Dimension-Free $L^p$-Maximal Inequalities in $\mathbb{Z}_{m+1}^N$

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, $$|y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,$$ and define the $L^1$-normalized indicator of the $r$-sphere, $$\sigma_r := \frac{1}{|\{|x| = r\}|} 1_{\{|x| =r\}}.$$ We study the $L^p \to L^p$ mapping properties of the maximal operator $$M^{N} f (x) := \sup_{r \leq N} | \sigma_r*f|$$ acting on functions defined on $\mathbb{Z}_{m+1}^N$. Specifically, we prove that for all $p>1$, there exist absolute constants $C_{m,p}$ so that $$\| M^{N} f \|_{L^p(\mathbb{Z}_{m+1}^N)} \leq C_{m,p} \| f \|_{L^p(\mathbb{Z}_{m+1}^N)}$$ for all $N$.