Guruswamy and Indyk [4] have shown that there exists an error-correcting code for which list-decoding from α (1 - ε) fraction of errors can be done in linear time. We present a binary code for which list-decoding from α (1/2 - ε) fraction of errors can be done in polylog time. The size of the list of candidates for the correct codeword is exponential but non-trivial and moreover is tight with respect to some known lower bounds. More precisely, for arbitrary constants ε > 0 and λ > 0 we present a code E:\{ 0,1\} ^n \to \{ 0,1\} ^{\overline n } such that \overline n = n^{0(\log ({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \varepsilon }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\varepsilon $}}))} and every ball in \{ 0,1\} ^{\overline n } of radius \{ \frac{1}{2} - \varepsilon \} \overline n (in the Hamming-distance sense) contains at most 2^{\lambda n} strings. Furthermore, the code E has encoding and list-decoding algorithms that produce each bit of their output in time polylog(n).