## Abstract

A code C ⊆ {0, 1}^{n}^{¯} is (s, L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n¯ − s symbols have at most L possible completions into a codeword of C. Non-explicitly, there exist binary ((1 − τ)¯ n, L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log _{τ}^{1} ). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [26, 24, 25]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ^{2}) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [24]. We construct an explicit binary ((1 − τ)¯ n, L) erasure list-decodable code having rate τ^{1+γ} (for any constant γ > 0 and small τ) and list-size poly(log _{τ}^{1} ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code. The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2 log(^{1}_{ε} ) or entropy-loss smaller than 2 log(^{1}_{ε} ), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1 + γ) log(^{1}_{ε} ) and entropy-loss O(log log ^{1}_{ε} ). The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz's extractor [39] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [27, 9, 13]) from achieving the above results.

Original language | English |
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Title of host publication | 35th Computational Complexity Conference, CCC 2020 |

Editors | Shubhangi Saraf |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771566 |

DOIs | |

State | Published - 1 Jul 2020 |

Event | 35th Computational Complexity Conference, CCC 2020 - Virtual, Online, Germany Duration: 28 Jul 2020 → 31 Jul 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 169 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 35th Computational Complexity Conference, CCC 2020 |
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Country/Territory | Germany |

City | Virtual, Online |

Period | 28/07/20 → 31/07/20 |

## Keywords

- Dispersers
- Erasure codes
- List decoding
- Ramsey graphs
- Two-source extractors