Let be the r-dimensional projective space over the finite field . A set of points of is a cutting blocking set if for each hyperplane Π of the set 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 , 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 of size as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of of size from seven lines of a Desarguesian line spread of . 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.
Funding statement: This work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM).
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