Positivity and negativity for $h$-bases for $n$

A set $A$ of integers is an $h$-basis for $n$ if every integer in $[0,n] = \{0,1,2,\ldots,n\}$ belongs to the sumset $hA$, that is, can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. The classical postage stamp problem asks for the largest integer $n_h(k)$ such th