Low-degree tests at large distances

TL;DR

This paper introduces efficient tests to distinguish boolean functions close to linear or quadratic polynomials from those far away, analyzing their properties using Gowers norms and applying results to coding theory.

Contribution

It develops nearly optimal property tests for linear and quadratic functions, including an inverse theorem for the third Gowers norm, with applications to Reed-Muller codes.

Findings

Functions with small Gowers norms behave randomly in hypergraph tests.

The inverse theorem for the third Gowers norm is proved.

Efficiently estimates distance from Reed-Muller codes beyond list-decoding radius.

Abstract

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius.