Constructing Minimal Blocking Sets Using Field Reduction

Abstract

AbstractWe present a construction for minimal blocking sets with respect to ‐spaces in , the ‐dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well‐known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS‐construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to ‐spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS‐constructionthen it is of Rédei‐type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.