Are there any known results about the following problem:


Given any integer $n\geqslant4$, how many $4$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any two $4$-elements subsets we have choosed has not more than $2$ element?


More generally, given any integer $n\geqslant2d$, how many $2d$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any two $4$-elements subsets we have choosed has not more than $d$ element?



How many $2d$-elements subsets with specific property at most can we choose from $\{1,2,\cdots,n\}$
This can be thought of as finding the largest binary code of constant weight $2d$, length $n$, minimal distant $2d$.


For $d=2$ the values are [A001843](https://oeis.org/A001843) in the OEIS. References/links on the OEIS page have some further information about both $d=2$ and other values of $d$. 


