Coarsest Fourier-reflexive Partitions for the Lee, Homogeneous and Subfield Metric

TL;DR

This paper investigates Fourier-reflexive partitions for various metrics over finite chain rings, establishing the coarsest partitions that preserve MacWilliams identities and enabling new bounds for code analysis.

Contribution

It characterizes the coarsest Fourier-reflexive partitions for Lee, homogeneous, and subfield metrics, revealing their structure and implications for weight enumerators.

Findings

The Lee partition is the coarsest Fourier-reflexive partition refining the weight partition.

New coarser partitions for homogeneous and subfield metrics still preserve MacWilliams identities.

Derived linear programming bounds for codes using these partitions.

Abstract

MacWilliams identities relate the weight enumerators of a code with those of its dual and are classically formulated with respect to the Hamming weight. For other metrics, however, these identities often fail when considering the weight partition of the ambient space. It is known that MacWilliams identities hold for enumerators associated with Fourier-reflexive partitions, and that orbits of subgroups of the linear isometry group always yield such partitions. This raises the question whether, for metrics beyond the Hamming metric, there exist meaningful partitions that lie strictly between the weight partition and the orbit partition: finer than the latter, yet still coarse enough to retain useful MacWilliams-type identities. In this work, we study this question for finite chain rings endowed with additive metrics. For the Lee metric, we show that the partition induced by the action of the full group of linear isometries is already the coarsest Fourier-reflexive partition refining the weight partition. In particular, no intermediate partition exists that is both finer than the Lee weight partition and Fourier-reflexive. We refer to this partition as the Lee partition and show that it allows the recovery of all additive weight enumerators over the ring. In contrast, for the homogeneous metric and for the subfield metric, we identify new, significantly coarser symmetrized partitions that remain Fourier-reflexive and still allow the recovery of the corresponding weight enumerators. We prove that these partitions are the coarsest such symmetrized partitions for which MacWilliams-type identities hold. As an application, we derive linear programming bounds based on the resulting MacWilliams identities.