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

Abstract

A subsetS of a finite projective plane of orderq is called a blocking set ifS meets every line but contains no line. For the size of an inclusion-minimal blocking setq+\(\sqrt q \)+≤∣S∣≤q\(\sqrt q \)+1 holds ([6]). Ifq is a square, then inPG(2,q) there are minimal blocking sets with cardinalityq\(\sqrt q \)+1. Ifq is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3q points. In the present note we show that inPG(2,q),q≡1 (mod 4) there are minimal blocking sets having more thanqlog2 q/2 points. The blocking sets constructed in this note contain the union ofk conics, wherek≤log2 q/2. A slight modification of the construction works forq≡3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog2 q for some constantc.

As a by-product we construct minimal blocking sets of cardinalityq\(\sqrt q \)+1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of\(\sqrt q \) parabolas, they are not classical.

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Szőnyi, T. Note on the existence of large minimal blocking sets in galois planes. Combinatorica 12, 227–235 (1992). https://doi.org/10.1007/BF01204725

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AMS subject classification code (1991)

  • 51 E 21