부산대학교 도서관

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).