Ritumoni Sarma

Ankit Yadav, Nilay Kumar Mondal, Ritumoni Sarma

In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find at least 32 new or improved quaternary linear codes as per the database \cite{aydin2022updated} of best-known quaternary codes, including codes from a Plotkin-optimal family. We also report 6 projective quaternary linear codes with best-known parameters that might outperform the currently reported best-known codes due to their projectivity. Further, we establish necessary and sufficient conditions for their Gray image to be linear, which in turn gives an infinite family of Griesmer codes and several infinite families of minimal binary linear codes.

Akanksha Tiwari, Ritumoni Sarma

Let $R^t$ denote the finite chain ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle},$ where $p$ is a prime and $t$ is a positive integer. In this article, for a prime $p$ and an automorphism $θ$ of $\mathbb{F}_{p^m}$, we give the structure of the left ideals of the ring $\frac{R^t[x,Θ]}{\langle f(x) \rangle},$ where $f(x)$ is in the center of the skew polynomial ring $R^t[x,Θ]$ and $Θ$ is an automorphism of $R^t$ that extends $θ$ with $Θ(u)=u$. These left ideals are also referred to as skew polycyclic codes associated to $f(x).$ In particular, when the central element \( f(x)\) is \(x^{np^s}-λ\), where $λ=λ_0+uλ_1+\cdots +u^{t-1}λ_{t-1}$ with $λ_0\ne0,$ and \( n=1,2 \), we give a more refined form of the left ideals (which are also called skew constacyclic codes). Moreover, the case $λ_1 \neq 0$ is analyzed in detail, yielding a simpler form of generators that reveals a more refined structural characterization of the left ideals. As an application, for $n=1,t=3$ and $n=2,t=2$ we give a full description of the left ideals by including certain necessary conditions that were omitted in available literature, preventing the different classes of left ideals from being mutually disjoint and in certain cases, we also compute $i$-th torsion codes.

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.

Anuj Kumar Bhagat, Ritumoni Sarma

For a natural number $n\ge2$ which is co-prime to Char$(\mathbb{F}_q)$, let $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ denote the cyclic codes of length $n$ over $\mathbb{F}_q$ generated by the $n$-th cyclotomic polynomial $Q_n(x)$ and the polynomial $Q_n(x)Q_1(x)$, respectively. In \cite{BHAGAT2025}, the minimum distances of the codes $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ were determined, and a conjecture regarding the minimum distances of their Euclidean duals was proposed. In this article, we completely describe the structure of these dual codes and as a consequence, we find their minimum distances explicitly as functions of $n$. In fact, we resolve the conjecture in \cite{BHAGAT2025} by proving that the minimum distance of the Euclidean dual of each of $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ is equal to $2^{ω(n)}$.

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