允许MacWilliams-type恒等式的偏序集-块空间的分类

MacWilliams identities for poset level weight enumerators of linear codes

Codes over various metrics such as Rosenbloom-Tsfasman (RT), Lee, etc. have been considered. Recently, codes over poset metrics have been studied. Poset metric is a great generalization of many metrics especially the well-known ones such as the RT and the Hamming metrics. Poset metric can be realized on the channels with localized error occurrences. It has been shown that MacWilliams identities are not admissible for codes over poset metrics in general [Kim and Oh, 2005]. Lately, to overcome this problem some further studies on MacWilliams identities over poset metrics has been presented. In t…

Isometries and MacWilliams Extension Property for Weighted Poset Metric

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…

Combinatorial metrics: MacWilliams-type identities, isometries and extension property

In this work we characterize the combinatorial metrics admitting a MacWilliams-type identity and describe the group of linear isometries of such metrics. Considering coverings that are not connected, we classify the metrics satisfying the MacWilliams extension property.

环上码的MacWilliams扩张定理和局部-整体性质

MacWilliams Extension Theorems and the Local-Global Property for Codes over Rings

研究了有限Frobenius环上各种权函数的MacWilliams扩张定理。这个问题是重新制定的一般线性群的子群的局部-全局性质。除其他事项外，它表明，扩展定理成立偏序集的权重，当且仅当底层偏序集是层次的。具体来说，向量码的Rosenbloom-Tsfasman权满足扩张定理，而矩阵码的Niederreiter-Rosenbloom-Tsfasman权不满足扩张定理。著名的齐次权的MacWilliams扩张定理的一个简短的特征理论证明。此外，它表明，扩展定理结转到直接产品的重量，但不对称产品。

Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

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 (ME…