On cutting blocking sets and their codes

Let PG(r, q) be the r-dimensional projective space over the finite field GF(q). A set X of points of PG(r, q) is a cutting blocking set if for each hyperplane Pi of PG(r, q) the set Pi boolean AND 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 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)) 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.