On Small Complete Arcs and Transitive ‐Invariant Arcs in the Projective Plane

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