List-decodable zero-rate codes
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1710.10663.pdf
Description
Submitted version
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314.97 KB
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cbde23423c9f5a18ba6d57e6790b819c
Author(s)
Alon, Noga
Bukh, Boris
Polyanskiy, Yury
Date Issued
March 2019
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Alon, Noga, Boris Bukh, and Yury Polyanskiy, "List-decodable zero-rate codes." IEEE Transactions on Information Theory 65, 3 (Mar. 2019): p. 1657-67 doi 10.1109/TIT.2018.2868957 ©2019 Author(s)
Version
Original manuscript
Abstract
We consider list decoding in the zero-rate regime for two cases: the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal τ ϵ [0,1] for which there exists an arrangement of M balls of relative Hamming radius τ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by L or more of them. As M → ∞ the maximal τ decreases to a well-known critical value T[subscript L]. In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is Θ (M-¹) when L is even, thus extending the classical results of Plotkin and Levenshtein for L=2. For L=3 , the rate is shown to be Θ (M -(2/3) ). For the similar question about spherical codes, we prove the rate is Ω (M-¹) and O([mathematical figure; see resource]). ©2019
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1109/TIT.2018.2868957
