## 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 F_{q} and an extension field Fqn, a property is a set of functions mapping Fqn to F_{q}. The property is said to be affine-invariant if it is invariant under affine transformations of Fqn, linear if it is an F_{q} -vector space, and sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. (2009) and followed by Kaufman & Lovett (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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Pages (from-to) | 37-77 |

Number of pages | 41 |

Journal | Computational Complexity |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015, Springer Basel.

## Keywords

- Affine invariance
- additive combinatorics
- locally testable codes
- sum-product estimates

## ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (all)
- Computational Theory and Mathematics
- Computational Mathematics