发表机构
Technical University of Denmark; UiT The Arctic University of Norway（丹麦技术大学; 挪威北极大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出多种从平展格、Orlik-Solomon代数及循环平展格计算拟阵广义重量多项式的方法，并给出线性码广义重量谱的应用及示例。
AI 中文摘要
我们提出了多种确定拟阵$M$的广义重量多项式的方法，并回顾了在给定这些多项式的情况下，如何通过码的任一生成矩阵所确定的拟阵来找到线性码的广义重量谱。一个主要目标是给编码理论研究者提供不同的方法来确定这些多项式。我们描述了如何直接从拟阵$M$的平展格中找到其广义重量多项式，并且我们还展示了如何从作为$M$的平展的收缩而出现的拟阵的相关的Orlik-Solomon代数的Poincare级数（在此情况下为多项式）中找到它们。这为利用拟阵的破圈信息来确定广义重量多项式开辟了途径。我们还回顾了拟阵的Orlik-Solomon代数与通过序同调得到的Whitney数之间的众所周知的关系，并利用它来展示如何用拟阵及其平展收缩的序同调中的Whitney数来描述重量多项式。我们简要回顾了在复数上可表示的拟阵$M$的Orlik-Solomon代数与从对应于$M$的超平面排列中若干超平面交集的补集得到的de Rham同调数之间的一个众所周知的关系。我们还描述了一种方法，用与拟阵的循环平展格相关的多项式来找到拟阵的广义重量多项式。在一个简单的运行示例中，我们展示了如何用不同的方法计算广义重量多项式，包括本文提出的方法以及早期论文中发展的选定方法。我们还包含一个不太简单的例子，即射影Reed-Muller码$PR_3(2,2)$。
英文摘要
We present various ways of determining generalized weight polynomials of a matroid $M$, and we recall how one can find the generalized weight spectra of a linear code, given these polynomials, for the matroid determined by any generator matrix of the code. A main goal is to give coding theorists different ways to determine these polynomials. We describe how one can find the generalized weight polynomials of any matroid $M$, directly from its lattice of flats, and we also show how one can find them from the Poincare series (in this case polynomials) of the associated Orlik-Solomon algebras of matroids arising as contractions of the flats of $M$. This opens for using information about broken circuits of the matroids to determine generalized weight polynomials. We also recall the well-known connection between the Orlik-Solomon algebra of a matroid, and Whitney numbers obtained by order homology, and use it to show how one can describe weight polynomials in terms of Whitney numbers from order homology of the matroid and its contraction of flats. We recall briefly a well-known relation between the Orlik-Solomon algebra of a matroid $M$, which is representable over the complex numbers, and de Rham homology numbers obtained from complements of intersections of hyperplanes in a hyperplane arrangement corresponding to $M$. We also describe a way to find the generalized weight polynomials of a matroid in terms of polynomials defined in connection with its lattice of cyclic flats. In a simple running example we show how one can calculate the generalized weight polynomials in different ways, including both the methods presented in this paper, and selected methods developed in earlier papers. We also include a less simple example with the projective Reed-Muller code $PR_3(2,2).$
Comments35 pages, 8 figures
