All $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$

TL;DR

This paper classifies all lpha+ueta -constacyclic codes of length np^s over the ring _{p^m}+u_{p^m}, providing explicit descriptions, enumeration, and duals, based on the factorization of x^n-lpha_0.

Contribution

It offers a complete characterization and explicit descriptions of all such constacyclic codes over a specific ring, including their structure, enumeration, and dual codes.

Findings

The residue ring is a chain ring when x^n-lpha_0 is irreducible.

Explicit ideals are given when x^n-lpha_0 is reducible.

The number of codewords and dual codes are determined.

Abstract

Let $\mathbb{F}_{p^{m}}$ be a finite field with cardinality $p^{m}$ and $R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$. We aim to determine all $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where $\alpha,\beta\in\mathbb{F}_{p^{m}}^{*}$, $n, s\in\mathbb{N}_{+}$ and $\gcd(n,p)=1$. Let $\alpha_{0}\in\mathbb{F}_{p^{m}}^{*}$ and $\alpha_{0}^{p^{s}}=\alpha$. The residue ring $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$ is a chain ring with the maximal ideal $\langle x^{n}-\alpha_{0}\rangle$ in the case that $x^{n}-\alpha_{0}$ is irreducible in $\mathbb{F}_{p^{m}}[x]$. If $x^{n}-\alpha_{0}$ is reducible in $\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of $R[x]/\langle x^{np^{s}}-\alpha-u\beta\rangle$. Besides, the number of codewords and the dual code of every $\alpha+u\beta$-constacyclic code are provided.