Four infinite series of $k$-configurations

We suggest an approach to the construction of $k$-configurations on the countable (or finite) set $X$. If $X$ is finite then $k$-configuration is a family of subsets in $X$ with the incidence matrix $L\in GL(|X|,2)$ such that $L$ and $L^{-1}$ have exactly $k$ ones in all rows and columns.