## 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* ∈ N there exists an explicit bijection *ψ*: {0, 1}^{n} → {*x* ∈ {0, 1}^{n+1} : |*x*| > *n*/2} such that for every *x* ≠ *y* ∈ {0, 1}^{n}, \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) 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 (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 requires ideas beyond the sensitivity-based structural results of Boppana (1997).

We study the mapping *ψ* further and show that it (and its inverse) are computable in DLOGTIME-uniform TC^{0}, but not in AC^{0}. Moreover, we prove that *ψ* is “approximately local” in the sense that all but the last output bit of *ψ* are essentially determined by a single input bit.

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Research supported by Israel Science Foundation (ISF) grant.

Research supported by ERC grant number 239985.

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Benjamini, I., Cohen, G. & Shinkar, I. Bi-Lipschitz bijection between the Boolean cube and the Hamming ball.
*Isr. J. Math.* **212, **677–703 (2016). https://doi.org/10.1007/s11856-016-1302-0

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