A CLASS OF LDPC CODE’S CONSTRUCTION BASED ON AN ITERATIVE RANDOM METHOD

This letter gives a random construction for Low Density Parity Check （LDPC） codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code＇s parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.     

A CLASS OF LDPC CODE''''S CONSTRUCTION BASED ON AN ITERATIVE RANDOM METHOD

This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code's parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.     

REGULAR LOW-DENSITY PARITY-CHECK CODES BASED ON SHIFTED IDENTITY MATRICES

This paper extends the class of Low-Density Parity-Check （LDPC） codes that can be constructed from shifted identity matrices. To construct regular LDPC codes, a new method is proposed. Two simple inequations are adopted to avoid the short cycles in Tanner graph, which makes the girth of Tanner graphs at least 8. Because their parity-check matrices are made up of circulant matrices, the new codes are quasi-cyclic codes. They perform well with iterative decoding.     

ON JUSTESEN''''S ALGEBRAIC GEOMETRY CODES

An isomorphism preserving Hamming distance between two algebraic geometry(AG)codes is presented to obtain the main parameters of Justesen's algebraic geometry(JAG)codes.To deduce a simple approach to the decoding algorithm,a code word in a“small”JAG codeis used to correspond to error-locator polynomial.By this means,a simple decoding procedureand its ability of error correcting are explored obviously.The lower and upper bounds of thedimension of AG codes are also obtained.     

A Belief Propagation Algorithm with Set-Breaking to Lower Error-Floors of Low-Density Parity-Check Codes

In order to more exactly describe the errorfloor phenomenon in the iterative decoding of Low-density parity-check （LDPC） codes, a modified concept of the stable trapping set is introduced. Based on this new concept, an improved belief propagation algorithm with setbreaking mechanism is proposed to lower the error-floors of LDPC codes. Message ranking of the bit nodes in the stable trapping sets will be greatly lowered than that of other bit nodes in the iterative decoding process. By using this characteristic to label the bit nodes in the set, the corresponding initial log likelihood ratios will be flipped to break the stable trapping set and restart to decode. Simulation results verify the validity of the proposed algorithm.     

NEW ITERATIVE SUPER-TRELLIS DECODING WITH SOURCE A PRIORI INFORMATION FOR VLCS WITH TURBO CODES

A novel Joint Source and Channel Decoding （JSCD） scheme for Variable Length Codes （VLCs） concatenated with turbo codes utilizing a new super-trellis decoding algorithm is presented in this letter. The basic idea of our decoding algorithm is that source a priori information with the form of bit transition probabilities corresponding to the VLC tree can be derived directly from sub-state transitions in new composite-state represented super-trellis. A Maximum Likelihood （ML） decoding algorithm for VLC sequence estimations based on the proposed super-trellis is also described. Simulation results show that the new iterative decoding scheme can obtain obvious encoding gain especially for Reversible Variable Length Codes （RVLCs）, when compared with the classical separated turbo decoding and the previous joint decoding not considering source statistical characteristics.     

Improved Low-Density Parity-Check Codes Based on Single Parity-Check Block and Their Performance Analysis on the Binary Erasure Channel

Based on some characteristics of the decoding and the error floors of Low-density parity-check （LDPC） codes, an approach of adding single parity-check blocks to original LDPC blocks over GF（2） is proposed, which can correct multiple block errors in a block group. When this approach is set to have correcting capacity of no more than two block errors in a code group, the performance of ensemble C（n, x^l-1,x^r-1） on the Binary erasure channel （BEC） with this approach is analyzed. The simulation results indicate that when the Block error probability （BEP） drops, the performance of the given LDPC codes can be improved exponentially. Also, the error floors of LDPC codes can be suppressed to much lower levels. Therefore, the short-length LDPC codes have a wide range of applications.     

An iterative reliability-based decoding for LDPC coded modulation system

In this paper, A Belief Propagation concatenated Orderd-Statistic Decoder （BP-OSD） based on accumulated Log-Likelihood Ratio （LLR） is proposed for medium and short lengths Low Density Parity-Check （LDPC） codes coded Bit-Interleaved Coded Modulation （BICM） systems. The accumulated soft output values delivered by every BP iteration are used as reliability values of Soft-Input Soft-Output OSD （SISO-OSD） decoder and the soft output of SISO-OSD is used as a priori probabilities of the demodulator for the next iteration. Simulation results show that this improved algorithm achieves noticeable performance gain with only modest increase in computation complexity.     

Bidirectional Viterbi Decoding Algorithm for OvTDM

Overlapped time domain multiplexing(OvTDM)is an innovative encoding scheme that can obtain high spectral efficiency.However,the intentional inter-symbol interference(ISI)caused by OvTDM will make the decoding process more complex.The computational complexity of maximum likelihood sequence detection increases exponentially with the growth of spectral efficiency in OvTDM.As a consequence of high complexity,the decoding effort for a given spectral efficiency may occasionally exceed the physical limitations of the decoder,leading inevitably to buffer overflows and information erasures.In this paper,we propose a bidirectional Viterbi algorithm(BVA)based on the bidirectional sequence decoding for OvTDM.With the BVA,the decoding operation starts simultaneously from the both ends of the corresponding trellis and stops at the middle of trellis.The simulation results show that compared with Viterbi algorithm(VA),the decoding time of BVA can be reduced by about half.And the memory space of two decoders in BVA are about half of that in VA,which means that the BVA has lower memory requirements for decoder.And the decoding performance of BVA is almost the same as VA.     

Layered Dynamic Schedulings for BP Decoding of LDPC Codes over GF（q）

A Layered dynamic scheduling （LDS） for Belief-propagation （BP） decoding of LDPC codes over GF（q） is presented, which is derived from the dynamic scheduling for the BP decoding of binary LDPC codes. In order to restrain the LDS from cycling in certain checknodes, a life-index for each check-node is adopted and the optimal value of the life-index is analyzed. Furthermore, in consideration of hardware implementation and decoding latency, a strategy, which allows many more checknodes to be updated in parallel, is introduced. Simulations show that the LDS with life-index speeds up the convergence rate and greatly improves the performance of the BP decoding at medium to high signal-to-noise ratio value, and the algorithm employing the LDS with life-index and the new strategy offers good trade-off between the performance and the decoding latency.     

基于二分图的乘积码迭代译码算法

该文给出了由汉明分量乘积码构造广义低密度(GLD)码的一般方法。基于所得稀疏矩阵的二分图,并结合分组码与低密度校验(LDPC)码的译码算法,设计出一种新颖的可用于乘积码迭代译码的Chase-MP算法。由于所得二分图中不含有长度为4和6的小环,因而大大减少图上迭代时外信息之间的相关性,进而提高译码性能。对加性高斯白噪声(AWGN)及瑞利(Rayleigh)衰落信道下,汉明分量 (63,57,3)2 乘积码的模拟仿真显示,该算法能够获得很好的译码性能。与传统的串行迭代Chase-2算法相比,Chase-MP算法适合用于全并行译码处理,便于硬件实现,而且译码性能优于串行迭代Chase-2算法。     

基于置信传播的优化译码算法研究

该文在对LDPC码的译码算法分析的基础上,针对校验矩阵中含有的环对译码算法的影响,提出了一种在置信传播算法基础之上的译码算法。该算法通过及时切断消息在环上的重传回路,可消除因校验矩阵中的环回传原始信息对译码造成的影响,保证优质的原始信息能尽可能地传播到其能传播的节点,从而提升了LDPC码的译码性能。仿真实验表明,在低信噪比的信道中,该算法具有相当于传统算法的性能和更低的计算复杂度;在良好的信道条件下可以取得比传统算法更优异的性能。     

Low-density parity check codes and iterative decoding for long-haul optical communication systems

A forward-error correction (FEC) scheme based on low-density parity check (LDPC) codes and iterative decoding using belief propagation in code graphs is presented in this paper. We show that LDPC codes provide a significant system performance improvement with respect to the state-of-the-art FEC schemes employed in optical communications systems. We present a class of structured codes based on mutually orthogonal Latin rectangles. Such codes have high rates and can lend themselves to very low-complexity encoder/decoder implementations. The system performance is further improved by a code design that eliminates short cycles in a graph employed in iterative decoding.     

Efficient algorithm for calculating short cycles in Tanner graph based on matrix computation

Loop distribution of Tanner graph affects the BER performance of low-density parity-check codes(LDPC) decoding.To count short cycles in the Tanner graph efficiently,a side by side recursion algorithm based on matrix computation was proposed.Firstly,5 basic graph structures were defined to realize recursive calculate in the implementation process.Compared with previous works,the algorithm provided many methods for counting the same length of cycles.The same result confirmed the correctness of the algorithm.The new algorithm could not only calculate the total number of cycles,but also gave the number each edge participating in fixed-length cycles.Its complexity was proportional to the product of D and square of N,where D was the average degree of variable nodes,and N denoted the code length.For LDPC codes,D was far less than N.For most of the LDPC codes,the calculation for numbers of cycle-length g、g+2、g+4 was only several seconds.     

Graph-based iterative decoding algorithms for parity-concatenatedtrellis codes

We construct parity-concatenated trellis codes in which a trellis code is used as the inner code and a simple parity-check code is used as the outer code. From the Tanner-Wiberg-Loeliger (1981, 1996) graph representation, several iterative decoding algorithms can be derived. However, since the graph of the parity-concatenated code contains many short cycles, the conventional min-sum and sum-product algorithms cannot achieve near-optimal decoding. After some simple modifications, we obtain near-optimal iterative decoders. The modifications include either (a) introducing a normalization operation in the min-sum and sum-product algorithms or (b) cutting the short cycles which arise in the iterative Viterbi algorithm (IVA). After modification, all three algorithms can achieve near-optimal performance, but the IVA has the least average complexity. We also show that asymptotically maximum-likelihood (ML) decoding and a posteriori probability (APP) decoding can be achieved using iterative decoders with only two iterations. Unfortunately, this asymptotic behavior is only exhibited when the bit-energy-to-noise ratio is above the cutoff rate. Simulation results show that with trellis shaping, iterative decoding can perform within 1.2 dB of the Shannon limit at a bit error rate (BER) of 4×10^-5 for a block size of 20000 symbols. For a block size of 200 symbols, iterative decoding can perform within 2.1 dB of the Shannon limit     

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。     

Selective avoidance of cycles in irregular LDPC code construction

This letter explains the effect of graph connectivity on error-floor performance of low-density parity-check (LDPC) codes under message-passing decoding. A new metric, called extrinsic message degree (EMD), measures cycle connectivity in bipartite graphs of LDPC codes. Using an easily computed estimate of EMD, we propose a Viterbi-like algorithm that selectively avoids small cycle clusters that are isolated from the rest of the graph. This algorithm is different from conventional girth conditioning by emphasizing the connectivity as well as the length of cycles. The algorithm yields codes with error floors that are orders of magnitude below those of random codes with very small degradation in capacity-approaching capability.     

LDPC码的改进译码算法

由于短帧长LDPC码存在很多环路,其译码性能不具有最优性.本文首先推导了有环路LDPC码的概率译码算法,然后在传统的概率译码算法引入了修正系数,从而减小了环路对译码性能的影响.仿真结果表明,采用改进的译码算法可以提高译码性能.