發(fā)布時間：2018-04-19 01:08

  本文選題：BCH碼 + 壓縮傳感矩陣　； 參考：《浙江大學》2016年博士論文

【摘要】：這篇論文考慮了代數(shù)編碼,組合設計和代數(shù)組合領域的若干理論問題.同時,也考慮了包括數(shù)字通信,信號處理和數(shù)據(jù)存儲等實際應用中提出的若干基礎性問題.本文的主旨在于利用包括代數(shù)數(shù)論,特征理論,指數(shù)和及代數(shù)函數(shù)域在內的多種數(shù)學工具,去考察這些理論和實際問題.在第2章,我們考慮壓縮傳感矩陣的確定性構造.由Candes, Donoho和Tao首倡,壓縮傳感的理論已成為信號處理領域過去十年來最重大的進展.壓縮傳感的一個核心問題是傳感矩陣的構造.注意到低相關值的矩陣給出性能良好的傳感矩陣,我們從編碼理論,組合設計和其它組合構型的角度出發(fā),構造了許多確定性傳感矩陣的無窮類.這些工作給出了基于相關值的最優(yōu)或近似最優(yōu)的傳感矩陣.在代數(shù)編碼和序列設計領域,許多問題可歸結為某些指數(shù)和及其值分布的計算.盡管這些計算總的來說是非常困難的,在第3章,我們通過引入新的思想取得了新的進展.具體來講,我們得到了一類Niho指數(shù)的循環(huán)碼的重量分布.我們計算了一個m-序列和它的特定的采樣序列的互相關分布.我們得到一類有任意多個非零點的循環(huán)碼的重量分層.在第4章,我們考慮一些組合設計的構造.劃分式差族是很多最優(yōu)構型背后的組合結構.我們提出一個組合的遞歸構造,統(tǒng)一了若干利用廣義分圓的代數(shù)構造.我們的新構造為推廣已有構造和生成新的劃分式差族的無窮類提供了很大的靈活性.可分組設計是組合設計理論的基本內容.由于缺乏合適的代數(shù)和幾何結構,型不一致的可分組設計的構造是一個非常具有挑戰(zhàn)性的問題.我們提出了一個新的構造,得到了型不一致可分組設計的若干新的無窮類.在第5章,我們考慮循環(huán)碼的理論和應用.作為實際中廣泛使用的循環(huán)碼,BCH碼是最重要的糾錯碼之一.注意到關于BCH碼的經典結果絕大部分考慮的是本原的BCH碼,我們首次系統(tǒng)研究了非本原的BCH碼.我們確定了幾類非本原BCH碼的參數(shù).作為量子信息處理的基礎,量子碼可由經典的糾錯碼導出.我們用偽循環(huán)碼構造了量子極大距離可分碼,統(tǒng)一了許多之前的構造且得到了新的無窮類.字符結對碼是用來糾正字符對讀取信道中錯誤的一種新的編碼方案.利用循環(huán)碼和擬循環(huán)碼,我們構造了三類極小結對距離為五或六的極大距離可分字符結對碼.此外,我們提出一個算法,得到了許多極小結對距離為七的極大距離可分字符結對碼.一個代數(shù)編碼和兩個代數(shù)組合領域的問題被收錄在附錄中.值得一提地,即使直接的計算看起來是不可能的,我們仍得出了一類有任意多個非零點的循環(huán)碼的重量分布.我們通過建立特定的指數(shù)和與一類圖的譜之間令人驚訝的聯(lián)系做到了這一點.此外,我們在一個有關差集的經典問題和一個有關偽平面函數(shù)的新興問題上取得了進展.前一個問題研究了不具有特征整除性質的差集,這是Jungnickel和Schmidt在1997年提出的公開問題.我們得到了不具備特征整除性質的差集的一些必要條件.后一個問題涉及與有限射影平面相關的一個新概念.這個工作豐富了偽平面函數(shù)的已知結果并建立了偽平面函數(shù)和結合方案之間的一個聯(lián)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.

【學位授予單位】：浙江大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O157.4
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本文編號：1770893