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Nupur Patanker

Publications and source records attributed to Nupur Patanker.

9 recordsLinked to original sources
arXiv2025

Code size constraints in b-symbol read channels: A bound analysis

In classical coding theory, error-correcting codes are designed to protect against errors occurring at individual symbol positions in a codeword. However, in practical storage and communication systems, errors often affect multiple adjacent symbols rather than single symbols independently. To address this, symbol-pair read channels were introduced \cite{Yuval2011}, and later generalized to $b$-symbol read channels \cite{yaakobi2016} to better model such error patterns. $b$-Symbol read channels generalize symbol-pair read channels to account for clustered errors in modern storage and communication systems. By developing bounds and efficient codes, researchers improve data reliability in applications such as storage devices, wireless networks, and DNA-based storage. Given integers $q$, $n$, $d$, and $b \geq 2$, let $A_b(n,d,q)$ denote the largest possible code size for which there exists a $q$-ary code of length $n$ with minimum $b$-symbol distance at least $d$. In \cite{chen2022}, various upper and lower bounds on $A_b(n,d,q)$ are given for $b=2$. In this paper, we generalize some of these bounds to the $b$-symbol read channels for $b>2$ and present several new bounds on $A_b(n,d,q)$. In particular, we establish the linear programming bound, a recurrence relation on $A_b(n,d,q)$, the Johnson bound (even), the restricted Johnson bound, the Gilbert-Varshamov-type bound, and the Elias bound for the metric of symbols $b$, $b\geq 2$. Furthermore, we provide examples demonstrating that the Gilbert-Varshamov bound we establish offers a stronger lower bound than the one presented in \cite{Song2018}. Additionally, we introduce an alternative approach to deriving the Sphere-packing and Plotkin bounds.

cs.IT↗
arXiv2025

On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices

Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of $(\mathbb{F}_q^*)^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of homogeneous monomially square-free polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d = 1$ has been determined by Jaramillo-Velez et al. in 2023, and has been determined in the cases where $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$, by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when $3 \leq d < s$ (respectively, $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$).

cs.IT↗
arXiv2025

Next-to-minimal weight of toric codes defined over hypersimplices

Toric codes are a type of evaluation codes introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of $(\mathbb{F}_q^*)^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of square-free homogeneous polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d = 1$ has been determined by Jaramillo-Velez et al. in 2023. In this work we use tools from Gr\"obner basis theory to determine the next-to-minimal weight of these codes for $d$ such that $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$.

cs.IT↗
arXiv2024

Codeword Stabilized Codes from m-Uniform Graph States

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d \ge m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]_2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[2^{2r}-1,2^{2r}-2r-3,3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2 - 32r-7, 5]]_2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.

quant-ph↗
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↗
arXiv2021

Generalization of trace codes to places of higher degree

In this note, we give a construction of codes on algebraic function field $F/ \mathbb{F}_{q}$ using places of $F$ (not necessarily of degree one) and trace functions from various extensions of $\mathbb{F}_{q}$. This is a generalization of trace code of geometric Goppa codes to higher degree places. We compute a bound on the dimension of this code. Furthermore, we give a condition under which we get exact dimension of the code. We also determine a bound on the minimum distance of this code in terms of $B_{r}(F)$ ( the number of places of degree $r$ in $F$), $1 \leq r < \infty$. Few quasi-cyclic codes over $\mathbb{F}_{p}$ are also obtained as examples of these codes.

cs.IT↗
arXiv2020

Generalized Hamming weights of toric codes over hypersimplices and square-free affine evaluation codes

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over $\mathbb{F}_{q}$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$ of any set of $r$ linearly independent square-free polynomials over $\mathbb{F}_{q}$ in $s$ variables, under certain conditions on $r$, $s$ and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluation codes, as defined in \cite{hyper}. Finally, we obtain the dual of these toric codes with respect to the Euclidean scalar product.

math.AC↗
arXiv2020

On geometric Goppa codes from Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$

Let $p$ be a prime number and $s> 0$ an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian $p$-extension of $\mathbb{F}_{p^{s}}(x)$. We determine their dimension and the exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list the exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for the self-duality and the quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field.

cs.IT↗