# Indistinguishable sceneries on the Boolean hypercube

## TL;DR

This paper proves that reconstructing scenery on the Boolean hypercube is generally impossible using locally biased or stable functions, providing super-polynomial lower bounds and extending results to other graphs.

## Contribution

It introduces constructive methods to show the impossibility of scenery reconstruction on the hypercube and other graphs, with super-polynomial lower bounds.

## Key findings

- Reconstruction on the hypercube is impossible in general.
- Super-polynomial lower bounds for locally biased and stable functions.
- Results extend to ^n and other graph classes.

## Abstract

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbors colored by $1$, and locally stable functions, in which every vertex has a constant fraction of neighbors colored by its own color. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for $\mathbb{Z}^n$ and other graphs, and offer several follow-up questions.