Helping Hadamard conjecture to become a theorem. Part 2

Abstract

Introduction:Hadamard conjecture about the existence of specific square matrices was formulated not by Hadamard but by other mathematicians in the early 20th century. Later, this problem was revised by Ryser together with Bruck and Chowla, and also by Hall, one of the founders of discrete mathematics. This is a problem of the boundary mixed type, as it includes both the continuous and discrete components. The combinatorial approach used in the framework of the discrete component has run its course by the end of the century. The article discusses an alternative based on both concepts.Purpose:To analyze the reasons why the conjecture about the existence of Hadamard matrices of all ordersn =4tis considered unproven, and to propose possible ways to prove it.Methods:Transition, by lowering the ordern =4t— 2, to two-level quasiorthogonal matrices with elements 1 and –bwhose existence on all specified orders is not a difficult problem due to the possible irrationality of their entries. Subsequent construction of a chain of transformations to matrix ordersn= 4t–1,n= 4t,n= 4t+ 1.Results:It is proved that Gauss points on anx2 + 2y2 +z2 =nspheroid are in one-to-one correspondence with symmetric Hadamard matrices (constructed on the basis of the Balonin — Seberry arrays), covering up the gaps on the unsolvable orders 140, 112, etc. known in Williamson’s array theory. Solution tables are found and systematized, which include so-called «best» three-block matricesL(p,q), wherep³qis the number of non-conjugated symmetric matrices of the order in question, andqis the number of block-symmetric matrices which coincide with Williamson’s solutions. The iterative Procrustes algorithm which reduces the norm of the maximum entry in a matrix is proposed for obtaining Hadamard matrices by searching for local and global conditional extremes of the determinant.Practical relevance:The obtained Hadamard matrices and quasi-orthogonal matrices of ordersn =4t– 2,n =4t– 1,n =4t +1 are of immediate practical importance for the problems of noise-resistant coding, compression and masking of video information.