### 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}, (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 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.

Original language | English (US) |
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Pages (from-to) | 677-703 |

Number of pages | 27 |

Journal | Israel Journal of Mathematics |

Volume | 212 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2016 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*212*(2), 677-703. https://doi.org/10.1007/s11856-016-1302-0