MacWilliams Extension Property With Respect to 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
$\Omega $
. We study the MacWilliams extension property (MEP) with respect to
$(\mathbf {P},\omega)$
-weight on
$\mathbf {H}$
, where
$\mathbf {P}=(\Omega,\preccurlyeq _{\mathbf {P}})$
is a poset and
$\omega:\Omega \longrightarrow \mathbb {R}^{+}$
is a weight function. We first give a characterization of the group of
$(\mathbf {P},\omega)$
-weight isometries of
$\mathbf {H}$
, which is then used to show that MEP implies the unique decomposition property (UDP) of
$(\mathbf {P},\omega)$
, which, for the case that
$\omega $
is identically 1, further implies that
$\mathbf {P}$
is hierarchical. When
$\mathbf {P}$
is hierarchical or
$\omega $
is identically 1, with some weak additional assumptions, we give necessary and sufficient conditions for
$\mathbf {H}$
to satisfy MEP with respect to
$(\mathbf {P},\omega)$
-weight in terms of MEP with respect to Hamming weight. With the help of these results, when
$S$
is a finite field, we compare MEP with various well studied coding-theoretic properties including the property of admitting MacWilliams identity (PAMI), reflexivity of partitions, UDP, transitivity of the group of isometries and whether
$(\mathbf {P},\omega)$
induces an association scheme; in particular, we show that MEP is always stronger than all the other properties.