We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming space and symmetric groups decays exponentially as their centers drift apart. To verify condition (iii), we prove some deviation inequalities `on the slice’ for functions with Lipschitz conditions.
We then use these estimates on intersection volumes to
obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjecture of Li and Wootters [IEEE Trans. Inf. Theory 2021]; and
improve sphere-covering bound from the 70s on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy [IEEE Trans. Inf. Theory 2004].
Our probabilistic point of view also offers a unified framework to obtain improvements on other sphere-covering bounds, giving conceptually simple and calculation-free proofs for q-ary codes, permutation codes, and spherical codes.
This is joint work with Jaehoon Kim and Hong Liu.
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