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Tangency sets in PG(3,q)

Klaus Metsch and Leo Storme (UGent)
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Abstract
A tangency set of PG(d, q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG(3, q) has at most q(2) + 1 points with equality only if it is an ovoid. We show that a tangency set of PG(3, q) with q(2) - 1, q >= 19, or q(2) points is contained in an ovoid. This implies the non-existence of minimal blocking sets of size q(2) - 1, q >= 19, and of q(2) with respect to planes in PG(3, q), and implies the extendability of partial 1-systems of size q(2)-1, q >= 19, or q(2) to 1-systems on the hyperbolic quadric Q(+)(5, q).

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Please use this url to cite or link to this publication:

MLA
Metsch, Klaus, and Leo Storme. “Tangency Sets in PG(3,q).” JOURNAL OF COMBINATORIAL DESIGNS 16.6 (2008): 462–476. Print.
APA
Metsch, Klaus, & Storme, L. (2008). Tangency sets in PG(3,q). JOURNAL OF COMBINATORIAL DESIGNS, 16(6), 462–476.
Chicago author-date
Metsch, Klaus, and Leo Storme. 2008. “Tangency Sets in PG(3,q).” Journal of Combinatorial Designs 16 (6): 462–476.
Chicago author-date (all authors)
Metsch, Klaus, and Leo Storme. 2008. “Tangency Sets in PG(3,q).” Journal of Combinatorial Designs 16 (6): 462–476.
Vancouver
1.
Metsch K, Storme L. Tangency sets in PG(3,q). JOURNAL OF COMBINATORIAL DESIGNS. 2008;16(6):462–76.
IEEE
[1]
K. Metsch and L. Storme, “Tangency sets in PG(3,q),” JOURNAL OF COMBINATORIAL DESIGNS, vol. 16, no. 6, pp. 462–476, 2008.
@article{682174,
  abstract     = {A tangency set of PG(d, q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG(3, q) has at most q(2) + 1 points with equality only if it is an ovoid. We show that a tangency set of PG(3, q) with q(2) - 1, q >= 19, or q(2) points is contained in an ovoid. This implies the non-existence of minimal blocking sets of size q(2) - 1, q >= 19, and of q(2) with respect to planes in PG(3, q), and implies the extendability of partial 1-systems of size q(2)-1, q >= 19, or q(2) to 1-systems on the hyperbolic quadric Q(+)(5, q).},
  author       = {Metsch, Klaus and Storme, Leo},
  issn         = {1063-8539},
  journal      = {JOURNAL OF COMBINATORIAL DESIGNS},
  language     = {eng},
  number       = {6},
  pages        = {462--476},
  title        = {Tangency sets in PG(3,q)},
  url          = {http://dx.doi.org/10.1002/jcd.20174},
  volume       = {16},
  year         = {2008},
}

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