Abstract
Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q h) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q h).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker R.D., Brown J.M.N., Ebert G.L., Fisher J.C.: bundles. Bull. Belg. Math. Soc. 3, 329–336 (1994)
Ball S., Blokhuis A., Mazzocca F.: Maximal arcs in Desarguesian planes of odd order do not exist. Combinatorica 17, 31–41 (1997)
Blokhuis A., Lavrauw M.: Scattered spaces with respect to a spread in PG(n,q). Geom. Dedicata 81(1–3), 231–243 (2000)
Bonoli G., Polverino O.: \({\mathbb {F}}\)-linear blocking sets in PG(2, q 4). Innov. Incidence Geom. 2, 35–56 (2005)
Donati G., Durante N.: On the intersection of two subgeometries of PG(n, q). Des. Codes Cryptogr. 46, 261–267 (2008)
Fancsali Sz.L., Sziklai P.: About maximal partial 2-spreads in PG(3m − 1, q). Innov. Incidence Geom. 4, 89–102 (2006)
Fancsali Sz.L., Sziklai P.: Description of the clubs. Annales Univ. Sci. Sect. Mat. (to appear).
Ferret S., Storme L.: Results on maximal partial spreads in PG(3, p 3) and on related minihypers. Des. Codes Cryptogr. 29, 105–122 (2003)
Glynn D.G.: Finite projective planes and related combinatorial systems. PhD thesis, Adelaide University (1978).
Harrach N., Metsch K.: Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points. Des. Codes Cryptogr. (submitted).
Harrach N., 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).
Hirschfeld J.W.P.: Projective geometries over finite fields, 2nd edition. Oxford University Press, New York, 1998. xiv + 555 pp.
Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometries. Nova Academic Publishers (to appear).
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. Combin. Theory Ser. A (submitted).
Lunardon G.: Normal spreads. Geom. Dedicata 75, 245–261 (1999)
Lunardon G., Polverino O.: Translation ovoids of orthogonal polar spaces. Forum Math. 16, 663–669 (2004)
Polito P., Polverino O.: On small blocking sets. Combinatorica 18(1), 133–137 (1998)
Polverino O.: Linear sets in finite projective spaces. Discrete Math. (2009). doi:10.1016/j.disc.2009.04.007.
Sziklai P.: On small blocking sets and their linearity. J. Combin. Theory Ser. A 115(7), 1167–1182 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Leo Storme.
Dedicated to the memory of A. Gács (1969–2009).
Rights and permissions
About this article
Cite this article
Lavrauw, M., Van de Voorde, G. On linear sets on a projective line. Des. Codes Cryptogr. 56, 89–104 (2010). https://doi.org/10.1007/s10623-010-9393-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-010-9393-9