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59
Stopping set distribution of LDPC code ensembles
 IEEE TRANS. INFORM. THEORY
, 2005
"... Stopping sets determine the performance of lowdensity paritycheck (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tannergraph ensembles, including the following. An expression for the normalized average stoppin ..."
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Cited by 70 (1 self)
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Stopping sets determine the performance of lowdensity paritycheck (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tannergraph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a I
On the application of LDPC codes to arbitrary discrete memoryless channels
"... Abstract—We discuss three structures of modified lowdensity paritycheck (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the wellknown binary LDPC codes following constructions proposed by Gallager and McEliece, the seco ..."
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Cited by 54 (2 self)
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Abstract—We discuss three structures of modified lowdensity paritycheck (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the wellknown binary LDPC codes following constructions proposed by Gallager and McEliece, the second is based on LDPC codes of arbitrary (ary) alphabets employing moduloaddition, as presented by Gallager, and the third is based on LDPC codes defined over the field GF (). All structures are obtained by applying a quantization mapping on a coset LDPC ensemble. We present tools for the analysis of nonbinary codes and show that all configurations, under maximumlikelihood (ML) decoding, are capable of reliable communication at rates arbitrarily close to the capacity of any discrete memoryless channel. We discuss practical iterative decoding of our structures and present simulation results for the additive white Gaussian noise (AWGN) channel confirming the effectiveness of the codes. Index Terms —ary lowdensity parity check (LDPC), belief propagation, coset codes, iterative decoding, LDPC codes, turbo codes. 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 32 (6 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.
Construction of Short Block Length Irregular LowDensity ParityCheck Codes
, 2004
"... We present a construction algorithm for short block length irregular lowdensity paritycheck (LDPC) codes. Based on a novel interpretation of stopping sets in terms of the paritycheck matrix, we present an approximate trellisbased search algorithm that detects many stopping sets. Growing the parit ..."
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Cited by 31 (11 self)
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We present a construction algorithm for short block length irregular lowdensity paritycheck (LDPC) codes. Based on a novel interpretation of stopping sets in terms of the paritycheck matrix, we present an approximate trellisbased search algorithm that detects many stopping sets. Growing the parity check matrix by a combination of random generation and the trellisbased search, we obtain codes that possess error floors orders of magnitude below randomly constructed codes and significantly better than other comparable constructions.
On decoding of lowdensity paritycheck codes over the binary erasure channel
 IEEE Trans. Inform. Theory
, 2004
"... Abstract—This paper investigates decoding of lowdensity paritycheck (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximumlikelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improv ..."
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Cited by 30 (0 self)
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Abstract—This paper investigates decoding of lowdensity paritycheck (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximumlikelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graphtheoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finitelength codes. Index Terms—Bipartite graphs, erasure channel, improved decoding, iterative decoding, lowdensity paritycheck (LDPC) codes, maximumlikelihood (ML) decoding, performance bound. I.
Capacityachieving codes with bounded graphical complexity on noisy channels
 in Proc. Allerton Conf. Commun., Control
, 2005
"... We introduce a new family of concatenated codes with an outer lowdensity paritycheck (LDPC) code and an inner lowdensity generator matrix (LDGM) code, and prove that these codes can achieve capacity under any memoryless binaryinput outputsymmetric (MBIOS) channel using maximumlikelihood (ML) de ..."
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Cited by 25 (3 self)
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We introduce a new family of concatenated codes with an outer lowdensity paritycheck (LDPC) code and an inner lowdensity generator matrix (LDGM) code, and prove that these codes can achieve capacity under any memoryless binaryinput outputsymmetric (MBIOS) channel using maximumlikelihood (ML) decoding with bounded graphical complexity, i.e., the number of edges per information bit in their graphical representation is bounded. We also show that these codes can achieve capacity for the special case of the binary erasure channel (BEC) under belief propagation (BP) decoding with bounded decoding complexity per information bit for all erasure probabilities in (0, 1). By deriving and analyzing the average weight distribution (AWD) and the corresponding asymptotic growth rate of these codes with a rate1 inner LDGM code, we also show that these codes achieve the GilbertVarshamov bound with asymptotically high probability. This result can be attributed to the presence of the inner rate1 LDGM code, which is demonstrated to help eliminate high weight codewords in the LDPC code while maintaining a vanishingly small amount of low weight codewords. 1
Smooth compression, Gallager bound and Nonlinear sparsegraph codes
"... Abstract — A data compression scheme is defined to be smooth if its image (the codeword) depends gracefully on the source (the data). Smoothness is a desirable property in many practical contexts, and widely used source coding schemes lack of it. We introduce a family of smooth source codes based on ..."
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Cited by 13 (0 self)
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Abstract — A data compression scheme is defined to be smooth if its image (the codeword) depends gracefully on the source (the data). Smoothness is a desirable property in many practical contexts, and widely used source coding schemes lack of it. We introduce a family of smooth source codes based on sparse graph constructions, and prove them to achieve the (information theoretic) optimal compression rate for a dense set of iid sources. As a byproduct, we show how Gallager bound on sparsity can be overcome using nonlinear function nodes. I.
On the growth rate of the weight distribution of irregular doublygeneralized LDPC codes
 in Proc. 2008 Allerton Conf. on Communications, Control & Computing
, 2008
"... In this paper, an expression for the asymptotic growth rate of the number of small linearweight codewords of irregular doublygeneralized LDPC (DGLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or ..."
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Cited by 10 (7 self)
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In this paper, an expression for the asymptotic growth rate of the number of small linearweight codewords of irregular doublygeneralized LDPC (DGLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or variable node minimum distance greater than 2 are shown to be asymptotically good, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linearweight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of DGLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of DGLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular DGLDPC codes.
Tight exponential upper bounds on the ML decoding error probability of block codes over fullyinterleaved fading channels
 IEEE Trans. on Communications
, 2003
"... We derive in this paper tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximumlikelihood decoded. It is assumed that the fading samples are statistically independent and that ..."
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Cited by 8 (1 self)
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We derive in this paper tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximumlikelihood decoded. It is assumed that the fading samples are statistically independent and that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of the Gallager bounds. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region exceeding the cutoff rate of the channel. By inserting interconnections between these bounds, we show that they are generalized versions of some reported bounds for the binaryinput AWGN channel.
Protograph LDPC codes over burst erasure channels
 IEEE Military Commun. Conf., MILCOM
, 2006
"... Abstract — In this paper we design high rate protograph based LDPC codes suitable for binary erasure channels. To simplify the encoder and decoder implementation for high data rate transmission, the structure of codes are based on protographs and circulants. These LDPC codes can improve data link an ..."
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Cited by 8 (1 self)
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Abstract — In this paper we design high rate protograph based LDPC codes suitable for binary erasure channels. To simplify the encoder and decoder implementation for high data rate transmission, the structure of codes are based on protographs and circulants. These LDPC codes can improve data link and network layer protocols in support of communication networks. Two classes of codes were designed. One class is designed for large block sizes with an iterative decoding threshold that approaches capacity of binary erasure channels. The other class is designed for short block sizes based on maximizing minimum stopping set size. For high code rates and short blocks the second class outperforms the first class. A scheme is proposed to use these LDPC codes over burst erasure channels. The proposed encoding method is also applicable to cases when packets are frequency hopped over channels with partial band jamming or frequency selective fading. Various LDPC codes are compared and simulation results are provided. I.