Расширение гипотезы Райзера на двуциклические структуры и разрешимость матриц Адамара орнаментом в виде бицикла с двойной каймой

Abstract

Purpose: Our goal is to expand the border of critical orders for Ryser's conjecture from circulant to bicirculant structures of quasi-orthogonal matrices with two values (levels) of the entries, and to investigate the resolvability of bicirculant structures with one or two borders for the known types of column/row orthogonal matrices. Results: We have shown that orthogonal bicirculant Euler matrices with levels a = 1, -b, where b =t/t+√2t, exist for all orders n = 4t - 2 and, with a border added, turn through an intermediate stage of real Mersenne matrices into integer Hadamard matrices, defining thereby a matrix structure of minimum complexity resolvable for all possible orders they have determined. In other words, the Hadamard matrix conjecture (well known by its irresolvability by non-combinatorial methods) is proved now through an appeal to “matrix transitions” from real matrix types (not limited by the ban to have irrational entries) to integer Hadamard matrices with entries 1, -1. We have demonstrated that maximum determinant matrices of orders n = 4t - 2 are related to orthogonal bicirculant matrices. They are essentially different from Euler matrices because their bicirculant structure, as well as the structure of bicirculant Hadamard matrices, is not always resolvable for their corresponding orders. We have also estimated the symmetry borders for various families of bicirculant maximum determinant matrices, including Hadamard matrices. Practical relevance: The algorithms of calculating bicirculant matrices have been used in developing research software. Matrices suboptimal by their determinant are the basis of Euler and Mersenne filters used for image compression and masking.