Pramod Kanwar

Akanksha Tiwari, Pramod Kanwar, Ritumoni Sarma

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 $R^{t,n}_δ:=\frac{R^t[x]}{\langle x^{np^s}-δ\rangle},$ where $δ= δ_0+uδ_1+\dots+u^{t-1}δ_{t-1}$ 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.

Akanksha Tiwari, Pramod Kanwar, Ritumoni Sarma

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 $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$.