Let $A$ be a set of $n$ positive integers with distinct $k$-sums. In other words, if $a_1\le\cdots\le a_k$ and $b_1\le\cdots\le b_k$ are elements of $A$ such that $a_1+\cdots+a_k=b_1+\cdots+b_k$, then $a_i=b_i$, $i=1,\ldots,k$. How small can $\max A$ be?

Let $S(n,k)$ be the smallest possible value of $\max A$.

Some trivial attempts: The construction $A=\lbrace 1,k,k^2,\ldots,k^{n-1}\rbrace$ gives $S(n,k)\le k^{n-1}$. On the other hand, there are ${k+n-1\choose n-1}$ sequences $a_1\le\cdots\le a_k$, which correspond to distinct $k$-sums whose values are at most $k\cdot\max(A)$. Thus $S(n,k)\ge\frac1k{k+n-1\choose n-1}$.

**Additional question:** How to explicitly construct the set $A$ so that $\max A$ is small? Thomas Bloom asserted below that $S(n,k)\asymp n^k$, but I will be satisfied with a construction where $\max A$ is a polynomial of $n$ (of order possibly larger than $k$).