Classification of Self-Dual Constacyclic Codes of Prime Power Length $p^s$ Over $\frac{\mathbb{F}_{p^m}[u]}{\left\langle u^3\right\rangle} $

TL;DR

This paper classifies and enumerates all self-dual cyclic codes of length 2^s over a specific ring extension, completing the classification for prime power lengths and correcting previous results.

Contribution

It provides a complete classification and enumeration of self-dual cyclic codes of length 2^s over the ring extension, and corrects earlier incomplete results.

Findings

Self-dual cyclic codes exist only when p=2.

Complete classification of these codes for length 2^s.

Corrections and improvements to previous classifications.

Abstract

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime number and $m$ is a positive integer. Self-dual constacyclic codes of length \( p^s \) over \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \) exist only when \( p = 2 \). In this work, we classify and enumerate all self-dual cyclic codes of length \( 2^s \) over \( \frac{\mathbb{F}_{2^m}[u]}{\langle u^3 \rangle} \), thereby completing the classification and enumeration of self-dual constacyclic codes of length \( p^s \) over \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \). Additionally, we correct and improve results from B. Kim and Y. Lee (2020) in \cite{kim2020classification}.