Optimally Resilient Codes for List-Decoding from Insertions and Deletions

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 result was that a $γ\leq 0.707$ fraction of insertions is tolerable by some binary code family. For any desired $ε> 0$, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of $γ$ fraction of insertions and $δ$ fraction of deletions as long as $γ+2δ\leq 1-ε$. On the other hand, for any $γ,δ$ with $γ+2δ=1$ 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 $q>2$ 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 $δ,γ$ 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.

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