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,
existence 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. 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, and 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 codes 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 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.
Audience: This volume will appeal to researchers and graduate students
whose work involves signal processing, coding, information security, and
applied discrete mathematics
