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

Zhao Hu, Liang Wang, Nian Li, Xiangyong Zeng, Xiaohu Tang

TL;DR

This work generalizes twisted generalized Reed-Solomon codes to the most general ($\mathcal{L},\mathcal{P}$)-TGRS form by fixing $\mathcal{L}=[n-k-1]$, $\mathcal{P}=[k-1]$ and a coefficient matrix $B$, and develops a universal method to study their fundamental properties. It proves a concise MDS criterion: ${\mathcal C}(\mathcal{L},\mathcal{P},B)$ is MDS if and only if $\det(I_k+BF_{\mathcal{T}}) \neq 0$ for all $k$-subsets $\mathcal{T}$, and it explicitly characterizes the parity-check matrices and a sufficient self-duality condition via $v_i^2=\lambda u_i$ and $B^T D B= N B + B^T N$. The paper then analyzes non-GRS properties through Schur-square arguments and a combinatorial criterion, obtaining large infinite families of non-GRS MDS codes and providing concrete constructions and counts. These results broaden the pool of MDS codes beyond GRS, with potential implications for cryptography and reliable data storage, and set up open questions on NMDS characterizations and further infinite families. The combination of determinant-based MDS criteria, explicit parity-check structure, and non-GRS proofs provides a cohesive framework for exploring twisted RS codes in broad settings.

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.

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

TL;DR

This work generalizes twisted generalized Reed-Solomon codes to the most general ()-TGRS form by fixing , and a coefficient matrix , and develops a universal method to study their fundamental properties. It proves a concise MDS criterion: is MDS if and only if for all -subsets , and it explicitly characterizes the parity-check matrices and a sufficient self-duality condition via and . The paper then analyzes non-GRS properties through Schur-square arguments and a combinatorial criterion, obtaining large infinite families of non-GRS MDS codes and providing concrete constructions and counts. These results broaden the pool of MDS codes beyond GRS, with potential implications for cryptography and reliable data storage, and set up open questions on NMDS characterizations and further infinite families. The combination of determinant-based MDS criteria, explicit parity-check structure, and non-GRS proofs provides a cohesive framework for exploring twisted RS codes in broad settings.

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 -TGRS codes, and provide a concise necessary and sufficient condition for -TGRS codes to be MDS, which extends the related results in the previous works. Secondly, we explicitly characterize the parity check matrices of -TGRS codes, and provide a sufficient condition for -TGRS codes to be self-dual. Finally, we conduct an in-depth study into the non-GRS property of -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.

Paper Structure

This paper contains 13 sections, 20 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GRS codes
  4. $({\mathcal{L}},{\mathcal{P}})$-TGRS codes
  5. Equivalence of linear codes
  6. MDS $({\mathcal{L}},{\mathcal{P}})$-TGRS codes
  7. The parity check matrices of $({\mathcal{L}},{\mathcal{P}})$-TGRS codes and the self-dual codes
  8. The parity check matrices of $({\mathcal{L}},{\mathcal{P}})$-TGRS codes
  9. The self-dual $({\mathcal{L}},{\mathcal{P}})$-TGRS codes
  10. The non-GRS properties of $({\mathcal{L}},{\mathcal{P}})$-TGRS codes
  11. Inequivalence based on the Schur square
  12. A combinatorial inequivalence argument
  13. Conclusions

Key Result

Lemma 1

Let $\boldsymbol{\alpha}=(a_1,a_2,...,a_n)\in\mathbb{F}_{q}^n$ with distinct $a_1,...,a_n$, $\boldsymbol{\nu}=(v_1,...,v_n)\in(\mathbb{F}_{q}^*)^n$ and $B$ be defined as in S8-B. Then we have the following:

Theorems & Definitions (48)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 38 more