Linear codes with covering radius 2, 3 and saturating sets in projective geometry

Infinite families of linear codes with covering radius R=2, 3 and codimension tR+1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.