High-entropy dual functions over finite fields and locally decodable codes

Abstract

We show that for infinitely many primes p, there exist dual functions of order k over F n p that cannot be approximated in L ∞ -distance by polynomial phase functions of degree k − 1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L ∞ -approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences.