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

TL;DR

This paper thoroughly investigates the most general form of twisted generalized Reed-Solomon codes, providing new conditions for their MDS and self-duality properties, and identifying large families of non-GRS MDS codes.

Contribution

It introduces a precise definition of $(\\mathcal{L},\\mathcal{P})$-TGRS codes, characterizes their parity check matrices, and explores their non-GRS properties using advanced techniques.

Findings

Provided necessary and sufficient conditions for $(\mathcal{L},\mathcal{P})$-TGRS codes to be MDS.

Characterized the parity check matrices of $(\mathcal{L},\mathcal{P})$-TGRS codes.

Constructed large families of non-GRS MDS codes.

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.