Local Maximum Determinant Matrices
Abstract
Purpose: Basic generalizations of Hadamard matrices are associated with maximum determinant matrices or matrices with
orthogonal columns non-optimal by determinant; quasi-orthogonal local maximum determinant matrices have been understudied. The
goal of this paper is to consider a theory of these matrices according to preliminary research results. Methods: Extreme solutions have
been found by minimization of maximum of matrices elements absolute values with their consequent classification according to an
amount and values of levels which depend on orders. Results: There has been substantiated a conjecture that there are only five nontrivial
and strongly optimal low-level matrices of odd order — less than 13. There have been identified and described by weighing
functions the main types of quasi-orthogonal local maximum determinant matrices (M-matrices) including Mersenne, Fermat and
Euler matrices. A conjecture concerning existence of all Mersenne matrices of odd order has been formulated. The issue of Mersenne
and Hadamard matrices existence has been considered. An example of Hadamard matrix of 668th order approximated by block array
with Williamson matrices based on Mersenne matrices has been given. M-Matrices determinants dependence on its orders has been
illustrated by graphs. Practical relevance: Algorithms to construct M-Matrices have been implemented while developing the research
software. Mersenne and Fermat Filters used for masking and image compression are based on matrices suboptimal by determinant.
