Is there any infinite family of $v$ for which all the $(v,k,\lambda)$-cyclic difference sets with $k-\lambda$ a prime power coprime to $v$ have been determined?

Let $C_v=\{D\mid D\text{ is a $(v,k,\lambda)$-cyclic difference set}, k-\lambda=p^f\text{ for some prime p}, \gcd(k-\lambda,v)=1\}$.

For a fixed $v$, $C_v$ is explicitly known if $v$ is not too large. My question actually is: have we already know $C_v$ for infinitely many $v$s'?

The above is like a community wiki question, and I would pose another question related to this: does there exist an $N$ such that for all $v>N$, $C_v$ contains at least one nontrivial difference set?