What complexity class is this ciruit problem? I'm exploring an algorithm that solves k-SAT.  It uses a ton of preprocessing, so I'm thinking that this will be a circuit bounds.

Without knowing the runtime, I speculate on how quickly it will solve k-SAT instances.  It will either solve all instances of k-SAT with probability $p$, or it won't be able to solve any instances.  The runtime is based on $p$.  Essentially, it runs in quasi-polynomial time.

So essentially, what we have is an algorithm that runs in time $2^{O((\log{n})^{c_1})}\cdot O(c_2 + q)$, and solves all instances with probability $\frac{2^q-1}{2^q}$, or it can't solve any instances.  Here, $c_1$ and $c_2$ are constants.

It also requires space polynomial in $q$.

I'm trying to find what complexity class this is, and what is known about it.

SOME NOTES

Just to ensure I've described this correctly, I have an algorithm that either solves all instances, or it can't solve any.  We can say it uses $2^{2^n}$ preprocessing.  Then, it runs in time that is quasi-polynomial multiplied by a function of $q$.

THE QUESTION, AGAIN

What complexity class is this, and what is known about it?
Assuming that the probability is over the randomized preprocessing and $p &gt; 0$, by fixing the correct random choice you get a circuit of size $S = O(2^{\log^{O(1)} n} q^{O(1)})$ for $k$-SAT. The corresponding complexity class is non-uniform circuits of size $S$. If, for example, $q$ is quasipolynomial ($2^{\log^{O(1)}n}$), the you get a non-uniform circuit of quasipolynomial size. 

We usually don't care too much about the time required to construct the circuit unless it is very low.