Deep holes of a class of twisted Reed-Solomon codes
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| Format: | Preprint |
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2025
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| _version_ | 1866912580601118720 |
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| author | Gu, Haojie Wang, Nan Zhang, Jun |
| author_facet | Gu, Haojie Wang, Nan Zhang, Jun |
| contents | The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes, we investigate the problem of deep holes of a class of twisted Reed-Solomon codes in this paper.
Firstly, we provide the necessary and sufficient conditions for $\boldsymbol{a}=(a_{0},a_{1},\cdots,a_{n-k-1})\in\mathbb{F}_{q}^{n-k}$ to be the syndrome of some deep hole of $TRS_{k}(\mathcal{A},l,η)$. Next, we consider the problem of determining all deep holes of the twisted Reed-Solomon codes $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$. Specifically, we prove that there are no other deep holes of $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$ for $\frac{3q+2\sqrt{q}-8}{4}\leq k\leq q-5$ when q is even, and $\frac{3q+3\sqrt{q}-5}{4}\leq k\leq q-5$ when q is odd. We also completely determine their deep holes for $q-4\leq k\leq q-2$ when $q$ is even. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08526 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep holes of a class of twisted Reed-Solomon codes Gu, Haojie Wang, Nan Zhang, Jun Information Theory The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes, we investigate the problem of deep holes of a class of twisted Reed-Solomon codes in this paper. Firstly, we provide the necessary and sufficient conditions for $\boldsymbol{a}=(a_{0},a_{1},\cdots,a_{n-k-1})\in\mathbb{F}_{q}^{n-k}$ to be the syndrome of some deep hole of $TRS_{k}(\mathcal{A},l,η)$. Next, we consider the problem of determining all deep holes of the twisted Reed-Solomon codes $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$. Specifically, we prove that there are no other deep holes of $TRS_{k}(\mathbb{F}_{q}^{*},k-1,η)$ for $\frac{3q+2\sqrt{q}-8}{4}\leq k\leq q-5$ when q is even, and $\frac{3q+3\sqrt{q}-5}{4}\leq k\leq q-5$ when q is odd. We also completely determine their deep holes for $q-4\leq k\leq q-2$ when $q$ is even. |
| title | Deep holes of a class of twisted Reed-Solomon codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2509.08526 |