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On cutting blocking sets and their codes

  • Daniele Bartoli , Antonio Cossidente , Giuseppe Marino and Francesco Pavese ORCID logo EMAIL logo
From the journal Forum Mathematicum

Abstract

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 Π𝒳 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,q3) of size 3(q+1)(q2+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.


Communicated by Manfred Droste


Funding statement: This work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM).

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Received: 2020-12-01
Published Online: 2022-01-06
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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