One Cyclic Codes over $\mathbb{F}_{p^k} + v\mathbb{F}_{p^k} + v^2\mathbb{F}_{p^k} + ... + v^r\mathbb{F}_{p^k}$

TL;DR

This paper studies cyclic codes over a specific finite ring, generalizing previous results, providing generator polynomials, idempotents, and exploring dual code properties with applications to Gray maps.

Contribution

It generalizes the structure of cyclic codes over a complex ring, offering explicit generators and solutions to an open problem from prior research.

Findings

Codes are principally generated over the ring

Explicit generator polynomials and idempotents are provided

Properties of dual codes and Gray map applications are analyzed

Abstract

In this paper, we investigate cyclic code over the ring $\mathbb{F}_{p^k} + v\mathbb{F}_{p^k} + v^2\mathbb{F}_{p^k} + ... + v^r\mathbb{F}_{p^k}$, where $v^{r+1}=v$, $p$ a prime number, $r>1$ and $\gcd(r,p)=1$, we prove as generalisation of P. Sol\'e et al. in 2015 that these codes are principally generated, give generator polynomial and idempotent depending on idempotents over this ring as response to an open problem related by J. QIAN et al. in 2005. we also give a gray map and proprieties of the related dual code.