Synchronization Strings and Codes for Insertions and Deletions—A Survey

Haeupler, Bernhard; Shahrasbi, Amirbehshad

doi:10.1109/tit.2021.3056317

Cited by 37 publications

(38 citation statements)

References 53 publications

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“…Also, 1 − p is a trivial upper bound on the possible rate, even for the simpler model of p fraction of erasures. Explicit constructions of p-deletion correcting of rate approaching 1 − p over an alphabet size independent of N were given in [HS18] based on synchronization strings, which is a very elegant tool that has since found several other applications (see the survey [HS21]).…”

Section: Deletion Correction In Related Modelsmentioning

confidence: 99%

“…This tantalizing question was implicit in early works on deletion codes, particularly in [Ull67], which gave bounds on the achievable tradeoffs between rate and deletion fraction, and was explicitly raised in [KMTU11]. Since then, this question has been mentioned in several works, including the work of Bukh and Ma [BM14] which showed that an upper bound of 1 2 − 1 poly log N on the correctable deletion fraction, and many recent works on deletion code constructions such as [Wan15, GL16, GW17, BGH17, GL20, GHS20], other works on coding theory [ZBJ20], as well as the recent surveys [CR21,HS21]. In the other direction, the best known lower bound p thr del > √ 2 − 1 is due to [BGH17], who constructed explicit binary codes of non-vanishing rate to correct a fraction of deletions approaching √ 2 − 1 (see [SZ99,KMTU11] for prior constructions).…”

Section: 9 Conclusion and Open Questions 1 Introductionmentioning

confidence: 99%

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The zero-rate threshold for adversarial bit-deletions is less than 1/2

Guruswami¹,

He²,

Li³

2021

Preprint

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We prove that there exists an absolute constant δ > 0 such any binary code C ⊂ {0, 1} N tolerating (1/2 − δ )N adversarial deletions must satisfy |C| 2 poly log N and thus have rate asymptotically approaching 0. This is the first constant fraction improvement over the trivial bound that codes tolerating N/2 adversarial deletions must have rate going to 0 asymptotically. Equivalently, we show that there exists absolute constants A and δ > 0 such that any set C ⊂ {0, 1} N of 2 log A N binary strings must contain two strings c and c ′ whose longest common subsequence has length at least (1/2 + δ )N. As an immediate corollary, we show that q-ary codes tolerating a fraction 1 − (1 + 2δ )/q of adversarial deletions must also have rate approaching 0. Our techniques include string regularity arguments and a structural lemma that classifies binary strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.

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Section: Deletion Correction In Related Modelsmentioning

confidence: 99%

Section: 9 Conclusion and Open Questions 1 Introductionmentioning

confidence: 99%

The zero-rate threshold for adversarial bit-deletions is less than 1/2

Guruswami¹,

He²,

Li³

2021

Preprint

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“…Proof. By Theorem 3.11, it is sufficient to verify that the inequality (12) holds assuming that the inequality (15) holds. Indeed (12) holds under our assumption since…”

Section: 3mentioning

confidence: 99%

“…Afterwards, various aspects of insdel codes such as bounds, constructions and decoding algorithms have been studied in literatures (see [23,2,20,31,24,27,1,3,12]).…”

Section: Introductionmentioning

confidence: 99%

Bounds and Constructions for Insertion and Deletion Codes

Liu¹,

Xing²

2021

Preprint

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Insertion and deletion (insdel for short) codes have recently attracted a lot of attention due to their applications in many interesting fields such as DNA storage, DNA analysis, race-track memory error correction and language processing. The present paper mainly studies limits and constructions of insdel codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on constructions of insdel codes.Although the insdel-metric Singleton bound has been derived before, it is still unknown if there are any nontrivial codes achieving this bound. Our first result shows that any nontrivial insdel codes do not achieve the insdel-metric Singleton bound. The second bound shows that every [n, k] Reed-Solomon code has insdel distance upper bounded by 2n − 4k + 4 and it is known in literature that an [n, k] Reed-Solomon code can have insdel distance 2n − 4k + 4 as long as the field size is sufficiently large. The third bound shows a trade-off between insdel distance and code alphabet size for codes achieving the Hamming-metric Singleton bound. In the second part of the paper, we first provide a non-explicit construction of nonlinear codes that can approach the insdel-metric Singleton bound arbitrarily when the code alphabet size is sufficiently large. The second construction gives two-dimensional Reed-Solomon codes of length n and insdel distance 2n − 4 with field size q = O(n 5 ).The non-explicit construction of insdel codes is based on constant-weight L 1 -codes that are introduced in this paper. We first establish a relation between constantweight L 1 -codes and insdel codes. Based on this relation, we construct constantweight L 1 -codes with reasonable parameters and subsequently give insdel codes approaching the insdel-metric Singleton bound. Via automorphism group of rational function field, we provide a necessary and sufficient condition under which a twodimensional Reed-Solomon code of length n has insdel distance 2n − 4. Based on this criterion, we present a construction of q-ary two-dimensional Reed-Solomon codes of length n and insdel distance 2n − 4 with q = O(n 5 ). Though this is worse than the current best field size, we provide a new angle to look into the problem.

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“…deletions is fundamental for the design of reliable DNA-based data storage systems with nanoporebased sequencing [11,12,13]. Second, understanding the structure of the mean trace of a string is by itself a natural information-theoretic problem which may lead to improved capacity bounds and coding techniques for channels with synchronization errors, both notoriously difficult problems (see the extensive surveys [14,15,7,16]).…”

Section: Introductionmentioning

confidence: 99%

Mean-Based Trace Reconstruction over Oblivious Synchronization Channels

Cheraghchi¹,

Downs²,

Ribeiro³

et al. 2021

Preprint

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Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that exp(O(n 1/3 )) traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon: exp(O(n 1/3 )) traces suffice for mean-based worst-case trace reconstruction over any memoryless channel that maps each input bit to an arbitrarily distributed sequence of replications and insertions of random bits, provided the length of this sequence follows a subexponential distribution.

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Synchronization Strings and Codes for Insertions and Deletions—A Survey

Cited by 37 publications

References 53 publications

The zero-rate threshold for adversarial bit-deletions is less than 1/2

The zero-rate threshold for adversarial bit-deletions is less than 1/2

Bounds and Constructions for Insertion and Deletion Codes

Mean-Based Trace Reconstruction over Oblivious Synchronization Channels

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