We give an explicit construction of length-$n$ binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$. This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size $Ω(2^n/n^4)$. Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size $Ω(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.
We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{Ω(1)}$. For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $…
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.
Levenshtein introduced the problem of constructing $k$-deletion correcting codes in 1966, proved that the optimal redundancy of those codes is $O(k\log N)$, and proposed an optimal redundancy single-deletion correcting code (using the so-called VT construction). However, the problem of constructing optimal redundancy $k$-deletion correcting codes remained open. Our key contribution is a solution to this longstanding open problem. We present a $k$-deletion correcting code that has redundancy $8k\log n +o(\log n)$ and encoding/decoding algorithms of complexity $O(n^{2k+1})$ for constant $k$.
This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to have information rate at most $1/2$ (achieved by the trivial 2-fold repetition code). Previously, it was (erroneously) reported that more generally no non-trivial linear codes correcting $k$ deletions exist, i.e., that the $(k+1)$-fold repetition codes and its rate of $1/(k+1)$ are basically optimal for any $k$. We disprove this and show…
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).
We first give a construction of binary $t_1$-deletion-$t_2$-insertion-burst correcting codes with redundancy at most $\log(n)+(t_1-t_2-1)\log\log(n)+O(1)$, where $t_1\ge 2t_2$. Then we give an improved construction of binary codes capable of correcting a burst of $4$ non-consecutive deletions, whose redundancy is reduced from $7\log(n)+2\log\log(n)+O(1)$ to $4\log(n)+6\log\log(n)+O(1)$. Lastly, by connecting non-binary $b$-burst-deletion correcting codes with binary $2b$-deletion-$b$-insertion-burst correcting codes, we give a new construction of non-binary $b$-burst-deletion correcting codes …
One of the main challenges in developing racetrack memory systems is the limited precision in controlling the track shifts, that in turn affects the reliability of reading and writing the data. A current proposal for combating deletions in racetrack memories is to use redundant heads per-track resulting in multiple copies (potentially erroneous) and recovering the data by solving a specialized version of a sequence reconstruction problem. Using this approach, $k$-deletion correcting codes of length $n$, with $d \ge 2$ heads per-track, with redundancy $\log \log n + 4$ were constructed. However…
In this paper, we construct systematic $q$-ary two-deletion correcting codes and burst-deletion correcting codes, where $q\geq 2$ is an even integer. For two-deletion codes, our construction has redundancy $5\log n+O(\log q\log\log n)$ and has encoding complexity near-linear in $n$, where $n$ is the length of the message sequences. For burst-deletion codes, we first present a construction of binary codes with redundancy $\log n+9\log\log n+γ_t+o(\log\log n)$ bits $(γ_t$ is a constant that depends only on $t)$ and capable of correcting a burst of at most $t$ deletions, which improves the Lenz-P…
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 investigate the problem of constructing codes capable of correcting two deletions. In particular, we construct a code that requires redundancy approximately 8 log n + O(log log n) bits of redundancy, where n is the length of the code. To the best of the author's knowledge, this represents the best known construction in that it requires the lowest number of redundant bits for a code correcting two deletions.
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…
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: …
This paper studies codes that correct bursts of deletions. Namely, a code will be called a $b$-burst-deletion-correcting code if it can correct a deletion of any $b$ consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically $\log(n)+b-1$, the redundancy of the best code construction by Cheng et al. is $b(\log (n/b+1))$. In this paper we close on this gap and provide codes with redundancy at most $\log(n) + (b-1)\log(\log(n)) +b -\log(b)$. We also derive a non-asymptotic upper bound on the size of $b$-burst-deletion-correcting codes an…
Deletion-Correcting Codes for the $\ell$-Symbol Read Channel
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n\times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $(t_r,t_c)$-criss-cross deletions if $t_r$ rows and $t_c$ columns are deleted, while a code correcting these deletion patterns is called a $(t_r,t_c)$-criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when $t_r=t_c$ the problems of correcting criss-cross deletions and criss-cross insertions are equival…
We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions $(δ=1)$ with an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions $δ$ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asympto…
In this work, we derive upper bounds on the cardinality of tandem duplication and palindromic deletion correcting codes by deriving the generalized sphere packing bound for these error types. We first prove that an upper bound for tandem deletions is also an upper bound for inserting the respective type of duplications. Therefore, we derive the bounds based on these special deletions as this results in tighter bounds. We determine the spheres for tandem and palindromic duplications/deletions and the number of words with a specific sphere size. Our upper bounds on the cardinality directly imply…
Permutation codes and multi-permutation codes have been widely considered due to their various applications, especially in flash memory. In this paper, we consider permutation codes and multi-permutation codes against a burst of stable deletions. In particular, we propose a construction of permutation codes correcting a burst stable deletion of length $s$, with redundancy $\log n+ 2\log \log n+O(1)$. Compared to the previous known results, our improvement relies on a different strategy to retrieve the missing symbol on the first row of the array representation of a permutation. We also general…
This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t \log n - \log(t!)$. Then, we present an existential construction of $t$-criss…
Motivated by applications in DNA-based storage and communication systems, we study deletion and insertion errors simultaneously in a burst. In particular, we study a type of error named $t$-deletion-$s$-insertion-burst ($(t,s)$-burst for short) which is a generalization of the $(2,1)$-burst error proposed by Schoeny {\it et. al}. Such an error deletes $t$ consecutive symbols and inserts an arbitrary sequence of length $s$ at the same coordinate. We provide a sphere-packing upper bound on the size of binary codes that can correct a $(t,s)$-burst error, showing that the redundancy of such codes …
In this paper, we show that the single deletion correcting Varshamov-Tenengolts code, with minor modifications, can also correct an ordered deletion-erasure pattern where one deletion and at most one erasure occur and the deletion always occurs before the erasure. For large code lengths, the constructed code has the same logarithmic redundancy as optimal codes.
We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that th…
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…
Codes for correcting sticky insertions/deletions and limited-magnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$-sticky deletions with $\ell$-limited-magnitude and propose a non-systematic code for correcting this type of error with redundancy $2t(1-1/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\fr…
Order-Optimal Systematic Permutation Codes for Correcting t Deletions
Beijing University of Posts and Telecommunications（北京邮电大学）；VinUniversity（文林大学）；Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications（北京邮电大学通用无线通信教育部重点实验室）；National University of Singapore（新加坡国立大学）
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.
Substring Edit Correcting Codes and Optimal Single Burst-Deletion Correcting Codes
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与 interdisciplinary 科学研究院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
Correcting Contextual Deletions in DNA Nanopore Readouts
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；Instituto de Telecomunicações（电信研究所）
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 …
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of either insertions or deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single in…
We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes, dating back to 1965, are zero-error single deletion $(δ=1)$ correcting codes, and have an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions remains an open problem. In this work, we relax the standard zero-error (i.e., worst-case) decoding requirement by assuming that the positions of the $δ$ deletions (or insertions) are independent of the codeword. Our contribution is a new family of explicit c…
Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of \emph{coded trace reconstruction}, the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called \emph{traces}) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i.d.\ deletions and constant deletion probability.…
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 substri…
We consider the problem of constructing binary codes for correcting deletions that are localized within certain parts of the codeword that are unknown a priori. The model that we study is when $δ\leq w$ deletions are localized in a window of size $w$ bits. These $δ$ deletions do not necessarily occur in consecutive positions, but are restricted to the window of size $w$. The localized deletions model is a generalization of the bursty model, in which all the deleted bits are consecutive. In this paper, we construct new explicit codes for the localized model, based on the family of Guess & Check…
In this work, we propose constructions that correct duplications of multiple consecutive symbols. These errors are known as tandem duplications, where a sequence of symbols is repeated; respectively as palindromic duplications, where a sequence is repeated in reversed order. We compare the redundancies of these constructions with code size upper bounds that are obtained from sphere packing arguments. Proving that an upper bound on the code cardinality for tandem deletions is also an upper bound for inserting tandem duplications, we derive the bounds based on this special tandem deletion error …
Construction of capacity achieving deletion correcting codes has been a baffling challenge for decades. A recent breakthrough by Brakensiek $et~al$., alongside novel applications in DNA storage, have reignited the interest in this longstanding open problem. In spite of recent advances, the amount of redundancy in existing codes is still orders of magnitude away from being optimal. In this paper, a novel approach for constructing binary two-deletion correcting codes is proposed. By this approach, parity symbols are computed from indicator vectors (i.e., vectors that indicate the positions of ce…
In this paper, we present a novel communication channel, called the absorption channel, inspired by information transmission in neurons. Our motivation comes from in-vivo nano-machines, emerging medical applications, and brain-machine interfaces that communicate over the nervous system. Another motivation comes from viewing our model as a specific deletion channel, which may provide a new perspective and ideas to study the general deletion channel. For any given finite alphabet, we give codes that can correct absorption errors. For the binary alphabet, the problem is relatively trivial and we …
We consider the problem of correcting insertion and deletion errors in the $d$-dimensional space. This problem is well understood for vectors (one-dimensional space) and was recently studied for arrays (two-dimensional space). For vectors and arrays, the problem is motivated by several practical applications such as DNA-based storage and racetrack memories. From a theoretical perspective, it is interesting to know whether the same properties of insertion/deletion correcting codes generalize to the $d$-dimensional space. In this work, we show that the equivalence between insertion and deletion …
Guess & Check (GC) codes are systematic binary codes that can correct multiple deletions, with high probability. GC codes have logarithmic redundancy in the length of the message $k$, and the encoding and decoding algorithms of these codes are deterministic and run in polynomial time for a constant number of deletions $δ$. The unique decoding properties of GC codes were examined in a previous work by the authors. In this paper, we investigate the list decoding performance of these codes. Namely, we study the average size and the maximum size of the list obtained by a GC decoder for a constant …
