The main aim of this paper is to introduce the relationship between the topic of coding theory and the projective plane of order three. The maximum value of size of code over finite field of order three and an incidence matrix with the parameters,  (length of code),  (minimum distance of code) and  (error-correcting of code ) have been constructed. Some examples and theorems have been given.

              In this paper, the packing problem for complete (  4)-arcs in  is partially solved. The minimum and the maximum sizes of complete (  4)-arcs in  are obtained. The idea that has been used to do this classification is based on using the algorithm introduced in Section 3 in this paper. Also, this paper establishes the connection between the projective geometry in terms of a complete ( , 4)-arc in  and the algebraic characteristics of a plane quartic curve over the field  represented by the number of its rational points and inflexion points. In addition, some sizes of complete (  6)-arcs in the projective plane of order thirteen are established, namely for  = 53, 54, 55, 56.

The goal of this paper is to construct an arcs of size five and six with stabilizer groups of type alternating group of degree five and degree six . Also construct an arc of degree five and size with its stabilizer group, and then study the effect of and on the points of projective plane. Also, find a pentastigm which has the points on a line. Partitions on projective plane of order sixteen into subplanes and arcs have been described.

This research is concerned with the study of the projective plane over a finite field . The main purpose is finding partitions of the projective line PG( ) and the projective plane PG( ) , in addition to embedding PG(1, ) into PG( ) and PG( ) into PG( ). Clearly, the orbits of PG( ) are found, along with the cross-ratio for each orbit. As for PG( ), 13 partitions were found on PG( ) each partition being classified in terms of the degree of its arc, length, its own code, as well as its error correcting. The last main aim is to classify the group actions on PG( ).

The main aims of this research is to find the stabilizer groups of a cubic curves over a finite field of order , studying the properties of their groups and then constructing the arcs of degree  which are embedding in a cubic curves of even size which are considering as the arcs of degree . Also drawing all these arcs.

  The purpose of this paper is  to find an arc of degree five in 31 ,29),(2, =qqPG , with stabilizer group of type dihedral group of degree five 5 D and arcs of degree six and ten with stabilizer groups of type alternating group of degree five 5 A ,  then study the effect of  5 D and 5A on the points of projective plane. Also, find a pentastigm which has collinear diagonal points.

       The basis of this paper is to study the concept of almost projective semimodules as a generalization of projective semimodules. Some of its characteristics have been discussed, as well as some results have been generalized from projective semimodules.

Suppose that  is a finite group and  is a non-empty subset of  such that  and . Suppose that  is the Cayley graph whose vertices are all elements of  and two vertices  and  are adjacent if and only if . In this paper, we introduce the generalized Cayley graph denoted by  that  is a graph with vertex set consists of all column matrices  which all components are in  and two vertices  and  are adjacent if and only if , where  is a column matrix that each entry is the inverse of similar entry of  and  is  matrix with all entries in  ,  is the transpose of  and . In this paper, we clarify some basic properties of the new graph and assign the structure of  when  is complete graph , complete bipartite graph  and complete

The aim of this paper is to construct cyclic subgroups of the projective general linear group over  from the companion matrix, and then form caps of various degrees in . Geometric properties of these caps as secant distributions and index distributions are given and determined if they are complete. Also, partitioned of  into disjoint lines is discussed.

In modules there is a relation between supplemented and Ï€-projective semimodules. This relation was introduced, explained and investigated by many authors. This research will firstly introduce a concept of "supplement subsemimodule" analogues to the case in modules: a subsemimodule Y of a semimodule W is said to be supplement of  a subsemimodule X if it is minimal with the property X+Y=W. A subsemimodule Y is called a supplement subsemimodule if it is a supplement of some subsemimodule of W. Then, the concept of supplemented semimodule will be defined as follows: an S-semimodule W is said to be supplemented if every subsemimodule of W is a supplemen