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Tuan Tran, Exponential decay of intersection volume with applications on list-decodability and sphere-covering bounds

Tuesday, December 14, 2021 @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)


Tuan Tran
University of Science and Technology of China

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.


Tuesday, December 14, 2021
4:30 PM - 5:30 PM KST
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Room B232
IBS (기초과학연구원)


Sang-il Oum (엄상일)
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IBS 이산수학그룹 Discrete Mathematics Group
기초과학연구원 수리및계산과학연구단 이산수학그룹
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