Let $K$ be a local field of positive characteristic. I'm looking for a $K$ which satisfies the following condition.

• Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$.

$K$ should be like the form $ \Bbb{F}_q((t))$, so I need to determine $q$.