### Abstract

A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a fraction δ of the coordinates are corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions. In this work we show a separation between 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form p ^{Ω(δd)} on the length of linear 2-query LCCs over double-struck F _{p}, that encode messages of length d. Our bound improves over the known bound of 2 ^{Ω(δd)} [9], [12], [8] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science. Corollaries of our main theorem are new incidence geometry results over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields.

Original language | English (US) |
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Title of host publication | Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |

Pages | 638-647 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 1 2011 |

Event | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States Duration: Oct 22 2011 → Oct 25 2011 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |
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Country | United States |

City | Palm Springs, CA |

Period | 10/22/11 → 10/25/11 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Keywords

- Sylvester-Gallai theorem
- additive combinatorics
- locally decodable codes

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## Cite this

*Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011*(pp. 638-647). [6108225] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2011.28