On Some Methods of Constructing Hadamard Matrices

الملخص

There are two methods , often used to produce examples of algebraic and combinatorial structures . One of these methods begins with at least one example of the desired structure at hand and then constructs further
structures of a like kind . We call such a construction method recursive
.
Another method (or methods) is to generate the desired structure simply after certain parameters regarding it have been specified . We shall call such a method of construction an initio ab
method .
Hadamard matrices are algebraic structures in the sense that they form an important subclass of the class of matrices and hence must conform to all the algebraic rules obeyed by matrices under the usual operations of addition and multiplication . On the other hand , Hadamard matrices are combinatorial structures as well since the entries +1 and -1 of which the matrix consists must follow certain patterns . Thus one expects that one should be able to utilize both type of constructions methods , recursive and ab initio , to construct Hadamard matrices . This is indeed the case and in this paper we review some of these construction methods for Hadamard matrices . In the second part we will introduce the concept of the Kronecker product and develop a recursive construction method for constructing Hadamard matrices based on it . Two important ab initio methods are discussed in the fourth part of this paper. These methods are due to Paley (1933) and is based on Galois fields . Hence some Galois field basics are presented in the third part also.