Constructing Thick $B_h$-sets

A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\cdots +a_h=b_1+\cdots +b_h$ (with $a_i,b_i\in A$) are the trivial solutions $\{a_1,\dots ,a_h\}=\{b_1,\dots ,b_h\}$ (as multisets). With $h=2$ and $X=\mathbb{Z}$, these sets are also known as Sidon sets, Golomb Rulers, and Babcock sets. In this work, we generalize constructions of Bose-Chowla and Singer and give the resultant bounds on the diameter of a $k$ element $B_h$ set in $\mathbb{Z}$ for small $k$. We conclude with a list of open problems.