Constructions of small complete arcs with prescribed symmetry

Abstract

We use arcs found by Storme and van Maldeghem in their classification of primitive arcs in ${\rm PG}(2,q)$ as seeds for constructing small complete arcs in these planes. Our complete arcs are obtained by taking the union of such a ``seed arc'' with some orbits of a subgroup of its stabilizer. Using this approach we construct five different complete 15-arcs fixed by $\Z_3$ in ${\rm PG}(2,37)$, a complete 20-arc fixed by $\S_3$ in ${\rm PG}(2,61)$, and two different complete 22-arcs fixed by $\D_5$ in ${\rm PG}(2,71)$. In all three cases, the size of complete arcs constructed in this paper is strictly smaller than the size of the smallest complete arcs (in the respective plane) known so far.