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183
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.
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.
Applications of LDPC Codes to the Wiretap Channel
, 2007
"... With the advent of quantum key distribution (QKD) systems, perfect (i.e. informationtheoretic) security can now be achieved for distribution of a cryptographic key. QKD systems and similar protocols use classical errorcorrecting codes for both error correction (for the honest parties to correct er ..."
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Cited by 72 (4 self)
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With the advent of quantum key distribution (QKD) systems, perfect (i.e. informationtheoretic) security can now be achieved for distribution of a cryptographic key. QKD systems and similar protocols use classical errorcorrecting codes for both error correction (for the honest parties to correct errors) and privacy amplification (to make an eavesdropper fully ignorant). From a coding perspective, a good model that corresponds to such a setting is the wire tap channel introduced by Wyner in 1975. In this paper, we study fundamental limits and coding methods for wire tap channels. We provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel. We also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes. In some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity. For the special case of a noiseless main channel and binary erasure channel, we consider encoder and decoder design for codes achieving secrecy on the wiretap channel; we show that it is possible to construct lineartime decodable secrecy codes based on LDPC codes that achieve secrecy.
Paritycheck density versus performance of binary linear block codes over memoryless symmetric channels
 IEEE Trans. on Information Theory
, 2003
"... Lowdensity paritycheck (LDPC) codes are efficiently encoded and decoded due to the sparseness of their paritycheck matrices. Motivated by their remarkable performance and feasible complexity under iterative messagepassing decoding, we derive lower bounds on the density of paritycheck matrices o ..."
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Cited by 63 (21 self)
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Lowdensity paritycheck (LDPC) codes are efficiently encoded and decoded due to the sparseness of their paritycheck matrices. Motivated by their remarkable performance and feasible complexity under iterative messagepassing decoding, we derive lower bounds on the density of paritycheck matrices of binary linear codes whose transmission takes place over a memoryless binaryinput outputsymmetric (MBIOS) channel. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes. For every MBIOS channel, we construct a sequence of ensembles of regular LDPC codes, so that an upper bound on the asymptotic density of their paritycheck matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of rightregular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative messagepassing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations
Asymptotic enumeration methods for analyzing LDPC codes
 IEEE Trans. Inform. Theory
, 2004
"... We show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of lowdensity paritycheck (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ense ..."
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Cited by 59 (2 self)
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We show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of lowdensity paritycheck (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ensembles. We then consider the binary erasure channel (BEC). Using these estimates we derive lower bounds on the error exponent, under iterative decoding, of LDPC codes used over the BEC. Both regular and irregular code structures are considered. These bounds are compared to the corresponding bounds when optimal (maximum likelihood) decoding is applied.
SparseGraph Codes for Quantum ErrorCorrection
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2004
"... We present sparsegraph codes appropriate for use in quantum errorcorrection. Quantum errorcorrecting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number ..."
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Cited by 48 (0 self)
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We present sparsegraph codes appropriate for use in quantum errorcorrection. Quantum errorcorrecting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number of quantum interactions associated with the quantum errorcorrection process small: a constant number per quantum bit, independent of the blocklength. Third, sparsegraph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum errorcorrecting codes.
Design and analysis of nonbinary LDPC codes for arbitrary discretememoryless channels
 IEEE TRANS. INFORM. THEORY
, 2005
"... We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discretememoryless channels (particularly nonbinary and asymmetric channels). We use a randomcoset analysis to produce an effect that is similar to outputsymmetry with binary channels ..."
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Cited by 43 (1 self)
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We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discretememoryless channels (particularly nonbinary and asymmetric channels). We use a randomcoset analysis to produce an effect that is similar to outputsymmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) paritycheck matrix induces a permutationinvariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q − 1 dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the AWGN channel, including results within 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz.
BlockLDPC: A practical LDPC coding system design approach
 IEEE Trans. Circuits Syst
, 2005
"... Abstract—This paper presents a joint lowdensity paritycheck (LDPC) codeencoderdecoder design approach, called BlockLDPC, for practical LDPC coding system implementations. The key idea is to construct LDPC codes subject to certain hardwareoriented constraints that ensure the effective encoder a ..."
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Cited by 33 (1 self)
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Abstract—This paper presents a joint lowdensity paritycheck (LDPC) codeencoderdecoder design approach, called BlockLDPC, for practical LDPC coding system implementations. The key idea is to construct LDPC codes subject to certain hardwareoriented constraints that ensure the effective encoder and decoder hardware implementations. We develop a set of hardwareoriented constraints, subject to which a semirandom approach is used to construct BlockLDPC codes with good errorcorrecting performance. Correspondingly, we develop an efficient encoding strategy and a pipelined partially parallel BlockLDPC encoder architecture, and a partially parallel BlockLDPC decoder architecture. We present the estimation of BlockLDPC coding system implementation key metrics including the throughput and hardware complexity for both encoder and decoder. The good errorcorrecting performance of BlockLDPC codes has been demonstrated through computer simulations. With the effective encoder/decoder design and good errorcorrecting performance, BlockLDPC provides a promising vehicle for reallife LDPC coding system implementations. Index Terms—Decoder, encoder, lowdensity parity check (LDPC), very largescale integration (VLSI) architecture. I.
Density Evolution for Asymmetric Memoryless Channels
 3rd International Symposium on Turbo Codes and Related Topics
"... Abstract — Density evolution is one of the most powerful analytical tools for lowdensity paritycheck (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied t ..."
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Cited by 31 (5 self)
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Abstract — Density evolution is one of the most powerful analytical tools for lowdensity paritycheck (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for nonsymmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. zchannels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels. Index Terms — Lowdensity paritycheck (LDPC) codes, density evolution, sumproduct algorithm, asymmetric channels, zchannels, rank of random matrices. I.