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

摘要

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 greater than or equal to 5 and m not equal 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 ([m/2] + 1). This confirms our conjecture in (Fiedler et al. (Adv. Geom. (2003) (Suppl.) S119)). For {p, q}-maps, we prove that if m greater than or equal to 7 and m not equal 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 [m/2] + 1 or [m/2] + 2. (C) 2004 Elsevier Inc. All rights reserved.