Directions in AG\((2,p^2)\).
Let \(U=\{(a_i,b_i) \mid i=1,2,\dots,q\}\) be a set of \(q\) points in \(AG(2,q),\) the Desarguesian plane of order \(q.\) We write \(D=\{\frac{a_i-a_j}{b_i-b_j}\mid i \neq j\}\) and call the elements of this set the \textit{directions determined by } \(U.\) This is a subset of \(GF(q)\cup\{\infty\}\) and consists of the slopes of the lines joining two points of \(U.\) Let \(N=|D|\) be the number of the determined directions. Embedding \(AG(2,q)\) into \(PG(2,q)\) and adding to \(U\) the points at infinity corresponding to the elements of \(D,\) we get a blocking set \(B\) of the projective plane. It contains a line if and only if either \(U\) is an affine line or \(U\) determines every direction. Then \(B\) has the property that there is a line (the line at infinity) missing exactly \(q\) points of \(B.\) It is easy to see that this property characterizes minimal blocking sets arising from the above construction; they are called blocking sets of Rédei type. If \(N<\frac{q+3}{2},\) then \(U\) is the graph of a \(GF(s)-\)linear function, where \(GF(s)\) is a subfield of \(GF(q)\) [see, e.g, \textit{S. Ball}, J. Comb. Theory, Ser. A 104, No. 2, 341--350 (2003; Zbl 1045.51004)]. In this paper, the authors have proved that if \(\frac{q+3}{2}\leq N\) and \(q=p^2\) is the square of a prime number \(p\), then either \(U\) is affinely equivalent to the graph of the function \(x \mapsto x^{\frac{q+1}{2}}\) or it determines at least \(\frac{q+\sqrt{q}}{2}+1\) directions and the bound is sharp.
scientific article; zbMATH DE number 5567862