环上码的广义秩重与扩展广义偏序重：一种Galois连接方法

中文摘要 AI 辅助

本文通过Galois连接方法研究环上码的广义秩重（GRWs）和扩展广义偏序重（EGPWs）。首先，我们展示了与广义重相关的各种编码理论性质，包括用于II型窃听信道的码的安全降级、Gabidulin码与其关联Delsarte码的广义重之间的联系、（广义）Singleton界、码的MDS偏差、MDS、近MDS、$i$-MDS、MRD、近MRD、$i$-MRD、（对偶）准MRD码的表征以及子空间的回避性质，都可以用Galois连接重新表述。其次，我们研究主理想环上模的GRWs和秩轮廓，特别是链环上的模。推广域上向量空间定义的GRWs，我们建立了Singleton界和Wei型对偶定理，表征了MRD、近MRD和对偶准MRD码并确定了它们的GRWs；此外，我们表征了$i$-MRD码，并为链环上的$(h,h)$-回避码建立了分散界，推广了有限域上向量空间的对应结果。最后，我们提出并研究了具有合成列的模的EGPWs和扩展偏序轮廓，它们实际上构成一个Galois连接。推广有限Galois环上模定义的EGPWs，我们为任意拟Frobenius环上的模建立了Wei型对偶定理，统一了文献[32]和[33]中导出的两个Wei型对偶定理。

英文摘要

In this paper, we study generalized rank weights (GRWs) and extended generalized poset weight (EGPWs) of codes over rings via a Galois connection approach. First, we show that various coding-theoretic properties related to generalized weights, including security drops of a code employed in wire-tap channel of type II, connections between generalized weights of a Gabidulin code and its associated Delsarte code, (generalized) Singleton bound, MDS discrepancy of a code, characterizations of MDS, near MDS, $i$-MDS, MRD, near MRD, $i$-MRD, (dually) quasi-MRD codes as well as evasive property of subspaces, can be reformulated in terms of Galois connections. Next, we study GRWs and rank profiles defined for modules over principal ideal rings, especially those over chain rings. Generalizing GRWs defined for vector spaces over fields, we establish a singleton bound and a Wei-type duality theorem, characterize MRD, near MRD and dually quasi-MRD codes and determine their GRWs; moreover, we characterize $i$-MRD codes and establish a scattered bound for $(h,h)$-evasive codes over chain rings, generalizing counterpart result established for vector space over finite fields. Finally, we propose and study EGPWs and extended poset profiles defined for modules with a composition series, which in fact form a Galois connection. Generalizing EGPWs defined for modules over finite Galois rings, we establish a Wei-type duality theorem for modules over arbitrary quasi-Frobenius rings, which unifies the two Wei-type duality theorems derived in both \cite{32} and \cite{33}.

发表机构

• Fudan University（复旦大学）
• National and Local Joint Laboratory of Cyberspace Security Technology（网络空间安全技术国家地方联合工程实验室）
• Yiwu Research Institute of Fudan University（复旦大学义乌研究院）
• The University of Hong Kong（香港大学）

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