the blocklength then also the resulting coupled ensembles have this property (assuming that the number of "sections" or copies of the underlying code is kept fixed). The equivalent result is true for stopping sets.…”

Section: Prior Results For the Binary Erasure Channelmentioning

confidence: 99%

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Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Kudekar

Richardson

Urbanke

2013

IEEE Trans. Inform. Theory

358

318

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Abstract-We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a-priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felström and Zigangirov, perform so well over this channel.We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble which fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in that ensemble have that property. The quantifier universal refers to the single ensemble/code which is good for all channels but we assume that the channel is known at the receiver.The key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble.We conclude by discussing some interesting open problems.

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Section: Prior Results For the Binary Erasure Channelmentioning

confidence: 99%

“…A representation of convolutional LDPC ensembles in terms of a protograph was introduced by Mitchell, Pusane, Zigangirov and Costello [64]. The corresponding representation for terminated convolutional LDPC ensembles was introduced by Lentmaier, Fettweis, Zigangirov and Costello [65].…”

Section: B Prior Work On Spatially Coupled Codesmentioning

confidence: 99%

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Kudekar

Richardson

Urbanke

2013

IEEE Trans. Inform. Theory

358

318

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“…With respect to the technique used in the proof we follow the lead of [15], [18] and [17], [22] which consider distance and pseudo-distance analysis of convolutional LDPC ensembles, respectively.…”

Section: A the (L R L) Ensemblementioning

confidence: 99%

Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC

Kudekar

Richardson

Urbanke

2010

2010 IEEE International Symposium on Information Theory

290

739

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Abstract-Convolutional LDPC ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles.We describe the fundamental mechanism which explains why "convolutional-like" or "spatially coupled" codes perform so well. In essence, the spatial coupling of the individual code structure has the effect of increasing the belief-propagation threshold of the new ensemble to its maximum possible value, namely the maximum-a-posteriori threshold of the underlying ensemble. For this reason we call this phenomenon "threshold saturation".This gives an entirely new way of approaching capacity. One significant advantage of such a construction is that one can create capacity-approaching ensembles with an error correcting radius which is increasing in the blocklength. Our proof makes use of the area theorem of the belief-propagation EXIT curve and the connection between the maximum-a-posteriori and beliefpropagation threshold recently pointed out by Méasson, Montanari, Richardson, and Urbanke.Although we prove the connection between the maximuma-posteriori and the belief-propagation threshold only for a very specific ensemble and only for the binary erasure channel, empirically a threshold saturation phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the "dynamical" threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms as well as to new techniques for analysis.

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“…Their convolutional counterparts, the LDPC convolutional codes, were invented in [2] and further developed in, e.g., [3,4]. Different constructions and analytical results have been provided in [3][4][5][6][7][8][9][10][11]. For example, Lentmaier et al [3] proposed a realization of LDPC convolutional codes and conjectured the capacity-achieving performance under belief-propagation (BP) decoding.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Method to Construct Parity-Check Matrices for Recursively Encoding Spatially Coupled LDPC Codes †

Wang

2016

Entropy

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Spatially coupled low-density parity-check (LDPC) codes have attracted considerable attention due to their promising performance. Recursive encoding of the codes with low delay and low complexity has been proposed in the literature but with constraints or restrictions. In this manuscript we propose an efficient method to construct parity-check matrices for recursively encoding spatially coupled LDPC codes with arbitrarily chosen node degrees. A general principle is proposed, which provides feasible and practical guidance for the construction of parity-check matrices. According to the specific structure of the matrix, each parity bit at a coupling position is jointly determined by the information bits at the current position and the encoded bits at former positions. Performance analysis in terms of design rate and density evolution has been presented. It can be observed that, in addition to the feature of recursive encoding, selected code structures constructed by the newly proposed method may lead to better belief-propagation thresholds than the conventional structures. Finite-length simulation results are provided as well, which verify the theoretical analysis.

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Asymptotically good LDPC convolutional codes based on protographs

Cited by 23 publications

References 19 publications

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC

An Efficient Method to Construct Parity-Check Matrices for Recursively Encoding Spatially Coupled LDPC Codes †

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