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.

Implementation of low-density parity-check codes decoder for CCSDS standard

The complexity/performance balanced decoder for low-density parity-check （LDPC） codes is preferred in practical wireless communication systems. A low complexity LDPC decoder for the Consultative Committee for Space Data Systems （CCSDS） standard is achieved in DSP. An ap- proximate decoding algorithm, normalized rain-sum algorithm, is used in the implementation for its low amounts of computation. To reduce the performance loss caused by the approximation, the pa- rameters of the normalized min-sum algorithm are determined by calculating and finding the mini- mum value of thresholds through density evolution. The minimum value which indicates the best per- formance of the decoding algorithm is corresponding with the optimized parameters. In implementa- tion, the memory cost is saved by decomposing the parity-check matrix into submatrices to store and the computation of passing message in decoding is accelerated by using the intrinsic function of DSP. The performance of the decoder with optimized factors is simulated and compared with the ideal BP decoder. The result shows they have about the same performance.

THE Q-MATRIX LOW-DENSITY PARITY-CHECK CODES

This paper presents a matrix permuting approach to the construction of Low-Density Parity-Check (LDPC) code. It investigates the structure of the sparse parity-check matrix defined by Gallager. It is discovered that the problem of constructing the sparse parity-check matrix requires an algorithm that is efficient in search environments and also is able to work with constraint satisfaction problem. The definition of Q-matrix is given, and it is found that the queen algorithm enables to search the Q-matrix. With properly permuting Q-matrix as sub-matrix, the sparse parity-check matrix which satisfied constraint condition is created, and the good regular-LDPC code that is called the Q-matrix LDPC code is generated. The result of this paper is significant not only for designing low complexity encoder, improving performance and reducing complexity of iterative decoding arithmetic, but also for building practical system of encodable and decodable LDPC code.

CONSTRUCTION OF NONSYSTEMATIC LOW-DENSITY PARITY-CHECK CODES BASED ON SYMMETRIC BALANCED INCOMPLETE BLOCK DESIGN

This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block codes is bounded by the average row weight of generalizedinverses of their generator matrices and code rate.Then a class of nonsystematic LDPC codes con-structed based on SBIBD is presented.Their characteristics include:both generator matrices andparity-check matrices are sparse and cyclic,which are simple to encode and decode;and almost arbi-trary rate codes can be easily constructed,so they are rate-compatible codes.Because there aresparse generalized inverses of generator matrices,the performance of the proposed codes is only0.15dB away from that of the traditional systematic LDPC codes.

Low-Complexity Optimization Algorithm for Irregular Low-Density Parity-Check Codes

A low-complexity algorithm is proposed in this paper in order to optimize irregular low-density parity-check （LDPC） codes.The algorithm proposed can calculate the noise threshold by means of a one-dimensional density evolution and search the optimal degree profiles with fast-convergence differential evolution,so that it has a lower complexity and a faster convergence speed.Simulation resuits show that the irregular LDPC codes optimized by the presented algorithm can also perform better than Turbo codes at moderate block length even with less computation cost.

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.

DESIGN OF LOW-DENSITY PARITY CHECK CODES FOR MIMO SYSTEMS WITH DOUBLY-ITERATIVE RECEIVERS

In this paper, the Multiple Input Multiple Output (MIMO) doubly-iterative receiver which consists of the Probabilistic Data Association detector (PDA) and Low-Density Parity-Check Code (LDPC) decoder is developed. The receiver performs two iterative decoding loops. In the outer loop, the soft information is exchanged between the PDA detector and the LDPC decoder. In the inner loop, it is exchanged between variable node and check node decoders inside the LDPC decoder. On the light of the Extrinsic Information Transfer (EXIT) chart technique, an LDPC code degree profile optimization algorithm is developed for the doubly-iterative receiver. Simulation results show the doubly-receiver with optimized irregular LDPC code can have a better performance than the one with the regular one.

Generalized Low-Density Parity-Check Coding Scheme with Partial-Band Jamming

In this study, a class of Generalized Low-Density Parity-Check （GLDPC） codes is designed for data transmission over a Partial-Band Jamming （PBJ） environment. The GLDPC codes are constructed by replacing parity-check code constraints with those of nonsystematic Bose-Chaudhuri-Hocquenghem （BCH）, referred to as Low-Density Parity-Check （LDPC）-BCH codes. The rate of an LDPC-BCH code is adjusted by selecting the transmission length of the nonsystematic BCH code, and a low-complexity decoding algorithm based on message- passing is presented that employs A Posteriori Probability （APP） fast BCH transform for decoding the BCH check nodes at each decoding iteration. Simulation results show that the LDPC-BCH codes with a code rate of 1/8.5 have a bit error rate performance of 1 x10-8 at signal-noise-ratios of -6.97 dB, -4.63 dB, and 2.48 dB when the fractions of the band jammed are 30%, 50%, and 70%, respectively.

Minimum distances of three families of low-density parity-check codes based on finite geometries

Three families of low-density parity-check （LDPC） codes are constructed based on the totally isotropic subspaces of symplectic, unitary, and orthogonal spaces over finite fields, respectively. The minimum distances of the three families of LDPC codes in some special cases are settled.

Construction of Rate-Compatible(RC) Low-Density Parity-Check(LDPC) Convolutional Codes Based on RC-LDPC Block Codes

In this paper,a family of rate-compatible(RC) low-density parity-check(LDPC) convolutional codes can be obtained from RC-LDPC block codes by graph extension method.The resulted RC-LDPC convolutional codes,which are derived by permuting the matrices of the corresponding RC-LDPC block codes,are systematic and have maximum encoding memory.Simulation results show that the proposed RC-LDPC convolutional codes with belief propagation(BP) decoding collectively offer a steady improvement on performance compared with the block counterparts over the binary-input additive white Gaussian noise channels(BI-AWGNCs).

Code Optimization, Frozen Glassy Phase and Improved Decoding Algorithms for Low-Density Parity-Check Codes

The statistical physics properties of low-density parity-cheek codes for the binary symmetric channel are investigated as a spin glass problem with multi-spin interactions and quenched random fields by the cavity method. By evaluating the entropy function at the Nishimori temperature, we find that irregular constructions with heterogeneous degree distribution of check （bit） nodes have higher decoding thresholds compared to regular counterparts with homo- geneous degree distribution. We also show that the instability of the mean-field caiculation takes place only after the entropy crisis, suggesting the presence of a frozen glassy phase at low temperatures. When no prior knowledge of channel noise is assumed （searching for the ground state）, we find that a reinforced strategy on normal belief propagation will boost the decoding threshold to a higher value than the normal belief propagation. This value is dose to the dynamicai transition where all local search heuristics fail to identify the true message （codeword or the ferromagnetic state）. After the dynamical transition, the number of metastable states with larger energy density （than the ferromagnetic state） becomes exponentially numerous. When the noise level of the transmission channel approaches the static transition point, there starts to exist exponentiaily numerous codewords sharing the identical ferromagnetic energy.

Multistep Linear Programming Approaches for Decoding Low-Density Parity-Check Codes

The problem of improving the performance of linear programming（LP） decoding of low-density parity-check（LDPC） codes is considered in this paper.A multistep linear programming（MLP） algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity.The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder.The MLP decoder result is shown to have the maximum-likelihood（ML） certificate property.Simulations with moderate block length LDPC codes suggest that the MLP decoder gives better performance than both the original LP decoder and the conventional sum-product（SP） decoder.

An improved low-complexity sum-product decoding algorithm for low-density parity-check codes

In this paper, an improved low-complexity sum-product decoding algorithm is presented for low-density parity-check （LDPC） codes. In the proposed algorithm, reduction in computational complexity is achieved by utilizing fast Fourier transform （FFT） with time shift in the check node process. The improvement in the decoding performance is achieved by utilizing an op- timized integer constant in the variable node process. Simulation results show that the proposed algorithm achieves an overall coding gain improvement ranging from 0.04 to 0.46 dB. Moreover, when compared with the sum-product algorithm （SPA）, the proposed decoding algorithm can achieve a reduction of 42%-67% of the total number of arithmetic operations required for the decoding process.

Quad-Level Cell NAND Design and Soft-Bit Generation for Low-Density Parity-Check Decoding in System-Level Application

QLC（Quad-Level Cell） NAND flash will be one of the future technologies for next generation memory chip after three-dimensional（3D） TLC（Triple-Level Cell） stacked NAND flash. In QLC device, data errors will easily occur because of 2~4 data levels in the limited voltage range. This paper studies QLC NAND technology which is 4 bits per cell. QLC programming methods based on 16 voltage levels and reading method based on ＂half-change＂ Gray coding are researched. Because of the probable error impact of QLC NAND cell＇s voltage change, the solution of generating the soft information after XOR（exclusive OR） the soft bits by internal read mechanism is presented for Low-Density Parity-Check（LDPC） Belief Propagation（BP） decoding in QLC design for its system level application.

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.

Improved parallel weighted bit-flipping algorithm

An improved parallel weighted bit-flipping（PWBF） algorithm is presented. To accelerate the information exchanges between check nodes and variable nodes, the bit-flipping step and the check node updating step of the original algorithm are parallelized. The simulation experiments demonstrate that the improved PWBF algorithm provides about 0. 1 to 0. 3 dB coding gain over the original PWBF algorithm. And the improved algorithm achieves a higher convergence rate. The choice of the threshold is also discussed, which is used to determine whether a bit should be flipped during each iteration. The appropriate threshold can ensure that most error bits be flipped, and keep the right ones untouched at the same time. The improvement is particularly effective for decoding quasi-cyclic low-density paritycheck（QC-LDPC） codes.

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.

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.