$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes

TL;DR

This paper establishes new bounds on $t$-fold $s$-blocking sets and projective $s$-minimal codes, generalizes the Ashikhmin-Barg condition, and constructs various examples of $s$-minimal codes.

Contribution

It introduces a stronger lower bound for $t$-fold $s$-blocking sets and extends the Ashikhmin-Barg condition to $s$-minimal codes, with numerous constructed examples.

Findings

New lower bound on size of $t$-fold $s$-blocking sets without $t \

Generalization of Ashikhmin-Barg condition for $s$-minimal codes

Construction of infinite families of $s$-minimal codes and examples

Abstract

Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t \leq q$, which is stronger than the classical result of Beutelspacher in 1983. Then a lower bound on lengths of projective $s$-minimal codes is also obtained. It is proved that $(s+1)$-minimal codes are certainly $s$-minimal codes. We generalize the Ashikhmin-Barg condition for minimal codes to $s$-minimal codes. Many infinite families of $s$-minimal codes satisfying and violating this generalized Ashikhmin-Barg condition are constructed. We also give several examples which are binary minimal codes, but not $2$-minimal codes.