Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions

TL;DR

This paper introduces a new class of twisted generalized Reed-Solomon codes with explicit constructions, analyzes their properties including self-orthogonality and NMDS conditions, and demonstrates their advantages over existing codes.

Contribution

It extends the structure of TGRS codes, provides explicit parity-check matrices, and constructs new LCD MDS codes with greater length flexibility, addressing open problems in the field.

Findings

Derived explicit parity-check matrices for TGRS codes.

Established necessary and sufficient conditions for self-orthogonality.

Constructed new families of LCD MDS codes with improved parameters.

Abstract

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.