Binary caps with many free pairs of points.

Journal of Combinatorial Designs 14 (2006), 490-499.

We introduce the following new viewpoint in the study of caps in PG(m,q). The objective is to maximize, among all caps of given cardinality in a given projective space, the number of free pairs of points, which we define as pairs of points not participating in any coplanar quadruple of points of the cap. We survey the known results (which were motivated by an application in the statistical experiment design) and then we improve the known lower bound on the number of free pairs of points for q=2 and the smallest cap sizes for which the maximization problem is nontrivial.