Abstract: In this talk we give some results on the reconstruction problem of permutations under Hamming metric. In the combinatorial context, the reconstruction problem is equivalent to finding the largest intersection of two metric balls of radius 𝑟. In particular, we discuss this problem for permutations of length 𝑛 distorted by single Hamming errors and determine the size of the largest intersection of two metric balls with radius 𝑟 whose centers are at distance 𝑑 = 2,3,4. Some other results and open problem are also discussed.