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arXiv · 2502.04746

On $(\mathcal{L},\mathcal{P})$-Twisted Generalized Reed-Solomon Codes

Zhao Hu·Liang Wang·Nian Li·Xiangyong Zeng·Xiaohu Tang
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Abstract

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.

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Zhao Hu, Liang Wang, Nian Li, Xiangyong Zeng, Xiaohu Tang. 2025-02-07. On $(\mathcal{L},\mathcal{P})$-Twisted Generalized Reed-Solomon Codes. https://arxiv.org/abs/2502.04746

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