On Mathon's Construction of Maximal Arcs in Desarguesian Planes

Date
2002
Journal Title
Journal ISSN
Volume Title
Publisher
Department of Mathematical Sciences
Abstract
We study the problem of determining the largest d of a non-Denniston max-imal arc of degree 2 d generated by a { p, 1 } -map in PG(2, 2 m ) via a recent construction of Mathon [M]. On one hand, we show that there are { p, 1 } -maps that generate non-Denniston maximal arcs of degree 2 m+1 2 , where m 5 is odd. Together with Mathon’s result [M] in the m even case, this shows that there are always { p, 1 } -maps generating non-Denniston maximal arcs of degree 2 b m+2 2 c in PG(2, 2 m ). On the other hand, we prove that the largest degree of a non-Denniston maximal arc in PG(2, 2 m ) constructed using a { p, 1 } -map is less than or equal to 2 m − 3 . We conjecture that this largest degree is actually 2 b m+2 2 c .
Description
Keywords
arc, linearized polynomial, maximal arc, quadratic form
Citation