The rise in maximum determinant matrix complexity

Introduction: In spite of the apparent relation between maximum determinant matrices of even (Hadamard matrices) and odd orders, the latter have particularly complex patterns of repetitive elements. This is what makes them unique and attractive for various applications in visual data processing, coding and masking. Purpose: Developing the theory of maximum determinant matrices, with the focus on using computer-aided analysis, and calculating unique matrices with unique pattern structures in their portraits. Results: We have found some peculiarities of maximum determinant matrices, outlined their families related to Fermat numbers, demonstrated the complication of patterns in other matrices as their orders grow. The presumption about the increasing complexity of structures as the matrix orders grow is confirmed by a chain of matrix portraits we demonstrate. As applied to orthogonal Belevitch matrices, it follows that they cannot be found even in small orders such as 66 or 86.