Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube).

Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set of elements all of which are at least distance $d$ from the identity, i.e. it has a set of generators all of which are large.

How many elements of this subgroup can be within $R$ of the identity?