Sequence Reconstruction over the Deletion Channel

The sequence reconstruction problem, first proposed by Levenshtein, models the setup in which a sequence from some set is transmitted over several channels, and the decoder receives the outputs from every channel. The channels are almost independent as it is only required that all outputs are different from each other. The main problem of interest is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the problem is equivalent to finding the maximum intersection between two balls of radius t, where the distance between their centers is at least d. The setup of this problem was studied before for several error metrics such as the Hamming metric, the Kendall-tau metric, and the Johnson metric. In this paper, we extend the study initiated by Levenshtein for reconstructing sequences over the deletion channel. While he solved the case where the transmitted sequence can be arbitrary, we study the setup, where the transmitted sequence belongs to a single-deletion-correcting code and there are t deletions in every channel. Under this paradigm, we study the minimum number of different channel outputs in order to construct a successful decoder.