On cutting blocking sets and their codes

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} be the r -dimensional projective space over the finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>GF</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{GF}(q)} . A set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="script">𝒳</m:mi></m:math> {\mathcal{X}} of points of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi mathvariant="normal">Π</m:mi><m:mo>∩</m:mo><m:mi mathvariant="script">𝒳</m:mi></m:mrow></m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>3</m:mn><m:mo>,</m:mo><m:msup><m:mi>q</m:mi><m:mn>3</m:mn></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(3,q^{3})} of size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>3</m:mn><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q -order subgeometries, and a cutting blocking set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>5</m:mn><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(5,q)} of size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>7</m:mn><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {7(q+1)} from seven lines of a Desarguesian line spread of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>5</m:mn><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\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.