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 {Picapmathcal{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 ) {operatorname{PG}(5,q)} of size 7 ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ( 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 minimal q-ary linear code having dimension 4 and 6.
On cutting blocking sets and their codes / Bartoli, D.; Cossidente, A.; Marino, G.; Pavese, F.. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 34:2(2022), pp. 347-368. [10.1515/forum-2020-0338]
On cutting blocking sets and their codes
Marino G.;
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 {Picapmathcal{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 ) {operatorname{PG}(5,q)} of size 7 ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ( 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 minimal q-ary linear code having dimension 4 and 6.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.