Arcs in Desarguesian nets

A trivial upper bound on the size k of an arc in an r-net is $k leq r + 1$. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case $k = r + 1$ cannot occur, and $k geq r - 1$ implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference $r - k$ does not exceed $sqrt{k/18}$. Moreover, it is proved that the same assumption ensures that the arc can be extended to an oval of the net.