arXiv subjects

Manish K Gupta

Publications and source records attributed to Manish K Gupta.

16 recordsLinked to original sources

arXiv2025

On Algebraic Approaches for DNA Codes with Multiple Constraints

DNA strings and their properties are widely studied since last 20 years due to its applications in DNA computing. In this area, one designs a set of DNA strings (called DNA code) which satisfies certain thermodynamic and combinatorial constraints such as reverse constraint, reverse-complement constraint, $GC$-content constraint and Hamming constraint. However recent applications of DNA codes in DNA data storage resulted in many new constraints on DNA codes such as avoiding tandem repeats constraint (a generalization of non-homopolymer constraint) and avoiding secondary structures constraint. Therefore, in this chapter, we introduce DNA codes with recently developed constraints. In particular, we discuss reverse, reverse-complement, $GC$-content, Hamming, uncorrelated-correlated, thermodynamic, avoiding tandem repeats and avoiding secondary structures constraints. DNA codes are constructed using various approaches such as algebraic, computational, and combinatorial. In particular, in algebraic approaches, one uses a finite ring and a map to construct a DNA code. Most of such approaches does not yield DNA codes with high Hamming distance. In this chapter, we focus on algebraic constructions using maps (usually an isometry on some finite ring) which yields DNA codes with high Hamming distance. We focus on non-cyclic DNA codes. We briefly discuss various metrics such as Gau distance, Non-Homopolymer distance etc. We discuss about algebraic constructions of families of DNA codes that satisfy multiple constraints and/or properties. Further, we also discuss about algebraic bounds on DNA codes with multiple constraints. Finally, we present some open research directions in this area.

cs.IT↗

arXiv2023

On the Classification of Weierstrass Elliptic Curves over $\mathbb{Z}_n$

The elliptic curves are beautiful mathematical object with several applications. They have been studied well over finite and infinite fields. In this work, we study Weierstrass elliptic curves over the finite ring $\mathbb{Z}_n$ through classification. Our study is supported with extensive computational data. We also present some conjectures.

cs.CR↗

arXiv2022

On Modular Gray Map

This paper introduces an isometry between the modular rings $\Z_{2^s}$ and $\Z_{2^{s-1}}$ with respect to the homogeneous weights. Certain product of these maps gives Carlet's generalised Gray map and also Vega's Gray map. For $s=2$ this reduces to popular Gray map. Several interesting properties of these maps are studied. Towards the end we list several interesting problems to work on.

cs.IT↗

arXiv2022

On the Classification of Codes over Non-Unital Ring of Order 4

In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E= \langle 2a=2b=0, a^2=a, b^2=b, ab=a, ba=b \rangle$ by presenting the classification of optimal and nice codes of length $n\leq7$ over $E$, along-with respective weight enumerators and complete weight enumerators.

cs.IT↗

arXiv2021

DNA Codes over the Ring $\mathbb{Z}_4 + w\mathbb{Z}_4$

In this present work, we generalize the study of construction of DNA codes over the rings $\mathcal{R}_θ=\mathbb{Z}_4+w\mathbb{Z}_4$, $w^2 = θ$ for $θ\in \mathbb{Z}_4+w\mathbb{Z}_4$. Rigorous study along with characterization of the ring structures is presented. We extend the Gau map and Gau distance, defined in \cite{DKBG}, over all the $16$ rings $\mathcal{R}_θ$. Furthermore, an isometry between the codes over the rings $\mathcal{R}_θ$ and the analogous DNA codes is established in general. Brief study of dual and self dual codes over the rings is given including the construction of special class of self dual codes that satisfy reverse and reverse-complement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound are established over the rings $\mathcal{R}_θ$. In addition to this, optimal class of codes are provided with respect to Singleton-like bound and Plotkin-like bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reverse-complement constraints using the Reed-Muller type codes over the rings $\mathcal{R}_θ$.

cs.IT↗

arXiv2019

On Conflict Free DNA Codes

DNA storage has emerged as an important area of research. The reliability of DNA storage system depends on designing the DNA strings (called DNA codes) that are sufficiently dissimilar. In this work, we introduce DNA codes that satisfy a special constraint. Each codeword of the DNA code has a specific property that any two consecutive sub-strings of the DNA codeword will not be the same (a generalization of homo-polymers constraint). This is in addition to the usual constraints such as Hamming, reverse, reverse-complement and $GC$-content. We believe that the new constraint will help further in reducing the errors during reading and writing data into the synthetic DNA strings. We also present a construction (based on a variant of stochastic local search algorithm) to calculate the size of the DNA codes with all the above constraints, which improves the lower bounds from the existing literature, for some specific cases. Moreover, a recursive isometric map between binary vectors and DNA strings is proposed. Using the map and the well known binary codes we obtain few classes of DNA codes with all the constraints including the property that the constructed DNA codewords are free from the hairpin-like secondary structures.

cs.IT↗

arXiv2018

Bounds on Fractional Repetition Codes using Hypergraphs

In the \textit{Distributed Storage Systems} (DSSs), an encoded fraction of information is stored in the distributed fashion on different chunk servers. Recently a new paradigm of \textit{Fractional Repetition} (FR) codes have been introduced, in which, encoded data information is stored on distributed servers, where encoding is done using a \textit{Maximum Distance Separable} (MDS) code and a smart replication of packets. In this work, we have shown that an FR code is equivalent to a hypergraph. Using the correspondence, the properties and the bounds of a hypergraph are directly mapped to the associated FR code. In general, the necessary and sufficient conditions for the existence of an FR code is obtained by using the correspondence. Some of the bounds are new and FR codes meeting these bounds are unknown. It is also shown that any FR code associated with a linear hypergraph is universally good.

cs.IT↗

arXiv2018

On Universally Good Flower Codes

For a Distributed Storage System (DSS), the \textit{Fractional Repetition} (FR) code is a class in which replicas of encoded data packets are stored on distributed chunk servers, where the encoding is done using the Maximum Distance Separable (MDS) code. The FR codes allow for exact uncoded repair with minimum repair bandwidth. In this paper, FR codes are constructed using finite binary sequences. The condition for universally good FR codes is calculated on such sequences. For some sequences, the universally good FR codes are explored.

cs.IT↗

arXiv2017

On Code Rates of Fractional Repetition Codes

In \textit{Distributed Storage Systems} (DSSs), usually, data is stored using replicated packets on different chunk servers. Recently a new paradigm of \textit{Fractional Repetition} (FR) codes have been introduced, in which, data is replicated in a smart way on distributed servers using a \textit{Maximum Distance Separable} (MDS) code. In this work, for a non-uniform FR code, bounds on the FR code rate and DSS code rate are studied. Using matrix representation of an FR code, some universally good FR codes have been obtained.

cs.IT↗

arXiv2016

DNA Image Pro -- A Tool for Generating Pixel Patterns using DNA Tile Assembly

Self-assembly is a process found everywhere in the Nature. In particular, it is known that DNA self-assembly is Turing universal. Thus one can do arbitrary computations or build nano-structures using DNA self-assembly. In order to understand the DNA self-assembly process, many mathematical models have been proposed in the literature. In particular, abstract Tile Assembly Model (aTAM) received much attention. In this work, we investigate pixel pattern generation using aTAM. For a given image, a tile assembly system is given which can generate the image by self-assembly process. We also consider image blocks with specific cyclic pixel patterns (uniform shift and non uniform shift) self assembly. A software, DNA Image Pro, for generating pixel patterns using DNA tile assembly is also given.

cs.ET↗

arXiv2016

On Optimal Family of Codes for Archival DNA Storage

DNA based storage systems received attention by many researchers. This includes archival and re-writable random access DNA based storage systems. In this work, we have developed an efficient technique to encode the data into DNA sequence by using non-linear families of ternary codes. In particular, we proposes an algorithm to encode data into DNA with high information storage density and better error correction using a sub code of Golay code. Theoretically, 115 exabytes (EB) data can be stored in one gram of DNA by our method.

cs.IT↗

arXiv2015

Computing Real Numbers using DNA Self-Assembly

DNA Self-Assembly has emerged as an interdisciplinary field with many intriguing applications such DNA bio-sensor, DNA circuits, DNA storage, drug delivery etc. Tile assembly model of DNA has been studied for various computational primitives such as addition, subtraction, multiplication, and division. Xuncai et. al. gave computational DNA tiles to perform division of a number but the output had integer quotient. In this work, we simply modify their method of division to improve its compatibility with further computation and this modification has found its application in computing rational numbers, both recurring and terminating, with computational tile complexity of $\mathcal{O} (1)$ and $\mathcal{O} (h)$ respectively. Additionally, we also propose a method to compute square-root of a number with computational tile complexity of $\mathcal{O} (n)$ for an n bit number. Finally, after combining tiles of division and square-root, we propose a simple way to compute the ubiquitously used irrational number, $π$, using its infinite series.

cs.ET↗

arXiv2014

3DNA: A Tool for DNA Sculpting

DNA self-assembly is a robust and programmable approach for building structures at nanoscale. Researchers around the world have proposed and implemented different techniques to build two dimensional and three dimensional nano structures. One such technique involves the implementation of DNA Bricks proposed by Ke et al., 2012 to create complex three-dimensional (3D) structures. Modeling these DNA nano structures can prove to be a cumbersome and tedious task. Exploiting the programmability of base-pairing to produce self-assembling custom shapes, we present a software suite 3DNA, which can be used for modeling, editing and visualizing such complex structures. 3DNA is an open source software which works on the simple and modular self assembly of DNA Bricks, offering a more intuitive better approach for constructing 3D shapes. Apart from modeling and envisaging shapes through a simple graphical user interface, 3DNA also supports an integrated random sequence generator that generates DNA sequences corresponding to the designed model. The software is available at www.guptalab.org/3dna

cs.ET↗

arXiv2014

RNA as a Permutation

RNA secondary structure prediction and classification are two important problems in the field of RNA biology. Here, we propose a new permutation based approach to create logical non-disjoint clusters of different secondary structures of a single class or type. Many different types of techniques exist to classify RNA secondary structure data but none of them have ever used permutation based approach which is very simple and yet powerful. We have written a small JAVA program to generate permutation, apply our algorithm on those permutations and analyze the data and create different logical clusters. We believe that these clusters can be utilized to untangle the mystery of RNA secondary structure and analyze the development patterns of unknown RNA.

q-bio.BM↗

arXiv2013

DNA Pen: A Tool for Drawing on a Molecular Canvas

DNA origami is an interdisciplinary area where DNA can be used as a building block for making useful stuff at nanoscale. This work presents an open source software DNA pen (based on the recent work of Peng Yin and his group) which can be used (using free hand and digital molecular canvas) to draw an object at nanoscale. Software generates error free DNA sequences which can be used in the wet lab to create the object at the nanoscale. Using DNA pen we have drawn several objects including the map of India and sanskrit letter "Om" from free hand molecular canvas and digital letter DNA using digitized molecular canvas.

cs.ET↗

arXiv2013

On Weak Dress Codes for Cloud Storage

In a distributed storage network, reliability and bandwidth optimization can be provided by regenerating codes. Recently table based regenerating codes viz. DRESS (Distributed Replication-based Exact Simple Storage) codes has been proposed which also optimizes the disk I/O. Dress codes consists of an outer MDS code with an inner fractional repetition (FR) code with replication degree $ρ$. Several constructions of FR codes based on regular graphs, resolvable designs and bipartite graphs are known. This paper presents a simple modular construction of FR codes. We also generalize the concept of FR codes to weak fractional repetition (WFR) codes where each node has different number of packets. We present a construction of WFR codes based on partial regular graph. Finally we present a simple generalized ring construction of both strong and weak fractional repetition codes.

cs.IT↗