# Additive cyclic codes over finite commutative chain rings

## TL;DR

This paper extends the study of additive cyclic codes to finite commutative chain rings, revealing new properties and dual code behaviors, with concrete examples illustrating these findings.

## Contribution

It generalizes previous work on Galois rings to broader chain rings and uncovers two types of additivity with distinct dual code properties.

## Key findings

- Two types of additivity identified in non-Galois chain rings
- Unusual dual code properties observed in one additivity type
- Concrete examples illustrating theoretical results

## Abstract

Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.