Minimal Complete Arcs in\( PG(2,q), q < 29 \)

In this paper, it has been verified, by a computer-based proof, that the smallest size of a complete arc is 12 in \( \text{PG}(2,27) \) and 13 in \( \text{PG}(2,29) \). Also, the spectrum of the sizes of the complete arcs of \( \text{PG}(2,27) \) has been found. The classification of the smallest complete arcs of \( \text{PG}(2,27) \) is given: there are seven non-equivalent 12-arcs, and for each of them, the automorphism group and some geometrical properties are presented. Some examples of complete 13-arcs of \( \text{PG}(2,29) \) are also described.