## Abstract

We consider the problem of testing if a given function f : double-struck F^{2}_{n} → double-struck F_{2} is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. Alon et al. [1] proposed and analyzed a natural 2^{d+1}-query test for this problem. This test turned out to be intimately related to the Gowers norm. Alon et. al. showed that this test accepts every degree d polynomial with probability 1, while it rejects functions that are Ω(1)-far with probability Ω(1/(d2^{d})). We give an asymptotically optimal analysis of this test, and show that it rejects functions that are (even only) Ω(2^{-d})-far with Ω(1)-probability (so the rejection probability is a universal constant independent of d and n). This implies a tight relationship between the (d + 1)^{st}-Gowers norm of a function and its maximal correlation with degree d polynomials, when the correlation is close to 1. Our proof works by induction on n and yields a new analysis of even the classical Blum-Luby-Rubinfeld [2] linearity test, for the setting of functions mapping double-struck F^{2}_{n} to double-struck F_{2}. The optimality follows from a tighter analysis of counterexamples to the "inverse conjecture for the Gowers norm" constructed by [3], [4]. Our result has several implications. First, it shows that the Gowers norm test is tolerant, in that it also accepts close codewords. Second, it improves the parameters of an XOR lemma for polynomials given by Viola and Wigderson [5]. Third, it implies a "query hierarchy"result for property testing of affine-invariant properties. That is, for every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [6] for graph properties.

Original language | English (US) |
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Title of host publication | Proceedings - 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 |

Publisher | IEEE Computer Society |

Pages | 488-497 |

Number of pages | 10 |

ISBN (Print) | 9780769542447 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

Event | 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 - Las Vegas, United States Duration: Oct 23 2010 → Oct 26 2010 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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### Conference

Conference | 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 |
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Country/Territory | United States |

City | Las Vegas |

Period | 10/23/10 → 10/26/10 |

## All Science Journal Classification (ASJC) codes

- Computer Science(all)

## Keywords

- Gowers norm
- Low-degree tests
- Property testing
- Reed-Muller codes
- Sublinear-time algorithms