The sequence reconstruction problem for permutations with the Hamming distance

TL;DR

This paper investigates the sequence reconstruction problem for permutations under Hamming distance, analyzing the properties of a Cayley graph and determining the maximum intersection of metric balls for specific parameters.

Contribution

It introduces a new model using a Cayley graph over the symmetric group and provides exact and bound results for the intersection of metric balls in this context.

Findings

Exact value of the largest intersection for d=2r

Lower bound for the largest intersection when d=2r-1

Properties of the Cayley graph related to permutation reconstruction

Abstract

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 $d=2r-1$.