This is the first book on higher-dimensional Hadamard matrices and their applications in telecommunications and information security．It is divided into three parts according to the dimensions of the Hadamard matrices treated．The first part stresses the classical 2-dimensional Walsh and Hadamard matrices．Fast algorithms，updated constructions，exitence results，and their generalised forms are presented．The second part deals with the lower-dimensional cases，e.g．3-，4-，and 6-dimensional Walsh and Hadamard matrices and transforms．The third part iS the key part．which investigates the N-dimensional Hadamard matrices of order 2，which have been proved equivalent to the well known H-Boolean functions and the perfect binary arrays of order 2．After introducing the definitions of the regular，proper，improper，and generalized higher-dimensional Hadamard matrices，many theorems about the existence and constructions are presented．Perfec binary arrays，generalized perfect arrays，and the orthogonal designs are also used to construct new higher-dimensional Hadamard matrices．The many open problems in the study ofthe theory ofhigher-dimensional Hadamard matrices which are also listed in the book will encourage further research． This volume will appeal to researchers and signal processing，coding，information security graduate students whose work involves and applied discrete mathematics．
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目录
Preface Part I Two-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 Orthogonal Designs and Hadamard Matrices 2.3.2 Existence Results Bibliography Part II Lower-Dimensional Cases Chapter 3 Three-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 Three-Dimensional Hadamard Matrices of Order 4k+2 3.3 Three-Dimensional Hadamard Matrices of Order 4k 3.3.1Recursive Constructions of Perfect Binary Arrays 3.3.2 Quasi-Perfect Binary Arrays 3.3.3 3-Dimensional Hadamard Matrices Based on PBA（2的m次方，2的m次方）and PBA（3.2的m次方，3.2的m次方） 3.4 Three-Dimensional Walsh Matrices 3.4.1 Generalized 2-Dimensional、valsh 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 Wlalsl-Hadamard Transforms 4.1.2 Definitions of 4-Dimensional Hadamard Matrices 4.2 Algebraical Theory of Higher-Dimensional Matrices 4.3 Multi-Dimensional Walsh-Hadamard Transforms 4.3.1 Tnansforms Based on 3-Dimensional Hadamard Matrices 4.3.2 nansforms Based on 4-Dimensional Hadamard Matrices 4.3.3 nansforms 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 2的n次方 Hadamard Matrices 5.1.1 Equivalence Between 2的n次方 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的n次方Hadamard Matrices 5.2.1 Classification of 2的4次方Hadamard Matrices 5.2.2 Enumeration of 2的5次方Hadamard Matrices 5.2.3 Enumeration of General 2的n次方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 Concluding Questions Index
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