Fourier-Reflexive Partitions Induced by Poset Metric

Let H be the cartesian product of a family finite abelian groups indexed by set Ω. A given poset (i.e., partially ordered set) P = (Ω, ≼P) gives rise to metric on H, which further leads partition Q(H, P) H. We prove that if is Fourier-reflexive, then its dual Λ coincides with Ĥ induced P, and moreover, necessarily hierarchical. This result establishes conjecture proposed Gluesing-Luerssen in [5]. also show some other assumptions, finer than P. In addition, we give necessary sufficient conditions for hierarchical, case an explicit criterion determining whether two codewords belong same block Λ. these results relating involved partitions certain polynomials, generalized version studied generalize aforementioned results.