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?

A subset $D=\{a_1,\ldots,a_k\}$ of $\mathbb{Z}/v\mathbb{Z}$ is said to be a $(v,k,\lambda)$-cycic difference set if for each nonzero $b\in\mathbb{Z}/v\mathbb{Z}$, there are exactly $\lambda$ ordered pairs $(a_s,a_t)\in D^2$ such that $a_s-a_t=b$. For a $(v,k,\lambda)$-difference set $D$, $k-\lambda$ is called the order.

Let $C_{v,n}$ be the set consisting of all cyclic difference sets of $\mathbb{Z}/v\mathbb{Z}$ with order $n$, and $$ C_v=\bigcup\limits_{\text{$n>1$ is a prime power coprime to $v$}}C_{v,n}. $$ For a fixed $v$, $C_v$ can be explicitly written down if $v$ is not too large. My question actually is: have we already know $C_v$ for infinitely many $v$s'?

I would pose another question related to this: does there exist an $N$ such that for all $v>N$, $|C_v|>0$?