Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes

In this paper, a simple and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists of starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance over the memoryless erasure channel, and suitable also for the iterative correction of single erasure bursts. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm. It executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed to achieve a good performance over the memoryless erasure channel, then the code obtained after the application of the algorithm provides a good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the algorithm effectiveness when applied to different categories of LDPC codes.     

Algebraic Construction of Quasi-Cyclic LDPC Codes for the AWGN and Erasure Channels

This paper is concerned with construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes for three different types of channels: the additive white Gaussian noise, the binary random erasure, and the binary burst erasure channels. Two algebraic methods for systematic construction of QC-LDPC codes are presented. Codes constructed perform well over all three types of channels.     

Generalized IRA Erasure Correcting Codes for Hybrid Iterative/Maximum Likelihood Decoding

The design of low-density parity-check (LDPC) codes under hybrid iterative / maximum likelihood decoding is addressed for the binary erasure channel (BEC). Specifically, we focus on generalized irregular repeat-accumulate (GeIRA) codes, which offer both efficient encoding and design flexibility. We show that properly designed GeIRA codes tightly approach the performance of an ideal maximum distance separable (MDS) code, even for short block sizes. For example, our (2048,1024) code reaches a codeword error rate of 10^-5 at channel erasure probability isin= 0.450, where an ideal (2048,1024) MDS code would reach the same error rate at isin = 0.453.     

RT Oblivious Erasure Correcting

An erasure correcting scheme is rateless if it is designed to tolerate any pattern of packet loss and reveal the transmitted information after a certain number of packets is received. On the one hand, transmission schemes that use rateless erasure correcting schemes do not usually use a feedback channel. However, they may require significant amount of additional processing by both the sender and the receiver. On the other hand, automatic repeated request protocols use a feedback channel to assist the sender, and do not usually require information processing. In this work we present a combined approach, where a lean feedback channel is used to assist the sender to efficiently transmit the information. Our Real-Time oblivious approach minimizes the processing time and the memory requirements of the receiver and, therefore, fits a variety of receiving devices. In addition, the transmission is real-time where the expected number of original packets revealed when a packet is received is approximately the same throughout the entire transmission process. We use our end-to-end scheme as a base for broadcast (and multicast) schemes. An overlay tree structure is used to convey the information to a large number of receivers. Moreover, the receivers may download the information from a number of senders or even migrate from one sender to another.     

A Finite-Length Algorithm for LDPC Codes Without Repeated Edges on the Binary Erasure Channel

This paper considers the performance, on the binary erasure channel, of low-density parity-check (LDPC) codes without repeated edges in their Tanner graphs. A modification to existing finite-length analysis algorithms is presented for these codes.      

Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.     

纠正随机错误与长突发删除的多进制乘积码研究

考虑多进制LDPC码的符号特性,以及对其残留错误和删除的分析,本文采用多进制LDPC码作为内码,相同Galois域下的高码率RS码作为外码来构造多进制乘积码;并提出了一种低复杂度的迭代译码方案,减少信息传输的各类错误。在译码时,只对前一次迭代中译码失败的码字执行译码,并对译码正确码字所对应的比特初始概率信息进行修正,增强下一次迭代多进制LDPC译码符号先验信息的准确性,减少内码译码后的判决错误,从而充分利用外码的纠错能力。仿真结果显示,多进制乘积码相较于二进制LDPC乘积码有较大的编码增益,并通过迭代进一步改善了性能,高效纠正了信道中的随机错误和突发删除。对于包含2%突发删除的高斯信道,在误比特率为10^-6时,迭代一次有0.4 dB左右的增益。     

Evaluation of Burst Error Correcting Codes Using a Simple Partitioned Markov Chain Model

The simple partitioned Markov chain model is used to evaluate the effectiveness of burst error correcting codes. A Massey diffuse convolutional code is analyzed as an example to illustrate the method. Calculated results and simulation results are presented.     

Adaptive Demodulation Using Rateless Erasure Codes

We introduce a rate-adaptive system in which the receiver demodulates only those bits that have a high probability of being correct, treating nondemodulated bits as erasures. Several sets of decision regions, derived using composite hypothesis testing, are proposed for 16-QAM and 16-phase-shift keying, which allow for the simple implementation of this demodulation strategy. We demonstrate that pre-encoding the data with a Raptor code allows for simple reconstruction of the message, regardless of the erasure pattern introduced from the nondemodulated bits. We prove the optimality of the proposed decision regions in selecting the most likely subset of bits from any received symbol in moderate-to-high signal-to-noise ratios, and we analyze the performance of demodulating with these decision regions over an additive white Gaussian noise channel. Also demonstrated is the strong performance of 16-QAM for this application, compared with other power-efficient constellations and the near-optimality of using Gray mapping, even under the proposed alternate sets of decision regions.     

Performance Analysis of Regular Low-Density Erasure Codes

We propose a simple method for calculating the thresholds of regular low-density erasure codes under erasure recovery algorithm. Based on this conclusion, we prove that among （l, r）-regular low-density erasure codes, （3, r）-regular codes perform best, for the given integer l ≥ 3 and any rate R with 1/4 〈R 〈 1. Numerical results show the correctness of this conclusion.     

New Constructions of Quasi-Cyclic LDPC Codes Based on Special Classes of BIBDs for the AWGN and Binary Erasure Channels

This paper presents new methods for efficiently constructing encodable quasi-cyclic low-density parity-check (LDPC) codes based on special balanced incomplete block designs (BIBDs). Codes constructed perform well over both the additive white Gaussian noise (AWGN) and binary erasure channels with iterative decoding.      

基于XOR纠删码的性能分析

基于异或的纠删码有一个不规则结构,这个结构可能允许它以汉明码码距或者超过汉明码码距的码距容错。为了能够完全描述基于异或纠删码的容错情况,本文提出一种基于异或纠删码的容错机制：最小删除列表。相对于其他基于异或纠删码的容错机制,最小纠删列表是一种有效机制。在本文我们还给出了一种有效确定纠删码的最小删除列表的算法。这个算法使用纠删码的结构去有效确定最小删除列表。试验结果表明,对于给定的纠删码,有超过最小删除列表数目的100倍的删除数目。     

一种基于 MDS-卷积码的LDPC码构造方法

近年来,结构化低密度奇偶校验(LDPC)码的构造方法受到了广泛地关注.本文提出了一种利用最大距离分割(MDS)编码构造结构化LDPC码的思路.该思路将基于两个信息符号的RS码构造LDPC码的方法扩展至适用于所有的MDS码.本文以具有MDS特性的卷积码为例详细描述该构造方法的细节,并构造了码长从255比特到4095比特的高码率LDPC码.由于卷积码的MDS定义不同于线性分组码,因此本文给出了一种对卷积码截短的方法及其必要的证明.仿真结果表明,本文构造MDS-Conv-LDPC码的性能优于随机构造的LDPC码.     

Reduced-Complexity Decoding of LDPC Codes

Reduced-Complexity Decoding of LDPC Codes Various log-likelihood-ratio-based belief-propagation (LLR- BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representation of the decoding computations are shown to achieve a reduction in complexity, by simplifying the check-node update or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from a performance, latency, computational complexity, and memory-requirement perspective.     

High Performance Non-Binary Quasi-Cyclic LDPC Codes on Euclidean Geometries LDPC Codes on Euclidean Geometries

This paper presents algebraic methods for constructing high performance and efficiently encodable non-binary quasi-cyclic LDPC codes based on flats of finite Euclidean geometries and array masking. Codes constructed based on these methods perform very well over the AWGN channel. With iterative decoding using a Fast Fourier Transform based sum-product algorithm, they achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision K?tter-Vardy algorithm. Due to their quasi-cyclic structure, these non-binary LDPC codes on Euclidean geometries can be encoded using simple shiftregisters with linear complexity. Structured non-binary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication or storage systems for combating mixed types of noise and interferences.     

LDPC码及其应用

先在阐述几种LDPC编码的基础上分析了多进制LDPC编码优点;接着介绍作为解释LDPC码和积译码算法的因子图;最后分析了LDPC码的优势以及它在通信领域中应用的可行性。     

速率无损失的空间耦合LDPC码

空间耦合（spatially coupled）LDPC码为通信系统提供了一种全新的接近容量限的方式,但在耦合过程中需要额外添加校验节点,因此存在一定的码率损失.为了消除码率损失,提出了一种新型的耦合方式,首位置变量节点的边展开方式不变,中间位置的边连接到前一个位置的校验节点并依次向后展开,末位置的边与首位置的边成对称分布,从而得到了一种无需添加额外校验节点,码率无损失的SC-LDPC码变体结构.在AWGN信道中通过外部信息转移（extrinsic information transfer）理论分析及误比特率性能仿真,结果表明：相对于SC-LDPC码的传统结构,本文提出的SC-LDPC码变体结构在链长较短时不仅能避免码率损失,并且具有更优的阈值和译码性能,在未来通信系统中更具优势.     

LDPC码在DSP上的实现

本文概述了LDPC码的编译码原理,重点论述在TI公司的DSP(TMS320C6416)上的(512,256)LDPC编译码器的算法实现,并给出其与(2,1,7)卷积码在AWGN信道条件下的纠错性能对比。对比表明(512,256)LDPC码比(2,1,7)软判决的卷积码在误码率为10-4时可具有1.5dB的编码增益。     

基于LDPC码的多中继HARQ系统研究

根据不同中继节点建立的通信信道衰落特性的不同,利用中继节点可增强无线信道空间分集,提高整个链路的传输性能。纠错码的使用可更有效地提高信道传输效率的特点。文中介绍了基于LDPC码的多中继HARQ系统模型,在不降低传输速率的条件下,可更高效地传输数据。利用Matlab仿真了不同中继个数对系统平均误码率、吞吐量以及平均时延产生的不同影响,仿真结果表明,采用该系统模型可降低系统的平均误码率和平均时延,提高了系统的吞吐量。同时在码长较长的情况下,采用LDPC码的多中继HARQ系统性能优于采用Turbo码的多中继HARQ系统性能。     

SPC乘积码用于纠删译码的性能分析

SPC乘积码是一种既可以纠错,又可以恢复删除的高效码。该码的最小码距是4,能够恢复所有单阶,两阶和三阶的删除模式,实际上这种码能够恢复更高阶的删除模式。传统上是通过最小码间距来评价SPC乘积码译码性能,若从其空间结构入手,能够推导出SPC乘积码迭代译码后删除率的新上界。仿真显示这个上界更紧。