On the Incompleteness of (k, n)-arcs in Desarguesian Planes of Order q Where n Divides q

We investigate the completeness of an ( nq − q + n − ε, n)-arc in the Desarguesian plane of order q where n divides q. It is shown that such arcs are incomplete for 0< ε ≤ n/2 if q/n3. For q = 2n they are incomplete for 0 < ε < 0.381n and for q = 3n they are incomplete for 0 < ε < 0.476n. For q odd it is known that such arcs do not exist for ε = 0 and, hence, we improve the upper bound on the maximum size of such a ( k, n)-arc.