Decoding Insertions/Deletions via List Recovery

In this work, we consider the problem of efficient decoding of codes from insertions and deletions. Most of the known efficient codes are codes with synchronization strings which allow one to reduce the problem of decoding insertions and deletions to that of decoding substitution and erasures. Our new approach, presented in this paper, reduces the problem of decoding insertions and deletions to that of list recovery. Specifically, any $(\rho, 2\rho n + 1, L)$-list-recoverable code is a $(\rho, L)$-list decodable insdel code. As an example, we apply this technique to Reed-Solomon (RS) codes, which are known to have efficient list-recovery algorithms up to the Johnson bound. In the adversarial insdel model, this provides efficient (list) decoding from $t$ insdel errors, assuming that $t\cdot k = O(n)$. This is the first efficient insdel decoder for $[n, k]$ RS codes for $k>2$. Additionally, we explore random insdel models, such as the Davey-MacKay channel, and show that for certain choices of $\rho$, a $(\rho, n^{1/2+0.001}, L)$-list-recoverable code of length $n$ can, with high probability, efficiently list decode the channel output, ensuring that the transmitted codeword is in the output list. In the context of RS codes, this leads to a better rate-error tradeoff for these channels compared to the adversarial case. We also adapt the Koetter-Vardy algorithm, a famous soft-decision list decoding technique for RS codes, to correct insertions and deletions induced by the Davey-MacKay channel.