On Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$ and their Cardinalities

TL;DR

This paper investigates the structure and cardinalities of polycyclic codes over a specific ring extension, providing explicit generators and torsion code properties, especially for the case when t=4.

Contribution

It introduces a method to generate all ideals in certain polynomial quotient rings and computes code cardinalities using torsional degrees for specific polynomial cases.

Findings

Derived generators for all ideals in the ring extension.

Computed torsion ideals and degrees for the case t=4.

Calculated the cardinality of polycyclic codes using torsional degrees.

Abstract

The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $\phi$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ ker(\phi)=\langle \pi\rangle$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots,a_n$ in $I$ such that $I=\langle a_1,a_2,\dots,a_n\rangle+\pi(I:\pi)$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x]}{\langle \omega(x)\rangle},$ where $\omega(x)\in \frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x] $. For the case when $\omega(x)=f(x)^{p^s}$, where $f(x)$ is an irreducible polynomial in $\mathbb{F}_{p^m}[x]$ and $s$ is a non-negative integer, we obtain several other results including computation of torsion ideals and their torsional degrees when $t=4$. We use the torsional degree to compute the cardinality of polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$.