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A decomposition of the universal embedding space for the near polygon ℍn

Bart De Bruyn (UGent)
(2012) DESIGNS CODES AND CRYPTOGRAPHY. 64(1-2). p.81-91
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Abstract
Let H-n, n >= 1, be the near 2n-gon defined on the 1-factors of the complete graph on 2n + 2 vertices, and let e denote the absolutely universal embedding of Hn into PG(W), where W is a 1/n((2n +2)(n + 1))-dimensional vector space over the field F-2 with two elements. For every point z of H-n and every i is an element of N, let Delta(i) (z) denote the set of points of H-n at distance i from z. We show that for every pair {x, y} of mutually opposite points of H-n, W can be written as a direct sum W-0 circle plus W-1 ... circle plus W-n such that the following four properties hold for every i is an element of {0,..., n}: (1) < e(Delta(i)(x) boolean AND Delta(n-i) (y))> = PG(W-i); (2) < e(boolean OR(j <= i)Delta(j)(x))> = PG(W-0 circle plus ... circle plus W-i); (3) < e(boolean OR(j <= i)Delta(j)(x))> = PG(Wn-i circle plus Wn-i+ 1 circle plus ... circle plus W-n); (4) dim(W-i) = |Delta(i) (x) boolean AND Delta(n-i)(y)| = ((n)(i)) - ((n)(i-1)).((n)(i+1)).
Keywords
Projective embedding, Universal embedding, Generating set, DUAL POLAR SPACES, GEOMETRIES

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MLA
De Bruyn, Bart. “A Decomposition of the Universal Embedding Space for the near Polygon ℍn.” DESIGNS CODES AND CRYPTOGRAPHY, vol. 64, no. 1–2, 2012, pp. 81–91, doi:10.1007/s10623-011-9533-x.
APA
De Bruyn, B. (2012). A decomposition of the universal embedding space for the near polygon ℍn. DESIGNS CODES AND CRYPTOGRAPHY, 64(1–2), 81–91. https://doi.org/10.1007/s10623-011-9533-x
Chicago author-date
De Bruyn, Bart. 2012. “A Decomposition of the Universal Embedding Space for the near Polygon ℍn.” DESIGNS CODES AND CRYPTOGRAPHY 64 (1–2): 81–91. https://doi.org/10.1007/s10623-011-9533-x.
Chicago author-date (all authors)
De Bruyn, Bart. 2012. “A Decomposition of the Universal Embedding Space for the near Polygon ℍn.” DESIGNS CODES AND CRYPTOGRAPHY 64 (1–2): 81–91. doi:10.1007/s10623-011-9533-x.
Vancouver
1.
De Bruyn B. A decomposition of the universal embedding space for the near polygon ℍn. DESIGNS CODES AND CRYPTOGRAPHY. 2012;64(1–2):81–91.
IEEE
[1]
B. De Bruyn, “A decomposition of the universal embedding space for the near polygon ℍn,” DESIGNS CODES AND CRYPTOGRAPHY, vol. 64, no. 1–2, pp. 81–91, 2012.
@article{3217992,
  abstract     = {{Let H-n, n >= 1, be the near 2n-gon defined on the 1-factors of the complete graph on 2n + 2 vertices, and let e denote the absolutely universal embedding of Hn into PG(W), where W is a 1/n((2n +2)(n + 1))-dimensional vector space over the field F-2 with two elements. For every point z of H-n and every i is an element of N, let Delta(i) (z) denote the set of points of H-n at distance i from z. We show that for every pair {x, y} of mutually opposite points of H-n, W can be written as a direct sum W-0 circle plus W-1 ... circle plus W-n such that the following four properties hold for every i is an element of {0,..., n}: (1) < e(Delta(i)(x) boolean AND Delta(n-i) (y))> = PG(W-i); (2) < e(boolean OR(j <= i)Delta(j)(x))> = PG(W-0 circle plus ... circle plus W-i); (3) < e(boolean OR(j <= i)Delta(j)(x))> = PG(Wn-i circle plus Wn-i+ 1 circle plus ... circle plus W-n); (4) dim(W-i) = |Delta(i) (x) boolean AND Delta(n-i)(y)| = ((n)(i)) - ((n)(i-1)).((n)(i+1)).}},
  author       = {{De Bruyn, Bart}},
  issn         = {{0925-1022}},
  journal      = {{DESIGNS CODES AND CRYPTOGRAPHY}},
  keywords     = {{Projective embedding,Universal embedding,Generating set,DUAL POLAR SPACES,GEOMETRIES}},
  language     = {{eng}},
  number       = {{1-2}},
  pages        = {{81--91}},
  title        = {{A decomposition of the universal embedding space for the near polygon ℍn}},
  url          = {{http://doi.org/10.1007/s10623-011-9533-x}},
  volume       = {{64}},
  year         = {{2012}},
}

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