Fourier-Reflexive Partitions Induced by Poset Metric

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