On the Inverse Construction of (k,n)-Arcs in the Projective 3-Space over GF(7)

Authors:Aidan Essa Mustafa Sulaimaan

Abstract: In this paper, we investigate the inverse construction of complete arcs in the three-dimensional projective space  over the Galois field  The method is based on systematically deleting selected points from maximal arcs of order m, where  and , with arc sizes restricted by . Using this approach, we construct a full hierarchy of complete arcs, ranging from the maximal case (400, 57) down to the minimal configuration. Furthermore, a geometric proof is provided to show that the smallest possible complete (k,n)-arc in  is uniquely realized as a arc. The results extend the known classifications of arcs in finite projective spaces and offer a systematic framework for their inverse construction and analysis.

Keywords: projective geometry; finite fields; (k,n)-arcs; maximal arcs; inverse construction