## Abstract

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed `, we construct a property P(`) ⊆ {0, 1}^{n} satisfying the following: Any testing algorithm for P(`) requires Ω(n) many queries, and yet P(`) has a constant query PCPP whose proof size is O(n · log(`) n), where log(`) denotes the ` times iterated log function (e.g., log^{(2)} n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n · polylog n). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed `, we construct a property that has a constant-query tester, but requires Ω(n/log(`)(n)) queries for every tolerant or erasure-resilient tester.

Original language | English |
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Title of host publication | 11th Innovations in Theoretical Computer Science Conference, ITCS 2020 |

Editors | Thomas Vidick |

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

ISBN (Electronic) | 9783959771344 |

DOIs | |

State | Published - Jan 2020 |

Externally published | Yes |

Event | 11th Innovations in Theoretical Computer Science Conference, ITCS 2020 - Seattle, United States Duration: 12 Jan 2020 → 14 Jan 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 151 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 11th Innovations in Theoretical Computer Science Conference, ITCS 2020 |
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Country/Territory | United States |

City | Seattle |

Period | 12/01/20 → 14/01/20 |

### Bibliographical note

Publisher Copyright:© Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Ron D. Rothblum.

## Keywords

- Coding theory
- Erasure resilient testing
- PCPP
- Property testing
- Randomized encoding
- Tolerant testing

## ASJC Scopus subject areas

- Software