Theory and Applications of Higher-Dimensional Hadamard Matrices

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 Hadarmard matrices. Fast algorithms, updated constructions, existence results, and their generalized forms are presented. The second part deals with the lower-dimensional cases. 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. The many open problems in the study of the theory of higher-dimensional Hadamard matrices which are also listed in the book will encourage further research.

【Main Contents】

Part I Two-Dimensional Cases
Chapter 1 Walsh Matrices
Chapter 2 Hadamard Matrices
Part II Lower-Dimensional Cases
Chapter 3 Three-Dimensional Hadamard Matrices
Chapter 4 Multi-Dimensional Walsh-Hadamard Transforms
Part III General Higher-Dimensional Cases
Chapter 5 n-Dimensional Hadamard Matrices of Order 2
Chapter 6 General Higher-Dimensional Hadamard Matrices