Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes

TL;DR

This paper explores Fourier-reflexive partitions of finite abelian groups and their duals, establishing conditions for MacWilliams identities in additive codes, with applications to Frobenius rings and poset structures.

Contribution

It introduces a framework for Fourier-reflexive partitions, characterizes when biduals coincide with primal partitions, and extends results to Frobenius rings and poset-based partitions.

Findings

Dual partitions are investigated via Fourier transform.

Conditions are provided for bidual and primal partitions to coincide.

The approach is extended to Frobenius rings and poset structures.

Abstract

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures.