Embeddings of Maximal Arc Type in Finite Projective Planes

Let \(D\) and \(\overline{D}^d\) be two designs such that there is a joint embedding \(D’\) and \(\overline{D}’\) of \(D\) and \(\overline{D}\) in a finite projective plane \(\pi\) of order \(n\) such that the points of \(D’\) and the lines of \(\overline{D}’\) are mutually all of the exterior elements of each other. We show that there is a tactical decomposition \(T\) of \(\pi\), two of the tactical configurations of which are \(D’\) and \(\overline{D}’\), and determine combinatorial restrictions on \(n\) and the parameters of \(D\) and \(\overline{D}^d\). We also determine the entries of the incidence matrices of \(T\).