The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors

TL;DR

This work tackles permutation sequence reconstruction under the Hamming metric by analyzing $N(n,r)$, the maximum overlap of radius-$r$ balls, and its connection to $I(n,d,r)$. It introduces a group-action framework to reduce computations to the small symmetric group $S_{2r}$ and derives exact polynomial formulas for $N(n,r)$ at $r\in\{5,6,7\}$, while also developing a character-theoretic approach (using irreducible characters of $S_n$) to compute $N(n,r)$ for larger parameters. The authors provide Algorithm 1 (group-action based) and Algorithm 2 (character-based), with GAP implementations and tabled results that corroborate the conjecture $N(n,r)=I(n,2,r)$ for the tested cases, including large instances like $N(43,8)$ and $N(24,14)$. Overall, the paper advances exact and scalable computation for permutation-based sequence reconstruction, with practical tools for researchers tackling related coding and reconstruction problems.

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 \(N(24,14)\).

Theorems & Definitions (21)

• Theorem 1.1
• Conjecture 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Example 3.2
• Lemma 3.3
• proof