arXiv subjects

Abhay Kumar Singh

Publications and source records attributed to Abhay Kumar Singh.

At least 19 recordsLinked to original sources

arXiv2026

Data Protection in Function-Correcting Symbol-Pair Codes: Redundancy Bounds and Protection Profiles

In several storage systems, including DNA storage and flash memory, errors affect neighbouring symbols jointly, and the Hamming metric does not adequately capture such error patterns. The symbol-pair read channel, introduced by Cassuto and Blaum~\cite{cassuto2011codes}, addresses this by reading consecutive pairs of symbols rather than individual symbols. Motivated by this, we introduce function-correcting symbol-pair codes with data protection (FCSPC-DP), which guarantee reliable recovery of a desired function of the message while simultaneously protecting the message itself against symbol-pair errors. We derive bounds on the optimal redundancy of such codes and establish a relationship with joint-pair distance matrices. We also give explicit constructions of FCSPC-DP for locally pair-bounded functions and symbol-pair weight functions. We introduce the pair-separation constant of a function, the minimum symbol-pair distance between messages sharing a function value, and show that when it is sufficiently large, data protection requires no additional redundancy: the optimal redundancy coincides with that of the corresponding code without data protection. Considering the symbol-pair analogue of the $α$-distance graph, we introduce two code invariants, the generation profile and the disconnection threshold, and use them to characterise a code's protection properties. Relating the two metrics through these invariants yields upper and lower bounds on the symbol-pair threshold in terms of its Hamming counterpart, both of which are attained. We further extend the classical Plotkin and sphere-packing bounds to this setting.

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arXiv2026

Function-Correcting Codes for Insertion-Deletion Channel

In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant importance for applications in DNA data storage and document exchange. Recently, a coding framework known as function-correcting codes has been proposed to address the challenge of minimising redundancy while preserving specific functions of the message. This framework has gained attention due to its potential applications in machine learning systems and long-term archival data storage. Motivated by the problem of redundancy optimisation for insertion and deletion channels, we propose a new framework called function-correcting codes for insdel channels. In this paper, we introduce the notions of function-correcting insertion codes, function-correcting deletion codes, and function-correcting insdel codes, and we show that these three formulations are equivalent. We then define insdel distance matrices and irregular insdel-distance codes, and derive lower and upper bounds on the optimal redundancy achievable by function-correcting codes for insdel channels. In addition, we establish Gilbert-Varshamov and Plotkin-like bounds on the length of irregular insdel-distance codes. Using the relation between optimal redundancy and the length of such codes, we obtain a simplified lower bound on optimal redundancy. Finally, we derive bounds on the optimal redundancy of function-correcting insdel codes for several classes of functions, including locally bounded functions, VT syndrome functions, the number-of-runs function, and the maximum-run-length function.

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arXiv2026

Construction of cyclic codes with large minimum distance from power functions over odd characteristic finite fields

Cyclic codes with dimensions exceeding half of the code length and minimum distance greater than the square root of the code length are of significant interest due to their high transmission efficiency and strong error-correcting capability. Such codes are well suited for demanding applications, including communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. Motivated by the work of Ding \cite{P3}, this paper extends the binary framework of Ding and Zhou \cite{P2} to a non-binary setting. By employing power functions with known differential uniformity over finite fields of odd characteristic, we present several infinite families of $q$-ary cyclic codes of length $q^m-1$ with dimensions exceeding $(q^m-1)/2$ and the lower bounds on the minimum distances greater than the square root of the code length, thereby achieving a favorable balance between code rate and error-correcting capability. We also determine the exact minimum distance of some of these codes. Furthermore, we partially resolve Open Problem $5.31$ posed by Ding in \cite{P3}.

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arXiv2026

Weight distributions of two classes of linear codes with few weights derived from Weil sums

Linear codes with few weights have been a subject of study for many years, as they have applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this article, two distinct classes of $p$-ary linear codes are constructed through the selection of two specific defining sets. Their weight distributions are completely determined for each case by detailed calculations on certain Weil sums. The constructed codes are shown to have only two, four, six, eight, and nine nonzero weights under different cases. In particular, we obtained an infinite family of two-weight optimal linear codes with respect to the Griesmer bound. Moreover, we observe that some of our newly constructed codes are minimal under certain conditions.

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arXiv2026

Function-Correcting Codes for Linear and Locally Bounded Functions Over a Finite Chain Ring

In this paper, we further extend the study of function-correcting codes in the homogeneous metric over a chain ring $\mathbb{Z}_{2^s}$ for broader classes of functions, namely, locally bounded functions and linear functions, and for weight functions, modular sum functions. e define locally bounded functions in the homogeneous metric over $\mathbb{Z}_{2^s}^k$ and investigate the locality of weight functions. We derive a Plotkin-like bound for irregular homogeneous distance code over $\mathbb{Z}_4$, which improves the existing bound. Using locality properties of functions, we establish upper and lower bounds on the optimal redundancy. We provide several explicit constructions of function-correcting codes for locally bounded functions, weight functions, and weight distribution functions. Using these constructions, we further discuss the tightness of the derived bound. We explicitly derive a Plotkin-like bound for linear function-correcting codes that reduces to the classical Plotkin bound when the linear function is bijective, we further discuss a construction of function-correcting linear codes over $\mathbb{Z}_{2^s}$.

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arXiv2026

On Function-Correcting Codes in the Lee Metric

Function-correcting codes are a coding framework designed to minimize redundancy while ensuring that specific functions or computations of encoded data can be reliably recovered, even in the presence of errors. The choice of metric is crucial in designing such codes, as it determines which computations must be protected and how errors are measured and corrected. Previous work by Liu and Liu [6] studied function-correcting codes over $\mathbb{Z}_{2^l},\ l\geq 2$ using the homogeneous metric, which coincides with the Lee metric over $\mathbb{Z}_4$. In this paper, we extend the study to codes over $\mathbb{Z}_m,$ for any positive integer $m\geq 2$ under the Lee metric and aim to determine their optimal redundancy. To achieve this, we introduce irregular Lee distance codes and derive upper and lower bounds on the optimal redundancy by characterizing the shortest possible length of such codes. These general bounds are then simplified and applied to specific classes of functions, including locally bounded functions, Lee weight functions, and Lee weight distribution functions. We extend the bounds established by Liu and Liu [6] for codes over $\mathbb{Z}_4$ in the Lee metric to the more general setting of $\mathbb{Z}_m$. Moreover, we give explicit constructions of function-correcting codes in Lee metric. Additionally, we explicitly derive a Plotkin-like bound for linear function-correcting codes in the Lee metric. As the Lee metric coincides with the Hamming metric over the binary field, we demonstrate that our bound naturally reduces to a Plotkin-type bound for function-correcting codes under the Hamming metric over $\mathbb{Z}_2$.

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arXiv2025

Function-Correcting $b$-symbol Codes for Locally $(λ, ρ,b)$-Functions

The family of functions plays a central role in the design and effectiveness of function-correcting codes. By focusing on a well-defined family of functions, function-correcting codes can be constructed with minimal length while still ensuring full error detection and correction within that family. In this work, we explore the concept of locally $(λ,ρ)$-functions for $b$-symbol read channels and investigate the optimal redundancy of the corresponding function-correcting $b$-symbol codes (FCBSC) by introducing the notions of locally $(λ,ρ,b)$-functions. First, we discuss the values of $λ$ and $ρ$ for which a function can be considered as a locally $(λ,ρ)$-function in $b$-symbol metric. The findings improve some known results in the Hamming metric and present several new results in the $b$-symbol metric. Then we investigate the optimal redundancy of $(f,t)$-FCBSCs for locally $(λ,ρ,b)$-functions. We establish a recurrence relation between the optimal redundancy of $(f,t)$-function-correcting codes for the $(b+1)$-symbol read and $b$-symbol read channels. We present an upper bound on the optimal redundancy of $(f,t)$-function-correcting $b$-symbol codes for general locally ($λ,ρ$, $b$)-functions by associating it to the minimum achievable length of $b$-symbol error-correcting codes and traditional Hamming-metric codes, given a fixed number of codewords and a specified minimum distance. We derive some explicit upper bounds on the redundancy of $(f,t)$-function-correcting $b$-symbol codes for locally $(λ,2t,b)$-functions. Moreover, for the case where $b=1$, we show that a locally ($3,2t,1$)-function achieves the optimal redundancy of $3t$. Additionally, we explicitly investigate the locality and optimal redundancy of FCBSCs for the $b$-symbol weight function and weight distribution function for $b\geq1$.

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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.

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arXiv2025

Binary cyclic codes from permutation polynomials over $\mathbb{F}_{2^m}$

Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding \cite{SETA5} from several known classes of permutation monomials and trinomials over $\mathbb{F}_{2^m}$. We present several infinite families of binary cyclic codes of length $2^m-1$ with dimensions larger than $(2^m-1)/2$. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters $[2^m-1,2^m-2-3m,8]$, where $m\geq 5$ is odd, according to the sphere-packing bound.

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arXiv2025

Function-Correcting Codes for b-Symbol Read Channels

Function-correcting codes are an innovative class of codes that are designed to protect a function evaluation of the data against errors or corruptions. Due to its usefulness in machine learning applications and archival data storage, where preserving the integrity of computation is crucial, Lenz et al. recently introduced function-correcting codes for binary symmetric channels to safeguard function evaluation against errors. Xia et al. expanded this concept to symbol-pair read channels over binary fields. The current paper further advances the theory by developing function-correcting codes for b-symbol read channels over finite fields. We introduce the idea of irregular b-symbol distance codes and establish bounds on their performance over finite fields. This concept helps in understanding the behavior of function-correcting codes in more complex settings. We also present a graphical approach of the problem of constructing function-correcting b-symbol codes. Furthermore, we apply these general concepts to specific classes of functions and compare the redundancy of function-correcting b-symbol codes with classical b-symbol codes. Our findings demonstrate that function-correcting b-symbol codes achieve lower redundancy while maintaining reliability.

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arXiv2024

A Symbol-Pair Decoder for CSS Codes

The relation between stabilizer codes and binary codes provided by Gottesman and Calderbank et al. is a celebrated result, as it allows the lifting of classical codes to quantum codes. An equivalent way to state this result is that the work allows us to lift decoders for classical codes over the Hamming metric to decoders for stabilizer quantum codes. A natural question to consider: Can we do something similar with decoders for classical codes considered over other metrics? i.e., Can we lift decoders for classical codes over other metrics to obtain decoders for stabilizer quantum codes? In our current work, we answer this question in the affirmative by considering classical codes over the symbol-pair metric. In particular, we present a relation between the symplectic weight and the symbol-pair weight and use it to improve the error correction capability of CSS-codes (a well-studied class of stabilizer codes) obtained from cyclic codes.

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arXiv2022

Construction of Multiple Constrained DNA Codes

DNA sequences are prone to creating secondary structures by folding back on themselves by non-specific hybridization among its nucleotides. The formation of secondary structures makes the sequences chemically inactive towards synthesis and sequencing processes. In this letter, our goal is to tackle the problems due to the creation of secondary structures in DNA sequences along with constraints such as not having a large homopolymer run length. In this paper, we have presented families of DNA codes with secondary structures of stem length at most two and homopolymer run length at most four. By mapping the error correcting codes over $\Z_{11}$ to DNA nucleotides, we obtained DNA codes with rates $0.5765$ times the rate of corresponding code over $\Z_{11}$, which include some new secondary structure free and better-performing codes for DNA based data storage and DNA computing purposes.

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arXiv2022

Vulnerability-CoVaR: Investigating the Crypto-market

This paper proposes an important extension to Conditional Value-at-Risk (CoVaR), the popular systemic risk measure, and investigates its properties on the cryptocurrency market. The proposed Vulnerability-CoVaR (VCoVaR) is defined as the Value-at-Risk (VaR) of a financial system or institution, given that at least one other institution is equal or below its VaR. The VCoVaR relaxes normality assumptions and is estimated via copula. While important theoretical findings of the measure are detailed, the empirical study analyzes how different distressing events of the cryptocurrencies impact the risk level of each other. The results show that Litecoin displays the largest impact on Bitcoin and that each cryptocurrency is significantly affected if an event of joint distress among the remaining market participants occurs. The VCoVaR is shown to capture domino effects better than other CoVaR extensions.

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arXiv2019

The Discussion on Hulls of Cyclic Codes over the Ring $\Re =\Z_4+v\Z_4$, $v^2=v$

For odd length $n$, the cyclic codes construction over $\Re= \Z_4[v]/ \langle v^2-v \rangle$ is provided. The hulls of cyclic codes over $\Re$ are studied. The average $2$-dimension $E(n)$ of the hulls of cyclic codes over $\Re$ is also conferred. Among these, the various examples of generators of hulls of cyclic codes over $\Re$ are provided, whose $\Z_4$-images are good $\Z_4$-linear codes with good parameters.

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arXiv2016

Construction of cyclic DNA codes over the Ring $\Z_4[u]/\langle u^2-1 \rangle $ Based on the deletion distance

In this paper, we develop the theory for constructing DNA cyclic codes of odd length over $R=\Z_4[u]/\langle u^2-1 \rangle$ based on the deletion distance. Firstly, we relate DNA pairs with a special 16 elements of ring $R$. Cyclic codes of odd length over $R$ satisfy the reverse constraint and the reverse-complement constraint are discussed in this paper. We also study the $GC$-content of these codes and their deletion distance. The paper concludes with some examples of cyclic DNA codes with $GC$-content and their respective deletion distance.

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arXiv2016

On Quantum Codes Obtained From Cyclic Codes Over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$

Let $R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ be a non-chain finite commutative ring, where $u^3=u$. In this paper, we mainly study the construction of quantum codes from cyclic codes over $R$. We obtained self-orthogonal codes over $\mathbb{F}_2$ as gray images of linear and cyclic codes over $R$. The parameters of quantum codes which are obtained from cyclic code over $R$ are discussed.

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arXiv2015

DNA Cyclic Codes Over The Ring $ \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle$

In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the ring $R = \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle$ which play an important role in DNA computing. We established a direct link between the element of ring $R$ and 64 codons by introducing a Gray map from $R$ to $R_1 = F_2 + uF_2, u^2 = 1$ where $R_1$ is the ring of four elements. The reverse constrain and the reverse-complement constraint codes over $R$ and $R_1$ are studied in this paper. Binary image of the cyclic codes over R also study. The paper concludes with some example on DNA codes obtained via gray map.

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arXiv2015

On cyclic DNA codes over the Ring $\Z_4 + u \Z_4$

In this paper, we study the theory for constructing DNA cyclic codes of odd length over $\Z_4[u]/\langle u^2 \rangle$ which play an important role in DNA computing. Cyclic codes of odd length over $\Z_4 + u \Z_4$ satisfy the reverse constraint and the reverse-complement constraint are studied in this paper. The structure and existence of such codes are also studied. The paper concludes with some DNA example obtained via the family of cyclic codes.

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