Sparse affine-invariant linear codes are locally testable

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

Research output: Contribution to journalArticlepeer-review


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 Fq and an extension field Fqn, a property is a set of functions mapping Fqn to Fq. The property is said to be affine-invariant if it is invariant under affine transformations of Fqn, linear if it is an Fq -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 languageEnglish
Pages (from-to)37-77
Number of pages41
JournalComputational Complexity
Issue number1
StatePublished - 1 Mar 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015, Springer Basel.


  • 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


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