On large minimal blocking sets in PG(2, q )

The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non‐square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non‐prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q 4/3  + 1 or q 4/3  + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non‐prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval $[4q\,{\rm log}\, q, q\sqrt q-q+2\sqrt q]$ . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.