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Keywords:

  • k-arcs;
  • Desarguesian projective plane;
  • exhaustive search;
  • isomorph-free;
  • finite projective plane

Abstract

A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 27 and 29 was obtained by computer. The resulting numbers of complete arcs are tabulated according to size of the arc and type of the automorphism group, and also according to the type of algebraic curve into which they can be embedded. For the arcs with the larger automorphism groups, explicit descriptions are given. The algorithm used for generating the arcs is an application of isomorph-free backtracking using canonical augmentation, an adaptation of an earlier algorithm by the authors. Part of the computer results can be generalized to other values of q: two families of arcs are presented (of size 12 and size 18) for which the symmetric group S4 is a group of automorphisms. Copyright © 2010 John Wiley & Sons, Ltd. 19:111-130, 2011