arXiv subjects

Shibsankar Das

Publications and source records attributed to Shibsankar Das.

10 recordsLinked to original sources

arXiv2026

Generalized Tan-Arlery-Rabaste-Lehmann-Ovarlez Lower Bound on Ambiguity Function of a Set of Sequences With Mismatched Filters

In this paper, a lower bound on the maximum ambiguity function (AF) sidelobes of a set of unimodular sequences is formulated for the desired low-ambiguity-zone (LAZ). Our main idea is to introduce a set of mismatched filters associated to a set of unimodular sequences and two weight vectors for the delay and Doppler shifts, respectively. The length of mismatched filter maybe different to the length of unimodular sequence. The proposed lower bound on the maximum AF sidelobes for the desired LAZ can be treated as an extension of Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound, published in 2020, which dealt with the conventional correlation of sequences.

cs.IT↗

arXiv2026

M-polynomial Based Mathematical Formulation of the Hyperbolic Sombor Index

The numerical values extracted from a graph that indicates its topology are called topological indices. A contemporary and efficient method is to compute a graph's topological indices using the graph polynomial that corresponds to it. This method of identifying degree-based topological indices involves the use of the M-polynomial. Very recently, in 2025, the hyperbolic Sombor index (HSO) was proposed and shows its chemical applicability for octane isomers and the structure sensitivity and abruptness for octane, nonane, and decane isomers, respectively. In this work, we establish the closed derivation formula for the above-mentioned index of a graph based on its M-polynomial. Additionally, we use our proposed derivation formula to calculate the hyperbolic Sombor index of a few standard graphs and chemical families. Moreover, we provide the numerical and graphical representations for the M-polynomial and the computed HSO index of the chemical families.

cs.DM↗

arXiv2025

On the Hyperbolic Sombor Index and Its Counterpart

For a graph $G$ with edge set $E$, let $d(w)$ denote the degree of a vertex $w$ in $G$. The hyperbolic Sombor index of $G$ is defined by $$HSO(G)=\sum_{uv\in E}(\min\{d(u),d(v)\})^{-1}\sqrt{(d(u))^2+(d(v))^2}.$$ If $\min\{d(u),d(v)\}$ is replaced with $\max\{d(u),d(v)\}$ in the formula of $HSO(G)$, then the complementary diminished Sombor (CDSO) index is obtained. For two non-adjacent vertices $v$ and $w$ of $G$, the graph obtained from $G$ by adding the edge $vw$ is denoted by $G+vw$. In this paper, we attempt to correct some inaccuracies in the recent work [J. Barman, S. Das, Geometric approach to degree-based topological index: hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94]. We establish a sufficient condition under which $HSO(G+vw) > HSO(G)$ holds, and also provide a sufficient condition guaranteeing $HSO(G+vw) < HSO(G)$. In addition, we give a lower bound on $HSO(G)$ in terms of the order and size of $G$. Furthermore, we obtain similar results for the CDSO index.

math.CO↗

arXiv2023

Two-Dimensional Z-Complementary Array Quads with Low Column Sequence PMEPRs

In this paper, we first propose a new design strategy of 2D $Z$-complementary array quads (2D-ZCAQs) with feasible array sizes. A 2D-ZCAQ consists of four distinct unimodular arrays satisfying zero 2D auto-correlation sums for non-trivial 2D time-shifts within certain zone. Then, we obtain the upper bounds on the column sequence peak-to-mean envelope power ratio (PMEPR) of the constructed 2D-ZCAQs by using specific auto-correlation properties of some seed sequences. The constructed 2D-ZCAQs with bounded column sequence PMEPR can be used as a potential alternative to 2D Golay complementary array sets for practical applications

cs.IT↗

arXiv2023

Root Cross Z-Complementary Pairs with Large ZCZ Width

In this paper, we present a new family of cross $Z$-complementary pairs (CZCPs) based on generalized Boolean functions and two roots of unity. Our key idea is to consider an arbitrary partition of the set $\{1,2,\cdots, n\}$ with two subsets corresponding to two given roots of unity for which two truncated sequences of new alphabet size determined by the two roots of unity are obtained. We show that these two truncated sequences form a new $q$-ary CZCP with flexible sequence length and large zero-correlation zone width. Furthermore, we derive an enumeration formula by considering the Stirling number of the second kind for the partitions and show that the number of constructed CZCPs increases significantly compared to the existing works.

cs.IT↗

arXiv2022

On DNA Codes Over the Non-Chain Ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=1$

In this paper, we present a novel design strategy of DNA codes with length $3n$ over the non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $64$ elements and $u^3=1$, where $n$ denotes the length of a code over $R$. We first study and analyze a distance conserving map defined over the ring $R$ into the length-$3$ DNA sequences. Then, we derive some conditions on the generator matrix of a linear code over $R$, which leads to a DNA code with reversible, reversible-complement, homopolymer $2$-run-length, and $\frac{w}{3n}$-GC-content constraints for integer $w$ ($0\leq w\leq 3n$). Finally, we propose a new construction of DNA codes using Reed-Muller type generator matrices. This allows us to obtain DNA codes with reversible, reversible-complement, homopolymer $2$-run-length, and $\frac{2}{3}$-GC-content constraints.

cs.IT↗

arXiv2021

Grouping Search Results with Product Graphs in E-commerce Platforms

Showing relevant search results to the user is the primary challenge for any search system. Walmart e-commerce provides an omnichannel search platform to its customers to search from millions of products. This search platform takes a textual query as input and shows relevant items from the catalog. One of the primary challenges is that this queries are complex to understand as it contains multiple intent in many cases. This paper proposes a framework to group search results into multiple ranked lists intending to provide better user intent. The framework is to create a product graph having relations between product entities and utilize it to group search results into a series of stacks where each stack provides a group of items based on a precise intent. As an example, for a query "milk," the results can be grouped into multiple stacks of "white milk", "low-fat milk", "almond milk", "flavored milk". We measure the impact of our algorithm by evaluating how it improves the user experience both in terms of search quality relevance and user behavioral signals like Add-To-Cart.

cs.IR↗

arXiv2019

Experimental Evaluation of Modified Decomposition Algorithm for Maximum Weight Bipartite Matching

Let $G$ be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of $G$ in $O(\sqrt{|V|}W'/k(|V|,W'/N))$-time by employing an algorithm designed by Feder and Motwani as a subroutine, where $|V|$ and $N$ denote the number of nodes and the maximum edge weight of $G$, respectively and $k(x,y)=\log x /\log(x^2/y)$. The parameter $W'$ is smaller than the total edge weight $W,$ essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case $W'=O(|E|)$ where $|E|$ be the number of edges of $G$ and in worst case $W'=W,$ that is, $|E| \leq W' \leq W.$ In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter $W'$. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.

cs.DM↗

arXiv2019

A Tight Lower Bound for the Weights of Maximum Weight Matching in Bipartite Graphs

Let $\Ga$ be the collection of all weighted bipartite graphs each having $σ$ and $m$, as the size of a vertex partition and the total weight, respectively. We give a tight lower bound $\lceil \frac{m-σ}σ \rceil+1$ for the set $\{\textit{Wt}(\textit{mwm}(G))~|~G \in \Ga\}$ which denotes the collection of weights of maximum weight bipartite matchings of all graphs in $\Ga$.

cs.DM↗

arXiv2019

New Optimal $Z$-Complementary Code Sets from Matrices of Polynomials

The concept of paraunitary (PU) matrices arose in the early 1990s in the study of multi-rate filter banks. So far, these matrices have found wide applications in cryptography, digital signal processing, and wireless communications. Existing PU matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice. Motivated by this, for the first time, we introduce a novel concept, called $Z$-paraunitary (ZPU) matrix, whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value. We show that there exists an equivalence between a ZPU matrix and a $Z$-complementary code set when the latter is expressed as a matrix with polynomial entries. Furthermore, we investigate some important properties of ZPU matrices, which are useful for the extension of matrix sizes and sequence lengths. Finally, we propose a unifying construction framework for optimal ZPU matrices which includes existing PU matrices as a special case.

cs.IT↗