The Levenshtein’s Sequence Reconstruction Problem and the Length of the List

In the paper, Levenshtein’s sequence reconstruction problem is considered in case where transmitted words are chosen from an e -error-correcting code, at most xmlns:xlink="http://www.w3.org/1999/xlink">t substitution errors occur each of xmlns:xlink="http://www.w3.org/1999/xlink">N channels and decoder outputs a list length xmlns:xlink="http://www.w3.org/1999/xlink">L . Previously, when = + ℓ word long enough, numbers required were determined for 1, 2 ℓ+1. Here we determine exact number cases 3,4,..., ℓ. This also provides size largest intersection balls radius (with respect to substitutions) centered with mutual Hamming distances least 1. Furthermore, aid covering codes, consider sizes xmlns:xlink="http://www.w3.org/1999/xlink">n rather small (improving previously known results). After that study how much can decrease use list-decoding codes. Finally, majority algorithm discussed decoding probabilistic set-up; particular, show output be verified one high probability.