Sparse Affine-Invariant Linear Codes Are Locally Testable

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 $\F_{q^n}$, a property is a set of functions mapping $\F_{q^n}$ to $\F_q$. The property is said to be affine-invariant if it is invariant under affine transformations of $\F_{q^n}$, 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.

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