The Period of Ducci Cycles on ${\mathbb{Z}}_{2^l}$ for Tuples of Length $2^k$

Let the Ducci function $D:{\mathbb{Z}}_m^n\to {\mathbb{Z}}_m^n$ be defined as $$D(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=(x_1+x_2\text{mod}m,x_2+x_3\text{mod}m,\mathrm{.}\mathrm{.}\mathrm{.},x_n+x_1\text{mod}m)$$ and let the Ducci sequence of $\mathbf{u}$ be the sequence $\{D^{\alpha }(\mathbf{u}){\}}_{\alpha =0}^{\mathrm{\infty }}$. %In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,\mathrm{.}\mathrm{.}\mathrm{.},0)$, as well as establish an upper bound for how many iterations it will take for this to happen. In this paper, we will provide another proof that for $n=2^k$ and $m=2^l$, that all Ducci sequences will end in $(0,0,\mathrm{.}\mathrm{.}\mathrm{.},0)$ and additionally prove that this will happen in at most $2^{k-1}(l+1)$ iterations of $D$.