A small complete arc in \(\text{PG}(2,q)\), \(q=p^2\), \(p\equiv 3\pmod 4\)
A complete \(k\)-arc in a projective plane is a set of \(k\) points of the plane, no three collinear, which can not be extended to a larger arc in that plane. Determining the smallest \(k\) for which there exists a complete \(k\)-arc is, in general, a difficult open problem. The values are known for some small order planes, and bounds on the size of \(k\) are known, but there are very few explicit constructions. These, for the most part, do not give values for \(k\) that are close to the known bounds. In this paper the authors give a construction for a \(k\)-arc which consists of selecting certain points from each of two related conics. Under the assumptions in the title, the arc has size \(k = 4({\sqrt q} - 1)\). They then consider the question of when this arc is complete. Due to the selection of the points, some automorphisms of the arc are easily identified. Using these automorphisms it is possible to reduce the problem of checking for completeness to a reasonable size. Finally, by using a computer to check the reduced problem, they were able to verify that the constructed arc is complete when \(q \leq 961\). Complete arcs containing the constructed arc, when it is not complete, were obtained for values of \(q\) up to and including \(16129\). This construction does not give the smallest complete arc in those cases where the size of the smallest complete arc is known.
scientific article; zbMATH DE number 1369788