Complete Arcs in a Projective Plane Over Galois Field

Abstract

The research is on Complete Arcs in a Projective Plane Over Galois Field. A(k,n)-arc in PG(2,p) is a set of k points no n+1 of which are collinear. A(k, 2)- arc is called k-arc which is a set of k points where no three of them are collinear. A k-arc is complete if it is not contained in a (k-1)-arc. The maximum number of points that a k-arc can have is (p+1) odd or (p+2) for p even. And k-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 k-arc in PG(2,11) over G(11). The aim of this research is to find a way to add a point to a k-arc in a projective plane PG(2, p) over field G(p) with p is odd number so that keeps k-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 fifth represent a conic. In order that the sixth point is successfully chosen it must satisfy the conic equition. We have found   p+1 conics of which p-2 non-degenerated in PG(2, p), hence p-2 complete arcs (oval) through any 4-arc certainly there is only one conic which contains any 5-arc.