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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2502.04746 |
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| _version_ | 1866915142123388928 |
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| author | Hu, Zhao Wang, Liang Li, Nian Zeng, Xiangyong Tang, Xiaohu |
| author_facet | Hu, Zhao Wang, Liang Li, Nian Zeng, Xiangyong Tang, Xiaohu |
| contents | Twisted generalized Reed-Solomon (TGRS) codes are an extension of the generalized Reed-Solomon (GRS) codes by adding specific twists, which attract much attention recently. This paper presents an in-depth and comprehensive investigation of the TGRS codes for the most general form by using a universal method. At first, we propose a more precise definition to describe TGRS codes, namely $(\mathcal{L},\mathcal{P})$-TGRS codes, and provide a concise necessary and sufficient condition for $(\mathcal{L},\mathcal{P})$-TGRS codes to be MDS, which extends the related results in the previous works. Secondly, we explicitly characterize the parity check matrices of $(\mathcal{L},\mathcal{P})$-TGRS codes, and provide a sufficient condition for $(\mathcal{L},\mathcal{P})$-TGRS codes to be self-dual. Finally, we conduct an in-depth study into the non-GRS property of $(\mathcal{L},\mathcal{P})$-TGRS codes via the Schur squares and the combinatorial techniques respectively. As a result, we obtain a large infinite families of non-GRS MDS codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $(\mathcal{L},\mathcal{P})$-Twisted Generalized Reed-Solomon Codes Hu, Zhao Wang, Liang Li, Nian Zeng, Xiangyong Tang, Xiaohu Information Theory Twisted generalized Reed-Solomon (TGRS) codes are an extension of the generalized Reed-Solomon (GRS) codes by adding specific twists, which attract much attention recently. This paper presents an in-depth and comprehensive investigation of the TGRS codes for the most general form by using a universal method. At first, we propose a more precise definition to describe TGRS codes, namely $(\mathcal{L},\mathcal{P})$-TGRS codes, and provide a concise necessary and sufficient condition for $(\mathcal{L},\mathcal{P})$-TGRS codes to be MDS, which extends the related results in the previous works. Secondly, we explicitly characterize the parity check matrices of $(\mathcal{L},\mathcal{P})$-TGRS codes, and provide a sufficient condition for $(\mathcal{L},\mathcal{P})$-TGRS codes to be self-dual. Finally, we conduct an in-depth study into the non-GRS property of $(\mathcal{L},\mathcal{P})$-TGRS codes via the Schur squares and the combinatorial techniques respectively. As a result, we obtain a large infinite families of non-GRS MDS codes. |
| title | On $(\mathcal{L},\mathcal{P})$-Twisted Generalized Reed-Solomon Codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2502.04746 |