Fourier-Reflexive Partitions Induced by Poset Metric
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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 <cit.>. 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.
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