Circles on lattices and maximum determinant matrices

Abstract

Introduction: The Hadamard conjecture about the existence of maximum determinant matrices in all orders multiple of 4 is closely related to Gauss's problem about the number of points with integer coordinates (Z3 lattice points) on a spheroid, cone, paraboloid or parabola. The location of these points dictates the number and types of extreme matrices. Purpose: Finding out how Gaussian points on sections of solids of revolution are related to the number and types of maximum determinant matrices with a fixed structure for odd orders. Specifying a precise upper bound of maximum determinant values for edged two-circulant matrices and the orders on which they prevail over simpler cyclic structures. Results: A newly proposed formula refines the overly optimistic Elich – Wojtas’ upper bound for the case of matrices with а fixed structure. Fermat numbers have a special role for orders of 4t + 1, and Barba numbers affect the formation of classes of maximum determinant matrices which occupy the areas of orders 4t + 3, successively replacing each other. For a two-circulant structure with an edge, the maximum order of an optimal symmetric solution is estimated as 67. It is proved that the determinant of edged block matrices is superior to the determinants of circulant matrices everywhere except for a special order 39. Practical relevance: Maximum (for a fixed structure) determinant matrices related to lattice points have a direct practical significance for noise-resistant coding, compression and masking of video data.