arXiv subjects

Eitan Yaakobi

Publications and source records attributed to Eitan Yaakobi.

At least 19 recordsLinked to original sources

arXiv2026

Making Every Edit Count: Efficient Binary Edit Correction at the Gilbert-Varshamov Bound

We construct binary codes that attain the classical Gilbert-Varshamov redundancy guarantee for a fixed number of edit errors, with deterministic polynomial-time construction, encoding, and decoding. For every fixed integer $t\ge 1$ and all sufficiently large block lengths $n$, our codes correct any combination of at most $t$ insertions, deletions, and substitutions in total, with redundancy at most $2t\log_2 n+O_t(1)$. All algorithms run in polynomial time in $n$ for fixed $t$, and decoding recovers the original information message. This realizes the classical counting guarantee algorithmically without enumerating the exponentially large space of binary words.

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arXiv2026

Representing Information on DNA using Patterns Induced by Enzymatic Labeling

Enzymatic DNA labeling is a powerful tool with applications in biochemistry, molecular biology, biotechnology, medical science, and genomic research. This paper contributes to the evolving field of DNA-based data storage by presenting a formal framework for modeling DNA labeling in strings, specifically tailored for data storage purposes. Our approach involves a known DNA molecule as a template for labeling, employing patterns induced by a set of designed labels to represent information. One hypothetical implementation can use CRISPR-Cas9 and gRNA reagents for labeling. Various aspects of the general labeling channel, including fixed-length labels, are explored, and upper bounds on the maximum size of the corresponding codes are given. The study includes the development of an efficient encoder-decoder pair that is proven to be optimal in terms of maximum code size under specific conditions. The codewords for this framework are codesets, i.e., each codeword is a collection of a few labels. Of special interest are codes whose codesets contain fixed-length labels. Their construction and analysis are tied to de Bruijn sequences and constrained coding.

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arXiv2026

On the Average List Size of VT Codes under Two Insertions or Two Deletions

Levenshtein's classical theorem shows that any code that can uniquely decode from $t$ deletions can also uniquely decode from $t$ insertions. This symmetry, however, does not generally hold in the list-decoding regime. Several prior results on the capacity and radii of list-decoding from insertions and deletions suggest that insertions are easier to handle than deletions. In an effort to better understand this asymmetry between insertions and deletions under list-decoding, we study an analogous version of McEliece's \emph{average list size problem}. More precisely, given a binary code $\mathcal{C}$ of length $n$ that uniquely corrects $t'$ insertions (or deletions), we are interested in the average number of codewords returned by a $t$-insertion (or deletion) list decoder, where $t' < t$. Focusing on the case of $t'=1$ and $t=2$, we find that under two insertions, this quantity is at most $(n+12)/4$ for any VT code of length $n$, whereas under two deletions it is at least $n/2 - 10$ when averaged over all length-$n$ VT codes. The nearly factor-of-two gap between these upper and lower bounds serves to confirm the relative ease of list-decoding of insertions compared to deletions from an average-case perspective.

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arXiv2026

Sequential Random Sampling PIR with Multiple Colluding Servers in DNA-Based Data Storage

As DNA-based data storage evolves, protecting user privacy during data retrieval has become increasingly important. We study sequential random sampling DNA private information retrieval (SRS DNA PIR) with multiple colluding random sampling servers, where the database is partitioned into servers of equal size. We investigate the tradeoff between the download cost, defined as the expected number of queries, and the privacy leakage, measured by mutual information. We derive lower bounds on this tradeoff, including a bound given by an optimization problem. This bound is tight when each server stores two files, and we construct schemes that attain it. For servers of any size, we construct schemes that apply a single-server scheme to a randomly selected subset of servers.

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arXiv2026

Optimal Codes for the Coverage Depth Problem and the Performance of Random Codes

DNA storage systems retrieve information by randomly sampling synthesized DNA strands, making the number of reads required for successful recovery a fundamental performance measure. This motivates the coverage depth problem: for given code parameters, determine a linear code that minimizes the expected number of randomly sampled columns required to span the entire information space. While MDS codes are known to be optimal whenever they exist, identifying optimal codes in parameter regimes where MDS codes do not exist remains largely open. In this work, we derive a general formula for the expectation in the coverage depth setting, and apply this result to establish the optimality of the $q$-ary simplex code for the parameters that allow its existence. Moreover, we show the optimality of the $q$-ary Hamming code by relating coverage depth to independent sets in the dual code and exploiting log-concavity properties of matroid basis-generating polynomials. We further analyze random linear codes, derive an exact expression for their expected coverage depth, and show that, in the constant-rate regime, their additive gap from the MDS benchmark remains bounded independently of the code length.

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arXiv2026

Efficient Synthesis for Two-Dimensional Strand Arrays with Row Constraints

In large-scale array-based DNA synthesis, optical and chemical coupling between nearby sites can limit simultaneous activations. Motivated by this constraint, we study strands synthesized according to a fixed global synthesis sequence, with at most one strand per row advancing in each cycle. We focus on the fundamental case of two strands in a single row and analyze the expected completion time of row-constrained synthesis. We introduce the laggard-first (LF) policy, a simple rule that always advances the strand with fewer synthesized symbols when a conflict arises, and establish that it is asymptotically optimal among online policies without look-ahead. In the binary case, one-symbol look-ahead strictly improves on the no-look-ahead bound. We further show that even complete advance knowledge does not eliminate the scheduling loss, as even a globally optimal schedule incurs an unavoidable expected overhead that grows linearly with the strand length. Finally, we complement these scheduling results with a dynamic programming algorithm for computing an optimal offline synthesis order and a constant-redundancy binary coding scheme that yields a deterministic worst-case synthesis time guarantee.

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arXiv2026

Correcting Tail Deletions in Rank Modulated Composite Encoding for Data Storage in DNA

We study the combination of two recent coding approaches, in the context of DNA based data storage. Composite DNA alphabets leverage properties of the DNA synthesis and sequencing process. A composite symbol does not represent a single nucleotide, but rather a designed mixture of DNA nucleotides. Using the high multiplicity that is intrinsic to synthesis and sequencing a composite symbol consists of frequencies in the mixture. Rank modulation codes use permutations to represent information. Combining the two, we construct encoding that uses permutations of nucleotide frequencies rather than the exact frequency values. Codes for this approach were addressed in previous work, under Kendall's tau distances. In this work we study deletion and insertion codes. We present bounds and constructions of efficient codes defined over partial permutations.

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arXiv2026

Random Access in DNA Storage: Algorithms, Constructions, and Bounds

As DNA data storage advances toward practical deployment, minimizing sequencing coverage depth is critical for reducing operational costs and retrieval latency. We study the random access problem of recovering a specific information strand from a DNA-based storage system. In this setting, $k$ information strands are encoded into $n$ strands using a generator matrix $G$, and each sequencing read returns one encoded strand sampled uniformly at random with replacement. We derive an exact formula for the expected number of samples required to recover a specific information strand, yielding an $O(n)$-time algorithm for fixed field size $q$ and dimension $k$. We further obtain explicit formulas for the average and maximum expected number of samples, enabling an efficient search for optimal generator matrices for small parameters. We present new constructions that improve the best-known upper bounds from $0.8815k$ to $0.8811k$ for $k=3$, and from $0.8637k$ to $0.8629k$ for $k=4$, for sufficiently large $q$. We also establish a tighter lower bound on the expected number of samples, which in particular proves the optimality of the simple parity code when $n=k+1$ over any field size $q$. Finally, for the non-random access setting, we derive new lower bounds and constructions that characterize the asymptotic behavior of the expected number of samples required to recover all information strands.

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arXiv2026

Error characterization and error correction approaches in combinatorial DNA-based storage

Data storage in DNA has recently emerged as a promising archival solution, offering space-efficient and long-lasting digital storage. Combinatorial DNA encoding enhances this potential by increasing the logical density through combinations of DNA shortmers, where each sequence position is represented by a set of predefined short DNA fragments, allowing more data to be encoded using fewer synthesis cycles. However, this method introduces unique synthesis and sequencing errors. In this study, we characterize errors in combinatorial DNA-based storage systems. We reveal that asymmetric combinatorial erasure errors, defined as the omission of a single shortmer from the set defining the combinatorial letter, are a prevalent error type, particularly in large-scale systems where read coverage is limited. In two previously published datasets, we observed a high frequency of erasure errors, where missing sequences obstruct the reconstruction of combinatorial letters. We conducted a large-scale experimental proof-of-concept and confirmed that erasure errors become increasingly prominent with reduced sequencing depth: below 50 reads per sequence, their frequency sharply increased. We developed an asymmetric error-correcting code for these errors, utilizing tensor-product codes to integrate standard erasure and substitution-correcting codes (such as Reed-Solomon (RS) codes) with asymmetric Varshamov-Tenengolts codes. We validated its performance in simulations and in a second large-scale experiment directly comparing it with the more straightforward 2D RS scheme. Our method consistently outperformed 2D RS, particularly in erasure-dominated scenarios, and demonstrated superior decoding accuracy under low coverage conditions where 2D RS struggled to decode the data. Our findings demonstrate the importance of error correction schemes tailored to the asymmetric nature of errors in combinatorial DNA.

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arXiv2026

More on Codes for Combinatorial Composite DNA

In this paper, we focus on constructions of unique-decodable/list-decodable on the recently studied $(t,e)$-composite-asymmetric error-correcting codes ($(t,e)$-CAECCs). Let $X$ be an $m\times n$ binary matrix, in which each row has Hamming weight $w$. When at most $t$ rows of $X$ suffer from errors and in each of these erroneous rows, there are at most $e$ $1 \to 0$ errors, we say that a $(t,e)$-composite-asymmetric-error occurs in $X$. For general $m,n,w,t,e$, we propose new constructions of $(t,e)$-CAECCs with redundancy at most $(t-1)\log(m)+O(1)$, where $O(1)$ is a number independent of the code-length $m$. In particular, this gives a class of $(2,e)$-CAECCs that are optimal in terms of their redundancy. %(in terms of redundancy, regarded as a function of the number of rows $m$) s. When $m$ is a prime power, the redundancy can be further reduced to $(t-1)\log(m)-O(\log(m))$. To further increase the size of these codes, we introduce a combinatorial object called a weak $B_e$-sets. When $e=w$, we show an efficient way to encode/decode our codes. At last, we investigate how much we can gain if we relax the requirement of uniquely decoding to list-decoding. It is shown that when the list size is $t!$ or an exponential function of $t$, there are list-decodable $(t,e)$-CAECCs with constant redundancy. When the list size is two, we show that there are list-decodable $(3,2)$-CAECCs with redundancy $\log(m)+O(1)$.

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arXiv2026

Bounds on Multiple $b$-Burst Deletion-Correcting Codes

Motivated by their applications in DNA-based storage systems, codes capable of correcting consecutive deletions have attracted significant attention. An important class of such codes consists of those that can correct multiple consecutive deletion errors, commonly referred to as multiple $b$-burst deletion-correcting codes. In this paper, we investigate the fundamental limits of multiple $b$-burst deletion-correcting codes. Specifically, we first characterize several structural properties of the associated deletion balls. Then, leveraging these properties, we derive several upper bounds and a combinatorial lower bound on the maximum size of such codes. As a consequence, our bounds improve upon the previously known results for general parameter regimes and are shown to be asymptotically optimal for certain cases.

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arXiv2026

Sequence Reconstruction for Substitution Channel: New Sufficient Conditions and Algorithms

In the sequence reconstruction problem, a codeword $\x$ is transmitted through several identical channels where each channel produces a noisy read of $\x$, and the problem is to analyze how to uniquely reconstruct $\x$ based on these noisy reads. Levenshtein has studied the minimum number of reads which guarantees unique reconstruction of $\x$, which is one sufficient condition for unique reconstruction. In this paper, we move on to a different perspective and propose a new framework for unique reconstruction. Our new sufficient condition for unique reconstruction takes both the number of reads and the distances among the reads into consideration. We offer both theoretical analysis and corresponding efficient reconstruction algorithms for our reconstruction framework.

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arXiv2026

Rank Modulated Composite Encoding for Data Storage in DNA

This paper studies two problems that are motivated by combining two novel approaches, namely DNA composite and rank modulation. The recent approach of composite DNA 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. Instead of considering all the possible composite symbols we are interested only in the rank of the motifs in the symbol. The first problem in this paper addresses the capacity of a channel that uses such symbols, while in the second, bounds and construction of such codes are studied.

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arXiv2026

Random Access Expectation in DNA Storage and Fountain Codes

Motivated by DNA data storage, we study the expected number of coded symbols drawn from a linear code until a desired information symbol can be decoded - the random access expectation. We focus on generator matrices with a type of symmetry, conjectured in prior work to be optimal, which we call fully symmetric. We point out an equivalence between binary fully symmetric codes and LT codes. Using this observation, we analyze the random access expectation of binary fully symmetric codes under a peeling decoder, in the large blocklength limit. Under these assumptions, the random access expectation, normalized by the number of information symbols, is at least $π/4 \approx 0.7854$, while a value of $\approx 0.7869$ is achievable.

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arXiv2026

Serving Every Symbol: All-Symbol PIR and Batch Codes

A $t$-all-symbol PIR code and a $t$-all-symbol batch code of dimension $k$ consist of $n$ servers storing linear combinations of $k$ information symbols with the following recovery property: any symbol stored by a server can be recovered from $t$ pairwise disjoint subsets of servers. In the batch setting, we further require that any multiset of size $t$ of stored symbols can be recovered from~$t$ disjoint subsets of servers. This framework unifies and extends several well-known code families, including one-step majority-logic decodable codes, (functional) PIR codes, and (functional) batch codes. In this paper, we determine the minimum code length for some small values of $k$ and $t$, characterize structural properties of codes attaining this optimum, and derive bounds that show the trade-offs between length, dimension, minimum distance, and $t$. In addition, we study MDS codes and the simplex code, demonstrating how these classical families fit within our framework, and establish new cases of an open conjecture from \cite{YAAKOBI2020} concerning the minimal $t$ for which the simplex code is a $t$-functional batch code.

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arXiv2026

Error-Correcting Codes for the Sum Channel

We introduce the sum channel, a new channel model motivated by applications in distributed storage and DNA data storage. In the error-free case, it takes as input an $\ell$-row binary matrix and outputs an $(\ell+1)$-row matrix whose first $\ell$ rows equal the input and whose last row is their parity (sum) row. We construct a two-deletion-correcting code with redundancy $2\lceil\log_2\log_2 n\rceil + O(\ell^2)$ for $\ell$-row inputs. When $\ell=2$, we establish an upper bound of $\lceil\log_2\log_2 n\rceil + O(1)$, implying that our redundancy is optimal up to a factor of 2. We also present a code correcting a single substitution with $\lceil \log_2(\ell+1)\rceil$ redundant bits and prove that it is within one bit of optimality.

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arXiv2026

The DNA Coverage Depth Problem: Duality, Weight Distributions, and Applications

The coverage depth problem in DNA data storage is about computing the expected number of reads needed to recover all encoded strands. Given a generator matrix of a linear code, this quantity equals the expected number of randomly drawn columns required to obtain full rank. While MDS codes are optimal when they exist, i.e., over large fields, practical scenarios may rely on structured code families defined over small fields. In this work, we develop combinatorial tools to solve the DNA coverage depth problem for various linear codes, based on duality arguments and the notion of extended weight enumerator. Using these methods, we derive closed formulas for the simplex, Hamming, ternary Golay, extended ternary Golay, and first-order Reed-Muller codes. The centerpiece of this paper is a general expression for the coverage depth of a linear code in terms of the weight distributions of its higher-field extensions.

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arXiv2026

Expected Recovery Time in DNA-based Distributed Storage Systems

We initiate the study of DNA-based distributed storage systems, where information is encoded across multiple DNA data storage containers to achieve robustness against container failures. In this setting, data are distributed over $M$ containers, and the objective is to guarantee that the contents of any failed container can be reliably reconstructed from the surviving ones. Unlike classical distributed storage systems, DNA data storage containers are fundamentally constrained by sequencing technology, since each read operation yields the content of a uniformly random sampled strand from the container. Within this framework, we consider several erasure-correcting codes and analyze the expected recovery time of the data stored in a failed container. Our results are obtained by analyzing generalized versions of the classical Coupon Collector's Problem, which may be of independent interest.

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