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

Hao Chen; Xu Pan; Conghui Xie

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arXiv · 2512.09457

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

Hao Chen·Xu Pan·Conghui Xie

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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.

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Hao Chen, Xu Pan, Conghui Xie. 2025-12-10. $t$-Fold $s$-Blocking Sets and $s$-Minimal Codes. https://arxiv.org/abs/2512.09457

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