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

Abstract: 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 $\mathcal{L}$. Moreover, it is assumed that the transmitted words are chosen from an $e$-error-correcting code $C \ (\subseteq {0,1}^n)$. Previously, when $t = e+\ell$ and the length $n$ of the transmitted word is large enough, the numbers of required channels are determined for $\mathcal{L} =1, 2 \text{ and } \ell+1$. Here we determine the exact number of channels in the cases $\mathcal{L} = 3, 4, \ldots, \ell$. 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 with high probability a decoder based on it is verifiably successful, i.e., the output word of the decoder can be verified to be the transmitted one.