The Erdős distinct subset sums problem in a modular setting

We prove the following variant of the Erdős distinct subset sums problem. Given $t\ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A\ge (\frac{1}{3}-o(1))N\mathrm{.}$$ Furthermore, we provide examples showing that the constant $\frac{1}{3}$ is best possible. For small values of $t$, we characterise the structure of all sumset-distinct sets modulo $N=2^n+t$ of cardinality $n$.