Large caps with free pairs in dimensions five and six (Q2455070)

In the \(N\)-dimensional projective space \(PG(N,q)\), a cap \(K\) is a point set containing at most \(2\) points from each line. Two points in \(K\) such that any plane containing them contains at most one other point from the cap are said to form a \textit{free pair of points} of \(K\). Caps with a free pair are of interest in the design of experiments in statistics, and therefore it is natural to seek the maximum size of such caps in a given projective space \(PG(N,q)\). In an earlier work [J. Geom. 85, No. 1--2, 35--41 (2006; Zbl 1110.51008)], the authors proved that for the size \(k\) of a cap with free pairs in \(PG(N,q)\), \[ k\leq q^{N-2}+q^{N-3}+\cdots+q+3 \tag{1} \] holds. They also showed the sharpness of (1) for \(N=3\) and \(N=4\). In the paper under review, some results in dimensions \(5\) and \(6\) are obtained. For \(N=5\), (1) is sharp provided that either \(q\) is even or \(q=3\), whereas for \(q>3\) odd (1) can be missed by at most \(q+1\). For \(N=6\), caps with free pairs of size \(q^4+q^2+q+2\) are constructed.