Generalized Belief Propagation to break trapping sets in LDPC codes

We propose a novel concatenated coding scheme combined Low Density Parity Check(LDPC) code and vertical Bell Labs layered space-time (VBLAST) code to improve the bit error rate (BER) performance for Free-space optical (FSO) communication links with On Off Keying(OOK) modulation. In the scheme, π-rotation encoding and belief-propagation (BP) decoding algorithm are employed in LDPC code, and minimum mean-square error (MMSE) soft-decision decoding algorithm is used for VBLAST code. Moreover, a joint iterative detection and decoding algorithm of VBLAST code and LDPC code is used to improve the overall link performance. At last, the BER of the proposed scheme over lognormal atmospheric turbulent fading channels is analyzed by using Monte Carlo method. The result shows that the BER of the proposed concatenated coding scheme is much improved than that only use VBLAST code scheme.

Several combinatorial properties of low-density parity-check (LDPC) codes, such as minimum distance, diameter, stopping number, girth and cycle-length distribution of the corresponding Tanner graph, are known to influence their performance under iterative decoding. Recently, a new class of combinatorial configurations, termed trapping sets, was shown to be of significant importance in determining the properties of LDPC codes in the error-floor region. Very little is known both about the existence/parameters of trapping sets in structured LDPC codes and about possible techniques for reducing their negative influence on the code's performance. In this paper, we address both these problems from an algorithmic and combinatorial perspective. We first provide a numerical study of the trapping phenomena for the Margulis code, which exhibits a fairly high error-floor. Based on this analysis, conducted for two different implementations of iterative belief propagation, we propose a novel decoding process, termed averaged decoding. Averaged decoding provides for a significant reduction in the number of incorrectly decoded frames in the error-floor region of the Margulis code. Furthermore, based on the results of the algorithmic approach, we suggest a novel combinatorial characterizations of trapping sets in the class of LDPC codes based on finite geometries. Projective geometry LDPC codes are suspected to have extremely low error-floors, which is a property that we may attribute to the non-existence of certain small trapping sets in the code graph.

Low Density Parity Check (LDPC) codes have capacity-approaching performance over several channels of interest. In practice, for good block-error rate performance, the girth of the Tanner graph of an LDPC code needs to be as high as possible. In theory, to show that block-error rate approaches zero for increasing block-lengths, the girth of the Tanner graph sequence needs to tend to infinity with block-length. To meet these requirements, we construct sequences of large-girth irregular LDPC codes for a given degree-distribution pair (DDP) by applying a general node splitting algorithm on large girth graphs. The obtained Tanner graph meets the required DDP up to a suitable approximation. By optimizing the node-splitting method and using suitable large-girth graphs, we show code constructions with smaller block length for the same girth, when compared to previous constructions. Similar gains in block length are observed in the construction of sequences of large-girth protograph LDPC codes. Simulations, over a binary erasure channel, confirm the gains in block-error rate obtained by the large girth construction.

This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code’s parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.

Low density parity-check (LDPC) codes are a class of linear block codes that are decoded by running belief propagation (BP) algorithm or log-likelihood ratio belief propagation (LLR-BP) over the factor graph of the code. One of the disadvantages of LDPC codes is the onset of an error floor at high values of signal to noise ratio caused by trapping sets. In this paper, we propose a two stage decoder to deal with different types of trapping sets. Oscillating trapping sets are taken care by the first stage of the decoder and the elementary trapping sets are handled by the second stage of the decoder. Simulation results on the regular PEG (504,252,3,6) code and the irregular PEG (1024,518,15,8) code shows that the proposed two stage decoder performs significantly better than the standard decoder.

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that nonzero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth.

Low-density Parity check codes (LDPC) are gaining interest for high data rate application in both terrestrial and spatial communications which requires low bit error rate. This paper proposes an algorithm through which the best parity check matrix is obtained for irregular low density parity check codes, which reduces the Bit Error Rate ,Frame Error Rate and computational complexity. Tanner graph also called a bipartite graph format in an Low Density Parity Check (LDPC) code design contains many short cycles; it will produce a computational complexity in a code design process and it degrades the code performance. To overcome this effect, we proposed an new algorithm which obtains a best final parity check matrix. Throughout the paper we will give the detailed description of an efficient algorithm proposed. The complexity is studied graphically and analytically. Using proposed algorithm, cycles of length four observed in corresponding matrix are removed. Each matrix is evaluated over a noisy Additive White Gaussian Noise (AWGN) channel. Bit Error rate and Frame error rate is calculated. The results are compared with randomly generated best parity check matrix. Simulation results shows that proposed method successfully overcomes the computational complexity and meets the near Shannon limit.

In this paper, we present an algorithm to find all low-weight codewords in a given quasi-cyclic (QC) low-density parity-check (LDPC) code with a fixed column-weight and girth. The main idea is to view a low-weight codeword as an (a, 0) trapping sets, and then show that each topologically different (a, 0) trapping set can be dissected into smaller trapping sets. The proposed search method relies on the knowledge of possible topologies of such smaller trapping sets present in a code ensemble, which enables an efficient search. Combined with the high-rate QC LDPC code construction which successively adds blocks of permutation matrices, the algorithm ensures that in the code construction procedure all codewords up to a certain weight are avoided, which leads to a code with the desired minimum distance. The algorithm can be also used to determine the multiplicity of the low-weight codewords with different trapping set structure.

In this paper, we propose a concatenated code by combining a Recursive Systematic Convolutional (RSC) code with a Low-Density Parity-Check (LDPC) code. The proposed concatenated code is encoded in parallel by an RSC encoder and an LDPC encoder without interleavers between them. When decoding the proposed concatenated code, the information bits are decoded by two decoding algorithms, and soft information is exchanged between the RSC decoder and the LDPC decoder. This allows for the elimination of wrong codewords output by each decoder. The use of an LDPC code allows us to halt the decoding process when the valid codeword is obtained. We evaluate the error rate performance of the proposed concatenated code by computer simulations. We show that the proposed concatenated code does not have a high error floor, and achieves better BLock Error Rate (BLER) performance than either conventional LDPC codes or conventional turbo codes in the high Eb/No region. We also show that since the proposed concatenated code can correct wrong codewords output by each decoder, it achieves better BLER than the regular LDPC code at the same Bit Error Rate (BER).

In this paper we propose an innovative channel coding scheme called 'Accumulate Repeat Accumulate codes' (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of Low Density Parity Check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded Repeat Accumulate (RA) code or as precoded Irregular Repeat Accumulate (IRA) code, where simply an accumulator is chosen as a precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known LDPC codes with the dame maximum node degree. Furthermore by puncturing the accumulators any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation that allows for high speed decoder implementation.

Turbo codes are attractive compared with Low Density Parity Check (LDPC) codes for Forward Error Correction (FEC) applications mainly due to their superior performance, especially at low Signal-to-Noise Ratio (SNR) such as are common in Powerline channels. For example, IEEE 1901-FFT PHY used the Turbo coding scheme defined in the HomePlug AV standards. However, patent fees are usually required for each turbo-code enabled manufactured device. The objective of this paper is to examine whether unlicensed LDPC codes, with optimized choices of block lengths, could be a viable alternative for future Powerline Communications (PLC) applications. The paper shows that the performance of the LDPC codes can approximate that of the Turbo codes with higher block lengths, on channels with typical and realistic PLC characteristics. The paper also shows that the additional complexity associated with this increase in block length can be mitigated by the use of Quasi-Cyclic LDPC (QC-LDPC) codes.

Big data storage demands larger cluster. The increasing size of the cluster may lead to failures of larger number of nodes. To provide reliable big data storage, replications are not cost-effective and does not provide a robust solution to prevent data loss. Traditional erasure codes applied in the RAID system such as Reed-Solomon (R-S) based solutions have limitations in providing high reliability. This is because higher reliability requires erasure codes with larger size. The computational cost of the R-S codes increase quadratically with the number of failures the R-S codes can tolerate for the same redundancy rate. It has been shown that Low Density Parity Check (LDPC) codes have lower computational cost and repair network traffic compared with R-S based solutions. Unfortunately, there does not exist a construction method for LDPC codes with larger size to control the computational cost and repair traffic. In this paper, a novel method is proposed to construct a family of LDPC codes - expanCodes with expandable sizes. The proposed expanCodes allows the encoding and decoding complexity remain unchanged with the increase of the size of the LDPC codes. As a result, increased reliability can be achieved without additional computation and repair traffic. The proposed expanCodes is integrated with the Hadoop system. Simulations show that more than 29% decrease in terms of encoding and decoding latency compared with R-S based solutions.

International audience

DOI: 10.1049/cp:20061337 ISBN: 0 86341 644 6 Location: Hangzhou, China Conference date: 6-9 Nov. 2006 Format: PDF The BER performance of presented quasi-cyclic (QC) low-density parity-check (LDPC) codes is not good as that of randomly constructed LDPC codes, while the lack of structure of randomly constructed LDPC codes implies serious disadvantages in terms of storing and accessing a large parity-check matrix. To solve the problems, this paper proposes a design of good quasi-cyclic (QC) low-density parity-check (LDPC) codes, the obtained algebraically structured codes have large minimum distances and good BER performance. The proposed design is based on index matrices, which determines the shifts of the circulant matrices of the sparse parity-check matrices. Compared with the presented design of QC LDPC codes, the designed QC LDPC codes are free from girth 4, sometimes free from girth 6, and have much larger minimum distances. Based on the algebraic code structure, the conditions of the girth and minimum distance of the codes are found. The BER performance of the designed QC LDPC block codes compares with that of randomly constructed LDPC codes for any block lengths. Better BER performance is obtained by increasing the circulant size of the base QC code. Simulations verify the construction of the QC LDPC codes to be valid. (4 pages) Inspec keywords: AWGN channels; error statistics; channel coding; algebraic codes; parity check codes; sparse matrices Subjects: Algebra; Other topics in statistics; Codes

This dissertation presents a systematic exposition on finite-block-length coding theory and practice. We begin with the task of identifying the maximum achievable rates over noisy, finite-block-length constrained channels, referred to as (e, n)-capacity Cen, with e denoting target block-error probability and n the block length. We characterize how the optimum codes look like over finite-block-length constrained channels. Constructing good, short-block-length error-correcting codes defined on sparse graphs is the focus of the thesis. We propose a new, general method for constructing Tanner graphs having a large girth by progressively establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) construction. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (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 an extension of the Erdos-Sachs bound. The PEG construction proves to be powerful for generating good, short-block-length binary LDPC codes. Furthermore, we show that the binary interpretation of GF(2b) codes on the cycle Tanner graph TG(2, dc), if b grows sufficiently large, can be used over the binary-input additive white Gaussian noise (AWGN) channel as good code for optimum and good code for iterative decoding. Codes on sparse graphs are often decoded iteratively by a sum-product algorithm (SPA) with low complexity. We investigate efficient digital implementations of the SPA for decoding binary LDPC codes from both the architectural and algorithmic point of view, and describe new reduced-complexity derivatives thereof. The unified treatment of decoding techniques for LDPC codes provides flexibility in selecting the appropriate design point in high-speed applications from a performance, latency, and computational complexity perspective.