V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the model where the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance $d$ from some code and there are at most $r$ errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius $r$, where the distance between their centers is at least $d$. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model we define a Cayley graph over the symmetric group, study its properties and find the exact value of the largest intersection of its two metric balls for $d=2r$. Moreover, we give a lower bound on the largest intersection of two metric balls for $d=2r-1$.

翻译：V. Levenshtein于2001年首次提出序列重建问题。该问题研究的是：来自某个集合的同一序列通过多个信道传输，解码器接收不同输出的模型。假设传输序列与某个码的距离为$d$，且每个信道中最多出现$r$个错误，则序列重建问题需要确定：为精确恢复传输序列所需的最小信道数，该数值需大于两个半径为$r$的度量球之间的最大交集（其球心距离至少为$d$）。本文研究了汉明距离下排列的序列重建问题。在该模型中，我们定义了对称群上的凯莱图，分析了其性质，并给出了$d=2r$时两个度量球最大交集的精确值。此外，我们给出了$d=2r-1$时两个度量球最大交集的下界。