AL-Rafidain Journal of Computer Sciences and Mathematics

      A (k ,r)-arc is a set of k points of a projective plane PG(2,q) such that some  r,
but no r + 1 of them, are collinear. The  (k ,r)-arc is complete if it is not contained in
a (k + 1,r)-arc.
     In this paper we give geometrical construction of  complete (k _r ,r)-arcs in PG(2,7),
 r = 2,3,…, 7, and the related projective [n,3,d]₇ codes.
 

A k-arc in a plane PG(2,q)  is a set of k point such that every line in the plane intersect it in at most two points and there is a line intersect it in exactly two points. A k-arc is complete if there is no k+1-arc containing it. This thesis is concerned with studies a k-arcs, k=4,5,….,10 and classification of projectively distinct k-arcs and distinct arcs under collineation. We prove by using  computer program that the only complete k-arcs is for, k= 6,10. This work take (150) hours computer time .