The sequence reconstruction of permutations with Hamming metric

In the combinatorial context, one of the key problems in sequence reconstruction is to find the largest intersection of two metric balls of radius r. In this paper we study this problem for permutations of length n distorted by Hamming errors and determine the size of the largest intersection of two metric balls with radius r whose centers are at distance \(d=2,3,4\). Moreover, it is shown that for any \(n\geqslant 3\) an arbitrary permutation is uniquely reconstructible from four distinct permutations at Hamming distance at most two from the given one, and it is proved that for any \(n\geqslant 4\) an arbitrary permutation is uniquely reconstructible from \(4n-5\) distinct permutations at Hamming distance at most three from the permutation. It is also proved that for any \(n\geqslant 5\) an arbitrary permutation is uniquely reconstructible from \(7n^2-31n+37\) distinct permutations at Hamming distance at most four from the permutation. Finally, in the case of at most r Hamming errors and sufficiently large n, it is shown that at least \({\varTheta }(n^{r-2})\) distinct erroneous patterns are required in order to reconstruct an arbitrary permutation.