On cutting blocking sets and their codes

Abstract

AbstractLetPG⁡(r,q){\operatorname{PG}(r,q)}be ther-dimensional projective space over the finite fieldGF⁡(q){\operatorname{GF}(q)}. A set𝒳{\mathcal{X}}of points ofPG⁡(r,q){\operatorname{PG}(r,q)}is a cutting blocking set if for each hyperplane Π ofPG⁡(r,q){\operatorname{PG}(r,q)}the setΠ∩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 ofPG⁡(3,q3){\operatorname{PG}(3,q^{3})}of size3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)}as a union of three pairwise disjointq-order subgeometries, and a cutting blocking set ofPG⁡(5,q){\operatorname{PG}(5,q)}of size7⁢(q+1){7(q+1)}from seven lines of a Desarguesian line spread ofPG⁡(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 minimalq-ary linear code having dimension 4 and 6.