On Mathon's construction of maximal arcs in Desarguesian planes II

Abstract

In a recent paper, Mathon (J. Combin. Theory (A) 97 (2002) 353) gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when m⩾5 and m≠9, the largest d of a non-Denniston maximal arc of degree 2 d in PG(2,2 m ) generated by a { p,1}-map is ( m 2 +1) . This confirms our conjecture in (Fiedler et al. (Adv. Geom. (2003) (Suppl.) S119)). For { p, q}-maps, we prove that if m⩾7 and m≠9, then the largest d of a non-Denniston maximal arc of degree 2 d in PG(2,2 m ) generated by a { p, q}-map is either m 2 +1 or m 2 +2 .