Constrained coding is a subfield of coding theory that tackles efficient communication under constraints. While fixed constraints (e.g., a fixed set of substrings may not appear in transmitted messages) have a general optimal solution, there is increasing demand for supportingparametricconstraints that are dependent on the message length and portray some property (e.g., no log(n)consecutive zeros). Several works have tackled such parametric constraints throughiterativealgorithms, yet they require complex constructions specific to each constraint to guarantee convergence throughmonotonic progression. In this paper, we propose a universal framework for tacklinganyparametric constraint problem through a new simple iterative algorithm. By reducing an execution of this iterative algorithm to an acyclic graph traversal, we prove a surprising result that guarantees convergence with low average time complexityeven without requiring any monotonic progression. We demonstrate the effectiveness of this universal framework, with much of our focus on the special case of single-symbol redundancy, while also considering a variety of bothlocalandglobalconstraints. We begin by exploring the local constraints involving illegal substrings of variable length, where the construction essentially iteratively replaces forbidden windows. This local algorithm is applied to various fundamental constraints, achieving state-of-the-art results through simple adaptations of the universal algorithm. We then continue by exploring global constraints, and demonstrate the effectiveness of the proposed construction on repeat-free encoding, reverse-complement encoding and DNA data storage. Overall, the proposed framework generates state-of-the-art constructions with significant ease while also enabling the simultaneous integration of multiple constraints for the first time.

We generalize the well-known Coupon Collector Problem (CCP) in combinatorics. Our problem is to find the minimum and expected number of draws, with replacement, required to recover n distinctly labeled coupons, with each draw consisting of a random subset of k different coupons and a random ordering of their associated labels. We specify two variations of the problem, Type-I in which the set of labels is known at the start, and Type-II in which the set of labels is unknown at the start. We show that our problem can be viewed as an extension of the separating system problem introduced by Rényi and Katona, provide a full characterization of the minimum, and provide a numerical approach to finding the expectation using a Markov chain model, with special attention given to the case where two coupons are drawn at a time.

Although the expenses associated with DNA sequencing have been rapidly decreasing, the current cost of sequencing information stands at roughly${\$}120$/GB, which is dramatically more expensive than reading from existing archival storage solutions today. In this work, we aim to reduce not only the cost but also the latency of DNA storage by initiating the study of the DNA coverage depth problem, which aims to reduce the required number of reads to retrieve information from the storage system. Under this framework, our main goal is to understand the effect of error-correcting codes and retrieval algorithms on the required sequencing coverage depth. We establish that the expected number of reads that are required for information retrieval is minimized when the channel follows a uniform distribution. We also derive upper and lower bounds on the probability distribution of this number of required reads and provide a comprehensive upper and lower bound on its expected value. We further prove that for a noiseless channel and uniform distribution, MDS codes are optimal in terms of minimizing the expected number of reads. Additionally, we study the DNA coverage depth problem under the random-access setup, in which the user aims to retrieve just a specific information unit from the entire DNA storage system. We prove that the expected retrieval time is at least k for$[n,k]$MDS codes as well as for other families of codes. Furthermore, we present explicit code constructions that achieve expected retrieval times below k and evaluate their performance through analytical methods and simulations. Lastly, we provide lower bounds on the maximum expected retrieval time. Our findings offer valuable insights for reducing the cost and latency of DNA storage.

This paper introduces a new solution to DNA storage that integrates all three steps of retrieval, namely clustering, reconstruction, and error correction.DNA-correcting codesare presented as a unique solution to the problem of ensuring that the output of the storage system is unique for any valid set of input strands. To this end, we introduce a novel distance metric to capture the unique behavior of the DNA storage system and provide necessary and sufficient conditions for DNA-correcting codes. We also establish bounds and constructions for these codes, including an exploration of the ℓ∞distance applied to permutations. Here, instead of interpreting permutation elements as numerical values and assessing absolute differences, we treat them as vectors and consider the Hamming distance to better model the DNA Storage System.

We investigate the fundamental limits of the recently proposedrandom access coverage depth problemfor DNA data storage. Under this paradigm, it is assumed that the user information consists ofkinformation strands, which are encoded intonstrands via a generator matrixG. 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 matrixG, and besides computing the expectation for different choices ofG, the goal is to construct matrices that minimize the maximum expectation over all possible requested information strands, denoted byTmax(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 determineTmax(G) for arbitrary generator matrices. The first formula depends on the linear dependencies between columns ofG, whereas the second formula takes into account recovery sets and their intersection structure. We also introduce the concept ofrecovery balanced codesand provide three sufficient conditions for a code to be recovery balanced. These conditions can be used to computeTmax(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 forTmax(G) are achieved with a specific combination of encoded strands and replication of the information strands.

DNA labelingis a powerful tool in molecular biology and biotechnology that allows for the visualization, detection, and study of DNA at the molecular level. Under this paradigm, a DNA molecule is beinglabeledby specifickpatterns and is then imaged. Then, the resulting image is modeled as a$(k+1)$-ary sequence in which any non-zero symbol indicates on the appearance of the corresponding label in the DNA molecule. The primary goal of this work is to study thelabeling capacity, which is defined as the maximal information rate that can be obtained using this labeling process. The labeling capacity is computed for almost any pattern of a single label and several results for multiple labels are provided as well. Moreover, we provide the optimal minimal number of labels of length one or two, over any alphabet of sizeq, that are needed in order to achieve the maximum labeling capacity of$\log _{2}(q)$. Lastly, we discuss the maximal labeling capacity that can be achieved using a certain number of labels of length two.

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 thek-deletion andk-insertion channels, where exactlyksymbols are deleted or inserted in the transmitted sequence, respectively. If all possible sequences are transmitted with the same probability, then studying the input and output entropies becomes equivalent. For both the 1-deletion and 1-insertion channels, it is shown that among all sequences with a fixed number of runs, the input entropy is minimized for sequences with a skewed distribution of run lengths, and it is maximized for sequences with a balanced distribution of 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 among all binary sequences. This conjecture is also verified for the 2-deletion channel, where it is proved that sequences with a single run minimize the input entropy.

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 maximal size of the corresponding codes are given. The study includes the development of an efficient encoder-decoder pair that is proven optimal in terms of maximum code size under specific conditions.

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 some generator matrix$G$. In the sequencing process, the strands are read uniformly at random, since 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) until it is possible to decode a specific information strand, which is requested by the user. The goal is to minimize the maximum expectation over all possible requested information strands, and this value is denoted by$T_{\max}(G)$. This paper introduces new techniques to investigate the random access coverage depth problem, which capture its combinatorial nature. We establish two general formulas to find$T_{\max}(G)$for arbitrary matrices. We introduce the concept of recovery balanced codes and combine all these results and notions to compute$T_{\max}(G)$for MDS, simplex, and Hamming codes. We also study the performance of modified systematic MDS matrices and our results show that the best results for$T_{\max}(G)$are achieved with a specific mix of encoded strands and replication of the information strands.

Constrained coding is a fundamental field in coding theory that tackles efficient communication through constrained channels. While fixed constraints (e.g., a fixed set of substrings may not appear in transmitted messages) have a general optimal solution, there is increasing demand for supporting parametric constraints that are dependent on the message length and portray some property that the substrings must satisfy (e.g., no log (n) consecutive zeros). Several works have tackled such parametric constraints through iterative algorithms following the sequence-replacement approach, yet this approach requires complex constraint-specific properties to guarantee convergence through monotonic progression. In this paper, we propose a universal framework for tackling any parametric constraint problem with far fewer requirements, through a simple iterative algorithm. By reducing an execution of this iterative algorithm to an acyclic graph traversal, we prove a surprising result that guarantees convergence with efficient average time complexity even without requiring any monotonic progression. We demonstrate how to apply this algorithm to the run-length-limited, minimal Hamming weight, local almost-balanced Hamming weight constraints, as well as repeat-free and secondary-structure constraints. Overall, this framework enables state-of-the-art results with minimal effort.

The Invertible Bloom Lookup Table (IBLT) is a probabilistic data structure for set representation, with applications in network and traffic monitoring. It is known for its ability to list its elements, an operation that succeeds with high probability for sufficiently large table. However, listing can fail even for relatively small sets. This paper extends recent work on the worst-case analysis of IBLT, which guarantees successful listing for all sets of a certain size, by introducing more general IBLT schemes. These schemes allow for greater freedom in the implementation of the insert, delete, and listing operations and demonstrate that the IBLT memory can be reduced while still maintaining successful listing guarantees. The paper also explores the time-memory trade-off of these schemes, some of which are based on linear codes and Bh-sequences over finite fields.

Although the expenses associated with DNA sequencing have been rapidly decreasing, the current cost stands at roughly $1.3K/TB, which is dramatically more expensive than reading from existing archival storage solutions today. In this work, we aim to reduce not only the cost but also the latency of DNA storage by studying the DNA coverage depth problem, which aims to reduce the required number of reads to retrieve information from the storage system. Under this framework, our main goal is to understand how to optimally pair an error-correcting code with a given retrieval algorithm to minimize the sequencing coverage depth, while guaranteeing retrieval of the information with high probability. Additionally, we study the DNA coverage depth problem under the random-access setup.

This paper introduces a new solution to DNA storage that integrates all three steps of retrieval, namely clustering, reconstruction, and error correction. DNA-correcting codes are presented as a unique solution to the problem of ensuring that the output of the storage system is unique for any valid set of input strands. To this end, we introduce a novel distance metric to capture the unique behavior of the DNA storage system and provide necessary and sufficient conditions for DNA-correcting codes. The paper also includes several upper bounds and constructions of DNA-correcting codes.

DNA labeling is a powerful tool in molecular biology and biotechnology that allows for the visualization, detection, and study of DNA at the molecular level. Under this paradigm, a DNA molecule is being labeled by specific k patterns and is then imaged. Then, the resulted image is modeled as a (k +1)-ary sequence in which any non-zero symbol indicates on the appearance of the corresponding label in the DNA molecule. The primary goal of this work is to study the labeling capacity, which is defined as the maximal information rate that can be obtained using this labeling process. The labeling capacity is computed for any single label and several results are provided for multiple labels as well. Moreover, we provide the optimal minimal number of labels of length one or two that are needed in order to gain labeling capacity of 2.

MOTIVATION: Optical genome mapping (OGM) is a technique that extracts partial genomic information from optically imaged and linearized DNA fragments containing fluorescently labeled short sequence patterns. This information can be used for various genomic analyses and applications, such as the detection of structural variations and copy-number variations, epigenomic profiling, and microbial species identification. Currently, the choice of labeled patterns is based on the available biochemical methods and is not necessarily optimized for the application. RESULTS: In this work, we develop a model of OGM based on information theory, which enables the design of optimal labeling patterns for specific applications and target organism genomes. We validated the model through experimental OGM on human DNA and simulations on bacterial DNA. Our model predicts up to 10-fold improved accuracy by optimal choice of labeling patterns, which may guide future development of OGM biochemical labeling methods and significantly improve its accuracy and yield for applications such as epigenomic profiling and cultivation-free pathogen identification in clinical samples. AVAILABILITY AND IMPLEMENTATION: https://github.com/yevgenin/PatternCode.

The rapid development of DNA storage has brought the deletion and insertion channel to the front line of research. When the number of deletions is equal to the number of insertions, theFixed Length Levenshtein(FLL) metric is the right measure for the distance between two words of the same length. Similar to any other metric, the size of a ball is one of the most fundamental parameters. In this work, we consider the minimum, maximum, and average size of a ball with radius one, in the FLL metric. The related minimum and the maximum size of a maximal anticode with diameter one are also considered.

We study the adversarial torn-paper channel. This problem is motivated by applications in DNA data storage where the DNA strands that carry information may break into smaller pieces which are received out of order. Our model extends the previously researched probabilistic setting to the worst-case. We develop code constructions for any parameters of the channel for which non-vanishing asymptotic rate is possible and show our constructions achieve asymptotically optimal rate while allowing for efficient encoding and decoding. Finally, we extend our results to related settings included multi-strand storage, presence of substitution errors, or incomplete coverage.

This paper introduces a new family of reconstruction codes which is motivated by applications in DNA data storage and sequencing. In such applications, DNA strands are sequenced by reading some subset of their substrings. While previous works considered two extreme cases in which all substrings of pre-defined lengths are read or substrings are read with no overlap for the single string case, this work studies two extensions of this paradigm. The first extension considers the setup in which consecutive substrings are read with some given minimum overlap. First, an upper bound is provided on the attainable rates of codes that guarantee unique reconstruction. Then, efficient constructions of codes that asymptotically meet that upper bound are presented. In the second extension, we study the setup where multiple strings are reconstructed together. Given the number of strings and their length, we first derive a lower bound on the read substrings’ length$\ell $that is necessary for the existence of multi-strand reconstruction codes with non-vanishing rates. We then present two constructions of such codes and show that their rates approach 1 for values of$\ell $that asymptotically behave like the lower bound.

This paper studies the adversarial torn-paper channel. This problem is motivated by applications in DNA data storage where the DNA strands that carry the information may break into smaller pieces that are received out of order. Our model extends the previously researched probabilistic setting to the worst-case. We develop code constructions for any parameters of the channel for which non-vanishing asymptotic rate is possible and show that our constructions achieve optimal asymptotic rate while allowing for efficient encoding and decoding. Finally, we extend our results to related settings included multi-strand storage, presence of substitution errors, or incomplete coverage.

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

The rapid development of DNA storage has brought the deletion and insertion channel, once again, to the front line of research. When the number of deletions is equal to the number of insertions, the Fixed Length Levenshtein$(FLL)$metric is the right measure for the distance between two words of the same length. The size of a ball is one of the most fundamental parameters in any metric. The size of the ball with radius one in the FLL metric depends on the number of runs and the length of the alternating segments of the given word. In this work, we find the minimum, maximum, and average size of a ball with radius one, in the FLL metric. The related minimum and maximum sizes of a maximal anticode with diameter one are also calculated.

This paper studies optimal decoding for a special case of the deletion channel, referred by the t-deletion channel, which deletes exactly$t$symbols of the transmitted word uniformly at random. The goal of the paper is to understand how such an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of a decoder for this setup is given for a channel that deletes one or two symbols. For$t$= 1 it is shown that when the code is the entire space, the decoder is the lazy decoder which simply returns the channel output. Similarly, for$t$= 2 it is shown that the decoder acts as the lazy decoder in almost all cases and when the longest run is significantly long, it prolongs the longest run by one symbol.

This paper studies the intersections of insertion and deletion balls. The t-insertion, t-deletion ball of a sequence x is the set of all sequences received by t insertions, deletions to x, respectively. While the intersection of either deletion balls or insertion balls has been rigorously studied before, the intersection of an insertion ball and a deletion ball has not been addressed so far. We find the maximum intersection size of any two insertion and deletion balls in the binary case. For the special case of one-insertion and one-deletion balls we find the intersection size for all pair of sequences. Then, we derive the largest and average values of this intersection size. Lastly, we present an algorithm that efficiently computes the intersection of any t1-insertion ball and t2-deletion ball.