Construction of nonbinary cyclic, quasi-cyclic and regular LDPC codes: a finite geometry approach

This paper presents five methods for constructing nonbinary LDPC codes based on finite geometries. These methods result in five classes of nonbinary LDPC codes, one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. Experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the AWGN channel and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates with either algebraic hard-decision decoding or Kotter-Vardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity.     

Low-density parity-check codes based on finite geometries: arediscovery and new results

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding     

An analysis of the block error probability performance of iterative decoding

Asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes N and the number of iterations I go to infinity. Three classes of codes are considered. These are Gallager's regular low-density parity-check (LDPC) codes, Tanner's generalized LDPC (GLDPC) codes, and the turbo codes due to Berrou et al. It is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability P/sub b/, but also the block (frame) error probability P/sub B/, goes to zero as N and I go to infinity.     

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.     

A Unified Approach to the Construction of Binary and Nonbinary Quasi-Cyclic LDPC Codes Based on Finite Fields

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.     

基于一些PBIBD的低密度校验码的构造

本文构造了两类部分平衡不完全区组设计.并利用它们构造了一类低密度校验码(LDPC码),其最小环长至少为6,码率的选取具有很大的灵活性,而且可以具有拟循环结构.计算机仿真结果表明这种方法构造的LDPC码,在加性高斯白噪声信道中BPSK调制下用和积迭代译码性能很好.     

Transactions Papers - Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach

This paper is concerned with construction of efficiently encodable nonbinary quasi-cyclic LDPC codes based on finite fields. Four classes of nonbinary quasi-cyclic LDPC codes are constructed. Experimental results show that codes constructed perform well with iterative decoding using a fast Fourier transform based q-ary sum-product algorithm and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard- decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm.     

High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2. The construction is similar to the derivation of Reed-Solomon (RS)-based LDPC codes presented by Chen and Djurdjevic Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes     

On algebraic construction of Gallager and circulant low-density parity-check codes

This correspondence presents three algebraic methods for constructing low-density parity-check (LDPC) codes. These methods are based on the structural properties of finite geometries. The first method gives a class of Gallager codes and a class of complementary Gallager codes. The second method results in two classes of circulant-LDPC codes, one in cyclic form and the other in quasi-cyclic form. The third method is a two-step hybrid method. Codes in these classes have a wide range of rates and minimum distances, and they perform well with iterative decoding.     

Low-floor decoders for LDPC codes

One of the most significant impediments to the use of LDPC codes in many communication and storage systems is the error-rate floor phenomenon associated with their iterative decoders. The error floor has been attributed to certain subgraphs of an LDPC code?s Tanner graph induced by so-called trapping sets. We show in this paper that once we identify the trapping sets of an LDPC code of interest, a sum-product algorithm (SPA) decoder can be custom-designed to yield floors that are orders of magnitude lower than floors of the the conventional SPA decoder. We present three classes of such decoders: (1) a bi-mode decoder, (2) a bit-pinning decoder which utilizes one or more outer algebraic codes, and (3) three generalized-LDPC decoders. We demonstrate the effectiveness of these decoders for two codes, the rate-1/2 (2640,1320) Margulis code which is notorious for its floors and a rate-0.3 (640,192) quasi-cyclic code which has been devised for this study. Although the paper focuses on these two codes, the decoder design techniques presented are fully generalizable to any LDPC code.     

LDPC编译码技术研究

LDPC码,即低密度奇偶校验码,本质上是一种线性分组码,其译码性能比Turbo码更接近香农限。文中首先介绍了LDPC码的定义及描述;其次对LDPC码快速编码方法进行分析,对可线性编码的LDPC码构造进行探讨;然后对LDPC的译码技术进行研究;最后对LDPC码的应用前景进行讨论。     

LDPC-Coded OFDM Transmission Over Graded-Index Plastic Optical Fiber Links

The possibility of 10-Gb/s transmission over graded-index plastic optical fiber (GI-POF) using the orthogonal frequency-division multiplexing (OFDM) and iteratively decodable codes is demonstrated by simulations. Several classes of low-density parity-check (LDPC) codes suitable for use in LDPC-coded OFDM transmission over GI-POF are presented as well. Several power efficient OFDM schemes are introduced     

Mapping interleaving laws to parallel turbo and LDPC decoder architectures

For high-data-rate applications, the implementation of iterative turbo-like decoders requires the use of parallel architectures posing some collision-free constraints to the reading/writing process from/into the memory. This consideration applies to the two main classes of turbo-like codes, i.e., turbo codes and low-density parity-check (LDPC) codes. Contrary to the literature belief, we prove in this paper that there is no need for an ad hoc code design to meet the parallelism requirement, because, for any code and any choice of the scheduling of the reading/writing operations, there is a suitable mapping of the variables in the memory that grants a collision-free access. The proof is constructive, i.e., it gives an algorithm that obtains the desired collision-free mapping. The algorithm is applied to two simple examples, one for turbo codes and one for LDPC codes, to illustrate how the algorithm works.     

低密度奇偶校验码及其应用研究

阐述了低密度奇偶校验码的发展、定义、编译码方法。通过对LDPC码和Turbo码的比较和分析,得出LDPC码的整体性能要优于Turbo码。文中还讨论了LDPC码的应用,主要是在STC-OFDM系统中的应用。最后,对低密度奇偶校验码的未来的研究方向进行了展望。     

有限平面LDPC码的停止集

有限平面LDPC码是一类重要的有结构的LDPC码,在利用和积算法(SPA)等迭代译码方法进行译码时表现出卓越的纠错性能。众所周知,次优的迭代译码不是最大似然译码,因而如何对迭代译码的性能进行理论分析一直是LDPC码的核心问题之一。近几年来,Tanner图上的停止集(stopping set)和停止距离(stopping distance)由于其在迭代译码性能分析中的重要作用而引起人们的重视。该文通过分析有限平面LDPC码的停止集和停止距离,从理论上证明了有限平面LDPC码的最小停止集一定是最小重量码字的支撑,从而对有限平面LDPC码在迭代译码下的良好性能给出了理论解释。     

Construction of low-density parity-check codes based on balanced incomplete block designs

This correspondence presents a method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatoric designs, known as balance incomplete block designs. Codes constructed by this method have girths at least 6 and they perform well with iterative decoding. Furthermore, several classes of these codes are quasi-cyclic and hence their encoding can be implemented with simple feedback shift registers.     

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。     

GB20600标准的LDPC码的改进

尽管LDPC码已经被GB20600标准采纳作为信道编码,与其它LDPC码相比,在同样码长和码率的情况下,GB20600 LDPC码误码率性能并非最佳;GB20600标准的LDPC码的码长达7493,存在编码复杂性问题,但是GB20600 LDPC码未采用基于校验矩阵的快速算法,这给GB20600 LDPC编解码器的硬件实现带来较大的困难。本文在现有GB20600 LDPC码的设计框架下,对GB20600中LDPC码的校验矩阵进行了修改,在此基础上提出一种有效的LDPC码的快速迭代算法,使编解码器的硬件易于实现。改进后的LDPC码的编码算法具有较低的实现复杂度。仿真结果表明,改进后的LDPC码的误包率性能优于现GB20600中LDPC码的误包率性能。      

一种高性能全分集LDPC码的构造方法

块衰落信道上全分集LDPC的构造与性能分析成为近期研究的热点。ML译码算法下全分集LDPC码可以通过设计列满秩的校验子矩阵来实现。然而,基于ML准则的全分集码字,采用迭代译码算法时,不能保证全分集。因此,该文通过设计特定结构的校验矩阵,实现了在迭代译码算法下能取得全分集的LDPC码,分析了其密度演化过程。 在此基础上,进一步研究了全分集LDPC码字结构与性能的关系,提出了提高全分集LDPC码编码增益的方法。仿真结果表明,该文构造的LDPC码不仅能够取得全分集,并且具有较高的编码增益。