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。

Combinatorial design-based quasi-cyclic LDPC codes with girth eight

This paper presents a novel regular Quasi-Cyclic (QC)Low Density Parity Check (LDPC)codes with columnweight three and girth at least eight.These are designed on the basis of combinatorial design in which subsets applied for the construction of circulant matrices are determined by a particular subset.Considering the nonexistence of cycles four and six in the structure of the parity check matrix,a bound for their minimum weight is proposed.The simtdations conducted confirm that without applying a masking technique,the newly implemented codes have a performance similar to or better than other well-known codes.This is evident in the waterfall region, while their error floor at very low Bit Error Rate (BER)is expected.

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.

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.

Low-Density Parity-Check Codes:Highway to Channel Capacity

Low-density parity-check(LDPC)codes are not only capacity-approaching,but also greatly suitable for high-throughput implementation.Thus,they are the most popular codes for high-speed data transmission in the past two decades.Thanks to the low-density property of their parity-check matrices,the optimal maximum a posteriori probability decoding of LDPC codes can be approximated by message-passing decoding with linear complexity and highly parallel nature.Then,it reveals that the approximation has to carry on Tanner graphs without short cycles and small trapping sets.Last,it demonstrates that well-designed LDPC codes with the aid of computer simulation and asymptotic analysis tools are able to approach the channel capacity.Moreover,quasi-cyclic(QC)structure is introduced to significantly facilitate their high-throughput implementation.In fact,compared to the other capacity-approaching codes,QC-LDPC codes can provide better area-efficiency and energy-efficiency.As a result,they are widely applied in numerous communication systems,e.g.,Landsat satellites,Chang’e Chinese Lunar mission,5G mobile communications and so on.What’s more,its extension to non-binary Galois fields has been adopted as the channel coding scheme for BeiDou navigation satellite system.

利用等差数列构造大围长准循环低密度奇偶校验码

针对准循环低密度奇偶校验(QC-LDPC)码中准循环基矩阵的移位系数确定问题,该文提出基于等差数列(AP)的确定方法。该方法构造的校验矩阵的围长至少为8,移位系数由简单的数学表达式确定,节省了编解码存储空间。研究结果表明,该方法对码长和码率参数的设计具有较好的灵活性。同时表明在加性高斯白噪声(AWGN)信道和置信传播(BP)译码算法下,该方法构造的码字在码长为1008、误比特率为510-时,信噪比优于渐进边增长(PEG)码近0.3 d B。

任意列重大围长QC-LDPC码的确定性构造

针对准循环低密度奇偶校验(Quasi-Cyclic Low-Density Parity-Check,QC-LDPC)码中准循环基矩阵的移位系数确定问题,提出基于等差数列的确定方法.该方法构造的校验矩阵围长为8,列重可任意选取,移位系数由简单的数学表达式确定,编码复杂度与码长呈线性关系,节省了编解码存储空间.研究结果表明,列重和围长是影响码字性能的重要因素.在加性高斯白噪声(Additive White Gauss Noise,AWGN)信道和置信传播(Belief Propagation,BP)译码算法下,该方法构造的码字在短码时可以获得与IEEE 802.11n、802.16e码相一致的性能,在长码时误比特率性能接近DVB-S2码.同时表明该方法对码长和码率参数的设计具有较好的灵活性.

高性能准循环低密度奇偶校验码的构造

提出了一种高性能QC-LDPC码的构造模式,通过扩展码长较短的LDPC码得到码长比较长的QC-LDPC码。仿真结果表明,该构造模式具有线性编码复杂度,编码性能和随机构造的LDPC码相当。

Modified Benes network architecture for WiMAX LDPC decoder

A modified Benes network is proposed to be used as an optimal shuffle network in worldwide interoperability for microwave access （WiMAX） low density parity check （LDPC） decoders, When the size of the input is not a power of two, the modified Benes network can achieve the most optimal performance. This modified Benes network is non-blocking and can perform any sorts of permutations, so it can support 19 modes specified in the WiMAX system. Furthermore, an efficient algorithm to generate the control signals for all the 2 × 2 switches in this network is derived, which can reduce the hardware complexity and overall latency of the modified Benes network. Synthesis results show that the proposed control signal generator can save 25.4% chip area and the overall network latency can be reduced by 36. 2%.

1 Gbit/s QC-LDPC码译码结构的设计

为了提高准循环奇偶校验(QC-LDPC)码的译码速率,提出一种改进的部分并行QC-LDPC译码结构(IPPD)。根据QC-LDPC校验矩阵的特点,对子矩阵进行平均分层,采用部分并行译码结构加快译码迭代速度。实验仿真采用IEEE802.16e中码长为864、码率为0.5的QC-LDPC码进行验证。实验结果表明,当最大迭代次数为15、系统时钟频率为107 MHz时,该译码结构的吞吐率可达1Gbit/s。

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.

Efficient construction and encoding of QC-LDPC codes by cyclic lifting of protographs

Quasi-cyclic low-density parity-check （QC-LDPC） codes can be constructed conveniently by cyclic lifting of protographs. For the purpose of eliminating short cycles in the Tanner graph to guarantee performance, first an algorithm to enumerate the harmful short cycles in the protograph is designed, and then a greedy algorithm is proposed to assign proper permutation shifts to the circulant permutation submatrices in the parity check matrix after lifting. Compared with the existing deterministic edge swapping （DES） algorithms, the proposed greedy algorithm adds more constraints in the assignment of permutation shifts to improve performance. Simulation results verify that it outperforms DES in reducing short cycles. In addition, it is proved that the parity check matrices of the cyclic lifted QC-LDPC codes can be transformed into block lower triangular ones when the lifting factor is a power of 2. Utilizing this property, the QC- LDPC codes can be encoded by preprocessing the base matrices, which reduces the encoding complexity to a large extent.

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.

1-GENERATOR QUASI-CYCLIC CODES

This paper discusses the enumeration of 1-generator quasi-cyclic codes and describes an algorithm which will obtain one, and only one, generator for each 1-generator quasi-cyclic code.

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.

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.

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.