Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

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, (Formula presented.), 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 TC⁰, but not in AC⁰. 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.