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 it holds that 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

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... aised 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 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.