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Om Prakash

Publications and source records attributed to Om Prakash.

At least 37 records · Page 2Linked to original sources
arXiv2025

Skew Generalized Polycyclic Codes with Derivations

In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;τ_1,δ_{τ_1}]$\\$[z_2;τ_2,δ_{τ_2}]$, where $\mathscr{R}$ is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring $\mathscr{R}$ and finite field $\mathbb{F}_q$, where $q=p^m$ for some positive integer $m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed.

cs.IT↗
arXiv2024

Commuting Jordan derivations over a Ring with idempotents

This paper explores the behaviour of commuting Jordan derivations over prime rings with non-trivial idempotents and demonstrates that they become zero maps. Further, it establishes this result for commuting Jordan higher derivations over prime rings under specific conditions. In fact, it introduces commuting generalized Jordan derivations over rings and establishes that the zero map is the only commuting generalized Jordan derivation over prime rings under certain assumptions.

math.RA↗
arXiv2024

Galois hulls of constacyclic codes over affine algebra rings

Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a square-free polynomial over $\mathbb{F}_q$. This paper studies the $k$-Galois hulls of $λ$-constacyclic codes over $\mathcal A$ regarding their idempotent generators. For this, first, we define the $k$-Galois inner product over $\mathcal A$ and find the form of the generators of the $k$-Galois dual and the $k$-Galois hull of a $λ$-constacyclic code over $\mathcal A$. Then, we derive a formula for the $k$-Galois hull dimension of a $λ$-constacyclic code. Further, we provide a condition for a $λ$-constacyclic code to be $k$-Galois LCD. Finally, we give some examples of the use of these codes in constructing entanglement-assisted quantum error-correcting codes.

cs.IT↗
arXiv2024

Roman domination number of zero-divisor graphs over commutative rings

For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \rightarrow \{0, 1, 2\}$ satisfies the property that if $f(v) = 0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)= 2$. The weight of a Roman dominating function $f$ is the value $f(V)= Σ_{u \in V} f(u)$, and the minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$, denoted by $γ_R(G)$. The main focus of this paper is to study the Roman domination number of zero-divisor graph $Γ(R)$ and find the bounds of the Roman domination number of $T(Γ(R))$.

math.CO↗
arXiv2024

There is no 290-Theorem for higher degree forms

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290-Theorem of Bhargava-Hanke.

math.NT↗
arXiv2024

No proper generalized quadratic forms are universal over quadratic fields

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.

math.NT↗
arXiv2024

Northcott property and universality of higher degree forms

Let $K$ be a totally real number field, $d$ a positive integer, and $Q$ a higher degree form over $K$. We prove that there are at most finitely many totally real extensions $L/K$ of degree $d$ such that $Q$ over $L$ is universal. Further, we show that there are no universal forms over totally real infinite extensions of $\mathbb{Q}$ having the Northcott property.

math.NT↗
arXiv2024

Highly Sensitive Label-free Biomolecular Detection Using Au-WS2 Nanohybrid Based SERS Substrates

Recent advancements in nanotechnology have led to the development of surface-enhanced Raman spectroscopy (SERS) based rapid and low-cost technologies for ultra-sensitive label-free detection and identification of molecular analytes. Herein, we utilized the synergistic plasmonic and chemical enhancement effects of Au-WS2 nanohybrids to attain the high-intensity Raman signals of targeted analytes. To develop these nanohybrids, a series of monodispersed Au nanoparticles (NPs) of varying diameters from 20 to 80 nm was chemically synthesized and successively blended with liquid-phase exfoliated WS2 nano-flakes of average lateral size 90 nm. They provided a maximum enhancement factor (EF) of ~1.80 109 corresponding to the characteristic peaks at 1364 cm-1 and 1512 cm-1 for R6G analyte molecules. Theoretical studies based on the finite-difference time-domain simulations on Au-WS2 nanohybrid systems revealed a huge field-intensity enhancement with an EF of more than 1000 at the plasmonic hotspots, which was induced by the strong coupling of individual plasmon oscillations of the adjacent Au NPs upon light interactions. These electromagnetic effects along with the chemical enhancement effects of WS2 nanoflakes were found to be mainly responsible for such huge enhancement in Raman signals. Furthermore, these hybrids were successfully employed for achieving highly sensitive detection of the E. coli ATCC 35218 bacterial strain with a concentration of 104 CFU/mL in phosphate-buffered saline media, indicating their real capabilities for practical scenarios. The findings of the present study will indeed provide vital information in the development of innovative nanomaterial-based biosensors, that will offer new possibilities for addressing critical public health concerns.

physics.app-ph↗
arXiv2024

Double skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$

In this study, in order to get better codes, we focus on double skew cyclic codes over the ring $\mathrm{R}= \mathbb{F}_q+v\mathbb{F}_q, ~v^2=v$ where $q$ is a prime power. We investigate the generator polynomials, minimal spanning sets, generator matrices, and the dual codes over the ring $\mathrm{R}$. As an implementation, the obtained results are illustrated with some good examples. Moreover, we introduce a construction for new generator matrices and thus achieve codes with improved parameters compared to those found in existing literature. Finally, we tabulate our obtained block codes over the ring $\mathrm{R}$.

cs.IT↗
arXiv2024

High-Frequency Stock Market Order Transitions during the US-China Trade War 2018: A Discrete-Time Markov Chain Analysis

Statistical analysis of high-frequency stock market order transaction data is conducted to understand order transition dynamics. We employ a first-order time-homogeneous discrete-time Markov chain model to the sequence of orders of stocks belonging to six different sectors during the USA-China trade war of 2018. The Markov property of the order sequence is validated by the Chi-square test. We estimate the transition probability matrix of the sequence using maximum likelihood estimation. From the heat-map of these matrices, we found the presence of active participation by different types of traders during high volatility days. On such days, these traders place limit orders primarily with the intention of deleting the majority of them to influence the market. These findings are supported by high stationary distribution and low mean recurrence values of add and delete orders. Further, we found similar spectral gap and entropy rate values, which indicates that similar trading strategies are employed on both high and low volatility days during the trade war. Among all the sectors considered in this study, we observe that there is a recurring pattern of full execution orders in Finance & Banking sector. This shows that the banking stocks are resilient during the trade war. Hence, this study may be useful in understanding stock market order dynamics and devise trading strategies accordingly on high and low volatility days during extreme macroeconomic events.

q-fin.ST↗
arXiv2023

Construction of $(σ,δ)$-cyclic codes over a non-chain ring and their applications in DNA codes

For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the finite field of characteristic $p$, and $\mathfrak{R}_l:=\mathbb{F}_{p^m}[v]/\langle v^l-v\rangle$ be a non-chain ring. In this paper, we study the $(σ,δ)$-cyclic codes over $\mathfrak{R}_l$. Further, we study the application of these codes in finding DNA codes. Towards this, we first define a Gray map to find classical codes over $\mathbb{F}_{p^m}$ using codes over the ring $\mathfrak{R}_l$. Later, we find the conditions for a code to be reversible and a DNA code using $(σ, δ)$-cyclic code. Finally, this algebraic method provides many classical and DNA codes of better parameters.

cs.IT↗
arXiv2023

$\mathbb{F}_q\mathcal{R}$-skew cyclic codes and their application to quantum codes

Let $p$ be a prime and $\mathbb{F}_q$ be the finite field of order $q=p^m$. In this paper, we study $\mathbb{F}_q\mathcal{R}$-skew cyclic codes where $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$ with $u^2=u$. To characterize $\mathbb{F}_q\mathcal{R}$-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over $\mathbb{F}_q\mathcal{R}$ and obtain their $\mathbb{F}_q$-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing $\mathbb{F}_q\mathcal{R}$-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.

cs.IT↗
arXiv2023

On ZpZp[u, v]-additive cyclic and constacyclic codes

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(α,β)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle x^α-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^β-1\rangle$ where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $β$ over $\mathbb{Z}_{p}[u,v]$ and $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length $(α,β)$ respectively. We show that the Gray image of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length $(α,β)$ is either a QC code of length $4α$ with index $4$ or a generalized QC code of length $(α,3β)$ over $\mathbb{Z}_{p}$. Moreover, some structural properties like generating polynomials, minimal generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic code with length $(α,p-1)$ are determined.

cs.IT↗
arXiv2023

Generalized Fruit Diophantine equation and Hyperelliptic curves

We show the insolvability of the Diophantine equation $ax^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1 \pmod {12}$ and $b=2^da-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we investigate the more general family with $b=2^da-3^r$, where $r$ is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over $\mathbb{Q}$. We conclude by providing some numerical evidence corroborating the main results.

math.NT↗
arXiv2023

Some generalized Jordan maps on triangular rings force additivity

In this paper, we show that a map $δ$ over a triangular ring $\mathcal{T}$ satisfying $δ(ab+ba)=δ(a)b+a τ(b)+δ(b)a+bτ(a)$, for all $a,b\in \mathcal{T}$ and for some maps $τ$ over $\mathcal{T}$ satisfying $τ(ab+ba)=τ(a)b+a τ(b)+τ(b)a+bτ(a)$, is additive. Also, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(ab)=T(a)b=aT(b)$, for all $a,b\in \mathcal{T}$, is additive. Further, we establish that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(ab)=2mD(a)b+2naD(b)$, for all $a,b\in \mathcal{T}$ and integers $m,n\geq 1$, then $D$ is additive.

math.RA↗
arXiv2023

Constacyclic codes of length $4p^s$ over the Galois ring $GR(p^a,m)$

For prime $p$, $GR(p^a,m)$ represents the Galois ring of order $p^{am}$ and characterise $p$, where $a$ is any positive integer. In this article, we study the Type (1) $λ$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $λ=ξ_0+pξ_1+p^2z$, $ξ_0,ξ_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $λ$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,λ)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-λ\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-δ\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+δ\rangle}$. In second, when $λ$ is not a square, we show that $\mathcal{R}_p(a,m,λ)$ is a chain ring whose ideals are $\langle (x^4-α)^i\rangle\subseteq \mathcal{R}_p(a,m,λ)$, for $0\leq i\leq ap^s$ where $α^{p^s}=ξ_0$. Also, we prove the dual of the above code is $\langle (x^4-α^{-1})^{ap^s-i}\rangle\subseteq \mathcal{R}_p(a,m,λ^{-1})$ and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) $λ$-constacyclic codes of length $4p^s$ are obtained when $λ$ is not a square.

cs.IT↗
arXiv2022

Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities

Let $\mathbb{Z}_p$ be the ring of integers modulo a prime number $p$ where $p-1$ is a quadratic residue modulo $p$. This paper presents the study of constacyclic codes over chain rings $\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langle u^2\rangle}$ and $\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}$. We also study additive constacyclic codes over $\mathcal{R}\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S}$ using the generator polynomials over the rings $\mathcal{R}$ and $\mathcal{S},$ respectively. Further, by defining Gray maps on $\mathcal{R}$, $\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S},$ we obtain some results on the Gray images of additive codes. Then we give the weight enumeration and MacWilliams identities corresponding to the additive codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$. Finally, as an application of the obtained codes, we give quantum codes using the CSS construction.

cs.IT↗