  Let PG$(r, q)$ be the $r$-dimensional projective space over the finite field
${\rm GF}(q)$. A set $\cal X$ of points of PG$(r, q)$ is a cutting blocking set
if for each hyperplane $\Pi$ of PG$(r, q)$ the set $\Pi \cap \cal X$ spans
$\Pi$. Cutting blocking sets give rise to saturating sets and minimal linear
codes and those having size as small as possible are of particular interest. We
observe that from a cutting blocking set obtained by Fancsali and Sziklai, by
using a set of pairwise disjoint lines, there arises a minimal linear code
whose length grows linearly with respect to its dimension. We also provide two
distinct constructions: a cutting blocking set of PG$(3, q^3)$ of size
$3(q+1)(q^2+1)$ as a union of three pairwise disjoint $q$-order subgeometries
and a cutting blocking set of PG$(5, q)$ of size $7(q+1)$ from seven lines of a
Desarguesian line spread of PG$(5, q)$. In both cases the cutting blocking sets
obtained are smaller than the known ones. As a byproduct we further improve on
the upper bound of the smallest size of certain saturating sets and on the
minimum length of a minimal $q$-ary linear code having dimension $4$ and $6$.
