Outer Strong Blocking Sets

TL;DR

This paper digs into the geometry of the concatenation method, introducing the concept of outer strong blocking sets and their coding theoretical counterpart, and improves the best-known upper bound on the minimum size of a strong blocking set.

Abstract

Strong blocking sets, introduced first in 2011 in connection with saturating sets, have recently gained a lot of attention due to their correspondence with minimal codes. In this paper, we dig into the geometry of the concatenation method, introducing the concept of outer strong blocking sets and their coding theoretical counterpart. We investigate their structure and provide bounds on their size. As a byproduct, we improve the best-known upper bound on the minimum size of a strong blocking set. Finally, we present a geometric construction of small strong blocking sets, whose computational cost is significantly smaller than the previously known ones.

Figures (1)

• Figure 1: Relations.

Theorems & Definitions (72)

• Definition 1.1
• Example 1.2
• Theorem 1.3: alfarano2022three
• Theorem 1.4: heger2021short
• Definition 1.5
• Definition 1.7
• Example 1.8
• Theorem 1.9: alfarano2019geometrictang2019full
• Lemma 1.10: AB condition ashikhmin1998minimal
• Theorem 1.11: alfarano2022three