Testing Reed-Muller codes

A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector´s bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n=2^m is a word in the rth-order Reed-Muller code R(r,m) of length n=2^m. For a given integer r≥1, and real ε>0, the algorithm queries the input vector υ at O(1/ε+r2^2r) positions. On the one hand, if υ is at distance at least εn from the closest codeword, then the algorithm discovers it with probability at least 2/3. On the other hand, if υ is a codeword, then it always passes the test. Our result is almost tight: any algorithm for testing R(r,m) must perform Ω(1/ε+2^r) queries.