New maximal arcs in Desarguesian planes
A maximal \(\{k, n\}\)-arc \(\mathcal{K}\) in a finite projective plane of order \(q\) is a subset of \(k = n q + n - q\) points such that every line meets \(\mathcal{K}\) in \(0\) or \(n\) points. Any point of a projective plane of order \(q\) is a maximal \(\{1, 1\}\)-arc in that plane, and the complement of any line is a maximal \(\{q^2, q\}\)-arc, (such maximal arcs are called ``trivial''). The set \(\mathcal{K}^\prime\) of lines external to a maximal arc \(\mathcal{K}\) is a maximal \(\{k^\prime, n^\prime\}\)-arc in the dual plane, where \(k^\prime = n^\prime q + n^\prime - q\) and \(n^\prime =q/n\). \(\mathcal{K}^\prime\) is called the dual of \(\mathcal{K}\). So a necessary condition for the existence of maximal arcs is that \(n\) divides \(q\). In [J. Comb. Theory 6, 317-319 (1969; Zbl 0167.49106)] \textit{R. H. F. Denniston} proved that in a Desarguesian projective plane \(PG(2, q)\) of order \(q\) this necessary condition is sufficient when \(q\) is odd. In 1998 [cf. Combinatorica 17, 31-41 (1997; Zbl 0880.51003)] \textit{S. Ball, A. Blokhuis} and \textit{F. Mazzocca} have shown that no maximal arc exists in \(PG(2, q)\) if \(q\) is odd. Maximal arcs give rise to many interesting incidence structures such as partial geometries, Steiner \(2\)-designs and linear codes. Also they can be used to construct \(2\)-weight codes. There are three basic general constructions for maximal arcs in \(PG(2, q)\), which give rise to three classes of maximal arcs: the maximal arcs of Denniston which are constructed using special pencils of conics in \(PG(2, q)\), \(q\) even, and two classes due to J. A. Thas which are constructed by using ovoids and spread in generalized quadrangle \(\mathcal{W}(q)\), or quadrics and spread in projective space, respectively. In the paper under review, new constructions are described of maximal arcs in Desarguesian projective planes using set of conics on a common nucleus in \(PG(2, q)\). Several new infinite families of maximal arcs in \(PG(2, q)\) are presented and a complete enumeration is carried out for \(PG(2, 16)\), \(PG(2, 32)\) and \(PG(2, 64)\). The author concludes with some conjectures and open problems.
scientific article; zbMATH DE number 1763788