Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification

We consider the problem of approximately locally listdecoding direct product codes. For a parameter k, the kwise direct product encoding of an N-bit message msg is an N^{k}-length string over the alphabet {0, 1}^k indexed by ktuples (i_1, . . . , i_k) \in {1, . . . ,N}^k so that the symbol at position (i_1, . . . , i_k) of the codeword is msg(i_1) . . . msg(i_k). Such codes arise naturally in the context of hardness amplification of Boolean functions via the Direct Product Lemma (and the closely related Yao�s XOR Lemma), where typically k \ll N (e.g., k = poly logN). We describe an efficient randomized algorithm for approximate local list-decoding of direct product codes. Given access to a word which agrees with the k-wise direct product encoding of some message msg in at least an \in fraction of positions, our algorithm outputs a list of poly(1/\in) Boolean circuits computing N-bit strings (viewed as truth tables of logN-variable Boolean functions) such that at least one of them agrees with msg in at least 1 - \delta fraction of positions, for \delta = O(k^{-0.1}), provided that \in =\Omega(poly(1/k)); the running time of the algorithm is polynomial in logN and 1/\in. When \in \ge e-^{k^{\alpha} } for a certain constant \alpha > 0, we get a randomized approximate listdecoding algorithm that runs in time quasi-polynomial in 1/\in (i.e., (1/\in)^{poly log 1/\in}). By concatenating the k-wise direct product codes with Hadamard codes, we obtain locally list-decodable codes over the binary alphabet, which can be efficiently approximately list-decoded from fewer than 1/2 - \in fraction of corruptions as long as \in = \Omega(poly(1/k)). As an immediate application, we get uniform hardness amplification for P^{NP_\parallel} , the class of languages reducible to NP through one round of parallel oracle queries: If there is a language in P^{NP_\parallel} that cannot be decided by any BPP algorithm on more that 1 - 1/n^{\Omega(1)} fraction of inputs, then there is another language in P^{NP_\parallel} that cannot be decided by any BPP algorithm on more than 1/2 + 1/n^{\omega(1)} fraction of inputs.