Abstract
A binary code Enc:{0,1}^k → {0,1}ⁿ is (1/2ε,L)list decodable if for every w ∈ {0,1}ⁿ, there exists a set List(w) of size at most L, containing all messages m ∈ {0,1}^k such that the relative Hamming distance between Enc(m) and w is at most 1/2ε. A qquery local listdecoder for Enc is a randomized procedure Dec that when given oracle access to a string w, makes at most q oracle calls, and for every message m ∈ List(w), with high probability, there exists j ∈ [L] such that for every i ∈ [k], with high probability, Dec^w(i,j) = m_i.
We prove lower bounds on q, that apply even if L is huge (say L = 2^{k^{0.9}}) and the rate of Enc is small (meaning that n ≥ 2^{k}):
 For ε = 1/k^{ν} for some constant 0 < ν < 1, we prove a lower bound of q = Ω(log(1/δ)/ε²), where δ is the error probability of the local listdecoder. This bound is tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of q = O(log(1/δ)/ε²) for the Hadamard code (which has n = 2^k). This bound extends an earlier work of Grinberg, Shaltiel and Viola (FOCS 2018) which only works if n ≤ 2^{k^ν} and the number of coins tossed by Dec is small (and therefore does not apply to the Hadamard code, or other codes with low rate).
 For smaller ε, we prove a lower bound of roughly q = Ω(1/(√ε)). To the best of our knowledge, this is the first lower bound on the number of queries of local listdecoders that gives q ≥ k for small ε.
Local listdecoders with small ε form the key component in the celebrated theorem of Goldreich and Levin that extracts a hardcore predicate from a oneway function. We show that blackbox proofs cannot improve the GoldreichLevin theorem and produce a hardcore predicate that is hard to predict with probability 1/2 + 1/𝓁^ω(1) when provided with a oneway function f:{0,1}^𝓁 → {0,1}^𝓁, where f is such that circuits of size poly(𝓁) cannot invert f with probability ρ = 1/2^√𝓁 (or even ρ = 1/2^Ω(𝓁)). This limitation applies to any proof by blackbox reduction (even if the reduction is allowed to use nonuniformity and has oracle access to f).
BibTeX  Entry
@InProceedings{ronzewi_et_al:LIPIcs.ITCS.2021.33,
author = {Noga RonZewi and Ronen Shaltiel and Nithin Varma},
title = {{Query Complexity Lower Bounds for Local ListDecoding and HardCore Predicates (Even for Small Rate and Huge Lists)}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {33:133:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13572},
URN = {urn:nbn:de:0030drops135724},
doi = {10.4230/LIPIcs.ITCS.2021.33},
annote = {Keywords: Local listdecoding, Hardcore predicates, Blackbox reduction, Hadamard code}
}
Keywords: 

Local listdecoding, Hardcore predicates, Blackbox reduction, Hadamard code 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 