Finding whether the language is CFL or regular
Suppose there exists a CFG whose language is $L$, and let $k$ be the pumping length. Consider the string $(0^k1)^k \in L$. Then, by pumping lemma, there exists a decomposition $(0^k1)^k = uvwxy$ where $|vwx| \leq k$, $vx \neq \varepsilon$, and for each $i \geq 0$, $uv^iwx^iy \in L$. Since $|vx| \leq |vwx| \leq k$, there are four types of values of $vx$: either $vx \in 0^+$, $vx \in 0^+1$, $vx \in 10^+$, or $vx \in 0^+10^+$. Try showing that in each of these cases, pumping $vx$ gives a string not in $L$, thus giving a contradiction and demonstrating that $L$ is not context-free.