On Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$ and their Cardinalities
Akanksha Tiwari, Pramod Kanwar, Ritumoni Sarma
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 $φ$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ ker(φ)=\langle π\rangle$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots,a_n$ in $I$ such that $I=\langle a_1,a_2,\dots,a_n\rangle+π(I:π)$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x]}{\langle ω(x)\rangle},$ where $ω(x)\in \frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x] $. For the case when $ω(x)=f(x)^{p^s}$, 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 $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$.