Large caps with free pairs in dimensions five and six

Abstract

A cap in PG$\left(N,q\right)$ is said to have a free pair of points if any plane containing that pair contains at most one other point from the cap. In an earlier paper we determined the largest size of caps with free pairs for $N=3$ and $4$. In this paper we use product constructions to prove similar results in dimensions $5$ and $6$ that are asymptotically as large as possible. If $q>2$ is even, we determine exactly the largest size of a cap in PG$\left(5,q\right)$ with a free pair. In PG$\left(5,3\right)$ we give constructions of a maximal size $42$-cap having a free pair and of the complete $48$-cap that contains it. Additionally, we give some sporadic examples in higher dimensions.