Dual and Covering Radii of Extended Algebraic Geometry Codes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918148328914944 |
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| author | Zhu, Yunlong Zhao, Chang-An |
| author_facet | Zhu, Yunlong Zhao, Chang-An |
| contents | Many literatures consider the extended Reed-Solomon (RS) codes, including their dual codes and covering radii, but few focus on extended algebraic geometry (AG) codes of genus $g\ge1$. In this paper, we investigate extended AG codes and Roth-Lempel type AG codes, including their dual codes and minimum distances. Moreover, we show that for certain $g$, the length of a $g$-MDS code over a finite field $\mathbb{F}_q$ can attain $q+1+2g\sqrt{q}$, which is achieved by an extended AG code from the maximal curves of genus $g$. Notably, for some small finite fields, this length $q+1+2g\sqrt{q}$ is the largest among all known $g$-MDS codes. Subsequently, we establish that the covering radius of an $[n,k]$ extended AG code has $g+2$ possible values. For the case of $g=1$, we prove that this range reduces to two possible values when the length $n$ is sufficiently large, or when there exists an $[n,k+1]$ MDS elliptic code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dual and Covering Radii of Extended Algebraic Geometry Codes Zhu, Yunlong Zhao, Chang-An Information Theory Many literatures consider the extended Reed-Solomon (RS) codes, including their dual codes and covering radii, but few focus on extended algebraic geometry (AG) codes of genus $g\ge1$. In this paper, we investigate extended AG codes and Roth-Lempel type AG codes, including their dual codes and minimum distances. Moreover, we show that for certain $g$, the length of a $g$-MDS code over a finite field $\mathbb{F}_q$ can attain $q+1+2g\sqrt{q}$, which is achieved by an extended AG code from the maximal curves of genus $g$. Notably, for some small finite fields, this length $q+1+2g\sqrt{q}$ is the largest among all known $g$-MDS codes. Subsequently, we establish that the covering radius of an $[n,k]$ extended AG code has $g+2$ possible values. For the case of $g=1$, we prove that this range reduces to two possible values when the length $n$ is sufficiently large, or when there exists an $[n,k+1]$ MDS elliptic code. |
| title | Dual and Covering Radii of Extended Algebraic Geometry Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2509.21773 |