加法MDS TRS码的新构造
New Constructions of Additive MDS TRS Codes
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中文总结 AI 辅助
本文研究加法TRS码，建立扭量t=2及扭量向量t=(1,2)的加法TRS码为加法MDS码的充要条件，证明相关码的存在性，确定奇偶校验矩阵，并通过舒尔平方技术得到其与加法RS码不等价的条件。
中文摘要 AI 辅助
有限域上的加法码是线性码的推广，而加法MDS码则是线性MDS码的自然扩展。本文研究加法扭化里德-所罗门（TRS）码，并得到加法MDS码的新构造。首先，针对扭量t=2且钩子任意的加法TRS码，我们建立了其为加法MDS码的充要条件，从而推广了[Jiayu Ma等人，《加法非里德-所罗门MDS码的新族》]第3节的结果。特别地，我们证明存在扭量t=2且钩子h=0的加法MDS TRS码，其长度比[Jiayu Ma等人，《加法非里德-所罗门MDS码的新族》]中t=2且h=k-1时得到的码更长。接下来，我们考虑扭量向量t=(1,2)且钩子向量h=(0,0)的加法TRS码，推导其为加法MDS码的充要条件，并进一步证明此类码的存在性。利用舒尔平方技术，我们得到所构造的族与加法里德-所罗门（RS）码不等价的温和条件。最后，确定本文所考虑的两类加法MDS码的奇偶校验矩阵。
英文摘要
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.
