Consider a finite group $\mathcal{G}$. Let $S_0$ be a $n$-tuple of elements of $\mathcal{G}$, and let $S_i$ be the cyclically shifted version of $S_0$ by $i$ indices to the right. So for example if $\mathcal{G}$ = $\text{GF}(2)$, $n = 3$, and $S_0 = (1,0,0)$, then $S_2 = (0,0,1)$.

Now let $\mathcal{S} = \{S_i : 0 \leq i \leq n-1\}$. One can add two $n$-tuples in $\mathcal{S}$ by adding the elements over the group $\mathcal{G}$. So in the above example, $S_0 + S_2 = (1,0,1)$.

The basic question is this: Pick any $g \in \mathcal{G}$. I want to choose a subset $\mathcal{H} \subseteq \mathcal{S}$ so that the number of times the element $g$ appears in the tuple $\sum_{h \in \mathcal{H}} h$ (let's call this the $g$-count) is maximum. I'm not really interested right now in the subset itself, but really in the maximum $g$-count.

Does anybody know what kind of results are known about this problem in general? Or what kind of techniques do you suspect are needed to tackle this problem?