List-decoding using the XOR lemma

We show that Yao´s XOR Lemma, and its essentially equivalent rephrasing as a Direct Product Lemma, can be re-interpreted as a way of obtaining error-correcting codes with good list-decoding algorithms from error-correcting codes having weak unique-decoding algorithms. To get codes with good rate and efficient list decoding algorithms, one needs a proof of the Direct Product Lemma that, respectively, is strongly derandomized, and uses very small advice. We show how to reduce advice in Impagliazzo´s proof of the Direct Product Lemma for pairwise independent inputs, which leads to error-correcting codes with O(n²) encoding length, 0^˜(n²) encoding time, and probabilistic 0^˜(n) list-decoding time. (Note that the decoding time is sub-linear in the length of the encoding.) Back to complexity theory, our advice-efficient proof of Impagliazzo´s hard-core set results yields a (weak) uniform version of O´Donnell results on amplification of hardness in NP. We show that if there is a problem in NP that cannot be solved by BPP algorithms on more than a 1 - 1/(log n)^c fraction of inputs, then there is a problem in NP that cannot be solved by BPP algorithms on more than a 3/4 + 1/(log n)^c fraction of inputs, where c > 0 is an absolute constant.