Minimal 1-saturating sets and complete caps in binary projective spaces

In binary projective spaces PG ( v , 2 ) , minimal 1-saturating sets, including sets with inner lines and complete caps, are considered. A number of constructions of the minimal 1-saturating sets are described. They give infinite families of sets with inner lines and complete caps in spaces with increasing dimension. Some constructions produce sets with an interesting symmetrical structure connected with inner lines, polygons, and orbits of stabilizer groups. As an example we note an 11-set in PG ( 4 , 2 ) called “Pentagon with center”. The complete classification of minimal 1-saturating sets in small geometries is obtained by computer and is connected with the constructions described.