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

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 $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$ and illustrate the result with some examples that verify the computed cardinality.