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

The sporadic complete $12$-arc in $\mathrm{PG}(2,13)$ contains eight points from a conic. In $\mathrm{PG}(2,q)$ with $q>13$ odd, all known complete $k$-arcs sharing exactly $\ha(q+3)$ points with a conic $\mathcal{C}$ have size at most $\frac{1}{2}(q+3)+2$, with only two exceptions, both due to Pellegrino, which are complete $(\frac{1}{2}(q+3)+3)$ arcs, one in $\mathrm{PG}(2,19)$ and another in $\mathrm{PG}(2,43)$. Here, three further exceptions are exhibited, namely a complete $(\frac{1}{2}(q+3)+4)$-arc in $\mathrm{PG}(2,17)$, and two complete $(\frac{1}{2}(q+3)+3)$-arcs, one in $\mathrm{PG}(2,27)$ and another in $\mathrm{PG}(2,59)$. The main result is Theorem 6.1 which shows the existence of a $(\frac{1}{2}(q^r+3)+3)$--arc in $\mathrm{PG}(2,q^r)$ with $r$ odd and $q\equiv 3 \pmod 4$ sharing $\frac{1}{2}(q^r+3)$ points with a conic, whenever $\mathrm{PG}(2,q)$ has a $(\frac{1}{2}(q+3)+3)$-arc sharing $\frac{1}{2}(q+3)$ points with a conic. A survey of results for smaller $q$ obtained with the use of the MAGMA package is also presented.