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The (+)-(L, P)-TGRS code

Zhonghao Liang, Chenlu Jia, Qunying Liao

Abstract

The construction of the non-Reed-Solomon (in short, non-RS) type linear code has been one of the research hotspots in recent years. In 2025, Hu et al. constructed some non-RS MDS codes by defining the (L, P)-twisted generalized Reed-Solomon code (in short, (L, P)-TGRS). In this paper, we focus on the (+)-(L, P)-TGRS code C. We firstly present a parity-check matrix. Secondly, we give a sufficient and necessary condition for C to be NMDS which partially answers two open problems proposed by Hu et al. in 2025, and prove that C is non-RS for 2k > n which partially improves the corresponding result given by Hu et al. in 2025,. Thirdly, we give a sufficient condition for C not to be self-dual or self-orthogonal, respectively, furthermore, we construct two classes of self-orthogonal codes which is a promotion of the corresponding result given by Ding et al. in 2025. Finally, some examples are given.

The (+)-(L, P)-TGRS code

Abstract

The construction of the non-Reed-Solomon (in short, non-RS) type linear code has been one of the research hotspots in recent years. In 2025, Hu et al. constructed some non-RS MDS codes by defining the (L, P)-twisted generalized Reed-Solomon code (in short, (L, P)-TGRS). In this paper, we focus on the (+)-(L, P)-TGRS code C. We firstly present a parity-check matrix. Secondly, we give a sufficient and necessary condition for C to be NMDS which partially answers two open problems proposed by Hu et al. in 2025, and prove that C is non-RS for 2k > n which partially improves the corresponding result given by Hu et al. in 2025,. Thirdly, we give a sufficient condition for C not to be self-dual or self-orthogonal, respectively, furthermore, we construct two classes of self-orthogonal codes which is a promotion of the corresponding result given by Ding et al. in 2025. Finally, some examples are given.

Paper Structure

This paper contains 10 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The parity-check matrix of the $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code
  4. The NMDS property of the $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code
  5. A sufficient condition for the $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code not to be self-dual
  6. The self-orthogonal $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code
  7. The construction for the self-orthogonal $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code
  8. A sufficient condition for the $(+)$-$(\mathcal{L},\mathcal{P})$-TGRS code not to be self-orthogonal
  9. The non-GRS property
  10. Conclusions

Key Result

Lemma 2.1

(A27, Lemma 2.3) Let $\boldsymbol{u} = (u_1, \ldots, u_n)$ with $u_j = -\prod\limits_{\substack{i=1 \\ i \neq j}}^n (\alpha_j - \alpha_i)$$(j = 1, \ldots, n)$. $(1)$ If $k \leq \frac{n}{2}$, then $\mathrm{GRS}_{k,n}(\boldsymbol{\alpha}, \boldsymbol{1}) \star \mathrm{GRS}_{k,n}(\boldsymbol{\alpha}, \

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Remark 2.1
  • Definition 2.3
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 12 more