The Maximum Length for Ducci Sequences on $Z_m^n$ when $n$ is Even

Let $D:{\mathbb{Z}}_m^n\to {\mathbb{Z}}_m^n$ be defined so $$D(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=(x_1+x_2\text{mod}m,x_2+x_3\text{mod}m,\mathrm{.}\mathrm{.}\mathrm{.},x_n+x_1\text{mod}m)\mathrm{.}$$ $D$ is known as the Ducci function and for $\mathbf{u}\in {\mathbb{Z}}_m^n$, $\{D^{\alpha }(\mathbf{u}){\}}_{\alpha =0}^{\mathrm{\infty }}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because ${\mathbb{Z}}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in ${\mathbb{Z}}_m^n$ to enter its cycle when $n$ is even.