Kreknin's function

Definition
Kreknin's function, denoted $$k$$, is a function from the set of natural numbers to the set of natural numbers defined as follows. $$k(n)$$ is the smallest number satisfying the following:

Any $$\mathbb{Z}/n\mathbb{Z}$$-graded Lie ring (see Lie ring graded over an abelian group) with degree zero component (denoted $$L_0$$) equal to the zero Lie ring has derived length at most $$k(n)$$.

Well definedness
To see that the above notion makes sense, see Kreknin's theorem on existence of Kreknin's function.

General bound
We have the following upper bound on Kreknin's function: $$k(n) \le 2^{n - 1}$$ for all natural numbers $$n$$.