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On Small Complete Arcs and Transitive A 5 ‐Invariant Arcs in the Projective Plane P G ( 2 , q )
Author(s) -
Pace Nicola
Publication year - 2014
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.21372
Subject(s) - mathematics , projective plane , collineation , combinatorics , invariant (physics) , projective linear group , prime power , transitive relation , blocking set , prime (order theory) , finite geometry , arc (geometry) , projective test , discrete mathematics , projective space , pure mathematics , geometry , correlation , mathematical physics
Let q be an odd prime power such that q is a power of 5 or q ≡ ± 1 (mod 10). In this case, the projective plane P G ( 2 , q ) admits a collineation group G isomorphic to the alternating group A 5 . Transitive G ‐invariant 30‐arcs are shown to exist for every q ≥ 41 . The completeness is also investigated, and complete 30‐arcs are found for q = 109 , 121 , 125 . Surprisingly, they are the smallest known complete arcs in the planes P G ( 2 , 109 ) , P G ( 2 , 121 ) , and P G ( 2 , 125 ) . Moreover, computational results are presented for the cases G ≅ A 4and G ≅ S 4 . New upper bounds on the size of the smallest complete arc are obtained for q = 67 , 97 , 137 , 139 , 151 .