Correction of samplable additive errors

We study the correctability of efficiently samplable errors. Specifically, we consider samplable additive-error channels, where unbounded-weight errors are sampled by a polynomial-time algorithm, and added to the channel input in an oblivious way. Assuming the existence of one-way functions, there are samplable distributions Z over {0, 1}
ⁿ
with entropy n
^ε
for 0 <; ε <; 1 that are not correctable by efficient coding schemes. Next, we show that there is an oracle relative to which there is a samplable Z with entropy ω(log n) that is not correctable by efficient syndrome decoding. For flat distributions Z with entropy m, we show that if Z forms a linear subspace, there is a linear code that corrects Z with rate R ≤ 1 - m/n. For general flat distributions Z, there is a linear code that corrects Z with error ε for rate R ≤ 1 - (m + O(log(1/ε)))/n, and no coding scheme can correct Z with error ε for rate R > 1-(m+log(1-ε))/n. Finally, we observe that small-biased distributions are not correctable by high-rate codes, and hence there is a small-biased Z with entropy m that is not correctable for rate R > 1-m/n+(2 log n+O(1))/n. To derive these results, we use relations between error-correcting codes and other notions such as data compression and randomness condensers.