论文标题
WEI型二元定理的Galois连接方法
A Galois Connection Approach to Wei-Type Duality Theorems
论文作者
论文摘要
在$ 1991 $中，WEI证明了双重性定理，该定理在线性代码的广义锤式权重与其双重代码的广义锤式权重之间建立了有趣的联系。此后，WEI的二元定理从不同的角度进行了广泛的研究，并扩展到其他设置。在本文中，从新的Galois连接角度来看，我们重新检查了WEI的双重定理及其各种扩展，此后称为Wei-Type二元定理。我们的方法是基于这样的观察结果，即广义锤击权重和线性代码的尺寸/长度轮廓形成了GALOIS连接。本文的主要结果是$ \ mathbb {z} $之间的两个Galois连接的一般wei型二元定理，可以从中恢复所有已知的Wei-Type二元定理。作为我们的中心成果的基础，我们证明了定义在有限集和$ w $ demi-demi-polymatroids上定义的$ w $ diality定理的新型二元定理，并在模块上定义了具有组成系列的模块，这使我们能够将所有已知的wei-type type duality theere theores neces enders enders建立，以便我们进一步统一并普遍化。
In $1991$, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei's duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei's duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result in this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb{Z}$, from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$-demimatroids defined over finite sets and $w$-demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.
