On List Decoding of Insertions and Deletions under the Reconstruction Model

The reconstruction model, first proposed by Levenshtein in 2001, assumes that a word is transmitted over multiple identical noisy channels that output distinct words. Given the channels' outputs, the transmitted word is guaranteed to be decoded uniquely only if the number of the channels is greater than some value. Otherwise, there could be several transmitted words leading to the same channels' outputs. Hence, these words should be found following the list decoding approach. Motivated by DNA storage systems, the present work takes the first steps towards studying list decoding for insertions and deletions under the reconstruction model. More specifically, it will be assumed that an arbitrary binary word is transmitted over m t-insertions (or t deletions) identical channels. For specific values of m, bounds on the largest list decoder size are provided. These bounds are mainly derived by investigating the largest intersection of m t- insertion (or t-deletion) balls surrounding arbitrary binary words in the space. Furthermore, all pairs of binary words achieving the largest intersection of their t-insertion balls are characterized.