本书是国际上第一本有关高维哈达玛矩阵及其在电信与信息安全领域中的应用专著《Theory and Applications of Higher Dimensional Hadamard Matrices》的修订版。根据高维哈达玛矩阵的维数与其实际应用，该修订版自然地分为三个部分。第一部分重点研究经典的2维Walsh矩阵和哈达玛矩阵，包括它们的快速算法、最新构造法、存在性结果及其一般性的推广。第二部分考虑的是低维情形，例如，3-维、4-维和6-维Walsh和哈达玛矩阵与变换。第三部分是全书的核心也是本书的独特之处。本部分研究了N-维2阶哈达玛矩阵，并证明了这类矩阵与著名的H-布尔函数和2阶最佳二进阵列是等价的，由此，推导出了一系列有关高维2阶哈达玛矩阵的计数结果。本书中还罗列了许多有关高维哈达玛矩阵理论研究和工程应用的公开问题。希望这些公开挑战问题有助于吸引更多的后起之秀投身于高维哈达玛矩阵的理论和应用研究。
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目录
Preface to the Second Edition Preface to the First Edition Part 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 Orthogonal Designs and Hadamard Matrices 2.3.2 Existence Results Bibliography Part 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 Bibliography Part 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 2n 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 Bibliography Part 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 On the 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 Bibliography Concluding Questions Index
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