Abstract
Let
PG
(
r
,
q
)
{\operatorname{PG}(r,q)}
be the r-dimensional projective space over the finite field
GF
(
q
)
{\operatorname{GF}(q)}
. A set
𝒳
{\mathcal{X}}
of points of
PG
(
r
,
q
)
{\operatorname{PG}(r,q)}
is a cutting blocking set if for each hyperplane Π of
PG
(
r
,
q
)
{\operatorname{PG}(r,q)}
the set
Π
∩
𝒳
{\Pi\cap\mathcal{X}}
spans Π. 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 in [20], 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
)
{\operatorname{PG}(3,q^{3})}
of size
3
(
q
+
1
)
(
q
2
+
1
)
{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
)
{\operatorname{PG}(5,q)}
of size
7
(
q
+
1
)
{7(q+1)}
from seven lines of a Desarguesian line spread of
PG
(
5
,
q
)
{\operatorname{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.
On cutting blocking sets and their codes
