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

TL;DR

The paper investigates MacWilliams-type identities for codes over finite chain rings under additive metrics beyond the Hamming weight. Using Fourier-reflexive partitions, it identifies the Lee partition as the coarsest refinement of the Lee weight partition that preserves identities, and constructs significantly coarser, yet Fourier-reflexive partitions for the homogeneous and subfield metrics that still determine the respective weight enumerators. These partitions enable explicit MacWilliams-type recurrences with Krawtchouk coefficients and support linear programming bounds for code size. The results illuminate a trade-off between partition granularity and the recovery of weight enumerators, with practical LP bounds for the Lee, homogeneous, and subfield metrics.

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.

Figures (2)

• Figure 1: Illustration of $\mathrm{Comp}^{\mathop{\mathrm{wt}}\nolimits}_{\pi}(\rho)$.
• Figure 2: Illustration of $\mathrm{Comp}^{\lambda}_{\pi}(\rho)$.

Theorems & Definitions (49)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Example 9
• Lemma 10: Schur Orthogonality over a Group