Efficient Linear and Affine Codes for Correcting Insertions/Deletions
| Autor: | Venkatesan Guruswami, Bernhard Haeupler, Xin Li, Kuan Cheng |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | SODA: ACM-SIAM Symposium on Discrete Algorithms SODA Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) |
| DOI: | 10.1137/1.9781611976465.1 |
| Popis: | 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 the existence of binary linear codes of length $n$ and rate just below $1/2$ capable of correcting $\Omega(n)$ insertions and deletions. This identifies rate $1/2$ as a sharp threshold for recovery from deletions for linear codes, and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically-good linear code for Hamming errors into an asymptotically-good linear code for insdel errors. Lastly, we show that the $\frac{1}{2}$-rate limitation does not hold for affine codes by giving an explicit affine code of rate $1-\epsilon$ which can efficiently correct a constant fraction of insdel errors. |
| Databáze: | OpenAIRE |
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