The Levenshtein's Sequence Reconstruction Problem and the Length of the List
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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 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 numbers of required channels are determined for ℒ =1, 2 and ℓ+1. Here we determine the exact 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 (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.
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