A novel construction method of QC-LDPC codes based on the subgroup of the finite field multiplicative group for optical transmission systems

According to the requirements of the increasing development for optical transmission systems, a novel construction method of quasi-cyclic low-density parity-check (QC-LDPC) codes based on the subgroup of the finite field multiplicative group is proposed. Furthermore, this construction method can effectively avoid the girth-4 phenomena and has the advantages such as simpler construction, easier implementation, lower encoding/decoding complexity, better girth properties and more flexible adjustment for the code length and code rate. The simulation results show that the error correction performance of the QC-LDPC(3 780,3 540) code with the code rate of 93.7% constructed by this proposed method is excellent, its net coding gain is respectively 0.3 dB, 0.55 dB, 1.4 dB and 1.98 dB higher than those of the QC-LDPC(5 334,4 962) code constructed by the method based on the inverse element characteristics in the finite field multiplicative group, the SCG-LDPC(3 969,3 720) code constructed by the systematically constructed Gallager (SCG) random construction method, the LDPC(32 640,30 592) code in ITU-T G.975.1 and the classic RS(255,239) code which is widely used in optical transmission systems in ITU-T G.975 at the bit error rate (BER) of 10^-7. Therefore, the constructed QC-LDPC(3 780,3 540) code is more suitable for optical transmission systems.     

Construction method of QC-LDPC codes based on multiplicative group of finite field in optical communication

In order to meet the needs of high-speed development of optical communication system, a construction method of quasi-cyclic low-density parity-check (QC-LDPC) codes based on multiplicative group of finite field is proposed. The Tanner graph of parity check matrix of the code constructed by this method has no cycle of length 4, and it can make sure that the obtained code can get a good distance property. Simulation results show that when the bit error rate (BER) is 10-6, in the same simulation environment, the net coding gain (NCG) of the proposed QC-LDPC(3 780, 3 540) code with the code rate of 93.7% in this paper is improved by 2.18 dB and 1.6 dB respectively compared with those of the RS(255, 239) code in ITU-T G.975 and the LDPC(3 2640, 3 0592) code in ITU-T G.975.1. In addition, the NCG of the proposed QC-LDPC(3 780, 3 540) code is respectively 0.2 dB and 0.4 dB higher compared with those of the SG-QC-LDPC(3 780, 3 540) code based on the two different subgroups in finite field and the AS-QC-LDPC(3 780, 3 540) code based on the two arbitrary sets of a finite field. Thus, the proposed QC-LDPC(3 780, 3 540) code in this paper can be well applied in optical communication systems.     

A novel QC-LDPC code based on the finite field multiplicative group for optical communications

A novel construction method of quasi-cyclic low-density parity-check(QC-LDPC) code is proposed based on the finite field multiplicative group,which has easier construction,more flexible code-length code-rate adjustment and lower encoding/decoding complexity.Moreover,a regular QC-LDPC(5334,4962) code is constructed.The simulation results show that the constructed QC-LDPC(5334,4962) code can gain better error correction performance under the condition of the additive white Gaussian noise(AWGN) channel with iterative decoding sum-product algorithm(SPA).At the bit error rate(BER) of 10-6,the net coding gain(NCG) of the constructed QC-LDPC(5334,4962) code is 1.8 dB,0.9 dB and 0.2 dB more than that of the classic RS(255,239) code in ITU-T G.975,the LDPC(32640,30592) code in ITU-T G.975.1 and the SCG-LDPC(3969,3720) code constructed by the random method,respectively.So it is more suitable for optical communication systems.     

光通信系统中基于BIBD对QC-LDPC码的研究

基于平衡不完全区组设计(BIBD),深入分析与研究了准循环低密度奇偶校验(QC-LDPC)码的一种新颖构造方法,并通过该构造方法构造了3种同码率不同码长的QC-LDPC码,通过对这3种QC-LDPC码的仿真分析表明,同码率下,码长越长性能越好。同时在BER=10-6时码率均为93.7%的情况下,所构造的BIBD-QC-LDPC(5392,5056)码的净编码增益(NCG)比已广泛应用于光通信系统中的经典RS(255,239)码和ITU-T G.975.1中的LDPC(32640,30592)码分别提高了约2.13dB和1.41dB。因而其纠错性能更强,更适用于高速长距离光通信系统。该新颖构造方法简单灵活且编译码更容易实现。     

A novel construction scheme of QC-LDPC codes based on the RU algorithm for optical transmission systems

A novel lower-complexity construction scheme of quasi-cyclic low-density parity-check (QC-LDPC) codes for optical transmission systems is proposed based on the structure of the parity-check matrix for the Richardson-Urbanke (RU) algorithm. Furthermore, a novel irregular QC-LDPC(4 288, 4 020) code with high code-rate of 0.937 is constructed by this novel construction scheme. The simulation analyses show that the net coding gain (NCG) of the novel irregular QC-LDPC(4 288,4 020) code is respectively 2.08 dB, 1.25 dB and 0.29 dB more than those of the classic RS(255, 239) code, the LDPC(32 640, 30 592) code and the irregular QC-LDPC(3 843, 3 603) code at the bit error rate (BER) of 10-6. The irregular QC-LDPC(4 288, 4 020) code has the lower encoding/decoding complexity compared with the LDPC(32 640, 30 592) code and the irregular QC-LDPC(3 843, 3 603) code. The proposed novel QC-LDPC(4 288, 4 020) code can be more suitable for the increasing development requirements of high-speed optical transmission systems.     

光传输系统中基于Bose第一类方法对QC-LDPC码的一种新颖构造方法

基于平衡不完全区组设计(BIBD)、循环矩阵分解和循环置换矩阵,提出了一种适用于光传输系统的新颖准循环低密度奇偶校验码(QC-LDPC)构造方法。利用Bose的第一类方法构造的低密度校验矩阵对其进行循环列分解得到相应的模板矩阵,再利用合适的循环置换矩阵对其进行扩展。采用该方法构造的QC-LDPC码具有良好的结构,且可根据实际需要来灵活地选择码长和码率。仿真结果表明:在误码率为10-6时其码率均为93.7%的情况下,该方法构造的novel-QC-LDPC(10992,10305)码比ITU-T G.975中RS(255,239)码的净编码增益(NCG)改善了约1.8d B。因此该构造方法所构造的QC-LDPC码具有更好的纠错性能,更适合高速长距离的光传输系统。     

光通信系统中一种低复杂度的AP-QC-LDPC码新颖构造方法

提出了一种低复杂度的具有等差数列(AP)特性的准循环低密度奇偶校验(QC-LDPC)码构造方法,该方法结构简单,节省了存储空间,可根据实际需要灵活地改变码长和码率.利用该方法构造出的AP-QC-LDPC(4599,4307)码的校验矩阵的每行元素为等差数列,且公差单调递增,所以该校验矩阵不含有4环.仿真结果表明:在误码率(BER)为10-6时,该AP-QC-LDPC(4599,4307)码比ITU-T G.975中的RS(255,239)码和ITU-T G.975.1中LDPC(32640,30592)码的净编码增益(NCG)分别改善了约2.19和1.48 dB,比基于有限域乘群的eIRA-QC-LDPC(4599,4307)码和QC-LDPC(3780,3540)码的净编码增益分别提高了约0.16和0.2dB.该方法构造的AP-QC-LDPC(4599,4307)码具有更好的纠错性能,能更好地适应光通信系统的需求.     

光通信中一种基于有限域的QC-LDPC码构造方法分析

针对光通信系统日益发展的需要,提出了一种的基于有限域准循环低密度奇偶校验(QC-LDPC)码的构造方法.该构造方法简单易行,可以有效避免四环的出现,具有良好的围长特性,同时能够自由地选择码长码率.仿真分析表明:通过该构造方法所构造的码率为93.7％的QC-LDPC(3780,3540)码的纠错性能优于利用随机构造方法所构造的SCG-LDPC (3969,3720)码和已广泛用于ITU-T G.975中的RS(255,239)码和ITU-T G.975.1中的LDPC(32640,30592)码,因而其更适合于光通信系统.     

光传输系统中LDPC码的新颖随机构造方法

分析光传输系统的传输特性与规则LDPC码的随机构造方法之后,提出LDPC码的一种新颖随机构造方法。基于该构造方法构造了一种新颖的码率为93.7%、冗余度为6.69%的规则SCG-LDPC(3969,3720)码,该规则LDPC码在将来的硬件实现中可相对地节省硬件存储空间和减少计算量。仿真分析表明:该新颖的SCG-LDPC(3969,3720)码在BER为10-8时,与同码率同冗余度的ITU-T G.975建议中RS(255,239)码和ITU-T G.975.1建议中LDPC(32640,30592)码相比,其净编码增益(NCG)和距离Shannon限都分别改善了约1.92dB和0.97dB。因而该规则LDPC码的纠错性能更为优越,更适用于光传输系统。     

一种高速长距离光通信系统中QC-LDPC码的构造方法

提出了一种新的准循环低密度奇偶校验(QC-LDPC)码的构造方法,给出了用该方法构造无环四QC-LDPC码的充分条件。并针对光通信系统的传输特点,用此方法构造了适用于高速长距离光通信系统的QC-LDPC(4 221,3 956)码。仿真结果分析表明:在码率为93.7%、误码率BER为10-6时,与广泛用于光通信系统中的经典RS(255,239)码相比,其净编码增益(NCG)提高了约1.8dB;比SCG-LDPC(3 969,3 720)码的NCG提高了约0.2dB,距离香农极限约1.4dB,且远低于PEG-LDPC(4 221,3 956)码的错误平层,这正满足光通信系统中低错误平层的要求。     

一种利用完备差集构造原模图QC-LDPC码的方法

利用组合数学中的完备差集,对原模图提出了一种新颖的准循环低密度奇偶校验码(Quasi-Cyclic Low-Density Parity-Check,QC-LDPC)扩展方法。该方法能大幅降低编译码的复杂度,所得到的校验矩阵中不存在四环。仿真结果表明:利用该方法构造出的P-CDS-QC-LDPC(798,399)码,在BER为10-4时,对比基于完备差集构造的同码率CDS-QC-LDPC(1092,546)码,其净编码增益提高约0.24dB。在BER为10-5时,对比基于渐近边增长(Progressive EdgeGrowth,PEG)算法构造的同码率PEG-LDPC(900,450)码,其净编码增益提高约0.15dB。     

基于大衍数列的规则QC-LDPC码构造方法

结合杨辉三角的结构形式,基于大衍数列提出了一种列重为3或4的规则准循环低密度奇偶校验(QC-LDPC)码的新构造方法.该方法构造的校验矩阵围长至少为6,码长可灵活变化,并且可节省存储空间.仿真结果表明:在相同的仿真参数下,当误码率(BER)为10-6时,所构造的列重为3的QC-LDPC(1260,620)码的净编码增益(NCG)比二次函数码改善了1 dB左右;列重为4的QC-LDPC(6056,3028)码相对于WMC-OCS、QC-OCS码分别有0.1 dB和0.2 dB的NCG提升.     

Novel construction of quasi-cyclic low-density parity-check codes with variable code rates for cloud data storage systems

This paper proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of medium to high code rates that can be applied in cloud data storage systems, requiring better error correction capabilities. The novelty of this method lies in the construction of sparse base matrices, using a girth greater than 4 that can then be expanded with a lift factor to produce high code rate QC-LDPC codes. Investigations revealed that the proposed large-sized QC-LDPC codes with high code rates displayed low encoding complexities and provided a low bit error rate (BER) of 10⁻¹⁰ at 3.5 dB E_b/N₀ than conventional LDPC codes, which showed a BER of 10⁻⁷ at 3 dB E_b/N₀. Subsequently, implementation of the proposed QC-LDPC code in a software-defined radio, using the NI USRP 2920 hardware platform, was conducted. As a result, a BER of 10⁻⁶ at 4.2 dB E_b/N₀ was achieved. Then, the performance of the proposed codes based on their encoding–decoding speeds and storage overhead was investigated when applied to a cloud data storage (GCP). Our results revealed that the proposed codes required much less time for encoding and decoding (of data files having a 10 MB size) and produced less storage overhead than the conventional LDPC and Reed–Solomon codes.     

Construction of a new regular LDPC code for optical transmission systems

A novel construction method of the check matrix for the regular low density parity check （LDPC） code is proposed. The novel regular systematically constructed Gallager （SCG）-LDPC（3969,3720） code with the code rate of 93.7% and the redundancy of 6.69% is constructed. The simulation results show that the net coding gain （NCG） and the distance from the Shannon limit of the novel SCG-LDPC（3969,3720） code can respectively be improved by about 1.93 dB and 0.98 dB at the bit error rate （BER） of 10^-8, compared with those of the classic RS（255,239） code in ITU-T G.975 recommendation and the LDPC（32640,30592） code in ITU-T G.975.1 recommendation with the same code rate of 93.7% and the same redundancy of 6.69%. Therefore, the proposed novel regular SCG-LDPC（3969,3720） code has excellent performance, and is more suitable for high-speed long-haul optical transmission systems.     

A new construction method of QC-LDPC codes with low error floor based on EETS and Zig-Zag

Aiming at the problem that quasi-cyclic low density parity check (QC-LDPC) codes may have the error floor in the high signal to noise ratio (SNR) region, a new construction method of the QC-LDPC codes with the low error floor is proposed. The basic matrix of the method is based on the progressive edge growth (PEG) algorithm and the improved eliminate elementary trapping sets (EETS) algorithm so as to eliminate the elementary trapping sets in the basic matrix, then the Zig-Zag method is used to construct the cyclic shift matrix which is used to extend the basic matrix in order to construct the parity check matrix. The method not only can improve the error floor in the high SNR region, but also can flexibly design the code length and code rate. The simulation results show that at the bit error rate of 10-6, the PEG-trapping-Zig-Zag (PTZZ)-QC-LDPC(3024,1512) codes with the code rate of 0.5, compared with the PEG-Zig-Zag (PZZ)-QC-LDPC(3024,1512) codes and the PEG-QC-LDPC(3024,1512) codes, can respectively improve the net coding gain of 0.1 dB and 0.16 dB. The difference among the bit error rate performance curves will become better with the increase of the SNR. In addition, the PTZZ-QC-LDPC(3024,1512) codes have no error floor above the SNR of 2.2 dB.     

光通信系统中基于伽罗华域乘群的QC-LDPC码的一种新颖构造方法

基于Galois域GF(q)乘群,提出了一种构造简单且编码容易实现的新颖准循环低密度奇偶校验(QC-LDPC)码构造方法,可灵活地调整码长、码率,且编译码复杂度低。用本文方法构造了适用于光通信系统的非规则QC-LDPC(3843,3603)码,仿真表明,与已广泛用于光通信系统中的经典RS(255,239)码相比,用本文方法构造的码具有更好的纠错性能,且其性能优于用SCG方法构造的LDPC码和规则的QC-LDPC(4221,3956)码,适合用于高速长距离光通信系统。     

基于基矩阵排列优化算法的非规则准循环低密度奇偶校验码构造

为了提升非规则准循环低密度奇偶校验(QC-LDPC)码的误码率性能、降低构造算法的复杂度,该文提出一种基于基矩阵排列优化算法的非规则QC-LDPC码构造方法。首先,利用基于外部信息传递(EXIT)图的阈值分析算法得到满足码率和列重要求的非规则QC-LDPC码的最优度分布,然后将围长和短环数量作为新的约束条件对具有最优度分布的码集进行分析,得到具有最优度分布和最少短环数量的最优基矩阵排列结构,最后,根据得到的基矩阵对规则指数矩阵进行置零操作得到目标非规则QC-LDPC码。该构造方法相对于随机构造方法具有更低的实现复杂度,同时可以通过改变算法的参数值实现码长和码率的灵活设计。仿真结果表明,与现有的一些构造方法相比,所提方法构造的非规则QC-LDPC码在加性高斯白噪声(AWGN)信道上具有更好的误码率性能。     

同类2FSK混合信号中微弱信号提取方法研究

针对光传输系统的特点与要求, 基于有限域提出了一种新颖的扩展非规则重复累积准循环低密度奇偶校验码(eIRA-QC-LDPC)构造方法, 该构造方法可根据实际需要灵活地调整码长码率, 利用此方法构造了一种码率为0.937的novel-eIRA-QC-LDPC(4599,4307)码。仿真结果表明: 在误码率(BER)为10-7且码率均为0.937的情况下, 该novel-eIRA-QC-LDPC(4599,4307)码比ITU-T G.975中RS(255,239)码和ITU-T G.975.1中LDPC(32640,30592)码的净编码增益(NCG)分别改善了约2.13和1.33dB, 比用SCG构造方法构造的SCG-eIRA-LDPC(3717,3481)码和基于阵列码构造方法构造的Array-eIRA-QC-LDPC(4560,4275)码的净编码增益分别提高了约0.24和0.41dB, 距离香农限约1.07dB。因而该构造方法所构造的eIRA-QC-LDPC码具有更好的纠错性能, 更适合高速长距离的光传输系统。     

基于FPGA有限域构造的QC-LDPC分层译码器设计

针对QC-LDPC码的Tanner图中存在的短环,尤其是4环,对迭代译码性能产生不利影响的问题,寻找到一种有限域乘群构造法,该方法构造的QC-LDPC码的Tanner图中不存在任何4环。基于此方法构造的码长为3 060,码率为的(3,12)规则QC-LDPC码,选用Altera公司StratixII系列的EP2S60F484C4器件,对其实现了分层译码器硬件结构的设计。实现结果表明,在最大迭代次数为5时,时钟频率最高可达35.38 MHz,吞吐量达到92.27 Mbit·s-1。     

利用完备差集构造QC-LDPC码

针对准循环低密度奇偶校验( QC-LDPC)码中循环置换矩阵的移位次数的确定问题,提出了一种利用组合设计中完备差集( PDF)构造QC-LDPC码的新颖方法。当循环置换矩阵的维度大于一定值时,该方法所构造的规则QC-LDPC码围长至少为6,具有灵活选择码长和码率的优点,且所需的存储空间更少,降低了硬件实现的复杂度。仿真结果表明：在误码率为10-5时,所构造的码率为3/4的PDF-QC-LDPC(3136,2352)与基于最大公约数(GCD)构造的GCD-QC-LDPC(3136,2352)码和基于循环差集(CDF)构造的CDF-QC-LDPC(3136,2352)码相比,其净编码增益(NCG)分别有0.41 dB和0.32 dB的提升;且在码率为4/5时,所构造的PDF-QC-LDPC (4880,3584)码比GCD-QC-LDPC(4880,3584)码和CDF-QC-LDPC(4880,3584)码的NCG分别改善了0.21 dB和0.13 dB。