# List Decoding of Insertions and Deletions

## TL;DR

This paper develops bounds and algorithms for list decoding of insertions and deletions in sequences, enabling efficient decoding beyond traditional limits, with specific focus on Levenshtein metric and VT codes.

## Contribution

It introduces a Johnson-like upper bound for list decoding in the Levenshtein metric and provides an efficient decoder for small numbers of insertions/deletions.

## Key findings

- Polynomial-time list decoding beyond half the Levenshtein distance is feasible.
- The maximum list size can grow exponentially with deletions for VT codes.
- An efficient decoder for two insertions/deletions with VT codes is presented.

## Abstract

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with with the well-known binary Varshamov-Tenengolts (VT) codes which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions.