Abstract

We generalize the Megyesi construction for Rédei minimal blocking sets by placing cosets of a multiplicative subgroup of $\mathsf{GF}\left(q\right)\setminus \left\{0\right\}$ on $n$ lines of the affine plane. These points together with the determined directions give a minimal blocking set $B$ with $\left|B\right|\ge \left(2-2∕9\right)q+O\left(\sqrt{q}\right)$. We also investigate some constructions in $\mathsf{PG}\left(2,q^h\right)$. We show that if there is a minimal blocking set of size $2q-x$ in $\mathsf{PG}\left(2,q\right)$, then minimal blocking sets of size $2q^h-x$ and $2q^h-x+1$ exist in $\mathsf{PG}\left(2,q^h\right)$, which are not necessarily of Rédei type.

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Mathematical Subject Classification 2000

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