## Abstract

A binary code Enc: {0, 1}^{k} → {0, 1}^{n} is (^{1}_{2} - 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 ^{1}_{2} - 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 _{ε}2^{/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 ^{1}_{2} + _{`ω}^{1}_{(1)} when provided with a one-way function f : {0, 1}^{`} → {0, 1}^{`}, where f is such that circuits of size poly(`) cannot invert f with probability ρ = 1/2 ^{√`} (or even ρ = 1/2^{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).

Original language | English |
---|---|

Title of host publication | 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) |

Editors | James R. Lee |

Place of Publication | Dagstuhl, Germany |

Pages | 33:1-33:18 |

Volume | 185 |

ISBN (Electronic) | 9783959771771 |

DOIs | |

State | Published - 1 Feb 2021 |

Event | 12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online Duration: 6 Jan 2021 → 8 Jan 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics (LIPIcs) |
---|---|

Publisher | Schloss Dagstuhl--Leibniz-Zentrum für Informatik |

### Conference

Conference | 12th Innovations in Theoretical Computer Science Conference, ITCS 2021 |
---|---|

City | Virtual, Online |

Period | 6/01/21 → 8/01/21 |

## Keywords

- Black-box reduction
- Hadamard code
- Hard-core predicates
- Local list-decoding

## All Science Journal Classification (ASJC) codes

- Software