$(1-2u^k)$-constacyclic codes over $\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_+u^{3}\mathbb{F}_{p}+\dots+u^{k}\mathbb{F}_{p}$

TL;DR

This paper investigates a class of constacyclic codes over a specific finite ring, analyzing their structure, generator polynomials, and properties using decomposition techniques.

Contribution

It introduces the study of $(1-2u^k)$-constacyclic codes over a complex ring structure and explores their algebraic properties and generator polynomials.

Findings

Structural properties of the codes are characterized.

Generator polynomials are explicitly described.

Decomposition theorem is applied to analyze the codes.

Abstract

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.