For $m \geq 2$, consider the group $\mathbb{Z}_{m}^N$, equipped with the Hamming metric, $|y| = |(y(1),\dots,y(N))|:= \#\{ 1 \leq i \leq N : y(i) \neq 0 \}.$ Using tools from harmonic analysis, we prove \emph{dimension independent} norm estimates on the spherical maximal functions $\mathcal{M}^N f(x) := \sup_{0 \leq k \leq N} \left| \frac{1}{| \{ |y| =k \}| } \sum_{|x-y| = k} f(y) \right|.$ A combinatorial application is the following: \begin{cor}[Combinatorial Application] Let $p > 1$ be arbitrary, and suppose $L \subset \Z_{m}^N$ satisfies $|L| =: \epsilon_L \cdot m^N,$ so the relative density of $L$ is $\epsilon_L$. Then there exists an $x \in \Z_{m}^N$ so that at every circle centered at $x$ the fraction of points which intersect $L$ is bounded by a constant multiple of $\epsilon_L^{1/p}$: $\frac{|\{ y \in \mathbb{Z}_m^N : |x-y| = k\} \cap L|}{|\{ |y| = k \}| } \leq C_{m,p} \epsilon_L^{1/p},$ where the constant $C_{m,p}$ may depend on $m,p$, but is independent of $N$.