Efficient Linear and Affine Codes for Correcting Insertions/Deletions
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This paper studies linear and 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 Ω(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 1/2-rate limitation does not hold for affine codes by giving an explicit affine code of rate 1-ϵ which can efficiently correct a constant fraction of insdel errors.
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