Constacyclic codes of length $np^s$ over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t\rangle}$: Torsions and Cardinalities

2026-05-12

AI Summary

This study investigates the structure of constacyclic codes of length $np^s$ over the finite chain ring $R = \mathbb{F}_{p^m}[u]/\langle u^t \rangle$, where $\gcd(n,p)=1$. By analyzing the ideal generators of the quotient ring $R[x]/\langle x^{np^s} - \delta \rangle$ and employing algebraic tools such as finite field extensions, polynomial factorization, and the module structure over local rings, the work provides a systematic characterization of these codes. The main contribution lies in the first complete classification of all constacyclic codes for the cases $n=1,2,3$ with $t=3$, explicitly describing their generator forms and precisely determining their torsion degrees and cardinalities, thereby extending the frontier of coding theory over finite chain rings.

Abstract

The purpose of this article is to study constacyclic codes of length $np^s$ over $R^t:=\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle },$ where $t$ is a natural number and $\gcd(n,p)=1$. We give generators of all the ideals of $R^{t,n}_δ:=\frac{R^t[x]}{\langle x^{np^s}-δ\rangle},$ where $δ= δ_0+uδ_1+\dots+u^{t-1}δ_{t-1}$ is a unit in $R^t$. For $n=1,\ 2, \ 3$ and $t=3$, we provide all types of ideals (constacyclic codes) and also give the torsional degrees as well as cardinalities of these codes.

Problem

• constacyclic codes
• finite chain rings
• ideal structure
• torsion degrees
• code cardinalities

Innovation

• constacyclic codes
• finite chain rings
• torsion degrees
• ideal generators
• code cardinalities