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 \({\mathbb{F}_q}\) and an extension field \({\mathbb{F}_{q^n}}\), a property is a set of functions mapping \({\mathbb{F}_{q^n}}\) to \({\mathbb{F}_q}\). The property is said to be affine-invariant if it is invariant under affine transformations of \({\mathbb{F}_{q^n}}\), linear if it is an \({\mathbb{F}_q}\) -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.