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 .