arXiv subjects

Manju Khan

Publications and source records attributed to Manju Khan.

6 recordsLinked to original sources

arXiv2026

A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the finite field of cardinality $p^m$, and let $ R_{u^2,v^2,p^m} =\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+v\mathbb{F}_{p^m} +uv\mathbb{F}_{p^m}, ~ u^2=v^2=0,\ uv=vu, $ be a finite non-chain ring. In this paper, we study skew polycyclic codes of length $lj$ associated with $f(x)^j$, where $f(x)$ is a central polynomial of degree $l$ in $R_{u^2, v^2, p^m}[x; \Theta],$ where $\Theta$ being an automorphism of $R_{u^2,v^2,p^m}$. We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with $x^{np^s}-\lambda$, where $\gcd(n,p)=1$ and $\Theta(\lambda)=\lambda$. This reduces the study of skew $(\lambda,\Theta)$-constacyclic codes of length $np^s$ to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;\Theta]}{\langle f(x)^j\rangle}, $ where $f(x)$ is a central irreducible divisor of degree $l$ of $x^{np^s}-\lambda$, for an invertible element $\lambda\in R_{u^2,v^2,p^m}$ and $j\in\mathbb{N}$. We then apply these results to skew $(\lambda,\Theta)$-constacyclic codes of length $p^s$ for different classes of units $\lambda$. Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when $\Theta$ is the identity automorphism, we study constacyclic codes of length $np^s$ over $R_{u^2,v^2,p^m}$, according as $x^n-\alpha_0$ is irreducible or reducible over $\mathbb{F}_{p^m}$. These results extend the work of \cite{CCDF18} and \cite{ZTG18} on constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ to the finite non-chain ring $R_{u^2,v^2,p^m}$.

cs.IT↗

arXiv2026

Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}

Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;\Theta]$, where $\Theta$ is an automorphism of $R_k$ such that $xa = \Theta(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - \lambda$ of degree $l$ and multiplicity $j$ in $R_k[x;\Theta]$, where $\lambda $ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length $np^s$ over $R_k$, which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial $f(x)^j$. Using the fact that skew polycyclic codes associated with a polynomial $f(x)^j$ can be described by the left ideal structure of the quotient ring $R_k[x;\Theta]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $\Theta$. In particular, if $\lambda$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included.

cs.IT↗

arXiv2025

Cyclic codes of length $n$ over finite chain rings

In this paper, the cyclic codes of length $n$, where $n$ is odd with certain restrictions, over a finite chain ring $R$, have been studied using the structure of group algebra approach. The primitive idempotents of $RG$ of a finite cyclic group $G$ have been determined, and then the number of codewords and the minimum weight corresponding to a cyclic code have been computed.

math.RA↗

arXiv2013

Units in $FD_{2p^m}$

In this note, we compute the order and provide the structure of the unit group $\mathcal{U}(FD_{2p^m})$ of the group algebra $FD_{2p^m}$, where $F$ is a finite field of characteristic 2 and $D_{2p^m}$ is the dihedral group of order $2p^m$ such that $p$ is an odd prime. Further, we obtain the structure of the unitary subgroup $\mathcal{U}_*(FD_{2p^m})$ with respect to canonical involution * and prove that it is a normal subgroup of the unit group $\mathcal{U}(FD_{2p^m})$.

math.RA↗

arXiv2013

Units in $FD_{2p}$

In this paper, we present the structure of the group of *-unitary units in the group algebra $FD_{2p}$, where $F$ is a finite field of characteristic $p > 2$, $D_{2p}$ is the dihedral group of order $2p$, and * is the canonical involution of the group algebra $FD_{2p}$. We also provide the structure of the maximal $p$-subgroup of the unit group $\mathscr{U}(FD_{2p})$ and compute a basis of its center.

math.RA↗

arXiv2012

UNITS IN $F_2D_{2p}$

Let $p$ be an odd prime, $D_{2p}$ be the dihedral group of order 2p, and $F_{2}$ be the finite field with two elements. If * denotes the canonical involution of the group algebra $F_2D_{2p}$, then bicyclic units are unitary units. In this note, we investigate the structure of the group $\mathcal{B}(F_2D_{2p})$, generated by the bicyclic units of the group algebra $F_2D_{2p}$. Further, we obtain the structure of the unit group $\mathcal{U}(F_2D_{2p})$ and the unitary subgroup $\mathcal{U}_*(F_2D_{2p})$, and we prove that both $\mathcal{B}(F_2D_{2p})$ and $\mathcal{U}_*(F_2D_{2p})$ are normal subgroups of $\mathcal{U}(F_2D_{2p})$.

math.RA↗