What's the best result about derandomizing BPP which based on some uniform assumptions?
For instance, is there some one proved that BPP can be simulate in subexp time if EXP \not\eq BPP?
What's the best result about derandomizing BPP which based on some uniform assumptions? For instance, is there some one proved that BPP can be simulate in subexp time if EXP \not\eq BPP? 


If $\mathrm{EXP}\ne\mathrm{BPP}$, then every problem in BPP can be solved deterministically in subexponential time on almost every input: http://dx.doi.org/10.1006/jcss.2001.1780. Basically, this says that you can trade nonuniformity in the assumption with approximability in the conclusion. There is a nice (although not quite recent) survey of various derandomization results by Kabanets: http://www.cs.sfu.ca/~kabanets/papers/chapter.ps. 


cseweb.ucsd.edu/~russell/iw1.ps "In other words, randomness never speeds computation by more than a polynomial amount unless nonuniformity always helps computation more than polynomially for problems with exponential time complexity." 

