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arXiv · 2511.03398

Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions

Zhonghao Liang·Chenlu Jia·Dongmei Huang·Qunying Liao·Chunming Tang
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Abstract

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.

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Zhonghao Liang, Chenlu Jia, Dongmei Huang, Qunying Liao, Chunming Tang. 2025-11-05. Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions. https://arxiv.org/abs/2511.03398

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