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

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