On the spectrum of non-Denniston maximal arcs in PG(2, 2\(^{h}\)).
A maximal arc of degree \(n\) of a finite projective plane \(P\) of order \(q\) is a set \(S\) of points such that every line of \(P\) meets \(S\) in either \(0\) or \(n\) points. For \(q\) even, Mathon introduced the notion of closed set of conics and proved that if \({\mathcal C}\) is a set of closed conics, then the union of the point sets of those conics together with their common nucleus form a maximal arc of degree \(| {\mathcal C}| + 1\). In the paper under review Hamilton and Mathon provide the following construction of closed sets of conics and hence of maximal arcs. Theorem. Let \(P = PG(2, q)\) be a finite projective plane of order \(q = 2^{2m}\) or \(q = 2^{2m+1}\) with \(m \geq 2\). Then there exist \(b_0, b_1, b_2 \in GF(q)\), \(b_2 \neq 0\) and \(A \subseteq GF(q)\) such that the set \(\{ x^2 + xy + (b_0 + b_1 \lambda + b_2 \lambda^3) y^2 + \lambda z^2 = 0 \mid \lambda \in A \}\) is a closed set of conics giving rise to a maximal arc of degree \(2^{m+1}\). Hamilton and Mathon also show that the above theorem implies that in \(PG(2, 2^h)\), \(h \geq 4\) there exist maximal arcs of degree \(n\) that are not of Denniston type for all \(n\) dividing \(q\) with the possible exceptions of \(n = 4\) or \(n = q/4\). Finally Hamilton and Mathon investigate the number of isomorphism classes of the so-obtained maximal arcs. Their result reads as follows: Theorem. In \(PG(2, q)\), \(q = 2^{2m+1}\), the number of isomorphism classes of maximal arcs of degree \(\sqrt{2q}\) that are not of Denniston type is bounded below by \(2 q^{m/4-10}\). In \(PG(2, q)\), \(q = 2^{2m}\), the number of isomorphism classes of maximal arcs of degree \(2 \sqrt{q}\) that are not of Denniston type is bounded below by \(4 q^{m/4-10}\).
scientific article; zbMATH DE number 2055710
scientific article