On cutting blocking sets and their codes
Daniele Bartoli, Antonio Cossidente, Giuseppe Marino, Francesco Pavese · Forum Mathematicum · 2022
Abstract Let PG ( r , q ) {\operatorname{PG}(r,q)} be the r -dimensional projective space over the finite field GF ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 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 of PG ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ( q + 1 ) ( q 2 + 1 ) {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 )