Maximum determinant two circulant matrices with border
Keywords:
Gauss points, Gauss problem, paraboloid lattice structure, orthogonal matrices, Hadamard matrices, maximum determinant matrices, bordered two circulant matricesAbstract
Introduction: The maximum determinant matrices are well known and studied for even orders 2t and 4t, where their structure is most often a bicycle, which is called Hadamard if the order is divided by 4. Odd orders have been studied much less due to the fact that the complexity of the structure of optimal matrices increases indefinitely. Purpose: To replace a deliberately complex object with a hyperobject consisting of a bicycle with a border, and being optimal on the set of matrices of fixed structure. To reveal the relationship of Gauss points (on the sections of bodies of revolution) with the number and types of maximum determinant matrices with a fixed structure for odd orders. To determine the upper and lower bounds for the values of the maxima of the determinant for bordered two circulant matrices. Results: We present the formulas that refine the overly optimistic Elich – Wojtas’ bound for the case of matrices of fixed structure with orders 4t+1 (adamarides) and 4t–1 (mersennides). In addition to the lower and upper bounds, piecewise-smooth quadratic approximations, which are closer to the values of the extremal matrix determinants are given. We present algorithms for mining (deep search) of matrices of the extended Hadamard family by several methods using orbits and compressions of binary sequences. Search results confirming the given bound estimates are also given. Practical relevance: The matrices of the maximum determinant are of direct practical importance for the problems of noise-correcting coding, compression and masking of video information.