Basic Definitions

This chapter presents some basic definitions concerning Hadamard matrices. These include finite fields, group rings, type 1 and 2 matrices, and Paley core matrices. Then the notations which are used in the chapter are provided in a table. The chapter also summarizes some known constructions of Hadamard Matrices that are based on algebraic and number theoretic methods. Hadamard matrices are of practical use in signal processing and design experiments among other applications. An algorithm which finds an Hadamard matrix is equivalent to one which has maximal excess shows the matrix is equivalent to a regular Hadamard matrix. The Williamson, Goethals–Seidel and Propus methods give constructions for Hadamard, skew Hadamard, and symmetric Hadamard ±1 matrices, using specific circulant matrices, satisfying the additive property.