## Abstract

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester. LTCs were initially studied as important components of probabilistically checkable proofs (PCP), and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "c3-LTCs", namely LTCs with constant rate, constant distance, and constant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.

Original language | American English |
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Title of host publication | STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Stefano Leonardi, Anupam Gupta |

Publisher | Association for Computing Machinery |

Pages | 357-374 |

Number of pages | 18 |

ISBN (Electronic) | 9781450392648 |

DOIs | |

State | Published - 6 Sep 2022 |

Event | 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 - Rome, Italy Duration: 20 Jun 2022 → 24 Jun 2022 |

### Publication series

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

### Conference

Conference | 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 |
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Country/Territory | Italy |

City | Rome |

Period | 20/06/22 → 24/06/22 |

### Bibliographical note

Funding Information:Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21. Shai Evra is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. Alexander Lubotzky’s research is supported by a grant from the Institute for Advanced Study at Princeton and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 882751). Shahar Mozes acknowledges support by ISF-Moked grant 2019/19. This material is based upon work supported by a grant from the Institute for Advanced Study.

Publisher Copyright:

© 2022 Owner/Author.

## Keywords

- error correcting codes
- expander codes
- locally testable codes