Minimal Length of Nontrivial Solutions of the Isometry Equation and MacWilliams Extension Property with Respect to Weighted Poset Metric

For $R \triangleq Ma{t_m}({\mathbb{F}})$, the ring of all m × m matrices over the finite field ${\mathbb{F}}$ with $|{\mathbb{F}}| = q$, and the left R-module $A \triangleq Ma{t_{m,k}}({\mathbb{F}})$ with m + 1 ⩽ k, by deriving the minimal length of solutions of the related isometry equation, Dyshko has proved in [3], [4] that the minimal code length n for A
ⁿ
not satisfying the MacWilliams extension property (MEP) with respect to Hamming weight is equal to $\prod\nolimits_{i = 1}^m {\left( {{q^i} + 1} \right)}$. In this paper, using the Möbius functions, we derive the minimal length of nontrivial solutions of the isometry equation for a finite lattice. For the finite vector space ${\mathbf{H}} \triangleq \prod\nolimits_{i \in \Omega } {{{\mathbb{F}}^{{k_i}}}}$, 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 [18]. For such a weight, we study the relations between the MEP and other properties including admitting MacWilliams identity, Fourier-reflexivity of involved partitions and the Unique Decomposition Property (UDP) defined for (P, ω). We give necessary and sufficient conditions for H to satisfy the MEP 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 [22].