A complete (48, 4)­arc in the projective plane over the field of order seventeen

: The article describes a certain computation method of (𝑘, 𝑛) -arcs to construct the number of distinct (𝑘, 4) -arcs in PG(2,17) for 𝑘 = 7, … ,48 . In this method, a new approach employed to compute the number of (𝑘, 𝑛) -arcs and the number of distinct (𝑘, 𝑛)– arcs respectively. This approach is based on choosing the number of inequivalent classes {𝜁 4 , 𝜁 3 , 𝜁 2 , 𝜁 1 , 𝜁 0 } of 𝑖 -secant distributions that is the number of 4-secant, 3-secant, 2-secant, 1-secant and 0-secant in each process. The maximum size of (𝑘, 4) -arc that has been constructed by this method is 𝑘 = 48 . The new method is a new tool to deal with the programming difficulties that sometimes may lead to programming problems represented by the increasing number of arcs. It is essential to reduce the established number of (𝑘, 𝑛) -arcs in each construction especially for large value of 𝑘 and then reduce the running time of the calculation. Therefore, it allows to decrease the memory storage for the calculation processes. This method’s effectiveness evaluation is confirmed by the results of the calculation where a largest size of complete (𝑘, 4) -arc is constructed. This research’s calculation results develop the strategy of the computational approaches to investigate big sizes of (𝑘, 𝑛)– arcs in 𝑃𝐺(2, 𝑞) where it put more attention to the study of the number of the inequivalent classes of 𝑖 -secants of (𝑘, 𝑛) -arcs 𝐾 𝑖 in 𝑃𝐺(2, 𝑞) which is an interesting aspect. Consequently, it can be used to establish a large value of 𝑘 .

In 1947, the notions of arcs and also complete arcs were presented.
In the projective plane (2, ) over the finite field F q , a (k,n)-arc is a set of k points such that some n are collinear but no n + 1 are collinear. Al so, a (k, n)-arc is complete if it is not subset of (k + 1, n)-arc. Moreover, a line contains two points of an arc is called a secant. The study of finite projective planes become one of the most essential problems in combinatorics and several associated areas. For instance, coding theory. In addition, many attempts have been made to determine the size of the maximum ( , )-arc in (2, ) to establish a good upper bound. For a more detailed to the sizes of (k, n)-arcs (1)(2)(3)(4)(5)(6)(7).
The main gaols in this research are to construct the number of ( , 4)-arcs, to classify the number of distinct ( , 4)-arcs in (2,17), to compute the stabiliser group for each distinct arc, and to discuss the group action of the most interesting stabiliser group on an associated arc. In this area of study ( , )-arcs and for n>2, many researchers have followed the method of constructing a particular ( , )-arc that contains many points. However, the technique used in this research is a classification method that classifies numbers of ( , )-arcs in (2, ). The spectrum of (k,4)-arcs has been calculated using a new technique which is based on the number of inequivalent classes of secant distribution in each construction process.
The start of this technique is fixing a set of six points = { 0 , 1 , 2 , , 1 , 2 }. Here, the points 0 , 1 , 2 , are the frame points and 1 , 2 are any two collinear points from the plane. (2,17), implies that the set containing four collinear points. For example, the first step begins from the set . Then, the second step is to construct the number of (7,4)-arcs. This construction is made by adding separately to all the points from the lines of (2, 17), = 1,…,307 that satisfy ∩ = 4. Then the -secant distribution for each arc is calculated and then among these classes, the distinct classes of -secant distribution with their associated arcs are chosen only. So, the number of distinct (7,4)-arcs is based on the number of distinct classes of secant distribution. All the details of this method are illustrated in section of research method.
This technique is a systematic method. It is used due to the increasing number of arcs especially for large value of . The maximum size of a (k, 4)-arc in (2,17) established by this method is k = 48. Also, a ( , 4)-arc of size 48 was also found in (8). This arc is found by prescribing the group generated and it has the same secant distribution of our arc. However, it has different stabiliser group. In conclusion, a new method established a large systematically based on the concept of the -secant distribution of arc. Also, it reduced the programming problems in terms of the constructed number of arcs, the executed time, and the memory storage. The programming language that used in this research is Gap (9). So, the problems of finding the number of ( , )-arcs and the number of distinct ( , )-arcs in (2, ) are still interesting problems (10). Thus, in the study of projective planes and for a certain value of q, some questions arise . (i) For given k and n, what is the number of projectively distinct (k, n)-arcs in (2, )? (ii)What is the maximum size m n (2, q) of a complete (k, n)-arc in (2, ) for given values of n and q?
The projective plane ( , ) The projective plane (2, ) is an incident structure of points and lines over the Galois field F q . It contains q 2 + q + 1 points and lines. Therefore, the line contains +1 points and there are +1 lines concurrent with a unique point. So, the number of points and lines in the plane (2,17) is 307, with 18 points on each line and 18 lines concurrent with a point as given in Tables 1 and 2. (11)(12)(13).   (14).  (14,15). Lemma (16) The following equations hold for a ( , )-arc:

Finite group:
A group is a finite group if it consists a finite number of elements (17).

Research aim:
The aim of this research is as follows: (1) What is the number of ( ,4)-arcs in (2,17)?.
(2) What is the number of distinct ( ,4)-arcs in (2,17)?. (3) What is the largest size of complete ( ,4)-arc in (2,17) can be obtained by the suggested method?. The answers to these questions are given in Tables  3-23.

Research Method:
The approach that has been used to classify the sets of ( ,4)-arcs in (2,17) is based on fixing a set of six points, that is = { 0 , 1 , 2 , , 1  In the same way that the set 0 was constructed. Iterate this method to construct the ( , 4)-arcs for >8. So, we can summarize the above method in the following steps.

Results:
From the review above, we describe the results of the classification of ( ,4)-arcs in (2,17) in the following subsections. These subsections summarises the findings and contributions made in tables.

Conclusion and suggestion:
In this research, the number of ( ,4)-arcs, the number of distinct ( ,4)-arcs, and the large size of ( ,4)-arc are discussed. These problems had an approach that addresses programming difficulties in terms of storing data and reducing the increasing numbers of arcs as well as the time of implementation. Therefore, The approach of thesecant distributions is considered in order to assure that the best result would be satisfied. From the previous calculation the following results are obtained : 1-The number of ( ,4)-arcs in (2,17) is classified.
In addition, the stabiliser group of this arc is computed. This group is GL (2, 3) ⋊ 2 . Also, the action of this group on the complete (48,4)-arc is described. The concept of secant distributions is a useful and substantial research which recommend to construct a large . But, it still needs to improve especially when the parameters , , and are large. For further study, super computers are needed.