COMPLETE (k,3)-ARCS CONTAINING COMPLETE QUADRANGLE VERTICES IN PG(2,4): STRUCTURAL ANALYSIS BASED ON DIAGONAL POINT INCLUSION

Abstract

In this study, complete (k,3)-arcs in the projective plane PG(2,4) that include all vertices of a complete quadrangle are systematically analyzed with respect to the inclusion of diagonal points. A computational algorithm developed in C# was employed to construct and classify such arcs. The results show that the complete quadrangle itself forms a complete (7,3)-arcs, and exactly 7560 such arcs exist depending on the choice of quadrangle point sets. Furthermore, three distinct types of complete (9,3)-arcs were identified: 24 arcs containing all four vertices and two diagonals, 48 arcs containing all four vertices and one diagonal point, and 480 arcs with all four vertices and no diagonal points. These findings reveal the combinatorial diversity of arc configurations in finite projective planes and provide new contributions to the classification of arcs. The methodology also demonstrates the effectiveness of algorithmic approaches in investigating geometric structures over finite fields.