On the List Size in the Levenshtein’s Sequence Reconstruction Problem

In the paper, the Levenshtein’s sequence reconstruction problem is considered in the case where at most t substitution errors occur in each of the N channels and the decoder outputs a list of length at most ℒ. Moreover, it is assumed that the transmitted words are chosen from an e-error-correcting code C (⊆ {0, 1}
ⁿ
). Previously, when t = e + ℓ and the length n of the transmitted word is large enough, the exact numbers of required channels is determined for ℒ = 1, 2 and ℓ + 1. Here we determine the number of channels in the cases ℒ = 3, 4,…, ℓ. Furthermore, with the aid of covering codes, we also consider the list sizes in the cases where the length n is rather small. Finally, the majority algorithm is discussed for decoding; in particular, we demonstrate that with high probability a decoder based on it, is verifiably successful, i.e., outputs a list (sometimes even of size one) such that it verifiably contains the transmitted word.