$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes
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| Format: | Preprint |
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2025
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| _version_ | 1866908704220119040 |
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| author | Chen, Hao Pan, Xu Xie, Conghui |
| author_facet | Chen, Hao Pan, Xu Xie, Conghui |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_09457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $t$-Fold $s$-Blocking Sets and $s$-Minimal Codes Chen, Hao Pan, Xu Xie, Conghui Information Theory 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. |
| title | $t$-Fold $s$-Blocking Sets and $s$-Minimal Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2512.09457 |