arXiv subjects

Eitan Yaakobi

Publications and source records attributed to Eitan Yaakobi.

At least 37 records · Page 2Linked to original sources

arXiv2025

Error-Correcting Codes for Labeled DNA Sequences

Labeling of DNA molecules is a fundamental technique for DNA visualization and analysis. This process was mathematically modeled in [1], where the received sequence indicates the positions of the used labels. In this work, we develop error correcting codes for labeled DNA sequences, establishing bounds and constructing explicit systematic encoders for single substitution, insertion, and deletion errors. We focus on two cases: (1) using the complete set of length-two labels and (2) using the minimal set of length-two labels that ensures the recovery of DNA sequences from their labeling for 'almost' all DNA sequences.

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arXiv2025

Coding for Ordered Composite DNA Sequences

To increase the information capacity of DNA storage, composite DNA letters were introduced. We propose a novel channel model for composite DNA in which composite sequences are decomposed into ordered standard non-composite sequences. The model is designed to handle any alphabet size and composite resolution parameter. We study the problem of reconstructing composite sequences of arbitrary resolution over the binary alphabet under substitution errors. We define two families of error-correcting codes and provide lower and upper bounds on their cardinality. In addition, we analyze the case in which a single deletion error occurs in the channel and present a systematic code construction for this setting. Finally, we briefly discuss the channel's capacity, which remains an open problem.

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arXiv2025

General Coverage Models: Structure, Monotonicity, and Shotgun Sequencing

We study coverage processes in which each draw reveals a subset of $[n]$, and the goal is to determine the expected number of draws until all items are seen at least once. A classical example is the Coupon Collector's Problem, where each draw reveals exactly one item. Motivated by shotgun DNA sequencing, we introduce a model where each draw is a contiguous window of fixed length, in both cyclic and non-cyclic variants. We develop a unifying combinatorial tool that shifts the task of finding coverage time from probability, to a counting problem over families of subsets of $[n]$ that together contain all items, enabling exact calculation. Using this result, we obtain exact expressions for the window models. We then leverage past results on a continuous analogue of the cyclic window model to analyze the asymptotic behavior of both models. We further study what we call uniform $\ell$-regular models, where every draw has size $\ell$ and every item appears in the same number of admissible draws. We compare these to the batch sampling model, in which all $\ell$-subsets are drawn uniformly at random and present upper and lower bounds, which were also obtained independently by Berend and Sher. We conjecture, and prove for special cases, that this model maximizes the coverage time among all uniform $\ell$-regular models. Finally, we prove a universal upper bound on the entire class of uniform $\ell$-regular models, which illuminates the fact that many sampling models share the same leading asymptotic order, while potentially differing significantly in lower-order terms.

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arXiv2025

Complex DNA Synthesis Sequences

DNA-based storage offers unprecedented density and durability, but its scalability is fundamentally limited by the efficiency of parallel strand synthesis. Existing methods either allow unconstrained nucleotide additions to individual strands, such as enzymatic synthesis, or enforce identical additions across many strands, such as photolithographic synthesis. We introduce and analyze a hybrid synthesis framework that generalizes both approaches: in each cycle, a nucleotide is selected from a restricted subset and incorporated in parallel. This model gives rise to a new notion of a complex synthesis sequence. Building on this framework, we extend the information rate definition of Lenz et al. and analyze an analog of the deletion ball, defined and studied in this setting, deriving tight expressions for the maximal information rate and its asymptotic behavior. These results bridge the theoretical gap between constrained models and the idealized setting in which every nucleotide is always available. For the case of known strands, we design a dynamic programming algorithm that computes an optimal complex synthesis sequence, highlighting structural similarities to the shortest common supersequence problem. We also define a distinct two-dimensional array model with synthesis constraints over the rows, which extends previous synthesis models in the literature and captures new structural limitations in large-scale strand arrays. Additionally, we develop a dynamic programming algorithm for this problem as well. Our results establish a new and comprehensive theoretical framework for constrained DNA, subsuming prior models and setting the stage for future advances in the field.

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arXiv2025

Correcting Multiple Substitutions in Nanopore-Sequencing Reads

Despite their significant advantages over competing technologies, nanopore sequencers are plagued by high error rates, due to physical characteristics of the nanopore and inherent noise in the biological processes. It is thus paramount not only to formulate efficient error-correcting constructions for these channels, but also to establish bounds on the minimum redundancy required by such coding schemes. In this context, we adopt a simplified model of nanopore sequencing inspired by the work of Mao \emph{et al.}, accounting for the effects of intersymbol interference and measurement noise. For an input sequence of length $n$, the vector that is produced, designated as the \emph{read vector}, may additionally suffer at most $t$ substitution errors. We employ the well-known graph-theoretic clique-cover technique to establish that at least $t\log n -O(1)$ bits of redundancy are required to correct multiple ($t \geq 2$) substitutions. While this is surprising in comparison to the case of a single substitution, that necessitates at most $\log \log n - O(1)$ bits of redundancy, a suitable error-correcting code that is optimal up to a constant follows immediately from the properties of read vectors.

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arXiv2025

Improved Constructions of Linear Codes for Insertions and Deletions

In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over $\mathbb{F}_{q^2}$, linear over $\mathbb{F}_q$, with rate $1/2 - δ- \varepsilon$ that can efficiently correct a $δ$-fraction of indel errors, where $q = O(\varepsilon^{-4})$. Additionally, we construct fully linear codes over $\mathbb{F}_q$ with rate $1/2 - 2\sqrtδ - \varepsilon$ that can also efficiently correct $δ$-fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code $C \subseteq \mathbb{F}^n$ linear over $\mathbb{E} \subset \mathbb{F}$ a subfield of $\mathbb{F}$, such that $C$ has the ability to correct $δ$-fraction of indels, the rate is bounded by $(1-δ)/2$.

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arXiv2025

A Combinatorial Perspective on Random Access Efficiency for DNA Storage

We investigate the fundamental limits of the recently proposed random access coverage depth problem for DNA data storage. Under this paradigm, it is assumed that the user information consists of $k$ information strands, which are encoded into $n$ strands via a generator matrix $G$. During the sequencing process, the strands are read uniformly at random, as each strand is available in a large number of copies. In this context, the random access coverage depth problem refers to the expected number of reads (i.e., sequenced strands) required to decode a specific information strand requested by the user. This problem heavily depends on the generator matrix $G$, and besides computing the expectation for different choices of $G$, the goal is to construct matrices that minimize the maximum expectation over all possible requested information strands, denoted by $T_{\max}(G)$. In this paper, we introduce new techniques to investigate the random access coverage depth problem, capturing its combinatorial nature and identifying the structural properties of generator matrices that are advantageous. We establish two general formulas to determine $T_{\max}(G)$ for arbitrary generator matrices. The first formula depends on the linear dependencies between columns of $G$, whereas the second formula takes into account recovery sets and their intersection structure. We also introduce the concept of recovery balanced codes and provide three sufficient conditions for a code to be recovery balanced. These conditions can be used to compute $T_{\max}(G)$ for various families of codes, such as MDS, simplex, Hamming, and binary Reed-Muller codes. Additionally, we study the performance of modified systematic MDS and simplex matrices, showing that the best results for $T_{\max}(G)$ are achieved with a specific combination of encoded strands and replication of the information strands.

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arXiv2025

Reed-Solomon Codes Against Insertions and Deletions: Full-Length and Rate-$1/2$ Codes

The performance of Reed--Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has attracted growing attention in recent literature. In this work, we further study this intriguing mathematical problem, focusing on two regimes. First, we study the question of how well full-length RS codes perform against insertions and deletions. For 2-dimensional RS codes, we provide a complete characterization of codes that cannot correct even a single insertion or deletion. Furthermore, we prove that for sufficiently large field size~$q$, nearly all full-length $2$-dimensional RS codes can correct up to $(1 - δ)q$ insertion and deletion errors for any $0 < δ< 1$. Extending beyond the 2-dimensional case, we show that for any $k \ge 2$, there exists a full-length $k$-dimensional RS code capable of correcting $q / (10k)$ insertion and deletion errors, provided $q$ is large enough. Second, we focus on rate $1/2$ RS codes that can correct a single insertion or deletion error. We present a polynomial-time algorithm that constructs such codes over fields of size $q = Θ(k^4)$. This result matches the existential bound given in \cite{con2023reed}.

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arXiv2025

Making it to First: The Random Access Problem in DNA Storage

In this paper, we study the Random Access Problem in DNA storage, which addresses the challenge of retrieving a specific information strand from a DNA-based storage system. In this framework, the data is represented by $k$ information strands which represent the data and are encoded into $n$ strands using a linear code. Then, each sequencing read returns one encoded strand which is chosen uniformly at random. The goal under this paradigm is to design codes that minimize the expected number of reads required to recover an arbitrary information strand. We fully solve the case when $k=2$, showing that the best possible code attains a random access expectation of $1+\frac{2}{\sqrt{2}+1}\approx 0.914\cdot 2$ for $q$ large enough. Moreover, we generalize a construction from~\cite{GMZ24}, specifically to $k=3$, for any value of $k$. Our construction uses $B_{k-1}$ sequences over $\mathbb{Z}_{q-1}$, that always exist over large finite fields. We show that for every $k\geq 4$, this generalized construction outperforms all previous constructions in terms of reducing the random access expectation.

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arXiv2025

Sequence Reconstruction over Coloring Channels for Protein Identification

This paper studies the sequence reconstruction problem for a channel inspired by protein identification. We introduce a coloring channel, where a sequence is transmitted through a channel that deletes all symbols not belonging to a fixed subset (the coloring) of the alphabet. By extending this to a coloring profile, a tuple of distinct colorings, we analyze the channel's information rate and capacity. We prove that optimal (i.e., achieving maximum information rate) coloring profiles correspond to 2-covering designs and identify the minimal covering number required for maximum information rate, as well as the minimum number for which any coloring profile is optimal.

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arXiv2025

The Coverage Depth Problem in DNA Storage Over Small Alphabets

The coverage depth problem in DNA data storage is about minimizing the expected number of reads until all data is recovered. When they exist, MDS codes offer the best performance in this context. This paper focuses on the scenario where the base field is not large enough to allow the existence of MDS codes. We investigate the performance for the coverage depth problem of codes defined over a small finite field, providing closed formulas for the expected number of reads for various code families. We also compare the results with the theoretical bounds in asymptotic regimes. The techniques we apply range from probability, to duality theory and combinatorics.

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arXiv2025

On The Decoding Error Weight of One or Two Deletion Channels

This paper tackles two problems that fall under the study of coding for insertions and deletions. These problems are motivated by several applications, among them is reconstructing strands in DNA-based storage systems. Under this paradigm, a word is transmitted over some fixed number of identical independent channels and the goal of the decoder is to output the transmitted word or some close approximation of it. The first part of the paper studies optimal decoding for a special case of the deletion channel, referred by the $k$-deletion channel, which deletes exactly $k$ symbols of the transmitted word uniformly at random. In this part, the goal is to understand how an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of an efficient optimal decoder for this setup, referred to as the maximum likelihood* (ML*) decoder, is given for a channel that deletes one or two symbols. The second part of this paper studies the deletion channel that deletes a symbol with some fixed probability $p$, while focusing on two instances of this channel. Since operating the maximum likelihood (ML) decoder, in this case, is computationally unfeasible, we study a slightly degraded version of this decoder for two channels and study its expected normalized distance. We observe that the dominant error patterns are deletions in the same run or errors resulting from alternating sequences. Based on these observations, we derive lower bounds on the expected normalized distance of the degraded ML decoder for any transmitted $q$-ary sequence of length $n$ and any deletion probability $p$. We further show that as the word length approaches infinity and the channel's deletion probability $p$ approaches zero, these bounds converge to approximately $\frac{3q - 1}{q - 1} p^2$. These theoretical results are verified by corresponding simulations.

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arXiv2025

Conditional Entropies of k-Deletion/Insertion Channels

The channel output entropy of a transmitted sequence is the entropy of the possible channel outputs and similarly the channel input entropy of a received sequence is the entropy of all possible transmitted sequences. The goal of this work is to study these entropy values for the k-deletion, k-insertion channels, where exactly k symbols are deleted, inserted in the transmitted sequence, respectively. If all possible sequences are transmitted with the same probability then studying the input and output entropies is equivalent. For both the 1-deletion and 1-insertion channels, it is proved that among all sequences with a fixed number of runs, the input entropy is minimized for sequences with a skewed distribution of their run lengths and it is maximized for sequences with a balanced distribution of their run lengths. Among our results, we establish a conjecture by Atashpendar et al. which claims that for the 1-deletion channel, the input entropy is maximized by the alternating sequences over all binary sequences. This conjecture is also verified for the 2-deletion channel, where it is proved that constant sequences with a single run minimize the input entropy.

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arXiv2025

Achieving DNA Labeling Capacity with Minimum Labels through Extremal de Bruijn Subgraphs

DNA labeling is a tool in molecular biology and biotechnology to visualize, detect, and study DNA at the molecular level. In this process, a DNA molecule is labeled by a set of specific patterns, referred to as labels, and is then imaged. The resulting image is modeled as an $(\ell+1)$-ary sequence, where $\ell$ is the number of labels, in which any non-zero symbol indicates the appearance of the corresponding label in the DNA molecule. The labeling capacity refers to the maximum information rate that can be achieved by the labeling process for any given set of labels. The main goal of this paper is to study the minimum number of labels of the same length required to achieve the maximum labeling capacity of 2 for DNA sequences or $\log_2q$ for an arbitrary alphabet of size $q$. The solution to this problem requires the study of path unique subgraphs of the de Bruijn graph with the largest number of edges. We provide upper and lower bounds on this value. We draw new connections to existing literature that let us prove an asymptotic result as the label length tends to infinity.

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arXiv2025

Optimizing the Decoding Probability and Coverage Ratio of Composite DNA

This paper studies two problems that are motivated by the novel recent approach of composite DNA that takes advantage of the DNA synthesis property which generates a huge number of copies for every synthesized strand. Under this paradigm, every composite symbols does not store a single nucleotide but a mixture of the four DNA nucleotides. The first problem studies the expected number of strand reads in order to decode a composite strand or a group of composite strands. In the second problem, our goal is study how to carefully choose a fixed number of mixtures of the DNA nucleotides such that the decoding probability by the maximum likelihood decoder is maximized.

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arXiv2025

Decoding Insertions/Deletions via List Recovery

In this work, we consider the problem of efficient decoding of codes from insertions and deletions. Most of the known efficient codes are codes with synchronization strings which allow one to reduce the problem of decoding insertions and deletions to that of decoding substitution and erasures. Our new approach, presented in this paper, reduces the problem of decoding insertions and deletions to that of list recovery. Specifically, any $(ρ, 2ρn + 1, L)$-list-recoverable code is a $(ρ, L)$-list decodable insdel code. As an example, we apply this technique to Reed-Solomon (RS) codes, which are known to have efficient list-recovery algorithms up to the Johnson bound. In the adversarial insdel model, this provides efficient (list) decoding from $t$ insdel errors, assuming that $t\cdot k = O(n)$. This is the first efficient insdel decoder for $[n, k]$ RS codes for $k>2$. Additionally, we explore random insdel models, such as the Davey-MacKay channel, and show that for certain choices of $ρ$, a $(ρ, n^{1/2+0.001}, L)$-list-recoverable code of length $n$ can, with high probability, efficiently list decode the channel output, ensuring that the transmitted codeword is in the output list. In the context of RS codes, this leads to a better rate-error tradeoff for these channels compared to the adversarial case. We also adapt the Koetter-Vardy algorithm, a famous soft-decision list decoding technique for RS codes, to correct insertions and deletions induced by the Davey-MacKay channel.

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arXiv2025

CertainSync: Rateless Set Reconciliation with Certainty

Set reconciliation is a fundamental task in distributed systems, particularly in blockchain networks, where it enables synchronization of transaction pools among peers and facilitates block dissemination. Traditional set reconciliation schemes are either statistical, offering success probability as a function of communication overhead and symmetric difference size, or require parametrization and estimation of that size, which can be error-prone. We present CertainSync, a novel reconciliation framework that, to the best of our knowledge, is the first to guarantee successful set reconciliation without any parametrization or estimators. The framework is rateless and adapts to the unknown symmetric difference size. Reconciliation is guaranteed whenever the communication overhead reaches a lower bound derived from the symmetric difference size and universe size. Our framework builds on recent constructions of Invertible Bloom Lookup Tables (IBLTs), ensuring successful element listing as long as the number of elements is bounded. We provide a theoretical analysis proving the certainty of reconciliation for multiple constructions. Our approach is validated by simulations, showing the ability to synchronize sets with efficient communication costs while maintaining guarantees compared to baseline schemes. To further reduce overhead in large universes such as blockchain networks, CertainSync is extended with a universe reduction technique. We compare and validate this extension, UniverseReduceSync, against the basic framework using real Ethereum transaction hash data. Results show a trade-off between lower communication costs and maintaining guarantees, offering a comprehensive solution for diverse reconciliation scenarios.

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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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