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1、国防科学技术大学 博士学位论文 准循环LDPC码的构造及其理论研究 姓名：许拔 申请学位级别：博士 专业：信息与通信工程 指导教师：张尔扬 2010-04 国防科学技术大学研究生院博士学位论文 第 i 页 摘 要 QC-LDPC 码是一类被广泛应用的结构化 LDPC 码，由于其奇偶校验矩阵的独特 结构，能够利用反馈移位寄存器实现线性复杂度的编码。本文主要针对 QC-LDPC 码 的构造及其理论分析展开研究，主要工作有以下几方面： 1、研究了基于 BIBD 的 QC-LDPC 码的构造算法。通过采用有限域上乘法群的 思想，将 BIBD 区组中元素的位置向量与元素在循环群中的幂次直接相对应，避免了
2、 大量的求幂与求模运算，极大地简化了位置向量的求解。从理论上证明了新构造的 QC-LDPC 码对应的 Tanner 图中围长至少为 6。仿真结果表明，构造的 QC-LDPC 码 与随机构造的 LDPC 码的性能相当，且迭代收敛快，错误平层低；译码性能较传统的 BIBD QC-LDPC 码有一定的改善。 2、通过对奇偶校验矩阵进行分解，构造出能够抵抗长突发删除错误的好码；根 据奇偶校验矩阵本身的结构特点，得出了纠长突发删除错误的能力；同时分析了矩阵 分解对码率 R 的影响，当 t 越大时，对码率的影响越小，码率本身越大。仿真结果表 明， 通过矩阵分解构造的码在 AWGN 和 BEC 信道中均有良
3、好的性能， 距离香农限非 常近。 3、研究了 IA-LDPC 码的最小汉明重量及其码字数目的计算。通过引出支撑矩阵 中每行满足 Cancel-Out 条件的等效条件，给出了码字的判断定理。在此基础上，详细 推导了 j 为 2、3 与4,5jq=时最小汉明重量及其码字数目；并且给出了5j =和 4,7jq=时的相应结果，这些结果通过计算机搜索得到。另外，通过分析IA-LDPC 的码字特点， 刻画了具有最小汉明重量的码字集合 0 A的组成结构， 并推导出 0 A与 1 A 的关系，其中 1 A表示支撑矩阵中包含全0列且具有最小汉明重量的码字集合。 4、提出了一种基于Dawson-Sankoff不等
4、式的差错概率下界算法。在重新证明 Dawson-Sankoff不等式的基础上，提出了对Dawson-Sankoff下界的改进算法，并对 算法合理性进行了严格的数学证明。分别针对AWGN信道与BSC信道，推导了基于 传统Dawson-Sankoff界的差错概率下界的表达式，并分析了该下界存在的问题，进 而提出了差错冗余事件的判断准则，得到了基于改进Dawson-Sankoff界的差错概率 下界算法。 仿真结果表明， 提出的下界较现有下界性能更紧， 与联合上界的距离更近。 关键词： QC-LDPC； BIBD； 位置向量； 围长； 支撑矩阵； 最大似然译码； IA-LDPC 国防科学技术大学研究生
5、院博士学位论文 第 ii 页 Abstract QC-LDPC code, which is a type of the structured LDPC code, is widely applied in several communication systems. QC-LDPC code can be efficiently encoded using simple shift registers with linear complexity, which is mainly owing to the unique structure of the parity-check matrix
6、. This dissertation studies the construction and theoretical analysis of QC-LDPC code, and stresses several tasks below. At first, a novel construction algorithm of QC-LDPC is proposed which is based on the traditional BIBD. Adopting the multiplicative group over finite fields, the algorithm replace
7、s the location vector of the BIBDs element by the exponent in the multiplicative group, which can avoid a mass of exponential and module calculations and thus simplify the calculation of the location vector. It is strictly proved that the new constructed QC-LDPC codes have girth at least 6 and exper
8、imental results show that the performance of these QC-LDPC codes is very close to the random LDPC and converge very fast. Moreover, these QC-LDPC codes have low error-floor and can outperform the traditional BIBD QC-LDPC codes. Secondly, good QC-LDPC codes for correcting erasure-bursts are construct
9、ed by the decomposition of the above parity-check matrixes. The ability of correcting erasure-bursts is obtained according to the structure characteristic of the parity-check matrix. In addition, the influence of the code rate caused by decomposition is analyzed, which indicates that the value of t
10、is larger, the influence is smaller and the rate is larger. Simulation results show that codes constructed by decomposition perform well over both AWGN and binary erasure channels and approach the Shannon Limit. Thirdly, the calculation of the minimum Hamming weight and the number of minimum weight
11、codewords of IA-LDPC codes are addressed. The condition, which is equivalent to the cancel-out condition holding for any row of the support matrix, is introduced to prove the judge theorem for the codeword. Based on the above discussion, the minimum Hamming weight and the number of minimum weight co
12、dewords of IA-LDPC codes with2,3j = or4,5jq= are deduced in detail, and the same results of IA-LDPC codes with5j = or4,7jq=are provided by computer searching. Moreover, the characteristic of the IA-LDPC codeword is analyzed, and the composing structure of the minimum weight codeword set A0 is descri
13、bed, then the relationship between the sizes of the set A0 and A1 is presented, where A1 represents the subset of A0 whose element has the all 0s column in its support matrix. At last, a new lower bound on the error probability is proposed, which is based on Dawson-Sankoff inequality. The bound are
14、obtained by applying a new lower bound on the probability of a union of events, derived by improving on Dawson-Sankoff inequality. The 国防科学技术大学研究生院博士学位论文 第 iii 页 improved Dawson-Sankoff inequality is strictly proved. For the AWGN and BSC channel, the expressions of the lower bound on the error proba
15、bility are presented, which are based on the traditional Dawson-Sankoff inequality and the disadvantages of these lower bounds are pointed out and analyzed. Furthermore, the judge rule of the redundant error events is brought forward and the improved lower bound is achieved. Experiment results show
16、that the new lower bound has tighter performance than the existed lower bounds and is very close to union upper bound. Key words ： Quasi-Cyclic Low-Density Parity-Check Codes ， Balanced Incomplete Block Designs，Location Vector，Girth，Support Matrix，Maximum Likelihood Decoding (ML Decoding)，Improper-Array LDPC 国防科学技术大学研究生院博士学位论文 第 IV 页 表 目 录 表 2.1 PEG 算法描述 19 表 2.2 比特填充算法描述. 21 表 2.3 2-D EG-LDPC 码的构造 24 表 2.4 2-D PG-LDPC 码的构造 26 表 2.5 BP 译码算法. 29 表 2.6 BP 译码算法的密度进化流程. 35 表 2.7 BP-Based 译码算法及
