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559
Design of capacityapproaching irregular lowdensity paritycheck codes
 IEEE TRANS. INFORM. THEORY
, 2001
"... We design lowdensity paritycheck (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the unde ..."
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Cited by 581 (6 self)
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We design lowdensity paritycheck (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a beliefpropagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.
On the design of lowdensity paritycheck codes within 0.0045 dB of the shannon limit
 IEEE Communications Letters
, 2001
"... Abstract—We develop improved algorithms to construct good lowdensity paritycheck codes that approach the Shannon limit very closely. For rate 1/2, the best code found has a threshold within 0.0045 dB of the Shannon limit of the binaryinput additive white Gaussian noise channel. Simulation results ..."
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Cited by 304 (6 self)
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Abstract—We develop improved algorithms to construct good lowdensity paritycheck codes that approach the Shannon limit very closely. For rate 1/2, the best code found has a threshold within 0.0045 dB of the Shannon limit of the binaryinput additive white Gaussian noise channel. Simulation results with a somewhat simpler code show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10 T using a block length of 10 U. Index Terms—Density evolution, lowdensity paritycheck codes, Shannon limit, sumproduct algorithm. I.
Analysis of sumproduct decoding of lowdensity paritycheck codes using a Gaussian approximation
 IEEE TRANS. INFORM. THEORY
, 2001
"... Density evolution is an algorithm for computing the capacity of lowdensity paritycheck (LDPC) codes under messagepassing decoding. For memoryless binaryinput continuousoutput additive white Gaussian noise (AWGN) channels and sumproduct decoders, we use a Gaussian approximation for message densi ..."
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Cited by 242 (2 self)
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Density evolution is an algorithm for computing the capacity of lowdensity paritycheck (LDPC) codes under messagepassing decoding. For memoryless binaryinput continuousoutput additive white Gaussian noise (AWGN) channels and sumproduct decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinitedimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a onedimensional problem of updating means of Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is IH. We show that by using the Gaussian approximation, we can visualize the sumproduct decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
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.
Extrinsic information transfer functions: A model and two properties
 in Proc. Conf. Info. Sci. Syst. (CISS
, 2002
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Efficient Encoding of LowDensity ParityCheck Codes
, 2001
"... Lowdensity paritycheck (LDPC) codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy: constrained random code ensembles and iterative decoding algorithms. In this paper, we consider the encoding problem for LDPC ..."
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Cited by 184 (3 self)
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Lowdensity paritycheck (LDPC) codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy: constrained random code ensembles and iterative decoding algorithms. In this paper, we consider the encoding problem for LDPC codes. More generally, we consider the encoding problem for codes specified by sparse paritycheck matrices. We show how to exploit the sparseness of the paritycheck matrix to obtain efficient encoders. For the @Q TAregular LDPC code, for example, the complexity of encoding is essentially quadratic in the block length. However, we show that the associated coefficient can be made quite small, so that encoding codes even of length IHH HHH is still quite practical. More importantly, we will show that “optimized” codes actually admit linear time encoding.
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.
Irregular Repeat Accumulate Codes
, 2000
"... In this paper we will introduce an ensemble of codes called irregular repeataccumulate (IRA) codes. IRA codes are a generalization of the repeataccumulate codes introduced in [1], and as such have a natural linear time encoding algorithm. We shall prove that on the binary erasure channel, IRA code ..."
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Cited by 150 (1 self)
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In this paper we will introduce an ensemble of codes called irregular repeataccumulate (IRA) codes. IRA codes are a generalization of the repeataccumulate codes introduced in [1], and as such have a natural linear time encoding algorithm. We shall prove that on the binary erasure channel, IRA codes can be decoded reliably in linear time, using iterater] sumproduct decoding,a# ra#SJ a#SJ8T a#SJ8 close tocha#T36 ca pa#J464 Asimila# resulta#u ea#S to be true on the AWGN channel, although we have no proof of this. We illustrate our results with numerical and experimenta# examples.