On cutting blocking sets and their codes
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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 Π of PG(r, q) the set Π∩ 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 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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