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.

Quantum quasi-cyclic low-density parity-check error-correcting codes

In this paper, we propose the approach of employing circulant permutation matrices to construct quantum quasicyclic （QC） low-density parity-check （LDPC） codes. Using the proposed approach one may construct some new quantum codes with various lengths and rates of no cycles-length 4 in their Tanner graphs. In addition, these constructed codes have the advantages of simple implementation and low-complexity encoding. Finally, the decoding approach for the proposed quantum QC LDPC is investigated.

Quaternary quasi-cyclic codes

Quasi-cyclic codes of length mn over Z4 are shown to be equivalent to A-submodules of A^n, where A = Z4[x]/（x^m - 1）. In the case of m being odd, all quasi-cyclic codes are shown to be decomposable into the direct sum of a fixed number of cyclic irreducible A-submodules. Finally the distinct quasi-cyclic codes as well as some specific subclasses are enumerated.

On the Girth of Tanner (5,7) Quasi-Cyclic LDPC Codes

The girth plays an important role in the design of LDPC codes. In order to determine the girth of Tanner(5,7) quasi-cyclic( QC) LDPC codes with length 7p for p being a prime with the form 35 m + 1,the cycles of lengths 4,6,8,and 10 are analyzed. Then these cycles are classified into sixteen categories,each of which can be expressed as an ordered block sequence,or a certain type. It is also shown that the existence of these cycles is equal to polynomial equations over Fpwho has a 35th unit root. We check if these polynomial equations have a 35th unit root and obtain the girth values of Tanner(5,7) QC LDPC codes.

Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs

This paper presents new half rate Quasi Cyclic Low Density Parity Check (QC- LDPC) codes formed on the basis of combinatorial designs. In these codes, circulant matrices of the parity check matrix are formed on the basis of subsets in which the difference between any two elements of a subset is unique with all differences obtained from the same or different subsets. This structure of circulant matrices guarantees non-existence of cycle-4 in the Tanner graph of QC-LDPC codes. First, an irregular code with girth 6 constituted by two rows of circulant matrices is proposed. Then, more criteria will be considered on the structure of subsets with the mentioned feature aiming to represent a new scheme of regular QC-LPDC codes with girth at least 8. From simulations, it is confirmed that codes have similar to or better performance than other well-known half rate codes, while require lower complexity in their design.

Quasi-cyclic Random Projection Code and Hardware Implementation

Random Projection Code (RPC) is a mechanism that combines channel coding and modulation together and realizes rate adaptation in the receiving end. Random projection code’s mapping matrix has significant influences on decoding performance as well as hardware implementation complexity. To reduce hardware implementation complexity, we design a quasi-cyclic mapping matrix for RPC codes. Compared with other construction approaches, our design gets rid of data filter component, thus reducing chip area 7284.95 um2, and power consumption 331.46 uW in 0.13 um fabrication. Our simulation results show that our method does not cause any performance loss and even gets 0.2 dB to 0.5 dB gain at BER 10-4.

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。

Z_p^(k+1)环上的Quasi-Cyclic码

近年来,剩余类环Zm上的编码理论有很大的发展,人们通过Gray映射建立了比较系统的Zm环上的编码理论。文章定义了对于n=n1pδ,环Zn1pk+1到环Zppkn1上的Gray映射,给出了该映射的几个性质,并由此得出了Zpk+1环上的指数为pδt长为n=pδn1的quasi_cyclic码与Zp环上的quasi_cyclic码的一一对应关系,这里的t|n1,(n1,p)=1,从而环Zpk+1上的quasi_cyclic码可以看作是环Zp上的quasi_cyclic码,也即看作是有限素域上的quasi_cyclic码。

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.

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.

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.

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。

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.

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.

基于Stanley序列的QC-LDPC码新颖构造方法

针对准循环低密度奇偶校验(quasi-cyclic low-density parity-check,QC-LDPC)码的短环结构会严重影响码字纠错性能的问题,基于Stanley序列(Stanley sequence,SS)提出一种围长至少为8的QC-LDPC码新颖构造方法。从Stanley序列中选取某些特定元素构成一个呈递增关系的集合,利用穷举算法搜索出满足无环4和环6条件的元素得到另一个递增集合,构造相应的指数矩阵,得到其奇偶校验矩阵。仿真结果表明,在误码率(bit error rate,BER)为10-6时,所构造的SS-QC-LDPC码与同码率码长的其他QC-LDPC码码型相比,其净编码增益均有一定提升,因而其纠错性能较好,且无错误平层现象。此外,该构造方法的计算复杂度较低。

基于原模图的8环QC-LDPC码构造方法

针对当前准循环低密度奇偶校验(Quasi-Cyclic Low-Density Parity-Check,QC-LDPC)码存在短环及纠错性能不够好的问题,基于原模图提出一种新颖的QC-LDPC码构造方法。该方法选择码长码率可灵活调整的原模图作为基矩阵,再结合具有特殊性质的卢卡斯数列和等差数列,通过原模图的低译码门限和数列的特殊性质,构造校验矩阵环长至少为8,且所需存储空间少,易于硬件实现。仿真结果表明:该方法构造的PLA-QC-LDPC(2400,1200)码与同等码长码率中基于卢卡斯数列和最大公约数序列的可快速编码的非规则LG-QC-LDPC码、基于素数和乘法表构造的PM-QC-LDPC码以及基于原模图和消除基本陷阱集的非规则PL-QC-LDPC码相比,净编码增益均有一定程度的提高。

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.

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.

NEW CONSTRUCTIONS OF LDPC CODES BASED ON CIRCULANT PERMUTATION MATRICES FOR AWGN CHANNEL

Two new design approaches for constructing Low-Density Parity-Check(LDPC) codes are proposed.One is used to design regular Quasi-Cyclic LDPC(QC-LDPC) codes with girth at least 8.The other is used to design irregular LDPC codes.Both of their parity-check matrices are composed of Circulant Permutation Matrices(CPMs).When iteratively decoded with the Sum-Product Algorithm(SPA),these proposed codes exhibit good performances over the AWGN channel.

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.