A local tester for an error correcting code CβΞ£n is a
tester that makes Q oracle queries to a given word wβΞ£n and
decides to accept or reject the word w. An optimal local tester is a local
tester that has the additional properties of completeness and optimal
soundness. By completeness, we mean that the tester must accept with
probability 1 if wβC. By optimal soundness, we mean that if the tester
accepts with probability at least 1βΟ΅ (where Ο΅ is small),
then it must be the case that w is O(Ο΅/Q)-close to some codeword
cβC in Hamming distance.
We show that Generalized Reed-Muller codes admit optimal testers with Q=(3q)βqβ1d+1ββ+O(1) queries. Here, for a prime power q=pk, the Generalized Reed-Muller code, RM[n,q,d], consists of the
evaluations of all n-variate degree d polynomials over Fqβ.
Previously, no tester achieving this query complexity was known, and the best
known testers due to Haramaty, Shpilka and Sudan(which is optimal) and due to
Ron-Zewi and Sudan(which was not known to be optimal) both required
qβqβq/pd+1ββ queries. Our tester achieves query
complexity which is polynomially better than by a power of p/(pβ1), which is
nearly the best query complexity possible for generalized Reed-Muller codes.
The tester we analyze is due to Ron-Zewi and Sudan, and we show that their
basic tester is in fact optimal. Our methods are more general and also allow us
to prove that a wide class of testers, which follow the form of the Ron-Zewi
and Sudan tester, are optimal. This result applies to testers for all
affine-invariant codes (which are not necessarily generalized Reed-Muller
codes).Comment: 42 pages, 8 page appendi