# A note on complete classification of $(\delta+\alpha u^2)$-constacyclic codes of length $p^k$ over $\F_{p^m}+u\F_{p^m}+u^2\F_{p^m}$

## TL;DR

This paper classifies and analyzes the structure of a specific class of constacyclic codes over a finite ring, focusing on their self-duality properties, for codes of length a power of a prime.

## Contribution

It provides a complete classification of $( ext{delta}+ ext{alpha} u^2)$-constacyclic codes over a finite ring of length $p^k$, including their self-dual variants.

## Key findings

- Complete classification of the codes
- Characterization of self-dual codes
- Structural insights into the code algebra

## Abstract

For units $\delta$ and $\alpha$ in $\F_{p^m}$, the structure of $(\delta+\alpha u^2)$-constacyclic codes of length $p^k$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ is studied and self-dual $(\delta+\alpha u^2)$-constacyclic codes are analyzed.