Good codes decodable by linear-sized circuits?
You should look at Tornado codes {1}, which, for any $R$ and $\epsilon&gt;0$ and large enough $n$ can be designed to recover (with high probability) from a loss of a $(1-R)(1-\epsilon)$ fraction of bits in time proportional to $n \ln \frac{1}{\epsilon}$ (see Theorem 1 in {1}).



{1} Luby, Michael G., et al. "Practical loss-resilient codes." Proceedings of the twenty-ninth annual ACM symposium on Theory of computing. ACM, 1997:  http://www.eecs.harvard.edu/~michaelm/NEWWORK/postscripts/losscodes.pdf