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Krishna Kaipa

Publications and source records attributed to Krishna Kaipa.

12 recordsLinked to original sources
arXiv2026

Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$

Let $\pi$ be a cuspidal representation of $GL(n,F)$ over a finite field $F$. Let $P=MN$ be the Levi decomposition of a maximal parabolic subgroup corresponding to the partition $(k,n-k)$ of $n$. Given a rank $r$ character $\psi_r$ of the unipotent radical $N$, the twisted Jacquet module $\pi_{N, \psi_r}$ is a representation of the subgroup $M_r$ of $M$ which stabilizes $\psi_r$. The problem we solve in this work is to determine the structure of $\pi_{N, \psi_r}$ as a $M_r$-module. This problem was first studied by D. Prasad, who solved it for the case $r=k=n/2$ by calculating the character of $\pi_{N, \psi_r}$ and matching it to a known representation of $M_r$. In this work, we solve the problem for all values of $(r,k,n)$ directly without calculating the character of $\pi_{N, \psi_r}$. Our solution depends on two other key conceptual advances: (i) We generalize the Bernstein-Zelevinsky framework for studying representations of the Mirabolic subgroup of $GL(n,F)$, to maximal parabolic subgroups $P$. In particular, we show that the twisted Jacquet functor which takes a representation of $P$ to its twisted Jacquet modules, gives an equivalence of categories between Rep$(P)$ and the direct sum $\oplus_r \text{Rep}(M_r)$. (ii) Using this, we construct a pair of recursively defined representations $\Pi_{k,n}, \Pi_{n-k,n}^\dagger$ of $P$, which generalizes to $P$, the representation of the Mirabolic subgroup obtained from the trivial representation by recursively applying the Bernstein-Zelevinsky $\Phi^+$ functor. Like the representation $(\Phi^+)^{n-1}(1)$ of the Mirabolic subgroup, the representation $\Pi_{n-k,n}^\dagger$ satisfies a universal property with respect to restrictions to $P$ of cuspidal representations of $GL(n,F)$. Our solution of the main problem is a simple consequence of this universal property.

math.RT↗
arXiv2025

Incidence of lines, points, and planes in $PG(3,q)$ with respect to the twisted cubic

We consider the orbits of the group $G=PGL_2(q)$ on the points, lines and planes of the projective space $PG(3,q)$ over a finite field $\mathbb F_q$ of characteristic different from $2$ and $3$. The points of $PG(3,q)$ can be identified with projective space of binary cubic forms, and the set $\mathcal L$ of lines of $PG(3,q)$ can be thought of as pencils of cubic forms. The action of $G$ on $PG(1,q)$ naturally induces an action of $G$ on binary cubic forms $f(X,Y)$. The points of $PG(3, q)$ decompose into five $G$ orbits. The $G$ orbits on $\mathcal L$ were recently obtained by the authors. Let $\mathcal I$ be the subset of $\mathcal L \times PG(3,q)$ consisting of pairs $(L,P)$ where $L$ is a line incident with the point $P$. The decomposition of $\mathcal L \times PG(3,q)$ into $G \times G$ orbits yields a partition of $\mathcal I$. The problem that we solve in this work is to determine the sizes of the corresponding parts of $\mathcal I$.

math.CO↗
arXiv2025

On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms in characteristic three

We consider the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $\mathbb{F}_q$ into orbits of the group $PGL_2(q)$ of linear symmetries of the twisted cubic $C$. The problem has been solved in literature in characteristic different from $3$, and in this work, we solve the problem in characteristic $3$. We reduce this problem to another problem, which is the classification of binary quartic forms into $PGL_2(q)$-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.

math.CO↗
arXiv2024

On the cardinality of matrices with prescribed rank and partial trace over a finite field

Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{\alpha}_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A,r}^{\alpha}$. We also solve the generalization of the problem to rectangular matrices.

math.RA↗
arXiv2024

Higher weight spectra of ternary codes associated to the quadratic Veronese $3$-fold

The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work.

math.CO↗
arXiv2023

TMAP: A Threat Modeling and Attack Path Analysis Framework for Industrial IoT Systems (A Case Study of IoM and IoP)

Industrial cyber-physical systems (ICPS) are gradually integrating information technology and automating industrial processes, leading systems to become more vulnerable to malicious actors. Thus, to deploy secure Industrial Control and Production Systems (ICPS) in smart factories, cyber threats and risks must be addressed. To identify all possible threats, Threat Modeling is a promising solution. Despite the existence of numerous methodological solutions for threat modeling in cyber-physical systems (CPS), current approaches are ad hoc and inefficient in providing clear insights to researchers and organizations involved in IIoT technologies. These approaches lack a comprehensive analysis of cyber threats and fail to facilitate effective path analysis across the ICPS lifecycle, incorporating smart manufacturing technologies and tools. To address these gaps, a novel quantitative threat modeling approach is proposed, aiming to identify probable attack vectors, assess the path of attacks, and evaluate the magnitude of each vector. This paper also explains the execution of the proposed approach with two case studies, namely the industrial manufacturing line, i.e., the Internet of Manufacturing (IoM), and the power and industry, i.e., the Internet of Production (IoP).

cs.CR↗
arXiv2023

On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor lies in any of its osculating planes. While the non-generic lines have been classified into $G$-orbits in literature, it has been an open problem to classify the generic lines into $G$-orbits. For a general field $F$ of characteristic different from $2$ and $3$, the twisted cubic determines a symplectic polarity on $\mathbb P^3$. In the Klein representation of lines of $\mathbb P^3$, the tangent lines of $C$ are represented by a degree $4$ rational normal curve in a hyperplane $\mathcal H$ of the second exterior power $\mathbb P^5$ of $\mathbb P^3$. Atiyah studied the lines of $\mathbb P^3$ with respect to $C$, in terms of the geometries of these two curves. Polar duality of lines on $\mathbb P^3$ corresponds to Hodge duality on $\mathbb P^5$, and $\mathcal H$ is the hyperplane of Hodge self-dual elements of $\mathbb P^5$. We show that $\mathcal H$ can be identified in a $PGL_2$-equivariant way with the space of binary quartic forms over $F$, and that pairs of polar dual lines of $\mathbb P^3$ correspond to binary quartic forms whose apolar invariant is a square. We first solve the open problem of classifying binary quartic forms over $GF(q)$ into $G$-orbits, and then use it to solve the main problem.

math.CO↗
arXiv2019

Deep Holes of Projective Reed-Solomon Codes

Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes --received words with maximum possible error distance-- for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy at most four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three in work arXiv:1612.05447

cs.IT↗
arXiv2018

An improvement of the asymptotic Elias bound for non-binary codes

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low relative minimum distance, where as the Plotkin bound is better at high relative minimum distance. In this work, we obtain a hybrid of these bounds which improves both. This in turn is based on the anticode bound which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture.

cs.IT↗
arXiv2016

Deep holes and MDS extensions of Reed-Solomon codes

We study the problem of classifying deep holes of Reed-Solomon codes. We show that this problem is equivalent to the problem of classifying MDS extensions of Reed-Solomon codes by one digit. This equivalence allows us to improve recent results on the former problem. In particular, we classify deep holes of Reed-Solomon codes of dimension greater than half the alphabet size. We also give a complete classification of deep holes of Reed Solomon codes with redundancy three in all dimensions.

cs.IT↗
arXiv2016

Counting generalized Reed-Solomon codes

In this article we count the number of generalized Reed-Solomon (GRS) codes of dimension k and length n, including the codes coming from a non-degenerate conic plus nucleus. We compare our results with known formulae for the number of 3-dimensional MDS codes of length n=6,7,8,9.

cs.IT↗