About matrices of the local maximum of determinant

The paper deals with the problem of basic generalizations of Hadamard matrices associated with maximal determinant matrices or not optimal by determinant matrices with orthogonal columns; quasi-orthogonal matrices of the local maximum of determinant have been studied not enough sufficiently. The goal of this paper is to develop theory of such matrices on the preliminary research results. Extreme solutions have been established by minimization of maximum of absolute values of the elements of the matrices followed its subsequent classification according to the quantity of levels and its values depending on orders. The conjecture that there are only five non-trivial and strong optimal low-quantity levels matrices of odd order less than 13 have been proposed. Main types of quasi-orthogonal matrices of the local maximum of determinant (M-matrices) including Mersenne, Fermat, and Euler matrices have been identified and described by its weighing functions. The conjecture accordingly existence of all Mersenne matrices of odd order have been formulated. The question of Mersenne and Hadamard matrices existence is observed Mersenne and Fermat Filters based on the suboptimal by determinant matrices have been used for the masking and image compression.