A new algorithm and a new type of estimate for the smallest size of complete arcs in

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 t 2 ( 2 , q ) of a complete arc in the projective plane P G ( 2 , q ) . We put t 2 ( 2 , q ) = d ( q ) q ln q , where d ( q ) < 1 is a decreasing function of q. The case d ( q ) < α / ln β q + γ , where α , β , γ are positive constants independent of q, is considered. It is shown that t 2 ( 2 , q ) < ( 2 / ln 1 10 q + 0.32 ) q ln q 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 P G ( 2 , q ) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that P G ( 2 , q ) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of P G ( 2 , q ) is not relevant.