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Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Jan De Beule (UGent) , Govaerts Patrick, Anja Hallez (UGent) and Leo Storme (UGent)
(2009) DESIGNS CODES AND CRYPTOGRAPHY. 50(2). p.187-201
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Abstract
Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342-349, 2008; Govaerts and Storme 4:279-286, 2004; Govaerts et al. 28:659-672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q(-)(5,3), to 56-caps contained in this elliptic quadric Q(-)(5,3).
Keywords
m-Covers, Generalized quadrangles, Tight sets, m-Ovoids, Polar spaces, Minihypers

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Chicago
De Beule, Jan, Govaerts Patrick, Anja Hallez, and Leo Storme. 2009. “Tight Sets, Weighted M-covers, Weighted M-ovoids, and Minihypers.” Designs Codes and Cryptography 50 (2): 187–201.
APA
De Beule, J., Patrick, G., Hallez, A., & Storme, L. (2009). Tight sets, weighted m-covers, weighted m-ovoids, and minihypers. DESIGNS CODES AND CRYPTOGRAPHY, 50(2), 187–201.
Vancouver
1.
De Beule J, Patrick G, Hallez A, Storme L. Tight sets, weighted m-covers, weighted m-ovoids, and minihypers. DESIGNS CODES AND CRYPTOGRAPHY. 2009;50(2):187–201.
MLA
De Beule, Jan et al. “Tight Sets, Weighted M-covers, Weighted M-ovoids, and Minihypers.” DESIGNS CODES AND CRYPTOGRAPHY 50.2 (2009): 187–201. Print.
@article{681347,
  abstract     = {Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342-349, 2008; Govaerts and Storme 4:279-286, 2004; Govaerts et al. 28:659-672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q(-)(5,3), to 56-caps contained in this elliptic quadric Q(-)(5,3).},
  author       = {De Beule, Jan and Patrick, Govaerts and Hallez, Anja and Storme, Leo},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keywords     = {m-Covers,Generalized quadrangles,Tight sets,m-Ovoids,Polar spaces,Minihypers},
  language     = {eng},
  number       = {2},
  pages        = {187--201},
  title        = {Tight sets, weighted m-covers, weighted m-ovoids, and minihypers},
  url          = {http://dx.doi.org/10.1007/s10623-008-9223-5},
  volume       = {50},
  year         = {2009},
}

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