The independence number of a subset of an abelian group

We call a subset $A$ of the (additive) abelian group $G$ {\it $t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal the sum of $k$ (not necessarily distinct) elements of $A$ unless $h=k$ and the two sums contain the same terms in some order. A {\it weakly $t$-independent} set satisfies this property for sums of distinct terms. We give some exact values and asymptotic bounds for the size of a largest $t$-independent set and weakly $t$-independent set in abelian groups, particularly in the cyclic group ${\mathbb Z}_n$.