Bounded independence plus noise fools products.

Let D be a b-wise independent distribution over {0, 1}(m). Let E be the "noise" distribution over {0,1}(m) where the bits are independent and each bit is 1 with probability eta/2. We study which tests f : {0, 1}(m) -> [-1,1] are epsilon-fooled by D + E, i.e., 1E[f (D + E)] - E[f (U)]vertical bar <= epsilon where U is the uniform distribution. We show that D + E epsilon-fools product tests f : ({0, 1}(n)k) -> [-1, 1] given by the product of k bounded functions on disjoint n-bit inputs with error epsilon = k(1 - eta)(Omega(b2/m)), where m = nk and b >= n. This bound is tight when b = Omega(m) and eta >= (log k)/m. For b >= m(2/3) log m and any constant eta the distribution D + E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed Solomon codes of dimension m/k where k = O(1), communication Omega(eta m) - O(log m) is required to decode one message symbol from a codeword with eta m errors, and communication O(eta m log m) suffices. Second, we obtain pseudorandom generators. We can epsilon-fool product tests f : ({0, 1}(n))(k) -> [-1, 1] under any permutation of the bits with seed lengths 2n + (O) over tilde (k(2) log(l/epsilon)) and O(n) + O (O) over tilde(root nk log 1/epsilon Previous generators have seed lengths >= nk/2 or >= n root/nk. For the special case where the k bounded functions have range {0,1} the previous generators have seed length >= (n + log k) log(l/epsilon).