Minimal Length of Nontrivial Solutions of the Isometry Equation and MacWilliams Extension Property with Respect to Weighted Poset Metric
For Rβ Mat_m(π½), the ring of all the mΓ m matrices over the finite field π½ with |π½|=q, and the left R-module Aβ Mat_m,k(π½) with m+1β©½ 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 β_i=1^m(q^i+1). In this paper, using the MΓΆbius functions, we derive the minimal length of nontrivial solutions of the isometry equation with respect to a finite lattice. For the finite vector space πββ_iβΞ©π½^k_i, a poset π=(Ξ©,βΌ_π) and a map Ο:Ξ©βΆβ^+ give rise to the (π,Ο)-weight on π, 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 (π,Ο). We give necessary and sufficient conditions for π to satisfy the MacWilliams extension property with the additional assumption that either π is hierarchical or Ο is identically 1, i.e., (π,Ο)-weight coincides with π-weight, which further allow us to partly answer a conjecture proposed by Machado and Firer in <cit.>.
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