Revised: June 11, 2013

Published: October 2, 2013

**Keywords:**locally testable codes, affine-invariant codes, Reed-Muller codes, query complexity

**Categories:**complexity theory, codes, locally testable codes, affine-invariant codes, Reed-Muller codes, query complexity, special issue, RANDOM, APPROX-RANDOM 2012 special issue

**ACM Classification:**F.2.2, F.1.3, G.1.6

**AMS Classification:**68Q17, 52C07, 11H06, 11H31, 05B40

**Abstract:**
[Plain Text Version]

Over a finite field $\F_q$, the $(n,d,q)$-Reed-Muller code is the code given by evaluations of $n$-variate polynomials of total degree at most $d$ on all points (of $\F_q^n$). The task of testing if a function $f:\F_q^n \to \F_q$ is close to a codeword of an $(n,d,q)$-Reed-Muller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all Reed-Muller codewords and rejecting words that are $\delta$-far from the code with probability $\Omega(\delta)$. (In this work we allow the constant in the $\Omega$ to depend on $d$.)

For codes over a prime field $\F_q$ the optimal query complexity is well-known and known to be $\Theta(q^{\lceil (d+1)/(q-1)\rceil})$, and the test consists of testing if $f$ is a degree-$d$ polynomial on a randomly chosen $(\lceil (d+1)/(q-1) \rceil)$-dimensional affine subspace of $\F_q^n$. If $q$ is not a prime, then the above quantity remains a lower bound, whereas the previously known upper bound grows to $O(q^{\lceil (d+1)/(q - q/p) \rceil})$ where $p$ is the characteristic of the field $\F_q$. In this work we give a new upper bound of $(c q)^{(d+1)/q }$ on the query complexity, where $c$ is a universal constant. Thus for every $p$ and sufficiently large $q$ this bound improves over the previously known bound by a polynomial factor.

In the process we also give new upper bounds on the “spanning
weight” of the dual of the Reed-Muller code (which is also
a Reed-Muller code). The spanning weight of a code is the smallest
integer $w$ such that codewords of Hamming weight at most $w$ span
the code.
The main technical contribution of this work is the design of tests
that test a function by *not* querying its value on an entire
subspace of the space, but rather on a carefully chosen
(algebraically nice) subset of the points from low-dimensional
subspaces.

An earlier version of this paper appeared in the Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM'12), pages 639-650, 2012.