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A new family of tight sets in Q+(5,q)

Jan De Beule (UGent) , Jeroen Demeyer (UGent) , Klaus Metsch (UGent) and Morgan Rodgers (UGent)
(2016) DESIGNS CODES AND CRYPTOGRAPHY. 78(3). p.655-678
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Abstract
In this paper, we describe a new infinite family of (q^2−1)/2-tight sets in the hyperbolic quadrics Q+(5,q), for q≡5 or 9 mod 12. Under the Klein correspondence, these correspond to Cameron–Liebler line classes of PG(3,q) having parameter (q^2−1)/2. This is the second known infinite family of nontrivial Cameron–Liebler line classes, the first family having been described by Bruen and Drudge with parameter (q^2+1)/2 in PG(3,q) for all odd q. The study of Cameron–Liebler line classes is closely related to the study of symmetric tactical decompositions of PG(3,q) (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when q≡9 mod 12 (so q=3^(2e) for some positive integer e), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler (in Linear Algebra Appl 46, 91–102, 1982); the nature of these decompositions allows us to also prove the existence of a set of type (1/2*(3^(2e)−3^e), 1/2*(3^(2e)+3^e)) in the affine plane AG(2, 3^(2e)) for all positive integers e. This proves a conjecture made by Rodgers in his Ph.D. thesis.
Keywords
CAMERON, NONEXISTENCE, M-OVOIDS, LIEBLER LINE CLASSES, Sets of type (m_n, Tactical decompositions, Tight sets, Cameron–Liebler line classes

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Citation

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Chicago
De Beule, Jan, Jeroen Demeyer, Klaus Metsch, and Morgan Rodgers. 2016. “A New Family of Tight Sets in Q+(5,q).” Designs Codes and Cryptography 78 (3): 655–678.
APA
De Beule, Jan, Demeyer, J., Metsch, K., & Rodgers, M. (2016). A new family of tight sets in Q+(5,q). DESIGNS CODES AND CRYPTOGRAPHY, 78(3), 655–678.
Vancouver
1.
De Beule J, Demeyer J, Metsch K, Rodgers M. A new family of tight sets in Q+(5,q). DESIGNS CODES AND CRYPTOGRAPHY. 2016;78(3):655–78.
MLA
De Beule, Jan, Jeroen Demeyer, Klaus Metsch, et al. “A New Family of Tight Sets in Q+(5,q).” DESIGNS CODES AND CRYPTOGRAPHY 78.3 (2016): 655–678. Print.
@article{5855765,
  abstract     = {In this paper, we describe a new infinite family of (q\^{ }2\ensuremath{-}1)/2-tight sets in the hyperbolic quadrics Q+(5,q), for q\ensuremath{\equiv}5 or 9 mod 12. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of PG(3,q) having parameter (q\^{ }2\ensuremath{-}1)/2. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter (q\^{ }2+1)/2 in PG(3,q) for all odd q. The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of PG(3,q) (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when q\ensuremath{\equiv}9 mod 12 (so q=3\^{ }(2e) for some positive integer e), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler (in Linear Algebra Appl 46, 91--102, 1982); the nature of these decompositions allows us to also prove the existence of a set of type (1/2*(3\^{ }(2e)\ensuremath{-}3\^{ }e), 1/2*(3\^{ }(2e)+3\^{ }e)) in the affine plane AG(2, 3\^{ }(2e)) for all positive integers e. This proves a conjecture made by Rodgers in his Ph.D. thesis.},
  author       = {De Beule, Jan and Demeyer, Jeroen and Metsch, Klaus and Rodgers, Morgan},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keyword      = {CAMERON,NONEXISTENCE,M-OVOIDS,LIEBLER LINE CLASSES,Sets of type (m\_n,Tactical decompositions,Tight sets,Cameron--Liebler line classes},
  language     = {eng},
  number       = {3},
  pages        = {655--678},
  title        = {A new family of tight sets in Q+(5,q)},
  url          = {http://dx.doi.org/10.1007/s10623-014-0023-9},
  volume       = {78},
  year         = {2016},
}

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