On Small Complete Arcs and Transitive A 5 ‐Invariant Arcs in the Projective Plane P G ( 2 , q )

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 .