Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

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...

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Veröffentlicht in:IEEE transactions on information theory Jg. 69; H. 1; S. 169 - 186
Hauptverfasser: Gabrys, Ryan, Guruswami, Venkatesan, Ribeiro, Joao, Wu, Ke
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.01.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Binary codes
Codes
Computer Science
Data storage
Decoding
Deletion
efficient encoding/decoding
Encoding
Engineering
Error correction
optimal codes
Redundancy
Sketches
Substitutes
Synchronization errors
Termination of employment
ISSN:0018-9448, 1557-9654
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Zusammenfassung: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: 1) We construct linear-time encodable and decodable length-<inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> non-binary codes correcting a single edit error with nearly optimal redundancy <inline-formula> <tex-math notation="LaTeX">\log n+O(\log \log n) </tex-math></inline-formula>, providing an alternative simpler proof of a result by Cai et al. (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. 2) We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy <inline-formula> <tex-math notation="LaTeX">\log n+O(\log \log n) </tex-math></inline-formula>. 3) We construct linear-time encodable and list-decodable binary codes with list-size 2 for one deletion and one substitution with redundancy <inline-formula> <tex-math notation="LaTeX">4\log n+O(\log \log n) </tex-math></inline-formula>. This matches the Gilbert-Varshamov existential bound up to an <inline-formula> <tex-math notation="LaTeX">O(\log \log n) </tex-math></inline-formula> additive term.
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USDOE
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3202856