Constructing Minimal Blocking Sets Using Field Reduction

Geertrui Van de Voorde
2015 Journal of combinatorial designs (Print)  
We present a construction for minimal blocking sets with respect to (k − 1)spaces in PG(n − 1, q t ), the (n − 1)-dimensional projective space over the finite field F q t of order q t . The construction relies on the use of blocking cones in the field reduced representation of PG(n − 1, q t ), 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
more » ... (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 [15] . Furthermore we show that every minimal blocking set with respect to the hyperplanes in PG(n − 1, q t ) can be obtained by applying field reduction to a minimal blocking set with respect to (nt − t − 1)-spaces in PG(nt − 1, q). 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 MPSconstruction, it is of Rédei-type whereas a small minimal blocking set arises from our cone construction if and only if it is linear.
doi:10.1002/jcd.21432 fatcat:yxolr5pllzdvvm5a45vt2q7ble