On cutting blocking sets and their codes

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 $6$.