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The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

Bart De Bruyn (UGent)
(2013) DESIGNS CODES AND CRYPTOGRAPHY. 68(1-3). p.259-284
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Abstract
In the paper "as reported by De Bruyn (Adv Geom, to appear)", we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 a parts per thousand currency sign s < a, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in "De Bruyn (Adv Geom, to appear)", we are able to give a complete classification of all pseudo-hyperplanes of . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of . We use the classification of the pseudo-hyperplanes of to obtain a list of all homogeneous pseudo-embeddings of .
Keywords
Pseudo-hyperplane, Generalized quadrangle, (Universal, homogeneous) pseudo-embedding, Pseudo-embedding rank, Tight set, m-Ovoid, DUAL POLAR SPACE, M-OVOIDS, TIGHT SETS, GEOMETRIES

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Citation

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Chicago
De Bruyn, Bart. 2013. “The Pseudo-hyperplanes and Homogeneous Pseudo-embeddings of the Generalized Quadrangles of Order (3, T).” Designs Codes and Cryptography 68 (1-3): 259–284.
APA
De Bruyn, B. (2013). The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t). DESIGNS CODES AND CRYPTOGRAPHY, 68(1-3), 259–284.
Vancouver
1.
De Bruyn B. The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t). DESIGNS CODES AND CRYPTOGRAPHY. 2013;68(1-3):259–84.
MLA
De Bruyn, Bart. “The Pseudo-hyperplanes and Homogeneous Pseudo-embeddings of the Generalized Quadrangles of Order (3, T).” DESIGNS CODES AND CRYPTOGRAPHY 68.1-3 (2013): 259–284. Print.
@article{4241831,
  abstract     = {In the paper {\textacutedbl}as reported by De Bruyn (Adv Geom, to appear){\textacutedbl}, we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 a parts per thousand currency sign s {\textlangle} a, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in {\textacutedbl}De Bruyn (Adv Geom, to appear){\textacutedbl}, we are able to give a complete classification of all pseudo-hyperplanes of . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of . We use the classification of the pseudo-hyperplanes of to obtain a list of all homogeneous pseudo-embeddings of .},
  author       = {De Bruyn, Bart},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keyword      = {Pseudo-hyperplane,Generalized quadrangle,(Universal,homogeneous) pseudo-embedding,Pseudo-embedding rank,Tight set,m-Ovoid,DUAL POLAR SPACE,M-OVOIDS,TIGHT SETS,GEOMETRIES},
  language     = {eng},
  number       = {1-3},
  pages        = {259--284},
  title        = {The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)},
  url          = {http://dx.doi.org/10.1007/s10623-012-9705-3},
  volume       = {68},
  year         = {2013},
}

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