Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-bit deletions with efficient encoding/decoding, for a fixed <inline-formula> <tex-math notation="LaTeX"...

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Veröffentlicht in:IEEE transactions on information theory Jg. 64; H. 5; S. 3403 - 3410
Hauptverfasser: Brakensiek, Joshua, Guruswami, Venkatesan, Zbarsky, Samuel
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.05.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
algebraic codes
Binary codes
Binary system
Codes
Coding theory
Decoding
Deletion
edit distance
Electronic mail
hashing
Linear codes
Protocols
pseudorandomness
Redundancy
synchronization errors
ISSN:0018-9448, 1557-9654
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Zusammenfassung:We consider the problem of constructing binary codes to recover from <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-bit deletions with efficient encoding/decoding, for a fixed <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with <inline-formula> <tex-math notation="LaTeX">\approx ~2^{n}/n </tex-math></inline-formula> codewords of length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, i.e., at most <inline-formula> <tex-math notation="LaTeX">\log n </tex-math></inline-formula> bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than <inline-formula> <tex-math notation="LaTeX">n^{\Omega (1)} </tex-math></inline-formula>. For any fixed <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>, we construct a binary code with <inline-formula> <tex-math notation="LaTeX">c_{k} \log n </tex-math></inline-formula> redundancy that can be decoded from <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> deletions in <inline-formula> <tex-math notation="LaTeX">O_{k}(n \log ^{4} n) </tex-math></inline-formula> time. The coefficient <inline-formula> <tex-math notation="LaTeX">c_{k} </tex-math></inline-formula> can be taken to be <inline-formula> <tex-math notation="LaTeX">O(k^{2} \log k) </tex-math></inline-formula>, which is only quadratically worse than the optimal, non-constructive bound of <inline-formula> <tex-math notation="LaTeX">O(k) </tex-math></inline-formula>. We also indicate how to modify this code to allow for a combination of up to <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> insertions and deletions. We also note that among linear codes capable of correcting <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> deletions, the <inline-formula> <tex-math notation="LaTeX">(k+1) </tex-math></inline-formula>-fold repetition code is essentially the best possible.
Bibliographie:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2017.2746566