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 work addresses the classification and enumeration of self-dual cyclic (constacyclic) codes of length $p^s$ over the ring $R_3 = \bF_{2^m}[u]/\langle u^3 \rangle$, showing that self-duality in this setting requires $p=2$ and focusing on length $2^s$. By exploiting the torsion-code framework and the structure of the chain ring $\mathcal{K}$, it fully classifies self-dual cyclic codes over $R_3$, deriving explicit generator forms for types 4–8 and establishing necessary and sufficient duality conditions. It provides comprehensive enumeration formulas, including new codes not captured by prior results (notably correcting and extending Kim & Lee 2020), with concrete counts in the binary-length case (e.g., length $8$ yields 18 such codes). The findings advance the theory of self-dual codes over finite chain rings and offer rigorous construction methods for structured, potentially robust error-correcting codes.

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}.

Theorems & Definitions (26)

• Proposition 1.1
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Lemma 2.6
• Lemma 2.7: sobhani2015complete
• Theorem 3.1
• proof