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 ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $ -distance by polynomial phase functions of degree $k-1$ . This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $ -approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.

Multiple correlation sequences not approximable by nilsequences

“…The idea of using Yekhanin's construction to give interesting examples in the additive combinatorics of higher-order correlations first arose in the finite field setting, in a joint work of the first author and Labib [5]. Those ideas have inspired the present work.…”

“…Remark. As mentioned previously, the idea of making this definition ultimately comes from coding theory (see [5] for further discussion). The important feature is that averages of f k (n)f k (n + d)f k (n + 2d) over n simplify substantially by using (4.1); we give the details of this in (4.5).…”

“…The idea behind the construction of these finitary examples ultimately comes from coding theory, and in particular a construction of Yekhanin [9]. We will only need the most basic form of these ideas; for instance, we can replace all the finite-field theory in Yekhanin’s work with the simple observation that the function defined by , , and periodic mod has the property that The idea of using Yekhanin’s construction to give interesting examples in the additive combinatorics of higher-order correlations first arose in the finite field setting, in a joint work of the first author and Labib [5]. Those ideas have inspired the present work.…”