Structured low-density parity-check codes

This article describes the different methods to design regular low density parity-check (LDPC) codes with large girth. In graph terms, this corresponds to designing bipartite undirected regular graphs with large girth. Large girth speeds the convergence of iterative decoding and improves the performance at least in the high SNR range, by slowing down the onsetting of the error floor. We reviewed several existing constructions from exhaustive search to highly structured designs based on Euclidean and projective finite geometries and combinatorial designs. We describe GB and TS LDPC codes and compared the BER performance with large girth to the BER performance of random codes. These studies confirm that in the high SNR regime these codes with high girth exhibit better BER performance. The regularity of the codes provides additional advantages that we did not explore in this article like the simplicity of their hardware implementation and fast encoding.     

On maximum-length linear congruential-sequences-based low-density parity-check codes

This letter extends a low-density parity-check code construction using maximum-length linear congruential sequences by Prabhakar and Narayanan. The corresponding bipartite graphs of their construction were guaranteed to have a girth larger than four by a sufficient condition. However, their sufficient condition was limited to regular codes and data-node degree equal to three. The extension in this letter allows arbitrary data-node degrees and is applicable to irregular codes. Further, simpler sufficient conditions are derived and larger girths are addressed.     

Analysis of low-density parity-check codes based on EXIT functions

We exploit extrinsic information transfer functions of single parity-check and repetition codes over the binary input additive white Gaussian noise (biAWGN) channel, derived by the authors, for asymptotic performance analysis of belief propagation decoding of low-density parity-check codes. The approach is based on a Gaussian approximation (GA) of the density evolution algorithm using the mutual information measure. We show that this method allows more accurate prediction of the decoding threshold in the biAWGN channel than the earlier known GA methods.     

Simple reconfigurable low-density parity-check codes

Low-density parity-check codes (LDPCC) have been recently investigated as a possible solution for high data rate applications, for both space and terrestrial wireless communications. A main issue is the research of low complexity encoding and decoding schemes. In this letter we present a class of reconfigurable LDPCC characterized by low encoding and decoding complexity: we call them generalized irregular repeat-accumulate (GeIRA) codes.     

Near-Shannon-limit quasi-cyclic low-density parity-check codes

This letter presents two classes of quasi-cyclic low-density parity-check codes that perform close to the Shannon limit.     

Incremental redundancy hybrid ARQ schemes based on low-density parity-check codes

We study the throughput of hybrid automatic retransmission request (H-ARQ) schemes based on incremental redundancy (IR) over a block-fading channel. We provide an information-theoretic analysis assuming binary random coding and typical-set decoding. Then, we study the performance of low-density parity-check (LDPC) code ensembles with iterative belief-propagation decoding, and show that, under the hypothesis of infinite-length codes, LDPCs yield almost optimal performance. Unfortunately, standard finite-length LDPC ensembles incur a considerable performance loss with respect to their infinite-length counterpart, because of their poor frame-error rate (FER) performance. In order to recover part of this loss, we propose two simple yet effective methods: using a modified LDPC ensemble designed to improve the FER; and using an outer selective-repeat protocol acting on smaller packets of information bits. Surprisingly, these apparently very different methods yield almost the same performance gain and recover a considerable fraction of the optimal throughput, thus making practical finite-length LDPC codes very attractive for data wireless communications based on IR H-ARQ schemes.     

数字调幅广播中基于LDPC码的多级编码方案

研究了DAMB系统中基于LDPC码的多级编码方案。首先,设计了MLC分量码的码率,从而给出一种具有自适应特性的以LDPC码为分量码的多级编码方案。提出了基于可靠信息传播的多级译码方案,该方案不会造成信息丢失,且不需要重编码。构造了一组性能优异的非正则LDPC码并在DAMB信道进行了差错率仿真,结果表明基于LDPC码的MLC方案全面优于标准中RCPC/MLC方案。     

Capacity-approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes

We design multilevel coding (MLC) and bit-interleaved coded modulation (BICM) schemes based on low-density parity-check (LDPC) codes. The analysis and optimization of the LDPC component codes for the MLC and BICM schemes are complicated because, in general, the equivalent binary-input component channels are not necessarily symmetric. To overcome this obstacle, we deploy two different approaches: one based on independent and identically distributed (i.i.d.) channel adapters and the other based on coset codes. By incorporating i.i.d. channel adapters, we can force the symmetry of each binary-input component channel. By considering coset codes, we extend the concentration theorem based on previous work by Richardson et al. ( see ibid., vol.47, p.599-618, Feb. 2001) and Kavc/spl caron/ic/spl acute/ et al.(see ibid., vol.49, p.1636-52, July 2003) We also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior. Next, we jointly optimize the code rates and degree distribution pairs of the LDPC component codes for the MLC scheme. The optimized irregular LDPC codes at each level of MLC with multistage decoding (MSD) are able to perform well at signal-to-noise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel. We also show that the optimized BICM scheme can approach the parallel independent decoding (PID) capacity as closely as does the MLC/PID scheme. Simulations with very large codeword length verify the accuracy of the analytical results. Finally, we compare the simulated performance of these coded modulation schemes at finite codeword lengths, and consider the results from the perspective of a random coding exponent analysis.     

Rate-compatible puncturing of low-density parity-check codes

In this correspondence, we consider puncturing of low-density parity-check (LDPC) codes for additive white Gaussian noise (AWGN) channels. We show that good puncturing patterns exist and that the puncturing can be performed in a rate-compatible fashion. Furthermore, rate-compatible puncturing results in a small loss of performance with respect to threshold, namely, the punctured code is good (in terms of threshold) across a range of rates when compared with the optimal codes for each rate. This allows one to implement a single "mother" encoder and decoder that is good across a wide range of rates.     

Graph-theoretic construction of low-density parity-check codes

This article presents a graph-theoretic method for constructing low-density parity-check (LDPC) codes from connected graphs without the requirement of large girth. This method is based on finding a set of paths in a connected graph, which satisfies the constraint that any two paths in the set are either disjoint or cross each other at one and only one vertex. Two trellis-based algorithms for finding these paths are devised. Good LDPC codes of practical lengths are constructed and they perform well with iterative decoding.     

Bootstrap decoding of low-density parity-check codes

An initial bootstrap step for the decoding of low-density parity-check (LDPC) codes is proposed. Decoding is initiated by first erasing a number of less reliable bits. New values and reliabilities are then assigned to erasure bits by passing messages from nonerasure bits through the reliable check equations. The bootstrap step is applied to the weighted bit-flipping algorithm to decode a number of LDPC codes. Large improvements in both performance and complexity are observed.     

Application of efficient Chase algorithm in decoding of generalized low-density parity-check codes

We consider the iterative decoding of generalized low-density (GLD) parity-check codes where, rather than employ an optimal subcode decoder, a Chase (1972) algorithm decoder more commonly associated with "turbo product codes" is used. GLD codes are low-density graph codes in which the constraint nodes are other than single parity-checks. For extended Hamming-based GLD codes, we use bit error rates derived by simulation to demonstrate this new strategy to be successful at higher code rates. For long block lengths, good performance close to capacity is possible with decoding costs reduced further since the Chase decoder employed is an efficient implementation.     

On ensembles of low-density parity-check codes: asymptotic distancedistributions

We derive expressions for the average distance distributions in several ensembles of regular low-density parity-check codes (LDPC). Among these ensembles are the standard one defined by matrices having given column and row sums, ensembles defined by matrices with given column sums or given row sums, and an ensemble defined by bipartite graphs     

Decoding low-density parity-check codes with probabilisticscheduling

We present a new message-passing schedule for the decoding of low-density parity-check (LDPC) codes. This approach, designated “probabilistic schedule”, takes into account the structure of the Tanner graph (TG) of the code. We show by simulation that the new schedule offers a much better performance/complexity trade-off. This work also suggests that scheduling plays an important role in iterative decoding and that a schedule that matches the structure of the TG is desirable     

Construction of low-density parity-check codes based on balanced incomplete block designs

This correspondence presents a method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatoric designs, known as balance incomplete block designs. Codes constructed by this method have girths at least 6 and they perform well with iterative decoding. Furthermore, several classes of these codes are quasi-cyclic and hence their encoding can be implemented with simple feedback shift registers.     

Efficient encoding of low-density parity-check codes

Low-density parity-check (LDPC) codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy: constrained random code ensembles and iterative decoding algorithms. We consider the encoding problem for LDPC codes. More generally we consider the encoding problem for codes specified by sparse parity-check matrices. We show how to exploit the sparseness of the parity-check matrix to obtain efficient encoders. For the (3,6)-regular LDPC code, for example, the complexity of encoding is essentially quadratic in the block length. However, we show that the associated coefficient can be made quite small, so that encoding codes even of length n≃100000 is still quite practical. More importantly, we show that “optimized” codes actually admit linear time encoding     

Improved low-density parity-check codes using irregular graphs

We construct new families of error-correcting codes based on Gallager's (1973) low-density parity-check codes. We improve on Gallager's results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance     

Efficient encoding of quasi-cyclic low-density parity-check codes

Quasi-cyclic (QC) low-density parity-check (LDPC) codes form an important subclass of LDPC codes. These codes have encoding advantage over other types of LDPC codes. This paper addresses the issue of efficient encoding of QC-LDPC codes. Two methods are presented to find the generator matrices of QC-LDPC codes in systematic-circulant (SC) form from their parity-check matrices, given in circulant form. Based on the SC form of the generator matrix of a QC-LDPC code, various types of encoding circuits using simple shift registers are devised. It is shown that the encoding complexity of a QC-LDPC code is linearly proportional to the number of parity bits of the code for serial encoding, and to the length of the code for high-speed parallel encoding.     

Construction of low-density parity-check codes by superposition

This paper presents a superposition method for constructing low-density parity-check (LDPC) codes. Several classes of structured LDPC codes are constructed. Codes in these classes perform well with iterative decoding, and their Tanner graphs have girth at least six.