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On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order
Author(s) -
Korchmáros Gábor,
Sonnino Angelo
Publication year - 2010
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20220
Subject(s) - conic section , mathematics , combinatorics , arc (geometry) , order (exchange) , affine transformation , magma , geometry , finance , volcano , seismology , economics , geology
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