← 研踪 PaperTrailarXiv:2511.03398 · v4
A Class of Self-Orthogonal, NMDS, or LCD MDS $(\mathcal{L}, \mathcal{P})$-Twisted Generalized Reed-Solomon Codes
首次提交 2025-11-05 20:05
版本更新 2026-11-12 10:22
雷达观测 2026-11-13 14:16
摘要
Maximum distance separable (in short, MDS), near MDS (in short, NMDS), linear complementary dual codes (in short, LCD) and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing schemes. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper focuses on a class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ characterized by $\ell+1$ twists. By considering a special class of parity-check matrices derived from existing results, we derive the corresponding necessary and sufficient conditions or sufficient conditions for $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ to be self-orthogonal, and then generalize the corresponding results given by Liu et al. (2021), Sui et al. (2021), and Zhang et al. (2025). As applications, leveraging these self-orthogonal structures, we can obtain several families of LCD MDS codes. Secondly, we provide a characterization of the NMDS property for the codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$, 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 the codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ are of non-GRS type when $2k > n>k+\ell+2$.
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v4 · 2026-11-12 10:22 · 最新观测版本
A Class of Self-Orthogonal, NMDS, or LCD MDS $(\mathcal{L}, \mathcal{P})$-Twisted Generalized Reed-Solomon Codes
Maximum distance separable (in short, MDS), near MDS (in short, NMDS), linear complementary dual codes (in short, LCD) and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing schemes. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper focuses on a class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ characterized by $\ell+1$ twists. By considering a special class of parity-check matrices derived from existing results, we derive the corresponding necessary and sufficient conditions or sufficient conditions for $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ to be self-orthogonal, and then generalize the corresponding results given by Liu et al. (2021), Sui et al. (2021), and Zhang et al. (2025). As applications, leveraging these self-orthogonal structures, we can obtain several families of LCD MDS codes. Secondly, we provide a characterization of the NMDS property for the codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$, 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 the codes $\mathcal{C}_{k-1}\left(\boldsymbolα,\boldsymbol{v},\boldsymbolη\right)$ are of non-GRS type when $2k > n>k+\ell+2$.
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