Efficient Linear and Affine Codes for Correcting Insertions/Deletions

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 having an information rate at most \(1/2\) (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial 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. Last, 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.