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

Abstract

The purpose of this article is to study polycyclic codes over the ring $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^t⟩},t\ge 1$, and their associated torsion codes. It is shown that if $\varphi$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ker(\varphi )=⟨\pi ⟩$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots ,a_n$ in $I$ such that $I=⟨a_1,a_2,\dots ,a_n⟩+\pi (I:\pi )$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^t⟩}[x]}{⟨\omega (x)⟩},$ where $\omega (x)\in \frac{{\mathbb{F}}_{p^m}[u]}{⟨u^t⟩}[x]$. 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 $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^4⟩}$ and illustrate the result with some examples that verify the computed cardinality.