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

TL;DR

This paper investigates the structure, generators, torsional degrees, and cardinalities of constacyclic codes of length $np^s$ over a specific finite ring extension, providing explicit classifications for small lengths and specific parameters.

Contribution

It offers a comprehensive description of all ideals (codes) over the ring $R^t$ for certain lengths, including generators, torsional degrees, and sizes, extending understanding of constacyclic codes over these rings.

Findings

Explicit generators of all ideals in the studied ring are provided.

Classifications of constacyclic codes for lengths 1, 2, 3 with $t=3$ are given.

Torsional degrees and cardinalities of these codes are explicitly calculated.

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 $R^{t,n}_{\delta}:=\frac{R^t[x]}{\langle x^{np^s}-\delta \rangle},$ where $\delta= \delta_0+u\delta_1+\dots+u^{t-1}\delta_{t-1}$ 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.