Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

TL;DR

This paper constructs an explicit bi-Lipschitz bijection between the Boolean cube and the Hamming ball, revealing new structural insights and implications for complexity theory and lower bounds in sampling distributions.

Contribution

It provides an explicit bi-Lipschitz bijection between the Boolean cube and Hamming ball, answering an open problem and analyzing its computational properties.

Findings

The bijection is computable in DLOGTIME-uniform TC0 but not in AC0.

The mapping is approximately local, with most output bits determined by a single input bit.

The Hamming ball is shown to be bi-Lipschitz transitive.

Abstract

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n there exists an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y) where distance(,) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions will require some new ideas beyond the sensitivity-based structural results of Boppana [IPL 97]. We study the mapping f further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f is "approximately local" in the sense that all but the last output bit of f are essentially determined by a single input bit.