On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

This is the authors' abstract. We don't add key points for this paper.

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2,q) withq>13 odd, all known completek‐arcs sharing exactly ½(q+3) points with a conic 𝒞 have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)‐arc in PG(2, 17), and two complete (½(q+3)+3)‐arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(qr+3)+3)‐arc in PG(2,qr) withrodd andq≡3 (mod 4) sharing ½(qr+3) points with a conic, whenever PG(2,q) has a (½(qr+3)+3)‐arc sharing ½(qr+3) points with a conic. A survey of results for smallerqobtained with the use of the MAGMA package is also presented.