Bounds on Achievable Rates of LDPC Codes Used Over the Binary Erasure Channel (original) (raw)
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Optimal rate and maximum erasure probability LDPC codes in binary erasure channel
2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2011
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
New Sequences of Capacity Achieving LDPC Code Ensembles Over the Binary Erasure Channel
2010
In this paper, new sequences (n ; n) of capacity achieving low-density parity-check (LDPC) code ensembles over the binary erasure channel (BEC) is introduced. These sequences include the existing sequences by Shokrollahi et al. as a special case. For a fixed code rate R, in the set of proposed sequences, Shokrollahi's sequences are superior to the rest of the set in that for any given value of n, their threshold is closer to the capacity upper bound 1 0 R. For any given , 0 < < 1 0 R, however, there are infinitely many sequences in the set that are superior to Shokrollahi's sequences in that for each of them, there exists an integer number n0, such that for any n > n0, the sequence (n ; n) requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within-neighborhood of the capacity upper bound 1 0 R. Moreover, it is proven that the check-regular subset of the proposed sequences are asymptotically quasi-optimal, i.e., their decoding complexity increases only logarithmically with the relative increase of the threshold. A stronger result on asymptotic optimality of some of the proposed sequences is also established. Index Terms-Asymptotically optimal sequences, binary erasure channel (BEC), capacity achieving sequences, check regular ensembles, low-density parity-check codes (LDPC).
Analysis of generalized LDPC codes with random component codes for the binary erasure channel
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.
Nonbinary spatially-coupled LDPC codes on the binary erasure channel
2013 IEEE International Conference on Communications (ICC), 2013
We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes, hence we conjecture that the BP threshold for large termination length approaches the MAP threshold of the underlying regular ensemble.
Upper bounds on the rate of LDPC codes
2002
Abstract We derive upper bounds on the rate of low-density parity-check (LDPC) codes for which reliable communication is achievable. We first generalize Gallager's (1963) bound to a general binary-input symmetric-output channel. We then proceed to derive tighter bounds. We also derive upper bounds on the rate as a function of the minimum distance of the code. We consider both individual codes and ensembles of codes.
Optimal regular LDPC codes for the binary erasure channel
IEEE Communications Letters, 2005
We prove that for any given R between 0 and 1 the best threshold value for a regular LDPC code of rate R with common variable degree v and common check degree c occurs when v is at least 3 and is minimal subject to the condition R=1-v/c.
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.
A decoding algorithm for LDPC codes over erasure channels with sporadic errors
2010
An efficient decoding algorithm for low-density parity-check (LDPC) codes on erasure channels with sporadic errors (i.e., binary error-and-erasure channels with error probability much smaller than the erasure probability) is proposed and its performance analyzed. A general single-error multipleerasure (SEME) decoding algorithm is first described, which may be in principle used with any binary linear block code. The algorithm is optimum whenever the non-erased part of the received word is affected by at most one error, and is capable of performing error detection of multiple errors. An upper bound on the average block error probability under SEME decoding is derived for the linear random code ensemble. The bound is tight and easy to implement. The algorithm is then adapted to LDPC codes, resulting in a simple modification to a previously proposed efficient maximum likelihood LDPC erasure decoder which exploits the parity-check matrix sparseness. Numerical results reveal that LDPC codes under efficient SEME decoding can closely approach the average performance of random codes.
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.