Sequence Reconstruction for Grassmann Graphs and Permutations

The sequence-reconstruction problem was first proposed by Levenshtein in 2001. This problem studies the model where the same word is transmitted over multiple channels. If the transmitted word belongs to some code of minimum distance $d$ and there are at most $r$ errors in every channel, then the minimum number of channels that guarantees a successful decoder (under the assumption that all channel outputs are distinct) has to be greater than the largest intersection of two balls of radius $r$ and with distance at least $d$ between their centers. This paper studies the combinatorial problem of computing the largest intersection of two balls for two cases. In the first part we solve this problem in the Grassmann graph for all values of $d$ and $r$. In the second part we derive similar results for permutations under Kendall's τ-metric for some special cases of ${d}$ and $r$.