On Mathonʹs construction of maximal arcs in Desarguesian planes II

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 2d in PG(2,2m) generated by a {p,1}-map is (m2+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 2d in PG(2,2m) generated by a {p,q}-map is either m2+1 or m2+2.