Isometries and MacWilliams Extension Property for Weighted Poset Metric

Let $\mathbf{H}$ be the cartesian product of a family of left modules over a ring $S$, indexed by a finite set $\mathrm{\Omega }$. We are concerned with the $(\mathbf{P},\omega )$-weight on $\mathbf{H}$, where $\mathbf{P}=(\mathrm{\Omega },{\preccurlyeq }_{\mathbf{P}})$ is a poset and $\omega :\mathrm{\Omega }⟶{\mathbb{R}}^{+}$ is a weight function. We characterize the group of $(\mathbf{P},\omega )$-weight isometries of $\mathbf{H}$, and give a canonical decomposition for semi-simple subcodes of $\mathbf{H}$ when $\mathbf{P}$ is hierarchical. We then study the MacWilliams extension property (MEP) for $(\mathbf{P},\omega )$-weight. We show that the MEP implies the unique decomposition property (UDP) of $(\mathbf{P},\omega )$, which further implies that $\mathbf{P}$ is hierarchical if $\omega$ is identically $1$. For the case that either $\mathbf{P}$ is hierarchical or $\omega$ is identically $1$, we show that the MEP for $(\mathbf{P},\omega )$-weight can be characterized in terms of the MEP for Hamming weight, and give necessary and sufficient conditions for $\mathbf{H}$ to satisfy the MEP for $(\mathbf{P},\omega )$-weight when $S$ is an Artinian simple ring (either finite or infinite). When $S$ is a finite field, in the context of $(\mathbf{P},\omega )$-weight, we compare the MEP with other coding theoretic properties including the MacWilliams identity, Fourier-reflexivity of partitions and the UDP, and show that the MEP is strictly stronger than all the rest among them.