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

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2, q ) with q >13 odd, all known complete k ‐arcs sharing exactly ½( q +3) points with a conic 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( q r +3)+3)‐arc in PG(2, q r ) with r odd and q ≡3 (mod 4) sharing ½( q r +3) points with a conic, whenever PG(2, q ) has a (½( q r +3)+3)‐arc sharing ½( q r +3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010