# How many $2d$-elements subsets with specific property at most can we choose from $\{1,2,\cdots,n\}$

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?

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 in the OEIS. References/links on the OEIS page have some further information about both $$d=2$$ and other values of $$d$$.