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.
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.
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