  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 $m\geq 5$ and
$m\neq 9$, 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 $(\floor {m/2} +1)$. This confirms our
conjecture in [FLX]. For {p,q}-maps, we prove that if $m\geq 7$ and $m\neq 9$,
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 $\floor {m/2} +1$ or $\floor{m/2}
+2$.
