Combinatorial metrics: MacWilliams-type identities, isometries and extension property

TL;DR

This paper characterizes combinatorial metrics that satisfy MacWilliams-type identities, describes their isometry groups, and classifies those with the extension property, advancing understanding of metric symmetries in coding theory.

Contribution

It provides a complete characterization of combinatorial metrics with MacWilliams identities and classifies metrics with the extension property, including non-connected coverings.

Findings

Identified conditions for combinatorial metrics to admit MacWilliams-type identities.

Described the group of linear isometries for these metrics.

Classified metrics satisfying the extension property, including non-connected cases.

Abstract

In this work we characterize the combinatorial metrics admitting a MacWilliams-type identity and describe the group of linear isometries of such metrics. Considering coverings that are not connected, we classify the metrics satisfying the MacWilliams extension property.