Approximate List-Decoding of Direct Product Codes and Uniform Hardness Amplification

Given a message $msg\in\{0,1\}^N$, its $k$-wise direct product encoding is the sequence of $k$-tuples $(msg(i_1),\dots,msg(i_k))$ over all possible $k$-tuples of indices $(i_1,\dots,i_k)\in\{1,\dots,N\}^k$. We give an efficient randomized algorithm for approximate local list-decoding of direct product codes. That is, given oracle access to a word which agrees with a $k$-wise direct product encoding of some message $msg\in\{0,1\}^N$ in at least $\epsilon\geqslant{poly}(1/k)$ fraction of positions, our algorithm outputs a list of ${poly}(1/\epsilon)$ strings that contains at least one string $msg'$ which is equal to $msg$ in all but at most $k^{-\Omega(1)}$ fraction of positions. The decoding is local in that our algorithm outputs a list of Boolean circuits so that the $j$th bit of the $i$th output string can be computed by running the $i$th circuit on input $j$. The running time of the algorithm is polynomial in $\log N$ and $1/\epsilon$. In general, when $\epsilon>e^{-k^{\alpha}}$ for a sufficiently small constant $\alpha>0$, we get a randomized approximate list-decoding algorithm that runs in time quasi-polynomial in $1/\epsilon$, i.e., $(1/\epsilon)^{{poly}\log1/\epsilon}$. As an application of our decoding algorithm, 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 than $1-1/n^{\Omega(1)}$ 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.