Arcs in cyclic affine planes (Q1022875)

In a projective plane \(\Pi\) of order \(n,\) a \textit{\(k-\)arc} \(\mathcal K\) \((k>2)\) is a set of \(k\) points no three of which collinear. An arc not contained in a larger one is said to be complete. The \((n+1)-\)arcs are also called \textit{ovals}. An affine plane \(A\) of order \(n\) is \textit{cyclic} if it admits a cyclic group \(G\) fixing one point and acting regularly on the set of the remaining points. Any such plane can be constructed from an affine difference set of order \(n.\) Let \(\Pi\) be the projective closure of an affine cyclic plane \(A\) of order \(n\equiv 1 ~(mod.~ 4).\) Using the construction of \(A\) from an affine difference set, the author provides a new \(k-\)arc \(\mathcal K\) of \(\Pi\) with the following properties: (a) \(k=\frac{n+7}{2},\) (b) \(\mathcal K\) shares \( \frac{n+3}{2}\) points with a suitable oval of \(\Pi.\)