Using higher-order Fourier analysis over general fields

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas. (i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code. Previously, this had been proved over prime fields [BL14] and for the case when |K|− 1 divides the order of the code [GKZ08]. (ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P : Kn → K can be decomposed as a particular composition of lesser degree polynomials. This had been previously established over prime fields [Bha14, BHT15]. (iii) For any fixed finite fieldK, we prove that all locally characterized affine-invariant properties of functions f : Kn → K are testable with one-sided error. The same result was known when K is prime [BFH13] and when the property is linear [KS08]. Moreover, we show that for any fixed finite field F, an affine-invariant property of functions f : Kn → F, where K is a growing field extension over F, is testable if it is locally characterized by constraints of bounded weight.