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Maximal Subset Sums in a Group
of Order a Power of $2$ John Greene and Clayton Higgins
Department of Mathematics and Statistics
University of Minnesota Duluth
Duluth, MN 55812
USA
jgreene@d.umn.edu
higgi503@d.umn.edu

Abstract:

If $G$ is an abelian group of order $2^{n}$, and $S = \{g_{1}, g_{2}, \ldots, g_{n} \}$ is a subset of $G$, we call $S$ a perfect cover for $G$ if every element of $G$ is the subset sum of elements in $S$. We prove that perfect covers always exist and count how many perfect covers there are for selected groups. In particular, we count the number of perfect covers for the groups $\mathbb{Z}_{2^{m}} \oplus \mathbb{Z}_{2^{n}}$ and $(\mathbb{Z}_{2})^{m} \oplus \mathbb{Z}_{2^{n}}$.