In this work, we construct complete (K, n)-arcs in the projective plane over Galois field GF (11), where 12 2 ≤ ≤ n  ,by using geometrical method (using the union of some maximum(k,2)- Arcs , we found (12,2)-arc, (19,3)-arc , (29,4)-arc, (38,5)-arc , (47,6)-arc, (58,7)-arc, (68,6)-arc, (81,9)-arc, (96,10)-arc, (109,11)-arc, (133,12)-arc, all of them are complete arc in PG(2, 11) over GF(11).  

  In this work, we construct and classify the projectively distinct (k,3)-arcs in PG(2,9), where k ≥ 5, and prove that the complete (k,3)-arcs do not exist, where 5 ≤ k ≤ 13. We found that the maximum complete (k,3)-arc in PG(2,q) is the (16,3)-arc and the minimum complete (k,3)-arc in PG(2,q) is the (14,3)-arc. Moreover, we found the complete (k,3)-arcs between them.

  The purpose of this work is to study the classification and construction of (k,3)-arcs in the projective plane PG(2,7). We found that there are two (5,3)-arcs, four (6,3)-arcs, six (7,3)arcs, six (8,3)-arcs, seven (9,3)-arcs, six (10,3)-arcs and six (11,3)-arcs.         All of these arcs are incomplete.         The number of distinct (12,3)-arcs are six, two of them are complete.         There are four distinct (13,3)-arcs, two of them are complete and one (14,3)-arc which is incomplete.         There exists one complete (15,3)-arc.
 

  The main purpose of this work is to find the complete arcs in the projective 3-space over Galois field GF(2), which is denoted by PG(3,2), by two methods and then we compare between the two methods

A (k,n)-arc A in a finite projective plane PG(2,q) over Galois field GF(q), q=pⁿ for same prime number p and some integer n≥2, is a set of k points, no n+1 of which are collinear.  A (k,n)-arc is complete if it is not contained in a(k+1,n)-arc.  In this paper, the maximum complete (k,n)-arcs, n=2,3 in PG(2,4) can be constructed from the equation of the conic.

   In this paper,we construct complete (kn,n)-arcs in the projective plane PG(2,11),  n = 2,3,…,10,11  by geometric method, with the related blocking sets and projective codes.
 

   The aim of this paper is to construct the (k,r)-caps in the projective 3-space PG(3,p) over Galois field GF(4). We found that the maximum complete (k,2)-cap which is called an                       ovaloid  , exists in PG(3,4) when k = 13. Moreover the maximum (k,3)-caps, (k,4)-caps and   (k,5)-caps. 

       In this thesis, some sets of subspaces of projective plane PG(2,q) over Galois field GF(q) and the relations between them by some theorems and examples can be shown.
 

A (k,n)-arc is a set of k points of PG(2,q) for some n, but not n + 1 of them, are collinear. A (k,n)-arc is complete if it is not contained in a (k + 1,n)-arc. In this paper we construct complete (kn,n)-arcs in PG(2,5), n = 2,3,4,5, by geometric method, with the related blocking sets and projective codes.

        The aim of this paper is to introduce the definition of projective 3-space over Galois field GF(q), q = p^m, for some prime number p and some integer m.

        Also the definitions of (k,n)-arcs, complete arcs, n-secants, the index of the point and the projectively equivalent arcs are given.

        Moreover some theorems about these notations are proved.