Duality for Modules and Applications to Decoding Linear Codes over Finite Commutative Rings

TL;DR

This paper extends syndrome decoding algorithms from linear codes over finite fields to those over finite commutative rings using module duality, simplifying the process for local rings with a control matrix.

Contribution

It generalizes syndrome decoding to finite commutative rings and introduces a simplified algorithm for local rings using control matrices.

Findings

Decoding over finite rings is feasible with module duality.

The algorithm is simplified for local rings.

Control matrices improve decoding efficiency.

Abstract

Using linear functional-based duality of modules, we generalize the syndrome decoding algorithm of linear codes over finite fields to those over finite commutative rings. Moreover, If the ring is local the algorithm is simplified by introducing the control matrix.