Sparse affine-invariant linear codes are locally testable

Eli Ben-Sasson, Noga Ron-Zewi, Madhu Sudan

Research output: Contribution to journalConference articlepeer-review

5 Scopus citations


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 Fq and an extension field double-struck Fqn, a property is a set of functions mapping double-struck Fqn to double-struck Fq. The property is said to be affine-invariant if it is invariant under affine transformations of double-struck Fqn, 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 languageEnglish
Article number6375335
Pages (from-to)561-570
Number of pages10
JournalProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
StatePublished - 1 Dec 2012
Externally publishedYes
Event53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States
Duration: 20 Oct 201223 Oct 2012


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

ASJC Scopus subject areas

  • General Computer Science


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