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. 

The aim of t his p aper is t o const ruct t he (k,r)-caps in t he p rojective 3-sp ace PG(3,p ) over Galois field GF(4). We found t hat t he maximum comp let e (k,2)-cap which is called an ovaloid, exist s in PG(3,4) when k = 13. Moreover t he maximum (k,3)-cap s, (k,4)-cap s and (k,5)-caps.

In this work, we construct the projectively distinct (k, n)-arcs in PG (3, 4) over Galois field GF (4), where k 5, and we found that the complete (k, n)-arcs, where 3 n 21, moreover we prove geometrically that the maximum complete (k, n)-arc in PG (3, 4) is (85, 21)-arc. A (k, n)-arcs is a set of k points no n+ 1 of which are collinear. A (k, n)-arcs is complete if it is not contained in a (k+ 1, n)-arcs

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

        Also, the definition of the plane in PG(3,q) is given and state the principle of duality.

        Moreover some theorems in PG(3,q) are proved.

        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.

    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).  

        The purpose of this work is to construct complete (k,n)-arcs in the projective 2-space PG(2,q) over Galois field GF(11) by adding some points of index zero to complete (k,n–1)arcs 3 ï‚£ n ï‚£ 11.         A (k,n)-arcs is a set of k points no n + 1 of which are collinear.         A (k,n)-arcs is complete if it is not contained in a (k + 1,n)-arc

  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.
 

  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.