Rédei blocking sets in finite Desarguesian planes
A blocking set in a projective plane is a set of points which intersects each line in at least one point. A blocking set is nontrivial when it does not contain a line. A blocking set is minimal when none of its proper subsets is still a blocking set. A blocking set \(B\) of size \(q+n \leq 2q\) which intersects at least one line in \(n\) points is called a Rédei-type blocking set. A line containing \(n\) points of a Rédei-type blocking set \(B\) of size \(q+n\) is called a Rédei-line of \(B\). A nontrivial blocking set \(B\) in \(PG(2,q)\) is called small when \(|B|<3(q+1)/2\). \textit{A. Blokhuis}, \textit{S. Ball}, \textit{A. E. Brouwer}, \textit{L. Storme} and \textit{T. Szönyi} [J. Comb. Theory, Ser. A 86, No. 1, 187-196 (1999; Zbl 0945.51002)] characterized almost all small nontrivial minimal Rédei-type blocking sets \(B\) in \(PG(2,q)\), \(q=p^h\), \(p\) prime. Let \(L\) be a Rédei-line of \(B\). Let \(AG(2,q) =PG(2,q) \setminus L\). They showed that every line intersects \(B\) in \(1 \pmod s\), \(s=p^e\), \(e >0\), points. They also characterized the small minimal Rédei blocking sets \(B\) which meet each line in \(1 \pmod{s}\) points for \(s>3\), and for \(s=3\) when \(|B|=q+q/3+1\), as being \(GF(p^e=s)\)-linear sets plus the Rédei-line. The author proves in detail that \(B\) is projectively equivalent to a set \(B=\{(x,p(x),1): x\in GF(q)\} \cup \{(1,p(x)/x,0): x \in GF(q)^*\}\), with \(B\) a linearized \(s\)-polynomial. He then describes these small nontrivial blocking sets via the profile of these sets. The profile is described in the following way. For every point \(r\) of \(B\) on a Rédei-line \(L\), every line through \(r\) different from \(L\) shares zero or a non-zero constant number, depending on \(r\), points with \(B\setminus L\). The profile of \(B\) is a list of \(n\) powers of \(s\), not including any ones, showing for each one of the \(n\) points of \(L \cap B\) this non-zero constant. The author shows the different possible profiles for \(q=s^2,s^3\) and \(q=s^4\). The results for \(q=s^4\) show that for the other possibilities for \(q\) and \(s\), a lot of different profiles will occur. He finishes the article by describing the links between the above mentioned results of A. Blokhuis et al. and results of a previous article [\textit{B. Sherman} J. Geom. 43, No. 1/2, 178-187 (1992; Zbl 0746.51018)] in which the author classified Rédei blocking sets having at least two Rédei lines.
scientific article; zbMATH DE number 1760685