$k$-wise independent probability spaces
For arbitrary $b$, Alon, Babai and Itai showed a lower bound on the probability space size of $m(n,\lfloor k/2 \rfloor)$ where
$$ m(n,k) = \sum\limits_{i=0}^k \binom{n}{i}$$

which is $\Omega(n^{k/2})$ for constant $k$.

They also gave a construction of size $O(n^{k/2})$ in the case of $b = 1$.

For $b=1$ there is a paper by Karloff and Mansour which shows lower bounds and upper bounds for arbitrary probabilities, i.e., for $p_1,\ldots,p_n$ with $p_i = P(Y_i = 1)$. E.g., there are probabilities $p_1,\ldots,p_n$ such that the probability space size is at least $m(n,k)$. They also say that $m(n,k)$ is also a upper bound for arbitrary probabilities.

I don't known any construction with a better upper bound than $O(n^k)$ which is given by the construction (see here) mentioned by Thomas as a comment.