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

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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≠ 9, the largest d of a non-Denniston maximal arc of degree 2ᵈ in PG(2,2ᵐ) 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≠ 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 m/2 +1 or m/2 +2.