Complete arcs in aprojective plane overgalois field

Abstract EN

A (X, n) - arc in PG(2, p) is a set of X points no n +1 of which are collinear.

A (X,2) - arc is called X - arc which is a set of X points where no three of them are collinear.

A X - arc is complete if it is not contained in a (X +1) - arc .

The maximum number of points that a X - arc can have is (p + 1) for p odd or (p + 2) for p even.

and X - arc with this number of points is an oval.

Hirchfeld,1979 [4] showed the construction and classification of k-arcs over Galois field with p < 9, and Rania, 1997[7] gave the construction and classification of X - arc in QG(2,11) over G(11).

The aim of the present research is to find a way to add a point to a X - arc in a projective plane QG(2, p) Over Galois field G(p) with p is odd number so that it keeps X - arc subject to addition of more points until we get maximum complete arc which is an oval.

We have found that at the beginning with 4 - arc, we can then add any point of the index zero.

The choice of the fifth point determines the method of choosing the other points, because 4 - arc with the fifth point represent a conic.

In order that the sixth point is successfully chosen it must satisfies the conic