On cutting blocking sets and their codes

TL;DR

This paper introduces new constructions of cutting blocking sets in projective spaces, leading to smaller sets and improved bounds for saturating sets and minimal linear codes, with implications for coding theory.

Contribution

It presents novel constructions of cutting blocking sets in PG(3, q^3) and PG(5, q), reducing their sizes compared to known sets, and improves bounds on saturating sets and minimal code lengths.

Findings

New smaller cutting blocking sets in PG(3, q^3) and PG(5, q)

Linear growth of minimal code length with respect to dimension

Improved upper bounds for saturating sets and minimal code lengths

Abstract

Let PG$(r, q)$ be the $r$-dimensional projective space over the finite field ${\rm GF}(q)$. A set $\cal X$ of points of PG$(r, q)$ is a cutting blocking set if for each hyperplane $\Pi$ of PG$(r, q)$ the set $\Pi \cap \cal X$ spans $\Pi$. 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 $6$.