The capacity of low-density parity-check codes under message-passing decoding

Richardson, Tanya; Urbanke, R.

doi:10.1109/18.910577

Cited by 2,523 publications

(2,306 citation statements)

References 15 publications

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“…An important idea due to Luby, Mitzenmacher, Shokrollahi and Spielman [LMSS01] was to use bipartite graphs with irregular degree distributions. These ideas were pushed further by Richardson and Urbanke [RSU01,RU01], resulting in nearly linear time decoding of codes with rates approaching channel capacity. (Our basic definition of rate applies only to a very simple model of noise.…”

Section: Error Correcting Codesmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,470

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Error Correcting Codesmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,470

1,355

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“…1. We define Tanner Using a concentration theorem, the authors of [15] have shown that for an ensemble of sufficiently long LDPC codes with given degree distribution of the graph nodes, the performance is concentrated at the average performance of the ensemble. However, at short to medium block lengths, performance of the randomly selected codes significantly deviate from the theoretical ensemble average performance.…”

Section: Code Graph Propertiesmentioning

confidence: 99%

Construction of Structured Nonbinary Low-Density Parity-Check Codes

Sulek¹

2013

IJEEEE

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Abstract-Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q) are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems. The nonbinary codes can achieve significantly better performance for short and moderate block lengths. A lot of works concerning "good" LDPC codes parity check matrix construction has been published so far. However, it is well known that efficient partially parallel hardware decoder architectures are allowed only for codes with blockwise partitioned structure of the parity check matrix, called structured codes. In this paper we present a versatile algorithm for construction of codes that are both nonbinary and structured. The proposed algorithm aims at optimizing the code graph (Tanner graph) by reducing the existence of small cycles with low external connectivity, while at the same time selecting appropriate nonzero coefficients from the Galois Field under interest. The algorithm can be used for code construction of any field order, block length and code rate.

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“…This is within 0.18 dB of Shannon's limit [30]. The density evolution and Gaussian approximation methods, which make use of the concentration theorem [28], can only be used to design the degree distributions for infinitely long LDPC codes. The concentration theorem states that the performance of cycle-free LDPC codes can be characterised by the average performance of the ensemble.…”

Section: Example 121 An Irregular Ldpc Code With the Following Degrementioning

confidence: 99%

LDPC Codes

Tomlinson¹,

Tjhai²,

Ambroze³

et al. 2017

Error-Correction Coding and Decoding

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LDPC codes are linear block codes whose parity-check matrix-as the name implies-is sparse. These codes can be iteratively decoded using the sum product [9] or equivalently the belief propagation [24] soft decision decoder. It has been shown, for example by Chung et al. [3], that for long block lengths, the performance of LDPC codes is close to the channel capacity. The theory of LDPC codes is related to a branch of mathematics called graph theory. Some basic definitions used in graph theory are briefly introduced as follows.Definition 12.1 (Vertex, Edge, Adjacent and Incident) A graph, denoted by G(V, E), consists of an ordered set of vertices and edges.• (Vertex) A vertex is commonly drawn as a node or a dot. The set V (G) consists of vertices of G(V, E) and if v is a vertex of G(V, E), it is denoted as v ∈ V (G). The number of vertices of V (G) is denoted by |V (G)|. • (Edge) An edge (u, v) connects two vertices u ∈ V (G) and v ∈ V (G) and it is drawn as a line connecting vertices u and v. The set E(G) contains pairs of elements of

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The capacity of low-density parity-check codes under message-passing decoding

Cited by 2,523 publications

References 15 publications

Expander graphs and their applications

Expander graphs and their applications

Construction of Structured Nonbinary Low-Density Parity-Check Codes

LDPC Codes

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