Cubic Curves Over the Finite Field of Order Twenty-Five

Abstract

An arc of degree three is a set of points in projective plane no four of which are collinear but some three are collinear, and a cubic curve is a non-singular projective plane cubic curve. There are cubic curves formed an arc of degree three over a finite field. The aims of this paper are to give the inequivalent cubic curves forms over the finite field of order twenty-five according to its inflexion points, and the incomplete curves have been extended to complete arcs of degree three. As a conclusion over F 25, the largest arc size of degree three constructed from the points of cubic curves is 36; that is, 36≤ m r (2,25) ≤51.