Optimally Resilient Codes for List-Decoding from Insertions and Deletions

TL;DR

This paper precisely characterizes the maximum fraction of insertions and deletions that binary and q-ary list-decodable codes can tolerate while maintaining positive rate, providing tight bounds and a new concatenation scheme.

Contribution

It establishes the exact feasibility region for list-decoding from insertions and deletions over binary and larger alphabets, including new bounds and an efficient decoding scheme.

Findings

Binary codes tolerate up to 1/2 insertions and deletions combined.

For q-ary codes, the feasibility region has a piecewise linear boundary with q-1 segments.

A concatenation scheme converts non-efficient codes into efficient, constant-rate list-decodable codes.

Abstract

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