TS-LDPC Codes: Turbo-Structured Codes With Large Girth

We consider turbo-structured low-density parity-check (TS-LDPC) codes-structured regular codes whose Tanner graph is composed of two trees connected by an interleaver. TS-LDPC codes with good girth properties are easy to construct: careful design of the interleaver component prevents short cycles of any desired length in its Tanner graph. We present algorithms to construct TS-LDPC codes with arbitrary column weight jges2 and row weight k and arbitrary girth g. We develop a linear complexity encoding algorithm for a type of TS-LDPC codes-encoding friendly TS-LDPC (EFTS-LDPC) codes. Simulation results demonstrate that the bit-error rate (BER) performance at low signal-to-noise ratio (SNR) is competitive with the error performance of random LDPC codes of the same size, with better error floor properties at high SNR     

一种具有较大围长的正则LDPC码构造方法

提出一种新的具有较大围长的正则LDPC码构造方法。首先介绍以矩阵分裂技术为基础的高围长正则LDPC码的构造方法,并在此基础上分析了设计围长时参数的选取方法。仿真表明,用这种方法构造的正则LDPC码围长可以达到12,并且在AWGN信道下的性能不差于相同参数、随机构造的LDPC码,在高信噪比时甚至优于相同参数的随机码。     

A Measure of the Girth Histogram for Improved Girth-Conditioning Construction of LDPC Codes

In this letter, we present a general quantitative measure of the girth histogram of low-density parity-check (LDPC) codes with joint consideration of the degrees and local girths of symbol nodes. We then propose a further girth-conditioning algorithm to optimize such a measure during the construction of LDPC codes. We show by simulation results that the algorithm improves not only the performance of irregular codes at high signal-to-noise ratios (SNR) considerably, but also the rates of regular codes     

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。     

Several properties of short LDPC codes

In this paper, we present several properties on minimum distance(d/sub min/) and girth(G/sub min/) in Tanner graphs for low-density parity-check (LDPC) codes with small left degrees. We show that the distance growth of (2, 4) LDPC codes is too slow to achieve the desired performance. We further give a tight upper bound on the maximum possible girth. The numerical results show that codes with large G/sub min/ could outperform the average performance of regular ensembles of the LDPC codes over binary symmetric channels. The same codes perform about 1.5 dB away from the sphere-packing bound on additive white Gaussian noise channels.     

一种用于短猝发通信的LDPC短码设计

介绍了用于构建大围长Tanner图的PEG构造方法及可用于构造准循环校验矩阵的改进算法。由于短环是影响LDPC短码性能的主要因素,必须尽量增大最小短环的围长并减少短环的数量,以获得最大的平均围长。针对短猝发通信需求,构造了一种基于PEG算法的码长较短的准循环LDPC码。计算机仿真表明其性能优异,在通用硬件平台上对其中频传输性能进行了测试,测试结果满足短猝发通信对误码率的要求。由于该短码性能较好,电路实现复杂度低,延时小,适用于短猝发通信。     

一种低密度奇偶校验码的环数统计方法

对于Tanner图中给定码长的序列,LDPC码的短环对码的性能有重要影响.本文在分析LDPc码在Tanner图中的环在校验矩阵中的形状的基础上,提出了一种统计LDPC码中不同环长的环数的方法.首先对校验矩阵中一定数目的行组合中的环数进行统计,然后将所有行组合中的环数相加即得到校验矩阵中的环数.该方法可根据LDPC码的短环分布情况对其性能进行评估.应用提出的方法分别对MacKay的随机码和Fossorier的准循环码进行了环数统计.BER性能显示,尽管随机码环数特性比准循环码要差,但它的误码率性能比准循环码要好.     

A combining method of quasi-cyclic LDPC codes by the Chinese remainder theorem

In this paper we propose a method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese remainder theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.     

基于FFT-BP算法的多元域LDPC码译码器设计与FPGA实现

GF(q)域上的LDPC码是二进制LDPC码的扩展,它具有比二进制LDPC码更好的纠错性能。FFT-BP算法是高效的LDPC码译码算法,本文在GF(4)域上探讨该算法的设计与实现。本文的创新之处在于,根据FFT-BP算法的特点设计了一种利用Tanner图进行信息索引的方式,简化了地址查询模块的设计。实验表明,在归一化信噪比为2.6dB时,译码器的误码率可达到10-6。     

Girth conditioning for construction of short block length irregular LDPC codes

A girth conditioning scheme for construction of short block length low-density parity-check (LDPC) codes is presented. The parity-check matrix is generated in a column-by-column fashion under the condition that a newly added column is chosen, among a number of trial columns, to maximise the average girth of variable nodes. The girth-conditioned LDPC codes are found to give a large performance gain especially at high SNRs.     

Iterative decoding for partial response (PR), equalized,magneto-optical (MO) data storage channels

Turbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research attention because of their remarkable near-capacity performance for additive white Gaussian noise (AWGN) channels. Previously, turbo code and LDPC code variants are being investigated as potential candidates for high-density magnetic recording channels suffering from low signal-to-noise ratios (SNR). We address the application of turbo codes and LDPC codes to magneto-optical (MO) recording channels. Our results focus on a variety of practical MO storage channel aspects, including storage density, partial response targets, the type of precoder used, and mark edge jitter. Instead of focusing just on bit error rates (BER), we also study the block error statistics. Our results for MO storage channels indicate that turbo codes of rate 16/17 can achieve coding gains of 3-5 dB over partial response maximum likelihood (PRML) methods for a 10^-4 target BER. Simulations also show that the performance of LDPC codes for MO channels is comparable to that of turbo codes, while requiring less computational complexity. Both LDPC codes and turbo codes with iterative decoding are seen to be robust to mark edge jitter     

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 column-weight 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 non-existence of cycles four and six in the structure of the parity check matrix, a bound for their minimum weight is proposed. The simulations 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.     

Partially Coherent EGC Reception of Uncoded and LDPC-Coded Signals over Generalized Fading Channels

This paper studies the bit-error rate (BER) performance of partially coherent equal-gain combining reception of uncoded and low-density parity-check (LDPC)-coded binary and quaternary phase-shift keying signals over generalized ?? ? ?? fading channels. For the uncoded signal detection case the obtained numerical and simulation results illustrate the BER performance degradation due to the imperfect reference signal recovery, receiver unbalancing and fading. It is demonstrated that imperfect cophasing causes an irreducible BER performance. Furthermore, we design large girth quasi-cyclic LDPC code with high code rate, suitable for use in communications over generalized fading channels. The proposed LDPC code does not exhibit the error floor phenomena, in the region of interest, even in the presence of imperfect cophasing and receiver unbalances, and outperforms standard convolutional code of lower code rate in the observed impairments. The effects of number of iterations in decoding algorithm and codeword length on BER performance are also investigated.     

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.     

一种具有最小距离下界的正则LDPC码的构造

主要提出一种新的计算规则LDPC(low-density parity-check)码的最小距离下界的方法。该方法是基于LDPC码的每个变量节点的独立树进行构造LDPC码。与随机构造的LDPC码和用PEG方法构造的方法比较,这个新的构造方法得到了更大的围长和最小距离下界。在AWGN信道中,在码长N=1 008和N=1 512时进行Matlab仿真,仿真结果表明随着信噪比的增加此方法构造的LDPC码有优异的误码率性能。     

Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel     

BER performance due to irregularity of row-weight distribution of the parity-check matrix in irregular LDPC codes for 10-gb/s optical signals

Forward-error correction (FEC) coding is theoretically investigated to improve bit-error-rate (BER) performance in a 10-Gb/s optical transmission system using randomly irregular low-density parity-check (LDPC) codes, regular LDPC codes, and the Reed-Solomon (RS) (255,239) code as a comparison. The irregular LDPC codes has different row-weight variances of a parity-check matrix from 10.9 to 18.8 and a row-weight mean of 60. Simulation is carried out under various conditions including the impairment factors such as dispersion, polarization-mode dispersion (PMD), and fiber nonlinearities. Results suggest that the irregular LDPC code with a low row-weight variance (=10.9) generally has better performance for the most impairment factors except for the factor of dispersion. On the other hand, for the factor of dispersion the irregular LDPC code performs better with a high row-weight variance (=18.8). A specific LDPC code can overcome the impairment limits in a deployed link.     

High Rate APPS and Bi-APPS LDPC Codes Design with Low Error Floor and Large Girth

In this paper, two new methods to construct low-density parity-check (LDPC) codes with low error floor and large girth are proposed. The first one is APPS-LDPC codes based on Arithmetic Progression theory and cycle classification, whose girth is at least eight. Based on the designed APPS-LDPC codes, we further construct Bi-diagonal APPS-LDPC codes with column degree 4, whose circulant permutation matrix is combined by two shifted identity matrix. The designed APPS-LDPC code has 0.25 and 0.2 dB coding gain compared to partition-and-shift (PS)-LDPC code and progressive-edge-growth (PEG)-LDPC code. And the Bi-APPS-LDPC code has similar performance to T2 LDPC code in CCSDS standard, but its effective structure is more suitable for high throughput decoder implementation on FPGA. Both codes have less construction complexity than PS-LDPC code and PEG-LDPC code.     

Design of Irregular LDPC Codes for BIAWGN Channels with SNR Mismatch

Belief propagation (BP) algorithm for decoding lowdensity parity-check (LDPC) codes over a binary input additive white Gaussian noise (BIAWGN) channel requires the knowledge of the signal-to-noise ratio (SNR) at the receiver to achieve its ultimate performance. An erroneous estimation or the absence of a perfect knowledge of the SNR at the decoder is referred to as ?SNR mismatch?. SNR mismatch can significantly degrade the performance of LDPC codes decoded by the BP algorithm. In this paper, using extrinsic information transfer (EXIT) charts, we design irregular LDPC codes that perform better (have a lower SNR threshold) in the presence of mismatch compared to the conventionally designed irregular LDPC codes that are optimized for zero mismatch. Considering that min-sum (MS) algorithm is the limit of BP with infinite SNR over-estimation, the EXIT functions generated in this work can also be used for the efficient analysis and design of LDPC codes under the MS algorithm.     

具有不等错误保护特性的LDPC 编码调制方案

针对LDPC（low-density parity-check）编码调制系统，提出了一种新的具有不等错误保护特性的调制方案，在一个码字内，利用不同的调制方式对于重要的比特给予较强的保护，对于次要的比特给予较弱的保护，该方案既适用于非规则LDPC码，也适用于规则LDPC码。计算机仿真结果表明，新方案的性能是传统16QAM及4QAM的折衷，当采用1／2码率时，其频带利用率与8PSK相同，但是误码率性能优于8PSK。新方案的性能优于现有文献中基于比特可靠性的调制映射方案，并采用EXIT（extrinsic information transfer）图对新方案的优异性能给出了解释。