On the Design of Fast Convergent LDPC Codes for the BEC: An Optimization Approach (original) (raw)
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In this paper, we present a novel way for solving the main problem of designing the capacity approaching irregular lowdensity parity-check (LDPC) code ensemble over binary erasure channel (BEC). The proposed method is much simpler, faster, accurate and practical than other methods. Our method does not use any relaxation or any approximate solution like previous works. Our method works and finds optimal answer for any given check node degree distribution. The proposed method was implemented and it works well in practice with polynomial time complexity. As a result, we represent some degree distributions that their rates are close to the capacity with maximum erasure probability and maximum code rate.
Deterministic Design of Low-Density Parity-Check Codes for Binary Erasure Channels
IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference, 2007
We propose a deterministic method to design irregular Low-Density Parity-Check (LDPC) codes for binary erasure channels (BEC). Compared to the existing methods, which are based on the application of asymptomatic analysis tools such as density evolution or Extrinsic Information Transfer (EXIT) charts in an optimization process, the proposed method is much simpler and faster. Through a number of examples, we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization. An important property of the proposed designs is the flexibility to select the number of constituent variable node degrees P. The proposed designs include existing deterministic designs as a special case with P = N-1, where N is the maximum variable node degree. Compared to the existing deterministic designs, for a given rate and a given δ > 0, the designed ensembles can have a threshold in δ-neighborhood of the capacity upper bound with smaller values of P and N. They can also achieve the capacity of the BEC as N, and correspondingly P and the maximum check node degree tend to infinity. Index Terms-channel coding, low-density parity-check (LDPC) codes, binary erasure channel (BEC), deterministic design. I. INTRDOUCTION Low-Density Parity-Check (LDPC) codes have received much attention in the past decade due to their attractive performance/complexity tradeoff on a variety of communication channels. In particular, on the Binary Erasure Channel (BEC), they achieve the channel capacity asymptotically [1-4]. In [1],[5],[6] a complete mathematical analysis for the performance of LDPC codes over the BEC, both asymptotically and for finite block lengths, has been developed. For other types of channels such as the Binary Symmetric Channel (BSC) and the Binary Input Additive White Gaussian Noise (BIAWGN) channel, only asymptotic analysis is available [7]. For irregular LDPC codes, the problem of finding ensemble
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2008
Recent works showed how low-density parity-check (LDPC) erasure correcting codes, under maximum likelihood (ML) decoding, are capable of tightly approaching the performance of an ideal maximum-distance-separable code on the binary erasure channel. Such result is achievable down to low error rates, even for small and moderate block sizes, while keeping the decoding complexity low, thanks to a class of decoding algorithms which exploits the sparseness of the paritycheck matrix to reduce the complexity of Gaussian elimination (GE). In this paper the main concepts underlying ML decoding of LDPC codes are recalled. A performance analysis among various LDPC code classes is then carried out, including a comparison with fixed-rate Raptor codes. The results show that LDPC and Raptor codes provide almost identical performance in terms of decoding failure probability vs. overhead. Index Terms-LDPC codes, Raptor codes, binary erasure channel, maximum likelihood decoding, ideal codes, MBMS, packet-level coding. • Wireless video/audio streaming. Link-layer coding is currently applied to the video streams in the framework of the DVB-H/SH standards. In such a context, erasure correcting codes take care of the fading mitigation, which is
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In this paper, a method for the asymptotic analysis of generalized low-density parity-check (GLDPC) codes on the binary erasure channel (BEC) is proposed. The considered GLDPC codes have block linear codes as check nodes. Instead of considering specific check component codes, like Hamming or BCH codes, random codes are considered, and a technique is developed for obtaining the expected check EXIT function for the overall GLDPC code. Each check component code is supposed to belong to an expurgated ensemble. Some GLDPC thresholds obtained by this technique are compared with those of GLDPC codes, with the same distribution and component codes lengths, using specific codes. Results obtained by combining our analysis with differential evolution tool are also presented.
IEEE Transactions on Information Theory, 2000
A method for the asymptotic analysis of doublygeneralized low-density parity-check (D-GLDPC) codes on the binary erasure channel (BEC) is described. The proposed method is based on extrinsic information transfer (EXIT) charts. It permits to overcome the impossibility to evaluate the EXIT function for the check or variable component codes, in situations where the information functions or split information functions for the component code are unknown. According to the proposed method, D-GLDPC codes where the check and variable component codes are random codes from an expurgated ensemble, are considered. A technique is then developed which permits to obtain the EXIT chart for the overall D-GLDPC code, by evaluating the expected EXIT function for each check and variable component code. This technique is then combined with differential evolution (DE) algorithm in order to generate some optimal D-GLDPC degree distributions. Numerical results on long, random codes, are presented which reveal how D-GLDPC codes can outperform standard LDPC codes in terms of both waterfall performance and error floor.
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IEEE Transactions on Information Theory, 2004
We derive upper bounds on the maximum achievable rate of low-density parity-check (LDPC) codes used over the binary erasure channel (BEC) under Gallager's decoding algorithm, given their right-degree distribution. We demonstrate the bounds on the ensemble of right-regular LDPC codes and compare them with an explicit left-degree distribution constructed from the given right degree.
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IEEE Communications Letters, 2005
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Binary linear-time erasure decoding for non-binary LDPC codes
2009 IEEE Information Theory Workshop, 2009
In this paper, we first introduce the extended binary representation of non-binary codes, which corresponds to a covering graph of the bipartite graph associated with the non-binary code. Then we show that non-binary codewords correspond to binary codewords of the extended representation that further satisfy some simplex-constraint: that is, bits lying over the same symbol-node of the non-binary graph must form a codeword of a simplex code. Applied to the binary erasure channel (BEC), this description leads to a binary erasure decoding algorithm of non-binary LDPC codes, whose complexity depends linearly on the cardinality of the alphabet. We also give insights into the structure of stopping sets for non-binary LDPC codes, and discuss several aspects related to upper-layer FEC applications.
Asymptotic and finite-length optimization of LDPC codes
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This thesis addresses the problem of information transmission over noisy channels. In 1993, Berrou, Glavieux and Thitimajshima discovered Turbo-codes. These codes made it possible to achieve a very good performance at low decoding complexity. In hindsight, it was recognized that the underlying principle of using sparse graph codes in conjunction with message-passing decoding was the same as the one proposed in Gallager's remarkable thesis of 1963, although Gallager's work had all but been forgotten in the intervening 30 years. In 1997, there occurred a breakthrough in the analysis of this type of codes. Luby, Mitzenmacher, Shokrollahi, Spielman and Stemann were able to give a complete characterization of the behavior of Low-Density Parity-Check code ensembles in the infinite blocklength case when used over the binary erasure channel. Soon thereafter, Richardson and Urbanke extended their results to binary-input memoryless symmetric channels. In this thesis we present tools to analyze the performance of these types of codes and ways to optimize their parameters. The optimization for the infinite blocklength case is relatively straightforward and we give a simple and efficient method of doing so. However, this does not really solve the problem in practice, since the asymptotic analysis has only limited relevance for the short or moderate blocklengths that are typically used in practice. This brings us to the main objective of this thesis, which is to bridge the gap between the asymptotic case and the practical finite-length case. We follow the lead of Luby et al. by considering the binary erasure channel. We show that the performance of LDPC codes obeys a well-defined scaling law as the blocklength increases. This scaling law refines the asymptotic analysis and provides a good way to understand and approximate the behavior of LDPC codes of short to moderate length. We show how to compute the scaling parameters involved and demonstrate how to use the resulting approximation as a design and optimization tool.
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2009
The optimal complexity-rate tradeoff for errorcorrecting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the joint optimization of rate and decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. Under certain mild conditions, the complexity measure is a convex function of the variable edge-degree distribution of the code, allowing an efficient design of complexity-optimized LDPC codes using convex optimization methods. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform thresholdoptimized codes at long block lengths.