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The complete k ‐arcs of PG(2, 27) and PG(2, 29)
Author(s) -
Coolsaet Kris,
Sticker Heide
Publication year - 2011
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.20261
Subject(s) - mathematics , automorphism , combinatorics , automorphism group , order (exchange) , group (periodic table) , type (biology) , algebraic number , equivalence (formal languages) , arc (geometry) , backtracking , discrete mathematics , geometry , algorithm , mathematical analysis , ecology , chemistry , organic chemistry , finance , economics , biology
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 S 4 is a group of automorphisms. Copyright © 2010 John Wiley & Sons, Ltd. 19:111‐130, 2011