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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| Published in: | Журнал математической физики, анализа, геометрии |
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| Date: | 2010 |
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| Format: | Article |
| Language: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/106633 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| _version_ | 1862536260998922240 |
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| author | Medianik, A.I. |
| author_facet | Medianik, A.I. |
| 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. |
| collection | DSpace DC |
| container_title | Журнал математической физики, анализа, геометрии |
| 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.
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| first_indexed | 2025-11-24T10:55:27Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-106633 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-11-24T10:55:27Z |
| publishDate | 2010 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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| spelling | Medianik, A.I. 2016-10-01T15:08:20Z 2016-10-01T15:08:20Z 2010 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 https://nasplib.isofts.kiev.ua/handle/123456789/106633 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. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Antipodal Polygons and Half-Circulant Hadamard Matrices Article published earlier |
| spellingShingle | Antipodal Polygons and Half-Circulant Hadamard Matrices Medianik, A.I. |
| title | 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_short | Antipodal Polygons and Half-Circulant Hadamard Matrices |
| title_sort | antipodal polygons and half-circulant hadamard matrices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106633 |
| work_keys_str_mv | AT medianikai antipodalpolygonsandhalfcirculanthadamardmatrices |