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

Abstract

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

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

“…The papers [70,71] contain inaccuracies indicated in the recent work [61]. Below we give a few results confirming the correctness of all entries on the existence of complete 1 2 (q +7)-arcs in [43, Table 2.4] and [17, Table 2] and, moreover, extending these tables.…”

“…In the first, we note that for q ≤ 125, the validity of the entries in [43, Table 2.4] and [17, Table 2] have checked by computer, see [61,Introduction]. In addition, in [46], for q ≡ 1 (mod 4), a construction of 1 2 (q + 7)-arcs in PG(2, q) is proposed and it is showed by computer that the arcs are complete if q ≤ 337, q = 17.…”

“…In addition, in [46], for q ≡ 1 (mod 4), a construction of 1 2 (q + 7)-arcs in PG(2, q) is proposed and it is showed by computer that the arcs are complete if q ≤ 337, q = 17. Also, in [61] it is proved that if q = ct − 1 ≥ 17, t odd prime, c ∈ {2, 4}, then in PG(2, q) there is a complete 1 2 (q + 7)-arc. So, for q = 19, 25,27,37,61,67,73,121,157,163, the complete arcs needed exist.…”

“…Also, in [61] it is proved that if q = ct − 1 ≥ 17, t odd prime, c ∈ {2, 4}, then in PG(2, q) there is a complete 1 2 (q + 7)-arc. So, for q = 19, 25,27,37,61,67,73,121,157,163, the complete arcs needed exist. Finally, by the method of [61], we obtained 1 2 (q + 7)-arcs for q = 97, 109, 139, 151, and checked by computer that they are complete.…”

“…So, for q = 19, 25,27,37,61,67,73,121,157,163, the complete arcs needed exist. Finally, by the method of [61], we obtained 1 2 (q + 7)-arcs for q = 97, 109, 139, 151, and checked by computer that they are complete. Now we give two explicit constructions of 1 2 (q + 7)-arcs in PG(2, q).…”