组合构型、指数和及其在信号处理、编码设计中的应用

发布时间：2018-04-19 01:08

  本文选题：BCH码 + 压缩传感矩阵　； 参考：《浙江大学》2016年博士论文

【摘要】：这篇论文考虑了代数编码,组合设计和代数组合领域的若干理论问题.同时,也考虑了包括数字通信,信号处理和数据存储等实际应用中提出的若干基础性问题.本文的主旨在于利用包括代数数论,特征理论,指数和及代数函数域在内的多种数学工具,去考察这些理论和实际问题.在第2章,我们考虑压缩传感矩阵的确定性构造.由Candes, Donoho和Tao首倡,压缩传感的理论已成为信号处理领域过去十年来最重大的进展.压缩传感的一个核心问题是传感矩阵的构造.注意到低相关值的矩阵给出性能良好的传感矩阵,我们从编码理论,组合设计和其它组合构型的角度出发,构造了许多确定性传感矩阵的无穷类.这些工作给出了基于相关值的最优或近似最优的传感矩阵.在代数编码和序列设计领域,许多问题可归结为某些指数和及其值分布的计算.尽管这些计算总的来说是非常困难的,在第3章,我们通过引入新的思想取得了新的进展.具体来讲,我们得到了一类Niho指数的循环码的重量分布.我们计算了一个m-序列和它的特定的采样序列的互相关分布.我们得到一类有任意多个非零点的循环码的重量分层.在第4章,我们考虑一些组合设计的构造.划分式差族是很多最优构型背后的组合结构.我们提出一个组合的递归构造,统一了若干利用广义分圆的代数构造.我们的新构造为推广已有构造和生成新的划分式差族的无穷类提供了很大的灵活性.可分组设计是组合设计理论的基本内容.由于缺乏合适的代数和几何结构,型不一致的可分组设计的构造是一个非常具有挑战性的问题.我们提出了一个新的构造,得到了型不一致可分组设计的若干新的无穷类.在第5章,我们考虑循环码的理论和应用.作为实际中广泛使用的循环码,BCH码是最重要的纠错码之一.注意到关于BCH码的经典结果绝大部分考虑的是本原的BCH码,我们首次系统研究了非本原的BCH码.我们确定了几类非本原BCH码的参数.作为量子信息处理的基础,量子码可由经典的纠错码导出.我们用伪循环码构造了量子极大距离可分码,统一了许多之前的构造且得到了新的无穷类.字符结对码是用来纠正字符对读取信道中错误的一种新的编码方案.利用循环码和拟循环码,我们构造了三类极小结对距离为五或六的极大距离可分字符结对码.此外,我们提出一个算法,得到了许多极小结对距离为七的极大距离可分字符结对码.一个代数编码和两个代数组合领域的问题被收录在附录中.值得一提地,即使直接的计算看起来是不可能的,我们仍得出了一类有任意多个非零点的循环码的重量分布.我们通过建立特定的指数和与一类图的谱之间令人惊讶的联系做到了这一点.此外,我们在一个有关差集的经典问题和一个有关伪平面函数的新兴问题上取得了进展.前一个问题研究了不具有特征整除性质的差集,这是Jungnickel和Schmidt在1997年提出的公开问题.我们得到了不具备特征整除性质的差集的一些必要条件.后一个问题涉及与有限射影平面相关的一个新概念.这个工作丰富了伪平面函数的已知结果并建立了伪平面函数和结合方案之间的一个联系.
[Abstract]:This paper considers some theoretical problems of encoding algebra, algebraic combinatorics and combinatorial design. At the same time, also included the digital communication, some basic problems of the practical application of signal processing and data storage etc.. The purpose of this paper is including the use of algebraic number theory, feature theory, index and algebraic function fields, and a variety of mathematical tools to study these theoretical and practical problems. In the second chapter, we consider the compressed sensing matrix to determine the structure. By Candes, Donoho and Tao initiated, compressed sensing theory has become the most important field of signal processing in the past ten years. A key problem is to construct the sensing matrix of compressed sensing note. The sensing matrix matrix gives good performance and low correlation value, we from the encoding theory, combinatorial design and other combination configuration angle, constructed many uncertain sensor Infinite matrices. These are the sensing matrix based on optimal or approximate optimal correlation value. In algebraic encoding and sequence design field, many problems can be attributed to some index and value distribution calculation. Although these calculations in general is very difficult, in the third chapter, we have made new progress by introducing new ideas. Specifically, we obtain the cyclic code weight distribution for a class of Niho index. We calculated the correlation distribution of a m- sequence and its specific sampling sequence. We obtain a class of any number of non zero cyclic code weight stratification. In the fourth chapter, we consider some combination of design structure. Division difference family is a composite structure behind many optimal configuration. We propose a combination of recursive structure, unified a number of points by using the generalized algebraic structure. We construct new round of Infinite generalized the known structure and generate a new partition type difference family provides great flexibility. Groupdivisibledesign is the basic content of combinatorial design theory. Due to the lack of algebraic and geometric structure, type inconsistent structure block design is a very challenging problem. We propose a the new structure, the type of inconsistent groupdivisibledesign several new infinite classes. In the fifth chapter, we consider the theory and application of cyclic codes. The cyclic code is widely used as a practice, BCH code is the most important one of the error correcting code. Note that BCH code on the classic results most is considered primitive BCH codes, we first studied non primitive BCH codes. We identified several types of non primitive BCH codes parameters. As the basis of quantum information processing, quantum codes from classical error correcting codes are derived. We use pseudo cyclic codes To construct quantum maximum distance separable code, unified structure and many before obtain infinite new character. In code is used to correct the character of a new encoding scheme for error reading channel. Using the cyclic codes and quasi cyclic codes, we construct three kinds of polar distance is the maximum distance of summary five or six pairs can be divided into character code. In addition, we propose an algorithm to get a lot of very great summary of distance distance of seven can be divided into character code. A pair of algebraic encoding and two algebraic combinatorial problem in the field is included in the appendix. It is worth mentioning, even if direct calculation looks is not possible, we still get a class of any number of non zero cyclic code weight distribution. We establish the specific index and with a kind of spectrum surprising connection to this point. In addition, we have a Progress has been made in the classical problem of difference set and a pseudo planar function emerging problems. A problem is studied with characteristics of divisibility of difference set, this is the open problem presented by Jungnickel and Schmidt in 1997. We obtain some necessary conditions do not have the characteristics of divisibility properties of difference sets. After a problem involving a new concept and a finite projective plane. This work enriches the pseudo plane function known results and a contact established pseudoplane function and combination between.

【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.4
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本文编号：1770893