Low-Density Parity-Check Codes: Research Status and Development Direction

In this paper, we conclude five kinds of methods for construction of the regular low-density parity matrix H and three kinds of methods for the construction of irregular low-density parity-check matrix H. Through the analysis of the code rate and parameters of these eight kinds of structures, we find that the construction of low-density parity-check matrix tends to be more flexible and the parameter variability is enhanced. We propose that the current development cost should be lower with the progress of electronic technology and we need research on more practical Low-Density Parity-Check Codes (LDPC). Combined with the application of the quantum distribution key, we urgently need to explore the research direction of relevant theories and technologies of LDPC codes in other fields of quantum information in the future.

DESIGN OF QUASI-CYCLIC LDPC CODES BASED ON EUCLIDEAN GEOMETRIES

A new method for constructing Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes based on Euclidean Geometry (EG) is presented. The proposed method results in a class of QC-LDPC codes with girth of at least 6 and the designed codes perform very close to the Shannon limit with iterative decoding. Simulations show that the designed QC-LDPC codes have almost the same performance with the existing EG-LDPC codes.

Quaternion Integers Based Higher Length Cyclic Codes and Their Decoding Algorithm

The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.

A NOVEL CONSTRUCTION OF QUANTUM LDPC CODES BASED ON CYCLIC CLASSES OF LINES IN EUCLIDEAN GEOMETRIES

The dual-containing (or self-orthogonal) formalism of Calderbank-Shor-Steane (CSS) codes provides a universal connection between a classical linear code and a Quantum Error-Correcting Code (QECC). We propose a novel class of quantum Low Density Parity Check (LDPC) codes constructed from cyclic classes of lines in Euclidean Geometry (EG). The corresponding constructed parity check matrix has quasi-cyclic structure that can be encoded flexibility, and satisfies the requirement of dual-containing quantum code. Taking the advantage of quasi-cyclic structure, we use a structured approach to construct Generalized Parity Check Matrix (GPCM). This new class of quantum codes has higher code rate, more sparse check matrix, and exactly one four-cycle in each pair of two rows. Ex-perimental results show that the proposed quantum codes, such as EG(2,q)II-QECC, EG(3,q)II-QECC, have better performance than that of other methods based on EG, over the depolarizing channel and decoded with iterative decoding based on the sum-product decoding algorithm.

Design of Irregular QC-LDPC Code Based Multi-Level Coded Modulation Scheme for High Speed Optical Communication Systems

In this paper, we focus on the design of irregular QC-LDPC code based multi-level coded modulation(MLCM) scheme by jointly optimizing the component code rate and the degree distribution of the irregular QC-LDPC component code. Firstly, the sub-channel capacities of MLCM systems is analyzed and discussed, based on which the optimal component code rate can be obtained. Secondly, an extrinsic information transfer chart based two-stage searching algorithm is proposed to find the good irregular QC-LDPC code ensembles with optimal component code rates for their corresponding sub-channels. Finally, by constructing the irregular QC-LDPC component codes from the designed ensembles with the aim of possibly enlarging the girth and reducing the number of the shortest cycles, the designed irregular QC-LDPC code based 16QAM and 64QAM MLCM systems can achieve 0.4 dB and 1.2 dB net coding gain, respectively, compared with the recently proposed regular QC-LDPC code based 16QAM and 64QAM MLCM systems.

Construction of protograph LDPC codes with circular generator matrices

The application of protograph low density parity check （LDPC） codes involves the encoding complexity problem. Since the generator matrices are dense, and if the positions of ＂1＂ s are irregularity, the encoder needs to store every ＂1＂ of the generator matrices by using huge chip area. In order to solve this problem, we need to design the protograph LDPC codes with circular generator matrices. A theorem concerning the circulating property of generator matrices of nonsingular protograph LDPC codes is proposed. The circulating property of generator matrix of nonsingular protograph LDPC codes can be obtained from the corresponding quasi-cyclic parity check matrix. This paper gives a scheme of constructing protograph LDPC codes with circulating generator matrices, and it reveals that the fast encoding algorithm of protograph LDPC codes has lower encoding complexity under the condition of the proposed theorem. Simulation results in ad- ditive white Gaussian noise （AWGN） channels show that the bit error rate （BER） performance of the designed codes based on the proposed theorem is much better than that of GB20600 LDPC codes and Tanner LDPC codes.

Design of good QC-LDPC codes without small girth in the p-plane

A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check （QC-LDPC） codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate （BER） performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

Ruling out small stopping sets and small girth in Tanner graph of QC-LDPC code

The existing constructions of quasi-cyclic low-density parity-check （QC-LDPC） codes do not consider the problems of small stopping sets and small girth together in the Tanner graph, while their existences will lead to the bit error rate （BER） performance of QC-LDPC codes being much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the specific chosen parity-check matrix of QC-LDPC codes without small stopping sets and small girth are proposed. A novel construction for QC-LDPC codes with long block lengths is presented by multiplying mmin or the multiple of mmin, which is the minimum order of the identity matrix for the chosen parity-check matrix. The simulation results show that the specific chosen parity-check matrix of QC-LDPC codes can effectively avoid specified stopping sets and small girth and exhibit excellent BER performance than random LDPC codes with the same longer codes length.

Weighted symbol-flipping decoding algorithm for nonbinary LDPC codes with flipping patterns

A novel low-complexity weighted symbol-flipping algorithm with flipping patterns to decode nonbinary low-density parity-check codes is proposed. The proposed decoding procedure updates the hard-decision received symbol vector iteratively in search of a valid codeword in the symbol vector space. Only one symbol is flipped in each iteration, and symbol flipping function, which is employed as the symbol flipping metric, combines the number of failed checks and the reliabilities of the received bits and calculated symbols. A scheme to avoid infinite loops and select one symbol to flip in high order Galois field search is also proposed. The design of flipping pattern＇s order and depth, which is dependent of the computational requirement and error performance, is also proposed and exemplified. Simulation results show that the algorithm achieves an appealing tradeoff between performance and computational requirement over relatively low Galois field for short to medium code length.

JOINT SOURCE-CHANNEL DECODING OF HUFFMAN CODES WITH LDPC CODES

In this paper, we present a Joint Source-Channel Decoding algorithm (JSCD) for Low-Density Parity Check (LDPC) codes by modifying the Sum-Product Algorithm (SPA) to account for the source redun-dancy, which results from the neighbouring Huffman coded bits. Simulations demonstrate that in the presence of source redundancy, the proposed algorithm gives better performance than the Separate Source and Channel Decoding algorithm (SSCD).

Efficient Early Termination Strategy for LDPC Codes in GPS Systems

By exploiting the structural features of L1C messages,a novel Early Termination( ET) strategy is proposed to speed up the decoding of low-density parity-check( LDPC) codes in the GPS system. The proposed strategy is based on the cyclic redundancy check( CRC) of the messages in the subframes 2 and 3. The simulation results show that average number of iterations of the proposed strategy is less than that of the standard ET strategy,with nearly no degradation in decoding performance. Besides,the proposed ET strategy can be efficiently implemented in a sequential or parallel manner. Thus,the proposed ET strategy is attractive for practical purposes.

Study on the Optimized Demodulation and Decoding of the BICM-ID system coded with LDPC

The demodulation and decoding solution commonly used in the bit interleaving code modulation and the LDPC coded demodulation and decoding system of the iterative decoding （BICM-ID） is to report the soft information output from the decoder of the receiver to the demodulator as priori information for completing the decoding. However, this will give rise to the reduction of the minimum Euclidean distances between codes, so that the performances of the system decline under non-fading Gaussian channel. According to this problem, an optimized iterative demodulation decoding solution is proposed and also is analyzed using simulation. The result shows that this solution can improve the demodulation and decoding performances of LDPC coded modulation and demodulation system.

基于Golomb Ruler的QC-LDPC码构造方法

针对准循环低密度奇偶校验(QC-LDPC)码中短环结构会影响其纠错性能的问题,基于Golomb Ruler提出了一种新颖的围长为8的QC-LDPC码构造方法。该方法先根据码长码率的需求,从Golomb Ruler中选择部分元素构造一个集合,结合指数矩阵中元素所在位置的四六环特性,通过搜索算法,依次找出符合无四六环条件的元素得到另一个集合,然后构造相应的指数矩阵,最后得到其奇偶校验矩阵。仿真结果表明:在误码率为10^(-6)时,所构造的GR-QC-LDPC码与同码率码长的其他4种QC-LDPC码的码型相比,其净编码增益均有一定的提高,且无明显错误平层现象。

基于阿贝尔群上的差分集构造LDPC码

本文给出了区组长度为3、4和5的平衡不完全区组设计,将设计的关联矩阵作为校验矩阵,从而构造出3种低密度奇偶校验码(LDPC,low-density parity-check code)。由于这些校验矩阵是准循环矩阵,所以构造的码是拟循环LDPC码(QC-LDPC,quasi-cyclic LDPC)。计算码的相关参数,并针对具体的例子通过与类随机码的译码性能进行分析比较,结果显示,所构造的码具有较好的性能。

无小环大列重QC-LDPC短码的显式构造

针对列重较大的无4环且无6环的准循环(Quasi-Cyclic,QC)低密度奇偶校验(Low-Density Parity-Check,LDPC)码,本文提出了三种新的显式构造方法.新方法的指数矩阵由两个整数序列完全定义,其中第一个序列是从0开始且公差为1的等差序列,第二个序列是由符合最大公约数约束的整数组成的特殊序列.对于现有显式方法只能提供较大循环块尺寸的多种行重类型,新显式构造方法在这些行重类型下均获得了相当小的循环块尺寸,从而将最小循环块尺寸降低到大约只有原来的一半.与近期提出的基于搜索的对称结构法相比,新的显式构造方法具有类似或更优的译码性能、极低的描述复杂度且不需要计算机搜索.

围长为8的较大列重准循环低密度奇偶校验码的行重普适代数构造

适合于任意行重(即行重普适(RWU))的无小环准循环(QC)低密度奇偶校验(LDPC)短码,对于LDPC码的理论研究和工程应用具有重要意义。具有行重普适特性且消除4环6环的现有构造方法,只能针对列重为3和4的情况提供QC-LDPC短码。该文在最大公约数(GCD)框架的基础上,对于列重为5和6的情况,提出了3种具有行重普适特性且消除4环6环的构造方法。与现有的行重普适方法相比,新方法提供的码长从目前的与行重呈4次方关系锐减至与行重呈3次方关系,因而可以为QC-LDPC码的复合构造和高级优化等需要较大列重基础码的场合提供行重普适的无4环无6环短码。此外,与基于计算机搜索的对称结构QC-LDPC码相比,新码不仅无需搜索、描述复杂度更低,而且具有更好的译码性能。

Quasi-cyclic LDPC codes with high-rate and low error floor based on Euclidean geometries

An improved Euclidean geometry approach to design quasi-cyclic （QC） Low-density parity-check （LDPC） codes with high-rate and low error floor is presented. The constructed QC-LDPC codes with high-rate have lower error floor than the original codes. The distribution of the minimum weight codeword is analyzed, and a sufficient existence condition of the minimum weight codeword is found. Simulations show that a lot of QC-LDPC codes with lower error floor can be designed by reducing the number of the minimum weight codewords satisfying this sufficient condition.

Design and Efficient Hardware Implementation Schemes for Non-Quasi-Cyclic LDPC Codes

The design of a high-speed decoder using traditional partly parallel architecture for Non-Quasi-Cyclic（NQC） Low-Density Parity-Check（LDPC） codes is a challenging problem due to its high memory-block cost and low hardware utilization efficiency. In this paper, we present efficient hardware implementation schemes for NQCLDPC codes. First, we propose an implementation-oriented construction scheme for NQC-LDPC codes to avoid memory-access conflict in the partly parallel decoder. Then, we propose a Modified Overlapped Message-Passing（MOMP） algorithm for the hardware implementation of NQC-LDPC codes. This algorithm doubles the hardware utilization efficiency and supports a higher degree of parallelism than that used in the Overlapped Message Passing（OMP） technique proposed in previous works. We also present single-core and multi-core decoder architectures in the proposed MOMP algorithm to reduce memory cost and improve circuit efficiency. Moreover, we introduce a technique called the cycle bus to further reduce the number of block RAMs in multi-core decoders. Using numerical examples, we show that, for a rate-2/3, length-15360 NQC-LDPC code with 8.43-d B coding gain for Binary PhaseShift Keying（BPSK） in an Additive White Gaussian Noise（AWGN） channel, the decoder with the proposed scheme achieves a 23.8%–52.6% reduction in logic utilization per Mbps and a 29.0%–90.0% reduction in message-memory bits per Mbps.

基于双序列反序的(4,L)规则girth-8 QC-LDPC短码

围长较大的短码长准循环(QC)低密度奇偶校验(LDPC)码的显式构造对于QC-LDPC短码的理论研究与工程应用具有重要意义。首先提出一种基于成对策略的贪婪搜索算法,并根据此算法在列重J为4时的经验结果,归纳总结出一种具有双序列反序特征的指数矩阵。随后证明了该指数矩阵对于任意行重L均对应于围长为8的QC-LDPC码。与现有的典型显式构造方法即最大公约数(GCD)方法相比,新QC-LDPC码提供的码长显著降低。最后,将指数矩阵的拆分拼接和掩膜处理技巧与新QC-LDPC码结合起来,设计出了译码性能在高信噪比区超过5G NR LDPC码的合成码。

Code Design and Shuffled Iterative Decoding of a Quasi-Cyclic LDPC Coded OFDM System

In multipath environments, the error rate performance of orthogonal frequency division multiplexing （OFDM） is severely degraded by the deep fading subcarriers. Powerful error-correcting codes must be used with OFDM. This paper presents a quasi-cyclic low-density parity-check （LDPC） coded OFDM system, in which the redundant bits of each codeword are mapped to a higher-order modulation constellation. The op- timal degree distribution was calculated using density evolution. The corresponding quasi-cyclic LDPC code was then constructed using circulant permutation matrices. Group shuffled message passing scheduling was used in the iterative decoding. Simulation results show that the system achieves better error rate performance and faster decoding convergence than conventional approaches on both additive white Gaussian noise （AWGN） and Rayleigh fading channels.