A General Construction for Blocking Sets in Finite Affine Geometries

Abstract

A t-fold affine blocking set is a set of points in $${{\,\mathrm{AG}\,}}(n,q)$$ intersecting each hyperplane in at least t points. In this paper we present a general construction of affine blocking sets in $${{\,\mathrm{AG}\,}}(n,q)$$ . The construction uses an arc in an r-dimensional subspace of $${{\,\mathrm{PG}\,}}(n,q)$$ and a blocking set in the affine part $$\cong {{\,\mathrm{AG}\,}}(n-r-1,q)$$ of its complementary subspace to produce a t-fold affine blocking set in $${{\,\mathrm{AG}\,}}(n,q)$$ . The infinite class of t-fold affine blocking sets with $$t=q-n+2$$ meeting Bruen’s bound is obtained as a special case of this construction. It gives also several optimal affine blocking sets whose cardinality meets the lower bound provided by Ball’s improvement of Bruen’s bound. These are the first examples for blocking sets meeting this new bound. The construction produces also many examples of affine blocking sets lying close to the lower bounds by Bruen, Ball-Blokhuis, and Ball.