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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2505.04315 |
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| _version_ | 1866909603439050752 |
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| author | Xu, Haojie Wu, Xia Lu, Wei Cao, Xiwang |
| author_facet | Xu, Haojie Wu, Xia Lu, Wei Cao, Xiwang |
| contents | Let $\mathcal{C}_{(q,q^m+1,3,h)}$ denote the antiprimitive BCH code with designed distance 3. In this paper, we demonstrate that the minimum distance $d$ of $\mathcal{C}_{(q,q^m+1,3,h)}$ equals 3 if and only if $\gcd(2h+1,q+1,q^m+1)\ne1$. When both $q$ and $m$ are odd, we determine the sufficient and necessary condition for $d=4$ and fully characterize the minimum distance in this case. Based on these conditions, we investigate the parameters of $\mathcal{C}_{(q,q^m+1,3,h)}$ for certain $h$. Additionally, two infinite families of distance-optimal codes and several linear codes with the best known parameters are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The minimum distance of the antiprimitive BCH code with designed distance 3 Xu, Haojie Wu, Xia Lu, Wei Cao, Xiwang Information Theory Let $\mathcal{C}_{(q,q^m+1,3,h)}$ denote the antiprimitive BCH code with designed distance 3. In this paper, we demonstrate that the minimum distance $d$ of $\mathcal{C}_{(q,q^m+1,3,h)}$ equals 3 if and only if $\gcd(2h+1,q+1,q^m+1)\ne1$. When both $q$ and $m$ are odd, we determine the sufficient and necessary condition for $d=4$ and fully characterize the minimum distance in this case. Based on these conditions, we investigate the parameters of $\mathcal{C}_{(q,q^m+1,3,h)}$ for certain $h$. Additionally, two infinite families of distance-optimal codes and several linear codes with the best known parameters are presented. |
| title | The minimum distance of the antiprimitive BCH code with designed distance 3 |
| topic | Information Theory |
| url | https://arxiv.org/abs/2505.04315 |