Innovations in Incidence Geometry

Small point sets of $\mathrm{PG}(n,q)$ intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points

Zsuzsa Weiner

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Abstract

The main result of this paper is that point sets in PG(n,q), q=p2h, q81, p>2, of size less than 3(qnk+1)2 and intersecting each k-space in 1 modulo q points (such point sets are always minimal blocking sets with respect to k-spaces) are either (nk)-spaces or certain Baer cones. The latter ones are cones with vertex a t-space, where max{1,n2k1}t<nk1, and with a 2((nk)t1)-dimensional Baer subgeometry as a base. Bokler showed that non-trivial minimal blocking sets in PG(n,q) with respect to k-spaces and of size at most (qnk+11)(q1)+ q(qnk1)(q1) are such Baer cones. The corollary of the main result is that we improve on Bokler’s bound. The improvement depends on the divisors of h; for example, when q is a prime square, we get that the non-trivial minimal blocking sets of PG(n,q) with respect to k-spaces and of size less than 3(qnk+1)2 are Baer cones.

Article information

Source
Innov. Incidence Geom., Volume 1, Number 1 (2005), 171-180.

Dates
Received: 10 December 2004
Accepted: 21 December 2004
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551206819

Digital Object Identifier
doi:10.2140/iig.2005.1.171

Mathematical Reviews number (MathSciNet)
MR2213957

Zentralblatt MATH identifier
1113.51003

Subjects
Primary: 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx] 51E21: Blocking sets, ovals, k-arcs

Keywords
blocking sets Baer subgeometries

Citation

Weiner, Zsuzsa. Small point sets of $\mathrm{PG}(n,q)$ intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points. Innov. Incidence Geom. 1 (2005), no. 1, 171--180. doi:10.2140/iig.2005.1.171. https://projecteuclid.org/euclid.iig/1551206819


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References

  • A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata 9 (1980), 425–449.
  • A. Blokhuis, On the size of a blocking set in PG$(2,p)$, Combinatorica 14 (1994), 273–276.
  • ––––, Blocking sets in Desarguesian planes, in: Paul Erdős is Eighty, vol. 2, eds. D. Miklós, V. T. Sós, T. Szőnyi, Bolyai Soc. Math. Studies. (1996), 133–155.
  • M. Bokler, Minimal blocking sets in projective spaces of square order, Des. Codes Cryptogr. 24 (2001), 131-144.
  • M. Bokler and K. Metsch, On the smallest minimal blocking sets in projective space generating the whole space, Contrib. Algebra Geom. 43 (2002), 43–53.
  • B. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald code, J. Combin. Theory 1 (1966), 96-104.
  • A. A. Bruen, Baer subplanes and blocking sets, Bull. Amer. Math. Soc. 76 (1970), 342–344.
  • ––––, Blocking sets and skew subspaces of projective space, Canad. J. Math. 32 (1980), 628–630.
  • U. Heim, Blockierende Mengen in endlichen projektiven Räumen, Mitt. Math. Semin. Giessen 226 (1996), 4–82.
  • J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979, 2nd edition, 1998.
  • J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: update 2001, in: Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference (Chelwood Gate, England, July 16-21, 2001) 3 (2001), 201–246.
  • G. Lunardon, Linear $k$-blocking sets, Combinatorica 21 (2001), 571–581.
  • K. Metsch, Blocking sets in projective spaces and polar spaces, J. Geom. 76 (2003), 216–232.
  • K. Metsch and L. Storme, 2-Blocking sets in ${\rm PG}(4,q)$, $q$ square, Beiträge Algebra Geom. 41 (2000), 247–255.
  • P. Polito and O. Polverino, On small blocking sets, Combinatorica 18 (1998), 133–137.
  • O. Polverino, Small minimal blocking sets and complete $k$-arcs in PG$(2,p^3)$, Discrete Math. 208/9 (1999), 469–476.
  • O. Polverino and L. Storme, Small minimal blocking sets in ${\rm PG}(2,q^3)$, European J. Combin. 23 (2002), 83–92.
  • L. Rédei, Lückenhafte Polynome über endlichen Körpern, Akadémiai Kiadó, Budapest, and Birkhäuser Verlag, Basel, 1970 (English translation: Lacunary Polynomials over Finite Fields, Akadémiai Kiadó, Budapest, and North Holland, Amsterdam, 1973).
  • L. Storme and Zs. Weiner, On $1$-blocking sets in PG$(n,q)$, $n\geq 3$, Des. Codes Cryptogr. 21 (2000), 235–251.
  • P. Sziklai, On small blocking sets and their linearity, Manuscript.
  • T. Szőnyi, Blocking sets in Desarguesian affine and projective planes, Finite Fields Appl. 3 (1997), 187–202.
  • T. Szőnyi and Zs. Weiner, Small Blocking sets in higher dimensions, J. Combin. Theory Ser. A 95 (2001), 88–101.
  • T. Szőnyi, A. Gács and Zs. Weiner, On the spectrum of minimal blocking sets in $\PG(2,q)$, J. Geom. 76 (2003), 256–281.