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

2014
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2014 IEEE 55th Annual Symposium on Foundations of Computer Science
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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

doi:10.1109/focs.2014.17
dblp:conf/focs/BenjaminiCS14
fatcat:3sj2ftzgrfbafhxnnefuc5l3ba