# How to improve this pseudorandom generator?

Let $$f$$ be a Boolean function and $$\varepsilon > 0$$.

There exists a pseudorandom generator $$G_f: \{0,1 \}^{n^{\varepsilon}} \to \{0,1 \}^n$$ with the following property. Let $$T$$ be a set and $$p(n)$$ a polynomial.
If $$|\Pr_{r \in \{0,1 \}^n}[r \in T] - \Pr_{r \in \{0,1 \}^{n^{\varepsilon}}}[G_f(x) \in T]| > \frac{1}{p(n)}$$ for all large $$n$$, then there exists a polynomial-size oracle circuit family $$\{C_n\}_{n \in \mathbb{N}}$$ with oracle $$T$$ that computes $$f$$ and queries $$T$$ non-adaptively.

Roughly speaking, if $$T$$ exposes generator $$G_f$$ then there exists a polynomial-size circuit with oracle $$T$$ that computes $$f$$.

For simplicity I will consider only oracles $$T$$ such that $$\Pr_{r \in \{0,1 \}^{n^{\varepsilon}}}[G_f(x) \in T] =0$$. For oracles $$L$$ and $$T$$ having this property we say that $$L$$ expose $$G_f$$ not worse than $$T$$ if $$T \subseteq L$$.

I want the following generalization of this result. If $$T$$ exposes generator $$G_f$$ then there exists a polynomial-size circuit $$C$$ such that $$C^L$$ computes $$f$$ for every oracle $$L$$ that is not worse than $$T$$.

Unfortunately, this generator $$G$$ (based on Nisan-Wigderson generator) does not have this property.

Is it possible to improve $$G$$ for achieving my goal?

Motivation. In the cited paper Power from random strings'' authors consider oracles containing strings with high Kolmogorov complexity, for example oracle $$R =\{x | C(x) \ge \frac{x}{2} \}$$ where $$C$$ is some variation of Kolmogorov complexity.

It is natural to consider a spoiled oracle instead of $$R$$ that answer yes' if we ask about string $$x$$ s.t. $$C(x) \ge \frac{x}{2} + B$$, answer no'' if we ask about string $$x$$ s.t. $$C(x) \le \frac{x}{2} - B$$ and can answer everything if $$\frac{x}{2} - B < C(x) < \frac{x}{2} + B$$. Here $$B$$ is bound for example constant.

If there exists a wishful generator which properties decribed above then some results from `Power from random strings'' can be generalized for such family of oracles.