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

Abstract

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.