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A full classification of the complete k ‐arcs of PG(2,23) and PG(2,25)
Author(s) -
Coolsaet K.,
Sticker H.
Publication year - 2009
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.20211
Subject(s) - mathematics , automorphism group , combinatorics , automorphism , backtracking , equivalence (formal languages) , order (exchange) , arc (geometry) , projective plane , type (biology) , discrete mathematics , algorithm , geometry , finance , economics , correlation , ecology , biology
A full classification (up to equivalence) of all complete k ‐arcs in the Desarguesian projective planes of order 23 and 25 was obtained by computer. The algorithm used is an application of isomorph‐free backtracking using canonical augmentation, as introduced by McKay, which we have adapted to the case of subset generation in Desarguesian projective planes. We have applied two variants of the same algorithm, and both techniques yield exactly the same results. Earlier (partial) results by other authors on k ‐arcs in PG(2, q ) with q ⩽25, are reproduced by our programs. We describe those parts of the algorithms which are relevant to the particular problem of generating k ‐arcs and which have made this project feasible. We also list the number of complete arcs in PG(2, 23) and PG(2, 25) according to size of the arc and type of the automorphism group. Explicit descriptions are given for the arcs with the larger automorphism groups. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 459–477, 2009