On (ℒ, 𝒫)-Twisted Generalized Reed-Solomon Codes

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 (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L,P</i>)-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 (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L,P</i>)-TGRS codes, and provide a sufficient condition for them to be self-dual. Finally, we investigate the non-GRS properties of (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L,P</i>)-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.