Conference Paper

# Constrained Coding with Error Control for DNA-Based Data Storage

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## Abstract

In this paper, we first propose coding techniques for DNA-based data storage which account the maximum homopolymer runlength and the GC-content. In particular, for arbitrary $\ell,\epsilon > 0$, we propose simple and efficient $(\epsilon, \ell)$-constrained encoders that transform binary sequences into DNA base sequences (codewords), that satisfy the following properties: • Runlength constraint: the maximum homopolymer run in each codeword is at most $\ell$, • GC-content constraint: the GC-content of each codeword is within $[0.5 − \epsilon, 0.5 + \epsilon]$. For practical values of l and ε, our codes achieve higher rates than the existing results in the literature. We further design efficient $(\epsilon,\ell)$-constrained codes with error-correction capability. Specifically, the designed codes satisfy the runlength constraint, the GC-content constraint, and can correct a single edit (i.e. a single deletion, insertion, or substitution) and its variants. To the best of our knowledge, no such codes are constructed prior to this work.

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... We believe that the studied model would be relevant to practical experiments where the GC-content of DNA strands is only required to be close to 50% (for example, between 45% to 55% in [5], or 40% to 60% in [8]). Via a modification of Knuth's method, we show that the number of redundant bits can be gracefully reduced from log n to O (1). Moreover, we also propose an efficient coding design that satisfy both RLL constraint and GC-content constraint. ...
... For DNA-based storage, we are interested in codewords that are -runlength limited and -balanced for 6, and sufficient small = o (1). ...
Article
We propose coding techniques that simultaneously limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given ℓ, ϵ > 0, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy all of the following properties: • Runlength constraint: the maximum homopolymer run in each codeword is at most ℓ, • GC-content constraint: the GC-content of each codeword is within [0.5 - ϵ; 0.5 + ϵ], • Error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. While various combinations of these properties have been considered in the literature, this work provides generalizations of codes constructions that satisfy all the properties with arbitrary parameters of ℓ and ϵ. Furthermore, for practical values of ℓ and ϵ, we show that our encoders achieve higher rates than existing results in the literature and approach capacity. Our methods have low encoding/decoding complexity and limited error propagation.
... There is no good way to solve this NP-complete problem because the number of DNA codes always decreases with increasing constraints. Recently, Nguyen et al. [47] designed constrained codes with error-correction capability, which were capable of correcting a single insertion / edited. wang et al. [48] presented novel similarity significance (SS) model to measure the similarity between DNA sequences. ...
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Metaheuristic algorithms have the drawback that local optimal solutions are prone to precocious convergence. In order to overcome the disadvantages of the whale optimization algorithm, we propose an improved selective opposition whale optimization algorithm (ISOWOA) in this paper. Firstly, the enhanced quasi-opposition learning (EQOBL) is applied to selectively update the position of the predator, calculate the fitness of the population before and after, and retain optimal individuals as the food source position; Secondly, an improved time-varying update strategy for inertia weight predator position is proposed, and the position update of the food source is completed by this strategy. The performance of the algorithm is analyzed by 23 benchmark functions of CEC 2005 and 15 benchmark functions of CEC 2015 in various dimensions. The superior results are further shown by Wilcoxon's rank sum test and Friedman's nonparametric rank test. Finally, its applicability is demonstrated through applications to the field of biological computing. In this paper, our aim is to achieve access to DNA files and designs high-quantity DNA code sets by ISOWOA. The experimental results show that the lower bounds of the multi-constraint storage coding sets implemented in this paper equals or surpasses that of previous optimal constructions. The data show that the amount of the DNA storage cods filtered by ISOWOA increased 2–18%, which demonstrates the algorithm's reliability in practical optimization tasks.
... To demonstrate the motivation for our work, let us examine an example as shown in Fig. 1, in which case the DNA-XL code, as well as many other DNA codes, fails to work when multiple errors occur within one block. Recently, Nguyen et al. [19] presented a simple and e cient (✏,`)-constrained encoder that can correct a single indel/edit error. Unfortunately, these existing DNA encoding methods still su↵er from high complexity. ...
Article
Deoxyribonucleic acid (DNA)-based data storage is a promising new storage technology which has the advantage of high storage capacity and long storage time compared with traditional storage media. However, the synthesis and sequencing process of DNA can randomly generate many types of errors, which makes it more difficult to cluster DNA sequences to recover DNA information. Currently, the available DNA clustering algorithms are targeted at DNA sequences in the biological domain, which not only cannot adapt to the characteristics of sequences in DNA storage, but also tend to be unacceptably time-consuming for billions of DNA sequences in DNA storage. In this paper, we propose an efficient DNA clustering method termed Clover for DNA storage with linear computational complexity and low memory. Clover avoids the computation of the Levenshtein distance by using a tree structure for interval-specific retrieval. We argue through theoretical proofs that Clover has standard linear computational complexity, low space complexity, etc. Experiments show that our method can cluster 10 million DNA sequences into 50 000 classes in 10 s and meet an accuracy rate of over 99%. Furthermore, we have successfully completed an unprecedented clustering of 10 billion DNA data on a single home computer and the time consumption still satisfies the linear relationship. Clover is freely available at https://github.com/Guanjinqu/Clover.
Preprint
Due to its higher data density, longevity, energy efficiency, and ease of generating copies, DNA is considered a promising storage technology for satisfying future needs. However, a diverse set of errors including deletions, insertions, duplications, and substitutions may arise in DNA at different stages of data storage and retrieval. The current paper constructs error-correcting codes for simultaneously correcting short (tandem) duplications and at most $p$ edits, where a short duplication generates a copy of a substring with length $\leq 3$ and inserts the copy following the original substring, and an edit is a substitution, deletion, or insertion. Compared to the state-of-the-art codes for duplications only, the proposed codes correct up to $p$ edits (in addition to duplications) at the additional cost of roughly $8p(\log_q n)(1+o(1))$ symbols of redundancy, thus achieving the same asymptotic rate, where $q\ge 4$ is the alphabet size and $p$ is a constant. Furthermore, the time complexities of both the encoding and decoding processes are polynomial when $p$ is a constant with respect to the code length.
Conference Paper
In this work, given n,ϵ > 0, two efficient encoding (decoding) methods are presented for mapping arbitrary data to (from) n×n binary arrays in which the weight of every row and every column is within [(1/2–ϵ)n, (1/2+ϵ)n], which is referred to as the ϵ-balanced constraint. The first method combines the divide and conquer algorithm and a modification of the Knuth’s balancing technique, resulting a redundancy of Θ(n) bits. On the other hand, for sufficiently large n, the second method uses the sequence replacement technique, which costs only one redundant bit. The latter method reduces significantly the redundancy of the best known encoder for two-dimensional p-bounded weight constrained codes from (n + 3) bits to a single bit.
Article
Due to its high data density and longevity, DNA is considered a promising medium for satisfying ever-increasing data storage needs. However, the diversity of errors that occur in DNA sequences makes efficient error-correction a challenging task. This paper aims to address simultaneously correcting two types of errors, namely, short tandem duplication and edit errors, where an edit error may be a substitution, deletion, or insertion. We focus on tandem repeats of length at most 3 and design codes for correcting an arbitrary number of duplication errors and one edit error. Because an edited symbol can be duplicated many times (as part of substrings of various lengths), a single edit can affect an unbounded substring of the retrieved word. However, we show that with appropriate preprocessing, the effect may be limited to a substring of finite length, thus making efficient error-correction possible. We construct a code for correcting the aforementioned errors and provide lower bounds for its rate. Compared to optimal codes correcting only duplication errors, numerical results show that the asymptotic cost of protecting against an additional edit is only 0.003 bits/symbol when the alphabet has size 4, an important case corresponding to data storage in DNA.
Conference Paper
Full-text available
In this work, given n, p>0 , efficient encoding/decoding algorithms are presented for mapping arbitrary data to and from n×n binary arrays in which the weight of every row and every column is at most pn. Such constraint, referred as p-bounded-weight-constraint, is crucial for reducing the parasitic currents in the crossbar resistive memory arrays, and has also been proposed for certain applications of the holographic data storage. While low-complexity designs have been proposed in the literature for only the case p=1/2 , this work provides efficient coding methods that work for arbitrary values of p . The coding rate of our proposed encoder approaches the channel capacity for all p .
Preprint
Due to its high data density and longevity, DNA is considered a promising medium for satisfying ever-increasing data storage needs. However, the diversity of errors that occur in DNA sequences makes efficient error-correction a challenging task. This paper aims to address simultaneously correcting two types of errors, namely, short tandem duplication and substitution errors. We focus on tandem repeats of length at most 3 and design codes for correcting an arbitrary number of duplication errors and one substitution error. Because a substituted symbol can be duplicated many times (as part of substrings of various lengths), a single substitution can affect an unbounded substring of the retrieved word. However, we show that with appropriate preprocessing, the effect may be limited to a substring of finite length, thus making efficient error-correction possible. We construct a code for correcting the aforementioned errors and provide lower bounds for its rate. Compared to optimal codes correcting only duplication errors, numerical results show that the asymptotic cost of protecting against an additional substitution is only 0.003 bits/symbol when the alphabet has size 4, an important case corresponding to data storage in DNA.
Article
Full-text available
We describe properties and constructions of constraint-based codes for DNA-based data storage which account for the maximum repetition length and AT/GC balance. Generating functions and approximations are presented for computing the number of sequences with maximum repetition length and AT/GC balance constraint. We describe routines for translating binary runlength limited and/or balanced strings into DNA strands, and compute the efficiency of such routines. Expressions for the redundancy of codes that account for both the maximum repetition length and AT/GC balance are derived.
Conference Paper
Full-text available
An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. We investigate codes that combat either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with 2log n + 2 redundant bits, while the other corrects a single indel with log n + 2 redundant bits. The latter encoder reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, exactly half of the symbols of a GC-balanced word are either C or G. Via a modification of Knuth’s balancing technique, we provide a linear-time map that translates binary messages into GC-balanced codewords and the resulting codebook is able to correct a single edit. The redundancy of our encoder is 3log n + 2 bits and this is the first known construction of a GC-balanced code that corrects a single edit.
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We analyze codes for DNA-based data storage which accounts for the maximum homopolymer repetition length and GC-AT balance. We present a new precoding method for translating words with a maximum run of k zeros into words with a maximum homopolymer run m = k + 1, which is atractive for securing GC-AT balance. Generating functions are presented for enumerating the number of n-symbol k-constrained codewords of given GC-AT balance Various efficient constructions are presented of block codes that satisfy a combined balance and maximum homopolymer run.
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We propose a coding method to transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy the following two properties • Run-length constraint. The maximum run-length of each symbol in each codeword is at most three; • GC-content constraint: The GC-content of each codeword is close to 0.5, say between 0.4 and 0.6. The proposed coding scheme is motivated by the problem of designing codes for DNA-based data storage systems, where the binary digital data is stored in synthetic DNA base sequences. Existing literature either achieve code rates not greater than 1.78 bits per nucleotide or lead to severe error propagation. Our method achieves a rate of 1.9 bits per DNA base with low encoding/decoding complexity and limited error propagation.
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Owing to its longevity and enormous information density, DNA, the molecule encoding biological information, has emerged as a promising archival storage medium. However, due to technological constraints, data can only be written onto many short DNA molecules that are stored in an unordered way, and can only be read by sampling from this DNA pool. Moreover, imperfections in writing (synthesis), reading (sequencing), storage, and handling of the DNA, in particular amplification via PCR, lead to a loss of DNA molecules and induce errors within the molecules. In order to design DNA storage systems, a qualitative and quantitative understanding of the errors and the loss of molecules is crucial. In this paper, we characterize those error probabilities by analyzing data from our own experiments as well as from experiments of two different groups. We find that errors within molecules are mainly due to synthesis and sequencing, while imperfections in handling and storage lead to a significant loss of sequences. The aim of our study is to help guide the design of future DNA data storage systems by providing a quantitative and qualitative understanding of the DNA data storage channel.
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Synthetic DNA is durable and can encode digital data with high density, making it an attractive medium for data storage. However, recovering stored data on a large-scale currently requires all the DNA in a pool to be sequenced, even if only a subset of the information needs to be extracted. Here, we encode and store 35 distinct files (over 200 MB of data), in more than 13 million DNA oligonucleotides, and show that we can recover each file individually and with no errors, using a random access approach. We design and validate a large library of primers that enable individual recovery of all files stored within the DNA. We also develop an algorithm that greatly reduces the sequencing read coverage required for error-free decoding by maximizing information from all sequence reads. These advances demonstrate a viable, large-scale system for DNA data storage and retrieval.
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We consider coding techniques that limit the lengths of homopolymer runs in strands of nucleotides used in DNA-based mass data storage systems. We compute the maximum number of user bits that can be stored per nucleotide when a maximum homopolymer runlength constraint is imposed. We describe simple and efficient implementations of coding techniques that avoid the occurrence of long homopolymers, and the rates of the constructed codes are close to the theoretical maximum. The proposed sequence replacement method for k-constrained q-ary data yields a significant improvement in coding redundancy than the prior art sequence replacement method for the k-constrained binary data. Using a simple transformation, standard binary maximum runlength limited sequences can be transformed into maximum runlength limited q-ary sequences, which opens the door to applying the vast prior art binary code constructions to DNA-based storage.
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DNA-based data storage is an emerging nonvolatile memory technology of potentially unprecedented density, durability, and replication efficiency. The basic system implementation steps include synthesizing DNA strings that contain user information and subsequently retrieving them via high-throughput sequencing technologies. Existing architectures enable reading and writing but do not offer random-access and error-free data recovery from low-cost, portable devices, which is crucial for making the storage technology competitive with classical recorders. Here we show for the first time that a portable, random-access platform may be implemented in practice using nanopore sequencers. The novelty of our approach is to design an integrated processing pipeline that encodes data to avoid costly synthesis and sequencing errors, enables random access through addressing, and leverages efficient portable sequencing via new iterative alignment and deletion error-correcting codes. Our work represents the only known random access DNA-based data storage system that uses error-prone nanopore sequencers, while still producing error-free readouts with the highest reported information rate/density. As such, it represents a crucial step towards practical employment of DNA molecules as storage media.
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For n >0, d &ges;0, n ≡ d (mod 2), let K ( n , d ) denote the minimal cardinality of a family V of ±1 vectors of dimension n , such that for any ±1 vector w of dimension n there is a v ∈ V such that | v - w |&les; d , where v - w is the usual scalar product of v and w . A generalization of a simple construction due to D.E. Knuth (1986) shows that K ( n , d )&les;[ n /( d +1)]. A linear algebra proof is given here that this construction is optimal, so that K ( n , d )-[ n /( d +1)] for all n ≡ d (mod 2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links
Portable and error-free DNAbased data storage
• S Yazdi
• R Gabrys
• O Milenkovic
DNA Codes with Run-Length Limitation and Knuth-Like Balancing of the GC Contents
• D Dubé
• W Song
• K Cai
DNA Codes with Run-Length Limitation and Knuth-Like Balancing of the GC Contents
• dubé