New sizes of complete (k, 4)-arcs in PG(2, 17)

In this paper, the packing problem for complete (k, 4)-arcs in PG(2, 17) is partially solved. The minimum and the maximum sizes of complete (k, 4)-arcs in PG(2, 17) 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 (k, 4)-arc K in PG(2, 17) and the algebraic characteristics of a plane quartic curve over the field F17 represented by the number of its rational points and inflexion points. In addition, some sizes of complete (k, 6)-arcs in the projective plane of order thirteen are established, namely for k = 53, 54, 55, 56.