Theory and Applications of Higher -Dimensional Hadamard Matrices（英文影印版）（硬皮精装）

　　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 of the theory of higher-dimensional hadamard matrices which are also listed in the book will encourage further research. .
　　 this volume will appeal to researchers and graduate students whose work involves signal processing, coding, information security and applied discrete mathematics. ...

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

　　Just over one hundred years ago, in 1893, Jacques Hadamard found 'binary' (±1) matrices of orders 12 and 20 whose rows (resp. columns) were pairwise orthogonal. These matrices satisfy the determinantal upper bound for 'binary' matrices. Hadamard actually proposed the question of seeking the maximal determinant of matrices with entries on the unit circle, but his name has become associated with the question concerning real (binary) matrices. Hadamard was not the first person to study these matrices. For example, J. J. Sylvester had found, in 1857, such row (column) pairwise orthogonal binary matrices of all orders of powers of two. Nevertheless, Hadamard proved that binary matrices with a maximal determinant could exist only for orders 1, 2, and 4t, t a positive integer. .
　　With regard to the practical applications of Hadamard matrices, it was M. Hall, Jr., L. Baumert, and S. Golomb who sparked the interest in Hadamard matrices over the past 30 years. They made use of the Hadamard matrix of order 32 to design an eight bit error-correcting code for two reasons. First, error-correcting codes based on Hadamard matrices have good error correction capability and good decoding algorithms. Second, because Hadamard matrices are (±1)-valued, all the computer processing can be accomplished using additions and subtractions rather than multiplication.
　　Walsh matrices are the simplest and most popular special kinds of Hadamard matrices. Walsh matrices are generated by sampling the Walsh functions, which are families of orthogonal complete functions. Based on the Walsh matrices, a very efficient orthogonal transform, called Walsh-Hadamard transform, was developed. The Walsh-Hadamard transform is now playing a more and more important role in signal processing and image coding.
　　P. J. Shlichta discovered in 1971 that there exist higher-dimensional binary arrays which possess a range of orthogonality properties. In particular, P. J. Shlichta constructed 3-dimensional arrays with the property that any sub-array obtained by fixing one index is a 2-dimensional Hadamard matrix. The study of higher-dimensional Hadamard matrices was mainly motivated by another important paper of P. J. Shlichta 'Higher-Dimensional Hadamard Matrices', which was published in IEEE Trans. on Inform., in 1979. Since then a lot of papers on the existence, construction, and enumeration of higher-dimensional Hadamard matrices have been reported. For example, J.Hammer and J. Seberry, found, in 1982, that higher-dimensional orthogonal designs can be used to construct higher-dimensional Hadamard matrices. To the author's knowledge much of the research achievements on higher-dimensional Hadamard matrices have been accomplished by S. S. Agaian, W. De Launey, J. Hammer, J. Seberry, Yi Xian Yang, K.J. Horadam, P. J. Shlichta, J. Jedwab, C. Lin, Y. Q. Chen, and others. Many new papers have been published, thus none can collect together all of the newest results in this area.
　　The book divides naturally into three parts according to the dimensions of Hadamard matrices processed. ..
　　The first part, Chapter 1 and Chapter 2, lay stress upon the classical 2-dimensional cases. Because quite a few books (or chapters in them) have been published which introduce the progress of (2-dimensional) Hadamard matrices, we prefer to present an introductory survey rather than to restate many known long proofs. Chapter I introduces Walsh matrices and Walsh transforms, which have been widely used in engineering fields. Fast algorithms for Walsh transforms and various useful properties of Walsh matrices are also stated. Chapter 2 is about (2-dimensional) Hadamard matrices, especially their construction, existence, and their generalized forms. The updated strongest Hadamard construction theorems presented in this chapter are helpful for readers to understand how difficult it is to prove or disprove the famous Hadamard conjecture.
　　The second part, Chapters 3 and 4, deals with the lower-dimensional cases, e.g., 3-, 4-, and 6-dimensional Walsh and Hadmard matrices and transforms. One of the aims of this part is to make it easier to smoothly move from 2-dimensional cases to the general higher-dimensional cases. Chapter 3 concentrates on the 3-dimensional Hadamard and Walsh matrices. Constructions based upon direct multiplication, and upon recursive methods, perfect binary arrays are introduced. Another important topic of this chapter is the existence and construction of 3-dimensional Hadamard matrices of orders 4k and 4k + 2, respectively. Chapter 4 introduces a group of transforms based on 2-, 3-, 4-, and 6-dimensional Walsh-Hadamard matrices and their corresponding fast algorithms. The algebraic theory of higher-dimensional Walsh-Hadamard matrices is presented also.
　　Finally, the third part, which is the key part of the book, consists of the last two chapters (Chapter 5 and 6). To the author's knowledge, the contents in this part (and the previous second part) have never been included in any published books. This part is divided into chapters according to the orders of the matrices (arrays) processed. Chapter 5 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. This equivalence motivates a group of perfect results about the enumeration of higher-dimensional Hadamard matrices of order 2. Applications of these matrices to feed forward networking, stream cipher, Bent functions and error correcting codes are presented in turn. Chapter 6, which is the longest chapter of the book, aims at introducing Hadamard matrices of general dimension and order. After introducing the definitions of the regular, proper, improper, and generalized higher-dimensional Hadamard matrices, many theorems about the existence and constructions are presented. Perfect binary arrays, generalized perfect arrays, and the orthogonal designs are also used to construct new higher-dimensional Hadamard matrices. The last chapter of the book is a concluding chapter of questions, which includes a list of open problems in the study of the theory of higher-dimensional Hadamard matrices. We hope that these research problems will motivate further developments.
　　In order to satisfy readers with this special interest, we list, at the end of each chapter, as many up to date references as possible.
　　I would like to thank my supervisors, Professors. Zhen Ming Hu and Jiong Pang Zhou for their guidance during my academic years at the Information Security Center of Beijing University of Posts and Telecommunications (BUPT). During my research years in higher-dimensional Hadamard matrices I benefited from Professors W. De Launey, J. Hammer, J. Seberry, K. J. Horadam, P. J. Shlichta, J. Jedwab. My thanks go to many of their papers, theses and communications. I was attracted into the area of higher-dimensional Hadamard matrices by P.J. Shhchta's paper 'Higher-Dimensional Hadamard Matrices' published in IEEE Trans. on Inform. Theory. My first journal paper was motivated by J. Hammer and J. Seberry's paper 'Higher-Dimensional Orthogonal Designs and Applications' published in IEEE Trans. on Inform. Theory. It is Dr. J. Jedwab's wonderful Ph.D thesis 'Perfect Arrays, Barker Arrays and Difference Sets' that motivated me to finish the first book on higher-dimensional Hadamard matrices. One of my main aims in this book is to motivate other authors to begin to publish more books on higher-dimensional Hadamard matrices and their applications, so that the readers in other areas can know what has been done in the area of higher-dimensional HaAamard matrices.
　　I specially thank my wife, Xin Xin Niu, and my son, Mu Long Yang, for their support. It is not hard to imagine how much they have sacrificed in family life during the past years. I would like to dedicate this book to my wife and son. Finally, I also dedicate this book to my parents, Mr. Zhong Quan Yang and Mrs. De Lian Wei for their love. ...