Theory and Applications of Higher-Dimensional Hadamard Matrices

This is the first book on higher-dimensional hadamard matrices and their application in telecommunictions and information security .It is divided into three parts according to the dimensional walsh and Hadamard matrices .Fast algorithms,updated constructions,existence results,and their generalised forms are presented. The second part deals with the lower-dimenstional cases,e.g.3-,4-,and 6- dimensional Walsh and Hadamard matrices and transforms.One of the aims of this part is to simplify moving smoothly from 2-dimensional cases to the general higher-dimensional cases.This part concentrates on the 3-dimensional Hadamard and Walsh matrices.Constructions based upon direct multiplication,and upon recursive methods,perfect binary arrays are also introduced.Another important topic is the existence and construction of 3-dimensional Hadamard matrices of orders 4k and 4k +2 respectively,and a group of transforms based on 2-,3-,4-and 6-dimensional walsh-Hadamard matrices and their corresponding fast algorithms.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.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 correction cods are presented in turn.after introducing the definitions of the regular,proper,improper,and generalised higher-dimensional Hadamard matrices ,many theorems about the existence and constructions are presented.Perfect binary arrays,generalised perfect arrays,and the orthogonal desgns 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.
Audience
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
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 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
3.3.1 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 Three-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 Algebraical 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
concluding Questions
Index