On the Intersection of Multiple Insertion (or Deletion) Balls and its Application to List Decoding Under the Reconstruction Model

In the reconstruction model, first proposed by Levenshtein in 2001, a word is transmitted over multiple identical noisy channels that output distinct erroneous words. Given the channels' outputs, unique decoding of the transmitted word is guaranteed to succeed only if the number of the channels is greater than a specific value. Otherwise, there may be several transmitted words that lead to the same channels' outputs. In this case, these words are recovered using a list decoder. Calculating the largest list size is a fundamental task when studying the list decoding problem. The present work takes the first steps towards studying list decoding of insertions and deletions under the reconstruction model. More specifically, it assumes that an arbitrary binary word is transmitted over m t-insertion (or t-deletion) identical channels, and provides the largest list size for specific values of m. These results are mainly achieved by investigating the largest intersection of m∼t -insertion (or t -deletion) balls surrounding arbitrary binary words in the space.