“Plug-In Template” Method: Williamson-Hadamard Matrices

Abstract

We have seen that one of the basic methods used to build Hadamard matrices is based on construction of a class of "special-component" matrices that can be plugged into arrays (templates) of variables to generate Hadamard matrices. Several approaches for constructing special-component matrices and templates have been developed. In 1944, Williamson first constructed "suitable matrices" (Williamson matrices) to replace the variables in a formally orthogonal matrix. Generally, the arrays into which suitable matrices are plugged are orthogonal designs, which have formally orthogonal rows (and columns) but may have variations, such as Goethals-Seidel arrays, Wallis-Whiteman arrays, Spence arrays, generalized quaternion arrays, Agayan (Agaian) families, Kharaghani's methods, and regular <i>s</i>-sets of regular matrices that give new matrices. This is an extremely prolific construction method. There are several interesting schemes for constructing the Williamson matrices and the Williamson arrays. In addition, it has been found that the Williamson-Hadamard sequences possess very good autocorrelation properties that make them amenable to synchronization requirements, and they can thus be used in communication systems. In addition, Seberry, her students, and many other authors have made extensive use of computers for relevant searches. For instance, Djokovic found the first odd number, <i>n</i> = 31, for which symmetric circulant Williamson matrices exist. There are several interesting papers concerning the various types of Hadamard matrix construction. A survey of the applications of Williamson matrices can be found in Ref. 78. In this chapter, two "plug-in template" methods of the construction of Hadamard matrices are presented. The first method is based on Williamson matrices and the Williamson array "template"; the second one is based on the Baumert-Hall array "template." Finally, we will give customary sequences based on construction of new classes of Williamson and generalized Williamson matrices. We start the chapter with a brief description of the construction of Hadamard matrices from Williamson matrices. Then we construct a class of Williamson matrices. Finally, we show that if Williamson-Hadamard matrices of order 4<i>m</i> and 4<i>n</i> exist, then Williamson-Hadamard matrices of order <i>mn</i>/2 exist.