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

TL;DR

This paper establishes fundamental lower bounds on the query complexity of local list-decoding algorithms, even for codes with small rate and large list sizes, advancing understanding of decoding limits in coding theory.

Contribution

It provides new lower bounds on query complexity for local list-decoding, extending previous results to codes with small rate and large list sizes, and introduces bounds for small error parameters.

Findings

Lower bound of q=Ω(log(1/δ)/ε^2) for ε ≥ 1/k^ν, tight with known upper bounds.

Lower bound of q=Ω(1/√ε) for smaller ε, first to show q ≥ k.

Black-box limitations on Goldreich-Levin hard-core predicate improvements.

Abstract

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.