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arXiv · 2601.02844

The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors

A. Abdollahi·J. Bagherian·H. Eskandari·F. Jafari·M. Khatami·F. Parvaresh·R. Sobhani
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Abstract

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 \(N(24,14)\).

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BibTeXRIS

A. Abdollahi, J. Bagherian, H. Eskandari, F. Jafari, M. Khatami, F. Parvaresh, R. Sobhani. 2026-01-06. The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors. https://arxiv.org/abs/2601.02844

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