Complete arcs in PG(2,25)PG(2,25): The spectrum of the sizes and the classification of the smallest complete arcs

In this paper it has been verified, by an exhaustive computer search, that in PG(2,25)PG(2,25) the smallest size of a complete arc is 12 and that complete 19-arcs and 20-arcs do not exist. Therefore, the spectrum of the sizes of the complete arcs in PG(2,25)PG(2,25) is completely determined. The classification of the smallest complete arcs is also given: the number of non-equivalent complete 1212-arcs is 606 and for each of them the automorphism group has been found and some geometrical properties have been studied. The exhaustive search has been feasible because projective equivalence properties have been exploited to prune the search tree and to avoid generating too many isomorphic copies of the same arc.