On algebraic characteristics of nonequivalent degree three arcs in PG(2,17)

Abstract

In this paper, the algebraic characteristics of nonequivalent arcs of degree three in  PG(2,17) are discussed. An approach to construct these sets is introduced. The approach focuses on obtaining a large size of complete degree three arc in PG(2,17). This work starts by fixing a set, X= {P_1,P_2,P_3} that have three points lie on the same line Ɫ_x  of the projective space of order seventeen, PG(2,17). This set is a (3;3)-Arc. Then, the process is continued to establish the sets of (ƙ;3)-Arcs by adding the points of the projective plane, PG(2,17) that satisfied the condition X ∩ lines Ɫ_i\ Ɫ_x  = 3, where Ɫ_i∈PG(2,17). So, the sets of (4;3)-Arcs, (5;3)-Arcs, (6;3)-Arcs, (7;3)-Arcs, (8;3)-Arcs, (9;3)-Arcs, (10;3)-Arcs, (11;3)-Arcs, (12;3)-Arcs, (13;3)-arcs, (14;3)-Arcs, (15;3)-Arcs, (16;3)-Arcs, (17;3)-Arcs, (18;3)-Arcs, (19;3)-Arcs, (20;3)-Arcs, (21;3)-Arcs, (22;3)-Arcs, (23;3)-Arcs, (24;3)-Arcs, (25;3)-Arcs, (26;3)-Arcs, (27;3)-Arcs, and (28;3)-Arcs are obtained. So that this approach gives the number of (ƙ;3)-Arcs in each construction for ƙ= 4,5,6,7,8,…28, and then the number of nonequivalent (ƙ;3)-Arcs for ƙ= 4,5,6,7,8,…,28 is given as well. This number is established according to the number of nonequivalent secant distributions of degree three arcs, (ƙ;3)-Arcs. Thus, the spectrum of nonequivalent arcs in each process is 2, 6, 16, 32, 49, 71, 97, 122, 149, 170, 192, 205, 220, 230, 233, 234, 229, 218, 190, 160, 101, 34, 4, 3, 1, respectively. Also, the associated stabilizer group for each constructed nonequivalent arc is computed. In addition, the action of each stabilizer group on the corresponding nonequivalent arc is discussed. As a result of these actions, there are different sizes of orbits. These sizes are one, two, three, four, and six. The largest size of degree three arc established in this process is  ƙ=28 .