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20
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.
Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Cited by 97 (1 self)
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Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofasimple lemma for linear codes over GF (p) used with plevel amplitude modulation. The relation between the combinatorial packing of solid bodies and the informationtheoretic “soft packing ” with arbitrarily small, but positive, overlap is illuminated. The “softpacking” results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda–Poltyrev result that spherically shaped lattice codes and adecoder that is unaware of the shaping can achieve the rate 1=2 log2 (P=N).
On the construction of some capacityapproaching coding schemes
, 2000
"... This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source ..."
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Cited by 82 (2 self)
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This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint sourcechannel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signaltonoise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on spacefilling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on spacefilling curves operates within 1.1 dB of Shannon’s ratedistortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the ratedistortion bound. The second scheme is based on lowdensity paritycheck (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution
On the performance of lossless joint sourcechannel coding based on linear codes
 in Proc. ITW 2006
, 2006
"... A general lossless joint sourcechannel coding scheme based on linear codes and random interleavers for multipleaccess channels is presented and then analyzed in this paper. By the informationspectrum approach and the codespectrum approach, it is shown that a linear code with good joint spectrum ..."
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Cited by 6 (4 self)
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A general lossless joint sourcechannel coding scheme based on linear codes and random interleavers for multipleaccess channels is presented and then analyzed in this paper. By the informationspectrum approach and the codespectrum approach, it is shown that a linear code with good joint spectrum can be used to establish limitapproaching joint sourcechannel coding schemes for correlated general sources and general multipleaccess channels, where the joint spectrum of the code is a generalization of the inputoutput weight distribution. Some properties of linear codes with good joint spectrums are investigated. A formula on the “distance ” property of good linear codes is derived, and based on this formula, it is further proved that the rate of a good systematic code can not be larger than a constant rate and sparse generator matrices can not yield good linear codes. The problem of designing variable rate coding schemes is also discussed. A novel idea called “generalized puncturing ” is presented, and hence one good low rate linear code is enough for the design of variable rate coding schemes. Finally, based on the results of this paper and previous literatures, the criteria and candidates of good linear codes for various problems of multipleaccess channels are reviewed in a unified framework by the codespectrum approach. Index Terms Code spectrum, correlated sources, multipleaccess channels, information spectrum, linear codes, lossless joint sourcechannel
Multistage computeandforward with multilevel lattice codes based on product constructions
 in Proc. IEEE ISIT
, 2014
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Some Properties of Bit Decoding Algorithms for Binary Linear Block Codes
, 2003
"... In this paper, we study certain properties of the bit decoding algorithms for the case of binary linear block codes. Our focus is on the Probability Density Function (pdf ) of the bit LogLikelihoodRatio (LLR). A general channel model with discrete input and discrete or continuous output is conside ..."
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Cited by 4 (4 self)
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In this paper, we study certain properties of the bit decoding algorithms for the case of binary linear block codes. Our focus is on the Probability Density Function (pdf ) of the bit LogLikelihoodRatio (LLR). A general channel model with discrete input and discrete or continuous output is considered. We prove that under a set of mild conditions on the channel, the pdf of the bit LLR of a specific bit position is independent of the transmitted codeword. It is also
CONSTRUCTIVE CONJUGATE CODES FOR QUANTUM ERROR CORRECTION AND CRYPTOGRAPHY
, 2007
"... A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding CalderbankShorSteane (CSS) quantum errorcorrecting code. It is known that conjugate code pairs are applicable ..."
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Cited by 4 (3 self)
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A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding CalderbankShorSteane (CSS) quantum errorcorrecting code. It is known that conjugate code pairs are applicable to quantum cryptography. In this work, a polynomial construction of conjugate code pairs is presented. The constructed pairs achieve the highest known achievable rate on additive channels, and are decodable with algorithms of polynomial complexity.
On the Basic Averaging Arguments For Linear Codes
"... Linear codes over F q are considered for use in detecting and in correcting the additive errors in some subset E of F q .(Themost familiar example of such an error set E is the set of all ntuples of Hamming weight at most t.) In this setup, the basic averaging arguments for linear codes are revie ..."
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Cited by 2 (0 self)
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Linear codes over F q are considered for use in detecting and in correcting the additive errors in some subset E of F q .(Themost familiar example of such an error set E is the set of all ntuples of Hamming weight at most t.) In this setup, the basic averaging arguments for linear codes are reviewed with emphasis on the relation between the combinatorial and the informationtheoretic viewpoint. The main theorems are (a correspondingly general version of) the VarshamovGilbert bound and a `randomcoding' bound on the probability of an ambiguous syndrome. These bounds are shown to result from applying the same elementary averaging argument to two different packing problems, viz., the combinatorial `sphere' packing problem and theprobabilistic `Shannon packing'. Some applications of the general bounds are indicated, e.g., hash functions and Euclideanspace codes, and the connection to Justesentype constructions of asymptotically good codes is outlined.
Invariance Properties of Binary Linear Block Codes over a Memoryless Channels with Discrete Input
, 2005
"... This work studies certain properties of the Probability Density Function (pdf) of the bit LogLikelihoodRatio (LLR) for binary linear block codes over a memoryless channel with discrete input and discrete or continuous output. We prove that under a set of mild conditions on the channel, the pdf of ..."
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Cited by 2 (2 self)
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This work studies certain properties of the Probability Density Function (pdf) of the bit LogLikelihoodRatio (LLR) for binary linear block codes over a memoryless channel with discrete input and discrete or continuous output. We prove that under a set of mild conditions on the channel, the pdf of the bit LLR of a specific bit position is independent of the transmitted codeword. It is also shown that the pdf of a given bit LLR when the corresponding bit takes the values of zero and one are symmetric with respect to each other (reflection of one another with respect to the vertical axis). For the case of channels with binary input, a sufficient condition for two bit positions to have the same pdf is presented.
Invariance Properties and Performance Evaluation of Bit Decoding Algorithms
, 2004
"... Certain properties of optimal bitwise APP (A Posteriori Probability) decoding of binary linear block codes are studied. The focus is on the Probability Density Function (pdf) of the bit LogLikelihoodRatio (LLR). A general channel model with discrete (not necessarily binary) input and discrete or c ..."
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Cited by 1 (0 self)
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Certain properties of optimal bitwise APP (A Posteriori Probability) decoding of binary linear block codes are studied. The focus is on the Probability Density Function (pdf) of the bit LogLikelihoodRatio (LLR). A general channel model with discrete (not necessarily binary) input and discrete or continuous output is considered. It is proved that under a set of mild conditions on the channel, the pdf of the bit LLR of a specific bit position is independent of the transmitted codeword. It is also shown that the pdf of a given bit LLR, when the corresponding bit takes the values of zero and one, are symmetric with respect to each other (reflection of one another with respect to the vertical axis). In the case of channels with binary inputs, a su#cient condition for two bit positions to have the same pdf is presented. An analytical method for approximate performance evaluation of binary linear block codes using an Additive White Gaussian Noise (AWGN) channel model with Binary Phase Shift Keying (BPSK) modulation is proposed. The pdf of the bit LLR is expressed in terms of the GramCharlier series expansion. This expansion requires knowledge of the statistical moments of the bit LLR. An analytical method for calculating these moments which is based on some recursive calculations involving certain weight enumerating functions of the code is introduced. It is proved that the approximation can be as accurate as desired, using enough numbers of terms in the GramCharlier series expansion. A new method for the performance evaluation of TurboLike Codes is presented. The method is based on estimating the pdf of the bit LLR by using an exponential model. The moment matching method is combined with the maximum entropy principle to estimate iii the parameters of the new model. A ...