Minimal Length of Nontrivial Solutions of the Isometry Equation and MacWilliams Extension Property with Respect to Weighted Poset Metric
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For R≜ Mat_m(𝔽), the ring of all the m× m matrices over the finite field 𝔽 with |𝔽|=q, and the left R-module A≜ Mat_m,k(𝔽) with m+1⩽ k, by deriving the minimal length of solutions of the related isometry equation, Dyshko has proved in <cit.> that the minimal code length n for A^n not to satisfy the MacWilliams extension property with respect to Hamming weight is equal to ∏_i=1^m(q^i+1). In this paper, using the Möbius functions, we derive the minimal length of nontrivial solutions of the isometry equation with respect to a finite lattice. For the finite vector space 𝐇≜∏_i∈Ω𝔽^k_i, a poset 𝐏=(Ω,≼_𝐏) and a map ω:Ω⟶ℝ^+ give rise to the (𝐏,ω)-weight on 𝐇, which has been proposed by Hyun, Kim and Park in <cit.>. For such a weight, we study the relations between the MacWilliams extension property and other properties including admitting MacWilliams identity, Fourier-reflexivity of involved partitions and Unique Decomposition Property defined for (𝐏,ω). We give necessary and sufficient conditions for 𝐇 to satisfy the MacWilliams extension property with the additional assumption that either 𝐏 is hierarchical or ω is identically 1, i.e., (𝐏,ω)-weight coincides with 𝐏-weight, which further allow us to partly answer a conjecture proposed by Machado and Firer in <cit.>.
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