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JOURNAL OF ELECTRONICS INFORMATION TECHNOLOGY

Construction of LDPC Codes Based on Circulant Permutation Matrices

Hua Qiao, Guan Wu, Mingke Dong, Hai-ge Xiang

Published Apr 12, 2011• 4 citations

Abstract

具有准循环结构的低密度奇偶校验码(QC-LDPC Codes)是目前被广泛采用的一类LDPC码。本文提出了一种结合PEG算法构造基于循环移位矩阵的QC-LDPC码的方法。该方法首先将QC-LDPC码传统的基于比特的二分图简化为基于Block的二分图，然后在该图中采用PEG算法遵循的环路最大原则确定每一个循环移位矩阵的位置，最后根据QC-LDPC码的环路特性为每一个循环移位矩阵挑选循环移位偏移量。利用该算法，本文构造了长度从1008bit到8064bit，码率从1/2到7/8各种参数的LDPC码。仿真结果表明，本文构造的LDPC码性能优于目前采用有限几何、两个信息符号的RS码、组合数学等常用的代数方法构造的QC-LDPC码。

DOI: 10.3724/sp.j.1146.2007.00526

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具有准循环结构的低密度奇偶校验码(QC-LDPC Codes)是目前被广泛采用的一类LDPC码。本文提出了一种结合PEG算法构造基于循环移位矩阵的QC-LDPC码的方法。该方法首先将QC-LDPC码传统的基于比特的二分图简化为基于Block的二分图，然后在该图中采用PEG算法遵循的环路最大原则确定每一个循环移位矩阵的位置，最后根据QC-LDPC码的环路特性为每一个循环移位矩阵挑选循环移位偏移量。利用该算法，本文构造了长度从1008bit到8064bit，码率从1/2到7/8各种参数的LDPC码。仿真结果表明，本文构造的LDPC码性能优于目前采用有限几何、两个信息符号的RS码、组合数学等常用的代数方法构造的QC-LDPC码。

In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.

Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QC-LDPC codes in minimum-distance, cycle distribution and error-rate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix A , if -1 ϵ, A, it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimum-distance, cycle distributions and bit-error-rate performances.

该文设计了一种特殊的高码率准循环低密度校验(QC-LDPC)码,其校验矩阵以单位矩阵的循环移位阵为基本单元,与随机构造的LDPC码相比可节省大量存储单元。利用该码校验矩阵的近似下三角特性,一种高效的递推编码方法被提出,它使得该码编码复杂度与码长成线性关系。另外,该文提出一种分析QC-LDPC码二分图中短长度环分布情况的方法,并且给出了相应的不含长为4环QC-LDPC码的构造方法。计算机仿真结果表明,新码不但编码简单,而且具有高纠错能力、低误码平层。

该文研究准循环多进制LDPC码的构造，给出多进制LDPC码的设计流程和构造方法。详细讨论了多进制LDPC码的性能影响因素，综合考虑了环长和环的连通性对性能的影响，研究了母矩阵扩展中偏移因子的选择以及GF(q)上非零元素替代。同时提出了次优解的搜索方法，以降低搜索复杂度。最后，将提出的方法用于不同阶数下LDPC码的构造。仿真结果表明，通过新方法构造得到的多进制LDPC码与二进制码相比，在BPSK调制方式下在误帧率10-4附近有0.2 dB的性能提升；在有限域阶数与调制阶数匹配的情况下，有更大的性能提升。与相近码长，相同码率的多进制循环码相比，该文构造得到的多进制LDPC码在误帧率10-4附近有0.25 dB的性能提升。