Small complete caps in \(PG(r,q)\), \(r\geq 3\) (Q1377807)

A \(k\)-cap in \(PG(r,q)\), is a set of \(k\) points such that no three of them are collinear and, a \(k\)-cap is complete if it is maximal with respect to set-theoretic inclusion. Determining the largest and the smallest size of a complete cap of \(PG(r,q)\) seems to be a very hard problem. The authors, after surveying the known results on the minimal size of a complete cap of \(PG(r,q)\), give some new examples of complete caps in \(PG(3,q)\) and \(PG(4,q)\), which are, for particular values of \(q\), smaller than the caps so far constructed.