부산대학교 도서관

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.