Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric
Yang Xu, Haibin Kan, Guangyue Han · IEEE Transactions on Information Theory · 2023
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 scenarios. Moreover, by examining the roots of the Krawtchouk polynomials, we give sufficient conditions for a partition of$\mathbf {H}$induced by combinatorial metric to be non-reflexive, and then give several examples of non-reflexive partitions. 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$\mathbf {H}$. More specifically, under 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 MEP is in fact an orbit partition induced by some subgroup of$\mathrm {Aut}\,_{\mathbb {F}}(\mathbf {H})$, which is necessarily reflexive. Furthermore, we show that the aforementioned non-reflexive partitions induced by combinatorial metric do not satisfy MEP, which further enables us to disprove a conjecture proposed by Pinheiro et al., (2019).