### Abstract

We prove that 3-query linear locally correctable codes over the Reals of dimension d require block length n > d^{2+λ} for some fixed, positive λ > 0. Geometrically, this means that if n vectors in Rd are such that each vector is spanned by a linear number of disjoint triples of others, then it must be that n > d^{2+λ}. This improves the known quadratic lower bounds (e.g. [20, 28]). While a modest improvement, we expect that the new techniques introduced in this work will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries. At a high level, our proof has two parts, clustering and random restriction. The clustering step uses a powerful geometric theorem of Barthe which finds a basis change (and rescaling) putting the code in nearly isotropic position. This together with the fact that any LCC must have many 'correlated' pairs of points, lets us deduce that the vectors must have a surprisingly strong geometric clustering, and hence also combinatorial clustering with respect to the spanning triples. In the restriction step, we devise a new variant of the dimension reduction technique used in previous lower bounds, which is able to take advantage of the combinatorial clustering structure above. The analysis of our random projection method reduces to a simple (weakly) random graph process, and works over any field.

Original language | English (US) |
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Title of host publication | STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 784-793 |

Number of pages | 10 |

ISBN (Print) | 9781450327107 |

DOIs | |

State | Published - Jan 1 2014 |

Event | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States Duration: May 31 2014 → Jun 3 2014 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Other

Other | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 |
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Country | United States |

City | New York, NY |

Period | 5/31/14 → 6/3/14 |

### All Science Journal Classification (ASJC) codes

- Software

## Cite this

*STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing*(pp. 784-793). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/2591796.2591818