Partitions of Frobenius Rings Induced by the Homogeneous Weight

TL;DR

This paper studies how the homogeneous weight induces partitions on finite Frobenius rings, characterizing when these partitions are reflexive or self-dual, with implications for algebraic coding theory.

Contribution

It provides a detailed analysis of partitions induced by the homogeneous weight on Frobenius rings and characterizes their duality properties.

Findings

Determined the homogeneous weight values for direct products of local Frobenius rings.

Characterized rings with reflexive and self-dual partitions induced by the homogeneous weight.

Connected the partition properties to dualization via character theory.

Abstract

The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.