Remarks on second and third weights of Projective Reed-Muller codes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909500974301184 |
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| author | Datta, Mrinmoy |
| author_facet | Datta, Mrinmoy |
| contents | Determining the weight distributions of the projective Reed-Muller codes is a very hard problem and has been studied extensively in the literature. In this article, we provide an alternative proof of the second weight of the projective Reed-Muller codes $\PRM (d, m)$ where $m \ge 3$ and $3 \le d \le \frac{q+3}{2}$. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on $d$. Furthermore, we compute the second weight of $\PRM (d, 2)$ for $3 \le d \le q-1$. Furthermore, we give an upper bound for the third weight of $\PRM(d, 2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on second and third weights of Projective Reed-Muller codes Datta, Mrinmoy Algebraic Geometry 14G05, 14G15, 11T06 Determining the weight distributions of the projective Reed-Muller codes is a very hard problem and has been studied extensively in the literature. In this article, we provide an alternative proof of the second weight of the projective Reed-Muller codes $\PRM (d, m)$ where $m \ge 3$ and $3 \le d \le \frac{q+3}{2}$. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on $d$. Furthermore, we compute the second weight of $\PRM (d, 2)$ for $3 \le d \le q-1$. Furthermore, we give an upper bound for the third weight of $\PRM(d, 2)$. |
| title | Remarks on second and third weights of Projective Reed-Muller codes |
| topic | Algebraic Geometry 14G05, 14G15, 11T06 |
| url | https://arxiv.org/abs/2406.07339 |