Abstract
This article discusses minimal s-fold blocking sets B in PG (n, q), q = ph, p prime, q > 661, n > 3, of size |B| > sq + c p q 2/3 - (s - 1) (s - 2)/2 (s > min (c p q 1/6, q 1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n ≥ 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Ball,On the size of a triple blocking set in PG (2,q), Europ. J. Combinatorics, Vol.17 (1996) pp.427–435.
S. Ball, Multiple blocking sets and arcs in nite planes, J. London Math. Soc.,Vol.54 (1996)pp.581–593.
S. Ball, On nuclei and blocking sets in Desarguesian spaces,J. Combin.Theory., Ser. A, Vol.85 (1999) pp.232–236.
S. Ball and A. Blokhuis,On the size of a double blocking set in PG 2,q), Finite Fields Appl.,Vol.2 (1996)pp.125–137.
S. Ball, A. Blokhuis and M. Lavrauw,Linear (q+1)-fold blocking sets in PG(2,q4), Finite Fields Appl.,Vol.6 (2000)pp.294–301.
A. Blokhuis, L. Storme and T. Szönyi,Lacunary polynomials,multiple blocking sets and Baer subplanes,J. London Math. Soc.,Vol.60,No.2 (1999)pp.321–332.
A.A. Bruen, Blocking sets in nite projective planes,SIAM J.of Appl.Math.,Vol.21 (1971)pp.380–392.
A.A. Bruen, Arcs and multiple blocking sets,Combinatorica,Symp.Math.,Vol.28 (1986)pp.15–29.
A.A. Bruen, Polynomial multiplicities over nite elds and intersection sets, J.Combin.Theory, Ser.A,Vol.60 (1992)pp.19–33.
A. Gács,The Rédei Method Applied to Finite Geometry, Ph.D.Thesis,Eötvös Loránd University, Budapest,Hungary (1997).
A. Gács and T. Szönyi,Double blocking sets and Baer subplanes, unpublished manuscript (1995).
J.H. Griesmer,A bound for error-correcting codes,IBM J.Res.Develop.,Vol.4 (1960)pp.532–542.
N. Hamada, Characterization of minihypers in a nite projective geometry and its applications to error-correcting codes, Bull.Osaka Women 's Univ.,Vol.24 (1987)pp.1–24.
N. Hamada and T. Helleseth,A characterization of some q-ary codes (q > (h-1) 2,h≥3) meeting the Griesmer bound,Math.Japonica, Vol.38 (1993)pp.925–940.
N. Hamada and F. Tamari,On a geometrical method of construction of maximal t-linearly independent sets,J.Combin.Theory,Ser.A, Vol.25 (1978)pp.14–28.
J.W.P. Hirschfeld, Projective Geometries Over Finite Fields, Clarendon Press, Oxford (1979).
L. Rédei,Lückenhafte Polynome über endlichen Körpern, Birkhäuser Verlag, Basel (1970).
G. Solomon and J.J. Stiffler, Algebraically punctured cyclic codes,Inform.and Control, Vol.8 (1965) pp.170–179.
L. Storme and Zs. Weiner,Minimal blocking sets in PG(n,q),n≥3,Des.Codes Cryptogr.,Vol.21 (2000)pp.235–251.
M. Sved,Baer subspaces in the n-dimensional projective space,Combinatorial mathematics,X (Adelaide,1982)pp.375–391.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barát, J., Storme, L. Multiple Blocking Sets in PG(n, q), n > 3. Designs, Codes and Cryptography 33, 5–21 (2004). https://doi.org/10.1023/B:DESI.0000032603.40135.6b
Issue Date:
DOI: https://doi.org/10.1023/B:DESI.0000032603.40135.6b