The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on \(n\) letters under the Hamming metric, this problem is closely related to the parameter $N(n,r)$, the maximum intersection size of two Hamming balls of radius $r$. While previous work has resolved \(N(n,r)\) for small radii (\(r \leq 4\)) and established asymptotic bounds for larger \(r\), we present new exact formulas for \(r \in \{5,6,7\}\) using group action techniques. In addition, we develop a formula for \(N(n,r)\) based on the irreducible characters of the symmetric group \(S_n\), along with an algorithm that enables computation of \(N(n,r)\) for larger parameters, including cases such as \(N(43,8)\) and \(N(24,14)\).

翻译：序列重构问题旨在从多个含噪声的副本中恢复原始序列，其中每个副本可能包含最多 $r$ 个错误。在汉明度量下对 $n$ 个字母的排列进行研究时，该问题与参数 $N(n,r)$——即两个半径为 $r$ 的汉明球的最大交集大小——密切相关。尽管先前的研究已解决了小半径（$r \leq 4$）情况下的 $N(n,r)$，并对更大的 $r$ 建立了渐近界，我们利用群作用技术给出了 $r \in \{5,6,7\}$ 时新的精确公式。此外，我们基于对称群 $S_n$ 的不可约特征，推导了 $N(n,r)$ 的一个计算公式，并提出了一个算法，使得能够计算更大参数下的 $N(n,r)$，包括诸如 $N(43,8)$ 和 $N(24,14)$ 等情形。