Dynamic generators of the quasiortogonal Hadamard matrix family

The problem of constructing non-linear and linear finite-field generators of quasi-orthogonal matrices of the Hadamard family with a small number of distinct values of their elements not exceeding by absolute value 1 and a global or local maximum of determinant is investigated. The properties of such dynamical systems are analyzed; the classification of the matrix families and their ornaments, obtained with their help, is described; the way of proving the existence of real and integer matrices different from the combinatorial approach is shown. The values to which the elements of the matrix are equal are called its levels. The concepts of the Hadamard norm and the determinant of a quasiorthogonal matrix are introduced. Levels, Hadamard norm and determinant play a fundamental role in the definitions of classes of generalized matrices of the Hadamard family. The classes of the Hadamard, Belevich (conference matrices), Seberri (weighing matrices), Mersenne, Euler, Odin (Seidel), Fermat matrices are described. The formulas for the values of their levels are given. Ornaments of Euler matrices answer to the question of the maximum complexity of Hadamard matrices — two border two circulant structure.