Antipodal Polygons and Half-Circulant Hadamard Matrices

Abstract

As known, the question on the existence of Hadamard matrices of order m = 4n, where n is an arbitrary natural number, is equivalent to the question on the possibility to inscribe a regular hypersimplex into the (4n ¡ 1)-dimensional cube. We introduced a class of Hadamard matrices of order 4n of half-circulant type in 1997 and a class of antipodal n-gons inscribed into a regular (2n-1)-gon. In 2004 we proved that a half-circulant Hadamard ma- trix of order 4n exists if and only if there exist antipodal n-gons inscribed into a regular (2n-1)-gon. On this background there was developed the method of constructing of the Hadamard matrices of order 4n, which is universal, i.e., it can be applied to any arbitrary natural number n, including a prime number case, that usually requires the individual approach to the construction of the Hadamard matrix of corresponding order. In the paper, there are obtained the necessary and su±cient algebraic-geometric conditions for the existence of antipodal polygons allowing to justify the inductive approach to be used to the proof of existence theorems for Hadamard matrices of arbitrary order 4n, n ≥ 3.