论文标题
由加权POSET度量和组合度量引起的分区的反射性
Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric
论文作者
论文摘要
令$ \ mathbf {h} $为有限的阿贝尔群体家族的笛卡尔产品。通过多项式方法，我们为加权Poset指标引起的$ \ mathbf {h} $的分区提供了足够的条件，这对于某些特殊情况也是必要的。此外，通过检查krawtchouk多项式的根源，我们建立了组合度量诱导的$ \ mathbf {h} $的非反射分区。当$ \ mathbf {h} $是有限字段$ \ mathbb {f} $上的矢量空间时，我们会考虑接纳MacWilliams Identity（PAMI）和MacWilliams Extension属性（MEP）的属性，用于$ \ Mathbf {H} $。有了一些不变性的假设，我们表明$ \ mathbf {h} $的两个分区允许MacWilliams身份当时，仅当它们是相互双重的和反思的，并且任何满足MEP的$ \ mathbf {h h} $的分区实际上是由某些轨道分区所引起的。 $ \ aut _ {\ Mathbb {f}}（\ Mathbf {h}）$，这必然是反射性的。作为上述结果的应用，我们建立了由组合指标引起的$ \ mathbf {h} $的分区，这些分区不满足MEP，这进一步使我们能够为Pinheiro，Machado，Machado和Firer提出的猜想提供反示例。
Let $\mathbf{H}$ be the Cartesian product of a family of finite abelian groups. Via a polynomial approach, we give sufficient conditions for a partition of $\mathbf{H}$ induced by weighted poset metric to be reflexive, which also become necessary for some special cases. Moreover, by examining the roots of the Krawtchouk polynomials, we establish non-reflexive partitions of $\mathbf{H}$ induced by combinatorial metric. When $\mathbf{H}$ is a vector space over a finite field $\mathbb{F}$, we consider the property of admitting MacWilliams identity (PAMI) and the MacWilliams extension property (MEP) for partitions of $\mathbf{H}$. With some invariance assumptions, we show that two partitions of $\mathbf{H}$ admit MacWilliams identity if and only if they are mutually dual and reflexive, and any partition of $\mathbf{H}$ satisfying the MEP is in fact an orbit partition induced by some subgroup of $\Aut_{\mathbb{F}}(\mathbf{H})$, which is necessarily reflexive. As an application of the aforementioned results, we establish partitions of $\mathbf{H}$ induced by combinatorial metric that do not satisfy the MEP, which further enable us to provide counter-examples to a conjecture proposed by Pinheiro, Machado and Firer in \cite{39}.
