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

Year 2025, Volume: 26 Issue: 4, 478 - 485, 25.12.2025

Elif Altıntaş Kahriman , Ayşe Bayar

https://doi.org/10.18038/estubtda.1754684

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.

Keywords

Projective plane , (k , 3)-arc , Complete quadrangle , Finite geometry , PG(2 , 4)

References

• [1] Hirschfeld JWP, Thas JA. General Galois Geometries. Springer Monographs in Mathematics. Springer- Verlag London, 2016.
• [2] Bayar A, Akca Z, Altintas E, Ekmekci S. On the complete arcs containing the quadrangles constructing the Fano planes of the left near field plane of order 9. New Trend Math. Sci 2016; 4(4), 266-266. http://dx.doi.org/10.20852/ntmsci.2016.113
• [3] Ekmekci S, Bayar A, Altintas E, Akca Z. On the Complete (k,2)- Arcs of the Hall Plane of Order 9. International Journal of Advanced Research in Computer Science and Software Engineering, 2016; 6 (10), 282-288. ISSN: 2277 128X.
• [4] Altıntaş Kahriman E, Bayar A. Investigating Incomplete (7,3)-Arcs and Their Extensions in PG(2,5): A Study On Secants And Complete Quadrangles. In: 5. Bilsel International World Scientific And Research Congress; 05-06 October 2024; İstanbul, Türkiye, 536-545.
• [5] Altıntaş Kahriman E, Bayar A. An Algorithm for Constructing (k,2)-Arcs Containing Triangle and Quadrangle in PG(2,4). In: 5. Bilsel International Gordion Scientific Researches Congress; 08-09 December, 2024; Ankara/ Türkiye, 987-997.
• [6] Coxeter HSM. Projective geometry. Springer, pp. 7; 95, 1987.
• [7] Hirschfeld JWP, Voloch J.F. Group-arcs of prime order on cubic curves, Finite Geometry and Combinatorics 2015; 191, 177-185.
• [8] Hirschfeld JWP, Pichanick EVD. Bounded for arcs of arbitrary degree in finite Desarguesian Planes. J Comb Des 2016; 24(4), 184-196.
• [9] Ekmekci S, Bayar A, Akca Z. On The Projective Planes in Projective Space PG(4,4). Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 2022; 38(3), 519-524.