Current status of Waring-Goldbach problem

http://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem

Wiki says that WG conjecture is that for every $k$, $\exists$ primes $p_{1}, p_{2}, \cdots p_{t}$ where $t$ is independent of $k$ such that $\sum_{i=1}^{t}p_{i}^{k} = N$ for large enough $N$.

Hua showed $t$ is atmost $O(k^{2}log{k})$. In his theorem, does anyone know the bounds on $N$ given $k$? Is it bounded below $O(2^{k})$ (that is for every $N > N_{0} = O(2^{k})$ does Hua's theorem hold)?