Minimax arcs (Q1182963)

Let \(C\) be an irreducible conic of \(\pi=\mathrm{PG}(2,q)\). Then the interior points of \(C\) together with \(C\) form a complete \(((q^ 2+q+2)/2,(q+3)/2))\)-arc, and the interior points together with one point of \(C\) form a complete \(((q^ 2-q+2)/2,(q+1)/2)\)-arc of \(\pi\). Both examples satisfy the relation \(k=(n-1)q+1\), and the points of these arcs have the property that the number of \(n\)-secants through each is either a maximum or a minimum. The author uses these conditions to give characterizations of these two classes of complete \((k,n)\)-arcs.