Caps with free pairs of points (Q860225)

A set \(C \subset \text{PG}(N,q)\) of cardinality \(n\) is called an \(n\)-cap iff no three points of \(C\) are collinear. A subset \(\{x,y\} \subseteq C\) is a free pair of points if for each \(z \in C \setminus \{x,y\}\) the plane generated by \(x,y,z\) does not contain any other point of \(C\). The number \(m_{2}^+(N,q)\) denotes the maximum cardinality of a cap in PG\((N,q)\) that contains a free pair of points. It is known that \(m_{2}^+(N,q) \leq (q^{N-1}+2q-3)/(q-1)\). For \(q=2\) this bound is sharp for any \(N\) [see \textit{H. Chen} and \textit{A. S. Hedayat}, J. Stat. Plann. Inference 75, No. 1, 147--158 (1998; Zbl 0938.62081)]. In the paper under review it is shown that this bound is sharp for \(N \leq4\) and any prime power \(q\).