Quasi-cyclic LDPC codes using overlapping matrices and their layered decoders

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have the parity-check matrices consisting of circulant matrices. Since QC LDPC codes whose parity-check matrices consist of only circulant permutation matrices are difficult to support layered decoding and, at the same time, have a good degree distribution with respect to error correcting performance, adopting multi-weight circulant matrices to parity-check matrices is useful but it has not been much researched. In this paper, we propose a new code structure for QC LDPC codes with multi-weight circulant matrices by introducing overlapping matrices. This structure enables a system to operate on dual mode in an efficient manner, that is, a standard QC LDPC code is used when the channel is relatively good and an enhanced QC LDPC code adopting an overlapping matrix is used otherwise. We also propose a new dual mode parallel decoder which supports the layered decoding both for the standard QC LDPC codes and the enhanced QC LDPC codes. Simulation results show that QC LDPC codes with the proposed structure have considerably improved error correcting performance and decoding throughput.     

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via LovÁsz's Local Lemma (LLL) and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum-likelihood (ML) decoding.      

LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.     

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 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     

一种高码率低复杂度准循环LDPC码设计研究

该文设计了一种特殊的高码率准循环低密度校验(QC-LDPC)码,其校验矩阵以单位矩阵的循环移位阵为基本单元,与随机构造的LDPC码相比可节省大量存储单元。利用该码校验矩阵的近似下三角特性,一种高效的递推编码方法被提出,它使得该码编码复杂度与码长成线性关系。另外,该文提出一种分析QC-LDPC码二分图中短长度环分布情况的方法,并且给出了相应的不含长为4环QC-LDPC码的构造方法。计算机仿真结果表明,新码不但编码简单,而且具有高纠错能力、低误码平层。     

Parity-Check Density Versus Performance of Binary Linear Block Codes: New Bounds and Applications

The moderate complexity of low-density parity-check (LDPC) codes under iterative decoding is attributed to the sparseness of their parity-check matrices. It is therefore of interest to consider how sparse parity-check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity. This issue was addressed by Sason and Urbanke, and it is revisited in this paper. The remarkable performance of LDPC codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum-likelihood (ML) decoding, and the additional loss which is imposed by the suboptimality of the decoder. These issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes, and lower bounds on the asymptotic density of their parity-check matrices as a function of the gap between their achievable rates and the channel capacity; these bounds are valid under ML decoding, and hence, they are valid for any suboptimal decoding algorithm. The new bounds improve on previously reported results by Burshtein and by Sason and Urbanke, and they hold for the case where the transmission takes place over an arbitrary memoryless binary-input output-symmetric (MBIOS) channel. The significance of these information-theoretic bounds is in assessing the tradeoff between the asymptotic performance of LDPC codes and their decoding complexity (per iteration) under message-passing decoding. They are also helpful in studying the potential achievable rates of ensembles of LDPC codes under optimal decoding; by comparing these thresholds with those calculated by the density evolution technique, one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms     

Finite field construction for quasi-cyclic LDPC convolutional codes with cyclic 2-D MDS codes

The parity-check matrices for quasi-cyclic low-density parity-check convolutional (QC-LDPC-C) codes have different characteristics of time-varying periodicity and need to realize fast encoding. The finite field construction method for QC-LDPC-C codes with cyclic two-dimensional maximum distance separable (2-D MDS) codes is proposed using the base matrix framework and matrix unwrapping, thus the constructed parity-check matrices are free of length-4 cycles. The unwrapped matrices are constructed respectively based on different cyclic 2-D MDS codes for the case of matrix period less than or greater than constraint block length, and construction examples are given. LDPC-C codes with different periodicity characteristics are compared with QC-LDPC-C codes constructed with the proposed method. Experimental results show that QC-LDPC-C codes with the proposed method outperform the other codes and have lower encoding and decoding complexity.

     

A class of generalized LDPC codes with fast parallel decoding algorithms

Given the Tanner graph of a generalized low-density parity-check (GLDPC) code, the decoding complexity is mainly dominated by the decoding algorithm of subcodes. In this paper, we propose a class of GLDPC codes with fast parallel decoding algorithm. The parity-check matrices of the newly constructed subcodes are composed of several square matrices, which can be viewed as elements of a finite field. Therefore the FFT-based a posteriori probability (APP) algorithm for nonbinary codes can be applied to decode the subcodes. When compared with the trellis-based APP algorithm, the FFT-based APP algorithm can be implemented in parallel and has lower complexity. Simulation results show that the proposed GLDPC codes perform well on AWGN channels.     

Quasi-cyclic LDPC codes for fast encoding

In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.     

一种基于增强奇偶校验码级联极化码的新型编译码方法

极化码作为一种纠错码,具有较好的编译码性能,已成为5G短码控制信道的标准编码方案。但在码长较短时,其性能不够优异。提出一种基于增强奇偶校验码级联极化码的新型编译码方法,在原有的奇偶校验位后设立增强校验位,对校验方程中信道可靠度较低的信息位进行双重校验,辅助奇偶校验码在译码过程中对路径进行修剪,以此提高路径选择的可靠性。仿真结果表明,在相同信道、相同码率码长下,本文提出的新型编译码方法比循环冗余校验(cyclic redundancy check,CRC)码级联极化码、奇偶校验(parity check,PC)码级联极化码误码性能更优异。在高斯信道下,当码长为128、码率为1/2、误码率为10-3时,本文提出的基于增强PC码级联的极化码比PC码级联的极化码获得了约0.3 dB增益,与CRC辅助的极化码相比获得了约0.4 dB增益。     

On the Stopping Redundancy of Reed–Muller Codes

The stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel. In this paper, we will consider the stopping redundancy of Reed-Muller codes and related codes. Let R(lscr,m) be the Reed-Muller code of length 2^m and order lscr. Schwartz and Vardy gave a recursive construction of parity-check matrices for the Reed-Muller codes, and asked whether the number of rows in those parity-check matrices is the stopping redundancy of the codes. We prove that the stopping redundancy of R(m-2,m), which is also the extended Hamming code of length 2^m, is 2m-1 and thus show that the recursive bound is tight in this case. We prove that the stopping redundancy of the simplex code equals its redundancy. Several constructions of codes for which the stopping redundancy equals the redundancy are discussed. We prove an upper bound on the stopping redundancy of R(1,m). This bound is better than the known recursive bound and thus gives a negative answer to the question of Schwartz and Vardy     

高性能时不变LDPC卷积码构造算法研究

该文基于由QC-LDPC码获得时不变LDPC卷积码的环同构方法,设计了用有限域上元素直接获得时不变LDPC卷积码多项式矩阵的新算法。以MDS卷积码为例,给出了一个具体的构造过程。所提构造算法可确保所获得的时不变LDPC卷积码具有快速编码特性、最大可达编码记忆以及设计码率。基于滑动窗口的BP译码算法在AWGN信道上的仿真结果表明,该码具有较低的误码平台和较好的纠错性能。     

围长至少为8的QC-LDPC码的新构造:一种显式框架

 构造围长较大的校验矩阵,是提高二进制和多进制QC-LDPC码译码性能的一种有效手段.本文提出一种不需要借助于任何计算机搜索步骤,能够直接构造出围长至少为8的QC-LDPC码的显式构造框架.该框架所构造的QC-LDPC码不仅满足围长至少为8的条件,而且还具有循环置换矩阵(CPM)尺寸可以连续变化的优点.该框架可以分为两个步骤:第一步是在无穷大CPM尺寸条件下利用确定性方法构造一个围长至少为8的校验矩阵;第二步是根据本文新发现的一个围长性质,从该校验矩阵的移位矩阵直接精确地计算出CPM尺寸连续变化的紧致下界.     

DC-free error-control block codes

DC-free codes and error-control (EC) codes are widely used in digital transmission and storage systems. To improve system performance in terms of code rate, bit-error rate (BER), and low-frequency suppression, and to provide a flexible tradeoff between these parameters, this paper introduces a new class of codes with both dc-control and EC capability. The new codes integrate dc-free encoding and EC encoding, and are decoded by first applying standard EC decoding techniques prior to dc-free decoding, thereby avoiding the drawbacks that arise when dc-free decoding precedes EC decoding. The dc-free code property is introduced into standard EC codes through multimode coding techniques, at the cost of minor loss in BER performance on the additive white Gaussian noise channel, and some increase in implementation complexity, particularly at the encoder. This paper demonstrates that a wide variety of EC block codes can be integrated into this dc-free coding structure, including binary cyclic codes, binary primitive BCH codes, Reed-Solomon codes, Reed-Muller codes, and some capacity-approaching EC block codes, such as low-density parity-check codes and product codes with iterative decoding. Performance of the new dc-free EC block codes is presented.     

Decoder Design for RS-Based LDPC Codes

This brief studies very large-scale integration (VLSI) decoder architectures for RS-based low-density parity-check (LDPC) codes, which are a special class of LDPC codes based on Reed-Solomon codes. The considered code ensemble is well known for its excellent error-correcting performance and has been selected as the forward error correction coding scheme for 10GBase-T systems. By exploiting the shift-structured properties hidden in the algebraically generated parity-check matrices, novel decoder architectures are developed with significant advantages of high level of parallel decoding, efficient usage of memory, and low complexity of interconnection. To demonstrate the effectiveness of the proposed techniques, we completed a high-speed decoder design for a (2048, 1723) regular RS-LDPC code, which achieves 10-Gb/s throughput with only 820 000 gates. Furthermore, to support all possible RS-LDPC codes, two special cases in code construction are considered, and the corresponding extensions of the decoder architecture are investigated.     

Multiple-Rate Low-Density Parity-Check Codes with Constant Blocklength

This paper describes and analyzes low-density parity-check code families that support variety of different rates while maintaining the same fundamental decoder architecture. Such families facilitate the decoding hardware design and implementation for applications that require communication at different rates, for example to adapt to changing channel conditions. Combining rows of the lowest-rate parity-check matrix produces the parity-check matrices for higher rates. An important advantage of this approach is that all effective code rates have the same blocklength. This approach is compatible with well known techniques that allow low-complexity encoding and parallel decoding of these LDPC codes. This technique also allows the design of programmable analog LDPC decoders. The proposed design method maintains good graphical properties and hence low error floors for all rates.     

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.     

低复杂度的LDPC码联合编译码构造方法研究

LDPC码因为其具有接近香农限的译码性能和适合高速译码的并行结构,已经成为纠错编码领域的研究热点。LDPC码校验矩阵的构造是基于稀疏的随机图,所以该类码字编码和译码的硬件实现比较复杂。以单位阵的循环移位阵为基本单元,构造LDPC码的校验矩阵,降低了LDPC码在和积算法下的译码复杂度。同时考虑到LDPC码的编码复杂度,给出了一种可以简化编码的结构。针对该方案构造的LDPC码,提出了消除其二分图上的短圈的方法。通过大量的仿真和计算分析,本文比较了这种LDPC码和随机构造的LDPC码在误码率性能,圈长分布以及最小码间距估计上的差异。     

TETRA中Reed-Muller码的大数逻辑译码方法

TETRA数字集群移动通信系统的物理层协议中采用了缩短Reed-Muller(RM)码,它与经典RM码的差异极大,无法采用Reed大数逻辑译码算法.根据正交校验矩阵的特点,提出了一种一般线性分组码的正交校验矩阵的穷举搜索算法.使用该算法搜索了缩短RM码的正交校验矩阵,对搜索速度进行了分析.证明了该码是两步完全可正交码,给出了它的Massey大数逻辑译码方法.仿真结果表明,无论是硬判决还是软判决,该译码方法的纠错性能都优于伴随式译码方法.