Wentu Song, Kui Cai, Tuan Thanh Nguyen
In this paper, we present an explicit construction of list-decodable codes for single-deletion and single-substitution with list size two and redundancy 3log n+4, where n is the block length of the code. Our construction has lower redundancy than the best known explicit construction by Gabrys et al. (arXiv 2021), whose redundancy is 4log n+O(1).
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This work constructs codes that are efficiently decodable from a constant fraction of \emph{worst-case} insertion and deletion errors in three parameter settings: (i) Binary codes with rate approaching 1; (ii) Codes with constant rate for error fraction approaching 1 over fixed alphabet size; and (iii) Constant rate codes over an alphabet of size $k$ for error fraction approaching $(k-1)/(k+1)$. When errors are constrained to deletions alone, efficiently decodable codes in each of these regimes were constructed recently. We complete the picture by constructing similar codes that are efficientl…
This paper presents general bounds on the highest achievable rate for list-decodable insertion-deletion codes. In particular, we give novel outer and inner bounds for the highest achievable communication rate of any insertion-deletion code that can be list-decoded from any $γ$ fraction of insertions and any $δ$ fraction of deletions. Our bounds simultaneously generalize the known bounds for the previously studied special cases of insertion-only, deletion-only, and zero-rate and correct other bounds that had been reported for the general case.
In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$.
We construct locally decodable codes (LDCs) to correct insertion-deletion errors in the setting where the sender and receiver share a secret key or where the channel is resource-bounded. Our constructions rely on a so-called "Hamming-to-InsDel" compiler (Ostrovsky and Paskin-Cherniavsky, ITS '15 & Block et al., FSTTCS '20), which compiles any locally decodable Hamming code into a locally decodable code resilient to insertion-deletion (InsDel) errors. While the compilers were designed for the classical coding setting, we show that the compilers still work in a secret key or resource-bounded set…
Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and decoding algorithms. In many applications, polynomial running time may still be prohibitively expensive, which has motivated the study of codes with super-efficient decoding algorithms. These have led to the well-studied notions of Locally Decodable Codes (LDCs) and Locally Correctable Codes (LCCs). Inspired by these notions, Ostrovsky and Paskin-Cherniavsky (…
Lower Bounds for all List-Decodable Deletion Codes
本文证明了所有列表可解码删除码的最优大小下界，改进已知结果，并给出首个非平凡下界，同时证明公共子序列和超序列数量的上界。
Characterization of Deletion/Substitution Channel Capacity for Small Deletion and Substitution Probabilities
本文扩展仅删除信道的分析方法，通过固定与随机删除/替换情形下的码本大小界，刻画了小概率二元 i.i.d. 删除/替换信道的渐近容量。
Explicit Construction of q-ary 2-deletion Correcting Codes with Low Redundancy
本文针对低冗余q元2-删除纠错码高效构造问题，以二进制2-删除码为黑盒，通过q元-二进制串转换加额外q元草图的方式，构造出冗余度低于现有同类码的显式q元2-删除纠错码及可高效列表译码的版本。
Linear List Decodable Edit-Correcting Codes with Rate Approaching $1$
本文针对纠正单个删除的线性码速率上限为 $1/2$ 的问题，构造了速率趋近于 $1$ 且列表大小合理的线性列表可译编辑纠错码，并实现了多项式时间的编码与解码。
Amortized Locally Decodable Codes for Insertions and Deletions
本文提出汉明到插入删除的编译器，将满足特定性质的摊销汉明LDC转换为摊销Insdel LDC，并在私钥和资源受限设置下构造理想摊销汉明LDC，最终获得常数摊销局部性、常数码率和常数容错能力的理想Insdel aLDCs。
Correcting insertions/deletions as well as substitution errors simultaneously plays an important role in DNA-based storage systems as well as in classical communications. This paper deals with the fundamental task of constructing codes that can correct a single insertion or deletion along with a single substitution. A non-asymptotic upper bound on the size of single-deletion single-substitution correcting codes is derived, showing that the redundancy of such a code of length $n$ has to be at least $2 \log n$. The bound is presented both for binary and non-binary codes while an extension to sin…
In this paper, two moment balancing schemes, namely a variable index scheme and a fixed index scheme, for either single insertion/deletion error correction or multiple substitution error correction are introduced for coded sequences originally developed for correcting substitution errors only. By judiciously flipping bits of the original substitution error correcting code word, the resulting word is able to correct either a reduced number of substitution errors or a single insertion/deletion error. The number of flips introduced by the two schemes can be kept small compared to the code length.…
One Burst of t-Deletion and One Burst of t-Substitution Error-Correcting Codes
针对通信系统中突发同步错误问题，提出了一种显式构造的纠错码，可纠正一次突发t删除错误和一次突发t替换错误，冗余度为O(log n)。
Recent work by Smagloy et al. (ISIT 2020) shows that the redundancy of a single-deletion $s$-substitution correcting code is asymptotically at least $(s+1)\log n+o(\log n)$, where $n$ is the length of the codes. They also provide a construction of single-deletion and single-substitution codes with redundancy $6\log n+8$. In this paper, we propose a family of systematic single-deletion $s$-substitution correcting codes of length $n$ with asymptotical redundancy at most $(3s+4)\log n+o(\log n)$ and polynomial encoding/decoding complexity, where $s\geq 2$ is a constant. Specifically, the encoding…
In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $Ω(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $τ_\mathsf{I} \geq 0$ and $τ_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $Ω(1)$ that are list-decodable from a $τ_\mathsf{I}$-fraction of insertions and $τ_\mathsf{D}$-fraction of deletions, where $q$ depends only on $τ_\mathsf{I}$ and $τ_\mathsf{D}$. We als…
Insdel errors occur in communication systems caused by the loss of positional information of the message. Since the work by Guruswami and Wang, there have been some further investigations on the list decoding of insertion codes, deletion codes and insdel codes. However, unlike classical Hamming metric or even rank-metric, there are still many unsolved problems on list decoding of insdel codes. The contributions of this paper mainly consist of two parts. Firstly, we analyze the list decodability of random insdel codes. We show that list decoding of random insdel codes surpasses the Singleton bo…
Binary Reconstruction Codes for Correcting One Deletion and One Substitution
本文研究可纠正一个删除和一个替换的二进制重构码，通过定义单删除单替换球函数并约束码字球交集大小，给出了在多种 N 取值下重构码冗余度的具体上界。
Coding for Composite DNA to Correct Substitutions, Strand Losses, and Deletions
本文针对复合DNA存储中的替换、链丢失和删除错误，提出编码技术，推导非渐近上界并给出可达界的显式构造。
We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in Σ'^n$ with $|Σ'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in Σ^n$, symbol-by-symbol, with $I$ results in a string $S' \in Σ''^n$ where $Σ'' = Σ\times Σ'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the e…
Two-Insertion/Deletion/Substitution Correcting Codes
本文面向DNA存储中的插入、删除与替换混合错误，利用高阶VT校验子和错误模式分类，将低冗余1-del 1-sub码推广为2-ins/del/sub纠错码。
This paper presents a coding scheme for an insertion deletion substitution channel. We extend a previous scheme for the deletion channel where polar codes are modified by adding "guard bands" between segments. In the new scheme, each guard band is comprised of a middle segment of '1' symbols, and left and right segments of '0' symbols. Our coding scheme allows for a regular hidden-Markov input distribution, and achieves the information rate between the input and corresponding output of such a distribution. Thus, we prove that our scheme can be used to efficiently achieve the capacity of the ch…
Robust Indexing for the Sliced Channel: Almost Optimal Codes for Substitutions and Deletions
针对无序字符串构成的分片信道，本文提出鲁棒索引技术，在建立顺序的同时嵌入信息，构造可纠正多种替换、删除和插入错误的阶最优码。
Maximum run-length limited codes are constraint codes used in communication and data storage systems. Insertion/deletion correcting codes correct insertion or deletion errors caused in transmitted sequences and are used for combating synchronization errors. This paper investigates the maximum run-length limited single insertion/deletion correcting (RLL-SIDC) codes. More precisely, we construct efficiently encodable and decodable RLL-SIDC codes. Moreover, we present its encoding algorithm and show the redundancy of the code.
On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel
本文研究了单删除双替换信道的序列重构问题，证明了在特定条件下错误球交集大小的上界，并表明该上界在加法常数范围内是紧的。
The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any eps > 0): (1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps) over an alphabet of size poly(1/eps); (2) Binary codes of rate 1-O~(sqr…
We give a complete answer to the following basic question: "What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?" This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a $γ\leq 0.707$ fraction of insertions is tolerable by some binary code family. For any desired $ε> 0$, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of $γ$ fraction of insertions and …
We study codes that can detect the exact number of deletions and insertions in concatenated binary strings. We construct optimal codes for the case of detecting up to $\del$ deletions. We prove the optimality of these codes by deriving a converse result which shows that the redundancy of our codes is asymptotically optimal in $\del$ among all families of deletion detecting codes, and particularly optimal among all block-by-block decodable codes. For the case of insertions, we construct codes that can detect up to $2$ insertions in each concatenated binary string.
Belief-Combining Framework for Multi-Trace Reconstruction over Channels with Insertions, Deletions, and Substitutions
本文提出一种信念结合框架，通过消息传递实现多轨迹重建，以降低计算复杂度并达到与联合最大后验估计相同的性能。
Locally Decodable Codes (LDCs) are error-correcting codes $C:Σ^n\rightarrow Σ^m$ with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson et al. showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-…
Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions
研究证明了针对插入和删除错误的局部可解和可纠正码的指数下界，揭示了汉明LDCs与Insdel LDCs之间的严格分离。
The list-decodable code has been an active topic in theoretical computer science.There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only …
We propose a new coding scheme, called the delayed coding (DC) scheme, for channels with insertion, deletion, and substitution (IDS) errors. The proposed scheme employs delayed encoding and non-iterative detection and decoding strategies to manage the transmission of multiple codewords in a linear code. In the DC scheme, a channel input sequence consists of subblocks of multiple codewords from the previous to current time instances. At the receiver side, the maximum a posteriori detection applies to the received sequences that contain information of the codeword at the current time instance, w…
Correcting One Deletion and One Substitution with a Constant Number of Reads
本文研究了设计(n,N;B)型重建码的问题，针对N={14,11,9,5}，通过减少冗余提高纠错能力，特别是当N=5时冗余降低至3logn+4。
Correcting Errors Through Partitioning and Burst-Deletion Correction
提出一种划分技术，将重叠的 t-删除 s-替换球分解为相交的 ≤t-突发删除球子对，并据此构造二进制与非二进制校正码，部分结果优于现有方法。
We consider the problem of constructing codes that can correct deletions that are localized within a certain part of the codeword that is unknown a priori. Namely, the model that we study is when at most $k$ deletions occur in a window of size $k$, where the positions of the deletions within this window are not necessarily consecutive. Localized deletions are thus a generalization of burst deletions that occur in consecutive positions. We present novel explicit codes that are efficiently encodable and decodable and can correct up to $k$ localized deletions. Furthermore, these codes have $\log …
We study codes that are list-decodable under insertions and deletions. Specifically, we consider the setting where a codeword over some finite alphabet of size $q$ may suffer from $δ$ fraction of adversarial deletions and $γ$ fraction of adversarial insertions. A code is said to be $L$-list-decodable if there is an (efficient) algorithm that, given a received word, reports a list of $L$ codewords that include the original codeword. Using the concept of synchronization strings, introduced by the first two authors [STOC 2017], we show some surprising results. We show that for every $0\leqδ<1$, e…
Sequence Reconstruction for the Single-Deletion Single-Substitution Channel
针对单删除单替换信道的序列重构问题，基于汉明距离至少为2的q元码，证明了两序列错误球交集大小的上界为2qn-3q-2-δ_{q,2}，并通过构造序列验证了该界的紧性。
In this paper, we present an efficiently encodable and decodable code construction that is capable of correction a burst of deletions of length at most $k$. The redundancy of this code is $\log n + k(k+1)/2\log \log n+c_k$ for some constant $c_k$ that only depends on $k$ and thus is scaling-optimal. The code can be split into two main components. First, we impose a constraint that allows to locate the burst of deletions up to an interval of size roughly $\log n$. Then, with the knowledge of the approximate location of the burst, we use several {shifted Varshamov-Tenengolts} codes to correct th…
Flow watermarks efficiently link packet flows in a network in order to thwart various attacks such as stepping stones. We study the problem of designing good flow watermarks. Earlier flow watermarking schemes mostly considered substitution errors, neglecting the effects of packet insertions and deletions that commonly happen within a network. More recent schemes consider packet deletions but often at the expense of the watermark visibility. We present an invisible flow watermarking scheme capable of enduring a large number of packet losses and insertions. To maintain invisibility, our scheme u…
Consider two remote nodes having binary sequences $X$ and $Y$, respectively. $Y$ is an edited version of ${X}$, where the editing involves random deletions, insertions, and substitutions, possibly in bursts. The goal is for the node with $Y$ to reconstruct $X$ with minimal exchange of information over a noiseless link. The communication is measured in terms of both the total number of bits exchanged and the number of interactive rounds of communication. This paper focuses on the setting where the number of edits is $o(\tfrac{n}{\log n})$, where $n$ is the length of $X$. We first consider the c…
We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: …
In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new Singleton type upper bounds on the sizes of codes with a given minimum Hamming distance. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes or list-decodable codes attaining the generalized Singleton bound are given. As an application of our general…
Binary Codes for Correcting Two Edits
本文给出二进制码可纠正两次编辑的充要条件，并据此构造低冗余纠错码与列表大小为二的列表译码码，显著改进已有冗余度上界。
We develop several analytical lower bounds on the capacity of binary insertion and deletion channels by considering independent uniformly distributed (i.u.d.) inputs and computing lower bounds on the mutual information between the input and output sequences. For the deletion channel, we consider two different models: independent and identically distributed (i.i.d.) deletion-substitution channel and i.i.d. deletion channel with additive white Gaussian noise (AWGN). These two models are considered to incorporate effects of the channel noise along with the synchronization errors. For the insertio…
Computationally Relaxed Locally Decodable Codes, Revisited
本文重新审视计算松弛局部可解码码，提出两个新构造：基于数字签名和Hamming码的简单Hamming crLDC，及其扩展处理插入删除错误的InsDel crLDC，均具多对数局部性和线性编码长度。
On the size of error ball in DNA storage channels
本文研究DNA存储信道中序列重建问题，针对最多三次编辑的多种错误类型信道，推导了单删除双替换、单删除双插入和单插入单替换错误球的大小。
We describe a strategy for constructing codes for DNA-based information storage by serial composition of weighted finite-state transducers. The resulting state machines can integrate correction of substitution errors; synchronization by interleaving watermark and periodic marker signals; conversion from binary to ternary, quaternary or mixed-radix sequences via an efficient block code; encoding into a DNA sequence that avoids homopolymer, dinucleotide, or trinucleotide runs and other short local repeats; and detection/correction of errors (including local duplications, burst deletions, and sub…
This paper deals with the state estimation problem in discrete-event systems modeled with nondeterministic finite automata, partially observed via a sensor measuring unit whose measurements (reported observations) may be vitiated by a malicious attacker. The attacks considered in this paper include arbitrary deletions, insertions, or substitutions of observed symbols by taking into account a bounded number of attacks or, more generally, a total cost constraint (assuming that each deletion, insertion, or substitution bears a positive cost to the attacker). An efficient approach is proposed to d…
Motivated by applications in DNA-based storage, we introduce the new problem of code design in the Damerau metric. The Damerau metric is a generalization of the Levenshtein distance which, in addition to deletions, insertions and substitution errors also accounts for adjacent transposition edits. We first provide constructions for codes that may correct either a single deletion or a single adjacent transposition and then proceed to extend these results to codes that can simultaneously correct a single deletion and multiple adjacent transpositions. We conclude with constructions for joint block…
In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that the encoder transmits $n$ binary symbols one-by-one over a channel, in which some symbols can be deleted and some additional symbols can be inserted. After each transmission, the encoder is notified about the insertions or deletions that have occurred within the previous transmission and the encoding strategy can be adapted accordingly. The goal is to design an encoder that is able to transmit error-free as much information as possible under the ass…
Codes for Correcting Asymmetric Adjacent Transpositions and Deletions
本文研究非对称相邻换位与删除错误，提出多种码构造方法，在多种情况下降低了冗余位，并设计了列表可解码码及纠正有限幅度0-删除块与相邻换位的非系统性与系统性码。
Order-Optimal Systematic Permutation Codes for Correcting t Deletions
本文提出两种系统置换码构造，分别以7t-1和4t个冗余标记纠正PID和SID模型下的t次删除，冗余为O(t log n)，可在n^{O(t)}时间内编解码，并扩展到多重置换情形。
Constant Query Local Decoding Against Deletions Is Impossible
本文证明了在插入/删除信道下，常数查询的局部可解码码（LDC）不存在，解决了Block等人的猜想，并推论局部可校正码也不存在。
Salami Slicing Trellis for Synchronization Errors in DNA Coding
针对DNA存储中同时存在替换、插入和删除错误的问题，提出萨拉米切片网格与跨链极化码结合的决策反馈译码方案，逼近信道容量。
Optimal Codes Detecting Deletions in Concatenated Binary Strings Applied to Trace Reconstruction
本文研究串联二进制串中删除的边界恢复问题，推导了删除检测编码冗余的下界，提出一种基于标记的渐近最优编码构造，并将其应用于轨迹重建，实现了只需常数数量轨迹的高效重建方案。
Correcting Contextual Deletions in DNA Nanopore Readouts
针对DNA纳米孔测序中上下文相关的删除错误，研究了基于游程长度阈值的编码方案，给出了最小冗余度的紧界并构造了高效编解码码。
This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li \cite{CGHL21} showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to $1$ fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to $1/2$ even over the binary alphabet. As shown in \cite{CGHL21}, these bounds are also the best possible. However, known explicit constructions in \cite{CGHL21}, and subsequent improved constructions by Con…
Decoding Insertions/Deletions via List Recovery
本文将插入/删除解码问题归约为列表恢复问题，并应用于 Reed-Solomon 码，首次实现 k>2 时的高效插入/删除解码，同时改进了 Davey-MacKay 信道下的速率-错误权衡。
Oblivious Deletion Codes
本文在遗忘错误模型下构造删除纠错码，通过素数模哈希减小冗余，实现显式与随机化构造，分别匹配或优于对抗性删除的存在性冗余界。
A Deep Learning Based Decoder for Concatenated Coding over Deletion Channels
本文提出一种基于深度学习的译码器，利用BI-GRUs作为LLR估计器和外码译码器处理删除/替换信道上的串行级联码，其错误率与MAP检测相当且支持广泛的信道参数。
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