# Permutation codes over finite fields

## TL;DR

This paper introduces permutation codes over finite fields, generalizing cyclic and quasi-cyclic codes, and provides examples of optimal codes over various alphabets, along with their algebraic structure.

## Contribution

It presents a new class of permutation codes, explores their structure as submodules over polynomial rings, and offers examples of optimal codes over different finite fields.

## Key findings

- Examples of optimal permutation codes over binary, ternary, and 5-ary fields
- Permutation codes generalize cyclic and quasi-cyclic codes
- Structural description as submodules over polynomial rings

## Abstract

In this paper we describe a class of codes called {\it permutation codes}. This class of codes is a generalization of cyclic codes and quasi-cyclic codes. We also give some examples of optimal permutation codes over binary, ternary, and $5$-ary. Then, we describe its structure as submodules over a polynomial ring.