最优和极小的 $p$ 元线性码：来自分层偏序集的广义序理想
Optimal and minimal $p$-ary linear codes from generalized order ideals of hierarchical posets
浏览论文内容
中文总结 AI 辅助
本文引入分层偏序集的广义序理想，构造了最优和极小的 $p$ 元线性码，确定了补码的重量分布，并获得了违反 Ashikhmin--Barg 条件的无限码族。
中文摘要 AI 辅助
Hyun、Kim、Wu 和 Yue 从具有两层结构的分层偏序集的序理想构造了最优和极小的二元线性码。已知有两种将底层反链（单纯复形）设置推广到奇特征的方法：在分量序下 $\nathbb{F}_p^n$ 的下集，以及 $\nathbb{F}_q^m$ 的支撑闭子集。但偏序集设置本身尚未出现推广。我们引入了阶为 $p-1$ 的偏序集的广义序理想，通过将 ${0,\dots,p-1}$ 中的重数附加到偏序集的元素上获得，并研究了在两层分层偏序集中出现的两种自然序理想概念。每当理想遇到上层时，所得定义集既不是下集也不是支撑闭的。我们确定了相关补码的重量分布，展示了一个 Griesmer 码族，其中上层元素携带任意重数，并通过广义序理想的特征函数，获得了一个长度为 $p^n-1$、维数为 $n+1$ 的无限族极小 $p$ 元码，这些码违反了 Ashikhmin--Barg 条件。
英文摘要
Hyun, Kim, Wu and Yue constructed optimal and minimal binary linear codes from order ideals of hierarchical posets with two levels. Two different generalizations of the underlying antichain (simplicial complex) setting to odd characteristic are known: down-sets of $\mathbb{F}_p^n$ under the componentwise order, and support-closed subsets of $\mathbb{F}_q^m$. No generalization of the poset setting itself has appeared. We introduce generalized order ideals of a poset of order $p-1$, obtained by attaching multiplicities in ${0,\dots,p-1}$ to the elements of a poset, and study the two natural notions of order ideal that arise for hierarchical posets with two levels. Whenever the ideal meets the upper level, the resulting defining sets are neither down-sets nor support-closed. We determine the weight distributions of the associated complement codes, exhibit a family of Griesmer codes in which the upper element carries an arbitrary multiplicity, and, via the characteristic function of a generalized order ideal, obtain an infinite family of minimal $p$-ary codes of length $p^n-1$ and dimension $n+1$ violating the Ashikhmin--Barg condition.
发表机构
- University of South Florida（南佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。
