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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

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1615-7168
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Volume 17, Issue 1

Issues

Tight sets in finite classical polar spaces

Anamari Nakić / Leo Storme
Published Online: 2017-02-16 | DOI: https://doi.org/10.1515/advgeom-2016-0034

Abstract

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r+1,q) and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where i<q5/8/2+1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

Keywords: Tight sets; finite classical polar spaces; minihypers; blocking sets; Hermitian varieties; symplectic polar spaces

MSC 2010: 05B25; 51E20

Communicated by: G. Korchmáros

References

  • [1]

    J. Bamberg, S. Kelly, M. Law, T. Penttila, Tight sets and m-ovoids of finite polar spaces. J. Combin. Theory Ser. A 114 (2007), 1293–1314. MR2353124 Zbl 1124.51003Google Scholar

  • [2]

    L. Beukemann, K. Metsch, Small tight sets of hyperbolic quadrics. Des. Codes Cryptogr. 68 (2013), 11–24. MR3046332 Zbl 1277.51009Google Scholar

  • [3]

    A. Blokhuis, On the size of a blocking set in PG(2, p). Combinatorica 14 (1994), 111–114. MR1273203 Zbl 0803.05011CrossrefGoogle Scholar

  • [4]

    A. Blokhuis, Blocking sets in Desarguesian planes. In: Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., 133–155, János Bolyai Math. Soc., Budapest 1996. MR1395857 Zbl 0849.51005Google Scholar

  • [5]

    A. Blokhuis, L. Lovász, L. Storme, T. Szőnyi, On multiple blocking sets in Galois planes. Adv. Geom. 7 (2007), 39–53. MR2290638 Zbl 1123.51012Google Scholar

  • [6]

    A. Blokhuis, L. Storme, T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc. (2) 60 (1999), 321–332. MR1724814 Zbl 0940.51007Google Scholar

  • [7]

    R. C. Bose, R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes. J. Combin. Theory Ser. A 1 (1966), 96–104. MR0197215 Zbl 0152.18106Google Scholar

  • [8]

    A. Bruen, Baer subplanes and blocking sets. Bull.Amer. Math. Soc. 76 (1970), 342–344. MR0251629 Zbl 0207.02601Google Scholar

  • [9]

    A. A. Bruen, Intersection of Baer subgeometries. Arch. Math. (Basel) 39 (1982), 285–288. MR682458 Zbl 0477.51008Google Scholar

  • [10]

    A. A. Bruen, J. W. P. Hirschfeld, Intersections in projective space. I. Combinatorics. Math. Z. 193 (1986), 215–225. MR856149 Zbl 0579.51010Google Scholar

  • [11]

    A. Cossidente, F. Pavese, On the geometry of unitary involutions. Finite Fields Appl. 36 (2015), 14–28. MR3396373 Zbl 1328.51001Google Scholar

  • [12]

    J. De Beule, A. Gács, Complete arcs on the parabolic quadric Q(4, q). Finite Fields Appl. 14 (2008), 14–21. MR2381472 Zbl 1137.51008Google Scholar

  • [13]

    J. De Beule, P. Govaerts, A. Hallez, L. Storme, Tight sets, weighted m-covers, weighted m-ovoids, and minihypers. Des. Codes Cryptogr. 50 (2009), 187–201. MR2469977 Zbl 1246.51006Google Scholar

  • [14]

    J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads of classical finite polar spaces. Serdica Math. J. 34 (2008), 689–714. MR2489960 Zbl 1224.05071Google Scholar

  • [15]

    S. De Winter, K. Thas, Bounds on partial ovoids and spreads in classical generalized quadrangles. Innov. Incidence Geom. 11 (2010), 19–33. MR2795055 Zbl 1266.51005Google Scholar

  • [16]

    G. Donati, N. Durante, On the intersection of two subgeometries of PG(n, q). Des. Codes Cryptogr. 46 (2008), 261–267. MR2372839 Zbl 1185.51010Google Scholar

  • [17]

    K. W. Drudge, Extremal sets in projective and polar spaces. PhD thesis, Univ. Western Ontario (1998). MR2698237Google Scholar

  • [18]

    V. Fack, S. L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes. Des. Codes Cryptogr. 46 (2008), 25–43. MR2361839 Zbl 1182.94062Google Scholar

  • [19]

    P. Govaerts, L. Storme, On a particular class of minihypers and its applications. II. Improvements for q square. J. Combin. Theory Ser. A 97 (2002), 369–393. MR1883871 Zbl 1009.51004Google Scholar

  • [20]

    P. Govaerts, L. Storme, H. Van Maldeghem, On a particular class of minihypers and its applications. III. Applications. European J. Combin. 23 (2002), 659–672. MR1924787 Zbl 1022.51005Google Scholar

  • [21]

    A. Hallez, Linear codes and blocking structures in finite projective and polar spaces. PhD thesis, Ghent University (2010).Google Scholar

  • [22]

    N. Hamada, Characterization of minihypers in a finite projective geometry and its applications to error-correcting codes. Bull. Osaka Women’s Univ. 24 (1987), 1–24.Google Scholar

  • [23]

    N. Hamada, T. Helleseth, A characterization of some q-ary codes (q > (h − 1)2, h ≥3) meeting the Griesmer bound. Math. Japon. 38 (1993), 925–939. MR1240296 Zbl 0786.05016Google Scholar

  • [24]

    M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual. Finite Fields Appl. 14 (2008), 1020–1038. MR2457544 Zbl 1153.51006Google Scholar

  • [25]

    G. Lunardon, Linear k-blocking sets. Combinatorica 21 (2001), 571–581. MR1863578 Zbl 0996.51003CrossrefGoogle Scholar

  • [26]

    K. Metsch, L. Storme, Partial t-spreads in PG(2t + 1, q). Des. Codes Cryptogr. 18 (1999), 199–216. MR1738667 Zbl 0964.51005Google Scholar

  • [27]

    P. Polito, O. Polverino, On small blocking sets. Combinatorica 18 (1998), 133–137. MR1645666 Zbl 0910.05017CrossrefGoogle Scholar

  • [28]

    O. Polverino, Small minimal blocking sets and complete k-arcs in PG(2, p3). Discrete Math. 208/209 (1999), 469–476. MR1725553 Zbl 0941.51008Google Scholar

  • [29]

    L. Rédei, Lückenhafte Polynome über endlichen Körpern. Birkhäuser 1970. MR0294297 Zbl 0321.12028Google Scholar

  • [30]

    M. Sved, Baer subspaces in the n-dimensional projective space. In: Combinatorial mathematics, X (Adelaide, 1982), volume 1036 of Lecture Notes in Math., 375–391, Springer 1983. MR731594 Zbl 0527.51021Google Scholar

  • [31]

    P. Sziklai, On small blocking sets and their linearity. J. Combin. Theory Ser. A 115 (2008), 1167–1182. MR2450336 Zbl 1156.51007Google Scholar

  • [32]

    T. Szőnyi, Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl. 3 (1997), 187–202. MR1459823 Zbl 0912.51004Google Scholar

  • [33]

    T. Szönyi, Z. Weiner, Small blocking sets in higher dimensions. J. Combin. Theory Ser. A 95 (2001), 88–101. MR1840479 Zbl 0983.51006Google Scholar

  • [34]

    Z. Weiner, Small point sets of PG(n, q) intersecting each k-space in 1 modulo q points. Innov. Incidence Geom. 1 (2005), 171–180. MR2213957 Zbl 1113.51003Google Scholar

About the article

Received: 2015-05-08

Accepted: 2015-12-15

Published Online: 2017-02-16

Published in Print: 2017-01-01


Anamari Nakić is supported in part by an STSM grant from the COST project Random network coding and designs over GF(q) (COST IC-1104) and in part by the Croatian Science Foundation under the project 1637


Citation Information: Advances in Geometry, Volume 17, Issue 1, Pages 109–129, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0034.

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