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.

翻译：最大距离可分码（简称MDS码）、近最大距离可分码（简称NMDS码）和自正交码在代数编码理论中发挥着关键作用，尤其在量子通信和秘密共享方案等应用中。近年来，非广义Reed-Solomon码（简称非GRS码）的构造已成为一个重要研究前沿。本文系统研究了一类具有 $\ell$ 个扭曲特征的广义 $(\mathcal{L}, \mathcal{P})$-扭曲广义Reed-Solomon码，扩展了Beelen等人和Hu等人先前提出的结构。我们首先通过分析对称多项式的性质，推导出这些码的显式校验矩阵。基于此代数框架，我们建立了所提出码自正交性的充要条件，推广了若干近期结果。利用这些自正交结构，我们构造了新的LCD MDS码族，其在码长方面相比现有文献提供了更大的灵活性。此外，我们给出了这些码NMDS性质的刻画，为Hu等人（2025）提出的关于一般 $(\mathcal{L}, \mathcal{P})$-TGRS码的公开问题提供了部分解答。最后，我们严格证明了当 $2k > n$ 时这些码属于非GRS类型，改进了先前的界。理论构造通过数值算例得到了验证。