New Constructions of Additive MDS TRS Codes

2026-08-19

AI Summary

本文研究了加性扭曲Reed-Solomon码，通过建立必要充分条件和使用Schur平方技术，构造了新的加性MDS码，并确定了其校验矩阵。

Abstract

Additive codes over finite fields generalize linear codes, and additive MDS codes provide a natural extension of linear MDS codes. In this article, we study additive twisted Reed--Solomon (TRS) codes and obtain new constructions of additive MDS codes. First, for additive TRS codes with twist $t=2$ and an arbitrary hook, we establish necessary and sufficient conditions for the codes to be additive MDS, thereby generalizing the results in Section 3 of [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. In particular, we show that the existence of an additive MDS TRS code with $t=2$ and hook $h=0$ yields codes of larger lengths than those obtained for $t=2$ and $h=k-1$ in [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. Next, we consider additive TRS codes with twist vector $\mathbf{t}=(1,2)$ and hook vector $\mathbf{h}=(0,0)$, and derive necessary and sufficient conditions for them to be additive MDS. We further establish the existence of such codes. Using the Schur square technique, we obtain mild conditions under which the constructed families are inequivalent to additive Reed--Solomon (RS) codes. Finally, we determine parity-check matrices for both families of additive MDS codes considered in this article.

Problem

• Additive MDS Codes
• Twisted Reed-Solomon Codes
• Necessary and Sufficient Conditions

Innovation

• additive MDS codes
• twisted Reed-Solomon (TRS) codes
• Schur square technique