This is a strengthening of an older question.
Is there a positive integer $c_0$ with the following property?
For every integer $n\geq c_0$ there is a function $f:\{0,1\}^{c_0}\to\{0,1\}$ such that the associated left-shift map $$\ell_f:\{0,1\}^n\to\{0,1\}^n$$ defined by $\big(x_0,\ldots,x_{n-1}\big)\mapsto \big(x_1\ldots,x_{n-1},f(x_0,\ldots,x_{c_0-1})\big)$ is a cyclic permutation of $\{0,1\}^n$.
Such map cannot exist for $n=c_0+(2^{c_0})!$.
Assume the contrary. Write an infinite word by starting with any $c_0$ digits; then, at each step, if the last $c_0$ digits are $x_1,\dots,x_{c_0}$, augment the word on the right with $f(x_1,\dots,x_{c_0})$.
This word is eventually periodic with some period length $k\leq 2^{c_0}$. Take a subword $w_0$ from the periodic part, say starting at the $d$-th position. The symbols at the positions $d+c_0$ and $d+n$ coincide, as $k\mid n-c_0$. Then $\ell_f(w)$ will be the subword starting at position $d+1$, and so on. This way, we will find a cycle in $\ell_f$ of length $k\leq 2^{c_0}<2^n$.
Remark. This still leaves a question whether such maps can exist for arbitrarily large $n$, with a fixed $c_0$.
