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

In a recent paper [M], Mathon 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 $m\ge 5$ and $m\mathrm{\ne }9$, 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 $(\text{floor}m\mathrm{/}2+1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $m\ge 7$ and $m\mathrm{\ne }9$, 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 $\text{floor}m\mathrm{/}2+1$ or $\floor{m/2} +2$.