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.