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 $6$.

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关于切割挡块及其代码

令PG $（r，q）$为有限域$ {\ rm GF}（q）$上的$ r $维投影空间。如果每个PG $（r，q）$的超平面$ \ Pi $的集合PG $（r，q）$的点的集合$ \ cal X $是切割阻塞集， $跨越$ \ Pi $。切割阻塞集会引起饱和集和最小的线性代码，并且具有尽可能小的尺寸的代码尤其令人关注。我们观察到，从Fancsali和Sziklai获得的切割阻塞集中，通过使用成对的不相交线的集合，出现了一个最小的线性代码，其长度相对于其尺寸线性增长。我们还提供了两种截然不同的结构：PG $（3，q ^ 3）$大小为$ 3（q + 1）（q ^ 2 + 1）$的切割阻塞集，是三个成对不相交的$ q $-阶的并集亚几何和PG $切割块集（5，来自PG $（5，q）$的Desarguesian线散布的7行的大小为$ 7（q + 1）$的q）$。在这两种情况下，获得的切削阻挡组均小于已知的切削阻挡组。作为副产品，我们进一步改善了某些饱和集的最小大小的上限，以及具有尺寸$ 4 $和$ 6 $的最小$ q $元线性代码的最小长度。