A new algorithm and a new type of estimate for the smallest size of complete arcs in PG(2,q)

In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, S. Marcugini and F. Pambianco, New types of estimate for the smallest size of complete arcs in a finite Desarguesian projective plane, preprint]. We propose a new type of upper bound for the smallest size t2(2,q) of a complete arc in the projective plane PG(2,q). We put , where d(q)<1 is a decreasing function of q. The case , where α,β,γ are positive constants independent of q, is considered. It is shown that if q⩽54881, q prime, or q∈R, where R is a set of 34 values in the region 55001…110017. Moreover, our results allow us to conjecture that this estimate holds for all q. An algorithm FOP using any fixed order of points in PG(2,q) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that PG(2,q) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of PG(2,q) is not relevant.