Query complexity lower bounds for local list-decoding and hard-core predicates (Even for small rate and huge lists)

A binary code Enc: {0, 1}^k → {0, 1}ⁿ is (¹₂ - ε, 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 ¹₂ - ε. A q-query local list-decoder 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) = mi. We prove lower bounds on q, that apply even if L is huge (say L = 2^k0.9) 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 _ε2/δ)}), where δ is the error probability of the local list-decoder. This bound is tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of q = O(log(1 _ε2^/δ)) 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^kν 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 list-decoders that gives q ≥ k for small ε. Local list-decoders with small ε form the key component in the celebrated theorem of Goldreich and Levin that extracts a hard-core predicate from a one-way function. We show that black-box proofs cannot improve the Goldreich-Levin theorem and produce a hard-core predicate that is hard to predict with probability ¹₂ + <sub>`ω</sub><sup>1</sup><sub>(1)</sub> when provided with a one-way function f : {0, 1}<sup>`</sup> → {0, 1}<sup>`</sup>, where f is such that circuits of size poly(`) cannot invert f with probability ρ = 1/2 ^{√`</sup> (or even ρ = 1/2<sup>Ω(`)}). This limitation applies to any proof by black-box reduction (even if the reduction is allowed to use nonuniformity and has oracle access to f).