Note on the existence of large minimal blocking sets in galois planes

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Abstract

A subset S of a finite projective plane of order q is called a blocking set if S meets every line but contains no line. For the size of an inclusion-minimal blocking set q+ {Mathematical expression}+≤{divides}S{divides}≤q {Mathematical expression}+1 holds ([6]). If q is a square, then in PG(2, q) there are minimal blocking sets with cardinality q {Mathematical expression}+1. If q is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3 q points. In the present note we show that in PG(2, q), q≡1 (mod 4) there are minimal blocking sets having more than qlog 2 q/2 points. The blocking sets constructed in this note contain the union of k conics, where k≤log 2 q/2. A slight modification of the construction works for q≡3 (mod 4) and gives the existence of minimal blocking sets of size cqlog 2 q for some constant c. As a by-product we construct minimal blocking sets of cardinality q {Mathematical expression}+1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of {Mathematical expression} parabolas, they are not classical.

Original languageEnglish
Pages (from-to)227-235
Number of pages9
JournalCombinatorica
Volume12
Issue number2
DOIs
Publication statusPublished - Jun 1992

Keywords

  • AMS subject classification code (1991): 51E21

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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