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223
Good ErrorCorrecting Codes based on Very Sparse Matrices
, 1999
"... We study two families of errorcorrecting codes defined in terms of very sparse matrices. "MN" (MacKayNeal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The ..."
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Cited by 741 (23 self)
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We study two families of errorcorrecting codes defined in terms of very sparse matrices. "MN" (MacKayNeal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sumproduct algorithm. We prove that these codes are "very good," in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binarysymmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binarysymmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes.
Raptor codes
 IEEE Transactions on Information Theory
, 2006
"... LTCodes are a new class of codes introduced in [1] for the purpose of scalable and faulttolerant distribution of data over computer networks. In this paper we introduce Raptor Codes, an extension of LTCodes with linear time encoding and decoding. We will exhibit a class of universal Raptor codes: ..."
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Cited by 567 (6 self)
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LTCodes are a new class of codes introduced in [1] for the purpose of scalable and faulttolerant distribution of data over computer networks. In this paper we introduce Raptor Codes, an extension of LTCodes with linear time encoding and decoding. We will exhibit a class of universal Raptor codes: for a given integer k, and any real ε> 0, Raptor codes in this class produce a potentially infinite stream of symbols such that any subset of symbols of size k(1 + ε) is sufficient to recover the original k symbols with high probability. Each output symbol is generated using O(log(1/ε)) operations, and the original symbols are recovered from the collected ones with O(k log(1/ε)) operations. We will also introduce novel techniques for the analysis of the error probability of the decoder for finite length Raptor codes. Moreover, we will introduce and analyze systematic versions of Raptor codes, i.e., versions in which the first output elements of the coding system coincide with the original k elements. 1
Regular and Irregular Progressive EdgeGrowth Tanner Graphs
 IEEE TRANS. INFORM. THEORY
, 2003
"... We propose a general method for constructing Tanner graphs having a large girth by progressively establishing edges or connections between symbol and check nodes in an edgebyedge manner, called progressive edgegrowth (PEG) construction. Lower bounds on the girth of PEG Tanner graphs and on the mi ..."
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Cited by 192 (0 self)
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We propose a general method for constructing Tanner graphs having a large girth by progressively establishing edges or connections between symbol and check nodes in an edgebyedge manner, called progressive edgegrowth (PEG) construction. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting lowdensity paritycheck (LDPC) codes are derived in terms of parameters of the graphs. The PEG construction attains essentially the same girth as Gallager's explicit construction for regular graphs, both of which meet or exceed the ErdosSachs bound. Asymptotic analysis of a relaxed version of the PEG construction is presented. We describe an empirical approach using a variant of the "downhill simplex" search algorithm to design irregular PEG graphs for short codes with fewer than a thousand of bits, complementing the design approach of "density evolution" for larger codes. Encoding of LDPC codes based on the PEG construction is also investigated. We show how to exploit the PEG principle to obtain LDPC codes that allow linear time encoding. We also investigate regular and irregular LDPC codes using PEG Tanner graphs but allowing the symbol nodes to take values over GF(q), q > 2. Analysis and simulation demonstrate that one can obtain better performance with increasing field size, which contrasts with previous observations.
Using linear programming to decode binary linear codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor grap ..."
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Cited by 184 (10 self)
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A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or paritycheck representation of the code. The resulting “LP decoder” generalizes our previous work on turbolike codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including “stopping sets, ” “irreducible closed walks, ” “trellis cycles, ” “deviation sets, ” and “graph covers.” The fractional distance ��— ™ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to ��— ™ P I errors and that there are codes with ��— ™ a @ I A. An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on lowdensity paritycheck (LDPC) codes between LP decoding and the minsum and sumproduct algorithms. Methods for tightening the LP relaxation to improve performance are also provided.
Lowdensity paritycheck codes based on finite geometries: A rediscovery and new results
 IEEE Trans. Inform. Theory
, 2001
"... This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and thei ..."
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Cited by 182 (7 self)
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This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth T. Finitegeometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasicyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finitegeometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finitegeometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.
Decoding ErrorCorrecting Codes via Linear Programming
, 2003
"... Abstract. Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original informa ..."
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Cited by 116 (5 self)
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Abstract. Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full errorcorrecting potential of the system is often very complex, especially for modern codes that approach the theoretical limits of the communication channel. In this thesis we investigate the application of linear programming (LP) relaxation to the problem of decoding an errorcorrecting code. Linear programming relaxation is a standard technique in approximation algorithms and operations research, and is central to the study of efficient algorithms to find good (albeit suboptimal) solutions to very difficult optimization problems. Our new “LP decoders ” have tight combinatorial characterizations of decoding success that can be used to analyze errorcorrecting performance. Furthermore, LP decoders have the desirable (and rare) property that whenever they output a result, it is guaranteed to be the optimal result: the most likely (ML) information sent over the
Graphcover decoding and finitelength analysis of messagepassing iterative decoding of LDPC codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are comp ..."
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Cited by 114 (16 self)
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The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graphcover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graphcover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and messagepassing iterative decoding. Namely, on the one hand it turns out that graphcover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like messagepassing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graphcover decoding can serve as a model to explain the behavior of messagepassing iterative decoding. Understanding the behavior of graphcover decoding is tantamount to understanding
Binary intersymbol interference channels: Gallager codes, density evolution and code performance bounds
 IEEE TRANS. INFORM. THEORY
, 2003
"... We study the limits of performance of Gallager codes (lowdensity paritycheck (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum–product messagepassing detector/d ..."
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Cited by 69 (8 self)
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We study the limits of performance of Gallager codes (lowdensity paritycheck (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum–product messagepassing detector/decoder, we prove two error concentration theorems. Our proofs expand on previous work by handling complications introduced by the channel memory. We circumvent these problems by considering not just linear Gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols. We compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify, by simulation, that the thresholds represent accurate predictions of the performance of the iterative sum–product algorithm for finite (but large) block lengths. We also demonstrate that for high rates, the thresholds are very close to the theoretical limit of performance for Gallager codes over ISI channels. If g denotes the capacity of a binary ISI channel and if g � � � denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables @g � � � gA, we prove that the maximum information rate achievable by the sum–product decoder of a Gallager (coset) code is upperbounded by g � � �. The last topic investigated is the performance limit of the decoder if the trellis portion of the sum–product algorithm is executed only once; this demonstrates the potential for trading off the computational requirements and the performance of the decoder.
Selective avoidance of cycles in irregular LDPC code construction
 IEEE Trans. Commun
, 2004
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