Antipodal Polygons and Half-Circulant Hadamard Matrices

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...

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Дата:2010
Автор: Medianik, A.I.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Назва видання:Журнал математической физики, анализа, геометрии
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/106633
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Antipodal Polygons and Half-Circulant Hadamard Matrices / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 56-72. — Бібліогр.: 10 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1066332016-10-02T03:02:46Z Antipodal Polygons and Half-Circulant Hadamard Matrices Medianik, A.I. 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. 2010 Article Antipodal Polygons and Half-Circulant Hadamard Matrices / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 56-72. — Бібліогр.: 10 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106633 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description 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.
format Article
author Medianik, A.I.
spellingShingle Medianik, A.I.
Antipodal Polygons and Half-Circulant Hadamard Matrices
Журнал математической физики, анализа, геометрии
author_facet Medianik, A.I.
author_sort Medianik, A.I.
title Antipodal Polygons and Half-Circulant Hadamard Matrices
title_short Antipodal Polygons and Half-Circulant Hadamard Matrices
title_full Antipodal Polygons and Half-Circulant Hadamard Matrices
title_fullStr Antipodal Polygons and Half-Circulant Hadamard Matrices
title_full_unstemmed Antipodal Polygons and Half-Circulant Hadamard Matrices
title_sort antipodal polygons and half-circulant hadamard matrices
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2010
url http://dspace.nbuv.gov.ua/handle/123456789/106633
citation_txt Antipodal Polygons and Half-Circulant Hadamard Matrices / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 56-72. — Бібліогр.: 10 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT medianikai antipodalpolygonsandhalfcirculanthadamardmatrices
first_indexed 2023-10-18T20:13:22Z
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