Classification of (k;4)-arcs up to projective inequivalence, for k < 10

In this paper, the classification of (k;4)-arcs up to projective inequivalence for k < 10 in PG(2,13) is introduced in details according to their inequivalent number, stabilisers, the action of each stabiliser on the associated arc, and the inequivalent classes N c of secant distributions of arcs. Here, the strategy is to start from the projective line PG(1,13) where there are three projectively inequivalent tetrads.


Finite groups
Definition .1 A group is an ordered pair (G, * ), where G is a non-empty set and * is a binary operation on G such that the following properties hold.
(2) There exists e ∈ G such that for all a ∈ G, a * e = a = e * a.
(3) For all a ∈ G, there exists b ∈ G such that a * b = e = b * a.

Group action on a set
Let G be a group acts on a set X if for each g ∈ G and x ∈ X an element gx ∈ X is defined, such that g 2 (g 1 x) = (g 2 g 1 )x and ex = x for all x ∈ X, g 1 , g 2 ∈ G.

The set
Orb is called the orbit of the element x. The stabilizer of an element x of X is the subgroup The fixed points set of an element g of G is the set defined as follows: Fix(g) = {x ∈ X | gx = x}.

The projective plane PG(2, q)
The projective plane PG(2, q) over F q contains q 2 + q + 1 points and lines. There are q + 1 points on each line and q + 1 lines passing through each point. The value of q that has been used in this work is q = 13. Therefore the projective plane PG(2, 13) has 183 points and lines, with 14 points on each line and 14 lines passing through each point. The point P(x 0 , x 1 , x 2 ) in the projective plane, PG(2, q), can be represented as a vector of three coordinates over F q as shown in Table 1.  P (x 0 , 1, 0) q P(1, 0, 0) 1 A line in PG(2, q) is a set of points P(x 0 , x 1 , x 2 ) satisfying the homogeneous linear equation with a, b, c ∈ F q not all zero; it is denoted by L(a, b, c). Thus, a projective plane is an incidence structure of points and lines with the following properties: (i) every two points are incident with a unique line; (ii) every two lines are incident with a unique point; (iii) there are four points, no three collinear.

General linear group of a vector space
Let F q is a finite field and let V (n, q) is a vector space of dimension n over F q , then the linear map V (n, q) −→ V (n, q), such that x −→ xA, for x ∈ V a row vector and A a non-singular n × n matrix over F q . The group consisting of all linear maps of V (n, q), that is, the group consisting of all non-singular n × n matrices over F q , is called the general linear group and is denoted by GL(n, q). The order of GL(n, q) is as follows: In addition, the subgroup SL(n, q) consisting of all matrices with determinant 1, and it is called the special linear group of degree n over F q . The group SL(n, q) contains a subgroup UT(n, q) consisting of those matrices with all entries below the main diagonal zero, and with the entries on the main diagonal equal to the identity. This subgroup is called the unitriangular group of degree n over F q .

The fundamental theorem in PG(2, q)
If φ : P −→ P is a bijective mapping from one projective plane, PG (2, q), to another, then there is a unique projectivity shifting any quadrangle, that is, a set of four points no three collinear, to another quadrangle.

Definition .2
A (k; n)-arc K in PG(2, q) is a set of k points such that no n + 1 of them are collinear but some n are collinear.

Projectively inequivalent (4; 4)-arcs
In this classification, the number of tetrads is constructed by fixing a triad, U 1 = {1, 2, 9}. There are eleven tetrads containing U 1 . The lexicographically least sets in the G-orbits of tetrads, where G = P GL(2, 13) took 2104 msec. Then among these canonical sets there are three projectively inequivalent tetrads; this took 1699 msec. Also, the three tetrads have sd-equivalent secant distribution. It took 1734. The statistics are shown in Table 2.

Remark
In Table 7, there are 11 types of the stabiliser groups as follows: These stabiliser groups of order at least two divide their corresponding projectively inequivalent (7; 4)-arcs into a number of orbits. All orbits of these groups are listed in Table 9.
The number of these groups is listed in Table 10. Also, the 7399 projectively inequivalent (8; 4)-arcs have eleven sd-inequivalent classes of secant distributions as shown in Table 11.
Note that the groups of order at least eight are as follows: These groups partition the associated projectively inequivalent (8; 4)-arcs into a number of orbits as shown below. (1

Projectively inequivalent (9; 4)-arcs
In PG(2, 13), the number of projectively inequivalent (9; 4)-arcs is 222536 according to the inequivalent lexicographically least set in the G-orbit of each (9; 4)-arc. These arcs have one of the groups 4 . In addition, the secant distribution of each of the 222536 projectively inequivalent arcs is calculated. There are 21 sd-inequivalent classes of secant distributions of the projectively inequivalent (9; 4)-arcs. The statistics are given in Tables  12, 13, and 14.

Remark
In