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.