Weak isometries of the Boolean cube

TL;DR

This paper classifies weak isometries of the Boolean cube that preserve certain Hamming distances but are not necessarily isometries, expanding understanding of distance-preserving transformations.

Contribution

It provides a complete classification of weak isometries of the Boolean cube that are not isometries, generalizing previous results on distance preservation.

Findings

Identifies conditions under which weak isometries differ from isometries.

Classifies all weak isometries of the Boolean cube.

Extends prior work on distance-preserving maps.

Abstract

Consider the metric space $\mathcal{C}$ consisting of the $n$-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of $\mathcal{C}$ is a permutation of $\mathcal{C}$ preserving a given subset of Hamming distances. In \cite{Krasin} Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of $\mathcal{C}$.