Small complete caps in PG(r,q), r ⩾ 3 Original Research Article

A very difficult problem for complete caps in PG(r,q) is to determine their minimum size. The results on this topic are still scarce and in this paper we survey the best results now known. Furthermore, we construct new interesting sporadic examples of complete caps in PG(3,q) and in PG(4,q) such that their size are smaller than the currently known. As a consequence, we get that the Pellegrinoʹs conjecture on the minimal size of a complete k-cap in PG(3,q), q odd, is in general false.