LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.     

Quasi-cyclic LDPC codes using overlapping matrices and their layered decoders

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have the parity-check matrices consisting of circulant matrices. Since QC LDPC codes whose parity-check matrices consist of only circulant permutation matrices are difficult to support layered decoding and, at the same time, have a good degree distribution with respect to error correcting performance, adopting multi-weight circulant matrices to parity-check matrices is useful but it has not been much researched. In this paper, we propose a new code structure for QC LDPC codes with multi-weight circulant matrices by introducing overlapping matrices. This structure enables a system to operate on dual mode in an efficient manner, that is, a standard QC LDPC code is used when the channel is relatively good and an enhanced QC LDPC code adopting an overlapping matrix is used otherwise. We also propose a new dual mode parallel decoder which supports the layered decoding both for the standard QC LDPC codes and the enhanced QC LDPC codes. Simulation results show that QC LDPC codes with the proposed structure have considerably improved error correcting performance and decoding throughput.     

Efficient encoding of quasi-cyclic low-density parity-check codes

Quasi-cyclic (QC) low-density parity-check (LDPC) codes form an important subclass of LDPC codes. These codes have encoding advantage over other types of LDPC codes. This paper addresses the issue of efficient encoding of QC-LDPC codes. Two methods are presented to find the generator matrices of QC-LDPC codes in systematic-circulant (SC) form from their parity-check matrices, given in circulant form. Based on the SC form of the generator matrix of a QC-LDPC code, various types of encoding circuits using simple shift registers are devised. It is shown that the encoding complexity of a QC-LDPC code is linearly proportional to the number of parity bits of the code for serial encoding, and to the length of the code for high-speed parallel encoding.     

Codes for iterative decoding from partial geometries

This paper develops codes suitable for iterative decoding using the sum-product algorithm. By considering a large class of combinatorial structures, known as partial geometries, we are able to define classes of low-density parity-check (LDPC) codes, which include several previously known families of codes as special cases. The existing range of algebraic LDPC codes is limited, so the new families of codes obtained by generalizing to partial geometries significantly increase the range of choice of available code lengths and rates. We derive bounds on minimum distance, rank, and girth for all the codes from partial geometries, and present constructions and performance results for the classes of partial geometries which have not previously been proposed for use with iterative decoding. We show that these new codes can achieve improved error-correction performance over randomly constructed LDPC codes and, in some cases, achieve this with a significant decrease in decoding complexity.     

误码条件下LDPC码参数的盲估计

针对非合作信号处理中LDPC码（Low-Density Parity-Check）的盲识别问题,提出了一种容错能力较强的开集识别算法.该算法通过对码字矩阵进行高斯约旦消元找到汉明重量较小的"相关列",并根据"相关列"中所包含的约束关系求得LDPC码的校验向量,然后剔除"相关列"中为"1"位置对应的错误码字.若根据高斯约旦消元求校验向量和剔除错误码字进行迭代无法得到更多校验向量,则对得到的这些校验向量进行稀疏化,再进行译码纠错.最后,综合利用校验向量的求解,错误码字的剔除,校验向量稀疏化,LDPC码译码进行迭代,实现LDPC码校验矩阵的有效重建.仿真结果表明,对于IEEE 802.16e标准中的（576,288）LDPC码,在误比特率为0.0022时,本文算法仍可以达到较好的识别效果.     

一种优化Girth分布的准循环LDPC码设计方法研究

在准循环LDPC码的构造中,校验矩阵拥有尽可能好的girth分布对于改善码的性能有着重要的意义。该文提出了构造准循环LDPC码的GirthOpt-DE算法,优化设计以获得具有好girth分布的移位参数矩阵为目标。仿真结果表明,该文方法得到的准循环LDPC码在BER性能和最小距离上均要优于固定生成函数的准循环LDPC码,Arrary码和Tanner码,并且使用上更为灵活,可以指定码长,码率及尽可能好的girth分布。     

Upper bounds on the rate of LDPC codes as a function of minimum distance

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.     

Quasi-Cyclic Low-Density Parity-Check Codes With Girth Larger Than 12

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.     

A lattice-based systematic recursive construction of quasi-cyclic LDPC codes

This paper presents a low-complexity recursive and systematic method to construct good well-structured low-density parity-check (LDPC) codes. The method is based on a recursive application of a partial Kronecker product operation on a given gamma x q, q ges 3 a prime, integer lattice L(gamma x q). The (n - 1)- fold product of L(gamma x q) by itself, denoted Lⁿ(gamma x q), represents a regular quasi-cyclic (QC) LDPC code, denoted (see PDF), of high rate and girth 6. The minimum distance of (see PDF) is equal to that of the core code (see PDF) introduced by L(gamma x q). The support of the minimum weight codewords in (see PDF) are characterized by the support of the same type of codewords in (see PDF). From performance perspective the constructed codes compete with the pseudorandom LDPC codes.     

Memory Access Optimized Implementation of Cyclic and Quasi-Cyclic LDPC Codes on a GPGPU

Software based decoding of low-density parity-check (LDPC) codes frequently takes very long time, thus the general purpose graphics processing units (GPGPUs) that support massively parallel processing can be very useful for speeding up the simulation. In LDPC decoding, the parity-check matrix H needs to be accessed at every node updating process, and the size of the matrix is often larger than that of GPU on-chip memory especially when the code length is long or the weight is high. In this work, the parity-check matrix of cyclic or quasi-cyclic (QC) LDPC codes is greatly compressed by exploiting the periodic property of the matrix. Also, vacant elements are eliminated from the sparse message arrays to utilize the coalesced access of global memory supported by GPGPUs. Regular projective geometry (PG) and irregular QC LDPC codes are used for sum-product algorithm based decoding with the GTX-285 NVIDIA graphics processing unit (GPU), and considerable speed-up results are obtained.     

Quasicyclic low-density parity-check codes from circulant permutation matrices

In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8,10, or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.     

LDPC codes from generalized polygons

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.     

Weight Distribution of Low-Density Parity-Check Codes

We derive the average weight distribution function and its asymptotic growth rate for low-density parity-check (LDPC) code ensembles. We show that the growth rate of the minimum distance of LDPC codes depends only on the degree distribution pair. It turns out that capacity-achieving sequences of standard (unstructured) LDPC codes under iterative decoding over the binary erasure channel (BEC) known to date have sublinearly growing minimum distance in the block length     

Optimal Overlapped Message Passing Decoding of Quasi-Cyclic LDPC Codes

Efficient hardware implementation of low-density parity-check (LDPC) codes is of great interest since LDPC codes are being considered for a wide range of applications. Recently, overlapped message passing (OMP) decoding has been proposed to improve the throughput and hardware utilization efficiency (HUE) of decoder architectures for LDPC codes. In this paper, we first study the scheduling for the OMP decoding of LDPC codes, and show that maximizing the throughput gain amounts to minimizing the intra- and inter-iteration waiting times. We then focus on the OMP decoding of quasi-cyclic (QC) LDPC codes. We propose a partly parallel OMP decoder architecture and implement it using FPGA. For any QC LDPC code, our OMP decoder achieves the maximum throughput gain and HUE due to overlapping, hence has higher throughput and HUE than previously proposed OMP decoders while maintaining the same hardware requirements. We also show that the maximum throughput gain and HUE achieved by our OMP decoder are ultimately determined by the given code. Thus, we propose a coset-based construction method, which results in QC LDPC codes that allow our optimal OMP decoder to achieve higher throughput and HUE.     

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 ha...     

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10^-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.     

Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.     

围长为8的QC-LDPC码的显式构造及其在CRT方法中的应用

对于任意码长PL(P≥3L2/4+L 1),利用完全确定的方式构造出一类围长为8的(4,L)QC-LDPC码。将这类码作为分量码,结合中国剩余定理(CRT)构造出一类围长至少为8且码长非常灵活的合成QC-LDPC码。在1/2码率和中等码长条件下的仿真结果表明,这种合成码在AWGN信道下具有优异的性能。     

Performance of low-density parity-check codes with linear minimum distance

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.     

Design of Low-Rate Irregular LDPC Codes Using Trellis Search

A simple design method using trellis search is proposed for good low-density parity-check (LDPC) codes with relatively low code rates. By applying a trellis search technique to the design of a pre-assigned part of the parity-check matrix that allows a simple encoding, we improve the distribution of cycles formed by the entries contained in the parity-check part of the parity-check matrix. In addition, we extend the proposed algorithm to a class of structured LDPC codes, which have been recently preferred in many practical applications. Simulation results show that the codes designed by the proposed method outperform those constructed by conventionally used greedy design algorithms.