Low-degree tests at large distances
Year: 2006
Type: preprint
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 thenumber of queries. In particular, we show that functions with small Gowers uniformity norms behave "ran-domly " with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem forthe third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficientlythe distance from the second-order Reed-Muller code on inputs lying far beyond its listdecoding radius.
Source: arXiv (Cornell University)
Author: Alex Samorodnitsky
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951307103