Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels

Kumar, Smita; Young, Andrew J.; Macris, Nicolas; Pfister, Henry D.

doi:10.1109/tit.2014.2360692

Cited by 87 publications

(77 citation statements)

References 47 publications

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“…Proof: For a large enough Γ, the pinning condition (25) and the variance symmetry (10) ensure that in the first SE iteration Σ 2 c (E (0) = 1) satisfies the following ordering along the positions: i) it is non-decreasing for all c ∈ {1, . .…”

Section: A Properties Of the Coupled Systemmentioning

confidence: 99%

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Barbier

Dia

Macris

2019

IEEE Trans. Inform. Theory

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Sparse superposition codes, or sparse regression codes, constitute a new class of codes which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity as one of the code parameter goes to infinity. Furthermore, we provide a closed form formula for the algorithmic threshold of the underlying code ensemble in terms of a Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing.

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Section: A Properties Of the Coupled Systemmentioning

confidence: 99%

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Barbier

Dia

Macris

2019

IEEE Trans. Inform. Theory

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“…Remark 5.6. The state evolution recursion was analyzed for a certain class of spatially coupled SPARCs by Barbier et al [8] using the potential method introduced in [120,76,35]. It is shown in [8] that the fixed points of the state evolution recursion (5.8)-(5.9) coincide with the stationary points of a suitably defined potential function.…”

Section: Asymptotic State Evolution Analysismentioning

confidence: 99%

Sparse Regression Codes

Venkataramanan

Tatikonda

Barron

2019

FNT in Communications and Information Theory

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Developing computationally-efficient codes that approach the Shannon-theoretic limits for communication and compression has long been one of the major goals of information and coding theory. There have been significant advances towards this goal in the last couple of decades, with the emergence of turbo codes, sparse-graph codes, and polar codes. These codes are designed primarily for discrete-alphabet channels and sources. For Gaussian channels and sources, where the alphabet is inherently continuous, Sparse Superposition Codes or Sparse Regression Codes (SPARCs) are a promising class of codes for achieving the Shannon limits.This monograph provides a unified and comprehensive over-view of sparse regression codes, covering theory, algorithms, and practical implementation aspects. The first part of the monograph focuses on SPARCs for AWGN channel coding, and the second part on SPARCs for lossy compression (with squared error distortion criterion). In the third part, SPARCs are used to construct codes for Gaussian multi-terminal channel and source coding models such as broadcast channels, multiple-access channels, and source and channel coding with side information. The monograph concludes with a discussion of open problems and directions for future work. v vi Chapter 1 A: β: 0, c 2 , 0, c L , 0, , 0 0, M columns M columns M columns Section 1 Section 2 Section L T n rows 0, c 1 , 0, 0, Figure 1.1: A Gaussian sparse regression codebook of block length n: A is a design matrix with independent Gaussian entries, and β is a sparse vector with one non-zero in each of L sections. Codewords are of the form Aβ, i.e., linear combinations of the columns corresponding to the non-zeros in β. The message is indexed by the locations of the non-zeros, and the values c 1 , . . . , c L are fixed a priori.

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“…For a comprehensive introduction to the spatially-coupled codes, see [24]- [27]. Below, we give a brief description of the construction of the spatially-coupled compound LDGM/ LDPC codes.…”

Section: Spatial Couplingmentioning

confidence: 99%

Spatially-coupled codes for write-once memories

Kumar

Vem

Narayanan

et al. 2015

2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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The focus of this article is on low-complexity capacity-achieving coding schemes for write-once memory (WOM) systems. The construction is based on spatially-coupled compound LDGM/LDPC codes. Both noiseless systems and systems with read errors are considered. Compound LDGM/LDPC codes are known to achieve capacity under MAP decoding for the closely related Gelfand-Pinsker problem and their coset decomposition provides an elegant way to encode the messages while simultaneously providing error protection. The application of compound codes to the WOM system is new. The main result is that spatial coupling enables these codes to achieve the capacity region of the 2-write WOM system with low-complexity message-passing encoding and decoding algorithms.

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Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels

Cited by 87 publications

References 47 publications

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Sparse Regression Codes

Spatially-coupled codes for write-once memories

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