Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911359835308032 |
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| author | Liang, Zhonghao Jia, Chenlu Huang, Dongmei Liao, Qunying Tang, Chunming |
| author_facet | Liang, Zhonghao Jia, Chenlu Huang, Dongmei Liao, Qunying Tang, Chunming |
| contents | Maximum distance separable (in short, MDS), near MDS (in short, NMDS), and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing scheme. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper presents a systematic investigation into a generalized class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes characterized by $\ell$ twists, extending the structures previously introduced by Beelen et al. and Hu et al.. We first derive the explicit parity-check matrices for these codes by analyzing the properties of symmetric polynomials. Based on this algebraic framework, we establish necessary and sufficient conditions for the self-orthogonality of the proposed codes, generalizing several recent results. Leveraging these self-orthogonal structures, we construct new families of LCD MDS codes that offer greater flexibility in code length compared to existing literature. Furthermore, we provide a characterization of the NMDS property for these codes, offering a partial solution to the open problem concerning general $(\mathcal{L}, \mathcal{P})$-TGRS codes posed by Hu et al. (2025). Finally, we rigorously prove that these codes are of non-GRS type when $2k > n$, providing an improvement over previous bounds. Theoretical constructions are validated through numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions Liang, Zhonghao Jia, Chenlu Huang, Dongmei Liao, Qunying Tang, Chunming Information Theory 94B05, 11T71 E.4 Maximum distance separable (in short, MDS), near MDS (in short, NMDS), and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing scheme. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper presents a systematic investigation into a generalized class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes characterized by $\ell$ twists, extending the structures previously introduced by Beelen et al. and Hu et al.. We first derive the explicit parity-check matrices for these codes by analyzing the properties of symmetric polynomials. Based on this algebraic framework, we establish necessary and sufficient conditions for the self-orthogonality of the proposed codes, generalizing several recent results. Leveraging these self-orthogonal structures, we construct new families of LCD MDS codes that offer greater flexibility in code length compared to existing literature. Furthermore, we provide a characterization of the NMDS property for these codes, offering a partial solution to the open problem concerning general $(\mathcal{L}, \mathcal{P})$-TGRS codes posed by Hu et al. (2025). Finally, we rigorously prove that these codes are of non-GRS type when $2k > n$, providing an improvement over previous bounds. Theoretical constructions are validated through numerical examples. |
| title | Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions |
| topic | Information Theory 94B05, 11T71 E.4 |
| url | https://arxiv.org/abs/2511.03398 |