# Do non-free LCD codes over finite commutative Frobenius rings exist?

## TL;DR

This paper investigates the existence and properties of LCD codes over finite commutative Frobenius rings, establishing non-existence of non-free LCD codes, and characterizing free LCD codes as reversible over finite chain rings.

## Contribution

It provides necessary and sufficient conditions for LCD code existence over finite Frobenius rings and characterizes free LCD codes as reversible over finite chain rings.

## Key findings

- Non-free LCD codes do not exist over finite commutative Frobenius local rings.
- Free LCD codes over finite chain rings are reversible.
- Constructed new optimal cyclic LCD codes over Z_4.

## Abstract

In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal $\mathbb{Z}_4$ codes of different lengths {which are} cyclic LCD codes over $\mathbb{Z}_4$.