发表机构
Fudan University; Yiwu Research Institute of Fudan University; The University of Hong Kong（复旦大学; 复旦大学义乌研究院; 香港大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究关于$\mathbf{P}$-支撑的$r$-极小码，通过切割$r$-阻塞映射等刻画，给出$r$-极小性条件，证明存在性结果，在$\mathbf{P}$分层时用$r$-极小汉明度量码刻画，还刻画相关切割$r$-阻塞集并回答问题。
AI 中文摘要
本文提出并研究关于$\mathbf{P}$-支撑的$r$-极小码，其中$\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$是在环境空间$\mathbf{H}$坐标集上定义的偏序集。$r$-极小$\mathbf{P}$-码是文献中广泛研究的汉明度量极小码的自然扩展。我们用所谓的切割$r$-阻塞映射来刻画$r$-极小$\mathbf{P}$-码，推广了极小汉明度量码与切割阻塞集之间的等价关系。还给出了基于$\mathbf{H}$上定义的$(\mathbf{P},\omega)$-权重的$r$-极小性的充要条件，推广了汉明度量极小码的阿希克明 - 巴尔格准则。接着证明了$r$-极小$\mathbf{P}$-码的两个存在性结果。当$\mathbf{P}$是分层的时，用$r$-极小汉明度量码刻画$r$-极小$\mathbf{P}$-码。最后刻画了由两级分层偏序集诱导的切割$r$-阻塞集，回答了文献[28]中提出的一个问题。
英文摘要
In this paper, we propose and study $r$-minimal codes with respect to $\mathbf{P}$-support, where $\mathbf{P}=(Ω,\preccurlyeq_{\mathbf{P}})$ is a poset defined on the coordinate set of the ambient space $\mathbf{H}$. $r$-Minimal $\mathbf{P}$-codes are natural extensions of Hamming metric minimal codes that have been extensively studied in the literature. We characterize $r$-minimal $\mathbf{P}$-codes in terms of the notion so called cutting $r$-blocking maps, which generalizes the well-known equivalence between minimal Hamming metric codes and cutting blocking sets. We also give a necessary and sufficient condition for $r$-minimality in terms of $(\mathbf{P},ω)$-weight defined on $\mathbf{H}$, where $ω:Ω\longrightarrow\mathbb{R}^{+}$ is an arbitrary weight function. This leads to a generalization of the well-known Ashikhmin-Barg criterion for Hamming metric minimal codes. We then prove two existence results for $r$-minimal $\mathbf{P}$-codes, both for general $\mathbf{P}$ and for the special case that $\mathbf{P}$ is a disjoint union of chains. When $\mathbf{P}$ is hierarchical, we characterize $r$-minimal $\mathbf{P}$-codes in terms of $r$-minimal Hamming metric codes. Finally, we characterize cutting $r$-blocking sets induced by hierarchical posets with two levels, which further enables us to answer a question raised in Hyun, Kim, Wu and Yue \cite{28}.
