Deep Holes of Twisted Reed-Solomon Codes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913866984718336 |
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| author | Fang, Weijun Xu, Jingke Zhu, Ruiqi |
| author_facet | Fang, Weijun Xu, Jingke Zhu, Ruiqi |
| contents | The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes.} The covering radius and a standard class of deep holes of twisted Reed-Solomon codes ${\rm TRS}_k(\mathcal{A}, θ)$ are obtained for a general evaluation set $\mathcal{A} \subseteq \mathbb{F}_q$. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes ${\rm TRS}_k(\mathbb{F}_q, θ)$. For even $q$, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of ${\rm TRS}_k(\mathbb{F}_q, θ)$ with $\frac{3q-4}{4} \leq k\leq q-4$. For odd $q$, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of ${\rm TRS}_k(\mathbb{F}_q, θ)$ with $\frac{3q+3\sqrt{q}-7}{4} \leq k\leq q-4$. In addition, for the boundary cases of $k=q-3, q-2$ and $q-1$, we completely determine their deep holes using results on certain character sums. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_11436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deep Holes of Twisted Reed-Solomon Codes Fang, Weijun Xu, Jingke Zhu, Ruiqi Information Theory The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes.} The covering radius and a standard class of deep holes of twisted Reed-Solomon codes ${\rm TRS}_k(\mathcal{A}, θ)$ are obtained for a general evaluation set $\mathcal{A} \subseteq \mathbb{F}_q$. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes ${\rm TRS}_k(\mathbb{F}_q, θ)$. For even $q$, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of ${\rm TRS}_k(\mathbb{F}_q, θ)$ with $\frac{3q-4}{4} \leq k\leq q-4$. For odd $q$, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of ${\rm TRS}_k(\mathbb{F}_q, θ)$ with $\frac{3q+3\sqrt{q}-7}{4} \leq k\leq q-4$. In addition, for the boundary cases of $k=q-3, q-2$ and $q-1$, we completely determine their deep holes using results on certain character sums. |
| title | Deep Holes of Twisted Reed-Solomon Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2403.11436 |