Optimal Testing of Reed-Muller Codes

Abstract

We consider the problem of testing if a given function $f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$ is close to any degree d polynomial in n variables, also known as the problem of testing Reed-Muller codes. We are interested in determining the query-complexity of distinguishing with constant probablity between the case where f is a degree d polynomial and the case where f is Ω(1)-far from all degree d polynomials. Alon et al. [AKK+05] proposed and analyzed a natural 2^d + 1-query test T ₀, and showed that it accepts every degree d polynomial with probability 1, while rejecting functions that are Ω(1)-far with probability Ω(1/(d 2^d)). This leads to a O(d 4^d)-query test for degree d Reed-Muller codes.

We give an asymptotically optimal analysis of T ₀, showing that it rejects functions that are Ω(1)-far with Ω(1)-probability (so the rejection probability is a universal constant independent of d and n). In particular, this implies that the query complexity of testing degree d Reed-Muller codes is O(2^d).

Our proof works by induction on n, and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping $\mathbb{F}_2^n$ to $\mathbb{F}_2$. Our results also imply a “query hierarchy” result for property testing of affine-invariant properties: For every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [GKNR08] for graph properties.

This is a brief overview of the results in the paper [BKS+09].