Small complete arcs in Galois planes

Abstract

In this paper we construct a class of k-arcs in PG(2, q), q=p ^h, h>1, p≠3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with

$$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern-\nulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$

Such complete k-arcs are the unique known complete k-arcs with

$$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern-\nulldelimiterspace} 4}.$$