Arcs and Curves over a Finite Field

In [11], a new bound for the number of points on an algebraic curve over a "nite "eld of odd order was obtained, and applied to improve previous bounds on the size of a complete arc not contained in a conic. Here, a similar approach is used to show that a complete arc in a plane of even order q has size q#2 or q!Jq#1 or less than q!2Jq#6. To obtain this result, "rst a new characterization of a Hermitian curve for any square q is given; more precisely, it is shown that a curve of su$ciently low degree has a certain upper bound for the number of its rational points with equality occurring in this bound only when the curve is Hermitian. Finally, another application is given concerning the degree of the curve on which a unital can lie. ( 1999 Academic Press