## Abstract

We show that sparse affine-invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field double-struck F_{q} and an extension field double-struck F_{q}^{n}, a property is a set of functions mapping double-struck F_{q}^{n} to double-struck F_{q}. The property is said to be affine-invariant if it is invariant under affine transformations of double-struck F_{q}^{n}, and it is said to be sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. [RANDOM 2009] and followed by Kaufman and Lovett [FOCS 2011]. The latter showed such a result for the case when q was prime. Extending to non-prime cases turns out to be non-trivial and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for affine-invariant linear properties.

Original language | English |
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Article number | 6375335 |

Pages (from-to) | 561-570 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: 20 Oct 2012 → 23 Oct 2012 |

## Keywords

- Additive Combinatorics
- Affine Invariance
- Locally Testable Codes
- Sum-product Estimates

## ASJC Scopus subject areas

- Computer Science (all)