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

【摘要】：這篇論文考慮了代數(shù)編碼,組合設(shè)計(jì)和代數(shù)組合領(lǐng)域的若干理論問(wèn)題.同時(shí),也考慮了包括數(shù)字通信,信號(hào)處理和數(shù)據(jù)存儲(chǔ)等實(shí)際應(yīng)用中提出的若干基礎(chǔ)性問(wèn)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(nèi)的多種數(shù)學(xué)工具,去考察這些理論和實(shí)際問(wèn)題.在第2章,我們考慮壓縮傳感矩陣的確定性構(gòu)造.由Candes, Donoho和Tao首倡,壓縮傳感的理論已成為信號(hào)處理領(lǐng)域過(guò)去十年來(lái)最重大的進(jìn)展.壓縮傳感的一個(gè)核心問(wèn)題是傳感矩陣的構(gòu)造.注意到低相關(guān)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(shè)計(jì)和其它組合構(gòu)型的角度出發(fā),構(gòu)造了許多確定性傳感矩陣的無(wú)窮類.這些工作給出了基于相關(guān)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(shè)計(jì)領(lǐng)域,許多問(wèn)題可歸結(jié)為某些指數(shù)和及其值分布的計(jì)算.盡管這些計(jì)算總的來(lái)說(shuō)是非常困難的,在第3章,我們通過(guò)引入新的思想取得了新的進(jìn)展.具體來(lái)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(jì)算了一個(gè)m-序列和它的特定的采樣序列的互相關(guān)分布.我們得到一類有任意多個(gè)非零點(diǎn)的循環(huán)碼的重量分層.在第4章,我們考慮一些組合設(shè)計(jì)的構(gòu)造.劃分式差族是很多最優(yōu)構(gòu)型背后的組合結(jié)構(gòu).我們提出一個(gè)組合的遞歸構(gòu)造,統(tǒng)一了若干利用廣義分圓的代數(shù)構(gòu)造.我們的新構(gòu)造為推廣已有構(gòu)造和生成新的劃分式差族的無(wú)窮類提供了很大的靈活性.可分組設(shè)計(jì)是組合設(shè)計(jì)理論的基本內(nèi)容.由于缺乏合適的代數(shù)和幾何結(jié)構(gòu),型不一致的可分組設(shè)計(jì)的構(gòu)造是一個(gè)非常具有挑戰(zhàn)性的問(wèn)題.我們提出了一個(gè)新的構(gòu)造,得到了型不一致可分組設(shè)計(jì)的若干新的無(wú)窮類.在第5章,我們考慮循環(huán)碼的理論和應(yīng)用.作為實(shí)際中廣泛使用的循環(huán)碼,BCH碼是最重要的糾錯(cuò)碼之一.注意到關(guān)于BCH碼的經(jīng)典結(jié)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(chǔ),量子碼可由經(jīng)典的糾錯(cuò)碼導(dǎo)出.我們用偽循環(huán)碼構(gòu)造了量子極大距離可分碼,統(tǒng)一了許多之前的構(gòu)造且得到了新的無(wú)窮類.字符結(jié)對(duì)碼是用來(lái)糾正字符對(duì)讀取信道中錯(cuò)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(gòu)造了三類極小結(jié)對(duì)距離為五或六的極大距離可分字符結(jié)對(duì)碼.此外,我們提出一個(gè)算法,得到了許多極小結(jié)對(duì)距離為七的極大距離可分字符結(jié)對(duì)碼.一個(gè)代數(shù)編碼和兩個(gè)代數(shù)組合領(lǐng)域的問(wèn)題被收錄在附錄中.值得一提地,即使直接的計(jì)算看起來(lái)是不可能的,我們?nèi)缘贸隽艘活愑腥我舛鄠�(gè)非零點(diǎn)的循環(huán)碼的重量分布.我們通過(guò)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(diǎn).此外,我們?cè)谝粋�(gè)有關(guān)差集的經(jīng)典問(wèn)題和一個(gè)有關(guān)偽平面函數(shù)的新興問(wèn)題上取得了進(jìn)展.前一個(gè)問(wèn)題研究了不具有特征整除性質(zhì)的差集,這是Jungnickel和Schmidt在1997年提出的公開問(wèn)題.我們得到了不具備特征整除性質(zhì)的差集的一些必要條件.后一個(gè)問(wèn)題涉及與有限射影平面相關(guān)的一個(gè)新概念.這個(gè)工作豐富了偽平面函數(shù)的已知結(jié)果并建立了偽平面函數(shù)和結(jié)合方案之間的一個(gè)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(xué)位授予單位】：浙江大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O157.4
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本文編號(hào)：1770893