  A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any
subset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all the
sums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of the
summands. Let $ \phi(h,n) $ denote the order of the smallest Abelian group
containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that \[ \lim_{h
\to \infty} \frac{ \phi(h,n) }{ h^n } = \frac{1}{n! \delta_L(\triangle^n)} , \]
where $ \delta_L(\triangle^n) $ is the lattice packing density of an $ n
$-simplex in Euclidean space. This determines the asymptotics exactly in cases
where this density is known ($ n \leq 3 $) and gives improved bounds on $
\phi(h,n) $ in the remaining cases. The corresponding geometric
characterization of bases of order $ h $ in finite Abelian groups in terms of
lattice coverings by simplices is also given.
