高維哈達(dá)瑪矩陣?yán)碚撆c應(yīng)用

前言

當(dāng)代數(shù)學(xué)在向縱深發(fā)展的同時(shí)，被空前廣泛地應(yīng)用于幾乎一切領(lǐng)域。一方面。它與其他學(xué)科交匯，形成了許許多多交叉學(xué)科（例如，信息科學(xué)、計(jì)算機(jī)科學(xué)、系統(tǒng)科學(xué)、數(shù)學(xué)物理、數(shù)學(xué)化學(xué)、生物數(shù)學(xué)、數(shù)學(xué)語(yǔ)言學(xué)、數(shù)量經(jīng)濟(jì)學(xué)、金融數(shù)學(xué)、復(fù)雜性科學(xué)、科學(xué)計(jì)算等）；另一方面，它又被應(yīng)用于高新技術(shù)的開(kāi)發(fā)（例如，信息安全、信息傳輸、圖像處理、語(yǔ)音識(shí)別、網(wǎng)絡(luò)、海量數(shù)據(jù)處理、網(wǎng)頁(yè)搜索、遙測(cè)遙感、交通管理、醫(yī)療診斷、手術(shù)方案、藥物檢驗(yàn)、商業(yè)廣告等方面），成為一些高新技術(shù)的核心。應(yīng)用數(shù)學(xué)的這種發(fā)展趨勢(shì)急劇地?cái)U(kuò)展了數(shù)學(xué)的疆界，也深刻地改變了數(shù)學(xué)的面貌。中國(guó)的經(jīng)濟(jì)正在迅猛發(fā)展，其中的科技含量也與日俱增。為了提高自主創(chuàng)新能力，我國(guó)已經(jīng)有不少數(shù)學(xué)工作者投身于這類應(yīng)用數(shù)學(xué)的研究中，還有更多的數(shù)學(xué)工作者則正在密切關(guān)注這方面的進(jìn)展，看好它的前景。愈來(lái)愈多的人希望了解這類應(yīng)用數(shù)學(xué)的現(xiàn)狀，尋找入門之徑?！稊?shù)學(xué)與現(xiàn)代科學(xué)技術(shù)叢書(shū)》是力圖反映這個(gè)發(fā)展趨勢(shì)的一套應(yīng)用數(shù)學(xué)叢書(shū)，它將較全面地向我國(guó)讀者介紹當(dāng)今數(shù)學(xué)在現(xiàn)代科學(xué)技術(shù)各個(gè)領(lǐng)域中應(yīng)用的狀況，通過(guò)必要的準(zhǔn)備知識(shí)，逐步把讀者引向相關(guān)的研究前沿。從事交叉學(xué)科研究和高新技術(shù)開(kāi)發(fā)的應(yīng)用數(shù)學(xué)家，除了要精通所需的數(shù)學(xué)知識(shí)外，還必須深入了解其所研究問(wèn)題的來(lái)龍去脈?！敖！笔菓?yīng)用數(shù)學(xué)研究實(shí)際問(wèn)題的關(guān)鍵。這也是一門數(shù)學(xué)藝術(shù)：從復(fù)雜的實(shí)際問(wèn)題中抽象出關(guān)鍵的“量的關(guān)系”，使得既能反映出問(wèn)題的基本特征，又能用現(xiàn)階段的數(shù)學(xué)工具加以處理。有鑒于此，這套叢書(shū)的一個(gè)特點(diǎn)就是：不但要介紹有關(guān)的數(shù)學(xué)理論和方法，還必須介紹問(wèn)題的來(lái)源與背景、數(shù)學(xué)建模以及如何運(yùn)用數(shù)學(xué)工具來(lái)解決實(shí)際問(wèn)題。本叢書(shū)適用于數(shù)學(xué)及相關(guān)專業(yè)的大學(xué)生和研究生，以及與數(shù)學(xué)有關(guān)的各專業(yè)科技工作者。

內(nèi)容概要

本書(shū)是國(guó)際上第一本有關(guān)高維哈達(dá)瑪矩陣及其在電信與信息安全領(lǐng)域中的應(yīng)用專著《Theory and Applications of Higher Dimensional Hadamard Matrices》的修訂版，分為三個(gè)部分。第一部分重點(diǎn)研究經(jīng)典的2維Walsh矩陣和哈達(dá)瑪矩陣，包括它們的快速算法、最新構(gòu)造法、存在性結(jié)果及其一般性的推廣。第二部分考慮的是低維情形，例如，3-維、4-維和6-維Walsh和哈達(dá)瑪矩陣與變換。第三部分是全書(shū)的核心也是本書(shū)的獨(dú)特之處，研究了N-維2階哈達(dá)瑪矩陣，并證明了這類矩陣與著名的H-布爾函數(shù)和2階最佳二進(jìn)陣列是等價(jià)的，由此，推導(dǎo)出了一系列有關(guān)高維2階哈達(dá)瑪矩陣的計(jì)數(shù)結(jié)果。本書(shū)中還羅列了許多有關(guān)高維哈達(dá)瑪矩陣?yán)碚撗芯亢凸こ虘?yīng)用的公開(kāi)問(wèn)題。

書(shū)籍目錄

Preface to the Second EditionPreface to the First EditionPart Ⅰ  2-Dimensional Cases　Chapter 1  Walsh Matrices 　 1.1  Walsh Functions and Matrices   　 1.1.1  Definitions　    1.1.2  Ordering　  1.2  Orthogonality and Completeness  　  1.2.1  Orthogonality　    1.2.2  Completeness　  1.3  Walsh Transforms and Fast Algorithms  　  1.3.1  Walsh-Ordered Walsh-Hadamard Transforms    　1.3.2  Hadamard-Ordered Walsh-Hadamard Transforms　  Bibliography　　Chapter 2  Hadamard Matrices　  2.1  Definitions  　  2.1.1  Hadamard Matrices　    2.1.2  Hadamard Designs  　  2.1.3  Williamson Matrices　  2.2  Construction  　  2.2.1  General Constructions  　  2.2.2  Amicable Hadamard Matrices　    2.2.3  Skew Hadamard Matrices    　2.2.4  Symmetric Hadamard Matrices　  2.3  Existence  　  2.3.1  Orth0gonal Designs and Hadamard Matrices　    2.3.2  Existence Results　    BibliographyPart Ⅱ  Lower-Dimensional Cases　Chapter 3  3-Dimensional Hadamard Matrices    3.1  Definitions and Constructions   　 3.1.1  Definitions　    3.1.2  Constructions Based on Direct Multiplications  　  3.1.3  Constructions Based on 2-Dimensional Hadamard Matrices   3.2  3-Dimensional Hadamard Matrices of Order 4k + 2   3.3  3-Dimensional Hadamard Matrices of Order 4k    3.3.1  Recursive Constructions of Perfect Binary Arrays    3.3.2  Quasi-Perfect Binary Arrays    3.3.3  3-Dimensional Hadamard Matrices Based on PBA(2m, 2m) and PBA(3.2m, 3.2m)  3.4  3-Dimensional Walsh Matrices    3.4.1  Generalized 2-Dimensional Walsh Matrices    3.4.2  3-Dimensional Walsh Matrices    3.4.3  3-Dimensional Pan-Walsh Matrices    3.4.4  Analytic Representations　  Bibliography　Chapter 4  Multi-Dimensional Walsh-Hadamard Transforms  　4.1  Conventional 2-Dimensional Walsh-Hadamard Transforms    　4.1.1  2-Dimensional Walsh-Hadamard Transforms　    4.1.2  Definitions of 4-Dimensional Hadamard Matrices  　4.2  Algebraic Theory of Higher-Dimensional Matrices　  4.3  Multi-Dimensional Walsh-Hadamard Transforms  　  4.3.1  Transforms Based on 3-Dimensional Hadamard Matrices　    4.3.2  Transforms Based on 4-Dimensional Hadamard Matrices  　  4.3.3  Transforms Based on 6-Dimensional Hadamard Matrices  BibliographyPart Ⅲ  General Higher-Dimensional Cases　Chapter 5  n-Dimensional Hadamard Matrices of Order 2　  5.1  Constructions of 2n Hadamard Matrices  　  5.1.1  Equivalence between 2n Hadamard Matrices and H-Boolean Functions    　5.1.2  Existence of H-Boolean Functions　    5.1.3  Constructions of H-Boolean Functions  　5.2  Enumeration of 2" Hadamard Matrices　    5.2.1  Classification of 24 Hadamard Matrices  　  5.2.2  Enumeration of 25 Hadamard Matrices    　5.2.3  Enumeration of General 2n Hadamard Matrices　  5.3 Applications  　  5.3.1  Strict Avalanche Criterion and H-Boolean Functions    　5.3.2  Bent Functions and H-Boolean Functions　    5.3.3  Reed-Muller Codes and H-Boolean Functions  　Bibliography　Chapter 6  General Higher-Dimensional Hadamard Matrices　  6.1  Definitions, Existences and Constructions  　  6.1.1  n-Dimensional Hadamard Matrices of Order 2k　    6.1.2  Proper and Improper n-Dimensional Hadamard Matrices  　  6.1.3  Generalized Higher-Dimensional Hadamard Matrices　  6.2  Higher-Dimensional Hadamard Matrices Based on Perfect Binary Arrays  　  6.2.1  n-Dimensional Hadamard Matrices Based on PBAs　    6.2.2  Construction and Existence of Higher-Dimensional PBAs　    6.2.3  Generalized Perfect Arrays　  6.3  Higher-Dimensional Hadamard Matrices Based on Orthogonal Designs　    6.3.1  Definitions of Orthogonality　    6.3.2  Higher-Dimensional Orthogonal Designs　    6.3.3  Higher-Dimensional Hadamard Matrices from Orthogonal Designs　  BibliographyPart Ⅳ  Applications to Signal Design and Analysis　Chapter 7  Design and Analysis of Sequences　  7.1  Sequences of Cryptographic Significance　    7.1.1  Enumerating Boolean Functions of Cryptographic Significance　    7.1.2  Constructing Boolean Functions of Cryptographic Significance  　  7.1.3  Correlation Immunity of Boolean Functions　    7.1.4  Entropy Immunity of Feedforward Networks  　7.2  Correlation Functions of Geometric Sequences　　    7.2.1  Onthe Correlation Functions of a Family of Gold-Geometric Sequences　    7.2.2  On the Correlation Functions of a Family of Generalized Geometric Sequences  　  7.2.3  On the Correlation Functions of p-Ary d-Form Sequences　  7.3  Sequence Pairs with Mismatched Filtering  　  7.3.1  Binary Sequences Pairs with Two-Level Autocorrelation Functions (BSPT)　    7.3.2  Difference Set Pairs　    7.3.3  Construction of BSPTs　    7.3.4  Periodic Complementary Binary Sequence Pairs　  7.4  Sequence Unusual Analysis  　  7.4.1  Boolean Neural Network Design　    7.4.2  Linear Complexity and Random Sequences with Period 2n　    7.4.3  Periodic Ambiguity Functions of EQC-Based TFHC　    7.4.4  Auto-, Cross-, and Triple Correlations of Sequences  　Bibliography　Chapter 8  Design and Analysis of Arrays　  8.1  Costas Arrays  　  8.1.1  Correlations of Costas Arrays　    8.1.2  Algebraically Constructed Costas Arrays  　  8.1.3  Enumeration Limitation of Costas Arrays　  8.2  Optical Orthogonal Codes  　  8.2.1  Parameters Bounds of Optical Orthogonal Codes　    8.2.2  Truncated Costas Optical Orthogonal Codes  BibliographyConcluding QuestionsIndex

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