论文标题
加权POSET度量标准的异构体和MacWilliams扩展属性
Isometries and MacWilliams Extension Property for Weighted Poset Metric
论文作者
论文摘要
令$ \ mathbf {h} $为圆环$ s $的左模块家族的笛卡尔产品，由有限套件$ω$索引。我们关注$（\ MathBf {p}，ω）$ - $ \ MathBf {H} $的重量，其中$ \ MathBf {p} =（ω，\ preccurlyeq _ {\ sathbf {\ mathbf {p of {p p o {p}}}） 功能。我们表征了$（\ Mathbf {p}，ω）$ - $ \ MATHBF {H} $的$ - 权重等法，并给出$ \ Mathbf {h} $的半简单子代码时的规范分解，当$ \ Mathbf {h} $时，当$ \ Mathbf {p} $是hierarcartical。然后，我们以$（\ Mathbf {p}，ω）$ - rige的$（\ Mathbf {p}，重量）研究MacWilliams扩展属性（MEP）。我们表明，MEP意味着$（\ Mathbf {p}，ω）$的唯一分解属性（UDP），这进一步暗示$ \ Mathbf {p} $是层次结构，如果$ω$相同$ 1 $。对于$ \ mathbf {p} $是层次结构或$ω$的$ 1 $，我们表明，$（\ Mathbf {p}，ω）$ - 重量的MEP可以以hamming重量的MEP为特征，并为$ \ Mathbf {H} $提供足够的条件，以满足$ \ Mathbf {H} $的足够条件。 $（\ mathbf {p}，ω）$ - $ s $是Artinian简单戒指（有限或无限）时。当$ s $是有限的字段时，在$（\ mathbf {p}，ω）$ - 重量的上下文中，我们将MEP与其他编码理论属性进行比较，包括MacWilliams的身份，分区和UDP的傅立叶反射性，并表明MEP严格比所有其余部分都更强。
Let $\mathbf{H}$ be the cartesian product of a family of left modules over a ring $S$, indexed by a finite set $Ω$. We are concerned with the $(\mathbf{P},ω)$-weight on $\mathbf{H}$, where $\mathbf{P}=(Ω,\preccurlyeq_{\mathbf{P}})$ is a poset and $ω:Ω\longrightarrow\mathbb{R}^{+}$ is a weight function. We characterize the group of $(\mathbf{P},ω)$-weight isometries of $\mathbf{H}$, and give a canonical decomposition for semi-simple subcodes of $\mathbf{H}$ when $\mathbf{P}$ is hierarchical. We then study the MacWilliams extension property (MEP) for $(\mathbf{P},ω)$-weight. We show that the MEP implies the unique decomposition property (UDP) of $(\mathbf{P},ω)$, which further implies that $\mathbf{P}$ is hierarchical if $ω$ is identically $1$. For the case that either $\mathbf{P}$ is hierarchical or $ω$ is identically $1$, we show that the MEP for $(\mathbf{P},ω)$-weight can be characterized in terms of the MEP for Hamming weight, and give necessary and sufficient conditions for $\mathbf{H}$ to satisfy the MEP for $(\mathbf{P},ω)$-weight when $S$ is an Artinian simple ring (either finite or infinite). When $S$ is a finite field, in the context of $(\mathbf{P},ω)$-weight, we compare the MEP with other coding theoretic properties including the MacWilliams identity, Fourier-reflexivity of partitions and the UDP, and show that the MEP is strictly stronger than all the rest among them.
