Abstract
The main result of this paper is that point sets of PG(n, q 3), q = p h, p ≥ 7 prime, of size less than 3(q 3(n−k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p 3), p ≥ 7 prime, of size less than 3(p 3(n−k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q 3) which meet every (n − 2)-space in 1 modulo q points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blokhuis A., Lavrauw M.: Scattered spaces with respect to a spread in PG(n, q). Geom. Dedicata 81, 231–243 (2000)
Bose R.C., Burton R.C.: A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald Codes. J. Comb. Theory 1, 96–104 (1966)
Harrach N.V., Metsch K., Szőnyi T., Weiner Zs.: Small point sets of PG(n, p 3h) intersecting each line in 1 mod p h points. J. Geom. (submitted).
Lavrauw M., Storme L., Van de Voorde G.: A proof for the linearity conjecture for k-blocking sets in PG(n, p 3), p prime. J. Comb. Theory A (submitted).
Lunardon G., Polverino O.: Translation ovoids of orthogonal polar spaces. Forum Math. 16, 663–669 (2004)
Polverino O.: Small minimal blocking sets and complete k-arcs in PG(2,p 3). Discrete Math. 208/209, 469–476 (1999)
Polverino O., Storme L.: Small minimal blocking sets in PG(2, q 3). Eur. J. Comb. 23, 83–92 (2002)
Storme L., Weiner Zs.: On 1-blocking sets in PG(n, q), n ≥ 3. Des. Codes Cryptogr. 21, 235–251 (2000)
Sziklai P.: A bound on the number of points of a plane curve. Finite Fields Appl. 14, 41–43 (2008)
Sziklai P.: On small blocking sets and their linearity. J. Comb. Theory A 115, 1167–1182 (2008)
Szőnyi T.: Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl. 3, 187–202 (1997)
Szőnyi T., Weiner Zs.: Small blocking sets in higher dimensions. J. Comb. Theory A 95, 88–101 (2001)
Weiner Zs.: Small point sets of PG(n, q) intersecting each k-space in 1 modulo \({\sqrt q}\) points. Innov. Incidence Geom. 1, 171–180 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Leo Storme.
Dedicated to the memory of András Gács (1969–2009).
Rights and permissions
About this article
Cite this article
Harrach, N.V., Metsch, K. Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points. Des. Codes Cryptogr. 56, 235–248 (2010). https://doi.org/10.1007/s10623-010-9407-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-010-9407-7