Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

Let H be the Cartesian product of a family finite abelian groups. Via polynomial approach, we give sufficient conditions for partition induced by weighted poset metric to reflexive, which also become necessary some special scenarios. Moreover, examining roots Krawtchouk polynomials, combinatorial non-reflexive, and then several examples non-reflexive partitions. When is vector space over field F, consider property admitting MacWilliams identity (PAMI) extension (MEP) partitions H. More specifically, under invariance assumptions, show that two admit if only they are mutually dual any satisfying MEP in fact an orbit subgroup Aut F (H), necessarily reflexive. Furthermore, aforementioned do not satisfy MEP, further enables us disprove conjecture proposed Pinheiro, Machado Firer [39].