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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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MLA
De Bruyn, Bart. “The Pseudo-Hyperplanes and Homogeneous Pseudo-Embeddings of the Generalized Quadrangles of Order (3, t).” DESIGNS CODES AND CRYPTOGRAPHY, vol. 68, no. 1–3, 2013, pp. 259–84, doi:10.1007/s10623-012-9705-3.
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. https://doi.org/10.1007/s10623-012-9705-3
Chicago author-date
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): 259–84. https://doi.org/10.1007/s10623-012-9705-3.
Chicago author-date (all authors)
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. doi:10.1007/s10623-012-9705-3.
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.
IEEE
[1]
B. De Bruyn, “The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t),” DESIGNS CODES AND CRYPTOGRAPHY, vol. 68, no. 1–3, pp. 259–284, 2013.
@article{4241831,
  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 .}},
  author       = {{De Bruyn, Bart}},
  issn         = {{0925-1022}},
  journal      = {{DESIGNS CODES AND CRYPTOGRAPHY}},
  keywords     = {{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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