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 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).