On the non-existence of Thas maximal arcs in odd order projective planes
In a projective plane of order \(q\) a maximal arc of degree \(m\), \(2 \leq m\leq q\), is a set of points such that every line meets the set in 0 or \(m\) points. One of the most exciting results in recent finite geometry is arguably the solution to the maximal arcs problem: maximal arcs do not exist in Desarguesian planes of odd order [\textit{S. Ball, A. Blokhuis} and \textit{F. Mazzocca}, Combinatorica 17, No. 1, 31-41 (1997; Zbl 0880.51003)]. For non-Desarguesian planes far less is known on the existence of maximal arcs. For certain translation planes there is a construction of maximal arcs due to Thas, which yields maximal arcs for fields of even characteristic [\textit{J. A. Thas}, Eur. J. Comb. 1, 189-192 (1980; Zbl 0449.51011)]. In the current paper it is shown that the Thas construction fails to produce maximal arcs in the case of fields of odd characteristic. As an extra it is shown that the all one vector is not contained in the binary code spanned by the incidence vectors of the tangents to an elliptic quadric in \(PG(3,q)\), with odd \(q\). The proofs to the theorems are elegant, although sometimes lacking details which could make the reading even more pleasant.
scientific article; zbMATH DE number 1199921