Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric
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Let 𝐇 be the cartesian product of a family of abelian groups indexed by a finite set Ω. A given poset 𝐏=(Ω,≼_𝐏) and a map ω:Ω⟶ℝ^+ give rise to the (𝐏,ω)-weight on 𝐇, which further leads to a partition 𝒬(𝐇,𝐏,ω) of 𝐇. For the case that 𝐇 is finite, we give sufficient conditions for two codewords to belong to the same block of Λ, the dual partition of 𝐇, and sufficient conditions for 𝐇 to be Fourier-reflexive. By relating the involved partitions with certain polynomials, we show that such sufficient conditions are also necessary if 𝐏 is hierarchical and ω is integer valued. With 𝐇 set to be a finite vector space over a finite field 𝔽, we extend the property of “admitting MacWilliams identity” to arbitrary pairs of partitions of 𝐇, and prove that a pair of 𝔽-invariant partitions (Λ,Γ) with |Λ|=|Γ| admits MacWilliams identity if and only if (Λ,Γ) is a pair of mutually dual Fourier-reflexive partitions. Such a result is applied to the partitions induced by 𝐏-weight and (𝐏,ω)-weight. With 𝐇 set to be a left module over a ring S, we show that each (𝐏,ω)-weight isometry of 𝐇 induces an order automorphism of 𝐏, which leads to a group homomorphism from the group of (𝐏,ω)-weight isometries to (𝐏), whose kernel consists of isometries preserving the 𝐏-support. Finally, by studying MacWilliams extension property with respect to 𝐏-support, we give a canonical decomposition for semi-simple codes C⊆𝐇 with 𝐏 set to be hierarchical.
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