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Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

Authors Ryan Gabrys , Venkatesan Guruswami , João Ribeiro , Ke Wu



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Author Details

Ryan Gabrys
  • ECE Department, University of California, San Diego, CA, USA
Venkatesan Guruswami
  • EECS Department, University of California, Berkeley, CA, USA
João Ribeiro
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Ke Wu
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

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Ryan Gabrys, Venkatesan Guruswami, João Ribeiro, and Ke Wu. Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 8:1-8:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.8

Abstract

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, who proved that binary Varshamov-Tenengolts codes correct one arbitrary edit error, i.e., one deletion 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: - We construct linear-time encodable and decodable length-n non-binary codes correcting a single edit error with nearly optimal redundancy log n+O(log log n), providing an alternative simpler proof of a result by Cai, Chee, Gabrys, Kiah, and Nguyen (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a new notion that may be of independent interest. - We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy log n+O(log log n). - We construct linear-time encodable and list-decodable binary codes with list-size 2 for one deletion and one substitution with redundancy 4log n+O(log log n). This matches the existential bound up to an O(log log n) additive term.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Synchronization errors
  • Optimal redundancy
  • Explicit codes

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References

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