Some sets of type \((m,n)\) in cubic order planes (Q1299894)

A set of type \((m,n)\) is a subset \(S\) of the points of a projective plane if every line meets \(S\) in exactly \(m\) or \(n\) points. The ``standard'' such sets in planes of order \(q\) are of order \((k,k+ \sqrt q)\) for some positive integer \(k\). The authors construct a (4,9) set in the plane of order \(5^3\) and a (4,11) set in the plane of order \(7^3\). The main tool consists of Singer cycles.