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 ≥ 5 and m 6= 9, the largest d of a non-Denniston maximal arc of degree 2 in PG(2, 2) generated by a {p, 1}-map is ( ⌊ m 2 ⌋ + 1). This confirms our conjecture in [FLX]. For {p, q}-maps, we prove that if m ≥ 7 and m 6= 9, then the largest d of a non-Denniston maximal arc of degree 2 in PG(2, 2) generated by a {p, q}-map is either ⌊ m 2 ⌋ + 1 or ⌊ m 2 ⌋ + 2.