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On Small Complete Arcs and Transitive math formula-Invariant Arcs in the Projective Plane math formula



Let q be an odd prime power such that q is a power of 5 or math formula (mod 10). In this case, the projective plane math formula admits a collineation group G isomorphic to the alternating group A5. Transitive G-invariant 30-arcs are shown to exist for every math formula. The completeness is also investigated, and complete 30-arcs are found for math formula. Surprisingly, they are the smallest known complete arcs in the planes math formula, and math formula. Moreover, computational results are presented for the cases math formula and math formula. New upper bounds on the size of the smallest complete arc are obtained for math formula.