On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order
✍ Scribed by Gábor Korchmáros; Angelo Sonnino
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 187 KB
- Volume
- 18
- Category
- Article
- ISSN
- 1063-8539
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✦ Synopsis
Abstract
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 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(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