Cubic Curves Over the Finite Field of Order Twenty Seven

Abstract

A cubic curve is a non-singular projective plane cubic curve. An (k; 3)-arc is a set of points no four are collinear but some three are linear. Most of the cubic curve can be regarded as an arc of degree three. In this paper, the projectively inequivalent cubic curves have been classified over the finite field of order twenty-seven with respect to its inflexions points and determined if they are complete or incomplete as arcs of degree three. Also the size of the largest arc of degree three that can be constructed form each incomplete cubic curve are given. The main conclusion that can be drawn is that, over F 27, the largest an arc of degree three can be constructed depending on the cubic curve is 38; that is, 38 ≤ m 3(2,27) ≤ 55.