We give a complete answer to the following basic question: ``What is the maximal fraction of deletions or insertions tolerable by q -ary list-decodable codes with non-vanishing information rate?''

This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best known results was that a 0 707 fraction of insertions is tolerable by some binary code family.For any desired 0"> 0 , we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of fraction of insertions and fraction of deletions as long as + 2 1 − . On the other hand, for any with + 2 = 1 list-decoding is impossible. Our result thus precisely characterizes the feasibility region of binary list-decodable codes for insertions and deletions.

We further generalize our result to codes over any finite alphabet of size q . Surprisingly, our work reveals that the feasibility region for 2"> q 2 is *not* the natural generalization of the binary bound above. We provide tight upper and lower bounds that precisely pin down the feasibility region, which turns out to have a ( q − 1 ) -piece-wise linear boundary whose q corner-points lie on a quadratic curve.

The main technical work in our results is proving the existence of code families of sufficiently large *size* with good list-decoding properties for any combination of within the claimed feasibility region. We achieve this via an intricate analysis of codes introduced by [Bukh, Ma; SIAM J. Discrete Math; 2014]. Finally we give a simple yet powerful concatenation scheme for list-decodable insertion-deletion codes which transforms any such (non-efficient) code family (with vanishing information rate) into an efficiently decodable code family with constant rate.