On (ℒ, 𝒫)-Twisted Generalized Reed-Solomon Codes
Zhao Hu, Liang Wang, Nian Li, Xiangyong Zeng, Xiaohu Tang · IEEE Transactions on Information Theory · 2025
Twisted generalized Reed-Solomon (TGRS) codes are an extension of generalized Reed-Solomon (GRS) codes, and have recently attracted significant attention due to their potential for constructing non-GRS MDS codes. This paper presents an in-depth and comprehensive investigation of TGRS codes in their most general form, allowing arbitrary twists at arbitrary positions. First, we introduce a more precise definition of TGRS codes, namely (L,P)-TGRS codes, and provide a concise necessary and sufficient condition for them to be MDS, thereby generalizing previous results. Second, we explicitly characterize the parity check matrices of (L,P)-TGRS codes, and provide a sufficient condition for them to be self-dual. Finally, we investigate the non-GRS properties of (L,P)-TGRS codes via two approaches: the dimensions of Schur squares and combinatorial techniques. As a result, we obtain an infinite family of non-GRS MDS codes.