Query complexity lower bounds for local list-decoding and hard-core predicates (even for small rate and huge lists)
A binary code Enc$:\{0,1\}^k \to \{0,1\}^n$ is $(0.5-\epsilon,L)$-list decodable if for all $w \in \{0,1\}^n$, the set List$(w)$ of all messages $m \in \{0,1\}^k$ such that the relative Hamming distance between Enc$(m)$ and $w$ is at most $0.5 -\epsilon$, has size at most $L$. Informally, a $q$-query local list-decoder for Enc is a randomized procedure Dec$:[k]\times [L] \to \{0,1\}$ that when given oracle access to a string $w$, makes at most $q$ oracle calls, and for every message $m \in \text{List}(w)$, with high probability, there exists $j \in [L]$ such that for every $i \in [k]$, 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 $n \ge 2^{k}$): 1. For $\epsilon \geq 1/k^{\nu}$ for some universal constant $0< \nu < 1$, we prove a lower bound of $q=\Omega(\frac{\log(1/\delta)}{\epsilon^2})$, where $\delta$ 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 $q=O(\frac{\log(1/\delta)}{\epsilon^2})$ 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 $n \le 2^{k^{\gamma}}$ for some universal constant $0<\gamma <1$, 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). 2. For smaller $\epsilon$, we prove a lower bound of roughly $q = \Omega(\frac{1}{\sqrt{\epsilon}})$. To the best of our knowledge, this is the first lower bound on the number of queries of local list-decoders that gives $q \ge k$ for small $\epsilon$. We also prove black-box limitations for improving some of the parameters of the Goldreich-Levin hard-core predicate construction.