Efficient Linear and Affine Codes for Correcting Insertions/Deletions

Abstract

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 (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate 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 -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.