MacWilliams Extension Property With Respect to Weighted Poset Metric

Abstract

<p>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.</p>