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

For $R\triangleq Mat_{m}(\mathbb{F})$, the ring of all the $m\times m$ matrices over the finite field $\mathbb{F}$ with $|\mathbb{F}|=q$, and the left $R$-module $A\triangleq Mat_{m,k}(\mathbb{F})$ with $m+1\leqslant k$, by deriving the minimal length of solutions of the related isometry equation, Dyshko has proved in \cite{3,4} that the minimal code length $n$ for $A^{n}$ not to satisfy the MacWilliams extension property with respect to Hamming weight is equal to $\prod_{i=1}^{m}(q^{i}+1)$. In this paper, using the M\"{o}bius functions, we derive the minimal length of nontrivial solutions of the isometry equation with respect to a finite lattice. For the finite vector space $\mathbf{H}\triangleq\prod_{i\in\Omega}\mathbb{F}^{k_{i}}$, a poset $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ and a map $\omega:\Omega\longrightarrow\mathbb{R}^{+}$ give rise to the $(\mathbf{P},\omega)$-weight on $\mathbf{H}$, which has been proposed by Hyun, Kim and Park in \cite{18}. 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 $(\mathbf{P},\omega)$. We give necessary and sufficient conditions for $\mathbf{H}$ to satisfy the MacWilliams extension property with the additional assumption that either $\mathbf{P}$ is hierarchical or $\omega$ is identically $1$, i.e., $(\mathbf{P},\omega)$-weight coincides with $\mathbf{P}$-weight, which further allow us to partly answer a conjecture proposed by Machado and Firer in \cite{22}.